Theoretical modelling
of nano-scaled systems
with heavy ions
Dissertation
zur Erlangung des Doktorgrades der Naturwissenschaften
(Dr. rer. nat.)
Kris Holtgrewe
Fachbereich 07
Institut für theoretische Physik
Gießen, Juli 2022
Dem Fachbereich 07 der Justus-Liebig-Universität Gießen
als Dissertation vorgelegt.
Tag der Disputation: 24. Oktober 2022
Erstgutachter: Prof. Dr. Simone Sanna
Zweitgutachter: Prof. Dr. Wolf Gero Schmidt
Equidem beatos puto, quibus deorum munere datum est aut facere scribenda aut
scribere legenda, beatissimos vero quibus utrumque.
— Gaius Plinius Caecilius Secundus (Plinius Minor), Epistulae VI.16
—
Ich jedenfalls glaube, dass diejenigen glücklich sind, denen als Geschenk der Götter die
Gabe zuteil geworden ist, entweder Dinge zu tun, die es Wert sind aufgeschrieben zu
werden, oder Dinge zu schreiben, die es Wert sind gelesen zu werden. Die Glücklichsten
sind wahrhaftig diejenigen, die beides erfüllen.
—
I, for one, believe that those are blissful who are gifted by the gods either to do things
worth writing down, or to write things worth reading. The most blissful, truly, are those
who fulfil both.
Abstract
Systems on the nanoscale are a hot topic in science and technology as they may
be applied in future concepts for microprocessors, memory storage and sensors.
This PhD thesis covers two nano-scaled systems which promise particularly advant-
ageous properties: rare-earth silicide nanowires on silicon surfaces, a quasi-one-
dimensional electronic system, and thin antimony layers on bismuth selenide, a two-
dimensional system with topologically protected surface states. These systems have
in common that their fascinating properties are due to heavy ions incorporated in
the structures. This thesis theoretically investigates them by means of density func-
tional theory (DFT). However, DFT encounters problems when describing such ma-
terials since the high atomic numbers of the involved elements give rise to exotic
phenomena, e.g. strongly correlated electronic subshells (the incomplete 4f shell of
the lanthanoids), strong relativistic effects (topologically non-trivial insulators) and
high contributions to the electronic long-range correlation (“van der Waals interac-
tions”). These problems are solved by approaches beyond DFT, including LDA+U,
spin-orbit coupling and dispersion corrections.
Both systems investigated in this work have higher-dimensional prototype struc-
tures, which are explored at first and then scaled down to the final nanostructures.
Different structure models are set up and optimised regarding the ionic positions.
Their stability is evaluated by means of ab initio thermodynamics and phase dia-
grams are derived. For the most stable structure models, the electronic properties
are calculated, including band structures, Fermi surfaces and simulated scanning
tunnelling microscopy. All theoretical findings on the structural and electronic prop-
erties are carefully compared with experimental reference. In this way, a conclusive
ab initio framework is established for these systems, which permits a deep under-
standing of the underlying physics. Furthermore, novel and fascinating phenomena
are identified. The rare-earth silicide nanowires on Si(557) show a unique dimen-
sional crossover, which gives rise to quasi-one-dimensional, metallic edge states. The
thin antimony layers are proven to underlie a complex interplay with the topolo-
gically protected surfaces states of the bismuth selenide surface, which involves an
intricate series of topological phase transitions.
iv
Zusammenfassung
Nanoskopische Systeme sind ein heißes Thema in Wissenschaft und Forschung, da sie
Anwendung in künftigen Konzepten für Mikroprozessoren, Datenspeicher und Sen-
soren finden könnten. Diese Dissertation behandelt zwei nanoskopische Systeme, die
besonders vorteilhafte Eigenschaften versprechen: Seltenerdsilizidnanodrähte auf
Siliziumoberflächen, ein quasi-eindimensionales elektronisches System, und dün-
ne Antimonlagen auf Bismutselenid, ein zweidimensionales System mit topologisch
geschützten Oberflächenzuständen. Diesen Systemen ist gemein, dass ihre faszinie-
renden Eigenschaften auf die Einbindung schwerer Ionen in die Strukturen zurück-
zuführen sind. Diese Dissertation untersucht sie mittels der Dichtefunktionaltheo-
rie (DFT). Jedoch ist die DFT problematisch in der Beschreibung solcher Materiali-
en, da die hohen Ordnungszahlen der beteiligten Elemente exotische Phänomene
verursachen, z.B. stark korrelierte elektronische Unterschalen (die inkomplette 4f-
Schale der Lanthanoide), starke relativistische Effekte (topologisch nicht-triviale Iso-
latoren) und hohe Beiträge zur langreichweitigen elektronischen Korrelation („van-
der-Waals-Wechselwirkungen“). Diese Probleme werden durch Methoden jenseits
der DFT gelöst, u.a. LDA+U, Spin-Bahn-Kopplung und Dispersionskorrekturen.
Für beide in dieser Arbeit untersuchten Systeme existieren höherdimensionale
Strukturprototypen, die zunächst erforscht und danach auf die finalen Nanostruktu-
ren herunter skaliert werden. Verschiedene Strukturmodelle werden aufgesetzt und
bezüglich der ionischen Positionen optimiert. Ihre Stabilität wird mittels ab initio-
Thermodynamik ausgewertet und Phasendiagramme werden aufgestellt. Für die sta-
bilsten Strukturmodelle werden die elektronischen Eigenschaften berechnet, darun-
ter Bandstrukturen, Fermiflächen und simulierte Rasterelektronenmikroskopie. Al-
le theoretischen Ergebnisse für die strukturellen und elektronischen Eigenschaften
werden sorgfältig mit experimenteller Referenz verglichen. Auf diese Weise wird für
diese Systeme ein geschlossenes Rahmenwerk konstruiert, das ein tiefes Verständ-
nis der zugrunde liegenden Physik ermöglicht. Des Weiteren werden neuartige und
faszinierende Phänomene identifiziert. In den Seltenerdsilizidnanodrähten wird ein
einzigartiger dimensionaler Übergang entdeckt, der quasi-eindimensionale metalli-
sche Kantenzustände erzeugt. Den dünnen Antimonlagen wird nachgewiesen, dass
sie einem komplexen Zusammenspiel mit den topologisch geschützten Oberflächen-
zuständen der Bismutselenidoberfläche unterworfen sind, das eine komplizierte Rei-
he von topologischen Phasenübergängen beinhaltet.
v
Contents
1 Introduction 1
1.1 The microprocessor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 The structure of this work . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Methods 9
2.1 Density functional theory . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 The LDA+U method . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.2 Spin-orbit coupling in DFT . . . . . . . . . . . . . . . . . . . . . 16
2.1.3 Van der Waals corrections . . . . . . . . . . . . . . . . . . . . . 17
2.2 The Bloch theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.1 Basis changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
I Rare-earth silicide nanowires on silicon surfaces 25
3 Chemical background 28
3.1 The rare-earth elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Silicon and silicides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4 The rare-earth silicide bulk phases 34
4.1 Structure prototypes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2 Structure optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2.1 Methodological details . . . . . . . . . . . . . . . . . . . . . . . 41
4.2.2 The stoichiometric RESi2 phases . . . . . . . . . . . . . . . . . . 45
4.2.3 The role of the 4f electrons in RESi2 . . . . . . . . . . . . . . . . 47
4.2.4 Vibrational contributions in RESi2 . . . . . . . . . . . . . . . . . 50
4.2.5 The vacancy-populated RESi2 – x phases . . . . . . . . . . . . . . 52
4.2.6 The CaSi2 phases . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3 Electronic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.4 Discussion and summary . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5 Monolayer films on silicon(111) 71
5.1 Structure optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.2 Electronic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.2.1 Band structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.2.2 Charge transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2.3 Fermi surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.3 Discussion and summary . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.3.1 Strain induced growth? . . . . . . . . . . . . . . . . . . . . . . . 89
vi
6 Nanowires on silicon(557) 91
6.1 Structure optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.2 Electronic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.2.1 Band structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.2.2 Fermi surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.3 Discussion and summary . . . . . . . . . . . . . . . . . . . . . . . . . . 103
II Thin antimony layers on bismuth selenide 105
7 Geometric phases 108
7.1 The Aharonov-Bohm effect . . . . . . . . . . . . . . . . . . . . . . . . . 108
7.2 The Berry phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.3 The electric polarisation in a crystal . . . . . . . . . . . . . . . . . . . . 113
7.4 Topological transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.4.1 The integer Hall effect . . . . . . . . . . . . . . . . . . . . . . . . 116
7.4.2 The two-dimensional ℤ2 insulator . . . . . . . . . . . . . . . . . 120
7.4.3 The three-dimensional ℤ2 insulator . . . . . . . . . . . . . . . . 128
8 The bulk phases of bismuth selenide and antimony 131
8.1 Bismuth selenide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
8.1.1 Structure optimisation . . . . . . . . . . . . . . . . . . . . . . . 133
8.1.2 Electronic properties . . . . . . . . . . . . . . . . . . . . . . . . 135
8.2 Antimony . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
8.2.1 Structure optimisation . . . . . . . . . . . . . . . . . . . . . . . 138
8.2.2 Electronic properties . . . . . . . . . . . . . . . . . . . . . . . . 139
8.3 Discussion and summary . . . . . . . . . . . . . . . . . . . . . . . . . . 142
9 The antimony on bismuth selenide heterostructure 143
9.1 Structure optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
9.1.1 The stacking sequence of the adlayer . . . . . . . . . . . . . . . 145
9.1.2 The effects from SOC and vdW on the heterostructure . . . . . 148
9.2 Electronic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
9.2.1 Band structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
9.2.2 Simulated STM images . . . . . . . . . . . . . . . . . . . . . . . 154
9.3 The topological phase transition . . . . . . . . . . . . . . . . . . . . . . 156
9.4 Discussion and summary . . . . . . . . . . . . . . . . . . . . . . . . . . 160
10 Conclusions 162
Appendix A1
Bibliography A15
Publication list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A24
Danksagung A25
vii
List of Figures
1.1 Performance development of the Top500 list . . . . . . . . . . . . . . . 6
3.1 Atomic radii of the REs. . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.1 Overview over the RESi2 structure prototypes. . . . . . . . . . . . . . 36
4.2 Possible configurations of ordered Si vacancies in RESi1.67. . . . . . . 36
4.3 Schematic phase diagram of the Tb–Si system. . . . . . . . . . . . . . . 39
4.4 Murnaghan fits of TbSi2, DySi2, HoSi2 and ErSi2 in the hex-AlB2 and the
tet-ThSi2 phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.5 Vibrational free energies of hex-AlB2-TbSi2, ort-AlB2-TbSi2 and tet-
ThSi2-TbSi2 and the corrected Tb chemical potentials 𝜇Tb(𝑇 ) calculated
in different supercells. . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.6 Structure models of vacancy-populated TbSi2 – b. . . . . . . . . . . . . . 54
4.7 Linear interpolation of Δ𝜇RE for TbSi2 – x and ErSi 12 – x between 𝑥 = 3 and
𝑥 = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.8 Models of TbSi2 in the ort-AlB2, tr3-CaSi2 and tr6-CaSi2 structures. . . 60
4.9 Band structures of tet-ThSi2-TbSi2 and hex-AlB2-TbSi2. . . . . . . . . . 63
4.10 Band structures of hex-AlB2-TbSi2 and ort-AlB2-TbSi2. . . . . . . . . . 65
4.11 Band structures of trivalent and f-valent, FMO ort-AlB2-RESi2. . . . . . 67
4.12 Band structures of hex-AlB2-TbSi2 – b and ort-AlB2-TbSi2. . . . . . . . . 68
4.13 Charge transfer in ort-AlB2-TbSi2 and hex-AlB2-RESi2 – b. . . . . . . . . 69
5.1 Sketch of the Si(111) slab and the TbSi2 monolayer structures. . . . . 74
5.2 Tb chemical potentials of the monolayer-Tb@Si(111) structures and de-
cision tree for the energy penalties upon structure variations. . . . . 76
5.3 Vertical positions and distances of the atomic layers in the monolayer-
Tb@Si(111) structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.4 Band structures of the T4-h-B structure and the clean Si(111) surface. 80
5.5 Band structures of the monolayer-Tb@Si(111) structures. . . . . . . . 83
5.6 Band folding of the bulk TbSi2 structures. . . . . . . . . . . . . . . . . 84
5.7 Charge transfer in T4-h-B. . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.8 Fermi surfaces of the monolayer-Tb@Si(111) structures. . . . . . . . . 87
5.9 Energy surfaces of T4-h-B at different energy levels. . . . . . . . . . . 87
6.1 Sketch of the Si(551) slab and the optimised TbSi2 nanowire structures. 93
6.2 Band structures of dense and sparse nanowires on Si(557) and the
monolayer on Si(111). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.3 Segmentation of the Brillouin zone of the monolayer. . . . . . . . . . . 99
6.4 Band structures of the nanowires and projected bands of the monolayer. . . 100
6.5 Fermi surfaces of the Tb@Si(ℎℎ𝑘) systems. . . . . . . . . . . . . . . . . 102
viii
7.1 Sketch of the ABE setting. . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.2 Path for calculating the electric polarisation as a geometric phase. . . . . . 115
7.3 Sketch of the directions of the integer Hall effect and path of 𝝃 on the
Brillouin torus for calculating the topological transport. . . . . . . . . 118
7.4 Sketch of the reciprocal space of the T-symmetric Bloch system and flow
chart of the connection between the partial polarisations 𝑷I and 𝑷II, the
total polarisation 𝑷+ and the time-reversal polarisation 𝑷− . . . . . . . 124
7.5 Sketch of the Laughlin-type gedankenexperiment and sketches of pos-
sible topologies of T-symmetric edge band structures. . . . . . . . . . 127
7.6 TRIMs 𝚪𝑖 in a cubic Brillouin zone. . . . . . . . . . . . . . . . . . . . . . . 128
8.1 Bulk structure of Bi2Se3. . . . . . . . . . . . . . . . . . . . . . . . . . . 133
8.2 Band structure of Bi2Se3 for the rhombohedrally centred DFT-D2+SOC
structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
8.3 Bulk structure of 𝛽-Sb. . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
8.4 Band structure of bulk 𝛽-Sb and isolated 𝛽-Sb sheets. . . . . . . . . . . 140
9.1 Sketch of the BS(0001) slab. . . . . . . . . . . . . . . . . . . . . . . . . 145
9.2 Sb chemical potentials 𝜇Sb and interlayer distances 𝑑1 and 𝑑2 of the
structure models for the Sb@BS system optimised with DFT-D2+SOC.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
9.3 Interlayer distances 𝑑1 and 𝑑2 of 1BL-Sb and 2BL-Sb optimised with dif-
ferent treatments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
9.4 Band structures of the clean Bi2Se3 surface and the heterostructures
1BL-Sb and 2BL-Sb optimised with DFT-D2+SOC. . . . . . . . . . . . . 150
9.5 Comparison between the theoretical band structures and ARPES images
of the heterostructures 1B-Sb and 2BL-Sb. . . . . . . . . . . . . . . . . 152
9.6 Spin texture of the clean Bi2Se3 surface and the heterostructures 1BL-Sb
and 2BL-Sb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
9.7 STM images of 1BL-Sb and 2BL-Sb. . . . . . . . . . . . . . . . . . . . . 155
9.8 Real-space isosurfaces of the wavefunctions at the Dirac points D and
D∗ of the clean substrate, (b) 1BL-Sb and (c) 2BL-Sb. . . . . . . . . . . . 156
9.9 Hexagonal Brillouin zone with TRIMs. . . . . . . . . . . . . . . . . . . . . 157
9.10 Topological phase transitions in the Sb@BS heterostructure. . . . . . 158
B.1 Band structures of hex-AlB2-TbSi2, hex-AlB2-Si2, ort-AlB2-TbSi2 and ort-
AlB2-Si2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A8
B.2 Band structures of trivalent and f-valent tet-ThSi2-TbSi2. . . . . . . . . A8
C.1 Band structures of 1BL-Sb and 2BL-Sb structure models with different
stacking optimised with DFT-D2+SOC. . . . . . . . . . . . . . . . . . . . A13
ix
List of Tables
3.1 Valencies of the lanthanoids occurring in aqueous solution and VEC of
the trivalent ions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.1 Bulk mechanical properties of TbSi2, DySi2, HoSi2 and ErSi2 in the hex-
AlB2 and the tet-ThSi2 phase. . . . . . . . . . . . . . . . . . . . . . . . 46
4.2 Lattice parameters and relative RE chemical potentials of TbSi2, DySi2,
HoSi2 and ErSi2 in the hex-AlB2 and the tet-ThSi2 phase. . . . . . . . . 46
4.3 Lattice parameters and relative Tb chemical potentials of TbSi2 in the
AlB2 and the ThSi2 phase under different f-electron treatments. . . . 48
4.4 Lattice parameters and relative Er chemical potentials of ErSi2 in the
AlB2 and the ThSi2 phase under different f-electron treatments. . . . 49
4.5 Lattice parameters and relative Tb chemical potentials of TbSi2 in the
AlB2 and the ThSi2 phase in orthorhombic Tb4Si8 cells. . . . . . . . . . 51
4.6 Lattice parameters and relative Tb chemical potentials of TbSi2 – x in the
stoichiometric and the vacancy-populated AlB2 and ThSi2 phases. . . 55
4.7 Si–Si nearest-neighbour distances in TbSi2 – x optimised with PBE. . . 58
4.8 Lattice parameters and relative Tb chemical potentials of stoichiomet-
ric TbSi2 in the AlB2 and CaSi2 phases. . . . . . . . . . . . . . . . . . . 61
5.1 Bond distances and angles in the monolayer-Tb@Si(111) structures. . 77
8.1 Lattice parameters of bulk Bi2Se3. . . . . . . . . . . . . . . . . . . . . . 135
8.2 Lattice parameters of bulk 𝛽-Sb. . . . . . . . . . . . . . . . . . . . . . . 139
9.1 Sb chemical potentials 𝜇Sb, basal lattice constants 𝑎 and interlayer dis-
tances 𝑑 of 2D and 3D 𝛽-Sb systems. . . . . . . . . . . . . . . . . . . . . 146
B.1 Lattice parameters and relative Er chemical potentials of ErSi2 – x in the
stoichiometric and the vacancy-populated AlB2 and ThSi2 phases. . . A6
B.2 Si–Si nearest neighbour distances in TbSi2 – x optimised with different
xc-functionals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A6
B.3 Si–Si nearest neighbour distances in ErSi2 – x optimised with different
xc-functionals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A7
B.4 Lattice parameters and relative RE chemical potentials of TbSi2 and
ErSi2 in the stoichiometric AlB2 and CaSi2 phases. . . . . . . . . . . . . A7
x
Acronyms
ABE Aharonov-Bohm effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
AMO antiferromagnetically ordered . . . . . . . . . . . . . . . . . . . . . . . . . 48
ARPES angle-resolved photoemission spectroscopy . . . . . . . . . . . . . . . . . 26
BL bilayer (layers consisting of two atomic sublayers) . . . . . . . . . . . . . . . 138
CB conduction band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
CBM conduction band minimum . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
CI conventional insulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
DFT density functional theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
DOS density of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
FMO ferromagnetically ordered . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
GGA generalised-gradient approximation . . . . . . . . . . . . . . . . . . . . . . 12
HPC high-performace computer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
IHE integer Hall effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
LDA local density approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
LDOS local density of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
LEED low-energy electron diffraction . . . . . . . . . . . . . . . . . . . . . . . . . 78
MEIS medium-energy ion scattering . . . . . . . . . . . . . . . . . . . . . . . . . 78
MOSFET metal-oxide-semiconductor field-effect transistor . . . . . . . . . . . . . 2
PAW projector-augmented wave . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
PBE Perdew-Burke-Ernzerhof functional . . . . . . . . . . . . . . . . . . . . . . 12
PBEsol Perdew-Burke-Ernzerhof functional revised for solids . . . . . . . . . . 12
pBZ prmitive Brillouin zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
PDOS partial density of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
QE Quantum Espresso (https://www.quantum-espresso.org/) . . . . . . . . . . . . 9
QL quintuple layer (layers consisting of five atomic sublayers) . . . . . . . . . . 132
RE rare-earth element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
xi
sBZ supercell Brillouin zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
SKP surface Kramers pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
SOC spin-orbit coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
STM scanning tunnelling microscopy . . . . . . . . . . . . . . . . . . . . . . . . 26
STS scanning tunnelling spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . 26
TI topological insulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
TRIM time-reversal-invariant momentum . . . . . . . . . . . . . . . . . . . . . . 121
TSS topologically protected surface state . . . . . . . . . . . . . . . . . . . . . . . 107
UHV ultra-high vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
VASP Vienna Ab initio Simulation Package (https://www.vasp.at/) . . . . . . . . 9
VB valence band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
VBM valence band maximum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
vdW van der Waals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
VEC valence electron configuration . . . . . . . . . . . . . . . . . . . . . . . . . . 28
XRD X-ray diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
ZKB Zintl-Klemm-Busmann concept . . . . . . . . . . . . . . . . . . . . . . . . . 33
xii
Erklärung
Ich erkläre: Ich habe die vorgelegte Dissertation selbstständig und ohne unerlaubte
fremde Hilfe und nur mit den Hilfen angefertigt, die ich in der Dissertation angege-
ben habe. Alle Textstellen, die wörtlich oder sinngemäß aus veröffentlichten Schrif-
ten entnommen sind, und alle Angaben, die auf mündlichen Auskünften beruhen,
sind als solche kenntlich gemacht. Ich stimme einer evtl. Überprüfung meiner Dis-
sertation durch eine Antiplagiat-Software zu. Bei den von mir durchgeführten und
in der Dissertation erwähnten Untersuchungen habe ich die Grundsätze guter wis-
senschaftlicher Praxis, wie sie in der „Satzung der Justus-Liebig-Universität Gießen
zur Sicherung guter wissenschaftlicher Praxis“ niedergelegt sind, eingehalten.
Ort, Datum Unterschrift
xiii

1 Introduction
Systems on the nanoscale are one of the most active scientific research fields. They
all have in common that they trap particles in less than three spatial dimensions:
surfaces (2D), nanowires (1D), quantum dots (0D) and systems with dimensions in
between. The reduced dimensionality gives rise to fascinating physical phenomena,
some of them are of pure quantum nature. The research efforts in the broad field
of nano-scaled systems are driven by two major motives. On the one hand, nano-
structured devices may be applied in new technologies, e.g. for processors, memory
storage and sensors. On the other hand, nano-scaled systems provide also the oppor-
tunity to observe and understand fundamental physical principles which are other-
wise predicted by theoretical frameworks.
This PhD thesis investigates two groups of low-dimensional systems. The rare-earth
silicide nanowires on silicon surfaces are ensembles of few nanometre wide and
several 100 nm long RESi2 strips which grow horizontally on silicon substrates. Since
bearing highly anisotropic metallic states, they are promising candidates for quasi-
one-dimensional electronic systems. The thin antimony layers on bismuth selen-
ide are van der Waals heterostructures consisting of a topologically non-trivial Bi2Se3
substrate and a few atomic layers thick Sb adsorbate. Such heterostructures consti-
tute a junction between materials of different dimensions and different topological
classes. Therefore, they bear a complex system of two-dimensional metallic surface
states, which have a unique spin texture and are protected by topology. Although the
two systems seem to be very different at first glance, they have in common that they
involve the heaviest non-radioactive elements of the periodic table: the lanthanoids
(La–Lu, 𝑍 ∈ [57, 71]) and bismuth (Bi, 𝑍 = 83), both located in the sixth and last stable
row. The high atomic numbers of the constituents are responsible for the unique
properties of the systems and give rise to exciting phenomena. A further common
property of the systems is the practical motivation that they both may play a role in
future processor technology. The nanowires are interesting from the point of view of
miniaturising the feature sizes. The Sb@Bi2Se3 heterostructures are interesting for
spintronics, an alternative concept of circuits based on spin densities.
Before the investigations of the above systems begin, the applicability-related motiv-
ation will be elaborated in more detail. In short, it is all about Moore’s law approach-
ing its limit. However, solely stating the emergence of general problems would be too
superficial to understand the technical and economic challenges impending in the
medium-term future. Therefore, a short overview of the microprocessor is given,
beginning with the success story of the computer and the silicon-based integrated
circuits. It is followed by the problems which the semiconductor and computer in-
dustry will encounter in the 2020s. These considerations clarify why so much effort
is put into developing concepts for novel electronics.
1
1 Introduction
1.1 The microprocessor
There are probably no technical component which has been affecting humankind
more than the microprocessor. Being part of nearly every device, microprocessors
are not only present in obviously computer-like devices, e.g. smartphones, laptops
and tablets. Also smart devices like watches, TVs, sound systems and kitchen equip-
ments contain small data-processing semiconductor chips, which run more or less
elaborate programs to facilitate our life. Since the breakthrough of home automa-
tion, formerly simple devices like thermostats, plug sockets and light switches have
been upgraded by processing units as well. They can be integrated into a “network of
smart things” where they communicate with each other, controlled by a computer-
based central control unit. Microprocessors do not only inspire intelligence into in-
dividual components. Also the internet, on which every job in every economic sector
depends, is based on servers, which are accumulations of microprocessors.
A striking example for how strong technological progress depends on powerful com-
puting units are cars. Despite their seemingly simple purpose to transport passengers
or freight, they are equipped with many microcontrollers and processors. The R&D
departments of the automotive industry are quite busy with developing computing
hardware and software as the abilities of a car become a growingly important sales
argument, beside CO2-neutral vehicle propulsion. Modern cars have an interface
which looks rather like a smartphone display than a traditional car interface. The
superficial applications like the board info, the navigation system or the smartphone
integration are only the visible output from the built-in microprocessors. Moreover,
there are advanced driver-assistance systems, such as adaptive cruise control and
lane centring, which are driven by processing units: the input from many sensors is
converted into output for the steering, braking and acceleration actuators. As Ger-
many licensed autonomous driving this year1, the future cars will require even more
powerful and, in particular, reliable processing units – on boards and off board – to
cope with the complexity of traffic.
A historical note
The history of the microprocessor is intricate as several parallel trends lead to those
silicon-based microprocessors which keep our modern world running. There were
two main threads: the development of the computer, which includes the fundament-
als on how to run a program on a machine, and the invention of the metal-oxide-
semiconductor field-effect transistor (MOSFET), which permitted miniaturising the
chip features. The history can be experienced in respective museums. The world’s
largest computer museum is the Heinz-Nixdorf-Forum in Paderborn, Germany, ded-
icated to Heinz Nixdorf, an important entrepreneur in the German computer in-
dustry. It offers many exhibits about mathematics, calculation machines and com-
puters from the past to the present, including original parts of famous computers
1Bald fahren autonome Fahrzeuge in Deutschland, https://www.faz.net/aktuell/wirtschaft/bald-
fahren-autonome-fahrzeuge-auf-deutschen-strassen-17827948.html (visited on 24/05/2022).
2
1.1 The microprocessor
which fundamentally changed the world2. The development of the transistor and
the microprocessor is comprehensively demonstrated as a timeline by the Computer
History Museum near San Jose, California3. The important milestones, which can be
found also on the museum websites, are summarised in the following paragraphs.
The development of the computer began with the emergence of relay-based calculat-
ors in the 1930s. Konrad Zuse developed the first programmable, program-controlled
machines in Germany. The most famous is the Z3 machine (1941), which was a purely
binary computer and, thus, capable for floating-point arithmetic. At the same time,
several relay-based, decimal machines were developed in the USA. Driven by their
participation in World War II, the USA needed fast machines for ballistic calculations
and for the development of nuclear weapons. The Harvard Mark I machine (1944)
by Howard H. Aiken was an early example, which, like the other machines, was slow
since based on mechanical relays. J. Presper Eckert and John Mauchly had the idea
to replace the slow relays with faster vacuum tubes. They presented the first purely
electronic computer at the University of Pennsylvania in 1946: the Electronic Nu-
merical Integrator and Computer (ENIAC). Although it was a 1000 times faster than
relay-based machines, the ENIAC had the drawback that “programming” consisted
in plugging cables, a procedure which could take several days. At that time, John
von Neumann, who worked in the atomic bomb projects, was looking for machines
capable of doing calculations on detonators and ignition heights. He soon got notice
of the ENIAC and joined the ENIAC team to develop a stored-program successor: the
Electronic Discrete Variable Automatic Computer (EDVAC, 1952). Its design, the von-
Neumann architecture comprising the arithmetic unit, the control unit, the memory,
input and output, is the basic principle for most of our modern computers2.
The basic ingredient for all computers is the transistor. In its simplest form, it has
three contacts: the source, the drain and the gate. In principle, it blocks the current
between the source and the drain, until the gate receives a signal. Relays and vacuum
tubes were the first “transistors”, which did their job, but could not noteworthy be
miniaturised. This inspired researchers to pursue the concept of solid-state transist-
ors, realised at the interface of materials with different carrier chemical potentials.
Although early ideas had already been proposed in the 1920s, it was not before 1947,
until a working point-contact transistor was invented by William Bradford Shockley,
John Bardeen and Walter Houser Brattain at Bell Labs. The fundamentality of their
invention was acknowledged with the Nobel Prize in Physics in 19564. Short after
the group at Bell Labs, Herbert Mataré and Heinrich Welker independently inven-
ted the point-contact transistor in Paris in 1948. In 1951, Gordon Teal and Morgan
Sparks from Bell Labs fabricated the first working 𝑛𝑝𝑛-transistor. It soon replaced
the vacuum tubes in many products, e.g. radios and clocks, until it dominated the
world electronics market less than ten years later. The fast growing transistor mar-
ket was accompanied by the foundation of several companies, of which some still
2THE INVENTION OF THE COMPUTER, https://www.hnf.de/en/permanent-exhibition/exhibition-
areas/the-invention-of-the-computer.html (visited on 14/04/2022)
3TIMELINE, https://www.computerhistory.org/siliconengine/timeline/ (visited on 15/04/2022)
4All Nobel Prizes in Physics, https://www.nobelprize.org/prizes/lists/all- nobel- prizes- in-
physics/ (visited on 04/03/2022).
3
1 Introduction
exist today: Intermetall (West Germany, today part of TDK-Micronas), Texas Instru-
ments (USA) and the Tokyo Telecommunications Company (Japan, renamed to Sony
in 1958). Of course, computers were upgraded with transistors as well. The first fully
transistorised computer was the TRAnsistor DIgital Computer (TRADIC), developed
at Bell Labs in 1954 and used by the U.S. Air Force.
In the first half of the 1950s, germanium transistors dominated the market as they
are easier to fabricate and can operate at higher frequencies than silicon transistors.
However, the operational temperature of germanium devices is limited to a range of
0 ◦C to 70 ◦C, whereas silicon devices work in a broader temperature range of −55 ◦C
to 125 ◦C. This excluded germanium devices in particular from aerospace technology,
which required devices operating reliably under extreme conditions. Although Mor-
ris Tanenbaum from Bell Labs succeeded in developing a working silicon transistor
in Jan. 1954, the laboratory misjudged the importance of his work and decided not
to pursue the silicon approach any more. Three months later, in Apr. 1954, Gordon
Teal, who had changed from Bell Labs to Texas Instruments by the time, succeeded in
the synthesis of a silicon transistor as well – unknowingly about Tanenbaum’s work.
After this breakthrough, silicon transistors quickly conquered the market and had
replaced the germanium transistors by the end of the decade. The 1950s were also
the decade in which important fabrication techniques were developed, including dif-
fusion processes, oxide masking and photolithography. These techniques combined
produced the planar integrated circuit, released in 1960. Finally, the MOSFET permit-
ted a fast miniaturisation of the features of integrated circuits. While the first patent
for a field-effect transistor was written by Julius Lilienfeld in 1928, the successful fab-
rication was not achieved before 1960 by Mohammed M. Atalla and Dawon Kahng at
Bell Labs. Since 1964, MOSFET-based integrated circuits have been sold and, since the
beginning of the 1970s, the central processing components of computers have been
integrated into central processing units (CPUs), called microprocessors today3 5.
Moore’s Law
Since their breakthrough, planar integrated circuits had smaller features with each
new generation so that more transistors could be assembled on a chip of fixed size.
Consequently, they became more and more powerful while their costs of produc-
tion and the energy demand decreased. In 1965, Gordon Moore stated his famous
law in an internal correspondence, telling that the number of transistors per chip
grows over time at exponential speed. He estimated the doubling time to 12 months
at first and corrected it to two years in 1975. Since then, Moore’s law has been a
self-fulfilling prophecy as the semiconductor industry benchmarks its new releases
5In the course of technical break-overs, the topography of companies underwent many changes. The
most important one was the foundation of the Shockley Semiconductor Laboratory by William
Shockley and Arnold Beckman in 1955. They recruited several capable people, among them Gor-
don Moore and Robert Noyce. After differences between Shockley and his employees, Moore, Noyce
and six other colleagues left the company to found the Fairchild Semiconductor Corporation in Palo
Alto, California, in 1957. This was the birth of Silicon Valley, named after the breakthroughs based
on the silicon transistor. Several startups and spin-offs of Fairchild still exist today, among them Intel
(from INTegrated ELectronics, founded by Moore and Noyce) and Advanced Micro Devices (AMD,
founded by Jerry Sanders, a sales manager at Fairchild who was singed off).
4
1.1 The microprocessor
to compete with Moore’s prediction. Although it has been valid for several decades,
it encountered problems, of which the first emerged in the 1980s: The growing com-
plexity of the chips required an increasing number of production steps and special-
ised machines which were distributed over a growing number of companies. This
made the costs of developing new chip generations explode. To maintain Moore’s
law, the US semiconductor industry initiated a board consisting of engineers from
several semiconductor companies worldwide. It developed the International Techno-
logy Roadmap for Semiconductors in 1998, which, updated every second year, should
facilitate the coordination, identify upcoming problems early and find solutions for
them so as to maintain the technological progress [1].
The first principal problem arose in the early 2000s, when the transistors became so
dense that the heat could not dissipate any more. The temperature problem forced
the manufacturers to limit the clock rates, which have not exceeded a few GHz un-
til today. In order to still gain computational power from feature miniaturisation,
the processor was divided into cores which each represent a closed processing unit.
Provided that a program is parallelisable, i.e. it can be divided into independent
chunks which can be distributed over the cores to be run in parallel, the total cal-
culative capability of a processor is proportional to the clock rate multiplied with
the number of cores [1]. The multi-core strategy maintained Moore’s law until the
second principal problem arose. When the feature sizes drop below a few 10 nm,
quantum mechanical effects like electron tunnelling produce undesirable leakage
currents. The complementary metal-oxide semiconductor (CMOS, a combination of
𝑝-type and 𝑛-type MOSFETs) was the last planar processor technology, which hit the
physical limits at the 22 nm technology node6in the 2010s. Smaller features were real-
ised with the fin field-effect transistor (FinFET), which exploits the third dimension
to a certain extent. In 2020, the most miniaturised processors belonged to the 5 nm
technology node with ∼100 million transistors per square millimetre. However, the
term “5 nm” is misleading as it refers to the minimum feature width instead of the av-
erage half pitch, which is in the range of 10 nm7. This marketing-driven redefinition
of the feature sizes cannot obfuscate that the development of the transistor density
clearly begins to deviate from Moore’s law8. Further miniaturisation might be phys-
ically possible and is actually planned. However, the development costs grow faster
than the transistor density as each new processor generation requires a new set of
more precise production facilities. Therefore, it is likely that Moore’s law will cease
for economical reasons before it reaches the physical limits [1].
The emergence of high-performance computing
Scientific research and technological progress have become crucially dependent on
the internet and on powerful computing facilities. Today’s science is based on the
solution of complex numerical problems and the evaluation of huge amounts of data,
622 nm refer to the half distance between repeating features, the half pitch; 22 nm process, https:
//en.wikipedia.org/wiki/22_nm_process (visited on 18/04/2022)
75 nm process, https://en.wikipedia.org/wiki/5_nm_process (visited on 18/04/2022).
8Intel Now Packs 100 Million Transistors in Each Square Millimeter, https://spectrum.ieee.org/
nanoclast/semiconductors/processors/intel- now- packs- 100- million- transistors- in- each-
square-millimeter (visited on 18/04/2022).
5
1 Introduction
0
Top
50
 
sum
#1
00
Figure 1.1: Performance devel- #5
opment of the Top500 list [W10].
Green→ sum of the list; Orange
→ leading machine; Blue→ last
machine. The lines are guides
for the eyes to demonstrate the
exponential growth and its re-
cent slowdown. Lists
which both exponentially grow in size. Consequently, the calculative demand of re-
search has exceeded the capability of local computers, which are limited in perform-
ance, even if equipped with multicore CPUs. Therefore, the numerics are outsourced
to large servers providing thousands of CPUs: the high-performace computers (HPCs)
or supercomputers. A very recent example for HPC-aided science is the research re-
lated to the SARS-CoV2 pandemic: HPC resources have been supporting the investig-
ations of the interaction between the spike protein of the virus and the human ACE2
receptors, the development of remedies and vaccines, and the analysis of the infec-
tion spread9. A further example is the large hadron collider (LHC) at CERN: Even
after filtering the experimental data, the LHC produces one petabyte (106 GB) of data
per day10, which can only be stored and analysed by huge computing facilities. In
addition to the numerics based on classical programs, artificial intelligence (AI) has
emerged as a tool for analysing scientific data. The growth in demand for HPC facil-
ities is thus expected to remain undiminished.
The fast growth of the world’s computing facilities can impressively be demonstrated
by the Top500 list of the world’s largest supercomputers11. In 1993, right after my
birth, the total peak performance of the Top500 amounted to 1.1 · 1012 floating-point
operations per second (FLOPS) or – with the appropriate prefix – 1.1 TFLOPS. To
classify this number, consider that a modern smartphone has a peak performance
of a few GFLOPS, so the top 500 smartphones in 2022 have more computational
power than the Top500 supercomputers in 1993. Today, almost 30 years later, the
Top500 peak performance has passed six orders of magnitude – or two prefixes –
9Mit Höchstleistungsrechnern und Data Analytics gegen das Coronavirus, https : / / www .
gesundheitsindustrie- bw.de/fachbeitrag/aktuell/mit- hoechstleistungsrechnern- und- data-
analytics-gegen-das-coronavirus (visited on 13/04/2022).
10Storage, https://home.cern/science/computing/storage (visited on 13/04/2022).
11PERFORMANCE DEVELOPMENT, https : / / top500 . org / statistics / perfdevel/ (visited on
13/04/2022).
6
Performance
1.1 The microprocessor
and amounts to 3.0 EFLOPS. It is a matter of time until performance on the exas-
cale is provided by the leading machine on its own. Fig. 1.1 shows that the total
performance of the Top500 list grows at exponential speed. From the 1990s to the
2010s, the tenfold-increase time was 4 years. In the last decade, however, the growth
has decelerated to a tenfold-increase time of approximately 8 years. Although there
is no definite reason for the decline, two points certainly contribute to arising dif-
ficulties in setting up more and more powerful supercomputers12. Firstly, Moore’s
law, having described the improvement of single processors (single- and multi-core)
for decades, approaches its limits. As a consequence, computational power cannot
notably improve at the processor level any more. Instead, the number of processors
in the HPC facilities grows, which, however, leads to the second problem: Because
each processor costs a certain amount of money and consumes a certain amount of
electric power, the costs for installing and running a supercomputer inflate with the
number of processors. A concrete example for the hugeness of the electric power
consumption of modern supercomputers is “Fugaku”13 in Japan, the leading super-
computer in 2021 with a peak performance of ∼500 PFLOPS. It is supplied with a
power of 30 MW, which corresponds to 260 GWh per annum. This equals the electric
power demand of 85 000 German average households in 201914.
The exploding operating costs put an economic limit on the growth of the world’s HPC
capacity based on the current technology. In particular, the electricity demand bears
a power-consumption wall which is already approaching and will become growingly
important within the 2020s (see the decline in growth, Fig. 1.1). The power prob-
lem even tapers as sustainability policies claim that the significantly growing energy
demand of the IT sector must be satisfied by regenerative sources – in addition, to de-
carbonising all other electric power generation. Thus, the maintenance of the compu-
tational progress will certainly be a tough challenge. As the recent chip crisis demon-
strates15, our modern economy is very sensitive towards the availability of more and
more capacious computing facilities. A stalling of their growth would entail disrup-
tions in every sector of our everyday life and seriously threaten the progress of sci-
ence and technology. In order to leave the silicon MOSFET technology, billions of
dollars are invested in the exploration of alternative concepts for integrated circuits
which permit miniaturising the features, increasing the clock rates and curbing the
energy consumption. The two concepts relevant for this work are the field of aniso-
tropic, metallic nanowires, to which the rare-earth silicide nanowires belong, and
spintronics, for which the Sb@Bi2Se3 heterostructures may be interesting. A break-
12TOP500 Meanderings: Sluggish Performance Growth May Portend Slowing HPC Market, https://www.
top500.org/news/top500-meanderings-sluggish-performance-growth-may-portend-slowing-hpc-
market/ (visited on 13/04/2022).
13TOP 10 Sites for November 2020, https://top500.org/lists/top500/2020/11/ (visited on 13/04/2022).
14Stromverbrauch der privaten Haushalte nach Haushaltsgrößenklassen, https://www.destatis.de/
DE / Themen / Gesellschaft - Umwelt / Umwelt / UGR / private - haushalte / Tabellen / stromverbrauch -
haushalte.html (visited on 24/05/2022).
15Halbleiter-Knappheit – Wie der Chipmangel überwunden werden kann, https://www.faz.net/aktuell/
wirtschaft/digitec/chipmangel- wie- das- problem- ueberwunden- werden- kann- 17682182.html
(visited on 24/05/2022), Ende der Chip-Engpässe könnte noch auf sich warten lassen, https://www.
faz.net/agenturmeldungen/dpa/ende- der- chip- engpaesse- koennte- noch- auf- sich- warten-
lassen-18027141.html (visited on 24/05/2022).
7
1 Introduction
through in novel processor technology would end the age of silicon-based MOSFET
processors, which have driven the scientific and economic progress for almost 60
years. The replacement of the vacuum tubes by solid-state transistors in the 1950s
proves how quick such an adaption can proceed.
1.2 The structure of this work
This thesis is organised in two parts, which are each dedicated to one of the two
above-mentioned nano-scaled systems. Both systems are derivable from higher di-
mensional structure prototypes, which are investigated as well since their properties
are necessary for understanding the lower dimensional systems. At the beginning
of each part, the respective systems are introduced in a general manner. The first
chapter of each part contains preparatory remarks which are necessary for under-
standing the particular methods employed in that part. The subsequent chapters
each treat one system, beginning with an overview over the literature and the for-
mulation of the scientific questions, and concluding with a discussion of the results.
In detail:
Part I treats the rare-earth silicide nanowires in four chapters: the chemical back-
ground of the constituents (Chap. 3), the bulk phases of RESi2 – x (Chap. 4), the mono-
layer TbSi2 film on Si(111) (Chap. 5), and the TbSi2 nanowires on Si(557) (Chap. 6).
Part II treats the thin antimony layers on Bi2Se3 in three chapters: the concept of geo-
metric phases (Chap. 7), the bulk phases of Bi2Se3 and Sb (Chap. 8), and the Sb@Bi2Se3
heterostructures (Chap. 9).
The thesis concludes with Chap. 10, where also an outlook on future work is given.
Part of the findings on the nano-scaled systems have already been published. The
own and co-authored publications are labelled by a separate indexing system, pre-
fixed by the letter “P”. Whenever this thesis addresses those publications, they are
referenced accordingly. The full publication list can be found on p. A24. Before the
investigations start, the general methods are shortly introduced in Chap. 2, with focus
on the density functional theory and its extensions.
8
2 Methods
The stationary condensed-matter problem consists in solving the eigenvalue equa-
tion for the Hamilton operator for 𝑁K nuclei with positions {𝑹𝑘} and momenta {𝑷𝑘},
and 𝑁e electrons with positions {𝒓𝑖} and momenta {𝒑𝑖}:
𝐻 = 𝑇K({𝑷𝑘}) + 𝑇e({𝒑𝑖}) +𝑉e–e({𝒓𝑖}) +𝑉K–K({𝑹𝑘}) +𝑉e–K({𝒓𝑖}, {𝑹𝑘}) (2.1)
where 𝑇K({𝑷𝑘}) are the kinetic energies of the nuclei; 𝑇e({𝒑𝑖}) are the kinetic ener-
gies of the electrons;𝑉e–e({𝒓𝑖}) are the electron-electron interactions;𝑉K–K({𝑹𝑘}) are
the nucleus-nucleus interactions; 𝑉e–K({𝒓𝑖}, {𝑹𝑘}) are the electron-nucleus interac-
tions. Calculating the eigenspace of Eq. (2.1) is a quantum-mechanical “𝑁 -particle
problem”, which is insoluble – even within numerics – as 𝑁 ∼ 1024 ≈ 1 mol in typical
solid-state systems. Therefore, methods are necessary which simplify the problem.
This chapter presents the main methods employed in this work: the density func-
tional theory (DFT) and the Bloch theorem. While the DFT is quite familiar to most
condensed-matter scientists, its extensions might be less known since they are rel-
evant only in special situations. The three approaches relevant for this work are
the LDA+U method, spin-orbit coupling (SOC) and van der Waals (vdW) corrections,
which are each shortly introduced. Both the DFT and the Bloch theorem are the fun-
dament of many plane-wave codes by means of which the properties of a solid-state
system can be calculated. The codes relevant for this work are the Vienna Ab initio
Simulation Package (VASP) [2, 3] and Quantum Espresso (QE) [4, 5].
2.1 Density functional theory
The first simplification of the solid-state problem consists in the Born-Oppenheimer
approximation. It is based on the assumption that the light electrons follow the move-
ment of the much heavier nuclei instantaneously, hence also called adiabatic approx-
imation. Consequently, the electronic subsystem can be separated from the dynamics
of the nuclei in that the nuclear positions {𝑹𝑘} enter the electronic Hamiltonian 𝐻e
merely as parameters. The electronic ground state energy 𝐸e({𝑹𝑘}) in turn determ-
ines the potential of the nuclear Hamiltonian 𝐻K [6, pp. 27ff].
𝐻e({𝑹𝑘}) = 𝑇e({𝒑𝑖}) + 𝑉e–e({𝒓𝑖〉}) +𝑉e–K({𝒓𝑖} | {𝑹𝑘}) +〉𝑉K–K({𝑹𝑘})
𝐻e({𝑹 𝑘}) Ψ({𝑹𝑘}) = 𝐸e({𝑹 }) 𝑘 Ψ({𝑹𝑘}) (2.2)
𝐻K = 𝑇K({𝑷𝑘}) + 𝐸e({𝑹𝑘})
9
2 Methods
As the electronic and the nuclear subproblems are disentangled, they can be treated
independently. A very common approach is to treat the electronic system quantum-
mechanically, while the nuclei (or ions if core electrons are attached) move in their
potentials 𝐸e({𝑹𝑘}) according to classical mechanics. The electronic system can be
simplified by exploiting symmetries. Most importantly, the translation symmetry of
crystals reduces the infinite number of electrons to a finite one, according to the
Bloch theorem (vide infra). Furthermore, the symmetries of the space group remove
redundancies in the unit cell. Although the symmetries can reduce the number of
interacting electrons to ∼100, the computational demand of directly solving the 𝑁 -
particle Schrödinger equation will be still huge. Walter Kohn, who was awarded with
the Nobel prize in chemistry for the DFT, impressively demonstrates this in his Nobel
prize lecture in 1999 [7, p. 1257]:
For an exemplary accuracy of O(10−2) of the total energy, each degree of freedom
contributes 3 ≤ 𝑝 ≤ 10 parameters to the Hilbert space of the multi-electron system,
resulting in a total number of parameters of
𝑀 = 𝑝3𝑁 . (2.3)
Even if 𝑝 is set to the lower boundary, the number of parameters for 𝑁e = 100 elec-
trons will exceed any imagination: 𝑀 = 33·100 ≈ 10143. Kohn states that he “cannot
foresee an advance in computer science which can minimize a quantity in a space
of 10150 dimensions” and referred to Eq. (2.3) as an “exponential wall” [7]. Today, 20
years later, one of the fastest supercomputers in the world is the Hawk machine in
Stuttgart, Germany. If its peak performance of 𝑃 = 26 PFLOPS [W16] was used for
calculating a quantity like the total energy depending on 𝑀 parameters, the compu-
tation time would be 𝑇 = 𝑀/𝑃 ≈ 10127 s at the minimum. For comparison, the age of
the universe amounts to 1018 s.
The basic idea of the DFT is that knowing observable quantities of a solid-state sys-
tem like the total energy 𝐸 and the ground state electron density 𝑛(𝒓) are sufficient
for many purposes. The detailed electronic many-particle wavefunction |Ψ⟩ is just a
means to an end. The bijective link between these three quantities is ensured by the
Hohenberg-Kohn theorem: 1) The electronic ground-state density 𝑛(𝒓) is uniquely
connected to the external potential 𝑣ext(𝒓) and 2) this ground state density minim-
ises the total energy 𝐸 of the system [7, 8]. The second part of the theorem leads to
the energy functional 𝐸[𝑛(𝒓)].
𝐸∫[𝑛(𝒓)] = 𝑇s [𝑛(𝒓)] +𝑉 [𝑛(𝒓)] +𝑈 [𝑛(𝒓)]∫+ 𝐸xc∫[𝑛(𝒓)] (2.4)with ′
𝑉 [𝑛(𝒓)] ( ) ( ) [ ( )] 1 ′ 𝑛(𝒓)𝑛(𝒓 )= d𝑟 𝑛 𝒓 𝑣ext 𝒓 and 𝑈 𝑛 𝒓 = d𝑟 d𝑟 2 𝒓 − 𝒓′
The total energy in Eq. (2.4) comprises four terms:
• the single-particle kinetic energy 𝑇s [𝑛(𝒓)] of the electrons
• the interaction between the electrons gas and the external potential 𝑉 [𝑛(𝒓)]
10
2.1 Density functional theory
• the classical Coulomb energy of the electron gas𝑈 [𝑛(𝒓)] (Hartree term)
• the exchange-correlation functional 𝐸xc [𝑛(𝒓)]
The last term, often abbreviated to xc-functional, contains any many-particle effects:
the part of the kinetic energy not covered by 𝑇s, the exchange energy between elec-
trons with parallel spins and the correlation due to electrostatic repulsion.
Eq. (2.4) can formally minimised by introducing the Lagrange parameter 𝜀 to reflect
the boundary cond∫ition of particle-number∫conservation[: ]
𝑁e − d𝑟 𝑛(𝒓) = 0 ⇒ 0 = d𝑟 𝛿𝑛(𝒓) ·
𝛿𝐸[𝑛(𝒓)]
( ) − 𝜀 (2.5)𝛿𝑛 𝒓
Kohn and Sham realised that Eq. (2.5) appears in the self-consistent Hartree scheme.
This scheme considers non-interacting electrons which move in an effective poten-
tial given by the external potential and their own electron gas. The correspond-
ing energy-density functional comprises 𝑇s [𝑛(𝒓)], 𝑉 [𝑛(𝒓)] and 𝑈 [𝑛(𝒓)]. Thus, the
Hartree scheme can be adapted to the Hohenberg-Kohn functional (Eq. (2.4)) by
adding the functional derivative of 𝐸xc [∑︁𝑛(𝒓)] to the effective potential [7, 9].𝑁e 2
[ 𝑛(𝒓) = 𝑓 𝑖 𝜓𝑖 (𝒓) ] (2.6a)𝑖=1
𝜀𝑖𝜓𝑖 (𝒓) = −1 22∇∫ + 𝑣ext(𝒓) + 𝑣H(𝒓) + 𝑣xc(𝒓) 𝜓𝑖 (𝒓) (2.6b)with
( ) 𝛿𝑈 [𝑛(𝒓)] ′ 𝑛(𝒓′)  ( ) 𝛿𝐸xc [𝑛(𝒓)]𝑣H 𝒓 = ( ) = d𝑟 ′ and 𝑣xc 𝒓 =𝛿𝑛 𝒓 𝒓 − 𝒓 𝛿𝑛(𝒓) (2.6c)
The coefficients 𝑓𝑖 ∈ [0, 1] in Eq. (2.6a) are occupation numbers which fill the ener-
getically lowest eigenstates according to a function of choice, e.g. finite-temperature
smearing. The Fermi energy is defined as the chemical potential of the fermions, coin-
ciding with the highest occupied level if the occupation function is a step function.
Eqs. (2.6), the Kohn-Sham equations, define a self-consistent scheme for obtaining the
electronic ground state:
1) Start with a trial charge density 𝑛(𝒓) and calculate the potentials 𝑣H(𝒓) and
𝑣xc(𝒓) with Eq. (2.6c).
2) Calculate the eigenvalues and eigenstates {(𝜀𝑖 , 𝜓𝑖 (𝒓))𝑖} of Eq. (2.6b).
3) With these states, calculate the new charge density with Eq. (2.6a) and iterate.
The above steps are looped until convergence is reached according to a stop criterion,
e.g. the constancy of the total energy 𝐸[𝑛(𝒓)] (Eq. (2.4)). The Kohn-Sham scheme
maps the many-particle problem of 𝑁e interacting electrons onto an effective system
of single-particle differential equations for 𝑁e non-interacting fermions. Therefore,
it considerably reduces the computational demand of the solution. The only term
which depends on the concrete system is the external potential 𝑣ext(𝒓), while all other
11
2 Methods
terms depend uniquely and universally on the charge density. Although the Kohn-
Sham scheme is formally exact, the unknown xc-functional 𝐸xc [𝑛(𝒓)] is a source of
uncertainty as it has to be approximated, despite its universality (vide infra).
All density-related quantities are, in principle, observable: the ground state density
𝑛(𝒓), the ground state energy 𝐸[𝑛(𝒓)] and, hence, all derived quantities, e.g. the ionic
forces. On the contrary, the Lagrange parameters 𝜀𝑖 , having the dimension of an en-
ergy, and the single particle orbitals 𝜓𝑖 (𝒓) are mere mathematical artefacts which
have no physical meaning a priori. In particular, they are not necessarily observable
and can differ considerably from real excitation energies, even if an exact represent-
ation of the xc-functional was given.
Several approximations for the xc-functional 𝐸xc [𝑛(𝒓)] exist. The best-known is the
local density approximation (LDA) which approximates 𝐸xc [𝑛(𝒓)] locally by the value
for a respective homogeneous electron gas. While systems with slowly varying dens-
ity can be described well, inhomogeneous systems can incur systematic failures. This
can be corrected (and is then likely to be overcorrected) by taking also the gradi-
ents of the local density into account, leading to the generalised-gradient approxim-
ation (GGA). Perdew et al. proposed a simple implementation of GGA, the Perdew-
Burke-Ernzerhof functional (PBE), which avoids several weaknesses of former GGA
approaches and in which all parameters are fundamental constants [10]. Although
PBE works quite well for atoms and molecules, it overcorrects the overbinding of
LDA in solids, so bulk lattice constants are slightly too large by ∼1 %. This issue is ad-
dressed by the Perdew-Burke-Ernzerhof functional revised for solids (PBEsol) [11].
The xc-functional is not restricted to scalar electronic densities, but it can be exten-
ded to spinorial densities. In this case, the density is a (2 × 2) matrix 𝑁 (𝒓). The
scalar electron density is then the trace of 𝑁 and the magnetisation densities are the
expectation values of the Pauli matrices {𝜎(𝑖}: )
𝑛(𝒓) = tr(𝑁 (𝒓) ) (2.7)
𝑚𝑖 (𝒓) = tr 𝜎𝑖𝑁 (𝒓)
The spinor components couple via the xc-functional, which depends on the scalar
electron density 𝑛(𝒓) and the total magnetisation density |𝒎(𝒓) |. As the derivatives
of the xc-functional yield a (2 × 2) potential, non-collinear magnetism emerges by
this means [12]. Although this approach allows the magnetic moments to assume
certain angles to each other, the magnetic structure is still isotropic, i.e. if all magnetic
densities are rotated by a rigid angle, the total energy remains the same. Magnetic
anisotropy follows from SOC (vide infra).
2.1.1 The LDA+U method
The approximative xc-functionals describe the total-energy-related quantities well if
the electrons are delocalised. In some cases, however, DFT fails in finding the cor-
rect ground state and yields considerably wrong energies and forces. In particular,
12
2.1 Density functional theory
systems with strong electronic correlation are problematic because DFT tends to de-
localise electrons and, thus, to weaken correlation effects. An example relevant for
this work consists in valence electronic multiplets which populate nearly atomic or-
bitals. The electronic correlation is then subjected to Hund’s rules, which force the
electrons into a certain configuration concerning spin and magnetic quantum num-
bers. In particular, the occupations of the single-particle subshell orbitals |𝜑𝑚⟩ are
integral, ergo 0 or 1. These on-site correlation effects prevail in localised d shells
and, even more pronounced, in localised f shells. The LDA+U method, on which Co-
coccioni wrote a comprehensive discussion [13], addresses this issue.
The approach is based on the orthogonality relations of quantum states, i.e. a loc-
alised subshell projects the delocalised electrons forming the chemical bonds out.
As a consequence, the intracorrelation of the subshell not covered by DFT is much
stronger than the intercorrelation between the subshell and the outer electrons. This
permits a selective correction of the former. A suitable ansatz is the Hubbard model,
which, being a tight-binding approach, is naturally constructed from a localised basis
set. In detail, each localised subshell orbital |𝜑𝐼𝑚⟩ with index 𝑚 and belonging to the
atomic site 𝐼 interacts with each subshell orbital |𝜑𝐼𝑚′⟩ on the same site. The sum of
the pairwise interactions, the Hubbard energy 𝐸Hub, is added to the DFT energy func-
tional 𝐸LDA. In order to avoid double-counting, the correlation energy 𝐸dc which DFT
already includes has to be subtracted. It is calculated from the mean-field interaction
of the electrons on the respective site, which depends only on their total number.
𝐸LDA+U [𝑛(𝒓)] = 𝐸 𝐼𝜎 𝐼𝜎LDA [𝑛(𝒓)] + 𝐸Hub [{𝑛𝑚𝑚′}] − 𝐸dc [{𝑁 }] (2.8)
where 𝑛(𝒓) is the DFT electron density and 𝜎 the spin. 𝑛𝐼𝜎𝑚𝑚′ is the on-site density
matrix defined by the projections of the Kohn-Sham eigenfunctions |𝜓𝜎⟩ (Eq. (2.6b))
𝑖
on the localised basis set {|𝜑𝐼𝑚⟩} of site 𝐼 . {𝑁 𝐼𝜎∑︁ } is the total number of electrons inthe subshell of site 𝐼 . 〈  〉〈  〉
𝑛𝐼𝜎 = 𝑓 𝜎 𝜓∑︁𝜎  𝜑𝐼 𝜑𝐼 𝜎𝑚𝑚′ ′ 𝑖 𝑖 𝑚 𝑚 𝜓𝑖 (2.9a)𝑖
𝑁 𝐼𝜎 = 𝑛𝐼𝜎𝑚𝑚 (2.9b)
𝑚
where 𝑓 𝜎 is the occupation function.
𝑖
In the one-band version of the Hubbard model, each site 𝐼 can be occupied by two
electrons with opposite spin. Only electron pairs on the same site increase the po-
tential energy of the Hamiltonian by the parameter𝑈 > 0 representing the repulsive
Coulomb interaction. In the limit of zero-hopping (total localisation), the Hubbard
total energy comprises only the 𝑈 contr∑︁ibutions. If the sites are inequivalent, theparameter𝑈 depends on 𝐼 .
𝐸 = 𝑈 𝐼𝑛𝐼↑𝑛𝐼↓Hub,1 (2.10)
𝐼
where 𝑛𝐼𝜎 is the occupation number of site 𝐼 and spin 𝜎. If two or more orbitals
interact on the same site, the total Coulomb interaction equals the sum of two-particle
13
2 Methods
interactions between all pairs of electrons with different orbital and/or spin quantum
numbers on that site. In this case, the total energy comprises not only the Coulomb
repulsion between each two electrons, which is modelled by the parameter𝑈 . It has
also a term describing the attractive exchange interaction between electrons with
parallel spin, which is modelled by a second parameter 𝐽 > 0. The Hubbard energy
functional 𝐸 [{𝑛𝐼𝜎Hub 𝑚∑︁𝑚′}[] is the∑︁n: ∑︁ ∑︁ ]𝐼 𝐼
𝐸 [{𝑛𝐼𝜎 }] 𝑈= [ 𝑛𝐼𝜎
′
𝑛𝐼𝜎 − 𝐽 𝑛𝐼𝜎 𝐼𝜎Hub 𝑚𝑚′ ′ ′ 𝑛 ′ ′∑︁ 2 𝑚𝑚 𝑚 𝑚 2 𝑚𝑚 𝑚 𝑚𝐼 𝑚,𝜎≠𝑚′𝜎′ 𝑚≠𝑚′ 𝜎 ]𝑈 𝐼 ∑︁ 𝐼 𝐼 ∑︁
= 𝐼𝜎 𝐼 (−𝜎)
𝑈 − 𝐽
𝑛 𝑛 𝐼𝜎 𝐼𝜎
2 𝑚𝑚 𝑚′𝑚′
+ 𝑛
2 𝑚𝑚
𝑛𝑚′𝑚′ (2.11)
𝐼,𝜎∑︁ [ 𝑚,𝑚′ ( 𝑚≠𝑚′ )]Eq. (2.9b) 𝑈 𝐼 𝐼 𝐼 ∑︁
= 𝑁 𝐼𝜎𝑁 𝐼 (−𝜎) + 𝑈 − 𝐽 𝑁 𝐼𝜎𝑁 𝐼𝜎 − 𝑛𝐼𝜎𝑚𝑚𝑛𝐼𝜎2 2 𝑚𝑚
𝐼,𝜎 𝑚
The double counting term 𝐸 [{𝑁 𝐼𝜎dc }] is approximated by the mean-field interaction
of the total number of electrons. If 𝑁↑ and 𝑁↓ are integers, there are 𝑁↑ ·𝑁↓ antipar-
allel spin pairs and 𝑁𝜎 · (𝑁𝜎 − 1)/2 parallel spin pairs for each spin. The former con-
tribute only repulsive 𝑈-terms, the latter repulsive 𝑈-terms and attractive 𝐽-terms.
This can be extrapolat∑︁ed t[o∑︁non-integral {𝑁
𝜎}∑︁. ]𝑈 𝐼 𝐼[{ 𝐼𝜎}] 𝐼𝜎 𝐼 (−𝜎) + 𝑈 − 𝐽 𝐼𝐸dc 𝑁 = 𝑁 𝑁 𝑁 𝐼𝜎 (𝑁 𝐼𝜎 − 1) (2.12)2 2
𝐼 𝜎 𝜎
Inserting Eqs. (2.11) and (2.12) into Eq. (2.8) and using Eq. (2.9b) gives the total cor-
rection of the DFT energy functional.∑︁ ( )𝐼 𝐼 ∑︁
𝐸LDA+U [𝑛(𝒓)] [ ( )] +
𝑈 − 𝐽
= 𝐸 𝐼𝜎LDA 𝑛 𝒓 ∑︁ 𝑁( − 𝑛
𝐼𝜎 𝐼𝜎
2 𝑚𝑚
𝑛𝑚𝑚
𝐼𝜎 𝑚 ) (2.13)(∗) 𝑈 𝐼 − 𝐽 𝐼
= 𝐸LDA [𝑛(𝒓)] + tr(?̂?𝐼𝜎) − tr( [?̂?𝐼𝜎]2)2
𝐼𝜎
Step (∗) uses that the atomic subshells have a finite and orthonormal basis, the spher-
ical harmonics. Thus, each configuration has an axis along which the density matri-
ces are diagonal. Replacing the them by the density operators ?̂?𝐼𝜎 renders Eq. (2.13)
invariant under unitary transformations of the localised basis set [13].
The Hubbard approach to correct the correlation in the DFT is called LDA+U method
since the𝑈 is the most important parameter. It is not restricted to LDA, but it can also
be adapted to other functionals like GGA. Eq. (2.13) is the rotationally invariant and
simplified LDA+U formulation introduced by Dudarev et al., where the parameters𝑈
and 𝐽 represent the spherically averaged Coulomb integrals [14, 15]. The more gen-
eral formulation employs explicitly calculated screened Coulomb integrals between
14
2.1 Density functional theory
the subshell wavefunctions for 𝐸Hub, as proposed by Liechtenstein et al. Neverthe-
less, the double-counting energy 𝐸dc has still to be parametrised by𝑈 and 𝐽 [16]. The
simplified formulation in Eq. (2.13) demonstrates important properties of the LDA+U
correction. The additive to the DFT functional is always positive if the density matrix
is not idempotent1. Conversely, if ?̂? is idempotent, i.e. all eigenvalues are 0 or 1, the
correction term vanishes. In other words, the LDA+U method penalises fractional oc-
cupations and drives the subshell into a state where all orbitals are fully occupied or
empty. Nota bene, the LDA+U correction in Eq. (2.13) does not explicitly encourage
spin polarisation (Hund’s second rule) as the effective (𝑈 − 𝐽)-term does not include
a magnetisation-dependent penalty.
An important deficiency of the LDA+U method is the introduction of parameters to
the DFT approach, which is otherwise ab initio and avoids empirical parameters.
The parameters𝑈 and 𝐽 for a particular system are often determined by tuning them
so that the materials properties match the experiment (e.g. band gaps, lattice con-
stants). However, this approach opens a gateway for systematic errors because it can
overcorrect other, erroneous effects and thus obfuscate weaknesses of the model. A
more sophisticated approach is to calculate the parameters self-consistently from lin-
ear response, as introduced by Cococcioni. In detail, a small penalty 𝛼𝐼 is applied to
the Kohn-Sham potential of a certain site 𝐼 and the effect on the on-site occupations
𝑁𝐾 = 𝑁𝐾↑ + 𝑁𝐾↓ is measured for sites 𝐾 . This leads to the susceptibility 𝜒.
d𝑁𝐾
𝜒𝐾𝐼 = (2.14)d𝛼𝐼
The inverse of 𝜒 basically equals the negative Hubbard𝑈 , according to the quadratic
dependence of the double-counting term on the on-site density 𝑁 𝐼 (Eq. (2.12)). How-
ever, a term from the electronic rehybridisation at constant total density, leading to
the susceptibility 𝜒0, has to be subtracted [13].
𝑈 𝐼 = (𝜒−10 − 𝜒−1)𝐼𝐼 (2.15)
The procedure of determining𝑈 can be summarised as follows:
1) Calculate the self-consistent DFT ground-state density of the system in question.
2) Apply an on-site penalty 𝛼𝐼 to site 𝐼 and calculate the respective on-site occupa-
tions 𝑁𝐾(𝛼𝐼) within a non-self-consistent calculation at the fixed ground-state
density. This yields 𝜒0 by Eq. (2.14) and finite differences.
3) Repeat 2), but let the charge relax (self-consistent charge density with penalty).
This yields 𝜒 by Eq. (2.14) and finite differences.
4) Calculate𝑈 𝐼 with Eq. (2.15).
1A quadratic matrix ?̂? is idempotent, if ?̂?2 = ?̂?.
Because tr(?̂?) = 1 and ?̂?∗ = ?̂?, the Cauchy-Schwarz inequality holds: tr(?̂?2) ≤ tr(?̂?).
15
2 Methods
2.1.2 Spin-orbit coupling in DFT
The steep potentials of nuclei with high atomic numbers give rise to relativistic ef-
fects, which must be considered if a material contains elements of the fifth row of the
periodic table and beyond. In particular, the elements of the sixth row, the last row
containing stable isotopes, are subjected to a considerable relativistic contraction of
the atomic shells. As a response to relativistic corrections, the 6s radius shrinks by a
few percent for the lighter elements, by ∼10 % for the heavier ones and by 15 % for
gold (Au, 𝑍 = 79), a local extremum [17, pp. 372f].
The fully relativistic version of quantum mechanics is based on the Dirac equation
which fulfils the prerequisite of invariance under Lorentz transformations. Being a
system of four coupled differential equations, the Dirac equation is demanding, even
for a single particle, and contains much redundant information for the purposes of
condensed-matter theory. The two approximations to make the relativistics suitable
for solid-state systems are the elimination of the small component and a truncated
series expansion of the resulting Hamilton operator2. The zeroth-order approxim-
ation of the correct expansion yields the following Hamilton operator, called CPD
Hamiltonian after Chang, Pelissier, and Durand [19, 20]: ( )
1 𝑉 (𝒓) −1
𝐻0 = 𝑉 (𝒓) + (𝝈 · 𝒑)𝜅(𝒓) (𝝈 · 𝒑) with 𝜅(𝒓) = 1 − (2.16)2𝑚0 2𝑚0𝑐2
where𝑉 (𝒓) is the potential, 𝑚0 the rest mass of the electron, 𝑐 the speed of light, and
𝒓 and 𝒑 the canonical variables. As 𝝈 = (𝜎𝑥 , 𝜎𝑦, 𝜎𝑧) is the vector of Pauli matrices, the
Hamilton operator in Eq. (2.16) acts on a 2-spinor. Test calculations on the uranium
atom (U, 𝑍 = 92), in which all core and valence states are affected by relativistics,
show that the zeroth order CPD Hamiltonian is quite accurate. It predicts the valence
levels very close to those calculated with the full Dirac equation and fails only in
predicting the levels of the most inner K and L shells [19].
The commutator relation [𝑟𝑖 , 𝑝 𝑗] = iℏ𝛿𝑖 𝑗 and the algebra of the Pauli matrices lead to
the following, more interpretable representation of 𝐻0:
(𝝈 · 𝒂) (𝝈 · 𝒃) (= 𝒂 · 𝒃 + i𝝈) · (𝒂 × 𝒃) (∗)
[ ( )] Eq. (A.93) −∇𝒑 𝑟𝑉 (𝒓) −iℏ, 𝜅 𝒓 = −iℏ(𝜅
′(𝒓) ) = 𝜅
2(𝒓) ∇
2 2 ( 2 ( 2 ) ) 𝑟𝑉 (𝒓) (∗∗)𝑚0𝑐 𝑚0𝑐(∗)
𝐻0 = 𝑉 (𝒓) +
1 · i𝒑 (𝜅(𝒓)𝒑) + 𝝈 · 𝒑 × 𝜅(𝒓)𝒑2𝑚0 2𝑚0 (2.17)(∗∗)
= 𝑉 (𝒓) + 1 𝒑 · 𝜅(𝒓) ℏ𝒑 + 𝜅2( )2 (𝒓) 𝝈 · (∇2 2 𝑟𝑉 (𝒓) × 𝒑)𝑚0 𝑚0𝑐
2An interesting issue of these expansions concerns in the convergence of the series. The traditional
expansion in (𝐸 − 𝑉 )/2𝑚 𝑐20 causes severe problems because electrons have non-zero probability
densities near the singularities of a Coulomb potential wherefore the expansion parameter diverges.
The expansion in 𝐸/(2𝑚0𝑐2 −𝑉 ) leads to a reliable and fast convergence [19].
16
2.1 Density functional theory
According to Eq. (2.17), the main relativistic effects can be summarised as two con-
tributions: an alteration in the kinetic energy which, in a simplifying manner, cor-
responds to the relativistic mass increase and the spin-orbit coupling (SOC) term. The
name of the latter name becomes obvious after inserting a spherically symmetric
potential as in the case of single atoms:
𝐻atom
1 ( )
0 = 𝑉 (𝒓) + 𝒑 · 𝜅(𝒓)𝒑 + 𝐻2 SOC𝑚0 (2.18)
ℏ 1 d𝑉
with 𝐻SOC = ( )2 𝜅
2(𝒓) 𝝈 · 𝑳 and 𝑳 = 𝒓 × 𝒑
2𝑚0𝑐 𝑟 d𝑟
Solving the eigenspace of Eq. (2.18) for solid-states is still demanding wherefore the
code package used in this work, VASP, uses a further approximation: As the relativist-
ics is expected to prevail solely in the vicinity of the atomic cores3, the SOC Hamilto-
nian can be assumed to act solely within the PAW spheres, which are introduced by
the projector-augmented wave (PAW) method [21, 22]. As a consequence, 𝐻SOC can
be expressed solely by means of the PAW projectors which mix the on-site densit-
ies of the pseudo-spinor [23]. Moreover, since the PAW projections are still resolved
with respect to orbital quantum numbers and spin, it is possible to track the atomic
orbitals upon forming chemical bonds even with SOC included.
2.1.3 Van der Waals corrections
Since the correlation part of 𝑣xc is local in the charge density or its gradients, DFT
bears the general weakness that it describes long-range electronic correlation insuf-
ficiently. This is less problematic for systems in which all bonding is covalent or ionic
as the atomic overlap lets the xc-functional account for the major part of correlation.
However, it can deteriorate the results for materials in which van der Waals interac-
tions dominate the bonding between closed-shell entities.
Van der Waals (vdW) interactions, more precisely the London dispersion effects4,
arise from fluctuations in charge distributions which polarise the environment. In
the demonstrative case of neutral molecules, the temporary dipole moment emerging
in one molecule induces dipole moments in the adjacent molecules, which results in
a net attractive force. The fundament of such fluctuations consists in the quantum
nature of charge, which is the probability density of quantum particles. The mutual
dependence of the temporary dipoles leads to their quasi-synchronised occurrence
and is thus a type of electronic correlation.
3If the potential vanishes, 𝜅(𝒓) is close to unity. Furthermore, the derivative of 𝑉 (𝑟) vanishes fast
for Coulomb-like 1/𝑟 potentials. Thus, Eq. (2.16) adopts the form of the non-relativistic Hamilton
operator 𝐻 = 𝑉 (𝒓) + 𝒑2/2𝑚0 on domains excluding the neighbourhood of the nuclei.
4The original work on interactions between neutral molecules by Johannes Diderik van der Waals
included attractive and repulsive forces of differing nature. One of these are the attractive inter-
actions from short-frequency perturbations between polarisable molecules, which were described
by Fritz London. He called these interactions the “dispersion effect” as he found that knowledge
about the frequency-dependence of the response of the charge densities towards short-frequency
perturbations is necessary for correctly calculating the same [24].
17
2 Methods
The common mistake to avoid when considering vdW interactions consists in the as-
sumption that these interactions are generally small. While this is indeed true for
the mutual attraction between isolated atoms, it proves to be completely wrong in
dense systems where vdW interactions dominate the intermolecular bonding. For ex-
ample, the boiling points of hydrogen halides increase with growing halogen atomic
numbers, despite the decrease of the static dipole moments. In this case, vdW in-
teractions obviously exceed the interactions between the static dipoles. Not only do
vdW interactions influence the intermolecular bonding, but they can also influence
the intramolecular bonding. For instance, the alkanes are often employed to demon-
strate the positive relation between vdW-related intermolecular attraction and chain
length. However, they are also subjected to intramolecular vdW interactions which
stabilise branched molecules with respect to the linear ones [25]. Further examples
where vdW interactions play an important role are “the structures of DNA and pro-
teins, the packing of crystals, the formation of aggregates, host-guest systems, or the
orientation of molecules on surfaces or in molecular films” [26]. Being a dynamic
long-range correlation effect, vdW interactions are complicated to describe and nu-
merically expensive. In particular their introduction into DFT is subject of a vivid
research activity which tries finding a compromise between numerical feasibility,
accuracy and avoiding empirical parameters. The large number of approaches can
be classified by their level of sophistication [25]. It is therefore obligatory to test the
different vdW schemes in order to find the method which is sufficiently accurate for
the respective problem while saving computational resources.
The DFT-D2 method introduced by Grimme will be the most important method in
this work. It is a semi-empirical approach which adds the 𝑅−6-dependent dipolar
dispersion correction for each pair of inter∑︁acting atoms [26]:𝐼 𝐽
𝐸DFT-D2
𝑠 𝐶
= − 6 6disp 6 𝑓 (𝑅2 𝐼 𝐽 ) (2.19)
≠ 𝑅𝐼 𝐽 𝐼 𝐽
where 𝑅𝐼 𝐽 is the distance between the atoms 𝐼 and 𝐽 and 𝑠6 is a universal scaling
constant which depends on the xc-functio√︃nal. The coefficients
𝐼 𝐽
𝐶 = 𝐶𝐼6 6 ·
𝐽
𝐶6
are the geometric mean of the 𝐶6 coefficients of the single atoms. These, in turn, are
proportional to the respective static dipolar polarisabilities, i.e. they describe how
susceptible the atoms are towards vdW interactions. 𝑓 (𝑅𝐼 𝐽 ) is a damping function
limiting the vdW interactions if the atoms are too close to each other. The paramet-
ers are fitted to match experimental reference and are considered universal for the
employed atomic species [26].
A bit more sophistication provides the DFT-D3 method by Grimme et al. [27]. This
method complements the 𝐶6 contributions from the dipole-dipole dispersion by the
𝐶8 contributions from the dipole-quadrupole dispersion. In addition, the coefficients
are not fixed any more, but they scale with the fractional coordination number of the
respective atoms to partially reflect different chemical environments. Moreover, the
18
2.2 The Bloch theorem
parameter fitting is more ab initio. The original DFT-D3 method employs a damp-
ing function which scales the dispersion to zero for small atomic distances (zero-
damping) [27]. Later, a Becke-Johnson-type damping was introduced, which reduces
the dispersion to a finite value instead of zero and enhances short vdW contacts [25,
28]. The last vdW scheme tested in this work is the method introduced by Tkatchenko
and Scheffler [29]. It is formally identical to the DFT-D2 method, however the vdW
coefficients reflect the variation of the atomic charges with respect to the neutral
atoms by means of Hirshfeld partitioning. This procedure explicitly considers the
influence of the chemical environment on the polarisabilities in an ab initio man-
ner. The original method [29], however, has problems with ionic systems, in which
charge transfer alters the atomic reference charges. The solution consists in shifting
the reference from the neutral atoms to the atoms in a charged state by means of an
iterative Hirshfeld partitioning scheme [30, 31].
The construction of the above semi-empirical methods shows why vdW interactions
become growingly important in materials involving elements with high atomic num-
bers: Since the 𝐶 coefficients are proportional to atomic polarisabilities, atoms with
larger electron shells contribute more dispersion to the total energy. Therefore, in
particular the elements of the last rows of the periodic table are expected to have
high vdW-related effects on the structures which incorporate them.
2.2 The Bloch theorem
Most solid-states are periodic, i.e. their properties are completely described on a
finite, microscopic domain which is replicated into all 𝑁 spatial directions. Tradi-
tional crystals, for example, are periodic in all three spatial dimensions with a small
replication unit representing the symmetry of the macroscopic material. Also non-
periodic systems can be described with a periodic ansatz. The unit cell has then to be
large enough, that the local conditions are well described and the influence from the
periodic replica vanishes. The domain of a periodic system is the generalised period
length, which can always be chosen as an𝑁 -dimensional parallelepiped (a line, a par-
allelogram and a parallelepiped for 𝑁 = 1, 2, 3, respectively). It contains all inform-
ation about the macroscopic properties, in particular the potential and the solutions
of the corresponding quantum-mechanical eigenvalue equation. Most importantly,
the finiteness of the domain ensures the integrability of the Hilbert space.
The wavefunctions of a particle in a periodic potential underlie boundary conditions,
which are summarised as Bloch’s theorem. It can be derived in several ways like
the common one considering the properties of translation operators or the abstract
one employing 𝑁 -dimensional Fourier series. The derivation presented here is based
on the properties of the translation group and connects the Bloch theorem with the
gauge degree of freedom of the wavefunctions. Furthermore, some identities are lis-
ted, which become important when the geometric phases are considered (Chap. 7).
In the following paragraphs, a single quantum particle moving in a periodic poten-
tial is assumed. Since the solutions of the Kohn-Sham equations fulfil this condition,
Bloch’s theorem is compatible with the DFT.
19
2 Methods
The direct lattice. A 𝑁 -dimensional crystal is defined by a set of 𝑁 linearly inde-
pendent (direct) basis vectors {𝑹𝑖 , 𝑖{∑︁= 1, . . . , 𝑁 }. The par}allelepiped
CR B 𝑟𝑖𝑹𝑖 | 𝑟𝑖 ∈ [0, 1] (2.20a)
𝑖
is the (direct) unit cell and the set {∑︁ }
R B 𝑛𝑖𝑹𝑖 | 𝑛𝑖 ∈ ℤ (2.20b)
𝑖
is the (direct) lattice of the crystal. An alternative unit cell, the Wigner-Seitz cell, is
defined as the set of points which are closer to 𝑹0 = 0 than to any other 𝑹 ∈ R.
The Bloch theorem as a gauge function
The physics of a charged particle in a crystal is determined by an electrostatic poten-
tial𝑉 (𝒙) which is periodic onR. Consequently, the corresponding Hamilton operator
𝐻 of a non-magnetic system (𝑨 ≡ 0) is invariant under translations by a lattice vector
𝑹 ∈ R, which are represented by the unitary translation operator𝑈𝑹.
𝑈𝑹 = exp(−i 1ℏ 𝑹 〉· 𝒑), 𝑈 𝐻〉 (𝒑, 𝒙)𝑈−1𝑹 𝑹 = 𝐻 (𝒑, 𝒙 −〉𝑹) = 𝐻 (𝒑 , 𝒙〉)⇒ 𝐻 (𝒑, 𝒙) Ψ𝑛 = 𝐸𝑛 Ψ ⇔ 𝐻 (𝒑, 𝒙)𝑈 𝑛 𝑹 Ψ𝑛 = 𝐸 𝑈 𝑛 𝑹 Ψ𝑛 (2.21)
Hence, if |Ψ𝑛⟩ is an eigenstate with eigenene rgy〉 𝐸𝑛, then𝑈𝑹 |Ψ𝑛⟩ is also an eigenstatewith the same energy. The effect of 𝑈𝑹 on Ψ𝑛 is thus equivalent to a gauge trans-
formation (Eq. (A.98)).
𝑈𝑹 = exp(−i𝜒(𝒙,𝑹)) (2.22)
The gauge function 𝜒(𝒙,𝑹) can be derived in two steps. Firstly, the group properties
of 𝑈𝑹 tell that 𝜒(𝒙,𝑹) is linear in 𝑹, so it can be written as a scalar product between
𝑹 and a vectorial function 𝒇 (𝒙).
𝜒(𝒙,𝑹) + 𝜒(𝒙, 𝑺) = 𝜒(𝒙,𝑹 + 𝑺), −𝜒(𝒙,𝑹) = 𝜒(𝒙,−𝑹) ⇒ 𝜒(𝒙,𝑹) = 𝑹 · 𝒇 (𝒙)
Secondly, because translations commute, 𝒇 (𝒙) has to be a constant vector.
exp(−i 1 𝑹′ℏ ·[𝒑) exp(−i𝑹 ]· 𝒇 (𝒙)) = exp(−i𝑹 · 𝒇 (𝒙)) exp(−i 1 ′ℏ 𝑹 · 𝒑)
⇒ 0 = − 𝒇 (𝒙),− 1 Eq. (A.93)ℏ𝒑 = i 𝒇
′(𝒙) ⇔ 𝒇 (𝒙) = 𝒌 ∈ ℝ𝑁
In summary, a crystal wavefunction which is translated by a lattice vector 𝑹 ∈ R
alters by a phase shift of −𝒌 · 𝑹, where 𝒌 is a constant, 𝑁 -dimensional vector. This
relation is called Bloch’s theoretions: 〉m and the wavefunc〉tions |Ψ𝑛,𝒌⟩ are called Bloch func-𝑈𝑹 Ψ𝑛,𝒌 = exp(−i𝒌 · 𝑹) Ψ𝑛,𝒌 ∀𝑹 ∈ R (2.23a)
Alternatively, in space representation:
Ψ𝑛,𝒌 (𝒙 + 𝑹) = exp(i𝒌 · 𝑹)Ψ𝑛,𝒌 (𝒙) (2.23b)
20
2.2 The Bloch theorem
The Bloch theorem proves that 𝒌 characterises the state |Ψ𝑛,𝒌⟩ like a quantum num-
ber, i.e. states with different 𝒌 (up to a reciprocal lattice vector, vide infra) are mutu-
ally 〈exclus ive. 〉 〈
Ψ 𝑛,𝒌 Ψ𝑛,𝒌′ = 〈Ψ𝑛,𝒌  −1 〉  〉 〈 𝑈 𝑈 Ψ ′ = Ψ   〉𝑹 𝑹 𝑛,𝒌 𝑛,𝒌 exp(i𝒌 · 𝑹) exp(−i𝒌′ · 𝑹) Ψ𝑛,𝒌′= Ψ𝑛,𝒌 Ψ𝑛,𝒌′ exp〈(i (𝒌 − 𝒌′) ·〉𝑹) ∀𝒌, 𝒌′ ∈ ℝ
𝑁 ,𝑹 ∈ R
⇒ Ψ𝑛,𝒌 Ψ𝑛,𝒌′ = 𝛿𝒌,𝒌′ (2.24)
The continuity of 𝒌 motivates the nota〉tion o f the B〉loch states as functions of 𝒌:Ψ 𝑛,𝒌 → Ψ𝑛(𝒌)
As the quantum-number nature of 𝒌 depends on the translation symmetry, it is also
called crystal momentum, analogously to momentum conservation following from
spatial homogeneity in classical mechanics.
The reciprocal lattice. The Bloch phase 𝒌 ·𝑹 in Eqs. (2.23) is periodic on the addition
of a vector 𝑮 ∈ ℝ𝑁 to 𝒌 with
𝑮 · 𝑹 = 2π𝑛, 𝑛 ∈ ℤ.
A basis for the 𝑮 vectors is defined by a set {𝑮 𝑗} with:
𝑮 𝑗 · 𝑹𝑖 = 2π𝛿𝑖, 𝑗 , 𝑹𝑖 ∈ R, 𝑖, 𝑗 = 1, . . . , 𝑁
Or more compactly written as matrices:
𝐺 B (𝑮1, . . . ,𝑮𝑁 ), 𝑅 B (𝑹1, . . . ,𝑹𝑁 ) ⇒ 𝐺ᵀ · 𝑅 = 2π𝐼 ⇔ 𝐺ᵀ = 2π𝑅−1
The column vectors of 𝐺 are the reciprocal basis vectors {𝑮 𝑗 , 𝑗 = 1, . . . , 𝑁 }, in ac-
cordance with their calculation fr{o∑︁m the (transposed) inverse of the matrix 𝑅. Theparallelepiped }
CG B 𝑠 𝑗𝑮 𝑗 | 𝑠 𝑗 ∈ [0, 1] (2.25a)
𝑗
is the reciprocal unit cell and the se{t∑︁ }
G = 𝑛 𝑗𝑮 𝑗 | 𝑛 𝑗 ∈ ℤ (2.25b)
𝑗
is the reciprocal lattice of the crystal. The Brillouin zone is the reciprocal analogon
of the Wigner-Seitz cell: the set of points being closer to 𝑮0 = 0 than to any other
𝑮 ∈ G.
Since the Bloch phase exp(−i𝒌 · 𝑹) is periodic on G, 𝒌 is only well defined within the
reciprocal unit cell CG (or within the〉Brillouin z〉one).Ψ𝑛(𝒌 +𝑮) = Ψ𝑛(𝒌) ∀𝑮 ∈ G (2.26)
21
2 Methods
Eq. (2.26) completes the classification of the eigenspace of a crystal Hamilton oper-
ator: Every eigenstate |Ψ𝑛(𝒌)⟩ has an 𝑁 -dimensional, continuous quantum number
𝒌 ∈ CG and a band index 𝑛 ∈ ℕ. Consequently, the eigenenergies are continuous func-
tions of 𝒌 ∈ CG. The band index 𝑛 is discrete for bound states, which exist in every
condensed-matter system. Hence, a band with fixed 𝑛 describes a hypersurface in
the 𝑁 + 1 dimensional 𝒌-energy space. As paper is only capable of displaying two-
dimensional plots, a section of the hypersurface can be drawn by plotting the bands
along a 𝒌-path in CG. The result is the band structure.
2.2.1 Basis changes
The Bloch states |Ψ𝑛(𝒌)⟩ are unhandy in practical applications since they are non-
periodic (Bloch theorem) and delocalized (they do not decay). These problems can
be encountered by basis changes which, of course, do not affect the physics. There
are different levels of transformations: those acting separately on each individual
state |Ψ𝑛(𝒌)⟩, those mixing states with different 𝒌-vectors at a fixed band 𝑛, and those
mixing different bands 𝑛 at a fixed 𝒌-vector. The two important bases presented here
are the Bloch factors and the Wannier functions, which solve the above-mentioned
problems about periodicity and localisation, respectively.
The Bloch factors
The first basis change exploits the quantum-number nature of 𝒌 and the gauge trans-
formation rules in Eqs. (A.99). The application of a 𝒌-dependent phase to the Bloch
functions transforms the stationary Schrödinger equation (2.21) into new eigenvalue
equation for the Bloch Ham〉iltonian ℎ(𝒑,〉𝒙 | 𝒌), which 𝒌 enters a〉s a parameter.𝑢𝑛(𝒌) B ?̃? 𝒌 Ψ𝑛(𝒌) = exp(−i𝒌 · 𝒙)Ψ𝑛(𝒌) (2.27a)
ℎ(𝒑, 𝒙 | 𝒌) B ?̃?𝒌𝐻 (𝒑,𝒙)
−1 Eq. (A.99b)?̃?𝒌 〉 = 𝐻 (𝒑 +〉ℏ𝒌, 𝒙) (2.27b)
⇒ ℎ(𝒑, 𝒙 | 𝒌) 𝑢𝑛(𝒌) = 𝐸 (𝒌)𝑛 𝑢𝑛(𝒌) (2.27c)
In contrast to the original states |Ψ𝑛(𝒌)⟩, the Bloch factors |𝑢𝑛(𝒌)⟩ are periodic on R.
This is a great simplification of the eigenvalue problem as techniques like Fourier
series become applicable. On the other hand, the periodicity in 𝒌 on G is lost and
phase-shifted b〉oundary conditions on CG hold instea〉d. That is ∀𝑹 ∈ R, 𝑮 ∈ G:
𝑈 𝑹 𝑢𝑛(𝒌) = 𝑈𝑹 exp(−i𝒌 · 𝒙)𝑈−1 𝑹 𝑈𝑹 Ψ𝑛(𝒌)
 〉 (2.28a) Eq. (2.23)  〉  〉= exp(−i𝒌 · (𝒙 −𝑹)) exp(−i〉𝒌 · 𝑹)Ψ𝑛(𝒌) = 𝑢𝑛(𝒌)〉𝑢𝑛(𝒌 +𝑮) = exp(−i (𝒌 +𝑮) · 𝒙)Ψ𝑛(𝒌 +𝑮) = exp(−i𝑮 · 𝒙)𝑢𝑛(𝒌) (2.28b)
The Wannier functions
Since the Bloch functions are periodic in 𝒌 on G, they have an 𝑁 -dimensional Fourier
series. The Fourier coefficients, which are indexed by the direct lattice vectors 𝑹 ∈ R
22
2.2 The Bloch theorem
(Eqs. (A.94), but with swapp〉ed space∫s), are the Wannier functio〉ns |𝑎 (𝑹)⟩.
𝑎 ( ) 1𝑹 〉B∑︁ d𝑘 exp(−i𝒌 〉· 𝑹)
𝑛
 𝑛 |C | Ψ𝑛(𝒌) (2.29a)G CGΨ𝑛(𝒌) = exp(i𝒌 · 𝑹)𝑎𝑛(𝑹) . (2.29b)
𝑹∈R
With respect to the Bloch theorem, the 𝑹th Wannier function results from shifting all
states of a fix〈edband 𝑛〉 into the 𝑹th u∫nit cel〈l and ta Eq. (2.23) 1 
king th〉e sum over 𝒌.
𝒙 𝑎 (𝑹) = 𝑛 |C | d𝑘 𝒙 − 𝑹 Ψ𝑛(𝒌) C 𝑎𝑛(𝒙 − 𝑹) (2.30)G CG
Therefore, each band 𝑛 has one Wannier function 𝑎𝑛(𝒙) which is centred at 0 and
defined on the entire space ℝ〈 𝑁 .〉This per∫mits the definition of a centre of charge:1
𝒙𝑛 = |C | d𝑥 𝒙
𝑎𝑛( 2𝒙) (2.31)
R ℝ𝑁
In order to show that Eq. (2.31) is well defined, the existence of the integral and the
gauge-invariance have to be proven.
The space integral in Eq. (2.31) can be partitioned into a series of unit-cell integrals.
The〈de〉finiti∑︁on of the∫cell-period ic Bloch fa∑︁ctors (E∫q. (2.27a)) then yields:1   𝒙𝑛 = |∫C | ∑︁d𝑥𝒙 𝑎𝑛( )
2 1  2𝒙 =
∈R R CR+𝑹   ∈R |CR∫|
d𝑥 (𝒙 − 𝑹) 𝑎𝑛(𝒙 − 𝑹)
𝑹 𝑹 CR
1 2 1 ∑︁  2= |C | ∫ d𝑥 𝒙 ∫𝑎𝑛(𝒙 −R CR 𝑹∈R
(∗) i ∫ 1 〈
𝑹) − |C〉R | 〈
d 𝑥 〉𝑹 𝑎𝑛(𝒙 − 𝑹)CR 𝑹∈R (2.32)
= |C | d𝑘 |C | d𝑥 𝑢𝑛(𝒌) 𝒙 ∇
′ 𝒙 𝑢 (𝒌′) 𝑘 𝑛 ′
G CG R C
𝒌 =𝒌
R
i 〈   〉
= d𝑘 𝑢 (𝒌) |C | 𝑛 ∇ 𝑘 𝑢𝑛(𝒌)G CG
In step∑︁(∗), the Parseval ident∫ity (Eqs〈. (A.95) an〉d (A.96)) was use∫d. ( − )2 1   ( ) 2 Eq. (2.27a) 1 〈  〉∑︁ 𝑎𝑛 𝒙 𝑹  = |C | ∫ d𝑘 〈 𝒙 Ψ𝑛 𝒌 〉 〈 = |C〉| d𝑘 𝒙 𝑢𝑛(𝒌)
2
𝑹∈R G CG G CG( − )2 −i𝑹 𝑎𝑛 𝒙∫ 𝑹 〈= |C |  〉 d𝑘 Ψ
 
〈  𝑛(𝒌) 𝒙〉∇𝑘
′ 𝒙 Ψ𝑛(∫𝒌′)  ′𝑹∈R G C 𝒌 =𝒌G
Eq. (2.27a) −i 1 〈  〉
= |C | d𝑘 𝑢 (𝒌)
 𝒙 ∇  ′   𝑛 𝑘′ 𝒙 𝑢𝑛(𝒌 ) 𝒌′=𝒌 + |C | d𝑘 𝒙 𝒙 𝑢𝑛(𝒌) 2G CG G CG
Eq. (2.32) is thus well defined because ⟨𝒙 | 𝑢𝑛(𝒌)⟩ is differentiable with respect to 𝒌.
23
2 Methods
To prove the gauge invariance, the centre of charge is calculated for Bloch factors
which are shifted by a 𝒌-dependent phase 𝜒(𝒌). In doing so, a variant of the Gauß
identity is used telling that the volume integral of a gradient field equals the surface
integral of the potential, 𝜒(𝒌) in this case. The surface integral can be evaluated with
the restriction on 𝜒(𝒌) that its values on two opposite faces of CG must differ by in-
teger multiples of 2π so as to retain boundary conditions of the Bloch factors (Eq.
(2.28b)).  〉  〉?̃?𝑛〈(𝒌)〉 = e〈xp〉(−i𝜒(𝒌))∫𝑢𝑛(𝒌) ∫1 〈 〉 ∑︁𝑁 𝑹⇒ ?̃?𝑛 = 〈𝒙𝑛〉 + |C | d ∇
𝑗
𝑘 𝑘𝜒(𝒌) = 𝒙𝑛 + d𝑠 · ∇𝑠𝜒(𝒔)
G CG [0,1]𝑁 2π𝑗=1
= 𝒙𝑛 + 𝑹, 𝑹 ∈ R
Thus, the centre of charge is only well defined modulo a direct lattice vector 𝑹. This is
expectable as translating all Wannier functions by 𝑹 corresponds to a reindexing.
24
Part I: Rare-earth silicide nanowires on
silicon surfaces
Rare-earth silicide nanowires are intriguing examples of systems on the nanoscale.
As the name suggests, they are derived from the rare-earth silicides, a compound of
silicon and the rare-earth elements (REs), which is often addressed by the chemical
formula “RESi2”, irrespectively of the exact stoichiometry. Horizontal thin streaks of
these hard, brittle metals form on several silicon surfaces by self-organised growth,
i.e. after RE deposition and successive annealing without further structuring meas-
ures. As their cross sections extend to only a few nanometres, while their lengths
can reach several 100 nm, these streaks are referred to as nanowires. As a result of
the small lateral dimensions, quantum confinement effects are expected to dominate
the electronic system. Hence, RESi2 nanowires are promising candidates for metallic
systems of one-dimensional electronic character.
The RESi2 nanowires establish a special class of nanowires as they combine a bunch
of unique and peculiar properties. In contrast to many other nanowire systems (e.g.
the not less interesting gold nanowires on Si(ℎℎ𝑘) [P13, P14]), RESi2 nanowires can
be considered a cut-out of a bulk structure, the bulk RESi2 prototypes in this case.
These cut-outs extend to heights and widths of a few unit cells and to lengths of hun-
dreds of unit cells. Such a structure bears several advantages: Firstly, since the in-
dividual streaks are subdivided into small building blocks, structure models of ar-
bitrary widths, heights and lengths can be set up in principle. Secondly, the streaks
incorporate a silicon network, which links to the substrate via strong Si–Si bonds
and guarantees stability against delamination. Thirdly, there is a vast number of
nanowire@silicon configurations because RESi2 occurs in two different phases and
silicon provides surfaces of very different morphologies. For these reasons, RESi2
nanowires provide an inexhaustible number of possibilities to synthesise and ex-
plore samples.
The research began in the 1980s when scientists experimented with metallic silicides
on pristine silicon. In particular, the integrated-circuit community was interested in
these systems as they establish metal-semiconductor contacts. Since epitaxy permits
the production of high-quality films, those silicides which grow epitaxially on silicon
attracted the most scientific attention. The REs stand out from the silicide-forming
elements as they produce very low Schottky barriers of 0.3 eV to 0.4 eV on 𝑛-type sil-
icon, the smallest known for this substrate in the 1980s [32]. Subsequent experiments
proved that the silicides of yttrium (Y, 𝑍 = 39) and all REs from gadolinium (Gd,
𝑍 = 64) to lutetium (Lu, 𝑍 = 71) are suitable for epitaxy on the Si(111) surface. This
is due to a match between the structure of the silicon-rich RESi2 bulk phases and the
geometry of the Si(111) surface: Both have a trigonal/hexagonal symmetry axis and
25
Rare-earth silicide nanowires on silicon surfaces
similar lattice constants, which deviate by −2.55 % to +0.83 % [33]. The thickness of
the RESi2 films on Si(111) can be tuned so that it spans only a few nanometres. In
particular, RE coverages corresponding to one monolayer lead to the formation of a
very thin and smooth silicide film with a thickness of one axial lattice constant of the
respective RESi2 bulk phase [34]. Such a film has an electronic band structure which
resembles that of a two-dimensional (semi-)metal with a Schottky barrier of ∼0.1 eV
[35]. It is remarkable that the experimental outcomes for different REs are almost
identical [P8, P12, 34–53].
The silicide epitaxy was also investigated on the Si(001) surface. In 1998, Kalka et al.
prepared a clean silicon sample with dysprosium (Dy, 𝑍 = 66) at a coverage corres-
ponding to one monolayer. Surprisingly and differently from the Dy@Si(111) system,
subsequent annealing at a few 100 ◦C lead to the formation of rectified, rectangular
islands instead of a smooth film [42]. Short after, Preinesberger et al. reduced the RE
coverage to sub-monolayer and observed the formation of strongly anisotropic DySi2
structures. Because the lateral dimensions measured a few nanometres, while their
lengths reached up to 200 nm, they referred to the structures as “nanowires” [54]. Not
only did they report the first fabrication of RESi2 nanowires, but they also assessed
the process to be quite simple as the nanowire formation was self-organised, con-
trolled mainly by macroscopic parameters like the RE coverage and the annealing
temperature. Since their discovery, RESi2 nanowires have been an active research
field in which many groups published fabrication details, topographical data (e.g.
STM), hypothetical structure models and electronic properties (e.g. ARPES or STS)
[P3, P6, 54–67].
RESi2 nanowires do not only grow on Si(001), as in the pioneering work. Also vicinal
Si(111) surfaces (Si(ℎℎ𝑘)) produce nanowires as the steps and terraces cause discon-
tinuities in the monolayer film [P8, 64, 68–70]. This part of the thesis is dedicated
to the theoretical exploration of RESi2 nanowires on the Si(557) surface (nanowire-
RE@Si(557)). Although structure models have been proposed based on the RESi2
monolayer on Si(111) and experimental findings [64, 68, 70], the real structure is
still unknown. Therefore, a major part of this work consists in setting up reasonable
structure models, optimising them by means of DFT and evaluating their stability
by means of ab initio thermodynamics. Once verified, the stable structure model
is used to explain the experimental outcomes of this system, particularly the elec-
tronic properties measured by ARPES (band structure and Fermi surfaces). Trivalent
terbium (Tb, 𝑍 = 65) is employed as a representative RE as it produces the desired
nanostructures in the experiment [70]. To a certain extent, the results are transfer-
able to other trivalent REs, in particular dysprosium (Dy, 𝑍 = 66) and erbium (Er,
𝑍 = 68), as proven by experimental evidence [64, 68], and holmium (Ho, 𝑍 = 67), as
it lies between Dy and Er in the lanthanoid series. The purpose of all investigations
is answering the following questions:
• Are the proposed structure models for the nanowire-Tb@Si(557) system stable?
• Does the electronic structure of the nanowire-Tb@Si(557) system have (quasi-)
one-dimensional electronic properties?
26
Rare-earth silicide nanowires on silicon surfaces
Before the RESi2 nanowires are addressed, some related systems are considered at
first. The respective insights will be absolutely crucial for understanding the physics
of the nanostructures. In the first instance, the chemical background of the constitu-
ent elements, the rare-earth elements and silicon, is collated in Chap. 3. The focus
of this literature research is on the role of the 4f electrons in chemistry and the gen-
eral properties of the silicide compound class. After that, the bulk phases of RESi2
are analysed in Chap. 4 since they are the three-dimensional bulk prototypes for the
nanostructures. Attention is paid to how the characteristic Si vacancies affect the
structural and electronic properties of the bulk phases. Furthermore, two important
approximations which are employed in the subsequent chapters are validated: the
frozen-core approach for treating the f electrons of Tb and the omission of the lattice
dynamics in ab initio thermodynamics.
Chap. 5 concerns the first nanostructure: the monolayer-Tb@Si(111) system, which
is the surface prototype for the nanowire-Tb@Si(557) system. Contrarily to the nano-
wires, the monolayer provides detailed experimental data on the lattice parameters.
Therefore, it serves as a test system for the approach. Furthermore, although the
RESi2 monolayers have already been treated in several experimental and theoret-
ical works (vide supra), the peculiarities about the established structure model are
not sufficiently understood. This in particular concerns the reason for the buckling
direction of the covering Si honeycomb, which is inverse to that of the substrate. A
detailed analysis of the structural and electronic properties illuminates the underly-
ing physics of the monolayer and explains the peculiarities. The chapter concludes
with a critical revision of the popular statement that mechanical strain is responsible
for the details of the established structure model. The nanowire-Tb@Si(557) system is
the matter of Chap. 6. The structure models are derived from those of the monolayer
and optimised by means of DFT. The RE coverage of the surface is varied between
dense and sparse so as to derive a corresponding phase diagram for the nanowire
system. For the stable structure models the electronic properties are calculated. The
chapter concludes with the proof of a unique dimensional crossover from 2D to 1D,
which has never been observed in this system before.
27
3 Chemical background
3.1 The rare-earth elements
The rare-earth elements are a group of 17 elements: scandium (Sc, 𝑍 = 21), yttrium (Y,
𝑍 = 39) and the 15 lanthanoids1. The latter are the first row of the f-block plus the first
5d element (inner transition metals), a series starting with lanthanum (La, 𝑍 = 57)
and terminating with lutetium (Lu, 𝑍 = 71) [71, p. 51]. The rare-earth elements are
often abbreviated to REE or, more like a chemical symbol, RE. The lanthanoids are
often denoted by the chemical symbol Ln = [La−Lu]. Since the individual members
of the REs are unfamiliar to most non-specialists, they are more than once addressed
by their name, chemical symbol and atomic number 𝑍 in this section.
The REs are very homogeneous concerning their chemical behaviour, while their
physical properties (in particular optical and magnetic properties) differ consider-
ably. This makes the REs important technology metals in nearly every high-tech
branch. “Though there are no uniform classifications for rare-earth applications, the
markets are commonly divided into nine sectors: catalysts, polishing, glass, phos-
phors and pigments, metallurgy, batteries, magnets, ceramics, and others” [72]. A
very prominent RE is neodymium (Nd, 𝑍 = 60) as the alloy Nd2Fe14B is the base for
strong permanent magnets used for efficient electric motors and generators [73, p.
78]. The high spin multiplicity of gadolinium (Gd, 𝑍 = 64) is exploited in contrast
agents for magnetic resonance tomography (MRT). The paramagnetic ions enhance
the relaxation times of the surrounding protons (hydrogen atoms) and thereby in-
crease the image contrast [73, p. 70]. Erbium (Er, 𝑍 = 68) is known from optical
devices, e.g. erbium-doped fibre amplifiers: The shielded 4f levels of the Er3+ ions
produce a near-infrared transition (from 𝐽 = 152 to 𝐽 =
13
2 ) with a long lifetime of
∼10 ms. Thus, most Er3+ ions in the fibre can be excited (“pumped”) so that optical
signals passing the fibre are amplified by stimulated emission [74, 75]. The chem-
ical similarity between the REs is due to analogous valence electron configurations
(VECs) which all yield the oxidation state III in most compounds [18, p. 2230], as well
as in aqueous solutions (Tab. 3.1). While this is expectable for scandium (Sc, 𝑍 = 21),
yttrium (Y, 𝑍 = 39) and lanthanum (La, 𝑍 = 57), which represent the third column of
the periodic table, the trivalence of the other 14 lanthanoids despite their consecut-
ive atomic numbers is an “aperiodic” behaviour. This phenomenon is unique in the
periodic table and due to the special role of the 4f electrons in chemistry.
1Most of the (older) literature refers to the elements from La to Lu as “lanthanides”. However, this
term is misleading since the suffix “-ide” denotes anionic constituents (cf. oxide O2 – , fluoride F – ).
In contrast, the suffix “-oid” indicates an instance similar to the word root. Therefore, the IUPAC
recommends the term “lanthanoids” for the group La–Lu, including La and Lu [71, p. 52].
28
3.1 The rare-earth elements
The role of f electrons in chemistry
The VEC can partially be derived from the shell model and the standard occupation
rules which lead to the periodic table. These rules correctly predict a trivalent VEC
of 4s23d1 and 5s24d1 for the non-lanthanoids scandium (Sc, 𝑍 = 21) and yttrium (Y,
𝑍 = 39), respectively. Furthermore, they predict that the first f shell, the 4f shell, is
empty up to the alkaline earth metal barium (Ba, 𝑍 = 56, VEC = 6s2) and fills up from
lanthanum (La, 𝑍 = 57) to ytterbium (Yb, 𝑍 = 70). Hence, the VEC of the 𝑛th lanthan-
oid would be 6s24f𝑛, 𝑛 ∈ [1, 14], according to the periodic table. The termination of
the lanthanoid series lutetium (Lu, 𝑍 = 71) would then be the first 5d-element with
a VEC of 6s25d14f14. Since the complete 4f shell is expected to remain inert, Lu is
correctly predicted to be trivalent with two s electrons and one d electron in the out-
ermost shells, analogously to Sc and Y. The trivalence of the other lanthanoids is not
obvious, in particular because the 4f levels are incomplete, except for ytterbium (Yb,
𝑍 = 70), and lie in the chemically relevant energy range near the Fermi level. Though,
the energy levels and occupations of a shell are not the only conditions for its ability
to participate in chemical bonding. In addition, the atomic orbitals have to spatially
overlap with those of another atom to form molecular orbitals. However, the spa-
tial distribution of the atomic 4f orbitals is enclosed by the shells with higher main
quantum numbers, in particular the 6s shell, the 5d shell and the 5p shell. Hence, the
4f electrons could only participate in chemical bonding, if all outer shells did also.
While this is the case for the 6s and 5d electrons, the 5p electrons are part of the
xenon noble gas configuration and thus inert. As a consequence, they screen the 4f
electrons from the environment and render them inert as well. Applying this to the
lanthanoids from lanthanum (La, 𝑍 = 57) to ytterbium (Yb, 𝑍 = 70) would imply a
divalent state from the two 6s electrons, not the observed trivalent state2.
Obviously, the rules of the periodic table cannot explain the trivalence of the lan-
thanoids. They even fail in predicting the correct atomic ground state for lanthanum
(La, 𝑍 = 57), cerium (Ce, 𝑍 = 58) and gadolinium (Gd, 𝑍 = 64). The apparent con-
tradictions are due to an oversimplification of the real electronic situation and can
be resolved by considering the ionisation energies of different VECs. A detailed ana-
lysis of this can be found in [75]. The key is that the atomic terms belonging to the
divalent VEC (6s25d04f𝑛) and the trivalent VEC (6s25d14f𝑛−1) are energetically very
close to each other for most of the lanthanoids. Therefore, one electron leaves the 4f
shell if the drop in the electronic chemical potential, e.g. caused by an electronegative
reaction partner, is large enough. The ability of the Ln2+ ions to donate the third elec-
tron depends on the third ionisation energies of the isolated Ln atoms. These grow
gradually from lanthanum (La, 𝑍 = 57) to europium (Eu, 𝑍 = 63) by +5.7 eV, show
a sharp drop at gadolinium (Gd, 𝑍 = 64) by −4.3 eV, grow again from gadolinium
(Gd, 𝑍 = 64) to ytterbium (Yb, 𝑍 = 70) by +4.4 eV, and finally drop at lutetium (Lu,
𝑍 = 71) by −4.1 eV [75]. The zigzag pattern is the manifestation of Hund’s second rule
2The terms “divalent” and “trivalent” denote the number of electrons in the outermost 6s and 5d shells
plus the charge state of the atom/ion. For example, the VEC of the neutral Gd atom (6s25d14f7) is
trivalent as well as that of the Gd3+ ion (4f7). On the other hand, the VEC of the neutral Yb atom
(6s25d04f14) is divalent as well as that of the Yb2+ ion (4f14). This nomenclature makes sense as the
only possibility for a formal valence electron not to take part in chemistry is to be part of the 4f shell.
29
3 Chemical background
La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu
2 2 2
(a) 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
4
(b) 4f0 4f1 4f2 4f3 4f4 4f5 4f6 4f7 4f8 4f9 4f10 4f11 4f12 4f13 4f14
Table 3.1: (a) “Valencies of the lanthanoids occurring in aqueous solution” [76]. (b) VEC of the
trivalent Ln3+ ion. Empty, half-filled and filled 4f shells are bold.
that the spin configuration of a specific shell tends to maximise its spin multiplicity.
More precisely, parallel spins occupying orbitals with different magnetic quantum
numbers reduce the total energy by their exchange energy. An overview over the
behaviour of the atomic terms and the third ionisations energies of the lanthanoids
can be found in the appendix (Sec. B.1).
Hund’s second rule is very helpful for predicting the valences. The trivalent state is
the most frequent for all lanthanoids (Tab. 3.1). If the electronic chemical potential
is not too low, e.g. if the reaction partners are not too electronegative, an additional
electron will go into the 4f shell to maximise the spin multiplicity. This is particularly
relevant for europium (Eu, 𝑍 = 63) and ytterbium (Yb, 𝑍 = 70) since their divalent
VEC provides a (half-)full 4f shell. For the left neighbours samarium (Sm, 𝑍 = 62)
and thulium (Tm, 𝑍 = 69), the trivalent state is already quite stable, so the electronic
chemical potential has to be high in order to push an additional electron into the
4f shell. On the other hand, if the electronic chemical potential is very low, cerium
(Ce, 𝑍 = 58) and (less likely) terbium (Tb, 𝑍 = 65) can oxidise to tetravalent Ln4+
ions [18, pp. 2300]. The valence of the REs has a large impact on the properties of
their compounds wherefore it is necessary to care about it. While isovalent RE com-
pounds may differ only slightly in most properties, different valences usually lead to
very different properties. A demonstrative example for this issue are the lanthanoid
monochalcogenides, whose chemical formulae Ln(S, Se, Te) suggest divalence. How-
ever, the properties of these compounds strongly depend on the atomic number of
the employed lanthanoid. While the monochalcogenides of Sm, Eu and Yb are salt-
like, those of the other lanthanoids are metallic. The reason for this is a deviation
between the formal oxidation state II and the actual valence of the Ln ions: They
are divalent in the salt-like compounds, described by the formula LnII(S, Se, Te) – II.
In the metallic compounds, the Ln ions lose an additional 4f electron to the conduc-
tion bands, whereupon their valence formally increments by +I and gives rise to an
“electride” LnIII(S, Se, Te) – IIe – [76].
The lanthanoid contraction
Although the 4f electrons are prevented from chemical bonding, their presence has
some indirect consequences on the chemistry. Most notably, despite the valences of
the REs are analogue, the atomic/ionic radii differ. For the first three REs without
f-electrons (Sc, Y and La), the ionic radius increases, as expected for columns in the
periodic table. On the contrary, the horizontal lanthanoid series from La to Lu shows
30
3.1 The rare-earth elements
220 Metal RE3+ ion, CN=6 RE3+ ion, CN=8
199.5
200 187.0 194.0182.5 182.0 181.4 181.0 180.2 178.7
180 176.3 175.2 174.3 173.4 172.4 171.8
160.6
160 177.6
140 130.0 128.3 126.6 124.9 123.3 121.9 120.6
115.9 119.3 118.0120 116.7 115.5 114.4 113.4 112.5 111.7
101.0
100 117.2 115.0 113.0 112.3 111.0104.0 109.8 108.7 107.8 106.3 105.2 104.1 103.0 102.0 100.8 100.1
80 88.5
Sc Y La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu
Figure 3.1: Atomic radii of the REs [17, Anhang IV]. Dashed lines between La and Lu highlight
the lanthanoid contraction. CN: coordination number of RE3+ ions.
the aperiodic trend of decreasing ionic radii for incrementing atomic numbers. The
lanthanoid contraction follows from two effects: the relativistic orbital contraction
(Sec. 8.1) and the filling of the core-like 4f shell: Because every additional 4f electron
screens the increment of the nuclear charge only partially, the fifth and sixth shells of
heavier REs feel steeper potentials whereupon they contract [17, p. 374]. This affects
the radii of the isolated atoms (with analogue VEC) as well as the radii of the trivalent
ions and the metallic radii (Fig. 3.1 (a)). The lanthanoid contraction is strong enough
to fully compensate the radius increase from the fifth to the sixth period at holmium.
Therefore, the light yttrium (Y, 𝑍 = 39) is sometimes indexed between holmium (Ho,
𝑍 = 67) and erbium (Er, 𝑍 = 68). The discontinuities of europium (Eu, 𝑍 = 63) and
ytterbium (Yb, 𝑍 = 70) in the otherwise smooth contraction trend of the metallic radii
are due to their divalent VECs in the metals.
If REs compounds with analogous VECs are compared, the RE radius has the largest
impact on the geometry. This can be very important in some cases, for example, if two
structural phases are energetically close to each other. Small differences in the ion
sizes, as caused by the lanthanoid contraction, then determine the favourability of
the one or the other phase. This proves to be a critical point when investigating the RE
silicides (Chap. 4). On the other hand, the smoothness of the lanthanoid contraction
permits the structural tuning of some RE compounds by varying the employed RE.
This is, for instance, utilised for the growth of RE-containing structures on surfaces.
The lanthanoid contraction is the f-analogon of the d-block contraction, which ex-
plains the otherwise unexpectedly small atomic radii of the post-transition metals
(in particular those following the 3d-elements: Ga and Ge in group 13 and 14)3. The
3Analogously to the post-transition elements, the post-lanthanoids have unexpectedly small atomic
radii. For the first post-lanthanoids, the f-block contraction is strong enough to fully compensate the
incrementing main quantum number of the outer shells from the fifth to the sixth row. The group-4
elements zirconium (Zr) and hafnium (Hf) have almost identical ionic radii, as well as the group-5
elements niobium (Nb) and tantalum (Ta). As a consequence, naturally occurring minerals of these
elements are often solid solutions between the respective two homologues. Niobium and Tantalum
were identified as different elements in the mid of the 19th century. Hafnium was discovered even
later in the 1920s, after having been predicted by the Bohr model [18, pp. 1809, 1831].
31
Radius (pm)
3 Chemical background
main difference between the f-block and the d-block contraction is that the lanthan-
oids have the same trivalent properties in most compounds and the same-period d-
elements not. In principle, the lanthanoid contraction should occur also in the actin-
oids, whose 5f shell fills up. In fact, the radii of isovalent actinoid ions contract for
incrementing atomic numbers. However, the trivalent state of the actinoids is less
stable because relativistic effects makes the inner s orbitals collapse, which in turn
expands the outer d and f orbitals. Consequently, the 5f orbitals can participate in the
chemistry [17, 18, pp. 372ff, 2230, 2320f]. The homologues neodymium (Nd, 𝑍 = 60)
and uranium (U, 𝑍 = 92) demonstrate this: While Nd is trivalent in most compounds,
U tends to employ all valence electrons and assumed a main valence of VI [18, p.
2323]. Hence, the lanthanoid contraction is indeed unique in the periodic table.
3.2 Silicon and silicides
Silicon (Si, 𝑍 = 14) is the second most abundant element in Earth’s crust (26.3 wt%),
after oxygen (48.9 wt%) [17, p. 1063]. It belongs to the third row and 14th group, the
carbon group. While carbon (C, 𝑍 = 6) is the basis of organic life on Earth, silicon
is mainly known as the most important technology material as it is employed in the
vast majority of integrated circuits (Sec. 1.1). However, silicon plays a role also in
physiology, e.g. in the hair and nails of animals [17, p. 1062], in the cell walls of
plants [77] and in the global oxygen generation (diatoms, “Kieselalgen”) [78].
Silicon shares several physical and chemical properties with its lighter homologue
carbon. They have analogous VECs (C: 2s22p2, Si: 3s23p2) with four valence electrons
to be paired for chemical bonding. The s and p orbitals tend to hybridise to four
equivalent sp3-hybrid orbitals in tetrahedral geometry, which is the origin of the al-
ternative name of the carbon group: the tetrels. In the elemental phases, the sp3
hybridisation leads to the diamond structure, a three-dimensional network in which
each atom has four equivalent, 𝜎-bound neighbours4. This structure is a stable phase
for both elements on a certain thermodynamic domain. On the other hand, the ele-
mental phases reveal an important difference between C and Si: C adopts the graph-
ite structure at normal pressures, a layered structure of flat honeycombs which stick
together by vdW interactions. Therein, the C atoms are sp2-hybridised and mutually
bind via sp𝜎 bonds in trigonal-planar geometry. The remaining p orbitals pointing
out of the plane (conventionally denoted p𝑧) connect with each other to a delocal-
ised system of p𝜋 bonds, which stabilises the individual sheets [17, pp. 997ff]. To a
certain extent, these are even stable in isolated form, the graphene sheets [79], for
which Andre Geim and Konstantin Novoselov were awarded with the Nobel Prize in
Physics in 20105. In contrast to C, Si avoids the graphite structure as well as p𝜋 bonds
4In fact, there are two possibilities of fourfold coordinated networks. The classical diamond structure
is based on the fcc lattice. Changing the layer stacking in the (111) direction from ABCABC to ABAB
leads to a related structure: the hexagonal diamond or lonsdaleite, which is based on the hcp lattice.
Being stable only under extreme thermodynamic conditions, lonsdaleite can be found in shockwave-
treated graphite and meteorites [17, p. 1002].
5All Nobel Prizes in Physics, https://www.nobelprize.org/prizes/lists/all- nobel- prizes- in-
physics/ (visited on 04/03/2022).
32
3.2 Silicon and silicides
in general. A demonstrative example of this are the properties of the oxides: While
CO2 with double bonds is a stable, gaseous molecule ( O C O ), SiO2 polymerises
to single-bonded [SiO4] tetrahedrons which connect via the O atoms to crystals under
normal conditions [17, pp. 1078ff, 1099ff]. For the same reason, the Si analogon of
graphene (“silicene”) is unstable as both flat (sp2) and buckled (sp3) honeycomb geo-
metries contain unpaired electrons, which render the allotrope highly reactive.
Zintl phases
A way to stabilise honeycomb-Si geometries is to charge the Si atoms and render
them isoelectronic to the right neighbour phosphorus (P, 𝑍 = 15). The sp2 hybridisa-
tion is then still avoided, but the five valence electrons give rise to a sp3 configuration
with three bonds and a non-binding electron pair. This allows the Si atoms to form a
buckled honeycomb with closed shells. Alkali (MI) and alkaline earth metals (MII) can
serve as electron donators since their electronegativities are smaller than that of Si by
>0.7 (except Mg and Be, Allred-Rochow scale, [17, Tafel III]). This leads to a partially
heterovalent compound MnSim, which belongs to the class of Zintl phases, named
after Eduard Zintl, who first investigated intermetallic compounds with high elec-
tronegativity differences. In 1963, W. Klemm and E. Busmann completed Zintl’s ideas
to the extended Zintl-Klemm-Busmann concept (ZKB concept): When an alkali or al-
kaline earth metal (non-noble component) reacts with a metallic/semiconducting p-
block element (noble component), the valence electrons of the non-noble component
are formally completely transferred to the noble one. Depending on the VEC after the
transfer, the noble component pairs the residual electrons by inter-se bonding and
forms the anionic partial lattice, into which the cations are embedded. The formal
strict charge transfer is then relaxed by introducing covalent bonds between the free
electron pairs of the anionic lattice and the empty valence orbitals of the cations
[80]. Since having four right neighbours, Si provides a particularly large variety of
anionic partial lattices (e.g. buckled honeycombs, zigzag chains, dumb-bells and isol-
ated atoms, [17, p. 1070]).
The ZKB concept can successfully explain some structural aspects of MnSim com-
pounds and will play an important role in explaining the inevitable vacancies of the
bulk phases of the RE silicides. Though, it does not predict the exact geometry and
the properties of the resulting compounds. Depending on the electronegativity differ-
ence of the reactants, the properties of Zintl phases can range from salt-like to metal-
lic. For instance, alkali/alkaline earth metal chalcogenides (selenides and tellurides)
are similar to the classical salts (oxides, halides) since the high electronegativity dif-
ference (>1.0, except Mg and Be, Allred-Rochow scale, [17, p. 158]) implies a strong
heterovalent (ionic) bonding. In particular, the reaction enthalpies are high and the
compounds are brittle and isolating/semiconducting. On the contrary, if the elec-
tronegativity difference is less pronounced, the bonding has a more covalent charac-
ter. Consequently, the bandgap decreases and the compound resembles more a metal
than a salt concerning the electronic and optical properties [17, 80]. Moreover, cova-
lent bonding can influence the partial anionic lattice because the empty d orbitals
of the heavier alkali/alkaline earth metal cations can be activated and exert a steric
effect on their binding partners [17, pp. 400ff].
33
4 The rare-earth silicide bulk phases
The RE silicides have been investigated since the end of the 1950s [81–108]. In 1959,
Perri et al. systematically analysed the “disilicides” of several REs by means of XRD
[81]. They found that the lighter REs crystallise in the tetragonal ThSi2 structure,
while the heavier REs (up to Dy) assume an orthorhombically distorted version of it.
The clear trend is that the orthorhombic distortion increases with higher RE atomic
numbers, corresponding to smaller ionic radii (lanthanoid contraction, Sec. 3.1). For
the heaviest RE silicide investigated in that work, DySi2, the distortion is considerable
with a 𝑏/𝑎 ratio of 97.8 %. Only Eu disrupts the otherwise monotone development as
it adopts the undistorted, tetragonal ThSi2 structure. Since coinciding with the dis-
continuity in the lanthanoid contraction, this is a clear evidence that Eu is divalent
in its silicide while all other investigated REs are trivalent [81]. Subsequent work
found that the orthorhombic distortion reversibly vanishes at temperatures of sev-
eral 100 ◦C, so all investigated RE silicides assume the tetragonal ThSi2 structure as
a high-temperature phase. On the other hand, the tetragonal silicides of Pr and Eu
distort at −120 ◦C and −150 ◦C, respectively [82]. Mayer et al. [83] completed the list
by the REs not investigated by Perri et al. They found that the silicides of Er and the
heavier REs crystallise in the hexagonal AlB2 structure, which is quite different from
the ThSi2 structure at first glance. Ho and all lighter REs join the findings of Perri et
al. as they adopt tetragonal/orthorhombically distorted ThSi2 structure, depending
on the ionic radius. The silicide of Y was found to adopt both structures depending
on the preparation temperature, in accordance with its ionic radius lying between
those of Ho and Er [83]. This dimorphism motivated the same group to test whether
the other RE silicides show the AlB2/ThSi2 transition as well. In fact, the silicides of Gd,
Tb, Dy, Ho and Y can assume either the tetragonal/orthorhombic ThSi2 structure or
the hexagonal AlB2 structure, the latter apparently being the low-temperature phase.
The REs lighter than Eu show only the tetragonal/orthorhombic ThSi2 structure, and
the heavier REs Er, Tm and Lu show only the hexagonal AlB2 structure [85].
The aim of this chapter is to illuminate the microscopic reasons for the peculiar be-
haviour of the RE silicides and to investigate their structure. At first, the prototypes
(ThSi2 and AlB2) are systematised in a novel and unconventional manner. This helps
to understand how the structures respond to specific variations, in particular con-
cerning vacant Si sites, which are inevitably present in all RE “disilicides” and which
have not been sufficiently investigated. Then, different structure models are optim-
ised and their stability is analysed by ab initio thermodynamics. The following points
are addressed: the stoichiometric structure models, the role of the 4f electrons, the
lattice dynamics and the Si vacancies. The findings on the structures are confirmed
by band structure analyses. The chapter concludes with a discussion of the mechan-
isms which determine the structure of the RE silicides.
34
4.1 Structure prototypes
4.1 Structure prototypes
The RE silicide structures can be classified into two prototypes: the ThSi2 structure
(Fig. 4.1 (a)) and the AlB2 structure (Fig. 4.1 (b)). Both are characterised by a skeleton
of mutually threefold coordinated Si atoms in trigonal-planar geometry, denoted M-
stars, into which the RE atoms are embedded. The simpler realisation of this is the
AlB2 structure. It consists of parallel, congruent sheets of flat-honeycomb-Si between
which hexagonal layers of RE atoms are embedded at hole position. The honeycomb
sheets can also be viewed as staggered rows of M-stars, as indicated by blue shad-
ows in (Fig. 4.1 (b)). The space group is 𝑃6/𝑚𝑚𝑚 (No. 191) and the atoms occupy
the Wyckoff positions1 RE → 1𝑎 and Si → 2𝑑, so the conventional unit cell contains
one formula unit RESi2. The ThSi2 structure is the alternative realisation. It can be
derived from the AlB2 structure by rotating the M-stars of every second row by 90◦
and staggering the rows to form a three-dimensional network. The RE atoms occupy
the interstices and are visible from both views along the basis. The space group is
𝐼41/𝑎𝑚𝑑 (No. 141) and the atoms occupy the Wyckoff positions RE→ 4𝑏 and Si→ 8𝑒,
so the conventional unit cell contains four formula units RESi2.
In order to clarify the relationship between the two structures, consider the building
blocks which both of them have in common [88]. These building blocks are cuboids
of width 𝑥, depth 𝑦 and height 𝑧 with Si atoms at the vertices and an RE atom in the
centre (orange boxes in Fig. 4.1). Their stacking in the z-direction determines the
structure. If stacked in a zigzag manner, i.e. alternately translated by ±𝑥2 in the 𝑥-
direction, the b√locks form the AlB2 structure. 𝑥 is then the lattice constant 𝑎 ([11.0]
direction), 𝑧 is 3𝑎/2 for a perfect hexagonal symmetry ([1̄1.0] direction), and 𝑦 is
the lattice constant 𝑐 ([00.1] direction). Alternatively, the blocks can be stacked in
a helical manner, i.e. alternately translated by ±𝑥2 in the 𝑥-direction and ±
𝑦
2 in the
𝑦-direction, which result is the ThSi2 structure. 𝑥 is then the lattice constant 𝑎 ([100]
direction), 𝑦 is 𝑏 ([010] direction) and 𝑧 is 𝑐/4 ([001] direction). The building blocks
demonstrate a further structural aspect concerning the ways how the Si atoms at
the vertices are connected. The splices between two block faces consist of parallel
rows of zigzag-chain-Si (different shades of blue in Fig. 4.1) which accrete to zigzag
planes. In the AlB2 structure, all zigzag chains point into the same [11.0] direction. In
the ThSi2 structure, the zigzag planes are alternately oriented in the [100] direction
(light blue) and the perpendicular [010] direction (dark blue).
Silicon vacancies
The stoichiometries of real RE silicides deviate considerably from RESi2 in that Si
atoms are missing. Hence, the structures in Fig. 4.1 should be considered only as pro-
totypes. The empty Si sites, the vacancies, in the real structures are unordered2and
their concentration depends on the fabrication technique, the temperature and the
atomic number of the RE. As the real stoichiometries range between RESi1.6 and
1The numbers preceding the letters denote the multiplicity of the positions.
2In thin RESi2 films of the AlB2 phase, w√hich √are epitaxially grown on Si(111), the vacancies are
ordered in-plane and give rise to a R30◦( 3 × 3) superstructure [36, 37, 39, 46, 50–52].
35
4 The rare-earth silicide bulk phases
(a)
γ
c c
z
y
x
a b
90°
[100] [11.0] [010] [00.1]
(b)
z
y
x
γ 90°
b
a
c
Figure 4.1: Overview over the RESi2 structure prototypes. (a) ThSi2 prototype; (b) AlB2 pro-
totype. Blue circles are Si atoms; Yellow circles are RE atoms; Circle sizes indicate the out-of-
plane positions. Left and centre: Views along perpendicular crystallographic axes. Red lines
mark conventional unit cells. Blue shadings mark threefold coordinated Si atoms in the Si
sublattice (M-stars). Right: Stacking of the cuboid building blocks (orange lines). Light and
dark blue Si atoms highlight Si zigzag planes.
(a) (b) 2
1 2 [100] 1
2 1
b 2
c a
1
a [11.0]
Figure 4.2: Possible configurations of ordered Si vacancies in RESi1.67. (a) ThSi2; (b) AlB2. Col-
ouring: Fig. 4.1. Solid orange circles mark the first set of vacancies, labelled dashed orange
circles possible second sets. Green lines mark the supercells.
36
AlB2 structure ThSi2 structure
[‾11.0] [001]
[001]
[‾11.0]
zigzag-planes Building blocks zigzag-planes
4.1 Structure prototypes
RESi1.9 [81, 85–93], the RE “disilicides” should always be referred to as RESi2 – x, where
𝑥 is the number of Si vacancies per RE atom. A value of 𝑥 = 0.33, as present in many
RE silicide preparations, implies that every sixth Si site is vacant. The large number
of vacancies affects the Si sublattice and, thus, the entire prototype in a considerable,
but predictable manner: Since experimental diffraction patterns do not indicate any
long-range ordering, the vacancies have to be assumed statistically distributed. A
uniform distribution retains the symmetries of the prototypes (AlB2 → hexagonal,
ThSi2→ tetragonal), while a non-uniform distribution is likely to distort them. In de-
tail, removing a Si atom makes the three neighbours of the respective M-star move
inwards, towards the vacant site. This combined with the topography of the zigzag
chains suggests how a given vacancy distribution alters the lattice constants. Fig. 4.2
shows exemplary snapshots of ordered vacancy configurations (orange circles) for
the ThSi2 and the AlB2 structure in RESi1.67 stoichiometry.
The conventional unit cell of the ThSi2 structure contains four alternately orthogonal
zigzag planes, of which every second is equivalent due to the body-centred unit cell
(highlighted by light and dark blue circles in Fig. 4.2 (a)). If the vacancies populate
both inequivalent zigzag planes to a similar extent (i.e every sixth Si site is vacant
in both the light and dark blue zigzag chains, not shown in Fig. 4.2), the basal lattice
constants 𝑎 and 𝑏 are expected to shrink equally. Also the axial lattice constant 𝑐 will
shrink as the zigzag planes move closer together. In summa, the cell volume will con-
tract while retaining the tetragonal symmetry of the prototype. The situation changes
if the vacancies concentrate in one of the two inequivalent zigzag planes (e.g. every
third Si site is vacant in the light blue zigzag chains, Fig. 4.2 (a)). Then, 𝑎 is expected
to shrink, as well as 𝑐, while 𝑏 will remain unchanged. In summa, the structure will
undergo an orthorhombic distortion along with a reduction of the symmetry. Such
an effect was confirmed in TbSi1.67 by Schobinger-Papamantellos et al., who meas-
ured a distortion of ≈2 % (neutron diffraction, [89]). If the vacancies populate the
zigzag planes of the AlB2 structure to an equal extent (configuration 1 in Fig. 4.2 (b)),
the basal lattice constants 𝑎 and 𝑏will equally contract while retaining the hexagonal
symmetry of the prototype. As a response to the squeezing of the basis, the interstitial
RE atoms will push the silicon layers apart and increase the axial lattice constant 𝑐.
On the other hand, if the vacancies concentrate in one of the two inequivalent zigzag
planes (e.g. configuration 2 in Fig. 4.2 (b)), an orthorhombic distortion is expected as
𝑎 and 𝑏 shrink differently. Such an effect was observed in hexagonal ErSi1.67 [91] and
TbSi1.67 [93] by Auffret et al. (neutron diffraction). However, the distortion of the AlB2
structure (<0.5 %) seems to be much weaker than that of the ThSi2 structure. While
the vacancies underlie a probability distribution in most RESi2 – x, Yb3Si5 c√an ad√opt
the ordered version of configuration 1 in the bulk, as indicated by a R30◦( 3 × 3)
superstructure (light green unit cell) (XRD, [87]).
As a final remark on the stoichiometry, the nomenclature of the RE silicides is quite
chaotic in the literature. Some publications refer to them as RESi2, irrespectively of
the large number of vacancies. Others refer to the silicides with a ThSi2-derived struc-
ture as RESi2 and to those with an AlB2-derived structure as RE3Si5. Although this
agrees with the relative positions in the temperature-composition phase diagrams
(vide infra), the stoichiometries are still incorrect. Moreover, it is rather confusing
37
4 The rare-earth silicide bulk phases
that the orthorhombically distorted ThSi2 structure is often referred to as the GdSi2
structure. On the one hand, the structure-determining vacancies are not ordered in
GdSi2 – x, so it has to be related to the ThSi2 structure anyway. On the other hand, GdSi2
is dimorphic and can adopt the AlB2 structure as well, depending on the preparation
conditions. In order to avoid misunderstandings, the following nomenclature is used
in this work. All RESi2 – x structures are assigned to one of the two prototypes: the AlB2
phase or the ThSi2 phase. If relevant, the presence or absence of orthorhombic distor-
tion is explicitly mentioned by a symmetry tag (“hex”: hexagonal, “tet”: tetragonal,
“ort”: orthorhombic). The chemical formula is RESi2 – b for the AlB2-type silicides and
RESi2 – a for the ThSi2-type silicides (as introduced by [102]), where 𝑏 is (slightly above)
0.33 and 𝑎 between 0.33 and 0. For example, orthorhombically distorted ThSi2-type
terbium silicide is denoted ort-ThSi2-TbSi2 – a and hexagonal AlB2-type erbium silicide
is denoted hex-AlB2-ErSi2 – b. The structure-neutral chemical formula is RESi2 – x.
The phase diagrams of the RE–Si systems
The silicides of the heavy REs always contain vacant Si sites, as the temperature-
composition phase diagrams of the RE–Si systems prove [102–107]. Summarising re-
views on the phase diagrams can be found for Gd, Tb, Dy, Ho and Er [109–113]. This
group is simply denoted by RE in the following paragraphs. Fig. 4.3 shows a sketch
of the silicon-rich half of the phase diagram of the Tb–Si system, to which those of
Gd, Dy, Ho and Er are similar. It furthermore contains a table with the respective
peritectic and eutectic temperatures.
The solubility of RE in Si and Si in RE is negligible (<1 at%) [102–104]. Several com-
pounds RExSiy exist, whose melting points are higher than those of elemental hcp-RE
and diamond-Si (𝑇REm > 1300 ◦C and 𝑇Sim = 1414 ◦C). All RE–Si systems have a com-
pound at a Si content of 50 at% (RE1Si1), which melts congruently at a high temperat-
ure (𝑇RESim ≈ 1850 ◦C). On the Si-rich side, all RE–Si systems have a compound at a Si
content of ≈62.5 at%, which transitions by a peritectic reaction into solid RE1Si1 and
liquidus at a peritectic temperature of𝑇1p ≈ 1650 ◦C . This compound is the AlB2 phase
with a stoichiometry of RESi2 – b (𝑏 ≈ 0.33, RESi1.67 or RE3Si5). In the Er–Si system,
AlB2-ErSi2 – b forms a eutectic with diamond-Si at a Si content of 81 at% and a eutectic
temperature of 𝑇e = 1210 ◦C. The Ho–Si, Dy–Si, Tb–Si and Gd–Si systems, in contrast,
have an additional, peritectic compound at a Si content between 62.5 at% (RESi2 – b)
and 66.7 at% (RESi2). The peritectic temperature 𝑇2p decreases monotonously from
Gd (1601 ◦C) to Ho (1290 ◦C). This compound is the ThSi2 phase with a stoichiometry
of RESi2 – a (𝑎 ∈ [0, 0.33], lower vacancy density than in AlB2-RESi2 – b). Similarly to
AlB2-ErSi2 – b in the Er–Si system, ThSi2-RESi2 – a forms a eutectic with diamond-Si at
Si contents between 82 at% and 87 at% and eutectic temperatures of 𝑇e ≈ 1200 ◦C.
Both the AlB2 and the ThSi2 phases show a reversible dimorphism (orthorhombic
distortion) at lower temperatures, in accordance with [82, 89, 91]. Furthermore, the
RESi2 – x phases have narrow homogeneity ranges of ≈1 at% [87, 104].
Several conclusions can be drawn from the phase diagrams. The first is that the form-
ation of the AlB2 and ThSi2 phases depend rather on the initial compositions than on
the temperature. In particular, the initial Si concentration of 62.5 at% is a critical
value beyond which AlB2-ErSi2 – b is saturated with silicon [87], and the silicon-richer
38
4.1 Structure prototypes
TbSi
Tm
liquidus
RE 𝑇1p (◦C) 𝑇2 ◦ ◦p ( C) 𝑇e ( C)1 Si
Tp Tm
Gd 1664 1601 1226
2
T Tb 1638 1517 1222p
Dy 1635 1437 1224
Ho 1620 1290 1210
Te Er 1620 — 1210
50% Si Composition 100% Si
Figure 4.3: Left: Schematic phase diagram of the Tb–Si system. Adapted from [111]. Right:
Peritectic and eutectic temperatures of some RE–Si systems [109–113].
ThSi2-RESi2 – a forms beside AlB2-RESi2 – b for the REs lighter than Er [88, 89, 93]. The
findings of Mayer et al. that ThSi2 seems to be the high-temperature phase and AlB2
the low-temperature phase [85] can be explained by the incompleteness of the reac-
tion if the temperature during the preparation is too low [108]. The second conclusion
concerns the peritectic transitions. For example, consider an initial preparation of
33 at% RE and 67 at% Si (RESi2). If the sample heats up to a temperature between
𝑇1p and 𝑇2p (∼1600 ◦C), AlB2-RESi2 – b crystals form beside a RE/Si melt on the liquidus
curve. Slow annealing drives the transition across the peritectic line 𝑇2p whereupon
the AlB2-RESi2 – b crystals vanish and solid ThSi2-RESi2 – a forms plus solid excess Si.
Such peritectic transitions require a particular care about the temperature variation
during annealing in order to obtain the desired single phase silicides. If the tem-
perature drops too fast, the product will contain considerable amounts of residual
AlB2-RESi2 – b which is kinetically inhibited from transitioning into the equilibrium
(ThSi2-RESi2 – a plus diamond-Si).
Moreover, the peritectic and eutectic temperatures𝑇1p , 𝑇2p and𝑇e show an interesting
dependence on the RE atomic numbers. While𝑇1p and𝑇e vary only slightly,𝑇2p shows a
rapid drop from Gd to Ho. In detail,𝑇2p lies short below𝑇1p in Gd–Si and approaches𝑇e
in Ho–Si. If the trend was extrapolated from Ho to Er, 𝑇2p would drop below𝑇e, which
coincides with the non-existence of ThSi2-ErSi2 – a. The behaviour of the peritectic
temperatures also confirms that the RE radius does not directly influence the stability
of the AlB2 structure, as stated in [85], but rather the stability of the ThSi2 structure.
The latter becomes unstable if the ratio between the atomic radii of RE and Si is too
small [84]. If the trend was extrapolated to the REs lighter than Gd, 𝑇2p would rise
beyond 𝑇1p , which implies that the ThSi2 phase melts congruently. This is indeed the
case for Sm–Si and Nd–Si [114, 115] (Eu left out because of its anomalous valence).
Hence, Gd, Tb, Dy and Ho are a unique quadruple concerning the peritectic transition
between ThSi2-RESi2 – a and AlB2-RESi2 – b.
Valence considerations
In early studies, the wrong assumption of vacancy-free, stoichiometric RESi2 lead
to the idea that the silicon sheets in the hex-AlB2 structure are analogous to the
39
Temperature
TbSi
TbSi2−b
TbSi2−a
diamond-Si
4 The rare-earth silicide bulk phases
carbon sheets in graphite, including the presence of delocalised 𝜋-electron systems.
For instance, Mayer et al. calculated a Si–Si bond distance of 2.16 Å from the basal
lattice constants of hex-AlB2-LuSi2. That the result is shorter than the single bond
length in diamond-Si (2.35 Å) indicates the presence of sesqui or double bonds. The
ratio between the two values (0.919) seems to confirm this as it is similar to the
bond-distance ratio of graphite-C and diamond-C (0.921) [83, 85]. However, as will
be proven, the Si–Si bond distance in hex-AlB2-RESi2 is close to that in diamond-Si.
Only the contraction of the basis due to the vacancies is such that a corresponding
vacancy-free honeycomb-Si sheet produces the above ratio by coincidence. Magaud
et al. were the first who theoretically investigated vacancy-populated YSi2 with DFT
(RE ionic radius similar to Ho and Er, no f electrons, [96]). It turned out that the
stoichiometry is decisive for the agreement between theoretical and experimental
lattice constants. Relatively to AlB2-YSi2, AlB2-Y3Si5 (ordered vacancies, configura-
tion 1 in Fig. 4.2 (b), light green unit cell) has an inverted 𝑐/𝑎 ratio as the basis of the
unit cell shrinks and the axial lattice constant grows (as predicted, vide supra). The
authors concluded that the vacancies allow the Si honeycomb to “release a part of
the constraint imposed by the two adjacent Y planes”, i.e. the vacancies are due to
mechanical strain [96]. However, as Zavala Aké et al. pointed out, both AlB2-YSi2 and
AlB2-Y3Si5 are unstrained zero-pressure structures and the Si–Si distances of respect-
ively 2.37 Å and 2.44 Å are quite close to that in diamond-Si (2.35 Å) (DFT, [100]). This
indicates that the Si atoms are singly bonded in both AlB2-YSi2 and AlB2-Y3Si5 and
that the removal of Si atoms causes the basal contraction, not vice versa.
The electronic density of states (DOS) gives an important hint at the true reason for
the stability of the vacancy-populated phases: The DOS of AlB2-YSi2 shows a peak at
the Fermi level, which is not present in AlB2-Y3Si5 [95, 100]. Obviously, the Si vacan-
cies partially deplete the conduction band and thereby reduce the total energy. In
order to find a simple explanation for this, the system will be illuminated within the
ZKB concept (p. 33). Since the anionic lattice of vacancy-free AlB2-YSi2 consists of
Si honeycomb sheets, the Si atoms have a formal valence of – I. However, each RE3+
cation donates three electrons into the system, of which two are accepted by two Si
atoms and one is spare. The compound formula can thus be written as an electride
REIIISi – Ie –2 , indicating that one electron per RE atom occupies the conduction band.
The removal of a Si atom primarily reduces the number of acceptors by one. On the
other hand, having lost their bonding partner, the three Si atoms adjacent to the va-
cant Si site can each accept an additional electron to complete the shells. In summa,
the capacity of the anionic partial lattice increases by two electrons for each vacancy.
The compound formula for RE3Si5 is thus REIII3 Si
−I −II −
2 Si3 e , so the number of conduc-
tion band electrons per RE atom is reduced to 1/3. Hence, the vacancies are expected
to improve the charge balance, which will be proven later.
According to the ZKB concept, the bonding in the vacancy-populated AlB2 structure
is as follows: The two Si – I atoms per unit cell bind each to the three Si – II atoms via in-
plane sp2-hybrid orbitals. The free electron pair of each Si – I atom thus occupies the
out-of-plane p orbital. The three Si – II𝑧 atoms per unit cell bind each to the two Si – I
atoms in an angled geometry. They are presumably sp3-hybridised and use the two
in-plane hybrid orbitals for the Si–Si bonds. The other two hybrid orbitals, which are
inclined out of the Si plane towards the vacancy, accept the two free electron pairs. Of
40
4.2 Structure optimisation
course, the charge transfer is formal and bonds have to be established between the
free electron pairs of the Si atoms and the empty orbitals of the RE atoms. While the
former have the above-mentioned geometry, the geometry of latter can be neglected
in first approximation as the s and d orbitals can hybridise in a quite flexible manner
[17, p. 396]. The electronegativity difference between Si and the REs from Gd to Er
(Δ𝜒 = 1.74−1.11 = 0.63, Allred-Rochow scale [17, Tafel III, Tafel V]) suggests a partial
ionic character of ca. 10 % [17, p. 158]. Thus, a covalent bonding between the RE-sd
orbitals and the Si-sp/Si-p𝑧 orbitals is expected to dominate the chemistry.
4.2 Structure optimisation
The preliminary considerations about the structure of bulk RESi2 – x are inspected by
means of DFT (Sec. 2.1). For this purpose, several structure models based on the AlB2
and ThSi2 prototypes are set up in suitable unit cells and optimised with respect to
the cell vectors and the ionic positions (mechanical equilibrium). After that, the ther-
modynamic properties are calculated in order to determine the most stable phases
under given conditions. The focus of the investigations is on the silicides of Tb since
Tb lies in the mid of the quadruple from Gd to Ho and is employed for the nanostruc-
tures in Chap. 5 and Chap. 6. At certain points, the TbSi2 – x are compared to ErSi2 – x
as it is the lightest RE not assuming the ThSi2 phase.
4.2.1 Methodological details
Volume relaxation
All bulk phases in this work are considered to be in mechanical equilibrium with
the vacuum, so the zero-pressure volumes of the structure models have to be found.
The first method to conduct the volume relaxations employs the code-level relaxation
routines which optimise the atomic positions, the volume and the shape of the unit
cell simultaneously. This is a quick way, but some systematic errors limit the reliab-
ility of the results. On the one hand, Pulay stress arises, i.e. artificial stress from the
link between the unit-cell volume and the Fourier sampling. This can be reduced by
iterative restarts of the relaxation. On the other hand, the results depend on the way
how the minimum is approached, which is due to the finite stop criterion (energy dif-
ference, forces, etc.). In particular, starting from volumes smaller and larger than the
optimum can result in different unit cells. These errors are avoided if sample unit
cells are optimised at fixed volumes and the Murnaghan equation of state is used
[116]. In detail, the free energy 𝐹 (𝑉,𝑇, 𝑁•), depending on the volume 𝑉 , the temper-
ature 𝑇 and t(he p)article number 𝑁• of the atomic spe[cies •, i(s m)inim]ised.𝐾′
( ) 𝜕𝑝 𝐾 𝑉 1 𝑉= − = + ′ ⇒ ( ) = + 0 · 0𝐾 𝑝 𝑉 𝐾0 𝐾 𝑝 𝐹 𝑉 𝐹0 ′ ′− + 1 −
𝐾0𝑉0
′− (4.33)𝜕𝑉 𝑇 𝐾 𝐾 1 𝑉 𝐾 1
where 𝐾 (𝑝) is the compressive modulus at 𝑝, 𝐾0 = 𝐾 (0) and 𝐾 ′ = 𝐾 ′(0).
41
4 The rare-earth silicide bulk phases
The function 𝐹 (𝑉 ) is fitted to the samples (𝑉, 𝐹 (𝑉 )), whereupon the fit parameters
finally yield the minimum (𝑉0, 𝐹0) as well as the bulk mechanical properties 𝐾0 and
𝐾 ′. As the prerequisite of Eq. (4.33) is the linear behaviour of 𝐾 (𝑝), the Murnaghan
equation of state is valid only on a small region around the minimum.
Thermodynamical framework
Since this chapter is preparatory work for the nanostructures on silicon surfaces,
the investigations concern only the Si-richest bulk RESi2 – x phases under Si-rich con-
ditions. In other words, the most stable structures are searched for a given amount of
RE and an infinitely large reservoir of Si provided by bulk diamond-Si. This translates
into the thermodynamic boundary conditions that the pressure 𝑝, the temperature 𝑇 ,
the RE amount 𝑁RE and the Si chemical potential 𝜇Si are constant. The situation is
best described by a thermodynamical potential 𝐺RE(𝑝,𝑇, 𝑁RE, 𝜇Si) derived from the
internal energy𝑈 (𝑉, 𝑆, 𝑁RE, 𝑁Si) by Legendre transformation.
𝐺RE(𝑝,𝑇, 𝑁RE, 𝜇Si) = 𝑈 (𝑉, 𝑆, 𝑁RE, 𝑁Si) + 𝑝𝑉 − 𝑇𝑆 − 𝜇Si𝑁Si (4.34)
where 𝑉 is the volume, 𝑆 the entropy, 𝑁• the particle number of the atomic species
• and 𝜇• the respective chemical potential. Under given boundary conditions, those
structures are stable which minimise 𝐺RE. According to Euler’s theorem, 𝐺RE com-
prises only the term 𝑁RE𝜇RE and can easily be transformed into an intensive quantity
by dividing by 𝑁RE. The remainder is then identical to the RE chemical potential.
1 𝜇RE𝑁RE
𝐺RE(𝑝,𝑇, 𝑁RE, 𝜇Si) = = 𝜇RE(𝑝,𝑇, 𝜇Si) (4.35)
𝑁RE 𝑁RE
The boundary conditions immediately replace some of the variables in Eq. (4.34)
with known quantities. The structure models determine {𝑁•}. 𝜇Si is fixed to 𝜇d-SiSi ,
the Si chemical potential of diamond-Si. The 𝑝𝑉 terms are zero because the struc-
ture models are relaxed into their zero-pressure volumes (vide supra). Concerning
𝑆 and 𝑇 , an approximation will be made, although its validity is not obvious a pri-
ori: The internal energy𝑈 is replaced by the DFT total energy 𝐸cellDFT and the 𝑇𝑆 terms
are omitted. This is based on the close relatedness between the RESi2 prototypes,
which are both characterised by cages of mutually three-fold coordinated Si atoms,
into which the RE atoms are embedded. Therefore, the vibrational contributions to
𝑈 and 𝑆 are expected to be similar in different RESi2 – x structures, so the 𝑇𝑆 terms
cancel out in energy differences. This is a common practice when related structures
are compared at low temperatures [53]. The validity of the 𝑇𝑆 cancellation will be
tested on two samples in Sec. 4.2.4.(With these repl)acements, Eq. (4.35) reduces to:
≈ 1 cell d-Si hcp-RE𝜇RE 𝐸
𝑁 DFT
− 𝜇Si 𝑁Si − 𝜇RE (4.36)
RE
Referring to hcp-RE𝜇RE 𝜇RE , the RE chemical potential of bulk hcp-RE, is necessary be-
cause only chemical potential differences have a physical meaning. As the reference
is arbitrary, hcp-RE𝜇RE could be substituted by the RE chemical potential of other phases,
e.g. by that of gaseous RE atoms. In the present form, Eq. (4.36) equals the free
42
4.2 Structure optimisation
formation enthalpy of RE + (2 – x) Si −−→ RESi2 – x under Si-rich conditions. As 𝜇RE is
intensive, structure models of different sizes and, particularly, containing different
numbers of RE and Si atoms can directly be compared regarding their stability. In
detail, if a set of RESi2 – x structures is given, only those are in chemical equilibrium
with diamond-Si which minimise hcp-RE𝜇RE. The 𝜇RE term cancels in 𝜇RE differences,
which shows that the reference is indeed arbitrary. If, in addition, the stoichiometry
of the two structures is equal, also the 𝜇d-SiSi term cancels, so knowledge about any
referential chemical potentials is not necessary in this case. A major part of the in-
vestigations concerns the question whether the AlB2 phase or the ThSi2 phase is more
stable under certain conditions (e.g. the Si vacancy density or the theoretical treat-
ment). Therefore, it is instructive to define the difference:
Δ𝜇T−ARE ≔ 𝜇
ThSi2 AlB2
RE − 𝜇RE (4.37)
where 𝜇ThSi2RE and 𝜇
AlB2
RE are the RE chemical potentials of the most stable ThSi2-RESi2 – x
and AlB2-RESi2 – x structure models at a certain 𝑥. Hence, Δ𝜇T−ARE < 0 implies that the
ThSi2 phase is more stable than the AlB2 phase and vice versa.
Vibrational contributions to the free energy
The omission of the lattice dynamics in Eq. (4.36) is based on the cancellation of the
vibrational contributions upon comparing similar structures. In order to check the
validity, it has to be proven that the lattice dynamics of the structures to be compared
yield similar free energies. Lattice dynamics covers a broad field of different effects
and methods which would exceed the purposes of this work. A quite feasible concept
for estimating lattice-dynamical effects at not too high temperatures is the frozen-
phonon method, which is applied in this work.
In principle, the dynamics of 𝑁K ions in a DFT system are described by the Hamilton
operator 𝐻K with the DFT total energy 𝐸DFT({𝑅𝑘}) acting as an effective potential.
The latter depends on the ionic positions {𝑅𝑘}, 𝑘 ∈ [1, 3𝑁K] (Born-Oppenheimer ap-
proximation, Eqs. (2.2); from now on, the positions are written component-wise).
For small displacements from the equilibrium, 𝐸DFT({𝑅𝑘}) can be approximated by
a second-order expansion, the harmonic approximation. The coefficient of the quad-
ratic term, the Hessian matrix of 𝐸DFT({𝑅𝑘}) or dynamical matrix, can be calculated
by finite differences. Its eigenvalues are (the square of) frequencies 𝜔𝑘 and its ei-
genvectors are harmonic vibrations. Hence, as long as the harmonic approximation
is valid, i.e. the temperature is not too high, the lattice dynamics is described by
an ensemble of 3𝑁K independent quantum-mechanical harmonic oscillators. Since
𝐸DFT({𝑅𝑘}) is periodic in crystals, an approach similar to the Bloch theorem (Sec. 2.2)
can be employed to reduce the vibrating ions to the𝑁 cellK ions of the unit cell. The band
structure of the eigenfrequencies gives rise to 3𝑁 cellK discrete branches, the phonons,
which depend on a continuous parameter 𝒒 confined to the Brillouin zone [6, pp.
41ff]. The “frozen-phonon” method consists in calculating the phonons at zero tem-
perature and using them for describing the lattice dynamics at finite temperatures.
The exact calculation of state variables of a phononic system involves q-integrations
over the Brillouin-zone and summations over discrete branches. In this work, the
43
4 The rare-earth silicide bulk phases
VASP-built-in phonon routines are employed, which calculate only the 𝑁 cellK phonons
at 𝒒 = 0. The sum over them corresponds to sampling the Brillouin-zone solely by
the Γ-point, which is insufficient particularly for small unit cells. To resolve this is-
sue, the unit cells are multiplied to supercells so that the Γ-point includes also 𝒒 ≠ 0
points according to band-folding. In doing so, symmetry-equivalent entries of the
dynamical matrix are identified so as to reduce the computational demand.
Once the eigenfrequencies 𝜔𝑘 are determined, the vibrational free energy 𝐹vib(𝑇 )
at finite temperatures 𝑇 can be calculated by the quantum statistics of independent
harmonic o∏scillators.3𝑁cell 3∏𝑁cellK K
𝑍 = tr[exp(−𝛽ℏ𝜔 1𝑘 (𝑛 + 2))] = (2 sinh(𝛽ℏ𝜔 /2))
−1
𝑘 , 𝛽 = 1/𝑘B𝑇
𝑘=1 ∑︁ 𝑘=13𝑁cellK 3∑︁𝑁cellK
⇒ 1 𝑇→0 ℏ𝜔𝑘𝐹vib(𝑇 ) = − ln(𝑍) = 𝑘B𝑇 ln(2 sinh(ℏ𝜔𝑘/2𝑘B𝑇 )) ≃ (4.38)
𝛽 2
𝑘=1 𝑘=1
Adding this to the electronic free energy ≈𝐸cellDFT yields the total free energy 𝐹 (𝑇 ).
𝐹 (𝑇 ) = 𝐸cellDFT + 𝐹vib(𝑇 ) (4.39)
In the low-temperature limit, 𝐹vib(𝑇 ) approaches the sum of the ground-state ener-
gies of the harmonic oscillators. Consequently, vibrational contributions can be rel-
evant even at low temperatures, at which the 𝑇𝑆 terms vanish.
The theoretical treatment of the rare-earth elements
As the considerations in Sec. 3.1 point out, the lanthanoids have a strongly correlated,
incomplete 4f shell. Therefore, pure DFT predicts the wrong ground state with frac-
tionally occupied 4f levels lying at the Fermi level. The first of two possible solutions
for this problem is the LDA+U method (Sec. 2.1.1), which works quite well for the 4f
electrons of the lanthanoids since they obey Hund’s rules. The alternative solution
makes use of the inertness of the 4f electrons and “freezes” them in the core during
the pseudopotential generation. The major a-priori decision to be made is then the
VEC of the Ln atom, i.e. how many 4f electrons are kept frozen in the core. Of course,
the second approach is only possible if the valence is known a priori.
The explicit treatment of the 4f electrons with LDA+U is a quite universal approach
as the valence of the Ln atoms is not set at the beginning, but rather develops self-
consistently in the respective structure model. However, this approach has import-
ant drawbacks. Firstly, the computational demand is significant, in particular for the
heavy trivalent lanthanoids, as 2 + 𝑛 electrons per Ln atom have to be considered,
while only 2 + 1 electrons determine the chemistry. Secondly, the convergence into
the ground state can be rather difficult, so many test calculations may have to be done
until the spin configuration is correct. The third drawback consists in the paramet-
ers 𝑈 and 𝐽 , which depend on the VEC and alter the total energy by a gauge. There-
fore, calculations with different𝑈 and 𝐽 and, hence, different VECs cannot directly be
44
4.2 Structure optimisation
compared. Instead, test calculations are necessary, from which the correct VEC can
be determined by comparing observable quantities with the experiment (cohesive
energies, bond lengths, lattice constants etc.). Since this limits the ab initio character
of the explicit f-electron treatment, the frozen-core approach often gives satisfying
results while saving computational demand. It will be the method employed in this
work, while the explicit f-electron treatment will be tested.
Computational parameters
The DFT calculations are carried out with VASP [2, 3] using PAW potentials and the
PBE xc-functional [10, 21, 22]. The 4f electrons of the REs are kept frozen in the core,
while the 5p electrons are treated explicitly as semi-core electrons. As the considered
REs are trivalent in their silicides [81, 83], the REs have a PAW valence of 5p65d16s2.
Si has a PAW valence of 3s23p2. The validity of freezing the 4f electrons in the core
will be tested for TbSi2 and ErSi2 (Sec. 4.2.3). The kinetic energy cutoff is 𝐸cut = 400 eV.
The Brillouin zone is sampled by a Γ-centred Monkhorst-Pack mesh [117] at a density
of 16 × 16 × 16 k-points for the hexagonal unit cell and 16 × 16 × 8 k-points for the
conventional tetragonal unit cell. For the supercells, the sampling density is scaled
accordingly. The Brillouin-zone integration is carried out by a Gaussian occupation
with a smearing of 20 meV for the ionic relaxations and by the tetrahedron method
with Blöchl corrections [118] for all static calculations. A convergence test proves that
increasing the energy cutoff or the sampling density alters the total energy by≈1 meV
per atom. The ionic forces are calculated according to the Hellmann-Feynman the-
orem [119]. The ions move along them towards the equilibrium positions and stop to
relax if the forces acting on each ion are smaller than 0.005 eV/Å.
4.2.2 The stoichiometric RESi2 phases
The mechanical equilibria of the vacancy-free prototypes of TbSi2, DySi2, HoSi2 and
ErSi2 are determined by Murnaghan relaxations. For this purpose, the unit cells of
AlB2-RESi2 and ThSi2-RESi2 are set up according to Fig. 4.1 and their zero-pressure
volumes 𝑉0 are estimated by the code-level routines. After that, eleven samples in
a ±10 % interval around the estimated 𝑉0 are set up and relaxed with respect to the
atomic positions and the shape of the unit cell, but at constant volume 𝑉 . The DFT
total energies are taken as free energies 𝐹, neglecting vibrational contributions (vide
supra). The samples (𝑉, 𝐹) are fitted by the Murnaghan equation of state 𝐹 (𝑉 ) (Eq.
(4.33)), which returns the bulk mechanical parameters. For the ThSi2 prototype, 𝑉
and 𝐹 are rescaled by 1/4 so that they become comparable with those of the AlB2
prototype. Furthermore, 𝐹 is reset to 𝐹0.
The Murnaghan fits excellently retrace the sample points for all silicides and phases
(Plots: Fig. 4.4, parameters: Tab. 4.1). The equilibrium volumes per formula unit 𝑉0
contract almost linearly towards heavier REs, in accordance with the lanthanoid con-
traction. Hex-AlB2-RESi2 is slightly denser than tet-AlB2-RESi2, i.e. 𝑉0 of the former
is smaller than that of the latter (−0.4 %, −0.5 %, −0.7 % and −0.9 % for TbSi2, DySi2,
45
4 The rare-earth silicide bulk phases
350 (a)  hex-AlB2 ErSi2 (b)  tet-ThSi2 ErSi2HoSi2 HoSi2
300 DySi2 DySi2
250 TbSi2 TbSi2
200
150
100
50
0
50 52 54 56 58 60 62 64 50 52 54 56 58 60 62 64
V − V0  (Å3) V − V0  (Å3)
Figure 4.4: Murnaghan fits of TbSi2, DySi2, HoSi2 and ErSi2 in (a) the hex-AlB2 and (b) the
tet-ThSi2 phase. Crosses: sample points, lines: Murnaghan fits. All extensive quantities are
scaled by the number of RE atoms per unit cell.
hex-AlB2 prototype tet-ThSi2 prototype
𝑉 (Å30 ) 𝐾0 (eV/Å3) 𝐾 ′ (eV/Å6) 𝑉0 (Å3) 𝐾0 (eV/Å3) 𝐾 ′ (eV/Å6)
Tb 57.901 0.524181 4.10229 58.141 0.540657 4.39081
Dy 57.308 0.527378 4.15985 57.617 0.543023 4.43436
Ho 56.756 0.530427 4.24312 57.162 0.547184 4.35291
Er 56.215 0.533331 4.38906 56.741 0.553545 4.19432
Table 4.1: Bulk mechanical properties of TbSi2, DySi2, HoSi2 and ErSi2 in the hex-AlB2 and the
tet-ThSi2 phase.
This work Exp. reference
𝑎 (Å) 𝑐 (Å) 𝑐/𝑎 Δ𝜇T−ARE (meV) 𝑎 (Å) 𝑐 (Å) 𝑐/𝑎
hex-AlB2 prototype
Tb 4.100 3.978 0.970 3.84 4.14 1.078
Dy 4.097 3.943 0.962 3.83 4.12 1.076
Ho 4.096 3.907 0.954 3.80 4.10 1.079
Er 4.090 3.881 0.949 3.78 4.08 1.079
tet-ThSi2 prototype
Tb 3.977 14.707 3.698 −49 4.05 (3.96) 13.38 3.304
Dy 3.953 14.747 3.731 −44 4.04 (3.95) 13.33 3.300
Ho 3.916 14.912 3.808 −40 4.03 (3.97) 13.31 3.303
Er 3.891 14.995 3.854 −37 — — —
Table 4.2: Lattice parameters and relative RE chemical potentials of TbSi2, DySi2, HoSi2 and
ErSi2 in the hex-AlB2 and the tet-ThSi2 phase. Numbers in parentheses indicate orthorhombic
distortion. Experimental reference: [82, 83, 85].
46
F − F0  (meV)
4.2 Structure optimisation
HoSi2 and ErSi2, respectively). The compressive moduli 𝐾0 of all RESi2 are compar-
able, as well as those of the AlB2 and the ThSi2 phase. They range from 0.52 eV/Å3
to 0.56 eV/Å3, i.e. ∼80 GPa. The rigidity slightly grows from the lighter to the heavier
REs and from the hex-AlB2 to the tet-ThSi2 phase.
The lattice parameters (Tab. 4.2) reveal that the volume contraction from TbSi2 to
ErSi2 is anisotropic. This can be linked to how the Si anionic sublattice is expected to
respond to shrinking RE cations. Concerning the hex-AlB2 structure, the honeycomb-
Si layers are quite rigid, so the shrinkage of the cations in between is mostly com-
pensated by the interlayer distance, the axial lattice constant 𝑐. From TbSi2 to ErSi2,
𝑐 shrinks by −2.4 %, while the basal lattice constant 𝑎 is almost constant (−0.2 %). In
summa, the volume contracts by −2.9 %. The situation in the tet-ThSi2 structure is
more complicated as the anionic sublattice comprises a three-dimensional network
of Si atoms. From TbSi2 to ErSi2, the 𝑎 shrinks by −2.2 %, while 𝑐 grows by +2.0 %. In
summa, the volume shrinks by −2.4 %, similarly to the hex-AlB2 phase. The chemical
potential differences Δ𝜇T−ARE (Eq. (4.37)) tell that the tet-ThSi2 phase is considerably
more stable than the hex-AlB2 phase for all stoichiometric RESi2. Although, the ThSi2
phase slightly destabilises from TbSi2 to ErSi2, the potential difference is still higher
than 𝑘B𝑇 at low and medium temperatures (𝑘B · 100 K = 8.6 meV). Therefore, the
formation of hex-AlB2-RESi2 is not possible.
The results for stoichiometric RESi2 conflict with several experimental observations.
The first contradiction is that the AlB2 phase is unstable, even for ErSi2, although
it is adopted by the silicides of all REs and the only prevailing phase in ErSi2 [85].
Furthermore, the lattice constants deviate considerably from the experimental ref-
erence, not only absolutely by several percent. Also the 𝑐/𝑎 ratios strongly differ by
more than 10 % and are even inverted for hex-AlB2-RESi2. The last major deviation
from the experiment is the missing orthorhombic distortion in ThSi2-RESi2 as 𝑎 and
𝑏 remain equal during relaxation, even if the symmetry of the starting configura-
tion is broken by hand. The origin of the discrepancies lies in the high amount of
Si vacancies in the real RESi2 – x structures, as it will be proven in Sec. 4.2.5. Before
addressing this, the validity of two technical approximations will be verified by test-
ing them on the stoichiometric RESi2 structures: The frozen-core approach of the 4f
electrons (Sec. 4.2.3) and the omission of vibrational contributions (Sec. 4.2.4).
4.2.3 The role of the 4f electrons in RESi2
In order to test if the 4f electrons have effects on the structures, the Murnaghan relax-
ations are repeated for the TbSi2 and ErSi2 structures with f-valent PAW potentials3.
The DFT calculations employ spin polarisation and the rotationally invariant, simpli-
fied LDA+U approach, proposed by Dudarev et al. [14, 15] (Sec. 2.1.1). Since only the
4f shell of the RE atoms must be corrected, one set of 𝑈 and 𝐽 parameters has to be
determined. 𝑈 is calculated with the self-consistent linear-response approach (Eqs.
(2.14) and (2.15), [13]). For this purpose, the unit cells from the previous section are
3Potentials which treat the 4f electrons explicitly as valence electrons. The corresponding calculations
are referred to as f-valent, in contrast to the trivalent ones above.
47
4 The rare-earth silicide bulk phases
multiplied to supercells ((2×2×2) for hex-AlB2-RESi2, (2×2×1) for tet-ThSi2-RESi2) so
that the penalised RE atom is separated from its periodic replica. The k-point mesh
is adapted to the supercells. The results for 𝑈 are 5.0 eV and 4.9 eV for AlB2-TbSi2
and ThSi2-TbSi2, respectively. Since the values are consistent, a common 𝑈 = 5.0 eV
is used for all f-valent TbSi2 calculations. AlB2-ErSi2 yields a self-consistent 𝑈 para-
meter of 7.4 eV, used for all f-valent ErSi2 calculations. According to Eq. (2.13), 𝑈
and 𝐽 enter the Dudarev correction only by their difference. The exact value of the
effective parameter (𝑈 − 𝐽) is irrelevant, though, once it passes a certain threshold
(idempotent occupation matrix, Hund’s rules). Therefore, (𝑈 − 𝐽) can safely be set
to 𝑈 . Once the individual RE atoms adopt the correct spin configuration, the atomic
magnetic moments provide different ways of ordering. Since magnetic order is not
the focus of this work, only orderings which are collinear and commensurate with
the conventional unit cells are considered. Containing one RE atom, the unit cell of
AlB2-RESi2 is described as ferromagnetically ordered (FMO). The four atoms in ThSi2-
RESi2 permit additional antiferromagnetically ordered (AMO) configurations, out of
which only that with alternating FMO (001)-planes is considered. Since the symmetry
group of the unit cell can prevent orbitals from splitting and artificially retain degen-
eracies, the symmetry reduction of the Brillouin zone is switched off and the lattice
of the initial unit cell is slightly distorted before relaxation.
The results of the Murnaghan relaxations for f-valent TbSi2 are compiled in Tab. 4.3.
As expected from the inertness of the 4f shell, the f-valent ThSi2-TbSi2 unit cell is
almost equal to the trivalent one (𝑎: +0.1 %, 𝑐: −0.7 %, 𝑉 : −0.6 %). In particular, it
retains the tetragonal symmetry. The structural deviations due to magnetic order-
ing are negligible. On the contrary, the f-valent AlB2-TbSi2 unit cell incurs a con-
siderable orthorhombic distortion with regard to trivalent hex-AlB2-TbSi2, resulting
in a new ort-AlB2-TbSi2 structure. In detail, the dashed orthorhombic envelope of
the primitive AlB2 unit cell in Fig. 4.1 (b) expands in the [1̄1.0]-direction and con-
tracts in the perpendicular [11.0]-direction. Thereby, the formerly isotropic basal
lattice constants split into a larger one in the [10.0]-direction (denoted 𝑎, 4.214 Å,
𝑎 𝑐 𝑐/𝑎 𝑉/𝑁Tb 𝛾 Δ𝜇Tb
(Å) (Å) (Å3) (meV)
trivalent potentials
hex-AlB2-TbSi2 4.100 3.978 0.970 57.90 120.0◦ 17
ort-AlB2-TbSi2 4.229 (3.878)𝑎 4.002 0.946 58.34 124.7◦ 0
tet-ThSi -TbSi 3.977 14.707 3.698 58.14 122.5◦2 2 −32
f-valent potentials
fmo-ort-AlB2-TbSi2 4.214 (3.886)𝑎 3.986 0.946 57.92 124.4◦ 0
fmo-tet-ThSi2-TbSi2 3.979 14.606 3.671 57.82 122.3◦ −31
amo-tet-ThSi2-TbSi2 3.978 14.617 3.674 57.83 122.2◦ −38
Table 4.3: Lattice parameters and relative Tb chemical potentials of TbSi2 in the AlB2 and
the ThSi2 phase under different f-electron treatments. The unit cells were optimised by
Murnaghan relaxations. 𝑎orthorhombic distortion.
48
4.2 Structure optimisation
+2.8 % compared to that of trivalent hex-AlB2-TbSi2) and a smaller one in the [11.0]-
direction (denoted 𝑏, 3.878 Å, −5.4 %). The orthorhombic distortion also redistributes
the three M-star angles of 120◦ to two larger ones of 𝛾 = 124.4◦ (cf. Fig. 4.1 (b)) and
a smaller one of 111.2◦. Remarkably, the axial lattice constant 𝑐, as well as the unit-
cell volume𝑉 remain constant (+0.2 % and ±0.0 %, respectively). In other words, the
orthorhombic distortion is an area-conserving deformation of the basis. In order to
clarify whether mechanisms other than the f-electron treatment are responsible for
this, the ort-AlB2-TbSi2 unit cell is again subjected to a Murnaghan relaxation, this
time with trivalent PAW potentials. Instead of relaxing back into hex-AlB2-TbSi2, it
retains the orthorhombic distortion (Tab. 4.3). All deviations in the lattice constants
of the trivalent and f-valent structures are below 0.5 % and the f-valent volume is
slightly smaller (−0.7 %), similarly to tet-ThSi2-TbSi2. Therefore, the orthorhombic
distortion of AlB2-TbSi2 is indeed physical and was merely inhibited by the fixed sym-
metry group in the previous trivalent calculations. It is remarkable that the volumes
per TbSi2 formula unit of ort-AlB2-TbSi2 and tet-ThSi2-TbSi2 are almost equal and that
the three angles of the M-stars assume similar values.
To facilitate the comparison, the Tb chemical potentials 𝜇Tb in Tab. 4.3 are reset by
that of ort-AlB2-TbSi2, respectively for the trivalent and the f-valent calculations.
Δ𝜇•Tb = 𝜇
•
Tb − 𝜇
ort-AlB2-TbSi2
Tb
Concerning the trivalent calculations, the orthorhombic distortion stabilises the AlB2-
TbSi2 structure by −17 meV. As a consequence, the AlB2 and the ThSi2 phase are en-
ergetically closer together (Δ𝜇T−ATb = −32 meV). Concerning the f-valent calculations,
tet-ThSi2-TbSi2 is more stable in the AMO than in the FMO configuration by −7 meV.
Comparing the two FMO structures, fmo-tet-ThSi2-TbSi2 is more stable than fmo-hex-
AlB2-TbSi2 by Δ𝜇T−ATb = −31 meV, in agreement with the trivalent calculations.
The results for f-valent ErSi2 may be found in Tab. 4.4. The overall trends are very
similar to those of TbSi2, including the well agreement between the trivalent and
𝑎 𝑐 𝑐/𝑎 𝑉/𝑁Er 𝛾 𝜇Er
(Å) (Å) (Å3) (meV)
trivalent potentials
hex-AlB2-ErSi2 4.090 3.881 0.949 56.22 120.0◦ 24
ort-AlB2-ErSi2 4.240 (3.840)𝑎 3.914 0.923 56.82 125.4◦ 0
tet-ThSi2-ErSi2 3.891 14.995 3.854 56.74 124.3◦ −13
f-valent potentials
fmo-ort-AlB2-ErSi2 4.234 (3.847)𝑎 3.912 0.924 56.67 125.2◦ (0)𝑏
fmo-tet-ThSi2-ErSi2 3.897 14.924 3.829 56.66 124.0◦ (−185)𝑏
amo-tet-ThSi2-ErSi2 — — — — — (−190)𝑏
Table 4.4: Lattice parameters and relative Er chemical potentials of ErSi2 in the AlB2 and
the ThSi2 phase under different f-electron treatments. The unit cells were optimised by
Murnaghan relaxations. 𝑎orthorhombic distortion. 𝑏not reliable due to numerical problems.
49
4 The rare-earth silicide bulk phases
f-valent lattices (<0.5 %) and the orthorhombic distortion of AlB2-ErSi2. The stabil-
ity gain from the orthorhombic distortion is higher for AlB2-ErSi2 (−24 meV) than
for AlB2-TbSi2 (−17 meV). Consequently, trivalent ort-AlB2-ErSi2 is energetically close
to trivalent tet-ThSi2-ErSi2 (Δ𝜇T−AEr = −13 meV). Although, the relaxation of f-valent
ThSi2-ErSi2 was performed only in the FMO configuration, the dependence of Δ𝜇Er
on the magnetic ordering can be estimated by the total energy of the FMO-relaxed
cell in the AMO configuration. The latter is more stable by −5 meV, similarly to the
TbSi2 case. The potential difference Δ𝜇T−AEr of the FMO structures (−185 meV) is much
higher than that of the trivalent structures (−13 meV), which is the only incongru-
ity. Since the deviation is not linked to any structure differences, it is likely that the
ground state configurations – at least that of fmo-ort-AlB2-ErSi2– were incorrect. In
detail, although the total on-site charge and magnetisation were as expected, the spin
ordering within the 4f shell could be wrong. Supporting evidence for this is that the
Murnaghan relaxations produced several classes of images which had equal on-site
charges, magnetisations and equilibrium volumes, but different energy offsets. The
nature of such problems is clearly numerical, so the f-valent Δ𝜇Er in Tab. 4.4 are not
reliable. That such problems were absent in f-valent TbSi2 agrees with the this as-
sumption as Tb has only one electron in the second half-shell, not four like Er.
In conclusion, neither the explicit treatment of the 4f electrons, nor the magnetic
ordering have major effects on the RESi2 structures, as expected. Also the relative
stabilities between the phases are retained, provided that the magnetic ordering is
analogue. Therefore, the frozen-core approach is safe for use and the trivalent poten-
tials are employed for all further RESi2 calculations (bulk, films and nanowires).
4.2.4 Vibrational contributions in RESi2
The lattice-dynamical effects on 𝜇Tb will be quantified for AlB2-TbSi2 and ThSi2-TbSi2
by calculating the vibrational free energies within the frozen-phonon approach and
the harmonic approximation (vide supra). The error due to the omitted thermal ex-
pansion can be estimated by the experimentally determined linear thermal expan-
sion coefficients of ≈15 · 10−6 K−1 for AlB2-RESi2 – x [98]. Since this approaches the
error bars for the lattice constants in this work (≈0.5 %, vide infra) at room tem-
perature, the approach is considered to be safe up to 300 K. The unit cells are mul-
tiplied to supercells so as to increase the effective Brillouin-zone sampling for the
phonons (vide supra). The comparability between the AlB2 and the ThSi2 structure
at different effective samplings is ensured by embedding the AlB2 structure into an
orthorhombic unit cell (the dashed red cell in Fig. Fig. 4.1 (b) doubled in the [1̄1.0]-
direction), which matches the tetragonal unit cell of the ThSi2 structure by the dimen-
sions. These Tb4Si8 cells, denoted ThSi2-type unit cells to demarcate them from the
conventional ones, are optimised by Murnaghan relaxations analogous to Sec. 4.2.2.
The results are compiled in Tab. 4.5. The lattice parameters of tet-ThSi2-TbSi2 are
equal to those in Tab. 4.3. The AlB2 phase splits into three slightly different struc-
tures. Two of these are hex-AlB2-TbSi2 (𝛾 = 120◦) and ort-AlB2-TbSi2 (𝛾 = 124.7◦),
whose lattice parameters are almost equal to those in Tab. 4.3. The third structure,
denoted ort2-AlB2-TbSi2 (𝛾 = 118.1◦), is new and represents the other possibility of
50
4.2 Structure optimisation
𝑎 (Å) 𝑏 (Å) 𝑐 (Å) 𝑉/𝑁 3Tb (Å ) 𝛾 Δ𝜇Tb (meV)
hex-AlB2-TbSi2 4.10 3.98 14.19 (4.10)𝑎 57.9 119.9◦ 18
ort-AlB2-TbSi2 3.88 4.00 15.04 (4.23)𝑎 58.4 124.7◦ 0
ort2-AlB2-TbSi2 4.20 3.98 13.87 (4.05)𝑎 58.0 118.1◦ 17
tet-ThSi2-TbSi2 3.98 3.98 14.71 58.1 122.6◦ −32
Table 4.5: Lattice parameters and relative Tb chemical potentials of TbSi2 in the AlB2 and
the ThSi2 phase in orthorhombic Tb4Si8 cells. The unit cells were optimised by Murnaghan
relaxations. The lattice constants 𝑎, 𝑏 and 𝑐 refer to the ThSi2-type unit cells4. 𝑎lattice constant
𝑎 in the conventional [10.0]-direction.
orthorhombic distortion, contrary to that in ort-AlB2-TbSi2, i.e. dilation in the [11.0]-
direction and contraction in the perpendicular [1̄1.0]-direction. This new structure is
discarded as it is not more stable than hex-AlB2-TbSi2. For the other three structures,
the (static) values for Δ𝜇Tb are equal to those in Tab. 4.3.
The Γ-point phonons are calculated in (1×1×1), (2×2×1) and (3×3×1) supercells
of the optimised ThSi2-type unit cells. This corresponds to a convergence test for the
effective sampling. The restriction of the multiplication to the basal plane is due to the
aspect ratio of the ThSi2-type unit cell. The (3×3×1) supercell is almost cubic so that
the effective sampling of the reciprocal space is nearly isotropic. The vibrational free
energy 𝐹vib(𝑇 ) of each supercell is calculated with Eq. (4.38), scaled by the respective
number of Tb atoms 𝑁Tb and plotted from 𝑇 = 0 K to 270 K (Fig. 4.5 (a) – (c)). While
the changes in 𝐹vib(𝑇 ) from the (1 × 1 × 1) to the (2 × 2 × 1) supercells are visible
(≈8 meV at𝑇 = 0 K), the free energies of the (2×2×1) and the (3×3×1) supercells are
almost equal (deviation <1 meV at 𝑇 = 0 K). Thus, the (3 × 3 × 1) supercells provide
well converged absolute vibrational free energies. The lattice dynamical effects can
be introduced into the Tb chemical potentials 𝜇Tb simply by adding the vibrational
free energies per Tb atom, since the stoichiometries of all structures are equal. They
are reset by 𝜇ort-AlB2-TbSi2Tb (𝑇 ) ≕ 𝜇
0
Tb(𝑇 ).
• • 0 0
Eq. (4.35) 𝐹 (𝑇 ) − 𝑁 𝜇Si 𝐹 (𝑇 ) − 𝑁 𝜇Si
Δ𝜇•Tb(𝑇 ) =
Si Si
𝑁•
−
𝑁0Tb Tb
Eq. (4.39) 1 ( )
= 𝐸• 0 • 0DFT − 𝐸DFT + 𝐹vib(𝑇 ) − 𝐹vib(𝑇 )𝑁Tb
The plots of Δ𝜇Tb(𝑇 ) for the (1 × 1 × 1) and the (3 × 3 × 1) supercells can be found in
Fig. 4.5 (d). Those of the (2×2×2) supercells (not shown) respectively lie in between.
Comparing the (1×1×1) and (3×3×1) curves of the same structure, the differences
in the (relative) Δ𝜇Tb(𝑇 ) (2 meV at 0 K, 4 meV at 270 K) are smaller than the respective
differences in the absolute free energies 𝐹vib/𝑁Tb (5 meV to 10 meV for all 𝑇 ). This in-
dicates an error cancellation which makes the relative free energies converge faster
than the absolute ones. In particular, the error in Δ𝜇Tb(𝑇 ) is near the overall DFT
error of this work (1 meV per atom).
4The mapping between the ThSi2-type and the conventional AlB2 unit cell holds (cf. Fig. 4.1 (b)):
𝑎ThSi2 = 𝑏AlB2 ([11.0]-direction), 𝑏 2ThSi2 = 𝑐AlB2 , (𝑐ThSi2/4) + (𝑎ThSi2/2)2 = 𝑎2AlB ([10.0]-direction)2
51
4 The rare-earth silicide bulk phases
100 hex-AlB2 ort-AlB tet-ThSi(a) hex-AlB 2 22 (1×1×1) (3×3×1) static
50 (1×1×1)
(2×2×1) 20
0 (3×3×1) (d)10
100 (b) ort-AlB2 0
50 (1×1×1)
(2×2×1)
0 (3×3×1) -10
100 (c) tet-ThSi -202
50 (1×1×1)
(2×2×1) -30
0 (3×3×1)
-40
0 50 100 150 200 250 0 50 100 150 200 250
T (K) T (K)
Figure 4.5: Vibrational free energies of (a) hex-AlB2-TbSi2, (b) ort-AlB2-TbSi2 and (c) tet-ThSi2-
TbSi2 and (d) the corrected Tb chemical potentials 𝜇Tb(𝑇 ) calculated in different supercells.
The line types correspond to the supercells and the colours to the structure. Dash-dotted
lines: 𝜇Tb without vibrational contributions (static).
Concentrating on the (3× 3× 1) results, the two AlB2 structures approach each other
by a bit (Δ𝜇Tb = +18 meV (static), +14 meV at 0 K and +10 meV at 270 K). Also ort-AlB2-
TbSi2 and tet-ThSi2-TbSi2 approach each other, though to a lesser extent (Δ𝜇T−ATb =
−32 meV (static), −31 meV at 0 K and −29 meV at 270 K). The shifts in 𝜇Tb do not
change the relative stabilities between the structures for 𝑇 < 300 K. It is remark-
able that the two AlB2 structures differ more from each other than ort-ThSi2-TbSi2
and tet-ThSi2-TbSi2 do, with respect to the vibrational free energies. This can be ex-
plained by the Si–Si bond distances and angles within the Si sublattices: While hex-
AlB2-TbSi2 provides distances of 2.37 Å, ort-AlB2-TbSi2 and tet-ThSi2-TbSi2 provide
each two shorter (2.38 Å) and one longer distance (2.42 Å and 2.41 Å, respectively).
Therefore, the local Si geometry seems to be more important for the lattice dynamics
than the global symmetry.
In conclusion, although the two TbSi2 prototypes look rather different, the vibra-
tional free energies in the most stable structures (ort-AlB2-TbSi2 and tet-ThSi2-TbSi2)
are almost equal. Thus, the free energies can safely be approximated by the DFT total
energies up to 300 K. These results are certainly transferable to all other structurally
related RESi2. Furthermore, they can be transferred to the nanostructures as well
(films and nanowires), provided that the local Si–Si geometries are comparable.
4.2.5 The vacancy-populated RESi2 – x phases
Although the above theoretical results for stoichiometric RESi2 clearly contradict the
experimental observations, they reproduce the previous theoretical findings for stoi-
chiometric hex-AlB2-YSi2 by Magaud et al. [96]. As already discussed above (pp. 39f),
52
Fvib/NRE  (meV)
μ  − μ ort-AlBTb 2Tb   (meV)
4.2 Structure optimisation
that work proved that the high amount of Si vacancies, as it occurs in the real AlB2-
YSi2 – x structures, is decisive for the agreement between the theoretical and the ex-
perimental lattice constants, in particular concerning the 𝑐/𝑎 ratios. The influence of
the vacancies on the AlB2 and the ThSi2 prototypes will be demonstrated for TbSi2 – b
and ErSi2 – b (one out of six Si sites is vacant). The vacancies are considered maxim-
ally ordered, so the conventional unit cells of the AlB2 and the ThSi2 phase have to
be tripled to map the exact stoichiometry. Only those configurations are considered
in which the vacancies are separated by at least two Si atoms from each other. Fur-
thermore, the experimental observation is included that the orthorhombic distor-
tion clearly affects the ThSi2 phase, while the AlB2 phase is at least approximately
hexagonal. This confines the planar vacancy distributions to configurations 1 and 2
of ThSi2-RESi2 – b and configuration 1 of AlB2-RESi2 – b (Fig. 4.2). The vacancy ordering
perpendicular to the plane of projection of Fig. 4.2 (ThSi2: the (010) plane, the AlB2:
(0001) plane) is congruent, i.e. the vacancies stack onto each other in that direction.
All other orderings would multiply the supercells in that direction, which would ex-
ceed the scope of this work. The energetic effects from out-of-plane ordering can be
estimated at∼30 meV per vacancy (in hex-AlB2-YSi2 – b [96]), which corresponds to 𝜇RE
variations of ∼10 meV.
The three RESi2 – b supercells to investigate comprise two (3 × 1 × 1)-ThSi2 supercells
describing Th√Si2-RE√Si2 – b (1) and ThSi2-RESi2 – b (2) (green cell in Fig. 4.2 (a))
5 and a
rotated R30◦( 3 × 3 × 1)-AlB2 supercell describing AlB2-RESi2 – b (light green cell
in Fig. 4.2 (b)). The structures of all supercells are optimised by code-level relaxa-
tions, which are not as accurate as Murnaghan relaxations, but whose uncertainty
is expected to be smaller than those introduced by other approximations, e.g. the
out-of-plane ordering of the vacancies. Pulay stress is reduced by iterating the relax-
ations until the supercells remain unchanged. In order to estimate the reliability of
the relaxation method, stoichiometric ort-AlB2-RESi2 and tet-ThSi2-RESi2 are optim-
ised at code level as well and compared to the respective Murnaghan results. The
deviations are very small. Only tet-ThSi2-TbSi2 shows a slightly different axial lattice
constant 𝑐 (∼1 %). As the Murnaghan structure is more stable by 5 meV per Tb atom,
the code-level relaxation is indeed less reliable. The confidence intervals can thus be
estimated to 1 % for the lattice constants and 5 meV for the RE chemical potentials
𝜇RE. The k-point meshes are adapted to the supercells. The first set of calculations
is conducted with the PBE xc-functional [10], a second set with PBEsol [11]. In addi-
tion to the RESi2 – x structures, the bulk phases of diamond-Si and elemental hcp-RE
are optimised as well so as to calculate 𝜇RE according to Eq. (4.36). The results for
TbSi2 – x are compiled in Tab. 4.6. Those for ErSi2 – x are in the appendix (Tab. B.1).
Fig. 4.6 shows images of the optimised TbSi2 – x structures.
Concentrating on TbSi2 – b, several predictions about the effects from the vacancies
are confirmed. The removal of a Si atom drives the three atoms at the corner of the re-
spective M-star towards its centre (light blue arrows in Fig. 4.6). As a consequence, the
basal lattice constant 𝑎 of the AlB2 phase contracts by −9.7 %, which, in turn, pushes
5Mind that for internal consistency, the vacancies populate the Si zigzag chains parallel to 𝑎, so the
expected orthorhombic distortion would result in 𝑎/𝑏 < 1. In contrast, most of the crystallographic
literature uses an interchanged notation for 𝑎 and 𝑏, so 𝑏/𝑎 < 1.
53
4 The rare-earth silicide bulk phases
(a) ThSi2-RESi2−b (1) (b) ThSi2-RESi2−b (2) (c) AlB2-RESi2−b (1)
d γva
γv da da
c d db c b √3a
dc dc γv
3a 3a
Figure 4.6: Structure models of vacancy-populated TbSi2 – b. Orange circles mark vacancies.
Dark red lines mark the supercells. Red dumb-bells mark Si–Si bond distances; Red angles
mark the bond angles 𝛾v near the vacancies. Light blue arrows indicate Si atoms moving
towards the vacancies. All 3D images in this work are rendered with XCrySDen [120].
the (0001)-planes apart and increases the axial lattice constant 𝑐 by +5.0 %. The sym-
metry remains hexagonal, contrarily to stoichiometric AlB2-TbSi2. The impact on the
ThSi2 prototype deviates from the predictions. The basal lattice constant 𝑎 does not
contract, but remains almost unchanged. Instead, the perpendicular basal lattice
constant 𝑏 grows by +1.3 % and +2.1 % for ThSi2-TbSi2 – b (1) and ThSi2-TbSi2 – b (2),
respectively. Consequently, the tetragonal symmetry breaks into an orthorhombic
distortion of 𝑎/𝑏 = 0.994 and 0.979, respectively. The axial lattice constant 𝑐 strongly
contracts by −9.4 % and −9.0 %, respectively.
The Tb chemical potentials 𝜇Tb (Eq. (4.36)) reveal that the Si vacancies clearly stabilise
the RESi2 prototypes (Tab. 4.6). In order to facilitate the stability considerations, the
𝜇Tb are reset by that of hex-AlB2-TbSi2 – b, the most stable TbSi2 – x structure.
Δ𝜇• ≔ 𝜇• − 𝜇hex-AlB2-TbSi2 – bTb Tb Tb (4.40)
Two further chemical potential differences are defined:
• Δ𝜇T−A = 𝜇ThSi2-TbSi2 – x AlB2-TbSi2 – xTb Tb −𝜇Tb : the difference between the most stable ThSi2
and AlB2 structures with the same stoichiometry (Eq. (4.37))
• Δ𝜇vac ≔ 𝜇•-TbSi2 – bTb Tb − 𝜇
•-TbSi2
Tb : the difference between TbSi2 – b and TbSi2 in the
same phase •, i.e. the energy gain per Tb atom upon introducing the vacancies.
The vacancies stabilise the AlB2 phase by Δ𝜇vacTb = −191 meV and the ThSi2 phase to a
lesser extent (−128 meV for ThSi2-TbSi2 – b (1)). As a consequence, the AlB2 phase be-
comes more stable than the ThSi2 phase (Δ𝜇T−ATb = +35 meV for TbSi2 – b and −28 meV
for TbSi2). However, the exact values have to be interpreted with caution because
they seem to strongly depend on the vacancy ordering, as indicated by the potential
difference of 42 meV between the two ort-ThSi2-TbSi2 – b configurations.
54
4.2 Structure optimisation
𝑎 𝑏 𝑐 𝑐/𝑎 𝑉/𝑁Tb 𝜇Tb Δ𝜇Tb
(Å) (Å) (Å) (Å3) (eV) (meV)
PBE
ort-AlB2-TbSi2 4.23 3.88 4.00 0.946 58.3 −1.636 191
hex-AlB2-TbSi2 – b 3.82 4.20 1.100 53.0 −1.827 0
tet-ThSi2-TbSi2 3.97 3.97 14.82 3.734 58.3 −1.664 163
ort-ThSi2-TbSi2 – b (1) 4.00 4.02 13.43 3.340 54.0 −1.792 35
ort-ThSi2-TbSi2 – b (2) 3.96 4.05 13.48 3.328 54.1 −1.750 77
PBEsol
ort-AlB2-TbSi2 4.17 3.88 3.92 0.940 56.2 −1.886 173
hex-AlB2-TbSi2 – b 3.76 4.16 1.107 51.0 −2.058 0
tet-ThSi2-TbSi2 3.93 3.93 14.57 3.709 56.2 −1.918 141
ort-ThSi2-TbSi2 – b (1) 3.93 3.98 13.27 3.335 51.9 −2.009 49
ort-ThSi2-TbSi2 – b (2) 3.90 4.01 13.35 3.333 52.1 −1.964 95
Exp. reference
hex-AlB2-TbSi2 – x 3.84 4.14 1.078 52.9
ort-ThSi2-TbSi2 – x 3.96 4.05 13.38 3.304 53.6
Table 4.6: Lattice parameters and relative Tb chemical potentials of TbSi2 – x in the stoi-
chiometric and the vacancy-populated AlB2 and ThSi2 phases. The unit cells were optimised
by code-level relaxations with different xc-functionals. Experimental reference (mind nota-
tion of 𝑎 and 𝑏 for ort-ThSi2): [83, 85]. ErSi2 – x: Tab. B.1.
The PBEsol results are similar to the PBE results. The most important difference
consist in the contraction of all lattice constants by −0.9 % to −1.9 %. Accordingly,
the unit-cell volumes contract by −3.5 % to −3.8 %. The Tb chemical potentials 𝜇Tb
shift relatively to each other by ∼20 meV, which does not change the stability or-
der, though. Notably, the stability of AlB2-TbSi2 – b relative to ThSi2-TbSi2 – b increases
(Δ𝜇T−A = +49 meV). The potential drop due to the vacancies is smaller (Δ𝜇vacTb Tb =
−173 meV and −92 meV for hex-AlB2-TbSi2 – b and ort-ThSi2-TbSi2 – b (1), respectively).
The absolute positions of 𝜇Tb are considerably lower by >200 meV for all structures.
The smaller lattice constants and the higher reaction enthalpies (𝜇Tb, vide supra)
within PBEsol are expectable [10, 11].
The ErSi2 – x results (Tab. B.1) are similar to the TbSi2 – x results. The unit-cell volumes
are smaller (RESi2 – b: ≈−4 %, RESi2: ≈−2.5 %) due to the lanthanoid contraction. The
lattice distortions upon the Si removal are more pronounced, in particular concern-
ing the axial lattice constant 𝑐 of the ThSi2 phase: While it is longer in tet-ThSi2-ErSi2
(15.0 Å) than in tet-ThSi2-TbSi2 (14.8 Å), it contracts so strongly due to the vacancies
that it is shorter in ort-ThSi2-ErSi2 – b (13.3 Å) than in ort-ThSi2-TbSi2 – b (13.4 Å). In
accordance with the higher structural impact, the potential drop due to the vacan-
cies is steeper in ErSi2 than in TbSi2 (Δ𝜇vacRE = −266 meV and −208 meV for hex-AlB2-
ErSi2 – b and ort-ThSi2-ErSi2 – b (1), respectively). As in TbSi2 – x, the relative stability
between the AlB2 and the ThSi2 phase changes sign upon introducing the vacancies
(Δ𝜇T−AEr = +43 meV for ErSi2 – b and −13 meV for ErSi2).
55
4 The rare-earth silicide bulk phases
The lattices of the vacancy-populated TbSi2 – b and ErSi2 – b structures compare very
well with the experimental reference [83, 85], much better than the respective stoi-
chiometric RESi2 prototypes. In particular, the symmetry groups match the experi-
mental observations, contrarily to the stoichiometric structures. Concerning the AlB2
phase, the 𝑐/𝑎-ratios correctly adopt values greater than 1. At first glance, the PBE
lattice parameters are closer to the experimental reference than the those obtained
with PBEsol, in particular concerning 𝑎 and 𝑉/𝑁RE. On the other hand, PBEsol de-
scribes 𝑐 better. It has to be kept in mind, though, that the RE silicide phases are ap-
proximated by ordered and frozen vacancies. Existing theoretical work hints that the
vacancies may show considerable diffusion along the Si zigzag chains at finite tem-
peratures [99]. Therefore, more sophisticated calculations including disorder and
diffusion may affect 𝑎 more than 𝑐. Concerning the ThSi2 phase, the lattice para-
meters of ort-ThSi2-TbSi2 – b reproduce the experimental reference up to 1 % within
PBE and are too small by up to −2 % within PBEsol. As in the AlB2 phase, disorder
and diffusion of the vacancies are likely to influence the lattice parameters. Further-
more, real ThSi2-TbSi2 – x contains more Si than TbSi2 – b does, so it is expectable that
the lattice constants of more realistic, Si-richer structures will be larger. In summary,
it is not possible to conclude on the suitability of the xc-functionals at this point. Fi-
nally, the volumes per atom of all investigated RESi2 – x structures are smaller than
the weighted atomic volumes of the constituents in the elemental phases. The con-
traction ratio ranges from 13 % to 22 %, independently of the xc-functional, and hints
at the partial heteropolar character of the compound [80].
Stability analysis
Both ErSi2 – x and TbSi2 – x favour the AlB2 phase in the RESi2 – b stoichiometry (𝑥 = 13 )
and the ThSi2 phase in the RESi2 stoichiometry (𝑥 = 0). In order to further analyse
this, the relative RE chemical potentials Δ𝜇•RE (Eq. (4.40)) for each RE and phase • are
linearly interpolated with respect to 𝑥 and plotted in Fig. 4.7. At the 𝑥 = 13 bound-
ary, Δ𝜇ThSi2RE (dashed line) is higher than Δ𝜇
AlB2
RE (solid line). The difference (Δ𝜇
T−A
RE ) is
similar for ErSi2 – b (green, PBE: +43 meV, PBEsol: +56 meV) and TbSi2 – b (yellow, PBE:
+35 meV, PBEsol: +49 meV). In other words, the relative stability of the AlB2 phase
with Si vacancies is independent of the RE radius. That RESi2 – b certainly adopts the
AlB2 phase agrees with the experimental observations and phase diagrams [89, 91,
93, 110, 111]. At the other boundary (𝑥 = 0), vacancy-free ErSi2 and TbSi2 both favour
the (tetragonal) ThSi2 phase over the (orthorhombic) AlB2 phase. The potential lines
intersect, as marked by black dotted lines. Because Δ𝜇T−ARE is, according to amount,
smaller in ErSi2 (PBE: −15 meV, PBEsol: −12 meV) than in TbSi2 (PBE: −28 meV, PBE-
sol: −32 meV), the intersection is closer to the Si-rich boundary for ErSi2 – x than for
TbSi2 – x. This permits the conclusion that ErSi2 – x needs a higher Si content to form
the ThSi2 phase than TbSi2 – x does. Furthermore, particularly within PBEsol, the sta-
bility range of ThSi2-ErSi2 – x so narrow that it is possible that more sophisticated cal-
culations shift the intersection beyond 𝑥 = 0, which would exclude the formation of
ThSi2-ErSi2 – x at all. Of course, the linear interpolation of Δ𝜇RE is a crude approxim-
ation as the real curves are convex6. Nevertheless, if similar shapes are assumed,
6It is expectable that 𝐺RE, defined by Eq. (4.34), has a minimum at a certain 𝑥 for a fixed phase.
56
4.2 Structure optimisation
300
(a)  PBE (b)  PBEsol
250 AlB2-ErSi2−x AlB2-ErSi2−x
ThSi2-ErSi2−x ThSi2-ErSi2−x
200 AlB2-TbSi2−x AlB2-TbSi2−x
ThSi2-TbSi2−x ThSi2-TbSi2−x
150
100
50
0
0.30 0.25 0.20 0.15 0.10 0.05 0 0.30 0.25 0.20 0.15 0.10 0.05 0
x x
Figure 4.7: Linear interpolation of Δ𝜇RE for TbSi2 – x (yellow) and ErSi2 – x (green) between 𝑥 =
1
3 and 𝑥 = 0. (a) PBE and (b) PBEsol. The line types correspond to the phases. Black dotted
lines mark the intersections (𝜇T−ARE = 0).
the real curves for the AlB2 and the ThSi2 phase will still be horizontally displaced
from each other and this displacement will be higher for ErSi2 – x than for TbSi2 – x,
reflecting the different 𝜇T−ARE at the boundaries.
As a last test concerning the vacancies, a further Si atom i√s rem√oved from the most
stable hex-AlB2-RESi2 – b structure. For this pur√pose, t√he ( 3 × 3 × 1) supercell is
doubled in both basal directions to obtain a (2 3 × 2 3 × 1) supercell, in which an
additional vacancy is created. The optimised RE12Si19 structures (RESi1.58) turn out to
be highly unstable against hex-AlB2-RESi2 – b (PBE: Δ𝜇Tb = +97 meV, Δ𝜇Er = +98 meV;
PBEsol: Δ𝜇Tb = +135 meV, Δ𝜇Er = +134 meV). Consequently, the optimal composition
under Si-rich conditions, at least for the AlB2 phase, indeed seems to be near the
RESi2 – b stoichiometry, in agreement with the phase diagrams.
Silicon bond length analysis
To complete the study of the RESi2 – x structures, the binding mechanisms are illu-
minated by analysing the geometry of the Si sublattices. Since the structures differ in
the arrangement of the Si–Si nearest neighbours, it is instructive to classify the Si–Si
bonds with respect to their environment by the following notation:
• Each Si atom is either twofold coordinated and adjacent to a vacancy, denoted
(v)-type, or threefold coordinated, denoted (m)-type.
• The nearest-neighbours pairs of Si atoms, the Si–Si dumb-bells, are classified
with respect to the types of the constituents, e.g. (mv)-type if the one Si atom is
(m)-type and the other (v)-type.
In hex-AlB2-RESi2 – b (Fig. 4.6 (c)), each (v)-type Si atom is coordinated by (m)-type Si
atoms and vice versa. Thus, all Si–Si dumb-bells are all equivalent and of (mv)-type
57
ΔμRE (meV)
4 The rare-earth silicide bulk phases
(length denoted 𝑑a). In ort-ThSi2-RESi2 – b (Fig. 4.6 (a) and (b)), the Si–Si dumb-bells
point into three different directions and are unequal. Their lengths are termed after
the lattice constant towards which they incline: 𝑑a, 𝑑b and 𝑑c. The stoichiometric
hex-AlB2-TbSi2, ort-AlB2-TbSi2 and tet-ThSi2-TbSi2 structures provide only (mm)-type
Si–Si dumb-bells, which have two different lengths for the latter two. The Si–Si bond
lengths of all PBE TbSi2 – x structures are compiled in Tab. 4.7. Those optimised with
PBEsol and those of the ErSi2 – x structures may be found in the appendix (Tab. B.2
and Tab. B.3). The diamond-Si structure serves as reference for the length of Si–Si
single bonds (PBE: 2.37 Å, PBEsol: 2.35 Å, tetrahedron angle: 109.5◦).
In stoichiometric hex-AlB2-TbSi2, the Si–Si bond length amounts to 𝑑a = 2.37 Å, ex-
actly the PBE bond length in diamond-Si. This clearly indicates the presence of single
bonds. The other two stoichiometric TbSi2 structures (ort-AlB2 and tet-ThSi2) have
(per formula unit) two Si–Si dumb-bells with lengths near the single bond (2.36 Å),
while the third dumb-bell is considerably longer (2.42 Å and 2.43 Å). In terms of the
zigzag planes of the Si sublattice (Fig. 4.1), the shorter bond length is that of the Si
zigzag chains (in-plane), while the larger is the connection between the planes. This
structural similarity underlines the above results that ort-AlB2-TbSi2 and tet-ThSi2-
TbSi2 merely differ in the way how the planes stack (unidirectional vs. alternately
perpendicular), but not in the local geometries. The explanation for the unequal bond
lengths may lie in the charge transfer from the RE ions to the Si sublattice: When the
zigzag planes move apart, the zigzag chains can accept more charge.
The bond lengths and angles in the vacancy-populated TbSi2 – b structures accord with
the charge transfer model as well. In hex-AlB2-TbSi2 – b, the (v)-type Si – II atom form-
ally has two binding and two free electron pairs in tetragonal coordination. However,
the tetrahedron is distorted because the two free pairs are localised closer to the Si
atom and push the two binding pairs away. This is reflected by the bond length of
𝑑a = 2.43 Å, which is longer than a single bond, and the bond angle of 𝛾v = 103.8◦,
which is smaller than the tetrahedron angle. On the other hand, the (m)-type Si – I
atom has three equivalent binding electron pairs in symmetric trigonal-planar geo-
metry, while the formal free electron pair occupies the out-of-plane p𝑧 orbital. Cor-
respondingly, the respective M-star has inner angles of exactly 120◦. The same mech-
anisms apply to the two ort-ThSi2-TbSi2 – b structures as well. All (mm)-type Si dumb-
PBE 𝑑a(Å) 𝑑b(Å) 𝑑c(Å)
hex-AlB2-TbSi2 2.37 (mm)
ort-AlB2-TbSi2 2.36/2.42 (mm)
hex-AlB2-TbSi2 – b 2.43 (mv)
tet-ThSi2-TbSi2 2.36 (mm) 2.36 (mm) 2.43 (mm)
ort-ThSi2-TbSi2 – b (1) 2.49 (vv) 2.44 (mv) 2.36 (mm) 2.39 (mv)
ort-ThSi2-TbSi2 – b (2) 2.50 (vv) 2.47 (vv) 2.37 (mm) 2.37 (mv)
Table 4.7: Si–Si nearest-neighbour distances in TbSi2 – x optimised with PBE. The notation ac-
cords with Fig. 4.6. Letters in parentheses indicate the types of the Si–Si dumb-bell. PBEsol:
Tab. B.2; ErSi2 – x: Tab. B.3.
58
4.2 Structure optimisation
bells have lengths near the single bond (2.36 Å to 2.37 Å). All (mv)-type Si dumb-bells
have larger bond lengths of 2.37 Å to 2.44 Å. If the Si dumb-bell connects two (v)-type
Si atoms, the bond length is even larger (2.47 Å to 2.50 Å), as expected from the double
repulsion from the free electron pairs at each end. The average bond angles at the
(v)-type Si atoms are clearly below the tetrahedron angle (𝛾 ◦ ◦v = 106.6 and 105.3 for
ort-ThSi2-TbSi2 – b (1) and ort-ThSi2-TbSi2 – b (2), respectively).
The Si–Si bond lengths of the ErSi2 – x structures (Tab. B.3) are quantitatively very
similar to those of TbSi2 – x, deviating by less than 1 %. This is surprising as the lattice
constants show larger differences of 1 % to 2 %. For instance, 𝑑a is exactly equal in
both hex-AlB2-TbSi2 – b and hex-AlB2-ErSi2 – b (2.43 Å). On the other hand, the basal lat-
tice constant 𝑎 of hex-AlB2-ErSi2 – b is smaller than that of hex-AlB2-TbSi2 – b by −1.4 %
(Tab. 4.6 and Tab. B.1). The lanthanoid contraction is thus not carried out by the Si–
Si bond distances, but rather by the bond angles at the vacancies, which are more
acute in hex-AlB2-ErSi2 – b (𝛾v = 101.4◦ vs. 𝛾v = 103.8◦). Similar considerations hold
for the ThSi2-ErSi2 – b structures. The analysis of the PBEsol structures (Tab. B.2 and
Tab. B.3) yields very similar results. All bond lengths are a bit shorter, as well as the
bond distance in diamond-Si (2.35 Å, experimental reference: 2.352 Å [17, p. 1065]).
The bond angles at the vacancies 𝛾v are more acute in PBEsol than in PBE.
The effects from the vacancies can be linked to the M-stars, which both structure
prototypes have in common. If vacancies are introduced into the AlB2 structure, the
remaining complete M-stars rotate and move closer together whereupon the basis
contracts. As a consequence, the RE atoms are squeezed out of the Si honeycombs,
which dilates the axis. Both deformations combined invert the 𝑐/𝑎 ratio. The ThSi2
structure behaves differently as its axis locally resembles the basal [1̄1.0] direction
of the AlB2 structure, while its basis is a mixture between the basal [11.0] and the
axial [00.1] direction of the AlB2 structure. Therefore, the basis of the ThSi2 structure
responds only weakly to the vacancies, while the axis considerably contracts.
4.2.6 The CaSi2 phases
The ZKB concept (p. 33) was introduced by means of the exemplary alkali and al-
kaline earth metal silicides (MmSin). In the particular case of CaIISi – I2 , the Si atoms
formally accept an electron and arrange themselves in buckled honeycombs between
which the Ca ions are embedded. Such a geometry reminds of the AlB2 phase and
raises the question whether the stoichiometric RE silicides might crystallise in one of
the CaSi2 phases.
As CaSi2 is polymorphic, several structures are discussed for this compound. Three
of them comprise alternating hexagonal Ca layers and buckled Si honeycombs. The
first structure is the simplest way of layer stacking as the Ca atoms arrange them-
selves in the “holes” of the buckled Si honeycombs. Denoted h1-CaSi2, this structure is
obviously analogous to the hex-AlB2-RESi2 structure, except that the Si honeycomb is
buckled in the former and flat in the latter. CaSi2 does not occur in the h1-CaSi2 struc-
ture, but CaGe2 does [121]. The second structure results from the following stacking:
The Ca atoms are placed above the lower Si atoms of the buckled honeycomb (T4
59
4 The rare-earth silicide bulk phases
(a) ort-AlB2-RESi2 (b) tr3-CaSi2-RESi2 (c) tr6-CaSi2-RESi2
c
c
c
a aa
Figure 4.8: Models of TbSi2 in the (a) ort-AlB2, (b) tr3-CaSi2 and (c) tr6-CaSi2 structures. Blue
circles are Si atoms; Yellow circles are RE atoms. Red lines mark the unit cells.
position), like eggs in a (hexagonal) egg box. The next Si honeycomb covers the Ca
layer so that the upper Si atoms are above the Ca atoms, like a turned egg box cover-
ing a filled egg box. Denoted tr3-CaSi2 with respect to the (ABC)-type sequence of the
Ca layers, this structure is an artificial phase of CaSi2 [122]. The third structure is a
mixture between h1-CaSi2 and tr3-CaSi2:
1) Ca atoms above the lower Si atoms of the buckled honeycomb (egg box).
2) Holes of the next Si honeycomb above the Ca atoms.
3) Ca atoms above the holes of the Si honeycomb.
4) Upper Si atoms of the next honeycomb above the Ca atoms (turned egg box).
Denoted tr6-CaSi2 with respect to the (AABBCC)-type sequence of the Ca layers, this
structure is one of the natural phases of CaSi2 [121].
To answer the above question, the h1-CaSi2, tr3-CaSi2 and tr6-CaSi2 structures are set
up with RE (Tb and Er) replacing Ca. The unit cells are optimised at code level with
the same parameters as above (Sec. 4.2.5), for PBE and PBEsol. The h1-CaSi2 structure
proved to transition into the ort-AlB2-RESi2 structure, which is chosen as reference.
Fig. 4.8 shows the optimised structure models. The PBE results for TbSi2 (lattice para-
meters and relative RE chemical potentials Δ𝜇RE) are compiled in Tab. 4.8, all other
results are in the appendix (Tab. B.4). For both RESi2, at least one of the two CaSi2
structures is more stable than ort-AlB2-RESi2 and even tet-ThSi2-RESi2. This is a very
surprising result, particularly because the differences in the RE chemical potentials
are considerable. They depend, however, strongly on the xc-functional: Switching
from PBE to PBEsol affects Δ𝜇RE of tr3-CaSi2-RESi2 to a large extent (+60 meV and
+77 meV for TbSi2 and ErSi2, respectively), and Δ𝜇RE of ort-AlB2-RESi2 (−18 meV and
−8 meV) and tr6-CaSi2-RESi2 (+15 meV and +22 meV) to an intermediate extent.
The basal lattice constants of TbSi2 and ErSi2 are very similar in the respective struc-
tures (deviations below 1 %). The axial lattice constant 𝑐 contracts from TbSi2 to ErSi2
by−1.5 % to−2.3 % (lanthanoid contraction). A similar result has already been found
for the hex-AlB2 structures (Sec. 4.2.2). In accordance with the bases, the Si–Si bond
lengths and angles of the respective TbSi2 and ErSi2 structures are very similar as
60
4.2 Structure optimisation
𝑎 𝑐 𝑉/𝑁RE 𝛾 𝑑a Δ𝜇RE
(Å) (Å) (Å3) (Å) (meV)
TbSi2, PBE
ort-AlB2 4.23 (3.88)𝑎 4.00 58.3 124.7◦ (110.6◦)𝑎 2.36 (2.42)𝑎 0
tr3-CaSi2 3.78 5.56 63.4 103.6◦ 2.41 −30
tr6-CaSi 3.94 9.34 60.9 104.9◦2 , 118.9◦ 2.49, 2.29 −67
Table 4.8: Lattice parameters and relative Tb chemical potentials of stoichiometric TbSi2
in the AlB2 and CaSi2 phases. The unit cells were optimised by code-level relaxations
with PBE. tr6-CaSi2-RESi2 has two independent M-star angles 𝛾 and Si–Si bond lengths 𝑑a.
𝑎orthorhombic distortion. Other results: Tab. B.4
well. Since determined by buckled Si honeycombs, the basis of tr3-CaSi2-RESi2 is
smaller than that of ort-AlB2-RESi2. On the other hand, the axial lattice constant 𝑐
is longer. In summa, the total volume per RE atom 𝑉/𝑁RE of tr3-CaSi2-RESi2 is larger
than that of ort-AlB2-RESi2 by ≈+9 %. Following the ZKB concept, the Si atoms of the
buckled honeycombs are sp3-hybridised and use three orbitals for inter-se bonding,
while the fourth accepts a free electron pair. This is reflected by the bond lengths
of 2.38 Å to 2.41 Å, which are longer than that of diamond-Si (within the respect-
ive xc-functional), and the bond angles, which are more acute than the tetrahedron
angle. The tr6-structure shows unexpected peculiarities: On the one hand, the upper
Si honeycomb of the unit cell (the tr3-like part) has very long bond lengths of 2.46 Å to
2.49 Å, which is near the length of a (vv)-type Si dumb-bell (vide supra). However, the
charge accumulation in a (vv)-type dumb-bell (formal Si2 – ion at each end) is much
larger than that in the bonds of the honeycomb (formal Si – ion at each end). On the
other hand, the lower Si honeycomb (the AlB2-like part) is almost flat (𝛾 ≈ 119◦) and
the bond lengths (2.26 Å to 2.28 Å) are considerably shorter than that in diamond-Si.
Since the AlB2-like part seems to be squeezed and the tr3-like part to be stretched, it
is even more surprising that tr6-CaSi2 is the most stable structure for TbSi2.
In conclusion, the AlB2 and ThSi2 prototypes of stoichiometric RESi2 are unstable
against at least one of the CaSi2 phases. This underlines that RESi2 without vacan-
cies are far away from the real structures. Furthermore, the stability of the buckled-
honeycomb geometries confirms that Si tends to avoid sp2 hybridisation, as already
discussed in Sec. 3.2. Since they seem to be concurring phases for Si-rich RESi2 – x,
the CaSi2 structures should be taken into account when RE–Si phase diagrams are
calculated by means of DFT. From the experimental phase diagrams (pp. 38ff), it is
expectable that the vacancies stabilise the ThSi2 and the AlB2 phase more than the
CaSi2 phases, so the latter do not occur. That the relative stabilities between the four
structures (tet-ThSi2, ort-AlB2, tr-3-CaSi2 and tr6-CaSi2) seem to strongly depend on
the xc-functional should be verified by Murnaghan relaxations. Moreover, the CaSi2
phases will be relevant for the monolayer-Tb@Si(111) system (Chap. 5) as they share
several structural properties with the latter.
61
4 The rare-earth silicide bulk phases
4.3 Electronic properties
The band structure 𝜀𝑛(𝒌) and the DOS are calculated non-self-consistently, i.e. the ei-
genvalues are calculated for a fixed ground-state density, which was self-consistently
determined on a (16 × 16 × 16) k-point mesh. The k-paths for the band structures
comprise densely sampled line segments between high-symmetry points of the Bril-
louin zones (according to [123, A.5. and A.10.]). The DOS is calculated on a dense,
Γ-centred (32 × 32 × 32) mesh, which produces very smooth curves while the non-
self-consistency considerably saves computation time. All other parameters are ana-
logous to those in Sec. 4.2
The band structures 𝜀𝑛(𝒌) become rather complicated even for the smallest RESi2
cells. Therefore, they are disentangled by projecting the wavefunctions |Ψ𝑛(𝒌)⟩ onto
the spherical harmonics |𝑌 𝑙𝑚,𝐼⟩ centred at ion 𝐼 , where 𝑙 and 𝑚 are the orbital and
magnetic quantum numbers. The square modulus of the projection is then a measure
for how strong a state is localised in the r〈espective ato〉m ic orbitals.
𝑃𝑙𝑚,𝐼 ( 2𝑛, 𝒌) =  𝑌 𝑙𝑚,𝐼 Ψ (𝒌) 𝑛 (4.41)
The projections are very convenient in the PAW method as they equal the PAW pro-
jections. Once determined, they define the following scheme for plotting the band
structures: Each band 𝜀𝑛(𝒌) is a function of 𝒌, which is plotted against the x-axis. The
line width is scaled locally in 𝒌 by the sum of the projections 𝑃𝑙𝑚,𝐼 (𝑛, 𝒌) of a certain
group {𝑙, 𝑚, 𝐼}. The portion of a subgroup in this group determines the line colour
according to a colour bar. The Brillouin-zone integration for obtaining the DOS can
be weighted by the p∑︁rojection∫s as well. This results in the partial DOS (PDOS).𝑁Band 1
DOS(𝐸) =
1∑︁|C |
d𝑘 𝛿(𝐸∫− 𝜀𝑛(𝒌)) (4.42a)𝑛= G 𝑁∑︁CGBand
PDOS(𝐸) 1= |C | d𝑘 𝛿(𝐸 − 𝜀𝑛(𝒌)) 𝑃
𝑙𝑚,𝐼 (𝑛, 𝒌) (4.42b)
𝑙,𝑚,𝐼∈group 𝑛=1 G CG
where 𝐶G is the reciprocal unit cell (Sec. 2.2). The PDOS is plotted against the energy
axis (y-axis) next to the band structures.
Throughout this section, Si portions (PDOS and bands) are coloured blue, Tb por-
tions yellow and Er portions green. As expected, the band structures of TbSi2 – x and
ErSi2 – x turned out to be almost identical. Therefore, only TbSi2 – x band structures
are analysed, while ErSi2 band structures are compared with those of TbSi2 when
the f-valent structures are considered. Moreover, the analysis concentrates on the
AlB2 structures since their bands are clearer than those of the ThSi2 structures.
The band structures of the RESi2 prototypes
Fig. 4.9 shows the band structures of the stoichiometric RESi2 prototypes (hex-AlB2-
TbSi2 and tet-ThSi2-TbSi2), each plotted along a k-path connecting all high-symmetry
62
4.3 Electronic properties
(a) (i) total (ii) s-orbitals (iii) p-orbitals (iv) d-orbitals
0
-2
-4
-6
-8
-10
-12
Γ XY Σ ΓZ Y1Σ1 Z Γ XY Σ ΓZ Y1Σ1 Z Γ XY Σ ΓZ Y1Σ1 Z Γ XY Σ ΓZ Y1Σ1 Z
Tb Z Y1
Σ tet-ThSi2-TbSi
A
2 H
1 Γ L
Σ Y X Γ
Si M K
hex-AlB2-TbSi(b) 2
0
-2
-4
-6
-8
-10
-12
Γ MK Γ A L H A Γ MK Γ A L H A Γ MK Γ A L H A Γ MK Γ A L H A
Figure 4.9: Band structures of (a) tet-ThSi2-TbSi2 and (b) hex-AlB2-TbSi2. The line width cor-
responds to the PAW projections of (i) all orbitals, (ii) the s orbitals, (iii) the p orbitals and (iv)
the d orbitals. The line colour indicates the Tb/Si portions, according to the colour bar (yellow
→ Tb, blue→ Si). The Brillouin zones and k-paths are shown in the middle. The respective
PDOSs are plotted against the y-axis to right of the band structures in arbitrary units.
points of the primitive Brillouin zone. The DOSs and PDOSs of tet-ThSi2-TbSi2 are
scaled by 1/2 so that they match those of hex-AlB 72-TbSi2 by scale . Panel (i) shows the
summed s, p and d projections of all atoms (line width, constant). Panels (ii), (iii), and
(iv) show the separate s, p and d portions in the PAW projections of all atoms. The
line colour indicates Tb/Si portions. On the right hand of each band structure, the
PDOS of the respective orbitals is plotted against the y-axis, where yellow and blue
portions correspond to Tb and Si, respectively (stack diagram). Thus, the sum of the
yellow and blue portions in panel (i) equals the total DOS.
The band structures of tet-ThSi2-TbSi2 and hex-AlB2-TbSi2 show strongly dispersing
bands. They look rather different from each other at first glance, but have several
common properties. The bands belonging to Si have s character at high binding ener-
gies8, gain p character at lower binding energies and are mostly p-like near the Fermi
level. Above−5 eV, strongly dispersive d bands associated with Tb arise, which finally
dominate the region between −3 eV and the Fermi level (and the empty conduction
bands above). The d contributions from Si are zero (since the Si PAW potential lacks
7The body-centred ThSi2 cell has two formula units RESi2. All DOSs and PDOSs have arbitrary units.
8The binding energy is exactly the opposite of the energy scale in Fig. 4.9.
63
E − EF  (eV) E − EF  (eV)
4 The rare-earth silicide bulk phases
d projectors) and the p and s contributions from Tb are small throughout the band
structure. The RESi2 prototypes furthermore have in common that the Tb-d states
mix with the Si-p states, so pure Tb states are hardly present. On the contrary, (al-
most) pure Si states exist in both structures.
The band structures agree with the findings of the structure analysis. The Si bands be-
low −3 eV are sp-like, do not interfere with the Tb states and amount to 3𝑁Tb bands
(6𝑁Tb electrons, more clearly visible in hex-AlB2-TbSi2). Hence, they represent the
Si–Si bonds and confirm that the bonding within the Si sublattice is single and elec-
tronically decoupled from the rest of the cell. The fourth electrons of the Si atoms
and the Tb valence electrons form rather complicated Si-Tb-hybrid orbitals, which
will be analysed in detail when the orthorhombic distortion is considered (vide in-
fra). Since these bands cut the Fermi level, the RESi2 prototypes are clearly metallic.
The band structure of hex-AlB2-TbSi2 has a peculiarity which is missing in that of tet-
ThSi2-TbSi2: Exactly at the Fermi level, the DOS has a rather sharp peak, which stems
from Si-sp and Tb-d states, as revealed by the PDOSs. This peak has already been
observed in hex-AlB2-YSi2 [95, 100]. Contrarily, the DOS of tet-ThSi2-TbSi2 is indented
right beneath the Fermi level.
Orthorhombic distortion
The band structures of hex-AlB2-TbSi2 and ort-AlB2-TbSi2 can be found in Fig. 4.10.
The k-path is restricted to the three spatial directions (ΓK ∥ e𝑥 , ΓM ∥ e𝑦 and ΓA ∥ e𝑧
(inset in Fig. 4.10 (b.iv)). The changes in the bands due to orthorhombic distortion
consist in a handful of features (highlighted by red markings) which agree with the
results from the structure analysis.
Both structures have in common that the lowest four bands (labelled 1 to 4) majorly
belong to the Si honeycomb. Band 1 is Si-s-like and decoupled from Tb. Bands 2 and
3 are each strongly dispersive in one of the basal directions and less dispersive in the
perpendicular directions. They are Si-s-like at lower energies and gain Si-p character
towards higher energies. Because having only weak Tb contributions, bands 1 to 3
represent exclusive Si–Si bonds (as already found above). The 𝑙𝑚-resolved projec-
tions (Fig. B.1, labels in Fig. 4.10) reveal that the p component of bands 2 and 3 points
into the direction of the more dispersive branch. In detail, the band being steep along
ΓM (the y-direction) has only p𝑦 projections (beside s projections at low energies).
The other band being steep along ΓK (the x-direction) has only p𝑥 projections. Both
bands are localised along ΓA (the z-direction). This accords with the expectation that
the electrons are mobile parallel to the bonds and less mobile perpendicular to them.
Band 4 is almost flat in the basal plane (the xy-plane), but steep along the crystal axis
(the z-direction). Accordingly, the Si projections are exclusively p𝑧. Differently from
band 2 and 3, the Si-s contributions are zero and, instead, Tb-d contributions arise
at the zone edges, more precisely Tb-d𝑥𝑧 and Tb-d𝑦𝑧. As these d orbitals are crosses
whose lobes point diagonally out of the basal plane, band 4 seems to represent a bond
between the Si-p𝑧 and the Tb-d orbitals.
The orthorhombic distortion affects only two of the four Si bands: band 2 and 3,
whose upper parts shift into opposite directions (feature C in Fig. 4.10). This is clearly
64
4.3 Electronic properties
(i) total (ii) s-orbitals (iii) p-orbitals (iv) d-orbitals
(a) 7,8 A B
0 2A A31 6
-2 54
-4 2,3 C
-6
-8
-10
1 1Å−1-12 hex-AlB2-TbSi2
M Γ K|Γ A M Γ K|Γ A M Γ K|Γ A M Γ K|Γ A
(b)
0 p dz z
2
-2
-4 px
-6 py
-8 M y
-10 x
1Å−1 K-12 ort-AlB2-TbSi Γ2
Figure 4.10: Band structures of (a) hex-AlB2-TbSi2 and (b) ort-AlB2-TbSi2. The panels and the
plotting mode are analogous to Fig. 4.9. Inset in (b,iv): k-path. Red markings highlight differ-
ences between (a) and (b). Green numbers enumerate the bands at the Γ-point.
linked to the distortion of the Si honeycomb and the concomitant splitting of the Si–
Si bond distances (Tab. 4.7). The dilation of the bonds in the y-direction gives rise to
a gain in binding energy, which becomes manifest in the downward movement of
the Si-p𝑦 band. The other two bonds, which form the zigzag chains and have their
major component in the x-direction, shorten a bit and lift the Si-p𝑥 band to lower
binding energies. The position of the Si-p𝑧 band, on the contrary, is insensitive to the
orthorhombic distortion, agreeing with the constancy of the axial lattice constant 𝑐.
Bands 5 and 6 cross the Fermi level and are, thus, partially occupied. Band 5 stems
from the d𝑧2 orbitals of the Tb atom, mixed with small in-plane Si-p contributions.
While it is shallow near the Γ-point on the in-plane k-paths, it is a very steep and
straight line crossing the Fermi level along the out-of-plane k-path. Band 6 is Si-s-
like below the Fermi level and its in-plane branch follows a quite narrow electronic
parabola with an effective mass of 0.15𝑚e. It abruptly bends into less steep branches
beyond the Fermi level, while the character simultaneously changes into a mixture
of in-plane Si-p and Tb-d. The out-of-plane branch is Si-s-like and completely below
the Fermi level. It follows a wider parabola with an effective mass of >1.5𝑚e, the
tenfold of the in-plane mass.
The orthorhombic distortion changes several features near the Fermi level. The Tb-
d band (band 5) slightly shifts at the K-point and the M-point, which flattens the in-
plane dispersion, but leaves the out-of-plane dispersion unchanged. In contrast, band
6 undergoes fundamental changes. The ΓM branch (y-direction) closes a gap with
a conduction band above (band 7) so that the light-electron parabola continues its
shape across the Fermi level. After the gap closing, it has an almost pure Si-sp char-
acter and crosses the second, Si-Tb-hybridised band without mixing (feature A1). The
ΓA branch (z-direction) opens a gap at the intersection with a double band (bands 7
65
E − EF  (eV) E − EF  (eV)
4 The rare-earth silicide bulk phases
and 8) dipping below the Fermi level halfway to the A-point. Thereby, band 6 hybrid-
ises with the band having Si-p𝑦-Tb-d𝑦𝑧 character and splits apart (feature A3), while
the other band having Si-p𝑥-Tb-d𝑥𝑧 character remains unchanged. The ΓK branch
(x-direction) retains its shape, but is pushed below the Fermi level (feature A2).
The band structures can be summarised as follows. The Si atoms are indeed sp2-
hybridised and bind to each other via the trigonal-planar hybrid orbitals, forming
three bands and, thus, hosting six electrons. The remaining p𝑧 orbitals are out of
plane and hybridise with out-of-plane Tb-d orbitals, forming one band (two elec-
trons). The remaining three electrons distribute over conduction bands near the
Fermi level. One of these bands stems from the inter-se hybridisation of the d𝑧2
orbitals of the Tb atoms, being rather flat in-plane, but strongly dispersive out-of-
plane. A second band is the s-like conduction band of the Si honeycomb, which is
strongly dispersive in-plane, and less dispersive out-of-plane. As long as the struc-
ture is hexagonal, this band transitions into Si-p-Tb-d states right at the Fermi level
along with an abrupt change in curvature. This is the origin of the DOS peak. The or-
thorhombic distortion rehybridises this band whereupon the DOS peak flattens out
(feature B). Since tet-ThSi2-TbSi2 is very similar to ort-AlB2-TbSi2 concerning the bond
lengths and angles, it lacks the DOS peak as well. The orbital-resolved partial atomic
charges reveal that the orthorhombic distortion transfers 0.1 electrons per unit cell
from the Si-p𝑦 orbitals to the Si-p𝑥 orbitals, in accordance with the shifting of band
2 and 3. This confirms the assumption that the orthorhombic distortion allows more
excess charge to accumulate in the zigzag chains, which stabilises the structure.
A last point to be emphasized is the incorrectness of the analogy between the Si hon-
eycomb in AlB2-RESi2 and a hypothetical graphite-like (or even graphene-like) Si allo-
trope, sometimes stated in the literature. In order to clarify this, the band structures
are calculated for the AlB2-TbSi2 structures (hexagonal and orthorhombic) after the
removal of the Tb atoms, which results in a layered “silicene” system (Si2, vide ap-
pendicem, Fig. B.1). The three lowest bands, representing the sp2 single bonds, are
very similar in TbSi2 and Si2, as well as the s-like conduction band right below the
Fermi level. The major difference consists in the p𝑧 band, whose in-plane branch is
flat in TbSi2 (bound to Tb-d→ localised in the basal plane) and dispersive in Si2 (de-
localised 𝜋-electron system). Thus, the decisive property of silicene, the delocalised
𝜋-electron system, is clearly missing in TbSi2. Interestingly, but not unexpectedly, the
effects on the Si bands from the orthorhombic distortion are similar in AlB2-TbSi2
and AlB2-Si2. Furthermore, the Si2 structures demonstrate the better charge storage
of the orthorhombically distorted structures. The total energy difference per Si atom
between ort-AlB2-Si2 and hex-AlB2-Si2 is Δ𝜇Si = +65 meV in the neutral state, so the
hexagonal version is more stable. The stability relations change when excess elec-
trons are added (Δ𝜇Si = −78 meV for +1 electron, Δ𝜇Si = −187 meV for +2 electrons).
This underlines that the orthorhombic distortion of AlB2-TbSi2 is driven by charge
balance issues due to the mismatch between the formal valences of Tb and Si.
Explicit f-electron treatment
Fig. 4.11 shows the band structures for f-valent fmo-ort-AlB2-TbSi2 and fmo-ort-AlB2-
ErSi2, beside trivalent ort-AlB2-TbSi2, which serves as reference. The plotting mode
66
4.3 Electronic properties
(i) ort-AlB2-TbSi2 (tri.) (ii) TbSi2 (f-up) (iii) TbSi2 (f-down) (iv) ErSi2 (f-up) (v) ErSi2 (f-down)
0
-2
-4
-6
-8
-10
-12
M Γ K|Γ A M Γ K|Γ A M Γ K|Γ A M Γ K|Γ A M Γ K|Γ A
Figure 4.11: Band structures of trivalent and f-valent, FMO ort-AlB2-RESi2. (i) trivalent TbSi2,
(ii) spin-up and (iii) spin-down of f-valent TbSi2, and (iv) spin-up and (v) spin-down of f-valent
ErSi2. The plotting mode is analogous to Fig. 4.9 (b,i). Colour: yellow/green→ Tb/Er, blue→
Si. Red arrows mark band shifts between spin-up and spin-down. ThSi2 structures: Fig. B.2.
and k-paths are analogous to Fig. 4.10 (i) (line width: sum of s, p, d and f contributions
→ constant; line colour: RE/Si portions). Because of the spin polarisation, each f-
valent RESi2 structure yields two band structures, one per spin component.
The most prominent features in the f-valent band structures are bold, horizontal lines
representing the 4f states. As expected, they are localised and occupy seven states
in the spin-up component and one (TbSi2) and four (ErSi2) states in the spin-down
component. Thus, the net spin polarisation is six (TbSi2) and three (ErSi2) electrons
per RE atom. The 4f levels of ErSi2 are deeper than those of TbSi2, reflecting the higher
atomic number of Er. Furthermore, the distance between the topmost spin-up and
spin-down levels is smaller in ErSi2 than that in TbSi2 as a consequence of the higher
exchange energy in the 4f shell of Er. The only non-f bands indirectly affected by the
4f electrons are the Tb-d𝑧2 states (band 5 in the trivalent case), which split apart by
0.41 eV in TbSi2 and 0.16 eV in ErSi2 (red arrows in Fig. 4.11). Being larger in TbSi2
than in ErSi2, this splitting is obviously linked to the net magnetic moment. All other
bands are only slightly affected by the presence of the 4f states (<0.1 eV).
Apart from the 4f levels, the band structures of ort-AlB2-TbSi2 and ort-AlB2-ErSi2 look
very similar. They merely differ by small band shifts of −80 meV to 170 meV, which
concentrate more in the RE bands than in the Si bands. This suggests that the band
shifts are caused by the differences between the TbSi2 and ErSi2 structures due to
lanthanoid contraction. The band structures of tet-ThSi2-TbSi2 and tet-ThSi2-ErSi2
differ by higher band shifts of −230 meV to 230 meV, in accordance with the higher
sensitivity of the ThSi2 structure to the cation size (vide appendicem, Fig. B.2).
Vacancy-populated structures
The band structure of hex-AlB2-TbSi2 – b optimised with PBEsol is shown in Fig. 4.12
(a). For comparison, the band structure of ort-AlB2-TbSi2 optimised with PBEsol is ad-
ded (Fig. 4.12 (b)). It shows merely tiny differences to the band structure of the PBE-
optimised unit cell (Fig. 4.10 (b)). The plotting mode is the same as that in Fig. 4.10
(line width → orbitals, colour → Tb/Si portion). In order to make potential d con-
tributions from the Si atoms visible, explicit projections on the spherical harmonics
are em√plo√yed instead of the PAW projections. As hex-AlB2-TbSi2 – b is embedded in a
R30◦( 3× 3×1) supercell of the primitive AlB2 cell, the k-path is chosen in such a way
67
E − EF  (eV)
4 The rare-earth silicide bulk phases
(i) total (ii) s-orbitals (iii) p-orbitals (iv) d-orbitals
(a)
0
-2
-4
-6
-8
-10
-12 1Å
−1
M K' Γ M' K|Γ A M Γ K|Γ A M Γ K|Γ A M Γ K|Γ A
(b)
0
-2
-4
-6
-8
-10
−1
-12 1Å
Figure 4.12: Band structures of (a) hex-AlB2-TbSi2 – b and (b) ort-AlB2-TbSi2. The unit cells
were optimised with PBEsol. The panels and the plotting mode are analogous to Fig. 4.9.
The k-path connects the high-symmetry points of the primitive Brillouin zone of the AlB2
structure. The high-symmetry points of the supercell Brillouin zone are marked red.
that it connects the high-symmetry points of the primitive Brillouin zone identified in
the reciprocal space of the supercell. The high-symmetry points of the supercell Bril-
louin zone M′ and K′ are marked red. All DOS and PDOS of the supercell are scaled
by 1/3 to reflect the different sizes of the unit cells.
Although the bands of hex-AlB2-TbSi2 – b are rather complicated due to band folding,
some differences to ort-AlB2-TbSi2 are apparent. The bands and the PDOS reveal that
the Si-s-like conduction bands disappear. The explicit projection scheme results in
small Si-d contributions in ort-AlB2-TbSi2, which are zero in hex-AlB2-TbSi2 – b. Fur-
thermore, the Fermi energy is lowered, so the binding energies of all bands are smal-
ler. Consequently, the Fermi level cuts the DOS in a region with a particularly low
density. The last difference consists in sharper peaks in the DOS along with flat bands
of Tb-Si character. All these differences accord well with the preliminary considera-
tions that the removal of a Si atom reduces excess charge. The strong Si-p-Tb-d con-
tributions to the DOS in the range from −1 eV to −4 eV, as well as the binding energy
reduction upon lower Si content agrees with experimental photoemission data [94,
97]. Although the DOS near the Fermi level is considerably reduced, hex-AlB2-RESi2 – b
is metallic with strongly dispersive bands cutting the Fermi level.
Charge transfer
The electronic properties are completed by a proof that RESi2 is at least partially het-
eropolar. For this purpose, the superposition of atomic charge densities is subtracted
from the ground state density of the unit cell, which yields the density of transferred
charge. The result is plotted as isosurfaces and tomographies in Fig. 4.13 for stoi-
chiometric ort-AlB2-TbSi2 and vacancy-populated hex-AlB2-TbSi2 – b, both optimised
68
E − EF  (eV) E − EF  (eV)
4.4 Discussion and summary
(a) ort-AlB2-TbSi2 (b) hex-AlB2-TbSi2−x
Figure 4.13: Charge transfer in (a) ort-AlB2-
TbSi2 and (b) hex-AlB2-RESi2 – b. Red→ elec-
tron gain; Blue → electron loss. The view
is along 𝑐. The bulbs are 3D isosurfaces of
the electron transfer. Tomographies across
(0001)-planes are inserted for three differ-
ence heights (relatively to 𝑐). Green mark-
ings highlight the zigzag chains in (a) and
the (v)-type Si atoms in (b). Black dotted
circles: vacancies.
with PBE. The tomographies are taken at three different heights along the axial lat-
tice constant 𝑐: the cut through the Si atomic plane at 𝑧 = 0.5𝑐, the cut through the
middle plane between the Si atoms and the Tb atoms at 𝑧 = 0.75𝑐, and the cut through
the Tb atomic plane at 𝑧 = 1.0𝑐 ≡ 0.0𝑐.
The isosurfaces of the ort-AlB2-TbSi2 structure show that charge accumulates in the
trigonal-planar bonds between the Si atoms, indicating sp2 hybridisation. In the hex-
AlB2-TbSi2 – b structure, the (m)-type Si atoms show three charge bulbs in trigonal-
planar geometry, indicating sp2 hybridisation. The (v)-type Si atoms show four bulbs,
indicating sp3 hybridisation. Two of them are placed on the bonds to the (m)-type Si
atoms and the other two are attached to the (v)-type Si atom, pointing towards the va-
cancy and tilted out of the plane by a large angle. According to the tomographies, the
Tb layer (𝑧 = 1.0) loses charge to the region between the Tb and the Si layer (𝑧 = 0.75).
The charge transfer is as expected and confirms the partial heteropolarity of RESi2 – x.
It is remarkable that the electrons accumulate mostly in sp2- or sp3-hybrid orbitals
around the Si atoms, and not in defined bonds between the Si and Tb atoms. Integrat-
ing the plane average of the transfer reveals that the Tb layer loses 0.14 electrons per
Tb atom in ort-AlB2-TbSi2 and 0.11 electron per Tb atom in hex-AlB2-TbSi2 – b. These
values have to be interpreted with caution as averaging over planes bears assign-
ment problems, in particular because the Tb planes and the Si honeycombs are quite
close together.
4.4 Discussion and summary
The treatment of TbSi2 – x and ErSi2 – x within DFT provided a detailed picture of the
chemistry and the physics of the RE silicides. The novel systematisation of the RESi2
prototypes revealed that the AlB2 and the ThSi2 structure are very similar to each
other and merely differ in the orientations of the characteristic planes of zigzag-
chain-Si (unidirectional→ AlB2, alternately perpendicular→ ThSi2). The structures
are determined by the Si networks, while the RE atoms are embedded in the inter-
stices. The Si–Si bond lengths are near the single-bond length of diamond-Si, dilated
69
z＝1.00 z＝0.75 z＝0.50
4 The rare-earth silicide bulk phases
by a few percent if charge accumulates. Structure variations are thus carried out
mainly by the Si–Si bond angles. That the Si sublattice behaves like a linkage of rigid
rods proved to be very helpful for understanding how Si vacancies and the RE ionic
radius alter the RESi2 – x structures.
The inspection of various RESi2 – x structures lead to the conclusion that both RESi2
prototypes are stabilised by vacant Si sites. Hence, stoichiometric RESi2 is unstable
against segregation of diamond-Si from a RESi2 – x remainder. The stability analysis
found that the AlB2 phase is stable for Si-poorer conditions, while the ThSi2 phase is
preferred at Si-rich conditions. Furthermore, the stability of the AlB2 phase seems to
be independent of the RE atomic number. On the contrary, the ThSi2 phase seems
to destabilise when switching from Tb to Er. This agrees with expected different
sensitivities of the Si sublattices towards the RE radius (layered for AlB2 → indif-
ferent, 3D network for ThSi2 → anisotropic deformation). The theoretical stability
analysis agrees with the experimentally determined temperature-composition phase
diagrams of the RE–Si systems. Moreover, the vacancies are crucial for the agreement
between the DFT-optimised lattice parameters and the experiment. They can also
change the symmetry group of the structures by an orthorhombic distortion, which
was traced back to an unequal vacancy concentration in the two inequivalent zig-
zag planes. The lattice parameters of the herein investigated structures match the
experimental reference up to 2 %, in spite of the approximated vacancy ordering.
The structural and electronic properties agree with the ZKB concept. As proven by
the band structures, the Si atoms bind to each other via bonds which are rather in-
dependent of the presence of the Tb atoms. The formal free electron pairs of the Si
anions occupy p𝑧 (m-type) or sp3-hybrid orbitals (v-type) and bind to the d orbitals of
the Tb atom in a predictable manner. All spare electrons occupy conduction bands,
which dominate the DOS in the Fermi-level region and consist of Tb-d states mixed
with Si-sp states. Since several bands cross the Fermi level, the RESi2 – x band struc-
tures are clearly metallic. The ZKB concept explains the inevitable presence of Si
vacancies by an improved charge balance, as confirmed by the electronic properties.
However, it fails to predict the exact stoichiometry as 1/3 electron per RE atom is still
spare in the most stable RESi2 – b structure (the saturation of all valence electrons re-
quires a stoichiometry of RESi1.5, which does not exist). The partial heteropolarity of
RESi2 – x was confirmed by the calculated charge transfer and the contraction of the
volumes upon compound formation.
The findings on the bulk RE silicides have important consequences for the RESi2
nanostructures on silicon. Firstly, the peritectic nature of the Si-rich RE silicides sug-
gests that also the RESi2 nanostructures might underlie a peritectic phase diagram,
which would provoke kinetically inhibited structures. Secondly, the vacancy-related
lattice distortions imply that one has to be very careful when employing lattice strain
to explain nanostructures which lack vacancies. In particular, the inverted 𝑐/𝑎 ra-
tio of the AlB2 phase is an issue in the RESi2 nanowires on Si(001) [P3, P6]. Sup-
posed strain-induced growth is discussed for the monolayer-Tb@Si(111) system as
well, which is the matter of the next chapter.
70
5 Monolayer films on silicon(111)
A silicon(111) substrate can be regarded as a stack of buckled Si honeycombs, some-
times referred to as Si bilayers, which also terminate the unreconstructed Si(111) sur-
face. Hex-AlB2-RESi2 – b, on the other hand, consists of alternating layers of hexagon-
ally arranged RE atoms and (defective) Si honeycombs. Hence, attaching a RESi2 – b
(0001) sheet to the Si(111) surface would be a natural continuation of the substrate.
As, in addition, the lateral lattice constants match up to a few percent, the silicides
of many REs are suitable for epitaxy on Si(111), among these the REs from Gd to Lu
and Y (lattice mismatch between −2.55 % and 0.83 % [33, 108]). The multilayer-RESi2
films on Si(111) are too thick for electronic quantum confinement, s√o thei√r electronic
properties are similar to those of bulk hex-AlB2-RESi2 – b. The R30◦( 3 × 3) surface
diffraction patterns indicate that the vacancies of the RESi2 films are ordered, dif-
ferently from the bulk phases [36, 37, 39, 46, 50–52]. In contrast, a RESi2 film with
a thickness of one bulk lattice constant, the monolayer-RE@Si(111) system, seems to
lack Si vacancies, as indicated by (1 × 1) surface diffraction patterns. More pecu-
liarly, the film is so thin that it confines the electronic states to a two-dimensional
(semi-)metal [P8, P12, 34, 35, 38–53].
In view of the nanowire-Tb@Si(557) system, this chapter recovers the properties
of the monolayer-Tb@Si(111) system. Tb can be considered a prototypical RE and
the results are expected to be transferable to the other monolayer-RE@Si(111) sys-
tems, at least if the RE is trivalent. This assumption is based on the knowledge about
the bulk-RESi2 phases (Chap. 4) and on the extensive, synoptic, joint-experimental-
theoretical work by Sanna et al., covering RE@Si(111) systems with thicknesses from
sub-monolayer to multilayer [53]. Although the structure model of the monolayer-
Tb@Si(111) system is well established, it bears peculiarities which have not been
understood yet. Therefore, the chapter begins with the analysis of the structure.
A systematic inspection of the configuration space of reasonable structure models
proves that the established structure model is the most stable. Moreover, satisfact-
ory microscopic explanations are found for all its peculiarities, e.g. the tendency
of the covering Si honeycomb to buckle into the opposite direction of the Si honey-
combs of the substrate. All findings on the structure are supported by the analysis
of the electronic properties. The band structures and Fermi surfaces are compared
to experimental ARPES measurements. The characteristic features are explained by
relating the bands of the monolayer to those of the structurally related CaSi2-TbSi2
phases according to a simple particle-in-a-box model. The charge transfer is determ-
ined as well. The chapter concludes with a critical revision of the popular statement
that surface-induced mechanical strain determines the morphological details of the
monolayer.
71
5 Monolayer films on silicon(111)
5.1 Structure optimisation
The slab method
Surfaces are problematic for plane-wave codes because they break the translation
symmetry. A solution for this consists in the slab method which employs a supercell
with several atomic layers modelling the substrate and empty space representing
the vacuum. The two interfaces between the vacuum layer and the substrate layer
then simulate the surfaces of the substrate. It is convenient to embed the slab into
a right-prismatic supercell so that the axis is parallel to the surface normal and the
basis equals the surface unit cell. Both the bulk and the vacuum region have to be
thick enough to prevent the states of opposite surfaces from interfering. The slab
method has two flavours: The asymmetric approach, which models the surface of
interest on the top side, while the bottom side is passivated with hydrogen; and the
symmetric approach, which models two identical copies of the surface on the top and
the bottom side of the slab. The latter is advantageous if passivation is not possible
or if the computational demand can be reduced by inversion symmetry.
The monolayer structures are set up on an asymmetric Si(111) slab with a thick-
ness of twelve atomic layers (six bilayers). The basis of the supercell represents
the trigonal (1 × 1) unit cell of the unreconstructed Si(111) surface. A 31 Å thick
vacuum layer separates the periodic replica of the slab in the axial direction (the
(111)-direction or z-direction). It reduces to a still sufficient thickness of 24 Å when
the slab bears the monolayer on the top side. The dangling bonds from the broken
sp3-hybrid orbitals on the bottom side are passivated by H atoms. The clean, unrecon-
structed slab is sketched in Fig. 5.1 (a). The lattice parameters of the substrate were
determined by a Murnaghan relaxation of the primitive unit cell of diamond-Si (PBE,
16 × 16 × 16 Monkhorst-Pack mesh). The relaxation yielded a conventional lattice
constant of 𝑎d-S√i = 5.469 Å, which translates into an Si(111) surface lat√tice constant of
𝑎Si(111) = 𝑎d-Si/ 2 = 3.867 Å and a Si–Si bond length of 𝑑d−Si = 𝑎d-Si · 6/4 = 2.368 Å.
These parameters slightly overestimate the corresponding experimental values by
+0.7 % (𝑎d-Si = 5.432 Å, 𝑎Si(111) = 3.841 Å, 𝑑d-Si = 2.352 Å [17, p. 1065]).
Thermodynamical framework
Following the experimental preparation, the monolayer forms in the annealing stage,
i.e. during heating the sample at a constant temperature after depositing a certain
amount of Tb. This translates into the thermodynamic boundary conditions of con-
stant temperature, pressure, Si chemical potential (determined by the presence of
the substrate) and amount of Tb (controlled by the Tb coverage). Similarly to the
case of bulk RESi2 – x (Sec. 4.2.1), the thermodynamic stability is determined by the Tb
chemical potential, defined an(alogously to Eq. (4.36): )
≈ 1 cell − d-Si − H-Sat − hcp-Tb𝜇Tb 𝐸DFT 𝜇Si 𝑁Si 𝜇H 𝑁H 𝜇𝑁Tb Tb
(5.43)
where 𝐸cell d−SiDFT is the DFT total energy of the supercell and 𝜇Si the Si chemical potential
in diamond-Si. Structures which minimise 𝜇Tb are stable. The H chemical potential
72
5.1 Structure optimisation
𝜇H-SatH refers to the H atoms, whose sole purpose is the saturation of the bottom side
of the slab. It is clear that they have no physical meaning for anything happening on
the top side, but their presence could have undesired effects on the thermodynamic
relations if they were ignored. Therefore, the H chemical potential is calculated from
a Si(111) slab which is symmetrically sa(turated by H atoms)at both sides.
𝜇H-Sat
1
= 𝐸H-Sat − 𝜇d−Si𝑁H-SatH 𝑁H-Sat DFT Si Si
(5.44)
H
By this means, all energetic effects from the H-saturated bottom side are stored in
𝜇H-SatH and thus removed from the thermodynamics. Of course, the H term cancels
out in differences between the Tb chemical potentials of structures having the same
bottom side. This is the case in all structures considered in this chapter.
Computational parameters
Many parameters are equal to those of the previous chapter (p. 45). The DFT calcula-
tions are carried out with VASP [2, 3] using PAW potentials and the PBE xc-functional
[10, 21, 22]. The Tb atoms are trivalent (5p65d16s2, 4f electrons frozen in the core). Si
and H have PAW valences of 3s23p2 and 1s1, respectively. The kinetic energy cutoff
is increased to 𝐸cut = 450 eV to meet the increased requirements of the H atoms.
The Brillouin zone is sampled by a Γ-centred surface Monkhorst-Pack mesh [117] at
a density of 15 × 15 × 1 k-points. The Brillouin-zone integration is carried out by a
Gaussian occupation with a smearing of 20 meV for the ionic relaxations of the metal-
lic systems and by the tetrahedron method with Blöchl corrections [118] for all other
calculations (static and relaxations of insulators). A convergence test proves that in-
creasing the energy cutoff or the sampling density alters the total energy by ≈1 meV
per atom. The ionic forces are calculated according to the Hellmann-Feynman the-
orem [119]. Differently from the relaxation of the bulk structures, the unit cell is
fixed. The lowest four atomic layers of Si are held fixed at the bulk positions during
all relaxations. The H atoms on the bottom side are held fixed at the equilibrium
positions of the symmetrically H-saturated slab. All other ions move within the unit
cell along the forces towards the equilibrium positions and stop to relax if the forces
acting on each ion are smaller than 0.005 eV/Å.
The Tb@Si(111) monolayer
The structure model of the monolayer-RE@Si(111) system (Fig. 5.1) is well established
by experimental analysis as well as by theoretical calculations. It is a sandwich struc-
ture of a hexagonal RE layer which is placed between the unreconstructed (1 × 1)-
Si(111) surface and a vacancy-free, buckled Si honeycomb. The monolayers of all
REs from Gd to Er and Y adopt this structure [39–41, 43, 45–48, 53]. The following no-
menclature is used in this work: terminating Si honeycomb or substrate termination
denotes the top Si bilayer of the substrate (Si3 and Si4) and covering Si honeycomb or
cover denotes the Si bilayer above the RE atoms (Si1 and Si2). In the established struc-
ture, the three valence electrons from each RE atom complement all incomplete Si
73
5 Monolayer films on silicon(111)
(a) vacuum (b) (c)A-type B-type [112‾]
31 Å
[112‾] ⊙ [111]
⊙ [11‾0]
t h
Si1
Si2
Si3 (d)
Si4
fixed in position S T4 H3 T1
Figure 5.1: (a) Sketch of the Si(111) slab and (b) – (d) the TbSi2 monolayer structures. Blue
circles are Si atoms; Yellow circles are Tb atoms; Red circles are H atoms. The light red rect-
angle in (a) encircles the subset of atoms fixed in position during the relaxations. Red lines
mark the unit cell. (b) shows the (11̄0) view and (c) the (111) view of the T4-h-B structure.
Orange markings indicate the symmetry positions of the Tb atom relative to the substrate
termination (T4, H3, T1 and S [39]) and to the cover (t-type and h-type). Small green arrows
indicate the buckling direction of the cover (A-type or B-type [39]). (d) shows the trigonal-
antiprismatic/ octahedral coordination of the Tb atom by the adjacent Si2 and Si3 layers.
octets, i.e. they pair the dangling bonds of Si1, Si2 and Si3. This closed-shell configura-
tion is reasonable a priori and holds for all structure models which are derived from
the established one by layer translation [39]. Although many publications concerning
different structure models exist, a rigorous and complete evaluation of the configur-
ation space of reasonable structures has not yet been performed. Since it helps to
understand why the established structure is stable in spite of some peculiarities, the
configuration space is systematically explored in the following paragraphs.
At first, the degrees of freedom to modify the established structure are identified
(Fig. 5.1 (b)). The first degree of freedom has already been addressed in many works,
e.g. Stauffer et al. [39]. It concerns the placement of the RE layer relatively to the
substrate. In reasonable structure models, the RE atom is located at one of the three
symmetry points of the unreconstructed Si(111)-(1 × 1) surface [39]:
• Above the lower Si atom (egg-box position), denoted T4
• Above the upper Si atom, denoted T1
• Above the hole (hole position), denoted H3
In addition, Stauffer et al. considered a structure where the RE atom substitutes the
upper Si atom, denoted S. This structure does not have a closed-shell configuration,
of course. The second degree of freedom consists in the placement of the covering
74
[111]
[11‾0]
5.1 Structure optimisation
Si honeycomb relative to the RE layer: holes over RE atoms (h-type, hole position) or
upper Si atoms over RE atoms (t-type, egg-box position). The symmetric placement
analogous to T1, i.e. lower Si atoms over RE atoms, is discarded due to the low num-
ber of Si atoms coordinating the RE atoms1. The cover placement has apparently
not been attended to so far as the literature concerns only h-type structure mod-
els. This probably originates in the premise that the monolayer-RE@Si(111) struc-
ture is based on the AlB2 structure. However, because the AlB2 phase is only stable
in vacancy-populated form, deriving the vacancy-free monolayer structure from the
AlB2 structure is not reasonable. Moreover, the CaSi2 structures, which are the stable
phases for (hypothetical) vacancy-free RESi2, incorporate h-type and t-type buckled
honeycomb-Si (Sec. 4.2.6). The monolayer structure will prove to be more related to
the CaSi2 structures than to the AlB2 structure, which underlines the need for invest-
igating t-type structure models. Finally, the third – and important – degree of freedom
is the direction into which the cover buckles: Conform with the underlying Si sub-
strate (A-type) or mirrored/rotated by 180◦ (B-type). According to this nomenclature,
the established structure is denoted T4-h-B.
The configuration space of the monolayer-Tb@Si(111) system is sampled by each four
T4 and H3 structure models (h-type/t-type × A-type/B-type) and each two T1 and S
structure models (h-type × A-type/B-type)2. All structure models are optimised with
respect to the atomic positions. The resulting Tb chemical potentials are plotted in
Fig. 5.2 (a). The T1 and S structures are highly unstable (𝜇Tb > 0). In the other struc-
tures, T4 and H3, the covering Si honeycomb prefers the h-type to the t-type position.
Interestingly, the favoured buckling direction of the cover depends on the relative
positions between the substrate, the Tb layer and the cover in an unobvious manner.
The most stable structure model is T4-h-B with 𝜇Tb = −806 meV, which agrees well
with the findings of Sanna et al. [53].
In order to quantify how varying the degrees of freedom affects 𝜇Tb, the respective
penalties are reckoned back in the order: buckling direction, cover placement and Tb
placement. The results are compiled as a decision tree in Fig. 5.2 (b), in which sum-
ming the values along a branch returns 𝜇Tb of the respective structure. Flicking the
buckling direction (lowest level) puts a penalty of+135 meV to+253 meV on the T4 and
H3 structures. The established T4-h-B structure lies amid the range with +189 meV,
matching the findings of [53] very well. Since the flicking penalties are much higher
than 𝑘B𝑇 at room temperature, the orientation of the cover is definite and experi-
mentally observable. Concerning the position of the cover, the T4 and H3 structures
are rather h-type than t-type (Δ𝜇Tb = +382 meV and +274 meV, respectively). Con-
cerning the position of the Tb layer, the T4 position is more favourable than the H3
position by +232 meV. The values agree very well with the difference of 225 meV in
the local minima of the potential energy surface (PES) of Si(111) for the adsorption
of a Dy atom [53]. The T1 site has a penalty of +1158 meV, also in agreement with the
PES in [53] and similar to the penalty of the substitutional S site (+1164 meV).
1With respect to the findings on bulk RESi2 – x (Chap. 4), the RE atom tends to favour Si coordination
numbers of about 8 to 10. However, placing the cover in a T1 manner would contribute only one
Si atom to the coordination polyhedron, leading to a maximum Si coordination number of five. For
the same reason, the T1 placement relatively to the substrate is strongly discouraged as well.
2The t-type structure models are ignored as T1 and S are per se strongly discouraged.
75
5 Monolayer films on silicon(111)
(a) (b)
400 425 −806 meV366 352 358
200 +232 +1164
0 +1158
0
-47 H3 T4 T1 S
-200
0 +274 0 +382
-289
-400 -300-353
-424 h t h t
-600
-617 -574 +253 0 0 +135 0 +66
-800 A B A B A B
-806
0 +221 +189 0 +14 0
A B A B A B
Figure 5.2: (a) Tb chemical potentials of the monolayer-Tb@Si(111) structures and (b) de-
cision tree for the energy penalties upon structure variations. Green boxes in (b) mark the
branch of the stable T4-h-B structure.
The tendency of the cover to buckle into a certain direction can be summarised as a
simple rule: The buckling direction is favourable if the upper Si atom of the substrate
termination (Si3) is not above the lower Si atom of the cover (Si2) (oblique, orange ar-
row in Fig. 5.1 (b)). This is the case in the stable T4-h-B structure, as well as in H3-h-A,
T4-t-A and H3-t-B. Conversely, the structure is penalised if Si2 is above Si3, as in the
case in T4-h-A, H3-h-B, T4-t-B and H3-t-A. The physical reason for this will be illumin-
ated by the band structures (Sec. 5.2). The stability relations of the other two degrees
of freedom depend on how the Tb atom interacts with the Si honeycombs. In order
to analyse this, the Si coordination numbers 𝑁C of the Tb atoms are compiled for all
structures in Tab. 5.1, as well as the Tb–Si bond distances and the bond distances
and angles of the two Si honeycombs Si1–Si2 and Si3–Si4. The vertical atomic-layer
distances, which immediately follow from the bond distances, are plotted in Fig. 5.3.
The energetically most stable T4-h-B structure is highlighted in each case. “Clean” de-
notes the clean, unreconstructed Si(111) surface and “H-Sat” the H-saturated Si(111)
surface (the same slab used for the calculation of 𝜇H-SatH , vide supra). For comparison,
the respective parameters of the bulk phases of TbSi2 are inserted as well.
Shifting the layers against each other changes the number of Si atoms coordinating
the Tb atom. In the hole position (cover: h-type, substrate: H3), the Tb atom is co-
ordinated only by Si2/Si3 in a threefold manner. The egg-box position (cover: t-type,
substrate: T4) adds the Si1/Si4 atom to the coordination polyhedron, incrementing
the coordination number by one. The coordinating Si–Tb bond distances range from
2.92 Å to 3.11 Å in the T4 and H3 structures. They follow the expectable trend that they
dilate for higher 𝑁C. In detail, their averages are 2.98 Å, 3.00 Å and 3.03 Å for 𝑁C = 6,
7 and 8, respectively. Depending on the relative positions, the Tb atom modifies the
Si honeycombs in the following manner: In the hole position, both the substrate ter-
mination and the cover (if buckled into the favourable direction) assume buckling
76
Δμ  − Δμ hcp-TbTb Tb  (meV)
T
4 -
T h-A
4 -h
T -B
4 -
T t-A
H4 -t-- B3H h-
3 - A
H h-B
3
H -t-A
T3 -t-B
1 -
T h-
1 - Ah
S -- Bh
S -- Ah-B
5.1 Structure optimisation
6.0 5.54 5.57 Si1
5.01 5.05 4.97
5.0 4.71 4.74 4.68 4.67 4.89 0.79 0.80 4.78 Si2
0.83
0.67 0.80 0.88 0.65 0.86 0.92 0.88 Tb
4.0 0.79 1.87 1.88 Si3
3.0 1.95 1.89 2.10 2.17 1.90 1.96 2.11 2.05 1.80 Si4
2.0 Si
d
4
2.88 2.89
1.0 2.09 2.05 2.03 2.05 1.98 2.06 2.00 1.92 2.10
0.0
0.76 0.63 0.89 0.91 0.90 0.89 0.78 0.81 0.76 0.74 0.81 0.81 0.90
-1.0
Figure 5.3: Vertical positions and distances of the atomic layers in the monolayer-Tb@Si(111)
structures. All vertical positions are relative to that of Si3. The numbers between two symbols
quantify the vertical distance between the respective layers. The numbers above the up-
triangles quantify the vertical distance between Si1 and Si3 layers and, thus, the height of the
monolayer. Sid4 marks the atomic layer distance in diamond-Si. Experimental reference for
the monolayer-Ho@Si(111) system: [43].
Si1–Tb Si2–Tb Si3–Tb Si4–Tb 𝑁C Si1–Si2 Si3–Si4
Clean 2.32 (113.0◦)
H-Sat 2.36 (110.1◦)
T4-h-A 3.44 2.96 3.06 2.98 7 2.33 (112.1◦) 2.40 (107.1◦)
T4-h-B 3.49 2.93 3.03 2.96 7 2.37 (109.3◦) 2.41 (106.7◦)
T4-t-A 2.98 3.07 3.02 2.92 8 2.40 (107.4◦) 2.41 (107.0◦)
T4-t-B 3.00 3.11 3.03 2.95 8 2.38 (108.5◦) 2.40 (107.1◦)
H3-h-A 3.50 2.93 2.99 3.55 6 2.37 (109.4◦) 2.37 (109.7◦)
H3-h-B 3.44 2.97 3.04 3.63 6 2.33 (112.5◦) 2.37 (109.1◦)
H -t-A 2.97 3.07 3.00 3.55 7 2.39 (107.9◦) 2.36 (110.2◦3 )
H3-t-B 2.97 3.03 2.94 3.47 7 2.41 (106.5◦) 2.35 (110.6◦)
T1-h-A 3.47 2.91 2.88 4.31 4 2.37 (109.5◦) 2.37 (109.0◦)
T1-h-B 3.49 2.92 2.89 4.32 4 2.37 (109.3◦) 2.37 (109.1◦)
diamond-Si 2.37 (109.5◦)
ort-AlB2-TbSi2 3.04 8 2.42, 2.36 (124.7◦, 110.6◦)
tr6-CaSi2-TbSi2 2.94 3.06 3.05 7 2.29 (118.9◦) 2.49 (104.9◦)
tr3-CaSi2-TbSi2 3.00 3.06 8 2.41 (103.6◦)
Table 5.1: Bond distances and angles in the monolayer-Tb@Si(111) structures. The distances
are given in Å; The angles are parenthesised. 𝑁C is the number of Si atoms coordinating each
Tb atom.
77
z (Å)
H-Sat
Clean
T
4 -h-A
T
4 -h-B
T
4 -t-A
T
4 -t-B
H
3 -h-A
H
3 -h-B
H
3 -t-A
H
3 -t-B
T
1 -h-A
T
1 -h-B
E
H xo pS .i2
5 Monolayer films on silicon(111)
parameters similar to those of diamond-Si (bond length: 2.37 Å, tetrahedron angle:
109.5◦). In the egg-box position, the Si–Si bond length is longer, while the bond angle
is more acute. This is expectable as a higher coordination enhances the bonding
between the Tb atom and the Si honeycomb at the expense of the Si–Si intra-bonds.
If the buckling flicks into the unfavoured direction, the cover flattens out a bit and,
thus, has more obtuse bond angles and shorter bond lengths. The H-saturated surface
has a bulk-like substrate termination, while that of the clean surface is considerably
flattened due to the low electron density of the dangling bond of Si3.
The layer distances follow the bond lengths by construction. The extending effect of
egg-box Tb atoms on the bonds of adjacent Si honeycombs is clearly visible. Particu-
larly in the stable T4-h-B structure, the cover is bulk-like, while the substrate termin-
ation is expanded. If the buckling direction is unfavourable, the cover as a whole is
elevated compared to the favourable case. In the t-type structures, the height of the
monolayer (Si1–Si3) is larger than that of the respective h-type structures by∼0.3 Å.
The calculated parameters of the most stable T4-h-B structure equal the results of
Sanna et al. [53] up to 0.1 Å, which is not surprising as the theoretical approaches
are very similar. Experimental data on the vertical layer distances are not available
for the monolayer-Tb@Si(111) system. Therefore, the results are compared to the
homologue monolayer-Ho@Si(111) system, for which Spence et al. measured com-
plete stratification data set by MEIS [43], plotted in the rightmost column in Fig. 5.3.
The calculated expansion of the substrate termination matches the experimental res-
ults of Spence et al. very well (0.91 Å vs. 0.90 Å). The distance between the cover and
the Tb layer (Si2–Tb) is greater than that of Spence et al. (1.89 Å vs. 1.80 Å). This not a
disagreement, tough, if the lanthanoid contraction is taken into account: It makes the
Si–RE layer distances contract by −0.2 Å to −0.3 Å per atomic-number increment [46,
53], hence adding up to −0.4 Å to −0.6 Å from Tb to Ho. The cover thickness (Si1–Si2)
and the distance between the Tb layer and the substrate (Tb–Si3) are smaller than
those of Spence et al. (0.80 Å vs. 0.88 Å and 2.05 Å vs. 2.10 Å, respectively). However,
they match the results of another study, a LEED 𝐼-𝑉 analysis on the same system [45]
(0.82 Å and 2.03 Å, respectively). In conclusion, the T4-h-B structure model optimised
with PBE agrees with experimental evidence within the error bars.
As the comparison between the Si-Tb coordination polyhedrons in the (T4 and H3)
monolayer structures and those in the vacancy-free bulk TbSi2 structures reveals,
that the monolayer is much more related with the CaSi2 structures than with the
AlB2 structure. In the bulk, the coordination polyhedrons are cuboids in ort-AlB2-
TbSi2, hexagonal prisms in hex-AlB2-TbSi2, trigonal antiprisms (distorted octahedra)
with both bases capped in tr3-CaSi2-TbSi2 and trigonal antiprisms with one basis
capped in tr6-CaSi2-TbSi2. In the monolayer, the Si2/Si3 coordination polyhedron is
a trigonal antiprism (octahedron) if the buckling direction of the cover is favour-
able and a regular trigonal prism otherwise. That the Tb atom seems to prefer the
trigonal-antiprismatic (octahedral) Si coordination hints at the microscopic explan-
ation for the buckling tendency of the cover (vide infra). The distinction between
T4 and H3 and, analogously, between t-type and h-type consists in the presence of
a cap at the respective basis. Accordingly, the coordination polyhedron in T4-t-A is
a doubly capped trigonal antiprism, exactly like that in tr3-CaSi2-TbSi2, and that in
78
5.2 Electronic properties
T4-h-B is a trigonal antiprism with capped bottom face (highlighted in Fig. 5.1 (d)),
similar to that in tr6-CaSi2-TbSi2. Thus, the T4-h-B structure agrees with the energetic
favourability of tr6-CaSi2-TbSi2 over tr3-CaSi2-TbSi2 and hex-AlB2-RESi2 within PBE
(Tab. 4.8). The analogy is underlined by an almost perfect lattice matching between
the basis of tr6-CaSi2-TbSi2 and the Si(111) surface (3.94 Å vs. 3.867 Å, +1.9 %). As the
same holds also for tr3-CaSi2-TbSi2 (3.78 Å, −2.2 %), it may be concluded that layered
structures of Tb atoms sandwiched between buckled Si honeycombs in general are
unstrained. In contrast to the CaSi2 phases, the basis of hex-AlB2-TbSi2 (4.10 Å) is too
large by +6.0 %, so this structure is even farther away from the monolayer structure
as already given by the geometry and the thermodynamical instability. This point
will be discussed in more detail in Sec. 5.3.1.
5.2 Electronic properties
5.2.1 Band structures
The band structure calculations are carried out within the same approach as that in
Sec. 4.3. The k-path in the surface Brillouin zone consists only of two line segments:
MΓ parallel to the [112̄] direction and ΓK parallel to the [11̄0] direction. A sketch of
it may be found in the inset of Fig. 5.4 (d,vi). For the calculation of accurate DOSs, the
sampling density of the Brillouin zone is increased to a (40 × 40 × 1) mesh.
The stable T4-h-B structure
Fig. 5.4 shows the band structures of (a) – (c) the most stable T4-h-B structure and (d)
the clean Si(111) surface. The line width corresponds to the portion of certain groups
in the PAW projections. The first row (a) shows the portions of the Tb portion, the top
four Si atoms Si1 – Si4 and bulk-Si3. The second row (b) shows the orbital-resolved
projections of the Tb atom, in which angular momenta with the same z-component
are combined (e.g. p𝑥 and p𝑦). The third row (c) shows the orbital-resolved projec-
tions of the cover (Si1 and Si2) and the top Si atom of the substrate Si3. The last row
(d) concerns the clean Si(111) surface, its panels showing the total portion of Si3, Si4
and bulk-Si, and the orbital-resolved projections of Si3. The line colour is fixed in
each panel, but indicates the displayed atom group. The energies are reset by the
bulk VBM 𝐸0 and the Fermi level of the slab is inserted as a red horizontal line. By
this means, the band structures of T4-h-B and clean Si(111) become comparable.
All Si bands in Fig. 5.4 have a characteristic set of three bands in common which
stem from the three in-plane sp3-hybrid bonds: a U-shaped band between −12 eV
and −8 eV and two bands between −8 eV and 0 eV which are shaped like a lying K
and intersect at the Γ-point. The U and K bands of bulk-Si (panels (a,vi) and (d,iii))
are rather concentrated shadows resulting from the slab projection combined with
3The average of the upper two fixed Si layers in the red box in Fig. 5.1 (a).
79
5 Monolayer films on silicon(111)
(a) T4-h-B (i) Tb (ii) Si1 (iii) Si2 (iv) Si3 (v) Si4 (vi) bulk-Si
2
0
-2
-4
-6
-8
-10
(b) (i) Tb-s (ii) px+py (iii) pz (iv) dxy+dx2−y2 (v) dxz+dyz (vi) dz2
2
0
-2
-4
(c) (i) (Si1+Si2)-s (ii) px+py (iii) pz (iv) Si3-s (v) px+py (vi) pz
2
0
-2
-4
-6
-8
-10
(d) Clean (i) Si3 (ii) Si4 (iii) bulk-Si (iv) Si3-s (v) px+py (vi) pz
2
0
-2
-4
-6 M y
-8 x
-10 1Å−1 Γ K
M Γ K M Γ K M Γ K M Γ K M Γ K M Γ K
Figure 5.4: Band structures of (a) – (c) the T4-h-B structure and (d) the clean Si(111) surface.
The line colour is fixed in each panel. The line width corresponds to: (a) the total PAW projec-
tions on (i) Tb, (ii) Si1, (iii) Si2, (iv) Si3, (v) Si4 and (vi) bulk Si; (b) the (i) s, (ii) (p𝑥 + p𝑦), (iii) p𝑧,
(iv) (d𝑥𝑦 + d𝑥2+𝑦2), (v) (d𝑥𝑧 + d𝑦𝑧) and (vi) d𝑧2 orbitals of the Tb atom; (c) the (i) s, (ii) (p𝑥 + p𝑦)
and (iii) p𝑧 orbitals of Si1 and Si2, and the (iv) s, (v) (p𝑥 + p𝑦) and (vi) p𝑧 orbitals of Si3; (d) the
total PAW projections on (i) Si3, (ii) Si4 and (iii) bulk Si, and the (iv) s, (v) (p𝑥 + p𝑦) and (vi) p𝑧
orbitals of Si3. All energies are reset by the VBM of the bulk-Si bands. The red horizontal lines
indicate the Fermi levels of the slabs. The inset in the bottom right panel shows the surface
Brillouin zone. Mind that the energy scale of (b) is different from that of the other rows as
there are no Tb states below −4 eV.
a low dispersion in the (111) direction. In contrast, the bands from the vertical sp3-
hybrid bonds are separated, diffuse, flat lines because they are less dispersive in-
plane, but strongly dispersive out-of-plane. The similarity between the bulk-Si bands
of the clean surface and T4-h-B proves that the slab is thick enough for the purposes
of this work. The difference between the VBM and the CBM of the bulk-Si bands, the
substrate band gap, amounts to 0.9 eV, which is smaller than the experimental band
80
E − E0  (eV) E − E0  (eV) E − E0  (eV) E − E0  (eV)
5.2 Electronic properties
gap of bulk silicon (1.1 eV), but higher than the DFT band gap of bulk silicon (0.7 eV).
Thus, the DFT-related underestimation of the band gap and the gap opening due to
the finiteness of the slab partially compensate each other.
Concerning the Si3 and Si4 bands of the clean Si(111) surface (panels (d,i) and (d,ii)),
the in-plane U and K bands look similar to the bulk-Si bands, which indicates a strong
coupling between the substrate termination the rest of the slab. In contrast, the dif-
fuse out-of-plane states of Si3 are much weaker than those of Si4. Instead, a surface
state of exclusive p𝑧 character appears as a single, isolated flat band just above the
VBM of the slab, but completely within the substrate band gap (panel (d,vi)). As ob-
served earlier [53], this state originates in the broken vertical sp3 bonds of Si3. That
the band is half-filled pins the Fermi level at +0.3 eV above the VBM. The Si3 and Si4
bands of the T4-h-B structure (panels (a,iv) and (a,v)) share several characteristics
with those of the clean surface. They are almost unaffected by the presence of the
TbSi2 monolayer. The only major difference consist in a splitting of the Si3 surface
state into an occupied part at the zone edges and unoccupied part at the centre.
The bands of Si1 and Si2 (panels (c,i) and (c,ii)) are similar to those the substrate ter-
mination with respect to the in-plane U and K bands. However, they are sharp in the
sense that the majority of their projections is concentrated in one copy. The out-of-
plane bands are sharp as well and resemble an O between −4 eV and +1 eV (panel
(c,iii)). Hence, the cover exclusively hosts surface states and is completely decoupled
from the bulk. The Tb states dominate the region a few eV below the Fermi level,
similarly to bulk TbSi2. They have strong Tb-d contributions at lower binding ener-
gies and hybridise with the dangling p𝑧 states of Si1, Si2 and Si3. The O-shaped double
band is the most prominent band. Its lower branch is majorly of Tb-Si2 character with
smaller Si1 portions. Its upper branch is isolated from the lower branch and lies com-
pletely within the band gap. Since the Fermi level is lifted to +0.7 eV, right below the
substrate CBM, the upper O band is almost completely filled. Below the Fermi level,
its character is a hybrid of Tb-d𝑥𝑦, Tb-d𝑥2− 𝑦2 (both in-plane), Si1-p𝑧 and (to a lesser
extent) Si2-p𝑧. At and above the Fermi level, the character changes to out-of-plane
Tb-d𝑧2 mixed with Si3-p𝑧. The upper O band is hole-like as it cuts the Fermi level
with a negative curvature. Near the M-points, a second, electron-like band crosses
the Fermi level. Its character is a hybrid of Tb-d𝑧2 and p𝑧 of Si1 and Si2. Since all
states crossing the Fermi level are confined to the surface and completely decoupled
from the rest of the substrate, the TbSi2 film gives rise to a clearly two-dimensional,
(semi-)metallic system.
Although the DFT eigenenergies are only crude approximations of real excitation
energies, the theoretical band structure in Fig. 5.4 compares well with experimental
ARPES images of the monolayer-Tb@Si(111) system [52] and the homologous systems
monolayer-Gd@Si(111) [47] and monolayer-Dy@Si(111) [49]. In all three works, the
O-shaped double band between −5 eV and 0 eV is clearly visible. Its branches are
separated by a smaller gap at the K-point and by a larger gap at the M-point so that
the hole-like upper branch is isolated from the substrate bands. In this work, the
binding energy of the upper O band at the zone boundary is ≈1.4 eV, which is close to
the experimental reference. Also the electron-like pockets at the M-points perfectly
match the experimental observations.
81
5 Monolayer films on silicon(111)
Comparison between the monolayer-Tb@Si(111) structures
In order to explain why T4-h-B is the most stable structure model for the monolayer-
Tb@Si(111) system, its band structure is compared with those of five other structure
models in Fig. 5.5. The plotting mode is the same as that in Fig. 5.4 (a). Compared
to T4-h-B (Fig. 5.5 (c)), all other structure models show characteristic changes in the
upper O band, which are marked orange.
The Fermi energy depends on the relative positions between the Tb layer and the
adjacent Si honeycombs in a quite predictable manner. As expected from bonding
efficacy, the Fermi energy falls if the Tb atom is in the egg-box position (T4 and t-type
structures) and rises if the Tb atom is in the hole position (H3 and h-type structures).
Quantitatively, the Fermi levels relative to the bulk VBM and averaged over the two
buckling directions are +0.58 eV for the T4-t structures, +0.76 eV for the T4-h and the
H3-t structures, and +0.81 eV for the H3-h structures (band structures not shown).
Hence, the T4-t structures minimise the Fermi energy.
Each variation from the stable T4-h-B structure gives rise to a characteristic band
changes near the Fermi level. From T4 to H3, the upper O band closes a gap with
the CBs at the Γ-point (feature A in Fig. 5.5 (a) and (b)). Furthermore, as indicated
by the sharpening, the in-plane bands of the substrate termination dehybridise from
the bulk states and concentrate at the surface, along with higher binding energies.
This accords with the contraction, ergo intra-bond strengthening, of the terminating
Si honeycomb upon switching from T4 to H3 (vide supra). From h-type to t-type, the
upper O band flattens and shows an electron-like furrow in the apex at the Γ-point
(feature B in Fig. 5.5 (e) and (f)). Furthermore, the bands of the cover shift to lower
binding energies, which accords with the expansion, ergo intra-bond weakening, of
the covering Si honeycomb upon switching from h-type to t-type. Flicking the buck-
ling direction from favoured to unfavoured entails a gap closing between the upper
O band and the CBs along ΓK (feature C in Fig. 5.5 (a), (d) and (f)). On closer inspec-
tion, both involved bands are characterised by strong Tb-d contributions beside p𝑧
contributions from Si1, Si2 and Si3. After the gap closing, the steeper band of the new
crossing is majorly of in-plane d𝑥𝑦 and d𝑥2− 𝑦2 character and the shallower band of
out-of-plane d𝑧2 character. This confirms that the steric impact of the Tb-d states is
the driving force behind the tendency of the cover to buckle into a certain direction.
All band variations increase the DOS at the Fermi level for different reasons, which
finally explains why T4-h-B is the most stable structure, despite not having the lowest
Fermi energy.
Relation to the bulk TbSi2 structures
That the TbSi2 monolayer seems to be electronically decoupled from the Si(111) sub-
strate suggests that the surface band structure might be derivable from the bands of
bulk TbSi2 by means of a simplified particle-in-box approach. For example, consider
a monolayer slice of the hex-AlB2-TbSi2 structure with a thickness of one axial lattice
constant 𝑐, which would be a layer of Tb atoms sandwiched between two Si honey-
combs. The lateral motion of the electrons may be undisturbed by the presence of the
boundary surfaces. The vertical distribution of the wave functions then determines
82
5.2 Electronic properties
(i) Tb (ii) Si1 (iii) Si2 (iv) Si3 (v) Si4
(a) 2 A C
0
-2
-4
H3-h-B
(b) 2 A
0
-2
-4
H3-h-A
(c) 2
0
-2
-4
T4-h-B
(d) 2 C
0
-2
-4
T4-h-A
(e) 2 B
0
-2
-4
T4-t-A
(f) 2 B C
0
-2 M y
-4 x
T4-t-B 1Å
−1 Γ K
M Γ K M Γ K M Γ K M Γ K M Γ K
Figure 5.5: Band structures of the monolayer-Tb@Si(111) structures. (a) H3-h-B, (b) H3-h-A, (c)
T4-h-B, (d) T4-h-A, (e) T4-t-A, (f) T4-t-B. The line width corresponds to the PAW projections of (i)
Tb, (ii) Si1, (iii) Si2, (iv) Si3 and (v) Si4. All energies are reset by the VBM of the bulk-Si bands.
The red horizontal lines indicate the Fermi levels. Orange markings highlight changes in the
upper O band relatively to the T4-h-B structure. The inset in the bottom right panel shows
the surface Brillouin zone.
83
E − E0  (eV) E − E0  (eV) E − E0  (eV) E − E0  (eV) E − E0  (eV) E − E0  (eV)
5 Monolayer films on silicon(111)
how the bulk bands have to be projected onto the surface Brillouin zone of the slice.
A reasonable ansatz is a first-order, fixed-end standing wave of wavelength 2𝑐 and
corresponding wavenumber π𝑐 . Such a standing wave can be constructed from the
solutions for the 3D crystal by superposing two countercurrent waves with equal lat-
eral and opposite axial wavenumbers π π𝑐 and − 𝑐 . Since these lie in the opposite bases
of the prismatic 3D Brillouin zone, the 2D band structure of the slice can be estim-
ated by calculating the bands of the 3D crystal along paths in one of these (identical)
bases. In the case of hexagonal unit cells, the surface k-path M–Γ–K corresponds to
the k-path L–A–H in the bulk Brillouin zone.
Out of the bulk TbSi2 structures in Sec. 4.2, three have a hexagonal basis. While the
band structure of hex-AlB2-TbSi2, has already been analysed (Fig. 4.10), those of tr3-
CaSi2-TbSi2 and tr6-CaSi2-TbSi2 are calculated from the PBE-optimised unit cells in
the same manner. Fig. 5.6 shows the bands of all three structures along (a) the main
path M–Γ–K|Γ–A and (b) the path for the slice projection L–A–H. In the bands of (iii)
tr6-CaSi2-TbSi2, the blue colour refers only to the buckled Si honeycomb, while the
contributions from the other, almost flat Si honeycomb are blended out. The Tb con-
tributions are divided by two so as to retain the Tb/Si ratio. In order to facilitate the
comparison, the slab band structures of (c) T4-h-B (related with tr6-CaSi2-TbSi2) and
(d) T4-t-A (related with tr3-CaSi2-TbSi2) are shown as well.
The L–A–H bands of hex-AlB2-TbSi2 (Fig. 5.6 (b,i)) have some features with the surface
bands of T4-h-B in common (Fig. 5.6 (c)). The in-plane U and K bands of the Si atoms
(i) hex-AlB2 (ii) tr3-CaSi2 (iii) tr6-CaSi2 (c) T4-h-B
(a)
0
-2
-4
-6
-8
-10
-12
M Γ K⏐Γ A M Γ K⏐Γ A M Γ K⏐Γ A (d) T4-t-A
(b)
0
-2
-4
-6
-8
-10
-12
L A H L A H L A H M Γ K
Figure 5.6: Band folding of the bulk TbSi2 structures: (i) ort-AlB2-TbSi2, (ii) tr3-CaSi2-TbSi2,
and (iii) tr6-CaSi2-TbSi2. The plotting mode is analogous to Fig. 4.9 (yellow→ Tb, blue→ Si).
K-paths: (a) M–Γ–K|Γ–A; (b) L–A–H. The two Si atoms of the almost flat honeycomb in tr6-
CaSi2-TbSi2 are suppressed. The bulk bands are reset by the Fermi level. The band structures
of (c) T4-h-B and (d) T4-t-A are plotted in an analogous manner, considering only Tb, Si1 and
Si2 (summed PAW projections→ line width, Tb/Si portion→ line colour). The slab bands are
reset by the VBM of the bulk-Si bands and red horizontal lines indicate the Fermi levels.
84
E − E0  (eV) E − E0  (eV)
5.2 Electronic properties
qualitatively match. Also the O-shaped double band of Tb-d and Si-p𝑧 character is
clearly visible. However, the band widths of hex-AlB2-TbSi2 differ considerably from
those of T4-h-B. The Si valence bands range over ≈8 eV (too small by ≈2 eV) and the
O-shaped double band ranges over ≈7 eV (too large by ≈2 eV). The L–A–H bands of
tr3-CaSi2-TbSi2 (Fig. 5.6 (b,ii)) resemble the surface bands of T4-h-B much closer. Not
only the band widths match, but also the shapes of the U and K bands. In particular,
the latter show a gap at ≈−5 eV due to hybridisation with the lower O band, which
missing in the bands of hex-AlB2-TbSi2. The upper O band is almost isolated, apart
from gap closings at the Γ-point and along ΓK. If these were relaxed, the upper O band
would resemble that in T4-t-A quite closely, including the furrow at the Γ-point. The
band structure of T4-h-B is derivable from the bulk bands without changing the band
order. Interestingly, also the electron pocket at the M-point is present, an important
feature missing in the bands of hex-AlB2-TbSi2.
Concerning the band structure of tr6-CaSi2-TbSi2, it has to be taken into account that
the cell consists of two Tb layers, so one unit cell would be a quasi-double slice. Con-
sequently, the surface band structure of a one-Tb-layer slice can be found in the
middle of the Brillouin zone, not at the base. The focus is thus on the M–Γ–K path
(Fig. 5.6 (a,iii)) instead of the L–A–H path. The bands are difficult to read because the
Tb atoms hybridise with two types of Si honeycombs: a buckled one in egg-box po-
sition and a rather flat one in hole position. Although the plotting mode suppresses
the contributions of the flat Si honeycomb, the respective Tb-Si hybrid states are still
visible as yellow lines. If these are ignored, i.e. only blue and greenish bands are
considered, several features become manifest: the U and K bands of the Si atoms
having a gap and the correct shape, the O-shaped double band whose upper branch
is deformed and dips into the Si valence bands, and the electron pocket at the M-
point. However, the projected bands of tr6-CaSi2-TbSi2 do not agree better with the
monolayers than the bands of tr3-CaSi2-TbSi2 do.
5.2.2 Charge transfer
The charge transfer in the T4-h-B structure is calculated analogously to pp. 68f and
shown in Fig. 5.7. Panel (a) shows the averages over Si(111) planes, where red por-
tions indicate electron gain and blue portions electron loss. Panel (b) shows tomo-
graphies of across three (01̄1)-planes: the cuts through Si1, Si2 and the Tb atom.
Fig. 5.7 (a) reveals that the plane-averaged charge transfer within the monolayer is
much smaller than that within the substrate upon formation of the covalent bonds.
The overall transfer into the covering Si honeycomb amounts to 0.01 electrons per
surface unit cell, if the boundary is set to the middle plane between the Tb layer and
the Si2 layer. The tomographies give insight into the three-dimensional redistribu-
tion. Most of the contrast is due to the expected charge accumulation in the Si–Si
intra-bonds. The Si atoms adjacent to the Tb atom (Si2 and Si3) show a pronounced
charge bulb at their dangling p𝑧 orbitals, whereas the charge above Si1 is almost like
that of the atomic state, as indicated by the faint colour. This leads to the conclusion
that the Tb atom transfers charge only to the adjacent Si atoms, whereas all other Si
85
5 Monolayer films on silicon(111)
(a) (b) Si1-cut Si2-cut Tb-cut
Si1
Si2
Tb
Si3
Si4
Si
Si56
Si7
Si8
Figure 5.7: Charge transfer in T4-h-B. Red→ electron gain; Blue→ electron loss. (a) average
over (111)-planes; (b) tomographies across (01̄1)-planes (parallel to a face of the surface unit
cell in Fig. 5.1 (d), which is titled away from the plane of projection by 30◦). Red ellipses mark
the charge bulbs of the Si atoms near the surface.
atoms do not accept additional charge. Since the charge bulbs of Si2 and Si3 clearly
penetrate the Tb layer, the plane average suffers from false charge assignments, ana-
logous to the problems discussed for the bulk TbSi2 – x structures (pp. 68f). A more
sophisticated method would be worth applying on this system in order to correctly
quantity the charge transfer.
5.2.3 Fermi surfaces
The analysis of the electronic properties concludes with the Fermi surfaces, which
correspond to the k-resolved DOS at the Fermi energy 𝐸F. They can thus be calculated
from the DFT eigenvalues at a dense s∑︁et of k-points with the following formula:𝑁Band
𝐷F(𝒌) = 𝛿(𝐸F − 𝜀𝑛(𝒌)) (5.45)
𝑛=1
The delta function is approximated as a Lorentzian with width of 𝛾 = 0.1 eV. The
Fermi surfaces of different structure models may be found in Fig. 5.8. Looking very
different from each other, they are a fingerprint for the respective structure. All
Fermi surfaces show the hole-pocket of the upper O band at the Γ-point and the elec-
tron pockets of the conduction band at the M-points. They have different shapes and
sizes, though, which is linked to the stability of the structures. The most stable T4-h-B
structure (Fig. 5.8 (a)) has the smallest pockets in accordance with the tendency to
reduce the DOS at the Fermi level.
Experimentally measured Fermi surfaces exist for the monolayer-Tb@Si(111) system
[52] as well as for the monolayer-Gd@Si(111) [47] and the monolayer-Dy@Si(111) [49]
systems. Only the T4-h-B structure reproduces the experimental reference almost
86
5.2 Electronic properties
(a) T4-h-B (b) H3-h-A (c) T4-t-A
K
1
0 Γ M
-1
(d) T4-h-A (e) H3-h-B (f) T4-t-B
1
0
-1
-1 0 1 -1 0 1 -1 0 1
k[112̄] (Å−1) k[112̄] (Å−1) k −1[112̄] (Å )
Figure 5.8: Fermi surfaces of the monolayer-Tb@Si(111) structures: (a) T4-h-B, (b) H3-h-A, (c)
T4-t-A, (d) T4-h-A, (e) H3-h-B, (f) T4-t-B. The axes point into the crystallographic directions
[112̄] (x-axis) and [1̄10] (y-axis). The Brillouin zone is marked white in (a). The colouring
corresponds to the colour bar on the right (a.u.).
(a) EF + 50 meV (b) EF (c) EF − 100 meV (d) EF − 200 meV
K
1
0 Γ M
-1
-1 0 1 -1 0 1 -1 0 1 -1 0 1
k −1[112̄] (Å ) k[112̄] (Å−1) k[112̄] (Å−1) k[112̄] (Å−1)
Figure 5.9: Energy surfaces of T4-h-B at different isoenergies: (a) 𝐸 = 𝐸F + 50 meV, (b) 𝐸 = 𝐸F,
(c) 𝐸 = 𝐸F − 100 meV and (d) 𝐸 = 𝐸F − 200 meV.
perfectly. In particular the experimental Fermi surface of the monolayer-Dy@Si(111)
system by Wanke et al. [49] looks like a copy of Fig. 5.8 (a), apart from the colouring.
Beside the Fermi surface, Wanke et al. measured also a series of energy surfaces at
different binding energies (0 meV, 100 meV and 200 meV). These measurements are
simulated by inserting corresponding isoenergies in Eq. (5.45). The result (Fig. 5.9)
shows that the electron pockets shrink to faint lines while the hole pocket grows and
acuminates for increasing binding energies, exactly as in the respective images of
Wanke et al. The theoretical energy surface at −50 meV binding energy (Fig. 5.9 (a))
worsens the agreement with the experiment. In detail, the ratio between the length
of the electron pockets and the width of the hole pocket is greater than 1 in Fig. 5.9
(a), whereas it is nearly 1 in Fig. 5.9 (b) and in [49]. This leads to the conclusion that
the Fermi levels of the simulated slab and the experiment are at the same position.
87
k  (Å−1[1̄10] )
k −1 −1[1̄10] (Å ) k[1̄10] (Å )
0 max
0 max
5 Monolayer films on silicon(111)
5.3 Discussion and summary
By means of a systematic inspection of the configuration space of reasonable struc-
ture models, the established T4-h-B model was found to be the most stable. Further-
more, all its peculiarities were explained. The inverse buckling direction of the cov-
ering Si honeycomb is due to a steric effect from the Tb-d orbitals, which clearly
favour an octahedral Si coordination irrespectively of caps (>100 meV per surface
unit cell). The preference of the Tb atoms to occupy the egg-carton positions of the
substrate termination (T4) and the hole positions of the cover (h-type) is based on two
countercurrent effects. On the one hand, the egg-carton position enhances the bond-
ing between the Tb layer and the respective Si honeycomb, which lowers the Fermi
level. On the other hand, this weakens the intra-bonding of the Si honeycomb. In the
case of the substrate termination, the second effect is compensated by the substrate
in that the weakened in-plane states hybridise with the bulk. On the contrary, the va-
cuum above the cover negates such a compensation, so the t-type structures cannot
profit from the Fermi level drop.
The band structures revealed that the Si atoms primarily bind to each other and the
Tb atoms bind via their d-orbitals to the dangling bonds the Si honeycombs. This
is remarkably analogous to the bulk TbSi2 – x structures and accords with the ZKB
concept. Moreover, the ZKB concept predicts a perfect charge balance in the mono-
layer structure as all valence electrons of the trivalent Tb atoms are compensated by
the incomplete shells of Si1, Si2 and Si3. The closed-shell configuration has already
been noted by Stauffer et al. [39]. However, the strength of the ZKB concept in this
case is that it explains in a comprehensible way why the monolayer lacks vacan-
cies, contrarily to bulk TbSi2. The Si3 plane sharply separates the electronic states
into two entities: All states below Si3 (including the in-plane sp3-hybrid bonds of Si3)
are three-dimensional substrate states and all states above (including the vertical
dangling bond of Si3) are vertically confined to the thin RESi2 monolayer. Therefore,
the monolayer is a clearly two-dimensional (semi-)metal. The hole pocket at the Γ-
point of the Fermi surface is due to hybrid states of out-of-plane Tb-d and Si3-p𝑧 The
electron pockets at the M-points are due to hybrid states of out-of-plane Tb-d, Si1-p𝑧
and Si2-p𝑧. The band structures and the Fermi surfaces very well reproduce experi-
mental ARPES images.
The structure analysis showed that the TbSi2 monolayer is much more related with
the CaSi2 phases than with the AlB2 phase concerning several morphological prop-
erties. Not only does the monolayer contain buckled honeycomb-Si instead of flat
honeycomb-Si, but also the coordination polyhedrons are very similar to those of the
CaSi2 phases. The lattice matching underlines the relatedness as the bases of the CaSi2
structures match the Si(111) surface very well, while that of hex-AlB2-TbSi2 is con-
siderably too large. In accordance with the structural relatedness, the surface band
structure of T4-h-B can be derived surprisingly well from the bulk band structures
of CaSi2-RESi2 by means of a simple particle-in-a-box consideration. The respective
projection of the hex-AlB2-TbSi2 band structure disagrees with the former. The struc-
tural and electronic properties agree very well with experimental data. Therefore,
the approach can be considered safe for the nanowire-Tb@Si(557) system, which is
88
5.3 Discussion and summary
the matter of the next chapter. Though, before treating these, the argument of strain-
induced growth will be critically revised in the following section.
5.3.1 Strain induced growth?
Many previous works argued on the mechanisms which determine the structures
of the monolayer- and multilayer-RE@Si(111) systems. The present findings that the
monolayer-Tb@Si(111) structure resembles the CaSi2-TbSi2 structures much closer
than the AlB2-TbSi2 structure is a novel point, which casts a different light on the
previous, heuristic explanations. In order to appraise their validity, the main points
of these explanations are summarised, in particular those stating that a strain release
of the Si honeycomb leads to the morphological details.
Epitaxists often consider the lattice matching between the substrate and the layer to
be grown so as to determine whether epitaxy is possible. If the lattice constants of
the respective planes match, the layer can reach a high thickness without structural
defects, and, otherwise, defect-free growth is limited. For the RE@Si(111) systems,
the following points are taken into account:
1) The surface lattice constant of the unreconstructed, trigonal Si(111) surface is
𝑎Lit.Si(111) = 3.841 Å [17, p. 1065].
2) The basis of the AlB2 phase of RESi2 – b (RE ∈ [Y,Gd − Lu]) matches the Si(111)
surface by the lattice constants, e.g. TbSi2 – b: 𝑎Lit.AlB -RESi = 3.847 Å [33].2 2 – b
3) Hypothetical flat honeycomb-Si would have a lattice constant of 4.074 Å if it ad-
opted the Si–Si bond distance of diamond-Si.
From 1) and 2), it follows that epitaxial growth of thick hex-AlB2-RESi2 – b layers on
Si(111) is possible.
While the growth of thick RESi2 – b layers can be embedded into simple geometrical
considerations, major misunderstanding concerns the formation of the monolayer
and thin-layer structures. Not few publications claim that the monolayer-RE@Si(111)
system is a one-unit-cell slice of the bulk hex-AlB2-RESi2 – b structure, which is at-
tached to the Si(111) surface. Since the covering Si honeycomb lacks vacancies, they
further argue that it buckles so as to release compressive strain exerted by the Si(111)
surface, following 1) and 3). More venturesome explications concern the hex-AlB2-
RESi2 – b structure, which, inserted between the monolayer and the substrate, leads
to multilayer-RE@Si(111) structures. Its Si layers supposedly contain vacant Si sites
because they are squeezed into the surface lattice constant. Instead of buckling, they
release strain by expelling every sixth Si atom so that the remaining Si atoms are al-
lowed to increase their bond distances. These argumentations, to read e.g. in [38, 39]
and from then inherited by subsequent publications, are problematic since they pos-
tulate that the AlB2 prototype is an “ideal” structure and that buckling and vacancies
are reconstructions or defects due to lattice mismatch. However, following this work
so far, it is already clear that both the buckling and the vacancies are of chemical
89
5 Monolayer films on silicon(111)
nature and that mechanical strain is not the reason for their occurrence. The ma-
jor mistake originates in the ignorance that AlB2-RESi2 incorporating flat, vacancy-
free honeycomb-Si is per se highly unstable. In order to rectify the physical picture
about the RE@Si(111) systems, the findings on the monolayer-RE@Si(111) and the
bulk RESi2 – x structures are brought together into the following conclusive theory.
The trivalent RE atoms in the monolayer-RE@Si(111) system each formally donate
one electron to the dangling bonds of the substrate termination and the other two
to a buckled, covering Si honeycomb. This structure is very convenient because,
firstly, the Si shells are all closed and, secondly, the buckled Si honeycomb is similar to
those composing the substrate. The reason for the buckling (sp3 hybridisation) of the
cover consists in the nature of Si to strongly avoid sp2 hybridisation and in a better
charge storage. This becomes manifest also in the high stability of CaSi2-RESi2 against
AlB2-RESi2 (constrained RESi2 stoichiometry). The multilayer structures can be de-
rived from the monolayer in an inductive manner. As further investigated in [53],
the (0001)-slices of bulk hex-AlB2-RESi2 – b, comprising one RE layer and a vacancy-
populated Si honeycomb, represent bulk building elements. Inserting these between
the substrate and the lowermost RE layer of the precedent structure increases the
thickness of the RESi2 – x film by one layer. The charge balance shows that the elec-
trons donated by one RE layer are exactly compensated by the substrate and the
vacancy-free, buckled covering Si honeycomb. Those of the other RE layers are com-
pensated by the other, vacancy-populated, flat Si honeycombs to an extent of 8/9.
Since the lateral lattice constants of the bulk building element and the Si(111) sub-
strate match, the stacking procedure can be repeated several times without running
the risk of strain-induced stacking faults. The reason for the presence of Si vacan-
cies is the same as that discussed for the bulk RESi2 – x structures, i.e. 1/6 vacancies
increase the electron capacity of the Si honeycomb from 2 to 5/3 per RE atom.
In summary, the monolayer- and multiplayer-RE@Si(111) systems can be general-
ised to a stack of building elements. The top building element (RE layer + vacancy-
free, buckled Si honeycomb) is always present and equals the monolayer if being
the only one. The bulk building elements (RE layer + vacancy-populated, flat Si hon-
eycomb) are facultative and increase the thickness of the RESi2 – x film, eventually
rendering it bulk-like. The morphologies of both building elements are solely due to
chemistry and independently match the Si(111) surface. Consequently, the statement
that surface-induced strain causes the buckling of the covering Si honeycomb or the
presence of Si vacancies is incorrect.
90
6 Nanowires on silicon(557)
After the detailed analysis of the monolayer-Tb@Si(111) system, the next step consists
in the investigation of the nanowire-Tb@Si(557) system. Derived from the Si(111) sur-
face by inclination towards the (001) plane, Si(ℎℎ𝑘) surfaces where ℎ < 𝑘 provide
Si(111)-like terraces which are separated by single Si(001)-like steps. The terrace
width is controlled by the inclination angle in that smaller angles leads to wider
terraces. If a Si(ℎℎ𝑘) substrate is prepared with RE at a coverage of approximately
one monolayer and successively annealed at a certain temperature, the monolayer-
RESi2 structure grows on the terraces, but is interrupted by the steps. In this manner,
strips of monolayer-RESi2 form by self-organisation and resemble nanowires from a
structural point of view. Experimental investigations of the structural and electronic
properties of nanowire-RE@Si(ℎℎ𝑘) systems can be found for Tb [70], Dy [68, 69] and
Er [68]. One of the key findings is that the electronic structure of densely covered
terraces is similar to that of the monolayer-RE@Si(111) system. In particular, it is of
two-dimensional character, in spite of the stripy morphology of the film.
This chapter covers the theoretical part of a joint-experimental-theoretical work on
the nanowire-Tb@Si(ℎℎ𝑘) system which goes a step further [P8]. It investigates how
the electronic dimensionality of the system alters with the Tb coverage, subjected to
the idea that lower coverages thin out the occupied terraces and separate the TbSi2
strips by semiconducting barriers. While the above stepped film, the system of dense
nanowires, is a two-dimensional electronic system, sparsely occupied terraces, the
sparse nanowires, are expected to reduce the dimensionality of the individual strips
so that they resemble a quasi-one-dimensional electronic system. Indeed, it turns
out that separating the TbSi2 strips by empty terraces induces anisotropy into the ef-
fective masses at the Fermi level, in accordance with the Heisenberg principle. In
addition, a new state arises at the edge of TbSi2 strips neighbouring an empty ter-
race. This edge state is of pure one-dimensional character and completely decoupled
from the rest of the strip. The corresponding features in ARPES images are more
pronounced, the narrower and thinner the nanowires are, as achieved by higher tilt
angles and/or lower Tb coverages. Since the agreement between the experimental
and theoretical findings is excellent, [P8] is a firm documentation of a fascinating
and unique dimensional crossover.
In addition to the results already published in [P8], this chapter provides a deeper
insight into the methodology and addresses the stability of the nanowire structures
under certain thermodynamic conditions. Furthermore, it is proven that the band
structures and Fermi surfaces of the nanowires can be derived from those of the
monolayer-Tb@Si(111) system by means of a simple particle-in-a-box model. This
highlights the differences between dense and sparse nanowires and illuminates the
origin of the one-dimensional edge state.
91
6 Nanowires on silicon(557)
6.1 Structure optimisation
Methodological details
The Si(557) substrate is modelled by tilted (5×1) (sketched in Fig. 6.1 (a)) and doubled
(10× 1) surface supercells of the Si(111) slab (Fig. 5.1 (a)). The slabs provide terraces
having a widths of five projected surface lattice constants (“5a”), which are separated
by Si(001)-like single steps. Similarly to the Si(111) slab, the Si(557) slab is twelve Si
layers thick, out of which the lowest four are fixed in position to simulate the bulk
of the substrate. The dangling bonds on the bottom side are saturated by H atoms,
which assume their equilibrium positions for a symmetrically saturated slab (exactly
that depicted in Fig. 6.1 (a)) and are then fixed in position for all further calculations.
The periodic replica in the (557)-direction (the z-direction) are separated by 27 Å of
vacuum, which reduces to 24 Å when the slab bears the nanowires.
Most computational parameters are equal to those of the monolayer-Tb@Si(111) cal-
culations (p. 73). The Brillouin-zone sampling is adapted to the (5× 1) supercell by a
Γ-centred (3 × 15 × 1) Monkhorst-Pack mesh so that 15 k-points are placed along the
wires and 3 k-points across. For the calculations of the (10 × 1) surfaces, the mesh
is adapted to (2 × 20 × 1). A convergence test proves that increasing the sampling
density alters the total energy by ≈1 meV per atom.
The dense nanowires
At first, the dense nanowires are analysed. For this purpose, the (5×1) slab is covered
with monolayer-TbSi2 so that each terrace bears a 5𝑎wide strip. Since it is a priori not
clear that the strips adopt the T4-h-B structure as the monolayer does, two structure
models are set up: the one derived from T4-h-B (Fig. 6.1 (b)) and the other derived
from H3-h-A (Fig. 6.1 (c)). They are named after the corresponding monolayer struc-
ture, prefixed by the width, “5a” in this case. The H chemical potential is recalibrated
with the symmetrically H-saturated slab according to Eq. (5.44).
The Tb chemical potential is calculated with Eq. (5.43) and determines the stability of
the structure during the annealing stage, i.e. when the Tb coverage and the temper-
ature are constant. The 5a-T4-h-B structure yields 𝜇Tb = −327 meV and the 5a-H3-h-A
structure 𝜇Tb = −53 meV, so the former is more stable than the latter. The 𝜇Tb dif-
ference between the nanowire structure models (274 meV) is a bit higher than that
between the respective monolayer structure models (232 meV). Concerning the geo-
metric details of the optimised structure models, the mid of the TbSi2 strip ranging
from Tb2 to Tb4, resembles the respective monolayer structure (Tab. 5.1). In particu-
lar, the Si–Si bond lengths of the Si honeycombs adjacent to the Tb layer are similar
to that of bulk Si and stretched if the Tb atom is in the egg-box position. Also the bond
distances between the Tb atoms and the coordinating Si atoms accord with those of
the respective monolayer structures, amounting to ⟨𝑑Tb{2,3,4}–Si⟩ ≈ 3.0 Å on average.
In contrast, the edges of the TbSi2 strips near Tb1 and Tb5 are considerably distorted.
The covering Si honeycomb at the left edge near Tb5 is flattened and, interestingly,
it adopts bond angles similar to the orthorhombically distorted Si honeycomb of the
92
6.1 Structure optimisation
(a) Si(557) slab (d) 5a-T4-h-B (Sparse NW)
[0 ⊙ [11‾0]01] [771‾0‾]
[112‾] 5 4 3 2 1
[771‾0‾]
⊙ [557]
fixed in position (e)
(b) 5a-T4-h-B (Dense NW) 40
D
5 4 20 en3 s2 e1  NW Sparse NW
0
-20 Monol
(c) 5a-H -h-A -40
aye
3 r
-60
5 4 3 2 1 -80 AlB2-TbSi2−x
-2.5 -2.0 -1.5 -1.0
μ  − μ hcp-TbTb Tb  (eV)
Figure 6.1: (a) Sketch of the Si(557) slab and (b) – (d) the optimised TbSi2 nanowire struc-
tures. Blue circles are Si atoms; Yellow circles are Tb atoms; Red circles are H atoms. The
light red polygon in (a) encircles the subset of atoms fixed in position during the relaxations.
Red lines mark the unit cells. Orange M-stars in (b) – (d) indicate mutually threefold coordin-
ated Si atoms near Tb. The green boxes and orange circles in (b) and (c) highlight the con-
nection between the covering Si honeycomb and the step. The orange circle in (d) marks
the stabilising Si atom. (e) shows the grand-potential-chemical-potential phase diagram of
the 5a-T4-h-B nanowire@Si(557) systems (solid, red → dense, orange → sparse). The T4-h-B
monolayer@Si(111) system is inserted as well (dashed, red) as it will be used for explaining
the details of the growth mechanism (vide infra). The dotted blue line sketches a hypothetical
phase with lower Tb density. All 𝛾(𝜇Tb) are reset to that of the clean Si(111) slab.
stoichiometric bulk ort-AlB2-TbSi2 structure, i.e. one angle is smaller and the other
two are larger than 120◦ (Tab. 4.3). In both structure models, the Tb5 edge is pinched
in that the coordinating Tb5–Si bond distances are shorter than those in the mid of the
strip (⟨𝑑Tb5–Si⟩ = 2.9 Å). In the 5a-H3-h-A structure, the other edge near Tb1 is pinched
as well (⟨𝑑Tb1–Si⟩ = 2.9 Å), while in the 5a-T4-h-B structure, it is similar to the mid of
the strip (⟨𝑑Tb1–Si⟩ = 3.0 Å). Whether the Tb1 edge is pinched or not is clearly linked
to how the covering Si honeycomb is connected to the step (green boxes in Fig. 6.1
93
[557]
[111]
γ − γ c-Si(111) (meV/Å2) [11‾0]
6 Nanowires on silicon(557)
(b) and (c)). In the 5a-T4-h-B structure, the Tb atom lies in the same (11̄0)-plane as
the Si(001)-like Si bridge of the step. Hence, the first Si atom of the covering Si honey-
comb lies in the adjacent (11̄0)-plane and can bind to the twofold coordinated tip of
the step (orange circle) in such a way that the tetrahedron of the latter completes. In
particular, the beginning of the cover and the tip of the step remain distinct Si atoms.
On the other hand, in the 5a-H3-h-A structure, the Tb and Si planes of the cover are
interchanged. Therefore, the tip of the step has to be the first atom of the covering
Si honeycomb at the same time, so the Tb1 edge is pinched. The different Tb1 edges
are the reason why the 𝜇Tb difference between the 5a-H3-h-A and the 5a-T4-h-B struc-
ture models is even larger than that between the respective monolayer structures.
In other words, the geometry of the stepped Si(557) surface per se favours the T4
position over the H3 position.
The balance of the valence electrons is facilitated by orange M-stars marking mutu-
ally threefold coordinated Si atoms in the structure models in Fig. 6.1. In 5a-T4-h-B,
all 15 valence electrons from the 5 Tb atoms are absorbed by 15 Si acceptors, so all
Si shells are closed. Although the same holds for 5a-H3-h-A, there is an imbalance as
the cover has 11 acceptors and the substrate termination 4, differently from the 2:1
ratio in the monolayer structures and the 5a-T4-h-B nanowires.
The sparse nanowires
The next step consists in the investigation of the sparse nanowires, which are mod-
elled on the doubled (10 × 1) slab providing two terraces per unit cell. Every second
terrace is occupied by a nanowire strip, while the others are saturated by H atoms so
as to remove artificial, metallic bands from the unreconstructed Si(111) planes. The
following paragraphs deal only with nanowires of the 5a-T4-h-B structure.
After emptying a terrace, a dangling bond remains at the covering Si honeycomb of
the nanowire on the right. Such a structure would be unstable against hcp-Tb se-
gregation (𝜇Tb = +100 meV) if not treated with an appropriate saturation. Therefore,
a stabilising Si atom is added t the structure so that the cover can smoothly link to
the Si bridge of the adjacent surface step. This lowers the Tb chemical potential to
−135 meV. The relaxed structure model of the sparse nanowires is depicted in Fig. 6.1
(d), where the stabilising Si atom is marked by an orange circle. The averaged co-
ordinating Tb–Si bond distances are equal to those in the dense nanowires for Tb1
to Tb4 (⟨𝑑Tb{1,2,3,4}–Si⟩ ≈ 3.0 Å). The coordination polyhedron of Tb5 has an average
distance of 3.0 Å as well, so it is more voluminous than that of the dense nanowires.
Hence, the Tb5-edge pinching of the dense nanowires is absent in the sparse nano-
wires, which is advantageous. The balance of the valence electrons shows that the
sparse nanowires have 5 Si acceptors in the substrate termination, similarly to the
dense ones. However, the cover has 12 Si acceptors and thus two in excess.
The phase diagram
Obviously, given a fixed number of Tb atoms, the dense arrangement of nanowires
is more advantageous than the sparse one as the Tb chemical potential of the former
94
6.1 Structure optimisation
is smaller than that of the latter (−327 meV vs. −135 meV). However, the thermo-
dynamic boundary condition of a constant Tb coverage is an idealised situation in
which the mobility of the Tb atoms on the substrate is infinitely high during the an-
nealing stage. Therefore, also the converse situation is considered, in which the Tb
atoms are in equilibrium with an infinite reservoir which fixes 𝜇Tb. This is described
by a thermodynamical potential Ω(𝑉,𝑇, 𝜇•) which results from the Legendre trans-
formation of the free energy with respect to the nu∑︁mber of atoms of all species •.
Ω(𝑉,𝑇, 𝜇•) = 𝐹 (𝑉,𝑇, 𝑁•) − 𝜇•𝑁• (6.46)•
Ω(𝑉,𝑇, 𝜇•) is also referred to as the grand potential. Dividing by𝑉 makes this quant-
ity intensive and returns the mechanical tension 𝑝 according to Euler’s theorem.
𝑝(𝑇, 𝜇•)
1
= Ω(𝑉,𝑇, 𝜇•) (6.47)
𝑉
The potential has to be adapted to the reduced dimensionality of surfaces by repla-
cing the volume𝑉 with the surface area 𝐴 and changing the symbol 𝑝 to 𝛾. Inserting
the bulk Si chemical potential and neglecting the lattice dynamics returns the follow-
ing ready-to-use f(orm:
( ) ≈ 1
)
𝛾 𝑇, 𝜇 𝐸cell − 𝜇d-Si• DFT Si 𝑁Si −
bot top top𝜇H 𝑁
bot c-Si(111)
H − 𝜇H 𝑁H − 𝜇Tb𝑁Tb − 𝛾 (6.48)𝐴
The discrimination between the H atoms of the bottom and the top side of the slab is
necessary because they play a different role. Setting 𝜇bot to 𝜇H-satH H (vide supra) guaran-
tees the cancellation of any effects from the bottom side. The H atoms of the top side,
only present in the sparse nanowires, have a more physical meaning as they interact
with the TbSi2 strips. If their chemical potential was set to 𝜇H-satH , i.e. if the H avail-
ability was so high that the empty Si terraces completely saturate, the slab carrying
the sparse nanowires would be highly preferable due to the H-saturated terraces, ir-
respectively of the (in-)stability of the TbSi2 strips. Though, this situation contradicts
the experiment because much effort is made to remove as much H from the vacuum
chamber as possible. Thus, the H atoms of the top side deserve a separate H chem-
ical potential top𝜇H . As they sit solely on the vertical dangling bonds of the Si(111)-like
terraces, top𝜇H is gauged in such a way that the top side of the H-saturated Si(557) slab
is in chemical equilibrium with the clean Si(111) surface. This approximates that the
availability of H is so small that the empty terraces remain clean.
Of course, the grand potential 𝛾 has to be referred to a reference system, which
is chosen to be the clean Si(111) surface (𝛾c-Si(111)) or, equivalently, the unsaturated
Si(557) surface, ascertained by the gauge of top𝜇H . Since the Tb chemical potential 𝜇Tb
is unknown, it has to be treated as a variable. 𝛾(𝜇Tb) then becomes a linear func-
tion, which can be plotted into phase diagrams. In doing so, the abscissa 𝜇Tb tunes
the experimental situation: higher and lower values for 𝜇Tb correspond to Tb-rich
and Tb-poor conditions, respectively. The slope of 𝛾(𝜇Tb) equals the areal Tb dens-
ity 𝑁Tb/𝐴. Consequently, phases with high Tb densities have steeper lines and are
principally favoured for Tb-rich conditions, and vice versa. By means of the phase
95
6 Nanowires on silicon(557)
diagrams, structure models with different Tb densities become comparable. The sta-
bility at a certain 𝜇Tb given by those which minimise 𝛾(𝜇 1Tb) .
Fig. 6.1 (e) shows the phase diagram of the sparse nanowires and the dense nano-
wires on Si(557) and the T4-h-B monolayer on Si(111), denoted simply monolayer in
the following paragraphs. The domain of 𝛾(𝜇Tb) is confined to Tb chemical poten-
tials smaller than that of hcp-Tb since higher values would precipitate elemental Tb.
The abscissa is reset by hcp-Tb𝜇Tb so that the domain is 𝜇Tb < 0. Concentrating on the
solid lines (red→ dense NW, orange→ sparse NW and black→ clean Si(111) / clean
Si(557)), there are stability regions for each phase, which are shaded in the respect-
ive colour. Very small values for 𝜇Tb make the Si substrate remain clean in the sense
that neither the sparse nor the dense nanowires form. Above ≈−2.0 eV, the sparse
nanowires are stable, superseded by the dense nanowires at ≈−1.5 eV. Of course,
the phase diagram is incomplete. There are probably other phases which are more
stable on certain regions, e.g. sparse nanowires in which the TbSi2 strips are sep-
arated by more than one empty terrace. An exemplary pathway of such a phase is
sketched in the phase diagram as a dotted blue line. Although not checked by cal-
culation, these phases would match the expectable trend that lower 𝜇Tb results in
sparser nanowires. In the other limit, there are nanowires of heights corresponding
to more than one monolayer which become stable at high 𝜇Tb. For very high 𝜇Tb, a
thick film assuming the hex-AlB2-TbSi2 – b structure would cover the substrate (vide
supra, Sec. 5.3.1), which manifests itself as a vertical line at 𝜇 = 𝜇hex-AlB2-TbSi2 – bTb Tb in
the phase diagram. A green arrow marks the position in Fig. 6.1 (e), employing the
value of the approximated bulk structure (−1.827 eV, Tab. 4.6). At first glance, this
would imply that dense nanowires would not occur at all. However, the thick film is
kinetically hindered as the Tb deposition is limited by experimental controlling and
because the Tb atoms are not infinitely mobile. That the silicide aggregation is locally
and temporally limited effectively constrains the height of the structures, one layer
in this case.
The phase diagram moreover reveals that the monolayer on Si(111) is considerably
more stable than the dense nanowires on Si(557). In terms of a constant Tb amount,
the difference between the Tb chemical potentials of the monolayer and the dense
nanowires amounts to: −806 meV + 327 meV = −479 meV. The nanowire formation
is thus solely due to presence of the steps of the Si(ℎℎ𝑘) surface, which hinder the
formation of a smooth monolayer. A second consequence of the high stability of the
monolayer with respect to the dense nanowires is that surfaces with wider terraces
and hence wider nanowires should be more stable than those with narrow terraces.
For a fixed tilt angle, a substrate can widen its terraces by step bunching, i.e. the
1The main difference between the grand potential in Eq. (6.48) and the Tb chemical potential in Eq.
(5.43) consists in the change of the thermodynamic boundary conditions from the constancy of the
Tb amount 𝑁Tb and the surface tension 𝛾 to the constancy of the Tb chemical potential 𝜇Tb and
the surface area 𝐴. In experimental terms, this corresponds to switching from the annealing stage,
where an infinite substrate with a fixed Tb coverage is held at constant temperature, to the de-
position stage, where a substrate of fixed area is exposed to vaporised Tb, the chemical potential
of which is controlled by the pressure of the Tb vapour. According to the Gibbs-Duhem relation, in-
creasing the pressure at constant temperature generally increases the chemical potential of a single-
species thermodynamic subsystem.
96
6.2 Electronic properties
formation of double or triple steps to the benefit of the terrace widths. The formation
of nanowires which are wider than the terraces of the single-stepped surface (>18 Å
for Si(557)) is indeed observed in the experiment, in particular, if nanowire samples
are “cleaned” and reused [P8, 70].
6.2 Electronic properties
6.2.1 Band structures
The band structures are calculated analogously to Sec. 5.2. Since the nanowires are
embedded in tilted supercells of the hexagonal (1 × 1)-Si(111) surface unit cell, the
supercell Brillouin zones (sBZs) of the nanowires are folded versions of the prmitive
Brillouin zone (pBZ) of the monolayer. In order to make the band structures compar-
able, the Γ-point, K-points and M-points of the pBZ are identified in the sBZs. How-
ever, it is impossible to find an exact mapping because the Si(557) and Si(111) sur-
faces are tilted against each other by 9.4◦ and the steps misalign the (557)-projected
(1 × 1) building elements of the terraces. Therefore, the M- and K-points of the pBZ
are approximately addressed by their Cartesian coordinates in the sBZs, although
they are not high-symmetry points in the latter. The band structures are calculated
along two k-paths: ΓK parallel to the nanowires (x-direction) and ΓM perpendicular
to the nanowires (y-direction). The points where the paths cross the boundaries of
the sBZs are labelled X and Y, respectively. Approximate sketches of the sBZs may be
found in Fig. 6.2 (c,iv). Accurate DOS calculations are carried out on dense Γ-centred
(40 × 8 × 1) and (40 × 4 × 1) meshes for the dense and sparse nanowires, respect-
ively. Fig. 6.2 shows the band structures of (a) the dense nanowires, (b) the sparse
nanowires and (c) the monolayer. The line width corresponds to the portions of (i)
all Tb atoms, (ii) all Si atoms of the cover (Si1 and Si2 in the monolayer), (iii) all top
Si atoms of the substrate termination (Si3 in the monolayer) and (iv) bulk-Si in the
PAW projections. The line colour is fixed in panels (ii) – (iv). In panel (i), it indicates
the Tb5 portion in the PAW projections of all Tb atoms according to the colour bar.
The energies are reset by the respective bulk VBMs 𝐸0. The Fermi level of the slab is
inserted as a red horizontal line.
Before the analysis can start, the band folding between Si(111) and Si(557) has to
be elaborated in more detail since the Si(557) supercells and the (1 × 1)-Si(111) unit
cell are incommensurable. It proves to be helpful to consider the unit cell of the
dense/sparse nanowires a (6 × 1)/(12 × 1) supercell of the (1 × 1) unit cell of the ter-
races. The increment of the symbolic notations (5× 1)/(10× 1) stems from the steps,
which approximately add a 1𝑎 advance to the 5𝑎 wide terraces. The new reconstruc-
tion symbols suggest that the ΓM path contains the ΓY path six and twelve times,
respectively. The band structures in Fig. 6.2 (a) and (b) confirm this by the periodic
features along ΓM having period lengths of 1/3 and 1/6, respectively. Conversely,
the substrate bands along the Si(557)-ΓY path are a superposition of respectively six
and twelve differently shifted images of the bands along the Si(111)-ΓM path. As a
consequence, the substrate bands along ΓM in Fig. 6.2 (a) and (b) are horizontally
97
6 Nanowires on silicon(557)
(i) Tb (ii) Si1 + Si2 (iii) Si3 (iv) Sibulk
(a) 1
4 5
3
0 2
1
-1
Dense NW
-2
M YΓ X K M YΓ X K M YΓ X K M YΓ X K
(b) 1
4 5
3
2
0
1
-1
Sparse NW
-2
M YΓ X K M YΓ X K M YΓ X K M YΓ X K
(c) 1 Tb5 portion
20 % 50 %
0 M
Y Γ
-1 K
X
Monolayer
-2 y
M Γ K M Γ K M Γ K x
Figure 6.2: Band structures of (a) dense and (b) sparse nanowires on Si(557) and (c) the mono-
layer on Si(111) (all based on the T4-h-B structure). The line width corresponds to the PAW
projections of (i) the Tb atoms, (ii) the Si atoms of the cover, (iii) the top Si atoms of the sub-
strate termination, and (iv) bulk-Si. All projections are normalised by the number of atoms
of the respective group. The line colour in (i) indicates the portion of Tb5 in all Tb projections
according to the colour bar: yellow→ 20 %, green→ 50 %. All energies are reset by the VBM
of the bulk-Si bands. The red horizontal lines indicate the Fermi levels of the slabs. (c,iv)
Brillouin zones: black→ML on (1× 1)-Si(111), red→ dense NW on (5× 1)-Si(557), orange→
sparse NW on (10 × 1)-Si(557).
blurred, particularly in the case of the dense nanowires. On the contrary, the perpen-
dicular Si(557)-ΓK direction shows projections of bands along different, equidistant
lines parallel to Si(111)-ΓK. After reduction by time-reversal symmetry, the multipli-
city is 6/2+1 = 4 for the dense and 12/2+1 = 7 for the sparse nanowires. As a result,
the substrate bands along ΓK in Fig. 6.2 (a) and (b) are vertically diffuse and, thus,
still distinguishable, in contrast to those along ΓM.
98
E − E0  (eV) E − E0  (eV) E − E0  (eV)
6.2 Electronic properties
The bands related to the nanowires, of which the Tb bands are the most important,
look different from the substrate bands. For both the dense and the sparse nano-
wires, they include five prominent, thick Tb bands, enumerated in Fig. 6.2 (a,i) and
(b,i). Along ΓK (parallel to the nanowires) they point downwards with a steepness
similar to that of the upper O band of the monolayer. Near the Γ-point and the K-point,
they bend oppositely, so they resemble each an inverted S. Along ΓM (perpendicular
to the nanowires) each Tb band is confined into a narrow energy range. Therein, the
dense and the sparse nanowires show an important difference: While the Tb bands
of the former are still wavy with strong dispersion, those of the latter are strictly flat
and, hence, completely localised in that direction. All thick Tb bands are hybridised
with the covering Si honeycomb on the entire Brillouin zone and with the substrate
termination near the Γ-point, which is remarkably similar to the upper O band of the
monolayer. While the projections of the four lowest thick Tb bands are distributed
over all Tb atoms, the topmost band 5 majorly belongs to Tb5 at the open edge of the
TbSi2 strips. Beside this band, there are further Tb5-associated bands at higher bind-
ing energies. They are thinner than band 5 and complemented by PAW projections
of the adjacent Si atoms, thus indicating strong Tb–Si hybridisation.
Relation to the monolayer Tb@Si(111) system
The analogy between the Tb bands of the nanowires and the upper O band of the
monolayer suggests that the former may possibly be related to the latter by projec-
tion. However, ordinary folding considerations like those above hold only for deloc-
alised states, e.g. the substrate bands. For the bands related to the nanowires, it has
to be taken into account that the TbSi2 strips extend to five projected (1 × 1) surface
lattice constants per terrace, while the sixth is the step. Therefore, it is likely that the
nanowire states originate in the monolayer states by (partial) localisation perpen-
dicular to the growth direction, i.e. by mixing countercurrent monolayer states with
wave vectors 𝒌 and −𝒌 to form standing waves. In order to test this, the hexagonal
Brillouin zone of the monolayer is segmented by equidistant parallel translations of
the ΓK path. Beginning at the x-axis, the path is shifted in the y-direction until it
reaches the edge of the Brillouin zone defined by KMK. As the path crosses the zone
y 1 2 3 4 5 6 7 8 9 10 11
x
1.0
0.5
0.0
-0.5
-1.0
Γ K M M K
Figure 6.3: Segmentation of the Brillouin zone of the monolayer. The band structure is plotted
along eleven equidistant paths parallel to ΓK, the sketches of which are shown above the
plots (red lines). The upper O band and the electron pocket, both associated with Tb, are
highlighted thick and yellow. All energies are reset to the VBM of bulk-Si. The red horizontal
lines indicate the Fermi level.
99
E − E0 (eV)
6 Nanowires on silicon(557)
boundary, it reappears in the lower half of the Brillouin zone after zone reduction.
The bands along 11 segments are plotted in Fig. 6.3 and the Brillouin zones with the
k-paths may be found above the respective panels. The upper O band and the con-
duction band forming the M-point electron pocket are highlighted thick and yellow.
From ΓK to MK+, the upper O band turns into an S shape similar to the Tb bands of
the nanowires. The band width decreases. If the path approaches an M-point, the
electron pocket appears (segments 6 and 11 in Fig. 6.3).
Sparse Dense Project. Fig. 6.4 shows an overlay of the equidistant seg-
NW NW ML
1 ments 1, 3, 5, 7 and 9 (right panel) and compares it
with the ΓK part of the band structures of the dense
nanowires (middle panel) and the sparse nanowires
0 (left panel), taken from Fig. 6.2 (a,i) and (b,i). The
right and middle panels look very similar to each
other. The lowest four Tb bands of the dense nano-
-1
Γ K Γ K Γ K wires follow the segments 3, 5, 7 and 9 of the upper O
band almost congruently, merely differing by smal-
Figure 6.4: Band structures of the ler shifts. Segment 1, the topmost band (coloured
nanowires and projected bands of green in Fig. 6.4), undergoes more shifting and re-
the monolayer. hybridisation to become the Tb5 band of the dense
nanowires. Hence, the band structure of the dense nanowires is related to that of
the monolayer by a simple zone projection in large part. It is astonishing that the
discontinuities from the steps only slightly affect the intercoupling between adja-
cent Tb strains, even at the edges of the TbSi2 strips. The close electronic related-
ness between the dense nanowires and the monolayer proves that the former are a
two-dimensional electronic system, in spite of the stripy structure.
The situation is different for the sparse nanowires (left panel). The bands related
to the inner Tb atoms are shifted to higher binding energies compared to the dense
nanowires and the monolayer projection, but they retain their morphology. Thus,
they are still subjected to the physics of the monolayer, concerning the intercoup-
ling of the Tb strains. However, the total localisation perpendicular to the nanowires
(vide supra) shows that these states are genuine standing waves in that direction.
Consequently, the TbSi2 strip can be considered a quantum well derived from the
TbSi2 monolayer. In contrast to the inner states, the green Tb5 band rehybridises so
strongly that it completely loses the comparability with segment 1 of the monolayer.
In other words, it represents a new state which cannot be derived from the mono-
layer any more. Because of its localisation at Tb5, it can be considered an edge state
and, thus, resembles a one-dimensional system within the sparse nanowires.
6.2.2 Fermi surfaces
The hypotheses on the dimensionality of the sparse and the dense nanowires are
tested by means of the Fermi surfaces, which are calculated analogously to Sec. 5.2.3.
Fig. 6.5 shows the Fermi surfaces of the monolayer and the nanowires. The Fermi
level of the sparse nanowires is aligned to that of the dense nanowires (+0.1 eV).
100
E − E0 (eV)
6.2 Electronic properties
The Fermi surface of the dense nanowires has stripy features, which are clearly
linked to a folding procedure from the hexagonal pBZ into the smaller sBZ, marked
white in Fig. 6.5 (a) and (b). Following the light blue dashed lines, the electron pock-
ets at the M-points generate the thin, bright stripe at the Γ-point as well as the broad,
weak stripes at the X-point. The thin, weak lines accompanying the middle bright
stripe (indicated by a light green arrow) stem from the points of the star-shaped hole
pocket, as clarified by the light green dashed line. These lines are the Fermi break-
throughs of band 5 (green arrow in Fig. 6.2 (a,i)). The relationship between them
and the hole pocket agrees with the above observation that band 5 originates from
the upper O band of the monolayer. Since all features are wavy or corrugated in the
(vicinal) [112̄] direction (perpendicular to the nanowires), the Fermi surface of the
dense nanowires has a two-dimensional character. At first glance, the Fermi surface
of the sparse nanowires looks similar to that of the dense nanowires. The major dif-
ference consists in a lack of corrugation, so all stripes are straight. Also the lines
related to band 5 straighten (green arrow), so they are completely separated from
the other stripes. Hence, the these states gain a purely one-dimensional character,
confirming the results from the band structure analysis.
In Fermi surface measurements, the electrons expelled from the sample carry a par-
allel momentum equal to the crystal momentum of the initial state. However, the
latter does not accord with the multiply reduced zone scheme of the supercells, but
with the Brillouin zone of the local periodicity of the primitive building elements, the
(1 × 1)-Si(111) terraces in the case. The Fermi surface in the reduced zone scheme
of any supercell can be unfolded into the extended zone scheme by weighting the
k-resolved DOS (Eq. (5.45)) with the Fourier components of the wave functions and
integrating it over the sBZ. In a second step, the unfolded Fermi surface can be folded
back into the pBZ [124]. In detail, the unfolded Fermi surface in terms of the primitive
Brillouin zone pBZ𝐷F (𝒌) holds:
𝑁∑︁Band ∫
pBZ
𝐷F (𝒌) = ∑︁d𝐾∑︁𝑊𝑛𝒌𝑲 𝛿(𝐸F − 𝜀𝑛(𝑲))𝑛=1 sBZ
𝑲 ∈ sBZ, 𝒌 ∈ pBZ, 𝑊 = 𝛿(𝑲 +𝑮 − (𝒌 + 𝒈))  𝑢 (𝑲)2 (6.49)𝑛𝒌𝑲 𝑛𝑮
𝒈 𝑮
where 𝑲 and 𝒌 are the k-vectors of the sBZ and pBZ, respectively, 𝑮 and 𝒈 are the
direct lattice vectors of the supercell and the primitive unit cell, respectively, and
𝑢𝑛𝑮 (𝑲) are the Bloch factors of the supercell (Eq. (2.27)). In practice, this procedure
works only if the super lattice vectors are linear combinations of the primitive ones
with integral coefficients. Therefore, the above Fermi surface of the dense nanowires
cannot be unfolded into the (1× 1) Brillouin zone since the terraces advance by half
a period in the parallel direction when passing a step. This issue could be resolved
by calculating the dense nanowires in a doubled (10×1) supercell, which is not done
in this work, though.
The sparse nanowires, on the other hand, are already modelled in a (10 × 1) super-
cell. Their unfolded Fermi surface is plotted in Fig. 6.5 (c). The unfolding procedure
recovers the M-point electron pockets from the thin stripe though the Γ-point and the
101
6 Nanowires on silicon(557)
(a) Monolayer (b) Dense nanowires
1.5
K (111) → (557)
1.0 + X
folding
0.5
M
0.0 Γ YΓ
-0.5
-1.0
-1.5
(c) Sparse NW (unfolded) (d) Sparse nanowires
1.5
K
1.0 X
0.5
M
0.0 Γ Γ Y
-0.5
-1.0
unfold
-1.5
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
k −1 −1[112̄] (Å ) k[112̄] (Å )
(e) Edge state
Figure 6.5: Fermi surfaces of the Tb@Si(ℎℎ𝑘) systems: (a) monolayer, (b) dense nanowires,
(c) and (d) sparse nanowires (all based on T4-h-B). The axes point into the directions [112̄]
(parallel to the NW, vicinal for NW) and [1̄10] (perpendicular to the NW). (c) is derived from
(d) by unfolding the sBZ to the local hexagonal pBZ of the Si(111) terraces. The Brillouin zones
are marked white. The colouring corresponds to the colour bar (arbitrary units, the ranges
are comparable for (b), (c) and (d), but not linked to that of (a)). Ellipses highlight the electron
pockets at the M-point. Green arrows indicate the new edge state. Dashed lines are guides
for the eyes. (e) Real-space distribution of the edge state. Reproduced from [P8].
broad stripes trough the X-points. However, the pocket shapes differ from those of
the monolayer: The two horizontal pockets have a considerably stretched aspect ra-
tio, which indicates higher effective masses perpendicular to the nanowires. Also the
electron pockets at the other four M-points are stretched and effectively rotate away
from the growth direction of the nanowires. Both effects indicate that the electronic
mobility perpendicular to the nanowires decreases. In contrast to the electron pock-
ets, the hole pocket is not recovered. Instead, the state linked to band 5 embraces the
102
k −1 −1[1̄10] (Å ) k[1̄10] (Å )
0 max
dense → sparse
6.3 Discussion and summary
region around Γ as two straight lines (green arrow). This confirms the above find-
ings that the band-5 state has undergone so many changes that the relationship with
the Tb states of the monolayer is lost. In order to clarify that this state is indeed an
edge state, as inferred from the PAW projections, the corresponding real-space par-
tial charge at the Fermi level is plotted as isosurfaces in Fig. 6.5 (e). Confirming all
expectations, this state is strongly confined to a narrow channel near Tb5.
6.3 Discussion and summary
The nanowire-Tb@Si(557) surface provided the opportunity to explore a fascinating
lower-dimensional quantum system. The terraces of the substrate are indeed a tem-
plate for a stripped TbSi2 film which resembles nanowires from a structural point
of view. The strips follow the stability relations of the monolayer-Tb@Si(111) film in
that they locally adopt the T4-h-B structure. Only the edges of the strips are subjected
to distortions. The geometric relations between the Si(001)-like steps and the Si(111)-
like terraces was shown to be very convenient for the T4-h-B structure since the TbSi2
strip can connect with the broken sp3 bonds of the step. This may be the reason why
these types of TbSi2 nanowires grow only on Si(ℎℎ𝑘) surfaces where ℎ < 𝑘 [70]. The
phase diagram simulating the deposition stage proved that the density of terraces
bearing a TbSi2 strip is smaller/higher for Tb-poor/Tb-rich conditions. It furthermore
hints that step-bunching leading to wider TbSi2 strips is favourable. As far as visible
from experimental STM images [P8], the narrow 5𝑎 wide TbSi2 strips do occur in
Tb@Si(557), but the surface is dominated by wider strips. TbSi2 nanowires on step-
bunched surfaces are certainly worth investigating in future work.
The dense nanowires are a two-dimensional electronic system, irrespectively of the
discontinuities from the steps. Their band structures and Fermi surfaces can be de-
rived from the monolayer by a simple zone projection. The electronic relatedness
between the dense nanowires and the monolayer agrees with experimental ARPES
and Fermi surface measurements [P8]. The sparse nanowires differ very much from
the dense nanowires regarding the electronic properties. The states associated with
the inner Tb atoms become genuine standing waves according to a simple particle-in-
a-box model. In other words, the isolated TbSi2 strips act as a quantum wells which
laterally trap the electrons. In doing so, the coupling within the strips remains unaf-
fected. The effective masses of the electron pockets agree with the Heisenberg prin-
ciple: The lateral confinement of the positions broadens the lateral crystal momenta
in the Fermi surface. Very similar effects were also observed in the experiment [P8].
In contrast, the edge of each isolated strip gives rise to an edge state which cannot be
derived from the monolayer states. This state is laterally confined to a narrow region
around the outermost Tb atom and has a clear one-dimensional character, as proven
by the unfolded Fermi surface. The edge state was also observed in the measure-
ments on the nanowire-Tb@Si(335) system [P8], whose terrace widths correspond
to the Si(557) substrate of this work (step bunching). In conclusion, the Tb@Si(557)
surface permits the observation of a fascinating dimensional crossover from a two-
dimensional film to a system of quasi-one-dimensional nanowires.
103

Part II: Thin antimony layers on bismuth
selenide
The discovery of topological order in solid-state physics was the door opener for
a new class of materials whose state underlies the mathematical concept of topo-
logy. Topologically non-trivial materials are band insulators with intrinsic properties
which are linked to the topological nature of the electronic wavefunctions. These spe-
cial properties show an exceptionally high stability against many perturbations since
topological non-triviality is much stronger than a plain symmetry protection.
The research of topology in materials science began in the 1980s when von Klitzing
et al. discovered the integer Hall effect (IHE) in the two-dimensional electron gas of
MOSFETs [125]. Their work was the first documentation of topological order in a
solid-state system. Because of the topological non-triviality, the IHE is insensitive to
deviations from the idealised assumption of a homogeneous, two-dimensional elec-
tron gas. Hence, it even works in the presence of impurities, the finiteness of the
sample and electron-electron interaction [126]. Topologically ordered solids soon at-
tracted the attention of the condensed-matter community, who had a great deal of
work published on different systems. In doing so, several types of topological order
were discovered, which are based on different topological invariants. The research
was so groundbreaking that is was awarded with three Nobel Prizes [W4]:
• Klaus von Klitzing “for the discovery of the quantized Hall effect” (1985)
• Robert B. Laughlin, Horst L. Störmer and Daniel C. Tsui “for their discovery of
a new form of quantum fluid with fractionally charged excitations” (1998)
• David J. Thouless, F. Duncan M. Haldane and J. Michael Kosterlitz “for theoret-
ical discoveries of topological phase transitions and topological phases of mat-
ter” (2016)
As von Klitzing pointed out in his Nobel Prize Lecture, nobody expected at that time
that semiconductor physics could produce any fundamental discoveries:
Semiconductor research and the Nobel Prize in physics seem to be contra-
dictory, since one may come to the conclusion that a complicated system
like a semiconductor is not useful for very fundamental discoveries. In-
deed, most of the experimental data in solid-state physics are analyzed on
the basis of simplified theories, and very often the properties of a semicon-
ductor device are described by empirical formulas since the microscopic
details are too complicated. Up to 1980 nobody expected that there ex-
ists an effect like the quantized Hall effect, which depends exclusively on
105
Thin antimony layers on bismuth selenide
fundamental constants and is not affected by irregularities in the semicon-
ductor like impurities. [126]
The topological class of materials is linked to certain quantities, the topological in-
variants. These quantities are fundamental properties of topological spaces and in-
dicate whether they can be transformed into each other or not. In detail, if two topo-
logical spaces have different topological invariants, they are not homeomorphic to
each other, i.e. they cannot be mapped onto each other in a continuous and bijective
manner. The reverse statement does not necessarily hold.
The simple example of Euler’s theorem of polyhedrons provides an easy approach
towards the matter of topology. As taught in many geometry lessons, it is well known
that all convex polyhedrons obey the rule that the number of vertices minus the
number of edges plus the number of faces equals 2. This holds, e.g., for the cube
(8−12+6 = 2), the tetrahedron (4−6+4 = 2) and the octahedron (6−12+8 = 2). The
theorem can even be generalised to closed surfaces which are not necessarily con-
vex or polyhedral, e.g. the smooth 𝕊2 sphere. In this case, a triangulation produces
countable vertices, edges and faces. The alternating sum of these is the Euler charac-
teristic, which defines a topological invariant of the surface. It is thus independent of
the details of the triangulation because different triangulations are homeomorphic
to each other. Since Euler’s theorem for polyhedrons is a special case of the Euler
characteristic, it is obvious that also the sphere has an Euler characteristic of 2 as it
can be mapped onto a cube, a tetrahedron or an octahedron.
The ring-shaped 𝕋2 torus is another example for a closed surface which clarifies the
usefulness of the Euler characteristic. A possible triangulation for this surface con-
sists in a right-angled, closed arc, similar to the shape of l’Arche de la Défense, Paris,
France2. Comprising 16 vertices, 32 edges and 16 faces, it yields an Euler character-
istic of 0, differently from the sphere. Thus, the torus and the sphere are so funda-
mentally different from each other that a homeomorphism between them does not
exist. In this case, the hole of the torus is the central property which discriminates the
topological classes. A consequence of the different topological classes which is indeed
relevant for materials scientists is the existence of a global map for the torus, while
a global map for the sphere does not exist3. This is the topological reason for why
a 2D crystal can be mapped onto a square (more generally a parallelogram), despite
being a closed torus due to the periodicity in two directions. The same holds for the
Brillouin zone, whose toroidal nature will be relevant at several points in Chap. 7.
In the case of solids, topological invariants can be calculated from the electronic
wavefunctions, provided that they can be separated into well-defined occupied and
unoccupied manifolds, i.e. the material of interest is a band insulator. Those insulat-
ors whose topological invariants are non-trivial are called topologically non-trivial
insulators or simply topological insulators (TIs). In terms of solid-state physics, the
2La Grande Arche de La Défense, https://fr.wikipedia.org/wiki/Arche_de_la_D%C3%A9fense#/media/
Fichier:Grande_Arche_F%C3%A9vrier_2020.jpeg (visited on 24/03/2022).
3For instance, the Earth has no single, global map. Instead, it has to be described by a set of local
maps which partially overlap: an atlas. More precisely, at least two local maps are necessary in
order to find any geographic point on at least one of the two maps. For example, the UNO logo lacks
Antarctica for this reason.
106
Thin antimony layers on bismuth selenide
above theorem of conserved topological invariants tells that the special properties of
TIs are retained as long as the band gap does not close. For this reason, the topological
order of a TI is insensitive to imperfections like disorder or defects.
The practical usefulness of TIs does not necessarily consist in the intrinsic proper-
ties. If insulators of different topological classes border each other, in particular if
a TI is connected to a topologically trivial insulator (conventional insulator, CI), the
band gap closes so as to allow the bands to reorder. In other words, there are metallic
edge states somewhere at the interface. Since the vacuum is also a CI, the surface of
every TI bears topologically protected surface states (TSSs), which are indestructible
by any perturbation, provided that the topological classes of the involved bulk ma-
terials are conserved. Of course, metallic surface states which exists regardless of the
morphological details are very attractive for surface scientists.
This part of the thesis is dedicated to the Sb@Bi2Se3 system, a van der Waals (vdW)
heterostructure consisting of thin layers of antimony (𝛽-Sb) which are adsorbed on
bismuth selenide substrates (Bi2Se3). This heterostructure is a particularly interest-
ing system because, firstly, the 𝛽-Sb adlayer adsorbs to the Bi2Se3 substrate by means
of vdW interactions and, secondly, the 𝛽-Sb adlayer is a CI, while bulk Bi2Se3 is a TI,
more precisely a ℤ2 topological insulator. Of course, TSSs emerge as the non-trivial
class of the substrate has to transition into the trivial class somewhere. However, to-
pology does not specify any details about the transition, e.g. the location of the TSSs.
A priori, the TSSs can be located anywhere, from being buried in the substrate to
rising up to the surface of the Sb film. That Bi2Se3 and 𝛽-Sb are vdW materials does
not simplify this issue as it is unclear how the adlayer and the substrate mutually
affect their electronic properties.
The aim of this part is to analyse the nature of the TSSs of the Sb@Bi2Se3 hetero-
structure. Since the interplay between the topological classes of the Bi2Se3 substrate,
the 𝛽-Sb adlayer and the vacuum is complex, elaborating the topological invariants
is inevitable for analysing the formation of the TSSs and for understanding topolo-
gical phase transitions. Therefore, Chap. 7 introduces the matter of geometric phases,
which finally leads to the definition of the ℤ2 invariant for time-reversal-symmetric
solids. Chap. 8 follows as a preparatory chapter which treats the constituents of the
heterostructure Bi2Se3 and 𝛽-Sb separately. The investigations serve two purposes:
testing the approach, which employs methods beyond DFT, in particular vdW and
SOC, and calculating the band structures and the ℤ2 invariants. The actual hetero-
structure is the matter of Chap. 9. At first, a stable structure model is derived, where
the focus is on the inversions of the Sb adlayer. After that the electronic properties
are calculated for the most stable structure models. The chapter concludes with the
analysis of the complex series of topological phase transitions, which occurs upon a
simulated adsorption process of the Sb adlayer.
107
7 Geometric phases
The concept of geometric phases is unfamiliar to many scientists, but fundamentally
important for understanding modern methods in condensed-matter theory. It is the
mathematical basis of the topological invariants in solid-state systems. The aim of
this chapter is thus to introduce the concept of geometric phases and then to derive
theℤ2 invariant step by step. The Aharonov-Bohm effect marks the beginning as it is
a manifestation of geometric phases, granting a more or less easy access to the matter.
After that, the geometric phases along with the respective geometric connections and
geometric potentials are formally defined, according to the approach of Berry [127].
The modern theory of polarisation according to the review of Resta [128] follows as
it is the basis of topological transport. This defines the IHE, which is introduced in
a generalised manner. Based on the IHE, the chapter concludes with the related ℤ2
topological invariant in two and three dimensions, the objective of this chapter.
7.1 The Aharonov-Bohm effect
According to the principle of minimal coupling (Sec. A.2), a gauge transformation of
the electromagnetic fields introduces a phase in the wavefunctions |Ψ⟩ of charged
particles. Consequently, the absolute phase of a quantum state is not a measurable
quantity since gauge transformations must not change the physical observations. On
the other hand, phase differences can be measured, e.g. in interference experiments.
This raises the question how the observability of phase differences accords with the
non-observability of gauge transformations. A phenomenon which is taught in every
quantum-mechanics class, but whose interpretation is still the matter of vivid discus-
sions, is the Aharonov-Bohm effect (ABE) [129–134]. After first hints by Franz [135] in
1939 and a more complete description by Ehrenberg and Siday [136] in 1949, Ahar-
onov and Bohm [129] developed a general gedankenexperiment in 1959 which clari-
fies the geometrical nature of gauge potentials.
The space-dependent, magnetic ABE is based on the magnetic vector potential and
can conceptually be realised by the model of an infinitely long solenoid with finite
radius 𝑅. The magnetic field may be constant inside the solenoid and point into the
z-direction. Outside, the magnetic field may vanish. In polar coordinates and in the
Coulomb g{auge, the vector potential 𝑨 holds:
𝐵𝜌 {
2 e𝜑 , 𝜌 < 𝑅( ) ⇒ ∇ × 𝐵e , 𝜌 < 𝑅𝑨 𝒙 = 𝐵𝑅2 𝑨 = 𝑩 =
𝑧 ∧ ∇ · 𝑨 = 0 (7.50)
2𝜌 e𝜑 , 𝜌 > 𝑅 0 , 𝜌 > 𝑅
108
7.1 The Aharonov-Bohm effect
Consider a particle with charge 𝑞 in the hollow-cylindrical field-free region Ω = {𝒙 |
𝜌 > 𝑅}. It may be in the quantum state |Ψ⟩ when the solenoid is switched off. Switch-
ing the solenoid on transforms |Ψ⟩ by a phase shift according to the principle of min-
imal coupling in combination with ∇ × 𝑨 = 0. This phase is proportional to the line
integral of 𝑨 〉along a〉𝐶1 curve 𝛾 ⊂ Ωs BΩ〉\ {𝒙 | 𝜑〉= 𝜑s ∈ [0, 2〉π)}. ( )  〉𝐻 (𝒑, 𝒙) Ψ = 𝐸 Ψ → 𝐻 (𝒑 − 𝑨 𝑞𝑞 , 𝒙) Ψ̃ = 𝐸Ψ̃ ⇒ Ψ̃ = exp i ℏ 𝜒(𝒙) Ψ
(7.51a)
with
∫𝛾 : [0, 1] → Ωs, 𝛾(0) = 𝒙0, 𝛾(1) = 𝒙
𝜒(𝒙) = 𝑨 · d𝛾 = 1 22𝐵𝑅 (𝜑 − 𝜑0) ⇔ 𝑨(𝒙) = ∇𝑥𝜒(𝒙) (7.51b)
𝛾
where 𝜑 and 𝜑0 denote the polar angles of 𝒙 and 𝒙0, respectively. The constraint on
Ω for integrating 𝑨 is necessary for the validity of Eq. (7.51b), as will be explained
below. In other words, 𝛾 must stay in a sector around the solenoid.
The phases in Eq. (7.51a) can be observed as an interference between two particle
beams running clockwise and anticlockwise around the solenoid. Both beams ori-
ginate in 𝒙0 ∈ Ω, the source, and meet again in 𝒙1 ∈ Ω, where the interference
measurement is performed. The points divide Ω into two parts Ω+ and Ω− in which
the paths of the beams are denoted 𝛾+ ⊂ Ω+ and 𝛾− ⊂ Ω− (Fig. 7.1). The superposed
probability amp lit〉ude ho+〉lds: 〉Ψ̃ = Ψ̃ + Ψ̃− [= exp(i 𝑞 +  +〉 𝜒 ) Ψ + e〉xp(i 𝑞 𝜒〉−]) 〉ℏ ℏ Ψ−
= exp(i 𝑞ℏ 𝜒
−) [exp(i 𝑞ℏ (𝜒+ − 𝜒〉−))Ψ+ 〉+] Ψ− (7.52)∫ = exp∮(i
𝑞
𝜒−) exp(i 𝑞∬𝜒 ) Ψ+ + Ψ−ℏ ℏ 𝛾with ∬
· d · d Stokes𝜒 = 𝑨 𝛾 = 𝑨 𝛾 = (∇ × 𝑨) · d𝑺 = 𝑩 · d𝑺 = 𝐵π𝑅2𝛾 = Φ0
𝛾+−𝛾− 𝛾 𝑆 𝑆
where Φ0 is the total magnetic flux through the area 𝑆 encircled by 𝛾+ and 𝛾− .
Eq. (7.52) leads to an astonishing conclusion: A magnetic field can influence the
phases of particles which are strictly prevented from crossing it. The phase differ-
ence 𝑞ℏ 𝜒𝛾 depends neither on the point 𝒙1 where the measurement is performed, nor
on the details of the paths, but it is proportional to the total magnetic flux through
the solenoid. The indirect effect of a magnetic field on charged particles via its vector
potential is the central result of Aharonov and Bohm [129]. As the field-free interac-
tion sounds spooky at first glance, the ABE causes arguments about its interpretation
until today [131–133]. The probably most important source of misunderstanding is
the seeming observability of the gauge transformation in Eq. (7.51b).
The misunderstandings arise if one infers the integrability of 𝑨 from its vanishing
curl ∇× 𝑨= 0. Since it would then have a scalar potential, all loop integrals of 𝑨
would be zero. Moreover, 𝑨 could be “gauged away” to 𝑨≡ 0, which would negate any
109
7 Geometric phases
action. However, curl-freeness is not sufficient for the integrability of a vector field,
but the domain has to be simply connected in addition [137, p. 264]. As long as simply
connected sectors of Ω, e.g. Ω+ or Ω− , are considered, 𝑨 is integrable and has a scalar
potential 𝜒, as explicitly shown in Eq. (7.51b). As a consequence, particles moving in
the same simply connected region do not show the ABE and the phase in Eq. (7.51a)
is indeed an unobservable gauge. This changes if the line integral runs around the
solenoid, as effectively realised by the two countercurrent beams. Since the solenoid
pierces a cylindrical hole into the field-free region, Ω is multiply connected and a
scalar potential for 𝑨 does not exist, despite ∇× 𝑨 = 0 1. Consequently, line integrals
in general depend on the integration curve and loop integrals can yield non-zero
values. Moreover, 𝑨 cannot be gauged away any more, but it encodes the magnetic
field inside the solenoid in any gauge. In mathematical terms: the equivalence class
[𝑨] defined by 𝑨 ∼ 𝑨 + ∇𝜒 is non-zero. While the phases in (7.51a) depend on the
gauge, the ABE phase in (7.52) is gauge invariant since additional gradient fields ∇𝜒
vanish in loop integrals. Therefore, it is a physical observable and can be measured
in principle.
The ABE is a fascinating illustration of the role of
Ω− x1 Ω+ fields and potentials in physics. Although it is not
obvious in this case, the ABE phase shift can be
explained solely by the magnetic field 𝑩, as well
as solely by its vector potential 𝑨, whose equi-
γ+ valence class is bijectively connected to 𝑩. Thus,
γ− the popular statement that 𝑨 contains “more in-R formation” than 𝑩 is not meaningful, as well as
the question whether the one or the other is more
important in physics. The ABE is a pure quantum
effect, for which a classical analogon does not ex-
x0 ist. For this reason, imagining a quantum particle
moving in a force field leads to confusions about
Figure 7.1: Sketch of the ABE setting.
+ − the ABE. Aharonov and Bohm pointed out that the𝛾 and 𝛾 connect 𝒙0 with 𝒙1 anti-
clockwise and clockwise around the idea of local force fields like 𝑩 determining the
solenoid (grey). physics of a quantum system applies nowhere in
quantum physics. Instead, the quantum state is
determined by a Hamilton operator which is constructed by locally acting potentials
(local in position and momentum operators) [130]. Non-local effects naturally arise
since the differential equations are solved for the entire domain of the operator. An
example for the absurdity of force fields acting locally on quantum particles is the
simple double-slit experiment: It is very confusing to imagine that a particle “going”
through the one slit “feels” whether the other slit is covered. Hence, the non-locality
of wavefunctions and the importance of path possibilities, as emerging in the ABE,
are not really exotic phenomena in quantum mechanics.
1One could have the idea to consider 𝜒(𝒓) = 𝜑 𝐵𝑅2/2 a scalar potential. Its gradient indeed yields the
correct vector field 𝑨(𝒓) of Eq. (7.50). However, the domain of 𝜒 is a simply connected sector of Ω,
e.g. the area {𝒓 | 𝜌 > 𝑅, 𝜑 ∈ [0, 2π)}. It is impossible to define 𝜒 continuously on Ω. More precisely,
at least two local definitions of 𝜒 are necessary in order to globally calculate 𝑨. One example for
these definitions are 𝜒+ and 𝜒− on the sectors Ω+ and Ω− above (with an appropriate overlap).
110
7.2 The Berry phase
Since the ABE phase depends only on the enclosed magnetic flux Φ0, and neither
on the point of measurement, nor on the details of the paths of the particle beams,
the ABE has already been observed in 1962: By means of the ABE, Möllenstedt and
Bayh determined the ratio ℎ/𝑒 with a precision of 14 % [138]. Many years followed,
in which experimental evidence for the ABE was still argued to be due to leaking
magnetic fields, until Tonomura et al. brought a convincing experimental proof in
1986 [139]. Obviously, the ABE phase is a fundamental property of the setup and
the wavefunction domains in that it is insensitive to continuous deformations of the
solenoid and of the beams. This phenomenon can be generalised by the concept of
geometrical phases, which is the matter of the next section.
7.2 The Berry phase
Geometric phases are also called Berry phases, as Berry introduced them in 1984 by
the following procedure [127].
Let 𝐻 be the Hamilton operator for a quantum system which depends parametric-
ally on a multidimensional parameter 𝝃 = (𝜉1, 𝜉 𝑁2, . . . , 𝜉𝑛) ∈ ℝ . The corresponding
stationary Schrödinger equation is then〉  〉
𝐻 (𝝃)𝑛(𝝃) = 𝐸 (𝝃)𝑛 𝑛(𝝃) . (7.53)
Furthermore, let 𝛾 : [0, 1] → 𝝃(𝑡) ⊂ ℝ𝑁 be a 𝐶1 curve which describes a continuous
parametric variation. It may be slow enough that the system stays in the eigenstate
|𝑛(𝝃)⟩ on the entire curve 𝛾, i.e. for every parameter 𝝃(𝑡), the state |𝑛(𝝃(𝑡))⟩ satis-
fies Eq. (7.53) and the eigenvalue 𝐸𝑛(𝝃) has the same index 𝑛. In other words, level
crossing during the parametric variation is explicitly forbidden.
The phase factor for |𝑛(𝝃(𝑡))⟩ at a given parameter 𝝃(𝑡) ≠ 𝝃(0) is not arbitrary like in
single-shot calculations, but it depends on the initial state |𝑛(𝝃(0))⟩. The connection
between the phases follows from considering 𝑡 as time and solving the Schrödinger
equation with〉 the initial condition |Ψ(0)⟩ B |𝑛(𝝃(0))⟩ and the following ansatz:Ψ(𝑡) = ︸︷1︷︸ e︸x    p    (  −     i   1ℏ    ∫   0 𝑡 ︷𝐸︷𝑛  (  𝝃    (   𝑡  ′  )   )         ′︸ ︸         〉d𝑡 ) exp(︷i𝜒︷𝑛   (  𝑡   )︸) 𝑛(𝝃(𝑡)) (7.54)
arb. phase dynamical phase geometric phase
The ansatz wavefunction |Ψ(𝑡)⟩ has three phase factors: the arbitrary phase (≡1), the
general dynamical phase from the temporal evolution, and a third term 𝜒𝑛(𝑡) which
bears all other phase shifts due to the parametric vari𝜕 ∫ 𝜕𝜒𝑛(𝑡)  ∫〉
ation.
𝜕𝝃(𝑡)  〉
iℏ |Ψ(𝑡)⟩ = 𝐻 (𝝃) |Ψ(𝑡)⟩ 〈 ⇒  ℏ 〉 𝑛(𝝃(𝑡)) = iℏ · ∇𝜉 𝑛(𝝃(𝑡))𝜕𝑡 𝜕𝑡 𝜕𝑡⇔ 𝜒𝑛 = i 𝑛(𝝃) ∇𝜉 𝑛(𝝃) · d𝝃 C A𝑛(𝝃) · d𝝃
𝛾 𝛾
111
7 Geometric phases
with 〈   〉
A𝑛(𝝃) = i 𝑛(𝝃)∇ 𝜉 𝑛(𝝃) (7.55)
As it is evident, the parametric variation does not only produce the normal dynamical
phase shift. There is an additional phase shift 𝜒𝑛 which originates in the parametric
variation itself: the geometric phase or Berry phase [127]. The integrand A𝑛(𝝃) is
the Berry connection, which describes how the phase of the eigenfunction |𝑛(𝝃)⟩ re-
sponds to infinitesimal variations of the parameter 𝝃.
If 𝛾 is a closed loop, the Berry phase assumes a cert∮ain value, which can be zero orfinite.
𝝃(1) = 𝝃(0) ⇒ 𝜒𝑛(1) = 𝜒B𝑛 = A𝑛(𝝃) · d𝝃 (7.56)
𝛾
Hence, a cyclic parametric variation of the Hamilton operator can shift the phases of
the eigenstates beside the usual dynamical phase shift. Since not depending on the
time lapse, the Berry phase arises e〉ven if the vaB riation〉 is instantaneous (Δ𝑡 = 0).𝑛final = exp(i𝜒𝑛) 𝑛initial
If the parameter space is three-dimensional, it is instructive to apply Stokes theorem
so as to obtain a surface integral: ∬
𝜒B𝑛 = B𝑛(𝝃) · d𝑺 (7.57)
𝑆
where 𝑆 is the area encircled by 𝛾 and
B𝑛(𝝃) = ∇𝜉 × A∑︁ 〈 〈 〉(𝝃) = i ∇𝜉𝑛(𝝃) |× |∇𝜉𝑛〉(𝝃) 〈 𝑛(𝝃) ∇ 𝐻 (𝝃) 𝑚(𝝃) × 𝑚(𝝃) ∇  〉
− 𝜉 𝜉
𝐻 (𝝃) 𝑛(𝝃) (7.58)
= Im ( ( ) − ( ))2 [127]≠ 𝐸𝑚 𝝃 𝐸𝑛 𝝃𝑚 𝑛
The way how the Berry connection (Eq. (7.55)) and the Berry potential (Eq. (7.57))
influence the phase of the parametrised system are remarkably analogue to the ac-
tion of a magnetic field and its vector potential on the phases of a charged quantum
particle in the ABE (Eq. (7.52)). Therefore, analogously to the ABE phase, the Berry
phase 𝜒𝑛(𝝃) is not observable for open curves as the absolute phase information is
gauge-dependent. Only if the system is compared with a reference system at the same
set of parameters, which effectively corresponds to a closed loop, or if the system is
directly subjected to a circular parametric variation, the Berry phase is, in principle,
observa〉ble. This holds becau〉se 𝜒
B
𝑛 is gauge invariant.
𝑛(𝝃) → exp(i𝜒(𝝃))𝑛(𝝃) ⇒ A𝑛(𝝃) → A𝑛(𝝃) − ∇𝜉𝜒(𝝃) ⇒ 𝜒B𝑛 → 𝜒B𝑛 + 0 (7.59)
In particular, the above choice of≡1 for the arbitrary phase has not affected the phys-
ical results. The proof that the ABE can indeed be formulated in terms of Berry phases
can be found in the appendix (Sec. C.1).
112
7.3 The electric polarisation in a crystal
The line integral of the Berry connection A𝑛(𝝃) is more handy for proving the con-
nection between physical quantities and for analytical problems. For the numeric
treatment, however, A𝑛(𝝃) is not easy to calculate because the derivatives in Eq.
(7.55) have to be replaced by finite differences with connected phase relations. In
order to avoid this problem, the surface integral of the Berry potential B𝑛(𝝃) can be
used as it depends on well-defined derivatives of 𝐻 (𝝃) [127].
7.3 The electric polarisation in a crystal
Geometric phases are applicable whenever the response of a quantum system to a
parametric variation is the quantity of interest. An example which actually neces-
sitates them for the correct description are null-field polarisation effects in a crystal,
e.g. ferroelectricity or piezoelectricity. A naive, but wrong formula for the macro-
scopic polarisation 𝑷 of a crystal would be the summation of all electrostatic dipole
moments created by the charge density 𝜌(𝒙∫) in the direct unit cell CR.
1
𝑷 = 𝑷ion + |C | d𝑥 𝒙𝜌(𝒙) (7.60)R CR
where 𝑷ion is the polarisation from the ions, |CR | the direct unit cell volume and 𝜌(𝒙)
the electronic charge density g∑︁iven by∫the mo〈dulus of t−𝑒 1  〉
h〈e Bloch fu〉nctions:
𝜌(𝒙) = |C | |C | d𝑘 Ψ𝑛(𝒌) 𝒙 𝒙 Ψ𝑛(𝒌)R 𝑛 G CG
where |CG | is the reciprocal unit cell volume. This definition works fine if all elec-
trons are localised within CR, e.g. in molecular crystals or ionic systems. In cova-
lent systems like semiconductors, however, the electrons are not confined to a spe-
cific cell, so Eq. (7.60) fails to correctly describe the macroscopic polarisation [140].
King-Smith and Vanderbilt (1993) and Resta (1994) wrote reviews on this, analysing
the quantum effects of electric polarisation and linking it to geometric phases of the
electronic wavefunctions [128, 141]. The key points of these reviews are retraced in
the following paragraphs as they are the basis of topological transport. From now
on, electrons in a crystal are considered. They are described by the Bloch Hamilto-
nian (vide supra, Eqs. (2.27)) and carry the negative elementary charge 𝑞 = −𝑒. Being
fermions, they occupy the states from the lowest band 𝑛 = 1 to the highest band 𝑛 = ?̄?
at zero temperature, making ?̄? electrons per unit cell. In the spin-degenerate case, the
bands are counted with respect to their algebraic duplicity. As will be proven later,
it is crucial that the system is insulating, i.e. the occupied and the unoccupied bands
are well separated by a band gap ∀𝒌 ∈ CG.
Eq. (7.60) fails because the phase information of the wavefunctions is omitted by
the modulus, so current terms between adjacent unit cells are missing. Beside the
mathematical reason, there is another, general point, why Eq. (7.60) cannot be well
defined: In real experiments, the bulk polarisation cannot be determined in a single
measurement. Instead, a series of measurements has to be conducted whereupon
113
7 Geometric phases
the polarisation changes can be deduced from the transported charge (current). The
same holds in solid-state theory since Berry phases are observable only as phase
differences. Resta begins his considerations with defining a charge redistribution
between two states (𝜆 = 0) and (𝜆 = 1) of a crystal, where 𝜆 represents the reac-
tion coordinate of the polarisation process. In the exemplary case of a ferroelectric,
𝜆 describes the transition from the unpolarised into the polarised state. In order to
resolve the delocalisation problem of the Bloch functions, he changes the basis to
the Wannier functions (vide supra, Eq. (2.29)). Since localised, the Wannier density
can safely be inserted into Eq. (7.60). As a result, the change in polarisation equals a
change in the centres of charge of the Wannier functions (𝜆 )𝑎𝑛 (𝒙) = ⟨𝒙 |
(𝜆 )
𝑎𝑛 (0)⟩, which
depend on the∑︁rea(ction coordinat)e 𝜆 via the Bloc∑︁h f∫unctions |
(𝜆 )
[ 𝑢𝑛 (𝒌)⟩.− ?̄? ?̄?𝑒 〈 〉 〈 〉 Eq. (2.31) −𝑒  ]
Δ𝑷 𝒙 (1) − 𝒙 (0) = d 𝒙  (1) 2  (0) B |C | 𝑛 ∫ 𝑛 |C |2 𝑥 𝑎 (𝒙)

𝑛 − 𝑎𝑛 (𝒙)2
R 𝑛=1 ∑︁ [〈  R  𝑛=1?̄?Eq. (2.32) −𝑒 ]= d (1)(𝒌) i∇ (1) 〉 〈 (0) 𝑘 𝑢 𝑢 (𝒌) − 𝑢 (𝒌)  i∇  (0) 〉(2π)3 𝑛 𝑘 𝑛 𝑛 𝑘 𝑢𝑛 (𝒌)
𝑛=1 CG
In order to extract the geometric phases, the expression for Δ𝑷 is evaluated by the
components with respect to the reciprocal basis {𝑮 𝑗}. For this purpose, let 𝑖 ∈ [1, 𝑁]
be a fixed index. The polarisation component Δ𝑃𝑖 along 𝑮𝑖 then holds:
Δ𝑃𝑖 B 𝑮𝑖 ·∑︁Δ𝑷?̄? ∫ [ ]−𝑒 〈 (1)  𝜕  (1) 〉 〈 (0)  𝜕  (0) 〉 (7.61)= |C | d𝑠 𝑢𝑛 (𝒔) i 𝑢𝑛 (𝒔) − 𝑢𝑛 (𝒔) i 𝑢 (𝒔)R [0,1]𝑁1 𝜕𝑠𝑖 𝜕𝑠 𝑛𝑛= 𝑖
where 𝒔 is the relative coordinate vector of the k-vector in the reciprocal unit cell2.
The integrand in Eq. (7.61) reminds of the Berry connection A (Eq. (7.55)). In order
to clarify the analogy, a two-dimensional parameter 𝝃 ∈ [0, 1]2 is introduced whose
first component is the reaction coordinate 𝜆 C 𝜉0. The second component is the
relative coordinate 𝑠𝑖 C 𝜉1 of 𝒌 along 𝑮𝑖 . The corresponding two-dimensional Berry
connection A 3𝑖∑︁then∏ho[l∫ds :?̄? 1 ] [〈 ]
A (𝝃) = d (
 
𝜆 ) 〉
𝑖 𝑠 𝑗 𝑢𝑛 (𝒔)  i∇  (𝜆 )𝜉 𝑢𝑛 (𝒔) with 𝝃 = (𝜆, 𝑠𝑖) (7.62)
𝑛=1 𝑗≠𝑖 0
2The reciprocal vector 𝒌 and its differential operator ∇𝑘 can be expressed in terms of 𝒔 ∈ [0, 1]𝑁 , the
relative coordinates of 𝒌∑︁in the recipr∑︁ocal b(asis {𝑮 𝑗}.𝑁 1 𝑁 ) 1 ∑︁𝑁 𝜕
𝒌 = 𝑮 𝑗𝑠 𝑗 = 𝑮 𝑗 𝑹2π 𝑗
· 𝒌 ⇒ ∇𝑘 = 𝑹2π 𝑗 𝜕𝑠
𝑗=1 𝑗=1 𝑗=1 𝑗
3The (𝑁−1)-dimensional integral in Eq. (7.62) traces all dimensions except 𝑖, so it is an ordinary state
summation like the 𝑛-sum over the bands. The 𝑖-th component of 𝒔 is excluded from the integration
and composes the variation parameter 𝝃 together with the reaction coordinate 𝜆.
114
7.3 The electric polarisation in a crystal
The line integral of A(𝝃) along the edge of its quadratic domain 𝛾 B 𝜕[0, 1]2 yields
the Berr∮y phase:
𝜒B𝑖 = ∫ A𝑖 (𝝃) · d𝝃𝛾 1 ∫ 1 ∫ 0 ∫ 0
= ︸       d     𝜆  ︷A︷0    (  𝜆    ,   0 ︸) +︸       d    𝑠    ︷A︷1    (  1    ,  𝑠   ︸) +︸       d     𝜆  ︷A︷0𝑖 𝑖 𝑖 𝑖 𝑖    (  𝜆    ,   1 ︸) +︸       d    𝑠   𝑖 ︷A︷1𝑖    (  0    ,  𝑠   𝑖︸) (7.63)0 0 1 1
I II III IV
where A 𝑗 (𝝃) denotes the 𝑗-th component of A𝑖 . The terms I and III of Eq. (7.63)𝑖
cancel each other because A𝑖 (𝜆, 0) = A𝑖 (𝜆, 1) ∀𝜆, reflecting the boundary conditions
of the Bloch factors (Eq. (2.28b)). The remaining terms II and IV reproduce the electric
polarisation in Eq. (7.61) up to a prefactor.
−𝑒 B −
∮
𝑒
Δ𝑃𝑖 = |C | 𝜒𝑖 = |C | A𝑖 (𝝃) · d𝝃 (7.64)R R 𝛾
Eq. (7.64) is the central result of [128]. It proves that the projection of the static electric
polarisation on a G-vector can indeed be considered a geometric phase of the para-
meter space spanned by the reaction coordinate and the k-path along that G-vector.
From Eq. (7.59), it is directly clear that Δ𝑃𝑖 is gauge-invariant under G-periodic𝑈 (1)
transformations, which preserve the band order. However, this restriction is a prob-
lem in solid-state crystals since the bands of the occupied subspace in general swap
indices when traversing the Brillouin zone or when progressing on the 𝜆-path. For
this reason, Resta generalised the proof for gauge invariance to more general 𝑈 (?̄?)
transformations, which mix states of the occupied subspace with different band in-
dices [128]. The proof can be found in the appendix (Sec. C.2).
As a consequence of the boundary conditions of the Bloch
factors, the parameter space is special in that its (𝜆, 𝑠𝑖 = 0) λ = 1 si
and (𝜆, 𝑠𝑖 = 1) edges are identical. In other words, the do-
main of the Berry connection A(𝝃) is a cylinder without
bases, like that sketched in Fig. 7.2. Its length corresponds λ
to the progressive nature of 𝜆 and its circumference re-
flects the periodicityA on the reciprocal lattice. One could λ = 0
have the idea that calculating only one part of Eq. (7.61), si
e.g. the 𝜆 = 1 part, might yield a physical quantity like
an “absolute” polarisation. It corresponds to integrating Figure 7.2: Path for calcu-lating the electric polarisa-
the geometric phase solely along one of the two circles in tion as a geometric phase.
Fig. 7.2. However, the single terms of Eq. (7.61) are ill-
defined because they are gauge-dependent. More precisely, according to Eq. (C.108)
and the boundary conditions of the Bloch factors (Eq. (2.28b)), the loop integrals
along the individual circles are only defined up to integer multiples of 2π. Con-
sequently, the single summands of Δ𝑃𝑖 are defined only modulo 2π −𝑒| C | , which cor-R
responds to an advance of the extensive polarisation |CR |𝑷 by −𝑒 · 𝑹𝑖 . This reflects
that the Wannier centres are defined only modulo a direct lattice vector [128].
115
7 Geometric phases
From a topological point of view, the subtraction of the two circles together with the
cancelling 𝜆-edges establishes a loop integral over the edge of a simply connected
rectangle which is wound up to a cylinder. In other words, the loop has an inner
area, so the integral assumes a distinct value according to Stokes. This is analogous
to the ABE, where only the complete knowledge about the gauge fields on a simply
connected region guarantees a well-defined ABE phase. Calculating only one of the
circles of the cylinder in Fig. 7.2 is ill-defined because this circle has no inner region
and is thus not simply connected. Although clear from the mathematics, it is quite
astonishing that the existence of the 𝜆-paths is crucial for the definition of the polar-
isation, despite their effective cancelling.
7.4 Topological transport
7.4.1 The integer Hall effect
The integer Hall effect (IHE) is based on topological transport, which is a straight-
forward application of the modern theory of polarisation. Only the points related to
topology are addressed in the following paragraphs, while the details about the IHE
as well as explicit calculations on the two-dimensional electron gas are passed.
Let {|𝑢𝑛(𝒌)⟩} be the occupied set of Bloch factors of an insulating, two-dimensional
Bloch system with direct basis vectors 𝑹1 and 𝑹2 and reciprocal basis vectors𝑮1 and
𝑮2. In order to calculate the response of the Bloch factors to an in-plane, constant,
homogeneous electric field 𝑬, consider Faraday’s law:
−𝜕𝑨(𝑡)𝑬 = ⇒ 𝑨(𝑡) = −𝑡 𝑬
𝜕𝑡
Without loss of generality, 𝑬 may be antiparallel to𝑮1 (Sketch in Fig. 7.3 (a)). Accord-
ing to the principle of minimal coupling4, the vector potential 𝑨(𝑡) couples into 𝒌 of
the Bloch Hamiltonian (Eq. 2.27):
ℎ(𝒑, 𝒙 | 𝒌) → ℎ(𝒑 + 𝑒𝑨(𝑡), 𝒙 | 𝒌) = ℎ(𝒑, 𝒙 | 𝒌 + 𝑒ℏ𝑨(𝑡)) (7.65)
Obviously, 𝑬 shifts the k-vector by a time-dependent term 𝑒ℏ𝑨(𝑡) = −
𝑒
ℏ 𝑡 𝑬. Because
of the parametric nature of 𝒌 in the Bloch formalism, this equals a time-dependent
parametric variation of the Bloch manifold. In particular, the k-vector advances once
across the Brillouin zone when the elapsed time is such that 𝑒ℏ𝑨(𝑡) = 𝑮1:
ℏ|𝑮1 | ℎ|𝑹2 |
Δ𝑡 = =
𝑒|𝑬 | 𝑒|𝑬 | |CR |
The field-related parametric variation permits the calculation of the concomitant
electric polarisation according to Sec. 7.3. For this purpose, a two-dimensional para-
4Mind that the charge 𝑞 is replaced by −𝑒
116
7.4 Topological transport
meter 𝝃 ∈ [0, 1]2 is introduced. The first component 𝜉1 represents the advance of the
k-vector due to the electric field. It equals 𝑠1, the coordinate of 𝒌 with respect to the
reciprocal basis vector 𝑮1, and corresponds to the reaction coordinate 𝜆 in Sec. 7.3.
The second component 𝜉2 arises from the Bloch-phase integration for obtaining the
Wannier centres. It equals the relative coordinate 𝑠𝑖 if the polarisation component
along 𝑮𝑖 is considered, the same 𝑠𝑖 as that in Sec. 7.3.
Integrating the Berry connection A𝑖 (𝝃) along the edge of its domain 𝛾 = 𝜕[0, 1]2
yields the Berry phase 𝜒B, which is p∑︁roportional to the polarisation 𝑮𝑖 · Δ𝑷.𝑖 ?̄? 〈   〉A𝑖 (𝝃) = 𝑢𝑛(𝝃) i∇∮ 𝜉 𝑢𝑛(𝝃) (7.66)𝑛=1
· −𝑒 B −𝑒𝑮𝑖 Δ𝑷 = |C | 𝜒𝑖 = |C | A𝑖 (𝝃) · d𝝃 (7.67)R R 𝛾
For the same reasons as those discussed in Sec. 7.3, Eq. (7.67) is invariant under𝑈 (?̄?)
transformations and, thus, the polarisation is observable in principle. The polarisa-
tion component along 𝑮1, parallel to the electric field, vanishes since the parameter
𝝃 has identical components (𝑠1, 𝑠1), so the line integral in Eq. (7.67) does not enclose
any finite area. Therefore, Δ𝑷 can be written in terms of the component perpendic-
ular to the electric field:
𝑹
𝑮 21 · Δ𝑷 = 0 ⇔ Δ𝑷 = 𝑃 |𝑹2 |
Along the other reciprocal basis vector 𝑮2, the polarisation is proportional to the
Berry phase 𝜒B C 𝜒B2 :
𝑮2 ·
2π −𝑒
Δ𝑷 = | |𝑃 =
B
𝑹2 |CR |
𝜒
In summary, each passage of Δ𝑡 pumps a certain amount of charge along 𝑹2, which
leads to a current on average. The corresponding current density 𝒋 is exactly the
quotient between the polarisation and the elapsed time:
Δ𝑷 −𝑒|𝑹 | 1 𝑒|𝑬 | |C | 𝑹 −𝑒2   1 𝑹
𝒋 = = 2 B R 2   B 2
Δ𝑡 |C | 𝜒 = 𝑬 𝜒 (7.68)R 2π ℎ|𝑹2 | |𝑹2 | ℎ 2π |𝑹2 |
If 𝜒B is positive, 𝒋 is turned by +90◦ relatively to 𝑬, as clarified by the sketch in Fig. 7.3
(a). Since the considered lattice is arbitrary, the right-angled relation between 𝒋 and
𝑬 can be summ( arised a)s Ohm’s law:
𝜎 𝜎 2
𝒋 = 𝜎𝑬 = 𝑥𝑥 𝑥𝑦
𝑒 1
𝑬 with 𝜎 B
𝜎 𝜎 𝑥𝑥
= 𝜎𝑦𝑦 = 0, 𝜎𝑥𝑦 = −𝜎𝑦𝑥 = − 𝜒 (7.69)
𝑦𝑥 𝑦𝑦 ℏ 2π
The astonishing detail about the above considerations becomes manifest if the loop
integral of the Berry connection A(𝝃) in Eq. (7.67) is written as a surface integral of
117
7 Geometric phases
(a) (b)
R2 R1
G2 G1
s1
E j
s2
Figure 7.3: (a) Sketch of the directions of the integer Hall effect and (b) path of 𝝃 on the Bril-
louin torus for calculating the topological transport.
the Berry pot∑︁ential B∬(𝝃) according to Stokes∬:1 ?̄? ?̄?B 1 d · B( ) [127] ∑︁ i 〈   〉𝜒 = 𝑺 𝝃 = d𝑺 · ∇ 𝑢 (𝝃) × ∇ 𝑢 (𝝃) (7.70)
2 𝜉 𝑛 𝜉 𝑛π
𝑛=1 2π 𝑆 1 2π𝑛= 𝑆
where 𝑆 denotes the surface enclosed by 𝛾.
As Berry noted, the integrand in Eq. (7.70) resembles a curvature whose surface in-
tegral yields the first Chern class of the Hermitian line bundle represented by the
eigenstates [127]. Moreover, and this is special about the Berry phase considered
here, the surface is a closed manifold since the reciprocal unit cell is homeomorphic
to the torus 𝕋2 (Fig. 7.3 (b)). As a consequence, the integral in Eq. (7.70) inclusive
the prefactor assumes integral values, leading to a quantisation of the conductivity
in Eq. (7.69):
𝑒2 𝑒2
𝜎𝑥𝑦 = − 𝑖 = 𝑖 𝜎0𝑥𝑦 with 𝑖 ∈ ℤ and 𝜎0𝑥𝑦 = − (7.71)ℎ ℎ
The above considerations show that the appliance of a lateral electric field 𝑬 on a
two-dimensional, insulating Bloch system raises a current density 𝒋 which is strictly
perpendicular to 𝑬. The corresponding off-diagonal conductivity 𝜎𝑥𝑦 is quantised
by the quantum conductivity 𝜎0𝑥𝑦 which depends only on natural constants, but not
on any details of the system. The quantum number 𝑖 depends on the system and
characterises it in the manner of a topological invariant. In other words, 𝑖, or equi-
valently 𝜎𝑥𝑦, is a fundamental property of Bloch systems and categorises them into
topological classes. The topological classification holds in general for all Bloch sys-
tems which fulfil the prerequisites of being two-dimensional and insulating. On the
contrary, topological classes are ill-defined for metallic systems. The Hall conduct-
ivity of a system is protected as long as the topological class does not change, which
means under all transformations which retain insulating state. This type of protec-
tion is much stronger than protection by mere symmetry since the latter permits a
continuous, gap-conserving destruction of the concomitant properties. An important
consequence of non-trivial Berry phases is that the Bloch functions cannot be defined
within a global gauge. Instead, there are overlapping patches on the torus in Fig. 7.3
118
7.4 Topological transport
(b) where the phase factors are continuously defined. A transition function permits
the change from one patch to the next by adding the necessary gauge. The integer in
Eq. 7.71 “is then related to the winding number of the phase of the transition function
around a non-contractable path” [142].
So far, the topologically invariant nature of 𝜎𝑥𝑦 is proven in a general manner. One
way to actually realise topologically non-trivial Bloch systems is the integer Hall ef-
fect (IHE), which occurs in a two-dimensional homogeneous electron gas subjected
to a strong, vertical magnetic field at a low temperature [125, 126]. In the model pic-
ture, the magnetic field introduces Landau levels, which, when integrally occupied,
render the electronic system insulating. In such a state, the Hall conductivity 𝜎𝑥𝑦 as-
sumes integer multiples of 𝜎0𝑥𝑦, hence the name integer Hall effect. The topological
state of this system can transition from 𝜎𝑥𝑦 to 𝜎 0𝑥𝑦 ± 𝜎𝑥𝑦 if the magnetic field varies.
However, this entails a gap closing. Such effects from topological phase transitions
are clearly visible in the IHE experiments as the parallel conductivity 𝜎𝑥𝑥 is zero if
the system is in an IHE state (→ insulating), and finite if the system changes the IHE
state (→metallic) [125, 126].
The IHE can be modelled by Laughlin’s gedankenexperiment, which will be discussed
when considering the ℤ2 insulators (vide infra). Therein, the topological transport
is explicitly derived from the wave functions of the Landau levels and Faraday’s law
of induction combined with an ABE phase shift (“flux threading”) [126]. However,
Laughlin’s gedankenexperiment bears problems if a periodic system is considered.
Thouless, Kohmoto, Nightingale and Nijs resolved them by theoretically investigat-
ing an explicit Bloch system involving a sinusoidal potential and a vertical magnetic
field. They calculated the Hall conductivity with the Kubo formula and obtained an
expression equivalent to Eq. (7.67). Although they did not extract the topological
nature, they argued that the emerging loop integral has to assume integer multiples
of 2π, which leads to a quantised conductivity [143]. As they were the first who es-
tablished a way of calculating the topological invariant for a quantum Hall state, the
integer 𝑖 in Eq. (7.71) is also called TKNN integer.
119
7 Geometric phases
7.4.2 The two-dimensional ℤ2 insulator
Theℤ2 insulators are closely related to the IHE insulators. They are classified by aℤ2
topological invariant, which, like the TKNN integer, describes topological transport,
not of charge, but of spin or, more precisely, of time-reversal polarisation (defined
later). Kane and Mele [144] were the first who demonstrated the existence of aℤ2 in-
variant in the quantum spin Hall phase constructed from two copies of the Haldane
model [145]. In the simplest case, the z-component of the spin 𝑠𝑧 is conserved, so the
copies represent each an independent IHE system with a TKNN integer classifying
the topological transport of charge. Since time-reversal symmetry (T-symmetry) is
presupposed, the TKNN integers of the two subsystems have equal magnitudes and
opposite signs, so the total TKNN integer of the system is trivial5. However, “the differ-
ence [between the two integers may be] non-zero and defines a quantized spin Hall
conductivity” [144]. The key point of [144] consists in the demonstration that the non-
trivial property of the system is retained if the 𝑠𝑧-conservation is explicitly broken.
The former topological transport of spin quanta 𝑠𝑧 then degrades to a shift in the
expectation value of the spin operator ⟨𝑆𝑧⟩, which was later denoted time-reversal
polarisation [142]. If the perturbation is strong enough, the system trivialises by a
change of its ℤ2 invariant to zero. However, the transition involves a closing of the
band gap at some point, as expected from the discontinuous nature of the change
between topological classes [144]. There are many different ways to derive the to-
pological ℤ2 invariant for two and three dimensions, which employ more or less ab-
stract topological field theories, summarised in [146]. The method presented here is
based on the IHE and demonstrates an alternative reason for the ℤ2 nature of the
invariant without going too deeply into the matter of topology.
Two-dimensional T-symmetric Bloch systems
In view of the three-dimensional ℤ2 topological insulators relevant for this work, it
is instructive to demonstrate the definition of the ℤ2 topological invariant for a two-
dimensional T-symmetric Bloch system at first and then to generalise the findings to a
three-dimensional crystal. This exactly follows the spirit of [142, 144, 147]. Hence, let
{|𝑢𝑛(𝒌)⟩} be the occupied set of Bloch factors of an insulating, T-symmetric 2D Bloch
system. It may host 2?̄? electrons per unit cell. The real-space domain is spanned
by the direct basis vectors 𝑹1 and 𝑹2 and the reciprocal unit cell by the reciprocal
basis vectors 𝑮1 and 𝑮2. Because what follows is slightly complicated, the lattice
geometry is simplified by the constraint that both R-vectors 𝑹1 and 𝑹2 have a length
of 1 and are orthogonal to each other so that they equal the euclidean coordinate
system. Consequently, the G-vectors𝑮1 and𝑮2 are also orthogonal to each other and
have a length of 2π. The Brillouin zone is thus a square defined by the points (±π,±π).
The analysis of the IHE (Sec. Sec. 7.4.1) shows that this simplification does not alter
the general findings.
5From an interpretational point of view, all Bloch states of a T-symmetric system occur in Kramer’s
pairs (defined later), so the Wannier centres of each pair move into opposite directions under any
T-symmetry-conserving variation. As a consequence, the transport of net charge is inhibited.
120
7.4 Topological transport
Time reversal is described by an antiunitary operator Θ [148], which for fermions
and in the representation of [142] holds: ( )
Θ = exp(i π 0 12𝜎𝑦)𝐾 = i𝜎𝑦𝐾 = −1 0 𝐾 (7.72)
where 𝜎𝑦 is the second Pauli matrix and 𝐾 is the complex-conjugation operator.
A T-symmetric Hamilton operator implies that each eigenstate |𝑛⟩ has the same ei-
genenergy 𝐸𝑛 as its time-reversed counterpart Θ|𝑛⟩. Hence, the eigenstates occur
in degenerate pairs, which are called Kramers pairs. The Kramers pairs of a Bloch
system comprise each two states with opposite k-vectors:
Θℎ(𝒑, 𝒙 | 𝒌)Θ−1 = Θ ?̃? 𝐻 (𝒑, 𝒙)?̃?−1 Θ−1𝒌 𝒌 = ?̃?−𝒌𝐻 (𝒑, 𝒙)?̃?
−1
−𝒌 = ℎ(𝒑, 𝒙 | −𝒌) (7.73)
where 𝐻 (𝒑, 𝒙) is the T-symmetric Hamilton operator and ℎ(𝒑, 𝒙 | 𝒌) the correspond-
ing Bloch Hamiltonian.
Since electronic Bloch systems host independent spin-12 fermions, the rotating action
of i𝜎𝑦 in Θ specifies the Kramers pairs to be linked components of the same spinor. In
detail, the occupied Bloch manifold of 2?̄? fermionic eigenstates groups into ?̄? spinors,
which permits identifying ?̄? states of each component I and II, spin-up and spin-down
in the 𝑠𝑧-conserving case. The ac〉tion of Θ then holds [142]: 〉𝑢I𝛼(−𝒌) 〉= − exp(i𝜒𝛼(𝒌)) Θ 𝑢II𝛼 (𝒌)𝑢II  〉 (7.74)𝛼 (−𝒌) = exp(i𝜒  I𝛼(−𝒌)) Θ 𝑢𝛼(𝒌)
The minus signs ensure that Θ2 = −1. The gauge functions exp(i𝜒𝛼(𝒌)) seem to be
arbitrary and are often replaced by 1 if topologically trivial systems are considered.
Though, they turn out to be essential forℤ2-insulating systems since the non-triviality
negates the existence of a global gauge for both components [142].
In the electronic band structure, two Kramers-paired bands obviously meet at the
Γ-point (𝒌 = 0) and form a twofold, same-k-vector degeneracy, i.e. the eigenspace
at 𝒌 = 0 is mapped onto itself under Θ. For this reason, the Γ-point is called a time-
reversal-invariant momentum (TRIM), denoted 𝚪0. In addition to 𝚪0, the periodicity
of the Bloch functions |Ψ𝑛(𝒌)⟩ on G produces three other TRIMs in CG where the
Kramers pairs meet agai{n t}o fo{rm a same-k degeneracy. This makes a total of fourirreducible TRIMs:
𝚪 1 1
}
𝑖 = 0, 2𝑮1, 2𝑮2, (
1
2𝑮1 +
1
2𝑮2) (7.75)
Partial, sum and difference polarisations
While T-symmetry expectedly inhibits the transport of net charge, the individual
components might still show a shifting of a non-zero quantity. This leads to the defini-
tion of partial polarisations 𝑷I and 𝑷II which arise in the components I and II, respect-
ively, and which are determined by a procedure analogous to the IHE case. Since all
types of polarisations 𝑷• emerge perpendicular to the applied vector potential shift,
they have only one non-vanishing component in the appropriate basis. The bold face
121
7 Geometric phases
of 𝑷• is retained so as to indicate its vectorial dimension. Whenever scalars are ad-
ded, it means that they are added to the non-vanishing component.
At first, the partial Berry connections A𝜎 (𝒌) are calculated from the Bloch factors of
the respective component. ∑︁ 〈   〉
A𝜎
?̄?
(𝒌) = 𝑢𝜎𝛼(𝒌)  i∇ 𝑘 𝑢𝜎𝛼(𝒌) (7.76)𝛼=1
where 𝜎 denotes the component and 𝛼 enumerates the Kramers pairs. Then, the
partial Berry phases 𝜒B,𝜎 are calculated by integrating A𝜎 (𝒌) along the edge of the
reciprocal unit cell. In order to exploit symmetries from the TRIM-nature of the Γ-
point, the integration loop is shifted to∮the edge of the quadratic Brillouin zone (bluesquare in Fig. 7.4 (a)).
𝜒B,𝜎 = A𝜎 (𝒌) · d𝒌 (7.77)
𝜕BZ
Finally, each traverse of the k-vector produces∮a partial polarisation 𝑷𝜎 proportionalto 𝜒B,𝜎6:
1
𝑷𝜎
1
= 𝜒B,𝜎
1 𝜎
− = A (𝒌) · d𝒌 (7.78)𝑒 2π 2π 𝜕BZ
The Kramers symmetry establishes the follo∑︁wing link between AI(𝒌) and AII(𝒌).
AI(−𝒌) = AII(𝒌) +∑︁ ?̄? (∇𝑘𝜒𝛼) (𝒌) (7.79a)𝛼=1
II I ?̄?A (−𝒌) = A (𝒌) − (∇ 𝜒 ) (−𝒌) (7.79b)
𝛼=1 𝑘 𝛼
The proof for these equations can be found in the appendix (Sec. C.2).
The sum of AI(𝒌) and AII(𝒌) results in the total Berry connection of the system,
denoted A+(𝒌). It describes the total charge polarisation 𝑷+ and is the same Berry
connection as that constructed for the IHE system (Eq. (7.56)). Inserting the Kramers
transformation (Eqs. (7.79)) and calculating the curl of A+(𝒌) shows that the total
Berry potential B+(𝒌) is antisymmetric with respect to∑︁the Γ-point:
A+
?̄?
(𝒌) B AI(𝒌) + AII(𝒌) = AI(𝒌) + AI(−𝒌) − (∇𝑘𝜒𝛼) (𝒌) (7.80a)
𝛼=1
B+(𝒌) = ∇𝑘 × A+(𝒌) = BI(𝒌) − BI(−𝒌) ⇒ B+(−𝒌) = −B+(𝒌) (7.80b)
For this reason, the surface integral of B+(𝒌) over the entire Brillouin zone vanishes,
as do the total Berry phase 𝜒B,+ and the tota∬l charge polarisation 𝑷+.
1
𝑷+
1
= 𝜒B,+ +− = B · d𝑺 = 0 (7.80c)𝑒 2π BZ
Eqs. (7.80) permits the following interpretation: The partial polarisations 𝑷I and 𝑷II
6Cf. Eq. (7.67) inserting the unit cell volume |CR | = 1 and the reciprocal basis vector |𝑮 | = 2π.
122
7.4 Topological transport
exactly cancel each other, implying that the Wannier centres of the Kramers pairs
move equally into opposite directions. This accords with the expectations from T-
symmetry and explains why the topological transport of net charge is inhibited. Of
course, the vanishing total polarisation is gauge-invariant, for the same reasons as
those discussed in Sec. 7.4.1.
Taking the difference between AI(𝒌) and AII(𝒌) results in the time-reversal Berry
connection A− (𝒌). It describes the time-reversal polarisation 𝑷− . Inserting the Kra-
mers transformation (Eqs. (7.79)) and calculating the curl of A− (𝒌) shows that the
time-reversal Berry potential B− (𝒌) is symmetric with∑︁respect to the Γ-point:
A−
?̄?
(𝒌) B AI(𝒌) − AII(𝒌) = AI(𝒌) − AI(−𝒌) + (∇𝑘𝜒𝛼) (𝒌) (7.81a)
𝛼=1
B− (𝒌) = ∇𝑘 × A− (𝒌) = BI(𝒌) + BI(−𝒌) ⇒ B− (−𝒌) = B− (𝒌) (7.81b)
Because of the centro-symmetry, the surface integral of B− (𝒌) over the entire Bril-
louin zone equals twice that over the positive half plane (shaded area in Fig. 7.4 (a)).
Therefore, the time-reversal Berry phase 𝜒B,− and, thus, the time-reversal polarisa-
tion 𝑷− are not necessarily zero∬. ∬
𝜒B,− = B− · d𝑺 = 2 B− · d𝑺 (7.81c)
BZ BZ+
Another representation for 𝜒B,− follows from the equality of the integrals of 𝐵I(−𝒌)
and 𝐵I(+𝒌) due to the centro-sym∬metry of the Brillouin zone:
𝜒B,− = 2 BI · d𝑺 = 2𝜒B,I = −2𝜒B,II (7.81d)
BZ
Eq. (7.81d) immediately implies that the time-reversal Berry phase 𝜒B,− is quantised:
Since each component 𝜎 represents a Bloch manifold, the respective partial Berry
phase 𝜒B,𝜎 assumes integer multiples of 2π, for the same reasons as those discussed
in Sec. 7.4.1. Consequently, the time-reversal Berry phase 𝜒B,− assumes integer mul-
tiples of 4π. In terms of polarisations, the partial polarisations 𝑷𝜎 are up to the sign
equal integer multiples of the unit polarisation, −𝑒 in this case. The time-reversal
polarisation 𝑷− thus assumes even multiples of −𝑒.
𝑷I = −𝑷II = −𝑖 𝑒, 𝑖 ∈ ℤ ⇒ 𝑷− = −2𝑖 𝑒 (7.82)
The link between these quantities is summarised in the upper part of Fig. 7.4 (b).
Although the time-reversal polarisation 𝑷− is quantised by a curvature integral, it
is not yet an observable as the gauge invariance remains to be proven. From Eq.
(C.108), it is clear that the individual partial Berry phases are invariant under 𝑈 (?̄?)
transformations which mix the states within a fixed component. If the components
are well-defined by good quantum numbers, 𝑷− is gauge invariant in the form of Eq.
(7.82) without further restrictions. However, if the components are not conserved,
i.e. unitary transformations are considered which mix them, 𝑷− obviously depends
123
7 Geometric phases
(a) (b) P+ = 0
⊕
π Γ2 Γ3 PI = −i e PII = +i e
I II
BZ+
∂CG ⊖
-π Γ π0 Γ1 P− = −2i e
y
−e
-π ∂BZ x U
⇄ +2e −2e
+e
Γ2' Γ3' P− += 4e
Figure 7.4: (a) Sketch of the reciprocal space of the T-symmetric Bloch system. The red lines
indicate the primitive reciprocal unit cell and the blue lines the Brillouin zone. The edges of
both can be chosen as the integration path for obtaining the Berry phases. 𝚪𝑖 are the TRIMs.
The positive half plane of the Brillouin zone is shaded grey. (b) Flow chart of the connection
between the partial polarisations 𝑷I and 𝑷II, the total polarisation 𝑷+ and the time-reversal
polarisation 𝑷− . The lower part shows how a unitary transformation𝑈⇄ flipping a Kramers
pair transfers one unit polarisation −𝑒 between 𝑷I and 𝑷II, which changes them by 2 unit
polarisations and increases 𝑷− by 4 unit polarisations.
on the division of the Kramers pairs into the two components and, thus, on the gauge.
The question is then which part of 𝑷− remains observable. For the answer, consider
the action of a unitary transformation𝑈⇄ which simply swaps the components of a
certain Kramers pair. According to Eq. (7.82), the transformation can transfer only
even multiples of the unit polarisation from 𝑷I to 𝑷II. At the same time, the oppositely
equal polarisation shifts from 𝑷II to 𝑷I, so the time-reversal polarisation 𝑷− changes
by multiples of 4 unit polarisations. The effect of𝑈⇄ is sketched in Fig. 7.4 (b).
The ℤ2 topological invariant
In summary, the time-reversal polarisation 𝑷− has two properties:
1) It is quantised into even multiples of the unit polarisation.
2) It is defined only modulo 4 unit polarisations for non-conserved components.
Analogously to the IHE, 𝑷− produces a current on average, provided that the com-
ponents are characterised by good quantum numbers, e.g. in spin-conserving Bloch
systems. This is exactly the quantum spin Hall effect, where an electric field gives
rise to a spin current at a spin Hall conductivity 𝜎−𝑥𝑦. Property 1) implies that 𝜎−𝑥𝑦 is
quantised into even multiples of the quantum conductivity.
𝜎−𝑥𝑦 = 2𝑖 𝜎
0
𝑥𝑦 = −2𝑖 𝑒2/ℎ, 𝑖 ∈ ℤ (7.83)
124
7.4 Topological transport
The integer 2𝑖 is analogous to the TKNN integer and defines a topological invariant
for the same reason.
For non-conserved components, the 4 modulus from property 2) restricts 𝜎−𝑥𝑦 to two
values: zero or twice the quantum conductivity.
𝜎−𝑥𝑦 = 2𝑖 𝜎
0
𝑥𝑦 = −2𝑖 𝑒2/ℎ, 𝑖 ∈ ℤ2 (7.84)
The cyclic structure renders 𝜎−𝑥𝑦 a ℤ2 invariant. All insulating T-symmetric 2D Bloch
systems belong either to the trivial class (𝑖 = 0) or to the non-trivial class (𝑖 = 1).
The latter is referred to as the class of ℤ2 topological insulators or simply ℤ2 insu-
lators. The arbitrariness of the sign of 𝜎−𝑥𝑦 negates the quantum spin Hall effect and
questions if there is anything observable at all. The key point is that there exists a
topologically non-trivial shifting in topologically non-trivial phases and that these
phases are distinct from the trivial systems in that they cannot be transformed into
each other without closing the band gap.
The analysis presented in this work shows in a systematic and comprehensible way
how the ℤ2 topological invariant emerges from the Kramers symmetry by applying
methods which are familiar from the IHE. Furthermore, it gives concise reasons for
why integrating the Berry connection over the edge of half the Brillouin zone is suf-
ficient for calculating the ℤ2 invariant. The original literature concerning this issue
[142, 144, 147] calculates the response of the edges by threading a half quantum flux
through a Laughlin cylinder. It employs multiply valued functions and branches of
logarithms and square roots in order to extract theℤ2 invariant. The following equa-
tions prove that the ℤ2 formulation presented here is equivalent to that of Fu and
Kane [142]. If dropping the factor 2, the remainder of Eq. (7.81c) is the time-reversal
polarisation due to the advance of the k-vector across half the Brillouin zone. This
can then be re∬formulated in ter∮ms of an edge inte∮gral of A− (𝒌). ∮
1 2π
𝑷− = B− · d Stokes𝑺 = A−− (𝒌) · d𝒌 = A
− (𝒌) · d𝒌 + A− (𝒌) · d𝒌
2 𝑒
BZ+ ∫ 𝜕BZ+ 𝚪′2∫𝚪′3𝚪1𝚪0 𝚪0𝚪1𝚪3𝚪2
(∗) 𝚪2 𝚪3
= − [(A∫− (𝒌) + A− (−𝒌)] · d𝒌 + [A− (𝒌) + A− (−𝒌)] · d𝒌𝚪0 𝚪1 )
Eq. (C.110) 𝚪2 [ ∑︁
= −∫ A+
]
(𝒌) − A+(−𝒌) · d𝒌 + 2∑︁ [𝜒𝛼(𝚪[ ] 2
) − 𝜒𝛼(𝚪0)]
𝚪0 𝛼
𝚪3
+ A+(𝒌) − A+(−𝒌) · d𝒌 + 2 [𝜒𝛼(𝚪3) − 𝜒𝛼(𝚪1)] (7.85)
𝚪1 𝛼
Step (∗) uses that the horizontal paths of 𝜕BZ+ in Fig. 7.4 (a) cancel each other because
of the boundary conditions of the Bloch factors. For the same reasons, the vertical
paths can be mapped onto the upper half plane where −𝚪′2𝚪 ′0 equals 𝚪2𝚪0 and −𝚪3𝚪1
equals 𝚪3𝚪1. The individual terms of Eq. (7.85) equal Eq. (3.21) in [142], from which
Fu and Kane derive the ℤ2 invariant as the product of the phases of the Pfaffians of
the time-reversal matrix.
125
7 Geometric phases
Edge states
As noted in the course of defining the ℤ2 invariant, a non-zero time-reversal con-
ductivity (Eq. (7.84)) raises the question whether there is an observable current at
all if the components are not conserved. Part of this question is answered in [144]:
A weak perturbation which breaks spin conservation alters the former current of
quantised spin to a reduced current of magnetisation density along the quantisation
axis. However, although the spin quantisation is lifted, the ℤ2 invariant remains the
same as long as the perturbation does not close the gap.
Another manifestation of the ℤ2 invariant is the behaviour of the edge states, as il-
lustrated by the Laughlin gedankenexperiment in [142, 147]: Consider a sample of
finite width and length which is wound up to a cylinder and has two edges at the top
and the bottom. The x-coordinate of the Brillouin zone 𝑘1 parallel to the circumfer-
ence may be the direction of the applied vector-potential shift. The y-coordinate 𝑘2
parallel to the axis is then the direction of the time-reversal polarisation. The vector
potential shift is equivalent to threading a magnetic flux Φ through the cylinder. If
the flux (divided by the circumference) equals the quantum flux ℎ/𝑒, the advance in
𝑘1 equals 𝑮1, the distance between the TRIM 𝚪0 and its copy 𝚪′0. A half quantum flux
ℎ/2𝑒 thus corresponds to an advance of𝑮1/2, the distance between the TRIMs 𝚪0 and
𝚪1. The setting is sketched in Fig. 7.5 (a).
Once the setting is clear, the gedankenexperiment is as follows. At zero flux (𝒌 = 𝚪0),
all Kramers pairs are same-k degeneracies and either doubly occupied or empty. This
holds also for any states arising at the edges of the cylinder. If 𝒌 now advances by 𝑮1
towards 𝚪′0, one of two possibilities occurs:
1) The band structure is trivial and the Kramers pairs associated with the edges
end up in the same configuration as the initial one, i.e. states which are Kramers
partners at 𝚪0 are also partners at 𝚪′0, as sketched in Fig. 7.5 (b). For example,
the Kramers pair denoted 1 splits up at 𝚪0, but finds together again at 𝚪′0. In
particular, if this Kramers pair is doubly occupied in the beginning, it will be
doubly occupied in the end. This accords with the prerequisite that the time-
reversal polarisation due to the traverse of the k-vector has to be zero.
2) The band structure is non-trivial and the Kramers pairs associated with the
edges end up in a configuration different from the initial one, i.e. the states
change partners, as sketched in Fig. 7.5 (c). For example, Kramers pair 1 splits
up at 𝚪0 and its upper branch (red) goes to Kramers pair 2 at 𝚪′0 where it finds a
new partner from the bulk conduction bands. The lower branch (blue) vanishes
in the bulk valence bands. In turn, the Kramers pair 1 at 𝚪′0 is restored by a blue
branch from Kramers pair 2 and a red branch from the bulk valence bands. As
a consequence of the partner change, if Kramers pair 1 is doubly occupied and
Kramers pair 2 is empty in the beginning, they both will be singly occupied in
the end, which corresponds to the transport of one electron. Since the oppos-
ite happens at the other edge, the net charge transport is zero. However, the
time-reversal polarisation amounts to 2 unit polarisations, which reflects the
non-trivial ℤ2 invariant.
126
7.4 Topological transport
(a) Φ (b) E trivial
bulk I
II
A 2 2 Edge
Edges 1 1 states
bulk
Γ0 Γ1 Γ0'
(c) E non-trivial
Γ2 Γ3 I
II
k ~ A 2 2
Γ0 Γ1 Γ0'
k 1 12
k1
G1 Γ0 Γ1 Γ0'
Figure 7.5: (a) Sketch of the Laughlin-type gedankenexperiment. The sample is wound up to a
finite cylinder and provides two edges. A magnetic flux Φ threading the cylinder corresponds
to a shifting of the vector potential 𝑨, which in turn makes the k-vector 𝒌 advance in that dir-
ection. (b) and (c): Sketches of possible topologies of T-symmetric edge band structures. Only
the bands of one edge are shown, where red and blue lines indicate the two components.
The bulk bands are shaded grey. At the TRIMs, red an blue bands have to form a same-k
Kramers degeneracies. In (b), the Kramers pairs at different TRIMs are trivially connected
(𝜎−𝑥𝑦 = 0), which allows the edge states to form a band gap. In (c), they are non-trivially con-
nected (𝜎−𝑥𝑦 = 2𝜎0𝑥𝑦), which strictly prohibits the formation of band gaps. Adapted from [147].
As it is evident, the non-trivial band structure in Fig. 7.5 (c) guarantees that the Fermi
level cuts at least one band associated with the edges of the cylinder. Furthermore, a
Laughlin gedankenexperiment with swapped roles for 𝑘1 and 𝑘2 obviously yields the
same results. This leads to the conclusion that the edges of a non-trivial ℤ2 insulator
in general host metallic edge states which are topologically protected, i.e. there is no
way to introduce a band gap within an adiabatic, T-symmetric transformation. On the
contrary, the trivial band structure in Fig. 7.5 (b) either provides a gap in the edge
states or the bands can be adiabatically shifted so that a gap opens.
The schematic band structures in Fig. 7.5 (b) and (c) moreover show that the topolo-
gical character of the edge states does not only emerge if the Laughlin-type gedanken-
experiment is performed for a full Brillouin-zone traverse of 𝒌 from 𝚪 ′0 to 𝚪0. Since
the T-symmetry strictly produces TRIMs halfway along the reciprocal basis vectors,
also the half traverse from 𝚪0 to 𝚪1 causes topologically distinctive partner changes.
In the trivial case, the states of each Kramers pair at 𝚪0 find together again at 𝚪1. In
the non-trivial case, the states of each Kramers pair at 𝚪0 find new partners at 𝚪1. The
Laughlin-type gedankenexperiment works also for a half quantum flux because the
time-reversal Berry potential is symmetric, so integrals over the half of the Brillouin
zone are sufficient for the calculation of the ℤ2 invariant.
127
7 Geometric phases
7.4.3 The three-dimensional ℤ2 insulator
The ℤ2 classification of a T-symmetric 3D Bloch system is based on the behaviour of
its surface states in the presence of a perturbing vector-potential shift. Analogously to
the edge band structure in the 2D case, a surface band structure is topologically non-
trivial if the surface states change partners when going from one TRIM to another.
However, differently from the 2D case, there are more than one possibilities to make
such considerations in three dimensions:
Firstly, instead of two edges in the 2D case, a 3D crystal has
three main surfaces which are defined by the reciprocal Γ2 Γ3
basis vectors 𝑮 𝑗 acting as surface normals. Secondly, in- Γ0 Γ1
stead of four TRIMs in the 2D case, the 3D Brillouin zone GΓ G16 Γ 37
provides eight TRIMs {𝚪𝑖}, which are determined analog- Γ4 Γ5
ously to Eq. (7.75) by taking also the third reciprocal basis
vector 𝑮3 into account. For the sake of simplicity, the dir-
ect and reciprocal unit cells of the T-symmetric 3D crystal Γ2 Γ3 G2
to be characterised are supposed to be both cubic. Then, Γ0 Γ1
the TRIMs define the vertices of a cube which is exactly an Figure 7.6: TRIMs 𝚪𝑖 in a
eighth of the cubic Brillouin zone (Fig. 7.6). cubic Brillouin zone.
Instead of one TRIM pair per edge in the 2D case, each surface of a T-symmetric 3D
crystal provides six different TRIM pairs between which the surface band structure
can be topologically trivial or non-trivial. The topological character is determined by
quantifying the topological transport on a square which is defined by the respective
TRIM pair and the two opposite TRIMs of the TRIM cube. In detail, the identification
of non-trivial phases is analogous to the following example:
• Consider the surface defined by the surface normal𝑮3 (shaded blue in Fig. 7.6).
• Consider the first surface k-path defined by 𝚪0𝚪1, which is parallel to 𝑮1 (red
labels in Fig. 7.6).
• Calculate the time-reversal polarisation along 𝑮3 due to a vector-potential shift
along 𝑮1 by integrating B− (𝒌) in the square spanned by 𝑮1 and 𝑮3. Accord-
ing to Eq. (7.85), this corresponds to an integration within the TRIM square
(𝚪0, 𝚪1, 𝚪5, 𝚪4) (red face in Fig. 7.6).
If the time-reversal polarisation in that square is zero, the surface band structure
along 𝚪0𝚪1 is trivial and looks like that Fig. 7.5 (b). Otherwise, it is non-trivial and
characterised by partner changes (Fig. 7.5 (c)). The above procedure can be repeated
for all surface paths defined by pairs out of {𝚪0, 𝚪1, 𝚪2, 𝚪3}. If the 𝑮3-surface is com-
pletely characterised, the above procedure can be performed also for the two other
surfaces defined by 𝑮1 and 𝑮2.
It is obvious that the above procedure has many redundancies. For example, if the
𝑮3-surface paths 𝚪0𝚪1 and 𝚪0𝚪2 are both non-trivial, the𝑮3-path 𝚪1𝚪2 must be trivial
because the ℤ2 invariant is only defined modulo 2. Moreover, the 𝑮3-surface path
𝚪0𝚪1 and the 𝑮1-surface path 𝚪0𝚪4 share the topological character because the TRIM
128
7.4 Topological transport
square for calculating the ℤ2 invariant is the same7. In order to remove those re-
dundancies, Fu and Kane simplified the characterisation by transforming Eq. (7.85)
into a form which relates the topological transport to a product of units 𝛿𝑖 (𝒌) which
are evaluated at each TRIM of the respective TRIM square. These units are the ratio
between the square root of the determinant of the time-reversal matrix 𝑤(𝒌) and
the Pfaffian of 𝑤(𝒌). In doing so, the branch of the square root has to be continuous
between TRIM pairs. For a T-symmetric 2D Bloch system, the ℤ2 invariant is non-
trivial if the produc∏t of the four 𝛿√︁𝑖 is −1. If the product is +1, the system is trivial[142]. 3 { }
(−1)𝜈
det(𝑤(𝚪𝑖))
= 𝛿𝑖 , 𝛿𝑖 = ( ( )) ∈ −1, +1 ⇒ 𝜈 ∈ ℤPf 2 (7.86)𝑤 𝚪
𝑖=0 𝑖
A T-symmetric 3D Bloch system has eight TRIMs and thus eight 𝛿𝑖 . Consequently,
there are still many possibilities of how the signs can be distributed over the cube.
However, five important cases can be distinguished:
1) All eight 𝛿𝑖 are positive.
2) Two 𝛿𝑖 on an edge of the cube are negative, while the others are positive.
3) Two 𝛿𝑖 on a face diagonal are negative, while the others are positive.
4) Two 𝛿𝑖 on a space diagonal are negative, while the others are positive.
5) One 𝛿𝑖 is negative, while the others are positive.
All other configurations can be transformed into one of the above cases by gauge
transformations, i.e. by changing the signs of four 𝛿𝑖 lying in the same plane. In
particular, case 5) is equivalent to all cases where an odd number of 𝛿𝑖 is negative.
Case 1) is the trivial case as each TRIM square yields a 𝛿𝑖 product of +1. Consequently,
each surface path supports a topologically trivial surface band structure. Cases 2), 3)
and 4) produce topologically non-trivial surface band structures for a differing num-
ber of surfaces and paths. These systems can be imagined as a stack of independent
2D ℤ2 insulators whose one-dimensional edges host topologically protected metallic
states. However, because the non-triviality is unstable against redefining the Bril-
louin zone, as it may occur in the presence of disorder, the edge states are not protec-
ted against all adiabatic, T-symmetric transformations. For this reason, these systems
are referred to as weak topological insulators [147].
The last case 5) is relevant for this work. Since only one 𝛿𝑖 is negative, any surface
contains one TRIM from which all surface paths produce non-trivial band structures.
Since case 5) is equivalent to all cases with an odd number of negative 𝛿𝑖 , a 3D ℤ2
invariant 𝜈0 ca be de∏fined via the√︁sign of the product of all eight 𝛿𝑖 .7
𝜈 det(𝑤(𝚪(−1) 0 𝑖
)) { }
= 𝛿𝑖 , 𝛿𝑖 = ∈ −1, +1 ⇒ 𝜈Pf ( ( )) 0 ∈ ℤ2 (7.87)𝑤 𝚪
𝑖=0 𝑖
7This is the reason why the T-symmetric 2D Bloch system with two edges and four TRIMs is character-
ised by only one ℤ2 invariant.
129
7 Geometric phases
As the non-triviality is stable against redefining the Brillouin zone, the metallic sur-
face states enjoy a topological protection which is similar to that of the metallic edge
states of the 2Dℤ2 insulator. For this reason, systems with non-trivial 𝜈0 are referred
to as strong topological insulators [147] and the subsequent surface states as topolo-
gically protected surface states (TSSs). As strongℤ2 TIs are the only TIs considered in
this work, the term topological insulator is used for them in an equalising manner.
Topological insulators with inversion symmetry
Although the definition and general properties ofℤ2 insulators are clear, the calcula-
tion of the ℤ2 topological invariant for a concrete system remains complicated. The
definition via the time-reversal Berry potential (Eq. (7.81c)) in principle resolves the
continuous-gauge problem since the Berry potential can be calculated locally by the
matrix elements of well-defined derivatives of the Hamilton operator (Eq. (7.58)). On
the other hand, the evaluation will be tedious in the 3D case because of the above-
mentioned combinatorial inflation due to the different surfaces and paths. The ap-
proach of [147] provides a simplification of the combinatorics by defining a single
3D ℤ2 invariant via the product of the 𝛿𝑖 of all TRIMs (Eq. (7.87)). Though, concrete
calculations still involve some exertion since the branches of the numerator and the
denominator of Eq. (7.87) have to be continuos over the entire TRIM cube.
Fortunately, there is one case in which the evaluation of Eq. (7.87) becomes partic-
ularly simple: If inversion symmetry (P-symmetry) is present, then the 𝛿𝑖 equal the
product of the parity eigenvalues of the respective Kramers pairs. Thus, if the solving
Bloch eigenspace is given, the 𝛿𝑖 can directly be determined from the Bloch factors
without need for caring about the conti∏nuity of the branches of Eq. (7.87).?̄?
𝛿𝑖 = 𝑝𝛼(𝚪𝑖) (7.88)
𝛼=1
where 𝑝𝛼(𝚪𝑖) denotes the parity eigenvalue of the Kramers pair with index 𝛼 at the
TRIM 𝒌 = 𝚪𝑖 [147]. Of course, Eq. (7.88) works both in the 2D and the 3D case.
As a last remark, the parity method does not only simplify the topological classific-
ation of insulators with P-symmetry. Also systems without P-symmetry are suitable
for it if they are proven to be adiabatically connected to a TI with P-symmetry. This
holds because any adiabatic transformation which retains the band gap conserves
the topological invariants. This considerably broadens the applicability of the parity
method and makes it the method of choice for identifying ℤ2 insulators [147].
130
8 The bulk phases of bismuth selenide
and antimony
Bismuth selenide and antimony, the materials composing the Sb@Bi2Se3 heterostruc-
tures, have two properties in common:
1) They involve heavy p-block elements: the medium-heavy fourth-row element
selenium (Se, 𝑍 = 34), the heavier fifth-row element antimony (Sb, 𝑍 = 51),
and the sixth-row element bismuth (Bi, 𝑍 = 83), the heaviest stable element
of the periodic table1. As a consequence of the high atomic numbers, strong
relativistic effects are expected to determine the physical properties of both the
Bi2Se3 substrate and the 𝛽-Sb adsorbate.
2) They are van der Waals materials, i.e. they comprise chemically saturated lay-
ers whose mutual bonding underlies vdW interactions.
These peculiarities make approaches beyond standard DFT necessary in order to cor-
rectly describe the Sb@Bi2Se3 heterostructures. Spin-orbit coupling (SOC) allows for
the most important relativistic effects in the valence shells (Sec. 2.1.2). The vdW inter-
actions are introduced by adding the London dispersion to the total energy (Sec. 2.1.3).
The first purpose of this chapter is to test the SOC and vdW approaches on the bulk
phases of Bi2Se3 and 𝛽-Sb, for which experimental lattice parameters exist, contrarily
to the heterostructures. Particularly the vdW corrections have to be tested because
there are several implementations with different degrees of empiricism. The inter-
layer distances are the central quantities to check since it is expectable that the vdW
corrections mostly influence those. Furthermore, the structural effects from SOC are
paid a special attention. In many previous investigations, SOC is only included for
calculating the final band structures, while it is omitted during the structure optim-
isations. It will turn out that SOC alters the interlayer distances in a non-negligible
manner.
The second purpose of this chapter concerns the electronic and topological properties
of Bi2Se3 and 𝛽-Sb, which are absolutely crucial for understanding the topological
phase transitions in the heterostructures. In detail, the band structures of bulk Bi2Se3
and 𝛽-Sb and, in addition, the two-dimensional band structures of free-standing thin
𝛽-Sb sheets are calculated and analysed. The ℤ2 invariant is explicitly calculated for
these systems with the parity method (Sec. 7.4.3).
1In fact, bismuth is radioactive. However, the decay is extremely slow with an half-life of 1.9 · 1019 yr.
It was not before 2003 until the instability of the most abundant isotope was proven [149].
131
8 The bulk phases of bismuth selenide and antimony
8.1 Bismuth selenide
Bismuth selenide is a compound of the pnictogen2 bismuth (Bi) and the chalcogen3
selenium (Se). Its structure belongs to the trigonal, P-symmetric 𝑅3̄𝑚 space group
(No. 166), in which the atoms occupy the Wyckoff positions Se→1𝑎, Se→2𝑐 and Bi→
2𝑐. Bi2Se3 belongs to the class of van der Waals materials or sparse matter. It is clearly
layered, consisting of covalently bound sheets which in turn stick together by vdW in-
teractions. In detail, each Bi2Se3 unit forms a quintuple layer (QL) of five atoms, which
arrange themselves as hexagonal atomic layers according to an (ABCAB) sequence.
The QLs in turn stack in such a way that the layer sequence continues, implying an
axial periodicity length of 15 atomic layers or 3 QL: (ABCAB)(CABCA)(BCABC). The
structure is sketched in Fig. 8.1 (a) and (b).
The microscopic reason for the layered structure is a closed-shell configuration of
the individual QLs [150]. However, the octet rule is obviously not fulfilled as the Bi
and Se atoms coordinate each other in (half-complete) octahedral geometry. Since Se
is more electronegative than Bi by Δ𝜒 = 2.48 − 1.67 = 0.81 according to the Allred-
Rochow scale [17, Tafel III], the bonding is expected to be covalent with a partial ionic
character of ca. 10 % [17, p. 158]. The Se atoms are in a negative oxidation state of
– II, which completes the octets. The Bi atoms, on the other hand, are in a positive
oxidation state of III, two less than the maximum of V for members of group 15.
The reasons for the stability of the III-valent state of Bi are numerous, but all based on
the low ability of the 6s electrons to participate in chemical bonding. A comparison
between the pnictogens (Pn) shows that P assumes the oxidation state V in many
compounds, while all other Pn prefer the oxidation state III. This manifests itself, for
instance, in the redox potentials of the Pn(V)/Pn(III) acidic systems: they are negative
for P, positive for N, As and Sb, and by far the highest for Bi [17, p. 962]. Four different,
countercurrent effects are responsible for this:
1) The higher main quantum number of heavier homologues loosens the valence
electrons. Therefore, P is more willing to donate its s electrons than N.
2) The d-block contraction binds the valence electrons closer the nucleus. There-
fore, As and Sb are less willing to donate their s electrons than P.
3) The f-block contraction or lanthanoid contraction binds the valence electrons
closer the nucleus. Therefore, Bi is even less willing to donate its s electrons
than As and Sb.
4) The relativistic effects contract mainly the s orbitals and bind them closer to the
nucleus. They are considerable for the heaviest elements, to which Bi belongs.
As a consequence of 2), 3) and 4), the 6s electrons of Bi are so tightly bound that
they behave almost like core states and become chemically inert. Hence, this effect
is called the inert-pair effect (or lone-pair effect) [17, pp. 335ff, 373ff, 961ff]. As only
the 6p orbitals of the Bi atoms are left for binding, the bonds arrange themselves in
2Also denoted pentels, group 15, the nitrogen group: N, P, As, Sb and Bi.
3Group 16, the oxygen group: O, S, Se, Te and Po.
132
8.1 Bismuth selenide
(a) (b) H3 T4 31.0 (c) 30.5T c (Å)1
30.0 30.4
C d
B a 29.0 28.9 28.8 28.8
w 28.4
28.7
A 28.6 28.7
C [‾11.0] 28.0 28.4SOC 28.2 28.3
28.5
B c
A no SOC
C 4.25
B QL (d) 4.21 a (Å) 3.2 (e) 3.144.20 d (Å)
A 4.20 4.19 3.0 3.09
C 4.19 4.20 4.18
B 4.15 4.17 4.15 4.16 2.8 2.65
A 4.14 4.13
2.63 2.63
4.14 2.6 2.58
C 4.10 4.12 2.58 2.55 2.49
B 2.4 2.47 2.47
2.38 2.44A 4.05 2.2
[‾11.0] no vdW D2 D3 D3+BJ TS TSH no vdW D2 D3 D3+BJ TS TSH
Figure 8.1: Bulk structure of Bi2Se3. Purple circles are Bi atoms; Light green circles are Se
atoms. (a) Side view; (b) Top view. The conventional unit cell is marked red. (a) shows the
rhombohedrally centred primitive unit cell as well. Orange stars in (b) mark some T1 posi-
tions of the (0001) surface. (c) – (e) Lattice parameters 𝑎, 𝑐 and 𝑑 of the structures optimised by
code-level relaxations within different approaches for the vdW corrections (x-axis, see text)
and SOC (empty and filled circles). The experimental reference [153] is inserted as dashed
lines.
an octahedral geometry with angles close to 90◦. This obviously holds in Bi2Se3 quite
strictly as Bi and Se coordinate each other in (half-)complete octahedral geometries.
That also the Se atoms obviously underlie p bonding without hybridising with the 4s
electron is presumably due to a steric effect from the p orbitals of the Bi atom. In the
lighter homologous compounds Sb2Se3 and As2Se3, the pnictogens coordinate the Se
atoms in a twofold and threefold manner [151, 152], not in a sixfold manner as in the
case of the Se atom in the middle of the QLs.
8.1.1 Structure optimisation
Computational parameters
The DFT calculations are carried out with VASP [2, 3] using PAW potentials and the
PBE xc-functional [10, 21, 22]. The Bi and Se atoms have PAW valences of 5d106s26p3
and 4s24p4, respectively. The kinetic energy cutoff is 𝐸cut = 400 eV. The Brillouin
zone is sampled by a Γ-centred Monkhorst-Pack mesh [117] at a density of 12×12×3
k-points for the hexagonal unit cell. The Brillouin-zone integration is carried out
by the tetrahedron method with Blöchl corrections [118] for all calculations as the
compound is a semiconductor. A convergence test proves that increasing the energy
cutoff or the sampling density alters the total energy by ≈1 meV per atom. The ionic
forces are calculated according to the Hellmann-Feynman theorem [119]. The ions
move along them towards the equilibrium positions and stop to relax if the forces
acting on each ion are smaller than 0.005 eV/Å.
133
[00.1]
[11.0]
8 The bulk phases of bismuth selenide and antimony
For different combinations of vdW approaches and SOC/no SOC, the volume of the
hexagonal unit cell is optimised by the code-level routines. For the combination re-
producing the experimental reference best, the volume optimisation is repeated by
the more exact Murnaghan equation of state (vide supra, p. 41). The six vdW ap-
proaches tested are: no vdW correction, DFT-D2 [26], DFT-D3 + zero-damping [27],
DFT-D3 + Becke-Johnson damping [28], the Tkatchenko-Scheffler method [29], and
the Tkatchenko-Scheffler method with iterative Hirshfeld partitioning [30, 31].
Volume optimisation
The relevant lattice parameters of the hexagonal unit cell of Bi2Se3 are: the basal
lattice constant 𝑎, the axial lattice constant 𝑐, the interlayer distance 𝑑, defined as
the difference between the z-coordinates of the outer Se atoms of adjacent QLs, the
QL width 𝑤 = 𝑐/3 − 𝑑, and the Wyckoff parameters of the rhombohedral centring 𝑢
(Bi atoms) and 𝑣 (outer Se atoms). The results for 𝑎, 𝑐 and 𝑑 may be found in Fig. 8.1
(c), (d) and (e), respectively. The experimental lattice parameters (XRD, [153]) are
inserted as dashed lines.
The vdW correction strongly affects the interlayer distance 𝑑, which is expectable
for a vdW material. Compared to the calculations without dispersion correction (no
vdW), the QLs move closer together by more than 0.5 Å when any vdW implementa-
tion is enabled. This corresponds to a shrinkage of 𝑑 by more than −15 %. Once vdW
is enabled, the different implementations alter 𝑑 to a lesser extent. All vdW-corrected
values vary by ca. 0.15 Å and approach the experimental reference up to a few per-
cent. The QL width 𝑤 and the basal lattice constant 𝑎 are almost indifferent to the
vdW implementation. Their values range around the experimental reference by ca.
1 %, even if vdW is disabled. As 𝑤 is rigid, the axial lattice constant 𝑐 congruently
follows the interlayer distances 𝑑. Concerning the effects from SOC, the geometry of
the individual QLs is almost indifferent. Both 𝑎 and 𝑤 slightly increase by ca. +0.3 %
upon enabling SOC, which is far less than the already small variations due to the vdW
implementation. In contrast, 𝑑 is surprisingly quite sensitive to whether SOC is en-
abled or not. It contracts by −1.5 % when vdW is disabled, and even stronger by ca.
−5 % when any vdW correction is enabled. The axial lattice constant 𝑐 inherits the
behaviour of 𝑑.
As the combination of DFT-D2 and SOC yields lattice parameters which are very
close to the experimental reference, the volume for this set is redetermined with
a Murnaghan relaxation. The resulting lattice parameters are compiled and com-
pared with the experimental reference [153] in Tab. 8.1. The Murnaghan-relaxed
DFT-D2+SOC structure is very close to the experimental one. The lattice constants
deviate by less than 0.5 %. The unit-cell volumes are even closer together, deviating
by only −0.1 %. The distribution of 𝑐 into 𝑑 and 𝑤 shows a bit more deviation (−0.9 %
and +0.8 %, respectively). The experimental and theoretical Wyckoff parameters 𝑢
and 𝑣 agree. The bond distances in the DFT-D2+SOC structure are 2.87 Å for the outer
Bi–Se bond and 3.06 Å for the inner Bi–Se bond. This accords with the coordination
polyhedrons: Because the inner Se atom is higher coordinated (six Bi atoms), the in-
dividual Bi–Se bonds are expected to be weaker and thus to be longer than those of
the lower coordinated outer Se atom (three Bi atoms). Another point of view on this
134
8.1 Bismuth selenide
𝑎 (Å) 𝑐 (Å) 𝑐/𝑎 𝑑 (Å) 𝑤 (Å) 𝑢 𝑣
Exp. [153] 4.143 28.636 6.91 2.579 6.966 0.4008 0.2117
DFT-D2 + SOC 4.134 28.732 6.95 2.556 7.021 0.4000 0.2111
Deviation −0.2 % +0.3 % −0.9 % +0.8 %
Table 8.1: Lattice parameters of bulk Bi2Se3. The unit cell was optimised by Murnaghan re-
laxations, including DFT-D2 and SOC. Experimental reference: [153].
is based on the ionic character of the Se–Bi bonds: The inner one is expected to be
more ionic wherefore it is closer to the sum of the respective ionic radii. On the con-
trary, the outer one is expected to be closer to the sum of the respective covalent radii
[153]. The bond angles are all close to 90◦: 92.1◦, 91.3◦, 84.9◦, 84.9◦ and 95.1◦ from
the outer to the inner angles. Hence, the Bi–Se octahedrons are almost regular, which
indicates a bonding majorly based on the p orbitals.
8.1.2 Electronic properties
The band structure of the bulk phase of Bi2Se3 is calculated in a manner which is
analogous to Sec. 4.3. All system-specific parameters are the same as those of the
relaxations above. In particular, the bands are calculated with SOC. The trigonal unit
cell of the DFT-D2+SOC structure is used. As the (0001) surface of the hexagonal unit
cell is of special interest later in this work, the corresponding symmetry points K, M
and Z are identified in the trigonal unit cell and chosen as vertices for the k-path4.
Fig. 8.2 shows the bands. The line colour corresponds to the colour bars above the
panels: (a) Bi/Se portions in all PAW projections, (b) s-orbital/p-orbital portions in all
PAW projections, (c) (p𝑥 + p𝑦)-orbital/p𝑧-orbital portions in all p-orbital projections.
The line widths in (a) and (b) are constant and that in (c) corresponds to the sum of
the p-orbital projections of all atoms.
The first conspicuous feature of the band structures consists in a strict separation of
the p bands lying near the Fermi level from the s bands lying at higher binding en-
ergies. The three lower, electron-like s bands accept the six Se-4s electrons. The two
higher, hole-like s bands accept the four Bi-6s electrons. The two sets of s bands are
separated by a gap of≈1.5 eV. The s bands in total are separated from the p bands by a
gap of ≈2.5 eV. That the deep and almost localised s electrons do not contribute much
to the bonding between the Bi and Se atoms perfectly agrees with the expectations
from the inert-pair effect. The nine p bands below the Fermi level host the 18 p elec-
trons from the constituents and are separated from the conduction bands by a global
band gap of ≈0.3 eV. The mixed Se-Bi character indicates a nearly covalent binding
between the p orbitals, which agrees well with the intermediate electronegativity
difference and the regular coordination octahedrons in the Bi2Se3 structure.
4The MΓ path and the ΓKM path in Fig. 8.2 are straight, perpendicular line segments. Since the two
M-points are not the same in the rhombohedral centring, they provide different eigenvalues. They
become identical in the hexagonal unit cell up to 𝐶3 and PT symmetry operations.
135
8 The bulk phases of bismuth selenide and antimony
(a) Bi Se (b) s p (c) px+py pz
0 1
-5 band gap~0.3 eV
Inert-pair effect 0
-10
-1
1Å−1
-15M Γ K M|ΓZ M Γ K M|ΓZ M Γ K M|ΓZ
Figure 8.2: Band structure of Bi2Se3 for the rhombohedrally centred DFT-D2+SOC structure.
The symmetry points correspond to the hexagonal unit cell. The line colour corresponds to
the colour bars above the panels. (a) Bi portion (purple) vs. Se portion (light green) in all PAW
projections; (b) s-orbital portion (red) vs. p-orbital portion (blue) in all PAW projections; (c)
(p𝑥 + p𝑦)-orbital portion (blue) vs. p𝑧-orbital portion (yellow) in the p-orbital projections of
all atoms (line width) round the Fermi level. The respective PDOSs are plotted against the
y-axis to right of the band structures in arbitrary units.
The ℤ2 invariant
The relativistic effects are so strong in Bi2Se3 that its bands distort into a ℤ2 insu-
lating system. From an interpretational point of view, SOC gives rise to a repulsion
between states having the same parity and total angular momentum, e.g. |p+𝑧 , ↑⟩ and
|p++i , ↓⟩. In Bi2Se3, the repulsion inverts the band order at the Γ-point in that a P-𝑥 𝑦
symmetric p𝑧 band and a P-antisymmetric p𝑧 band swap occupations. At the other
TRIMs, the band order remains the same [154]. The remainder of the band inversion
is visible in Fig. 8.2 (c). Near the Γ-point, the CBM and the VBM are of p𝑧 character.
The valence band shows a clear furrow and the conduction band is apparently flat-
ter than a parabola. This is the result of overlaying two parabolae and introducing
gaps at the intersections. A comparison between the Bi2Se3 band structures with and
without SOC which clarifies this issue can be found in [154]. The new point about the
band structures presented here consists in the colouring which proves that the band
inversion at the Γ-point indeed happens in the p𝑧 states, as schematised in [154].
In order to verify that Bi2Se3 is indeed a TI, the ℤ2 topological invariant is explicitly
calculated by means of the parity method. For this purpose, the Quantum Espresso
(QE) package [4, 5] is employed because the parities are difficult to extract from the
VASP output. The pseudopotentials are taken from the pslibrary (straightforward
valency + d electrons, PAW, PBE, full-relativistic). The ground-state density of the
above rhombohedrally centred DFT-D2+SOC structure is obtained at a cutoff energy
of 60 Ry and a k-point mesh of 12×12×12. For this ground-state density, the eigenval-
ues and eigenfunctions are calculated at the eight TRIMs {𝚪𝑖}. Then 𝛿𝑖 is calculated
for each TRIM by extracting the parities of the Kramers pairs and inserting them into
Eq. (7.88). The product of all eight 𝛿𝑖 returns the ℤ2 invariant by Eq. (7.87). The
rhombohedral unit cell of Bi2Se3 has 78 valence electrons (30 Se-3d + 20 Bi-5d + 6
Se-4s + 4 Bi-6s + 12 Se-4p + 6 Bi-6p). Including all the bands returns 𝛿0 = −1 for 𝚪0,
136
E − EF (eV)
8.2 Antimony
the Γ-point, and 𝛿𝑖 = +1 for all other TRIMs. Thus, Bi2Se3 is a topologically non-trivial
ℤ2 insulator. According to the classification by Fu and Kane [147], the topological
class is 1;(000), where the first integer indicates the 3D ℤ2 invariant. Repeating the
procedure for the s and p electrons only, leaving the d electrons out, yields the same
result, as expected from the non-binding nature of the complete d shells. A more
interesting case consists in calculating the ℤ2 invariant for the p electrons only and
leaving the s electrons out. This is rectified as the two submanifolds are separated by
a finite band gap. The s bands yield a trivial ℤ2 invariant, while the p bands yield a
non-trivialℤ2 invariant. This underlines the inert-pair effect and that the topological
non-triviality is solely due to the ordering of the p bands. Repeating the procedure
for the hexagonal unit cell yields the same results, as expected from the stability of
the 3D ℤ2 invariant against band folding.
8.2 Antimony
Antimony denotes the elemental phase of the pnictogen Sb5, the lighter, fifth-row
homologue of Bi. Before analysing its structure, also in view of the heterostruc-
tures, it is instructive to compare the different bulk phases of the pentels: In order
to complete the octets, the pentels tend to cross-link in mutual trigonal-pyramidal
coordination. There are several possibilities for saturating all bonds, the allotropes,
whose stability depends on the thermodynamic boundary conditions. Phosphorus,
for instance, adopts two metastable phases: white phosphorus (P4 tetrahedrons) and
violet phosphorus (cross-linked tubes). Mediated by pressure, they transition into
the orthorhombic black phase, which is stable under normal conditions. Under high
pressure (>80 000 bar), the black phase reversibly transitions into a further, rhombo-
hedral modification [17, pp. 849ff]. Both the orthorhombic, low-pressure phase (the
𝛼-phase6) and the rhombohedral, high-pressure phase (the 𝛽-phase) are vdW mater-
ials consisting of few-atoms layers wherefore they are of special interest in this work.
Bulk As adopts both the 𝛼-phase and the 𝛽-phase under normal conditions, with the
difference to P that the 𝛼-phase (semiconducting, black) is only metastable and the
𝛽-phase (semimetallic, grey-lustrous) is stable. For Sb and Bi, the 𝛼-phase becomes
5There are two Latin words for the mineral stibnite Sb2S3: the medieval word antimonium and the
ancient word stibium. While the origin of the former is unclear, the latter stems from an ancient
cosmetic product. Today, antimony refers only to the elemental metal, while for derivates, the root
stib- should be used. Of course, the latter name is the origin of the chemical symbol Sb.
Heilen mit Antimon: Von der Chemiatrie zur Chemotherapie, https://www.pharmazeutische-zeitung.
de/inhalt-10-2000/titel-10-2000/ (visited on 21/05/2022).
The ancient Romans used “stimi” or “stibi” for pharmaceutical and cosmetic purposes. Plinius Maior
wrote in his famous encyclopedia “Naturalis Historia” that the drying and tightening properties of
stibium were used by women, who covered the area round their eyes with stibnite additives so as
to widen them. The literal translation of “platyophthalmon” exactly means that.
Gaius Plinius Secundus (Plinius Maior), Naturalis Historia, Liber XXXIII.101f
6The terminology is not unique as, in fact, 𝛼-P and 𝛽-P denote modifications of white phosphorus
P4 [17, p. 850]. The nomenclature used in this work concentrates on the distinction between the
isolated layers from the two layered phases (black phosphorus and high-pressure phosphorus). It is
common in most works dealing with thin Pn sheets [P5, P10, 155–157].
137
8 The bulk phases of bismuth selenide and antimony
(a) C (b) T
H
w T 1
3 (c) 2.4
2.31
B 4 2.3 d (Å)
A d
C c 2.2 2.24
B 2.14 2.22
A BL a
2.13
2.1
d d [‾11.0] SOC no vdW D2 D3 D3+BJ
[‾11.0] 1 2
Figure 8.3: Bulk structure of 𝛽-Sb. Blue circles are Sb atoms. (a) Side view; (b) Top view. The
conventional unit cell is marked red. (a) shows the rhombohedrally centred primitive unit
cell as well. Yellow stars in (b) mark some T1 positions of the (0001) surface. (c) Interlayer
distance 𝑑 of the structures optimised by Murnaghan relaxations with SOC and different vdW
schemes (see text). The experimental reference [158] is inserted as a dashed line.
entirely unstable and only the 𝛽-phase remains stable under normal conditions. 𝛽-Sb
is a semimetal like 𝛽-As, while 𝛽-Bi is a metal [17, pp. 943ff].
Like Bi2Se3, 𝛽-Sb belongs to the trigonal, P-symmetric𝑅3̄𝑚 space group (No. 166). The
Sb atoms occupy the Wyckoff position 2𝑐. 𝛽-Sb consists of covalently bound sheets of
buckled-honeycomb-Sb which stack in a manner that the lower Sb atom of the next
sheet is over the H3 position of the previous one. The individual sheets are considered
as bilayers (BLs), which stack according to a layer sequence with an axial periodicity
of 6 atomic layers or 3 BL: (AB)(CA)(BC). The structure of 𝛽-Sb is sketched in Fig. 8.3 (a)
and (b). Obviously, the BLs come so close together that the distance between them is
not much larger than their width. In terms of bond distances, 𝑑1, the Sb–Sb distance
within the BLs, is not much larger than 𝑑2, the distance across. This is a trend in the
pnictogens as the 𝑑2/𝑑1 ratio decreases from P to Bi so that the 𝛽-phase approaches
the cubic primitive structure [17, p. 943].
8.2.1 Structure optimisation
Computational parameters
The DFT calculations are carried out with VASP [2, 3] using PAW potentials and the
PBE xc-functional [10, 21, 22]. The Sb atoms have a PAW valence of 5s25p3. The kin-
etic energy cutoff is 𝐸cut = 300 eV. The Brillouin zone is sampled by a Γ-centred
Monkhorst-Pack mesh [117] at a density of 16 × 16 × 16 k-points for the rhombo-
hedrally centred unit cell. The Brillouin-zone integration is carried out by a Gaus-
sian occupation with a smearing of 5 meV for the ionic relaxations and by the tet-
rahedron method with Blöchl corrections [118] for all static calculations. A conver-
gence test proves that increasing the energy cutoff or the sampling density alters the
total energy by ≈1 meV per atom. The ionic forces are calculated according to the
Hellmann-Feynman theorem [119]. The ions move along them towards the equilib-
rium positions and stop to relax if the forces acting on each ion are smaller than
0.005 eV/Å. SOC is enabled in all calculations. For different vdW approaches, the
volume of the rhombohedrally centred unit cell is optimised by the Murnaghan equa-
tion of state. The four approaches tested are: no vdW correction, DFT-D2 [26], DFT-D3
+ zero-damping [27], and DFT-D3 + Becke-Johnson damping [28]
138
[00.1]
[11.0]
8.2 Antimony
Volume optimisation
The relevant lattice parameters of the hexagonal unit cell are: the basal lattice con-
stant 𝑎, the axial lattice constant 𝑐, the interlayer distance 𝑑, the BL width 𝑤, and the
Wyckoff parameter of the rhombohedral centring 𝑧. The results for 𝑑 may be found
in Fig. 8.3 (c). The experimental reference (XRD, [158]) is inserted as a dashed line.
As in the case of Bi2Se3, the vdW correction makes the BLs come closer together.
Though, 𝑑 does not shrink as strongly as in Bi2Se3, only by −0.1 Å to −0.2 Å (−4 %
to −8 %) upon activating any vdW implementation. In fact, disabling the vdW cor-
rections already produces an interlayer distance which lies only +3 % above the ex-
perimental reference. DFT-D2 and DFT-D3+BJ even underestimate 𝑑 by −4 %, while
DFT-D3 yields a good value for 𝑑 lying only−0.7 % below the reference. However, 𝑑 is
not the only vdW-sensitive lattice parameter. Also 𝑎 varies up to 2 %, where DFT-D2
and DFT-D3+BJ yield good values and DFT-D3 a value which is too large by +1.4 %. In-
terestingly, all vdW approaches as well as disabling vdW overestimate 𝑤 by ca. +2 %.
The unit-cell volumes𝑉 agree with the experiment in the order: DFT-D3+BJ (−0.6 %),
DFT-D2 (−1.2 %), DFT-D3 (+3.1 %) and no vdW (+6.5 %). The different qualities of the
approaches can be summarised by the Sb–Sb bond distances 𝑑1 and 𝑑2 (Tab. 8.2 (a)):
DFT-D2 describes 𝑑1 better, DFT-D3 describes 𝑑2 better and omitting the vdW correc-
tion overestimates both distances. Since DFT-D2+SOC yielded good results for Bi2Se3
and will be chosen as the central approach for the heterostructures, the complete set
of lattice parameters of the respective 𝛽-Sb unit cell are compiled in Tab. 8.2 (b).
(a) Exp. [158] no vdW DFT-D2 DFT-D3 DFT-D3+BJ
𝑑1 (Å) 2.90 2.95 (+1.8 %) 2.92 (+0.7 %) 2.95 (+1.5 %) 2.94 (+1.2 %)
𝑑2 (Å) 3.34 3.43 (+2.5 %) 3.29 (−1.7 %) 3.36 (+0.5 %) 3.28 (−1.8 %)
(b) 𝑎 (Å) 𝑐 (Å) 𝑉 (Å3) 𝑑 (Å) 𝑤 (Å) 𝑧
Exp. [158] 4.301 11.222 179.75 2.238 1.503 0.2336
DFT-D2 + SOC 4.311 11.033 177.59 2.144 1.534 0.2362
Deviation +0.2 % −1.7 % −1.2 % −4.2 % +2.1 %
Table 8.2: Lattice parameters of bulk 𝛽-Sb. (a) Bond distances 𝑑1 and 𝑑2 under different vdW
treatments. (b) Lattice parameters optimised with DFT-D2 and SOC. The unit cell was optim-
ised by Murnaghan relaxations. Experimental reference: [158].
8.2.2 Electronic properties
The band structure of the DFT-D2+SOC structure of 𝛽-Sb is plotted in Fig. 8.4 (a) and
(b), where the plotting procedure is analogous to that in Fig. 8.2 (b) and (c). In ad-
dition, the bands of the isolated Sb sheets are inserted in (c)→ 1 BL and (d)→ 2 BL.
Their structures are the peeled-off structures from the respective heterostructures
in Sec. 9.1, i.e. the heterostructure optimised with DFT-D2+SOC after removal of the
substrate. In this manner, the bands of the isolated sheets can directly be compared
139
8 The bulk phases of bismuth selenide and antimony
(a) s p (b) px+py pz (c) 1BL (d) 2BL
2
0
0
-5
-2
unhyb-
-10 ridiseds orbital -4
-15 -6M Γ K M|Γ Z M Γ K M|Γ Z M Γ K M Γ K
Figure 8.4: Band structure of (a) and (b) bulk 𝛽-Sb and (c) and (d) isolated 𝛽-Sb sheets. For
bulk 𝛽-Sb, the rhombohedrally centred DFT-D2+SOC structure was used. The 𝛽-Sb sheets are
the peeled-off structures of the respective Sb@Bi2Se3 heterostructures. The symmetry points
correspond to the hexagonal unit cell. The line colour corresponds to the colour bars above
the panels. (a) s-orbital portion (red) vs. p-orbital portion (blue) in all PAW projections for
bulk 𝛽-Sb; (b) – (d) (p𝑥 + p𝑦)-orbital portion (blue) vs. p𝑧-orbital portion (yellow) in the p-
orbital projections of all atoms (line width) for (b) bulk 𝛽-Sb, (c) an isolated 1 BL sheet and
(d) an isolated 2 BL sheet of 𝛽-Sb. The respective PDOSs are plotted against the y-axis to right
of the band structures in arbitrary units.
with the Sb bands in the layer-desorption-and-readsorption procedure for tracking
the topological phase transitions in Sec. 9.3. Since the bands of the isolated sheets
thematically fit better to this chapter, they are presented here.
𝛽-Sb is a semimetal in which the valence bands and the conduction bands are sep-
arated locally in 𝒌, but not by a global band gap. In other words, the band gap is
negative, which will be important for the ℤ2 classification. There are two s bands,
which host the four 5s electrons from the two Sb atoms. The other three valence
bands are of p character and host the six 5p electrons. The s bands and the p bands
are clearly separated from each other at every k-point, which reminds of the Bi2Se3
case. Though, the band width of the s bands is larger than that of Bi2Se3. That the s
orbitals do not hybridise with the p orbitals indicates a certain unwillingness of the s
electrons to take part in chemistry and agrees with the nearly right-angled geometry
of the BLs (95.0◦ bond angle within the BLs).
While the local band gap is large in-plane, the valence bands and the conduction
bands approach each other near the out-of-plane Z-point. The p𝑧-resolved bands
show a peculiarity there: The uppermost valence band is of p𝑧 character and has
a positive slope on most of the Γ𝑍 path. Near the Z-point, however, it bends down
and changes to a hybrid between p𝑧 and (p𝑥 + p𝑦). That the opposite happens in
the lowermost conduction band hints at a band inversion at that point. This might
produce a topologically non-trivial state if the negativeness of band gap is ignored.
The ℤ2 invariant
The supposed topologically non-trivial state due to the apparent band inversion can
easily be checked by calculating the ℤ2 invariant. In doing so, the negative band
gap is ignored, i.e. the bands are assumed to be separated into a well-defined set of
140
E − EF (eV)
8.2 Antimony
valence and conduction bands by making the local band gaps global. The procedure
is completely analogous to the calculation of theℤ2 invariant of Bi2Se3. Again the QE
package [4, 5] is employed with a 5s25p4-valent, full-relativistic PAW-PBE potential
for Sb. The rhombohedrally centred DFT-D2+SOC structure used as input. All other
parameters are the same as those in the Bi2Se3 case (vide supra).
Applying the parity method on all ten bands returns a set of five positive and three
negative 𝛿𝑖 . Their negative product proves that the valence manifold is topologically
non-trivial wherefore 𝛽-Sb can be considered a 3D topologically non-trivialℤ2 semi-
metal. A series of gauge transformations on the TRIM cube reduces the number of
negative 𝛿𝑖 to one, which then sits at the vertex 𝚪7, the space-diagonally opposite of
𝚪0. According to the classification of Fu and Kane [147], the full topological class is
1;(111), in agreement with earlier results [159]. Repeating the ℤ2 classification for
the s bands only yields a trivial character, while the p-band subspace produces the
same results as those above. In particular, the position of the non-trivial 𝛿𝑖 at 𝚪7 im-
plies that the band inversion happens the Z-point of the hexagonal unit cell, which
confirms the presumed band inversion in Fig. 8.4.
The classification of bulk 𝛽-Sb as a topologically non-trivial semimetal is per se not
very useful due to the fractional occupation. The practical consequences emerge in
thin films: Since the band widths of a bulk material decrease for thin slices, the band
gaps increase and, in particular, semimetals (negative band gap) can adiabatically
turn into semiconductors/insulators (positive band gap). For (0001)-slices of 𝛽-Sb,
this happens at a thickness of ∼20 BL [160]. In doing so, the valence manifold of the
𝛽-Sb slice inherits the topological character from the bulk semimetal as the bands do
not reorder. Therefore, such films of 𝛽-Sb are 3D ℤ2 TIs with two surfaces bearing
the corresponding TSSs. A further reduction of the slice thickness below 8 BL opens
a small band gap in the TSSs as the bulk character vanishes. However, being a 2D
insulator, the thin slice permits the calculation of the 2Dℤ2 invariant, with the result
that it is a 2D ℤ2 TI. Finally, at a thickness of 3 BL, the gap closes and reopens at a
thickness of 2 BL. This is a topological phase transition which renders 𝛽-Sb sheets
with thicknesses of 2 BL and 1 BL topologically trivial 2D insulators [160]. This is
confirmed in this work, as the 𝛿𝑖 are negative for all four (2D) TRIMs. As a last remark,
the band structures of the isolated sheets in Fig. 8.4 (c) and (d) already hint at the
band inversion in bulk 𝛽-Sb. In the 1 BL structure, the VBM is at the Γ-point and the
three p bands resemble a rehybridised superposition of a blue, hole-like parabola, a
yellow p𝑧 band shaped like a lying B and a deeper, greyish, W-shaped band. When a
second BL is added to the structure, one of the B bands is shifted to the Fermi level
and approaches the conduction bands. Adding further BLs is likely to invert the band
order at the Γ-point. If the band gap reopens after such a reordering, the system will
be in a topologically non-trivial state. This exactly agrees with the observations in
[160] and furthermore confirms the band inversion of the bulk band structure at the
Z-point, which coincides with the Γ-point of the (0001) surface.
141
8 The bulk phases of bismuth selenide and antimony
8.3 Discussion and summary
The investigations of the bulk phases proved that it is absolutely crucial to take vdW
corrections into account when treating a vdW material like Bi2Se3 within DFT. Omit-
ting them results in too large interlayer distances. The DFT-D2 approach in combin-
ation with SOC produced lattice parameters for Bi2Se3 which agree with the experi-
mental reference [153] up to 1 %. The unit cell volumes deviate by only 0.1 %. While
the strong attractive effect between the QLs from the vdW corrections is expectable,
the intermediate, also attractive effect from SOC is uncommon. There are not many
detailed works on the first-principles structure of Bi2Se3, probably because not giving
attention to all peculiarities of the material results in lattice constants which consid-
erably deviate from the experiment. An extensive DFT work testing more vdW ap-
proaches than this work was published by Luo et al. [161]. Their DFT-D2+SOC lattice
constants (QE) are very close to those obtained in this work (VASP). Luo et al. invest-
igated the SOC-related effects on the structure as well. Although they found that SOC
reduces the interlayer distances, in agreement with this work, they doubt the phys-
ical relevance (“it is still unclear if SOC is truly important for structural optimization”
[161]). There is, however, a good reason why SOC is likely to exert physical effects on
the structure: Since Bi2Se3 is a TI, the orbital occupation changes at the Γ-point when
SOC is enabled. This point is certainly worth investigating in more detail, e.g. by
analysing partial charges with and without SOC.
While the results for Bi2Se3 agree with the experiment very well, those obtained for
𝛽-Sb show slightly larger deviations. Although DFT-D3 produces better interlayer
distances, DFT-D2 is considered more suitable for describing 𝛽-Sb for two reasons.
Firstly, the bond distance within the BLs is better predicted by DFT-D2. Secondly, the
unit-cell volume within DFT-D2 is close to the experimental one, contrarily to that
of the DFT-D3 structure. This will be important for the heterostructures since the
Bi2Se3 substrate strains the thin Sb adlayer, so part of the substrate-adlayer distance
is due to anisotropic deformation. In this case, a correct description of the volumes
is obligatory. Aktürk et al. had a work published which investigates the structure of
𝛽-Sb with VASP [155]. Although, they used a parameter set very similar to that of this
work (PBE, cutoff energy, k-mesh), they obtained DFT-D2+SOC lattice constants which
are closer to the experiment [153] than those of this work. Thus, it is not excluded
that the approach of this work was not free of errors. It is certainly worth testing
further vdW functionals on 𝛽-Sb in order to explain the deviances.
𝛽-Sb and Bi2Se3 have in common that the bond angles are close to the right angle
(regular coordination octahedrons in the QLs and right-angled trigonal-pyramidal
coordination in the BLs). This accords very well with the electronic properties as the
major part of the bonding is due to the p orbitals, while the s orbitals are proven
not to hybridise with them. This is much pronounced in Bi2Se3, where the inert-
pair effect prevents the s electrons from chemical bonding. The topological classes
were calculated and agree with previous knowledge. The band inversion in the bulk
phases is clearly visible from the p𝑧-resolved bands. In the topologically trivial, isol-
ated 1 BL and 2 BL sheets of 𝛽-Sb, the beginning band inversion at the surface Γ-point
was demonstrated.
142
9 The antimony on bismuth selenide
heterostructure
The (0001) surface of Bi2Se3, denoted BS(0001), results from cleaving the material ex-
actly between two QLs, in accordance with the layered structure of the vdW material
[157, 162]. Since the atomic shells of the QLs are closed, the surface morphology is de-
termined by a unreconstructed buckled honeycomb. The upper atoms are the outer
Se atoms of the terminating QL and the lower atoms are Bi atoms (vide supra, Fig. 8.1
(b)). The metallic TSSs of BS(0001) give rise to a single Dirac cone at the Γ-point, as
confirmed by DFT calculations and ARPES measurements [P7, P9, 154, 163–166].
The adsorption of Sb on BS(0001) produces the Sb@BS system, a class of vdW het-
erostructures consisting of a Sb adlayer which binds to the Bi2Se3 substrate by vdW
interactions. The thinnest films occur in two different phases depending on the ther-
modynamic boundary conditions: the 𝛼-phase and the 𝛽-phase, named after the two
layered bulk phases of the pnictogens (vide supra, Sec. 8.2). The films can indeed be
considered as single sheets from the respective bulk phases, which are attached to
the substrate. The 𝛼-Sb adlayers are incommensurate with BS(0001) due to differ-
ent 2D symmetry groups (orthorhombic vs. trigonal). In contrast, the 𝛽-Sb adlayers
have the same symmetry as BS(0001) and, in addition, similar basal lattice constants
(𝑎𝛽-Sb = 4.30 Å and 𝑎Bi2Se3 = 4.14 Å, vide supra). Therefore, they are commensurate
with the substrate and can reach heights of several BLs by vdW epitaxy [157, 166,
167]. The 𝛼-phase can be transformed into the 𝛽-phase by a temperature-induced
phase transition [P5, P10].
Beside the structural peculiarities of the system, the electronic properties attract sci-
entific attention as well. The interplay between the TSSs of the substrate and the 2D
states of the 𝛽-Sb adlayer (CIs for thicknesses of 1 BL and 2 BL, vide supra) is complex.
Although the adsorption is based on vdW interactions, the electronic band structure
of the heterostructure looks rather different from that of the constituents. Theoret-
ical work on the heterostructures has been published [166, 168], though the origin
of the features has not yet been satisfyingly understood. Furthermore, high-quality
band structures and ARPES images are missing and the connection between the band
structures and the structural details has not been investigated yet.
The aim of this chapter is thus to understand the electronic properties of commen-
surate 𝛽-Sb films with thicknesses of 1 BL and 2 BL by means of DFT. For this pur-
pose, structure models are set up and subjected to ionic relaxations. The focus of
the structure analysis is on the way how the Sb BLs stack on the substrate, which
has not yet been paid sufficient attention, but proves to be thermodynamically rel-
evant. Also the effects from vdW corrections and SOC are illuminated. In particu-
143
9 The antimony on bismuth selenide heterostructure
lar, the effects from SOC are paid a special attention since one tends to deactivate
SOC during the relaxations to save computation time. For the thermodynamically
stable structure models, the electronic band structures are calculated and compared
with experimental ARPES measurements. Furthermore, ab initio STM simulations
are performed and the results are compared with experimental STM measurements.
The last section contains a simulated desorption-and-readsorption process for the
𝛽-Sb adlayer which permits tracking the evolution of the TSSs upon Sb layer adsorp-
tion. By this means, a complex series of topological phase transitions is illuminated.
The deep analysis finally explains all details of the electronic band structures of the
heterostructures. The results have already been published [P5, P7, P9].
9.1 Structure optimisation
The Bi2Se3 slab
The topological non-triviality of Bi2Se3 discourages the asymmetric-slab approach
since the passivation of the metallic surface states is impossible. Therefore, the Bi2Se3
substrate is modelled by a symmetric, right-prismatic slab, which provides identical,
hexagonal surfaces on the top and the bottom side. The lattice parameters of the
slab are taken from the Bi2Se3 unit cell optimised with DFT-D2 and SOC (vide supra,
Tab. 8.1). The thickness of the slab amounts to 6 QL, which is thick enough so that the
TSSs of the two surfaces do not interfere1. The periodic replica in the z-direction are
separated from each other by 60 Å of vacuum, which reduces to 40 Å when the slab
bears the Sb adlayers. The slab is sketched in Fig. 9.1 (a) and (b).
Computational details
The computational parameters for the heterostructure are similar to those used for
the constituents (Chap. 8). The DFT calculations are carried out with VASP [2, 3] us-
ing PAW potentials and the PBE xc-functional [10, 21, 22]. The Bi, Se and Sb atoms
have PAW valences of 5d106s26p3, 4s24p4 and 5s25p3, respectively. The kinetic en-
ergy cutoff is 𝐸cut = 400 eV. The Brillouin zone is sampled by a Γ-centred surface
Monkhorst-Pack mesh [117] at a density of 12 × 12 × 1 k-points. The Brillouin-zone
integration is carried out by a Gaussian occupation with a smearing of 5 meV for the
ionic relaxations and by the tetrahedron method with Blöchl corrections [118] for
all static calculations. The ionic forces are calculated according to the Hellmann-
Feynman theorem [119]. The unit cell is fixed. The atoms of the inner four QLs are
held fixed at the bulk positions during all relaxations. All other ions, the top QL right
below the surface and the Sb adlayer, move within the unit cell along the forces to-
wards the equilibrium positions and stop to relax if the forces acting on each ion are
smaller than 0.005 eV/Å. Unless otherwise noted, SOC is explicitly included also in the
relaxations and the DFT-D2 method [26] is employed for the vdW contributions.
1Test calculations show that the real-space distributions of the wavefunctions near the Dirac points of
the two surfaces overlap up to 5 QL, although the band gap already closes at a thicknesses of 3 QL.
144
9.1 Structure optimisation
(a) (b) 60 Å (c) P01 P10 P0120 P0102
vacuum H3 T1 BL2
T1 H3 T4 w
tQL T1 H3
T H d4 3 T4 T 24 BL1
H3 T4 H3 H3
QL d1
tQL
[‾11.0] + symmetriclower half [‾11.0]
Figure 9.1: Sketch of the BS(0001) slab. Purple circles are Bi atoms; Light green circles are Se
atoms; Blue circles are Sb atoms. (a) Top view of the slab. (b) Upper half slab in side view.
The lower half results from inversion symmetry. Dark red lines mark the unit cells. The light
red rectangle in (b) encircles the subset of atoms fixed in position during the relaxations.
(c) Examples for Sb adlayers with heights of 1 BL and 2 BL. The symmetry positions of the
substrate T1, T4 and H3 are indicated in (a) and (c). Reproduced from [P9].
9.1.1 The stacking sequence of the adlayer
The structure models for a vdW heterostructure in general have many degrees of
freedom, in particular if the adlayer is not commensurate with the substrate. In the
case of Sb@BS, the experimentally confirmed commensurateness [P5, 157] constrains
the configuration space to those structures where the Sb atoms of the first BL sit at the
symmetry positions of the terminating buckled honeycomb of the BS(0001) surface:
T1, T4 and H3 (Fig. 9.1 (a)). The adsorption of multiple BLs can be understood in a
successive manner as the BLs themselves are buckled honeycombs, which provide
each a new triple of symmetry positions for the next BL.
The bulk phases of both Bi2Se3 and 𝛽-Sb underlie a stacking which consists in the
strict repetition of the atomic-layer sequence ABC with differing vdW-layer interrup-
tions (Chap. 8). In order to transfer this to the Sb@BS heterostructure, let tQL, BL1
and BL2 be the top QL of the substrate, the first BL of 𝛽-Sb and the second BL of 𝛽-Sb,
respectively. If BL1 continues the sequence of tQL (ABCAB) by (CA), the lower Sb atom
is above H3 and the upper Sb atom above T4. BL2 can proceed with the indices (BC).
Though, this is not the only possibility to stack the Sb adlayer in a reasonable manner:
An inverted BL1 having the atomic-layer indices (AC), i.e. lower Sb atom above T4 and
upper Sb atom above H3, would fit into the same two surface symmetry positions as
the non-inverted (natural) BL1. Of course, the combinatorics inflate the configuration
space with growing thicknesses of the Sb adlayer. Therefore, the configurations are
systematised by the following nomenclature: The three original symmetry positions
of the BS(0001) surface are mapped onto integers: H3→ 0, T4→ 1 and T1→ 2. A Sb@BS
structure model is denoted “P” plus the sequence of integers which the adsorbed Sb
atoms occupy. Thus, the single, natural BL is called “P01”, the single, inverted BL is
called “P10” and two natural BLs are called “P0120”. The nomenclature simplifies the
identification of inverted structures as it reduces to permutations between integers.
145
[11.0]
fixed in 
position
[00.1]
9 The antimony on bismuth selenide heterostructure
Fig. 9.1 (c) shows examples for the nomenclature. Furthermore it defines the inter-
layer distance between tQL and BL1 𝑑1 and that between BL1 and BL2 𝑑2, which will
be used for the analysis of the structures.
In order to compare the different structure models regarding their thermodynamical
stability, consider the following boundary conditions: The Sb adlayer forms in the
annealing stage, which translates into a constant temperature𝑇 , a constant pressure
(𝑝 = 0, UHV) and a constant amount of Sb 𝑁Sb. If Sb diffusion into the Bi2Se3 substrate
is negligible, the amounts of Bi and Se can be considered constant, as well as their
chemical potentials. Consequently, the substrate contributes a constant term to the
DFT total energy of the Sb@SB slabs 𝐸Sb@BSDFT . If this term is set to the DFT total energy of
the “clean” slab without adsorbates 𝐸Clean and subtracted from 𝐸Sb@BSDFT DFT , the remainder
approximately equals the internal energy of the Sb adsorbate only. Transforming it
into the Gibbs free energy and dividing it by 𝑁Sb results in the Sb chemical potential
𝜇Sb. This is the quantity to minimise and permits the direct comparison between
structure models with different amounts of Sb [P5, P9, P10, 157]. The Sb chemical
potential of (natural) bulk 𝛽-Sb 𝜇b-natSb is chosen as the reference point.
𝐺adlayer ( )
𝜇 = − 𝜇b-nat ≈ 1 𝐸Sb@BS − 𝐸Clean b-natSb
𝑁 Sb 𝑁 DFT DFT
− 𝜇Sb (9.89)
Sb Sb
Analogously to the RESi2 case, the phononic contributions to the free energy are neg-
lected (vide supra, Sec. 4.2.1). This approximation is based on the similar, layered
structures, suggesting that each Sb BL has similar lattice-dynamical contributions.
Since the following analysis concentrates on the adsorption of different 𝛽-Sb adlay-
ers, it is instructive to consider hypothetical phases of elemental 𝛽-Sb:
• An alternative bulk phase in which every second BL is inverted (b-inv). The
stacking sequence is thus (AB)(CB). Its structure is optimised at code level.
• Free-standing 𝛽-Sb sheets, whose structures are optimised by the 2D-version of
the Murnaghan equation of state: the 1 BL sheet (f1BL) and the 2 BL sheet with
natural (f2BL-nat) and inverted stacking (f2BL-inv).
Tab. 9.1 shows the Sb chemical potentials 𝜇Sb, the basal lattice constants 𝑎 and the
interlayer distances 𝑑 for these structures and for the stable bulk phase (b-nat).
System Abbr. 𝜇Sb (meV) 𝑎 (Å) 𝑑 (Å)
Bulk (natural stacking) b-nat 0 4.31 2.14
Bulk (inverted stacking) b-inv 40 4.12 2.84
Free-standing 1BL sheet f1BL 276 4.05 —
Free-standing 2BL sheet (nat. stacking) f2BL-nat 178 4.16 2.43
Free-standing 2BL sheet (inv. stacking) f2BL-inv 173 4.08 2.97
Table 9.1: Sb chemical potentials 𝜇Sb, basal lattice constants 𝑎 and interlayer distances 𝑑 of
2D and 3D 𝛽-Sb systems. Reproduced from [P9].
146
9.1 Structure optimisation
(a) 200 180 184
150
100 89 76
55 65 36 39 45 Figure 9.2: (a) Sb chemical potentials50 33 µb-invSb 𝜇Sb and (b) interlayer distances 𝑑1
0 (squares) and 𝑑2 (circles) of the struc-
(b) 3.8 ture models for the Sb@BS system
3.6 d1 optimised with DFT-D2+SOC. The two
3.4 d2 bulk phases of 𝛽-Sb b-nat (stable) and3.2
3.0 b-inv (hypothetical) are inserted asi i
2.8 db-inv dashed lines. Red “i” symbols mark in-
2.6 i i verted Sb BLs. Reproduced from [P9].
2.4
2.2 db-nat
01 12 20 10 21 02 20 02 1 2P P P P P P 1 1 02 01
P0 P0 P1 P1
At first, 1 BL thick 𝛽-Sb adlayers (1BL-Sb) are investigated by optimising the structure
of all six adsorption possibilities. The results for 𝜇Sb and 𝑑1 are plotted in Fig. 9.2
(blue). P01 is the most stable structure with 𝜇Sb = 55 meV. Hence, it is indeed favour-
able if BL1 continues the stacking sequence of the Bi2Se3 substrate. Surprisingly, P10,
the inverted BL, is only slightly less stable by Δ𝜇Sb = +10 meV. Since the difference is
below 𝑘B𝑇 at room temperature, the two structures are energetically so close to each
other that a conclusion about their relative stability is not possible. The next stable
pair of structures are P02 and P12, where the upper Sb atom is above T1. Their 𝜇Sb
are higher than that of P01 by +21 meV and +34 meV, also in the order of 𝑘B𝑇 at room
temperature. Only P20 and P21, where the lower Sb atom is above T1, are clearly
unstable against P01 by Δ𝜇Sb > +100 meV.
The interlayer distances 𝑑1 congruently follow 𝜇Sb. This permits the conclusion that
the binding energy between the substrate and BL1 depends on how close the former
lets the latter approach. Continuing the stacking sequence of the substrate, P01 has
the smallest interlayer distance of all 1BL-Sb structures (𝑑1 = 2.40 Å, +0.25 Å com-
pared to 𝑑b-nat), which minimises 𝜇Sb. In contrast, in P20 and P21, the top Se atom
of the substrate pushes the BL quite far away (𝑑1 > 3.5 Å). This reduces the binding
energy to such an extent that the BL is energetically closer to the free-standing 1 BL
sheet (f1BL) than to P01. In all other structures, 𝑑1 is close to 𝑑b-inv.
The results for 2 BL thick 𝛽-Sb adlayers (2BL-Sb) are plotted in Fig. 9.2 (green). Us-
ing the 1BL-Sb results, only those 2BL-Sb structures are considered where both BLs
successively occupy the convenient H3 and T4 positions. These are P0120 (BL1 nat,
BL2 nat), P0102 (BL1 nat, BL2 inv), P1021 (BL1 inv, BL2 nat) and P1012 (BL1 inv, BL2
inv). Similarly to the 1BL-Sb case, it is favourable if the BLs continue the stacking
sequence of the substrate (P0120). However, the 𝜇Sb of the other three structures are
surprisingly close to that of P0120 (Δ𝜇Sb ≤ 12 meV ∼ 𝑘B𝑇 ). Hence, it remains unclear
which structure model represents the real 2BL-Sb structure. The interlayer distances
clearly conform with the layer inversions: If the BL is natural/inverted, the corres-
ponding 𝑑𝑖 is closer to 𝑑b-nat/𝑑b-inv (red “i” symbols in Fig. 9.2).
147
d (Å) μSb (meV)
9 The antimony on bismuth selenide heterostructure
Tab. 9.1 shows that in both cases of natural and inverted stacking, the basis of bulk
𝛽-Sb is larger than that of a free-standing 2 BL sheet, which in turn is larger than that
of a free-standing 1 BL sheet. This suggests that a free-standing 𝛽-Sb sheet laterally
expands with growing thickness. In the bulk limit, the basal lattice constant of b-
inv is smaller than that of b-nat (−4.4 %). The behaviour of the sheets has important
consequences for the lattice matching between the Sb adlayer and the Bi2Se3 sub-
strate (surface lattice constant 𝑎BS = 4.13 Å): Considerations based on natural bulk
𝛽-Sb would lead to the wrong conclusion that the Sb adsorbate underlies compress-
ive strain (−4.0 %). On the contrary, the strain is even tensile in the 1BL-Sb hetero-
structure (+2.1 % w.r.t. f1BL), while in 2BL-Sb, the lattice matching is almost perfect
(f2BL-nat: −0.7 %, f2BL-inv: +1.2 %). For thick, bulk-like Sb adlayers, the inverted
stacking (b-inv) produces an almost perfect lattice matching (+0.2 %), while a natur-
ally stacked layer (b-nat) has to be compressed by−4.0 % (vide supra). Since adapting
the bases to BS(0001) makes 𝜇b-nat b-invSb approach 𝜇Sb , it is not clear which of the two
stackings occurs even for thick Sb adlayers [P9].
The free-standing 2 BL sheets demonstrate why the inversion of BL1 and BL2 in the
2BL-Sb heterostructure has such a low impact on the stability. Firstly, 𝜇Sb is almost
equal in both free-standing 2 BL sheets, differing only by 5 meV. Secondly, both sheets
match the substrate equally well, so the adsorption energy per atom depends only on
the relation between BL1 and the substrate. Thirdly, BL1 is indifferent to inversions
in the 1BL-Sb case. These three points combined explain why all four investigated
2BL-Sb structures are energetically so close together that a conclusion about the real
structure is not possible. In spite of this, the subsequent work concentrates on the
heterostructures with natural stacking: P01 and P0120.
9.1.2 The effects from SOC and vdW on the heterostructure
Although DFT-D2+SOC is the approach of choice in this chapter, it is interesting to
investigate how SOC and the vdW corrections affect the heterostructures. For this
purpose, the structure models for 1BL-Sb and 2BL-Sb are optimised with disabled cor-
rections: pure PBE, DFT-D2 only, SOC only. All other parameters remain unchanged.
The results for the interlayer distances 𝑑1 and 𝑑2 are plotted in Fig. 9.3. Similarly to
bulk Bi2Se3, both vdW and SOC make the adlayer move closer to the substrate. In
1BL-Sb, 𝑑1 (blue squares) shrinks by a similar percentage if either vdW or SOC is en-
abled (−15 % and −10 %, respectively). If both are enabled, the shrinkage amounts
to −20 %. The overall contractive effect and the importance of SOC are more pro-
nounced in the heterostructure than in bulk Bi2Se3. In 2BL-Sb, 𝑑1 (green squares) is
generally smaller, but the relative shrinkage is similar (−17 % if enabling vdW and
SOC). 𝑑2 (green circles) behaves differently from 𝑑1 as it contracts upon enabling vdW
(−5 % to −6 %), but varies much less due to SOC (+1 % to +2 %).
Back to DFT-D2+SOC, while 𝑑1 and 𝑑2 vary considerably, the other lattice parameters
are close to the respective bulk values. The thickness of tQL and the interlayer dis-
tance beneath tQL are very close to 𝑤 and 𝑑 in bulk Bi2Se3, deviating by <0.5 % and
<1 %, respectively. The Sb–Sb bond distances in the adsorbed BLs (2.88 Å to 2.89 Å)
are close to the respective bond distance in bulk 𝛽-Sb (2.92 Å, −1.4 % to −1.0 %).
148
9.2 Electronic properties
3.0
2.98 1BL-Sb 2BL-Sb
d1 d1
2.8 d2 Figure 9.3: Interlayer distances 𝑑1 (squares)2.78
2.68 and 𝑑2 (circles) of 1BL-Sb (blue) and 2BL-Sb
(green) optimised with different corrections.
2.6 2.59 bulk Bi2Se3: 2.58 Å
2.53 Treatments: 1) pure PBE, 2) SOC only, 3) DFT-2.58 2.57
2.46 D2 only, 4) DFT-D2 and SOC. The experimental
2.42 interlayer distance of the bulk phases are inser-
2.4 2.382.41 ted as horizontal dashes lines: purple→ Bi2Se3
[153]; blue→ 𝛽-Sb [158].
bulk β-Sb: 2.24 Å 2.30
2.2
PBE SOC D2 D2+SOC
9.2 Electronic properties
The band structures of the three slabs (clean Bi2Se3, 1BL-Sb and 2BL-Sb, all optim-
ised with DFT-D2+SOC) are calculated analogously to Sec. 4.3 and may be found in
Fig. 9.4. All system-specific parameters are the same as those of the relaxations above.
The surface k-path comprises two line segments: KΓ ∥ (112̄0) in the x-direction and
ΓM ∥ (1̄100) in the y-direction (sketch in Fig. 9.4 (b)). The bands associated with the
surface are separated from the bands of the simulated bulk by means of the PAW
projections. In detail, the summed PAW projections of the atoms of the upper half
slab divided by the sum over all PAW projections of the slab minus 0.5 defines the
line width. Thus, a band which is totally localised in the upper half slab is displayed
at the maximum line width, while bands localised to less than 50 % in the upper half
are displayed as hair lines. The colour corresponds to whether the bands are associ-
ated with the substrate (Bi and Se atoms, blue) or with the adlayer (Sb atoms, red).
The projected bands of bulk Bi2Se3 are shaded grey.
9.2.1 Band structures
The bands of the clean substrate (Fig. 9.4 (a)) show that the topologically protected
surface states arising from the bulk (TSSBS) manifest themselves as a single Dirac
cone at the Γ-point. The Dirac point D is right above the VBM and defines the Fermi
energy of the slab. The presence of an odd number of Dirac cones located at the
surface Γ-point confirms the expectations from the parities of bulk Bi2Se3, according
to the extended topological classification of TIs with inversion symmetry [159]. The
theoretical band structure compares qualitatively well with ARPES measurements
[P7, 165, 166, 169]. However, there are two minor, quantitative discrepancies: Firstly,
the experimental binding energy of D is positive in ARPES, e.g. 0.2 eV in [166], while
the theoretical one is zero. The reason for this deviation is that Bi2Se3 samples are
often 𝑛-type due to Se vacancies [166, 169]. Furthermore, ARPES measurements on
TIs suffer from the problem that the inherent UV illumination varies the binding
energy of D by 0.2 eV over the time of measurement [165]. The second quantitative
149
d (Å)
9 The antimony on bismuth selenide heterostructure
K Γ M D* PSb
0.3
(a) Clean Bi2Se (b)3
0.8 0.2
0.1
0.4 0.0
D TSSBS -0.1
0.0
-0.2
-0.4 PBE SOC D2 D2+SOC
Structural optimisation scheme
-0.8 M y
BS Sb K xΓ
0.8 0.4 0.0 0.4 0.8
K Γ M K Γ M
(c) 1BL-Sb (d) 2BL-Sb
0.8 0.8
0.4 0.4
TSSSb D* TSSSb D*
0.0 0.0
PSb RBS B
-0.4 -0.4
B PSb RBS
-0.8 -0.8
0.8 0.4 0.0 0.4 0.8 0.8 0.4 0.0 0.4 0.8
k  (Å−1) k  (Å−1ǁ ǁ )
Figure 9.4: Band structures of (a) the clean Bi2Se3 surface and the heterostructures (b) 1BL-
Sb and (d) 2BL-Sb optimised with DFT-D2+SOC. All energies are reset by the Fermi levels
of the respective slabs. The line colour corresponds to the colour bar: blue → substrate;
red→ Sb adlayer. The line width corresponds to the localisation in the upper half slab (see
text). The projected bands of bulk Bi2Se3 are shaded grey. Arrows highlight special features.
(b) Positions of the Γ-point features (red uptriangles→ PSb; red crosses→ D∗; grey boxes→
bulk conduction bands) of the 1BL-Sb structure optimised with different schemes (pure PBE,
PBE+SOC, PBE+D2, PBE+D2+SOC). Reproduced from [P7].
discrepancy consists in differing shapes of the Dirac cone. The lower part is rather
flat in the theoretical band structure, while it is more X-like in ARPES [P7, 169]. The
origin of this lies in the many-body effects which are insufficiently accounted for
by DFT. The correction of the self-energy of the quasiparticles within GW partially
reverts the band inversion so that the furrow at the Γ-point bulges out [170]. As a
further consequence, the band gap decreases, which is exactly the opposite of what is
expectable from GW corrections in other, topologically trivial semiconductors [170].
Concerning the surface, GW renders the shape of the Dirac cone more X-like, which
then compares very well to the ARPES bands [163, 164].
When the first Sb BL adsorbs to the substrate, the band structure undergoes funda-
mental changes (Fig. 9.4 (c)). The Fermi level rises into the CBs of bulk Bi2Se3. The
150
E − EF (eV) E − EF (eV)
E − EF (eV)
9.2 Electronic properties
original Dirac cone is not recognisable any more and new surface features emerge
instead. Above the first set of bulk CBs, two X-shaped bands form a Dirac point D∗ at
−0.2 eV binding energy. They are interrupted by the bulk CBs and have a visible gap in
the ΓM direction, while the gap along ΓK vanishes. If they are nevertheless continued
across the bulk CBs, they represent a set of surface bands which are more associated
with the Sb adlayer than with the substrate. Since it seems that they preserve the
topological non-triviality of the surface band structure, they are denoted TSSSb. A
detailed analysis of the topological properties can be found below (Sec. 9.3). Beside
TSSSb, there are three further important surface features: A band resembling a lying
B, a hole-like double parabola PSb with apex at 0.1 eV binding energy and Rashba-like
bands RBS at the bulk CBs. The B band is more associated with the substrate than
with the adlayer. PSb is completely localised in the adlayer and represents the Sb–Sb
bonds as indicated by the strong (p𝑥 + p𝑦) character (not shown). The RBS states are
completely localised in the surface region of the substrate, more precisely mostly in
tQL. Since they are right and left shifted versions of the bulk CBs, they seem indeed
to represent a Rashba splitting at tQL.
The addition of a second BL alters the band structure further (Fig. 9.4 (d)), though,
the qualitative features of the 1BL-Sb band structure are all still present. Compared
to 1BL-Sb, the Fermi level rises by 0.1 eV. TSSSb becomes completely localised in the
adlayer and overall steeper. It has a clear gap where the bulk CBs pass, in both along
ΓM and ΓK. The lower part of the former X-shaped band around D∗ bends upwards.
The B band is steeper and its apices are near the Fermi level. In contrast, PSb shifts to
higher binding energies so that it is right above the bulk VBM. The Rashba splitting
RBS is much stronger. An additional feature, which is less pronounced in 1BL-Sb,
consists in the wavy bands in the ΓM direction at −0.2 eV binding energy.
The interlayer distance 𝑑1 of 1BL-Sb influences the alignment of the bands, which
becomes visible in the bands of the structure models which were optimised with
different corrections disabled (Sec. 9.1.2). Fig. 9.4 (b) shows the schematic alignment
between the bulk CBs, PSb and D∗ for these structures. The more the adlayer moves
away from the surface (from right to left in the diagram), the closer PSb and D∗ are
together, until D∗ finally dips into the CBs. It confirms the importance of accounting
for both vdW corrections and SOC during the relaxations. A detailed explanation for
the dependence between D∗, PSb and 𝑑1 can be found below (Sec. 9.3).
Comparison with ARPES
The theoretical band structures are in well agreement with ARPES measurements,
which were performed by the experimental collaborators of the joint-experimental-
theoretical work [P7] (Fig. 9.5). The surface states TSSSb and B are clearly visible in
ARPES and conform with the theory regarding the relative binding energies, slopes
and curvatures. Also the quantitative differences between 1BL-Sb and 2BL-Sb and
between the directions of the k-path are well reproduced. The ARPES image of 2BL-
Sb clearly confirms the Rashba splitting RBS. In the ARPES image of 1BL-Sb, RBS is
difficult to recognise as the splitting is weaker and superposed by blurry bulk CBs.
The peak PSb is missing because in-plane p𝑥 and p𝑦 bands are generally suppressed
by matrix-element effects [P7].
151
9 The antimony on bismuth selenide heterostructure
(a) 1BL-Sb (b) 2BL-Sb
-M Γ M -M Γ M -M Γ M -M Γ M
0.0 0.0
0.2 0.2
0.4 0.4
0.6 0.6
0.8 0.8
-0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4
-K Γ K -K Γ K -K Γ K -K Γ K
0.0 0.0
0.2 0.2
0.4 0.4
0.6 0.6
0.8 0.8
-0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4
k (Å−1) k  (Å−1) k  (Å−1ǁ ǁ ǁ ) k  (Å−1ǁ )
Figure 9.5: Comparison between the theoretical band structures (right, cut out from Fig. 9.4)
and ARPES images (left, features highlighted) of the heterostructures (a) 1B-Sb and (b) 2BL-
Sb. Top row: mid part of -MΓM; Bottom row: mid part of -KΓK. Reproduced from [P7].
The good agreement between the ARPES image of the 2BL-Sb system and the respect-
ive theoretical band structure supports the P0120 structure model. The energetically
competitive, inverted structure models P0102, P1012 and P1021 yield band structures
which look similar at first glance, but differ in the details. The band structures can
be found in the appendix (Fig. C.1). Along ΓM, the inwards bending of TSSSb towards
the Γ-point and the crossing between the B band and RBS are unique for P0120. Along
ΓK, P0120 produces a slope for TSSSb which agrees with the ARPES image, while the
slopes of all other structure models are too steep. The two BL-Sb band structures are
too similar to each other to evaluate them by the ARPES image. The main distinctive
feature is the distance between P and D∗Sb , which is 0.3 eV for P01 and 0.2 eV for P10.
However, both points are not visible in ARPES. The changes in the band structures
upon layer inversion can partially be attributed to the larger interlayer distances, as
further elaborated in Sec. 9.3.
The spin texture
The magnetic projections𝑚𝑙𝑚,𝐼 (𝑛, 𝒌〈) are analog〉ous to〈the band projections 𝑃
𝑙𝑚,𝐼 (𝑛, 𝒌)
𝑖
in Eq. (4.41): 
𝑚𝑙𝑚,𝐼
〉
(𝑛, 𝒌) 1= Ψ𝑛(𝒌) 𝑌 𝐼𝑙𝑚 · 𝜎
𝐼 
𝑖 2 𝑖
· 𝑌𝑙𝑚 Ψ𝑛(𝒌) (9.90)
where 𝜎𝑖 are the Pauli matrices, 𝑖 is the spatial direction, |Ψ𝑛(𝒌)⟩ is the spinor at
band index 𝑛 and k-point 𝒌, and |𝑌 𝐼 ⟩ are the spherical harmonics centred at atom
𝑙𝑚
𝐼 . The sum over 𝑙𝑚 of the magnetic projections of a specific atom 𝐼 for a specific 𝒌
and 𝑛 is simply denoted atomic magnetisation 𝑚𝐼 (𝑛, 𝒌). The atomic magnetisations
𝑖
permit a band colouring which is completely analogous to that based on the PAW
projections. In order to obtain the local spin texture, the upper half slab is divided
into two groups: the substrate and the adsorbate. For each group, the magnetisation
𝑚𝑖 (𝑛, 𝒌) is obtained by summing the 𝑚𝐼 (𝑛, 𝒌) of the respective atoms. The line col-𝑖
our corresponds to a colour bar: blue→ negative magnetisations and red→ positive
152
Binding energy (eV)
Binding energy (eV)
9.2 Electronic properties
K Γ M K Γ M
(a) z y M 0.8 1BL-Sb (b) 2BL-Sb (c)
K 0.4
x
0.0
Γ
m -0.4 (a.u.)
-0.5 +0.5 -0.8 my mx my mx
K Γ M 0.8 0.4 0 0.4 0.8 0.8 0.4 0 0.4 0.8
0.8 Clean (d) (e) (f)
0.4
0.0
-0.4
-0.8 my mx my mx my mx
0.8 0.4 0 0.4 0.8 0.8 0.4 0 0.4 0.8 0.8 0.4 0 0.4 0.8
K Γ -K K Γ -K K Γ -K
0.8 (g) (h) (i)
0.4
0.0
-0.4
-0.8 mz mz mz mz mz mz
0.8 0.4 0 -0.4 -0.8 0.8 0.4 0 -0.4 -0.8 0.8 0.4 0 -0.4 -0.8
k  (Å−1 −1ǁ ) kǁ(Å ) kǁ (Å−1)
Figure 9.6: Spin texture of (d), (g) the clean Bi2Se3 surface and the heterostructures (b), (e),
(h) 1BL-Sb and (c), (f), (i) 2BL-Sb. The line width and Fermi levels are analogous to Fig. 9.4.
The line colour corresponds to the colour bar in (a): blue→ negative magnetisations; red→
positive magnetisations. (b), (c): 𝑚 𝑦 along ΓK and 𝑚𝑥 along ΓM of the Sb adlayer: (d) – (f):
the same for the Bi2Se3 substrate only; (g) – (h): 𝑚𝑧 along ΓK and Γ-K. (a) Brillouin zone with
a schematic helical spin texture. Reproduced from [P7].
magnetisations. The line width corresponds to the localisation in the upper half slab
(as in Fig. 9.4) and suppresses bands of the simulated bulk and the other surface. The
results are shown in Fig. 9.6. The three columns represent the clean substrate, 1BL-
Sb and 2BL-Sb. The first row shows 𝑚 𝑦 (𝑛, 𝒌) along ΓK (the x-direction) and 𝑚𝑥 (𝑛, 𝒌)
along ΓM (the y-direction) for the Sb adlayer only. The second row shows the same for
BS only. The third row shows the z-component 𝑚𝑧 (𝑛, 𝒌) along KΓ-K (the x-direction)
for the entire half slab. The Kramers degeneracy (Eq. (7.74)) tells that the magnet-
isations along Γ-𝑋 and Γ𝑋 , 𝑋 ∈ [K,M], are equal up to the sign, which was explicitly
confirmed (not shown). All other components, e.g. 𝑚𝑥 (𝑛, 𝒌) along ΓK, vanish.
153
E − EF (eV) E − EF (eV)
E − EF (eV)
BS (+ Sb) BS only Sb only
9 The antimony on bismuth selenide heterostructure
The Dirac cone of the clean substrate (panel (d)) is characterised by a magnetisation
which is strictly perpendicular to the k-vector. The in-plane magnetisations rotate
clockwise in the upper cone (sketched in (a)) and anticlockwise in the lower cone. The
perpendicular relation between the magnetisation (“spin”) and the k-vector (“mo-
mentum”) is called spin-momentum locking: In the model of nearly free electrons,
the spins of surface states underlying strong SOC are perpendicular to the momenta,
as the CPD Hamiltonian shows (Eq. (2.17), ∇𝑉 ∥ e𝑧 ⇒ 𝝈 ⊥ 𝒑 ∼ 𝒌, [171]). The
spin-momentum locking is confirmed by the band structures of this work as all mag-
netisations parallel to 𝒌 and associated with the surface remain white (not shown).
The T-symmetry causes the rotation (Kramers degeneracy). Panel (g) shows that the
magnetisation is in-plane near the Γ-point, but gains a finite out-of-plane component
𝑚𝑧 along ΓK. This component positive for three 𝐶3-related K-points, negative for the
other three K-points and zero along ΓM. The wave-like out-of-plane tilting of the spin
when circling D is due to the hexagonal warping effect, as arising from higher-order
𝒌 · 𝒑 theory combined with the spatial symmetries of the surface [172].
In 1BL-Sb, the Dirac cone around D∗ (b) has the same spin texture as the original
Dirac cone of the clean substrate (d). However, progressing on the ΓK path, the spin
of TSSSb rotates from (𝑚 𝑦 > 0, 𝑚𝑧 = 0) via (𝑚 𝑦 = 0, 𝑚𝑧 < 0) to (𝑚 𝑦 < 0, 𝑚𝑧 < 0)
((b) and (h)). The B band behaves similarly, but with opposite signs and an additional
sign change when it connects to double peak PSb. The latter has thus a coloured up-
per branch, while the lower branch shows only weak magnetisations. The overall
𝑚𝑧 contributions are considerably stronger than those of the clean substrate, indic-
ating a stronger hexagonal warping effect. Along ΓM, the magnetisations of TSSSb, B
and PSb remain strictly in plane and are rather constant due to hexagonal warping.
The magnetisations furthermore show that the substrate (e) the adlayer (b) equally
contribute to TSSSb and the B band, while PSb is only barely visible in the substrate
bands. The magnetisations of RBS are completely in-plane and localised in the sub-
strate (e). They resemble a double parabola, whose branches are shifted leftwards
(red) and rightwards (blue). This agrees with the expectations from the Rashba effect.
The spin texture of 2BL-Sb looks very similar to that of 1BL-Sb. The states around D∗
have the same spin texture as the original Dirac cone and are mainly localised in
the adlayer (c). The in-plane magnetisation of TSSSb is weaker and the out-of-plane
component very strong (i). Also the Rashba splitting is stronger than in 1BL-Sb (f).
9.2.2 Simulated STM images
Scanning tunnelling microscopy STM sheds light on the morphology of surfaces. For
the above systems, theoretical STM images are simulated according to the theorem
of Tersoff and Hamann [173]. For this purpose, the local density of states (LDOS) is
integrated over an energy frame from the Fermi level to the STM bias voltage 𝑉bias.
The experimental STM images of 1BL-Sb and 2BL-Sb were taken by the collaborators
of [P7] at 𝑉bias = +0.2 V in constant-current mode, where the positive sign indicates
the probing of empty states (for details, see [P7]). Therefore, the STM simulation
consists in summing the partial charges of all states between 𝐸F and 𝐸F + 0.2 eV. The
resulting isosurfaces are shown in Fig. 9.7.
154
9.2 Electronic properties
The experimental STM image of 1BL-Sb (panel (a)) is characterised by a hexagonal
pattern of bright spots. The theoretical image (b) reproduces this very well and the
underlying structure (c) shows that these spots indeed stem from charge bulbs above
the upper Sb atoms. Since the lower Sb atoms are invisible due to the strong signal
from the upper Sb atoms, the symmetry of the STM image is hexagonal, despite the
trigonal symmetry of the atomic arrangement. The STM image of 2BL-Sb (d) looks
different from 1BL-Sb, despite the same isocurrents and same geometry of the up-
permost atomic layer. The bright spots are triangular and connected by bridges.
Consequently the symmetry of the image is clearly trigonal. The theoretical image
(e) reproduces this very well. An astonishing detail is that the triangular spots point
upwards, not downwards as the geometry of the top BL (f) suggests. In other words,
the apparent buckled-honeycomb pattern is exactly inverted with respect to the real
buckled Sb honeycombs: The bright spots are above the upper Sb atoms of the top
BL, the bridges above the holes and the apparent holes above the lower Sb atoms.
The reason for this is that the charge bulbs of the upper Sb atoms of BL1 penetrate
the holes of BL2 and are thus probed beside the bright charge bulbs of the upper Sb
atom of BL2. This demonstrates that STM only measures charge densities, which can
deviate from real atomic arrangements.
max. (a) (b) (c) (d) (e) (f)
min. 1nm 1nm
Figure 9.7: STM images of (a) – (c) 1BL-Sb and (d) – (f) 2BL-Sb. (a) and (d) are experimental
images taken at𝑉bias = +0.2 V and 𝐼 = 20 nA (empty states). (b) and (e) are theoretical images
simulated at the same voltage. (c) and (f) show the underlying structure of the simulated
images. Grey circle are Se atoms; Blue/purple circles are the Sb atoms of BL1/BL2. Red rhombi
indicate the surface unit cell. Reproduced from [P7].
The last point of this section concerns the real-space distribution of the Dirac points
D and D∗. As these points are fourfold degenerated due to T-symmetry and the P-
symmetry of the entire slab, the DFT code produces states which are artificially loc-
alised at both surfaces, although the partial charges still show a thick region with
vanishing density in the simulated bulk. Averaging all four states of the Dirac points
solves this symmetrisation problem. Fig. 9.8 shows the isosurfaces of the partial
charges of the Dirac points in real space. For the clean substrate, the isosurface (panel
(a)) as well as the plane average (d) show that the partial charge of D extends deeply
into the substrate, far beyond tQL. This explains why the slab has to be so thick in
order to entirely disentangle the simulated surfaces. The highest peak in the plane
average coincides with the bond between the top Se atom and the Bi atom below
(grey dashed line). The partial charge of D∗ in 1BL-Sb (b) is more localised in the sur-
face region as it already decays to near zero in tQL. The two largest peaks in (d) stem
from charge bulbs at the top Sb atom (blue dashed lines), which look like p𝑧 orbitals.
The PAW projections indeed confirm that TSSSb has strong Sb-p𝑧 contributions (not
155
height (a.u.)
9 The antimony on bismuth selenide heterostructure
(a) Clean (b) 1BL-Sb (c) 2BL-Sb (d) xy-average
Figure 9.8: Real-space isosurfaces of D*
the wavefunctions at the Dirac points D*
D and D∗ of (a) the clean substrate, (b)
1BL-Sb and (c) 2BL-Sb. Grey circles D
are Bi and Se atoms; Blue/purple
circles are the Sb atoms of BL1/BL2.
(d) Corresponding plane averages.
Reproduced from [P7].
shown). In 2BL-Sb (c) the partial charge even more concentrates in the top Sb layer.
As in 1BL-Sb, the p𝑧 orbitals of the top Sb atom contribute most to the plane aver-
age (purple dashed lines). The comparison between the three systems shows that the
TSSs are transported to the surface upon Sb layer adsorption and that the vertical
confinement of the states strengthens from the clean substrate to the 2BL-Sb hetero-
structure. This clearly hints that the transition between different topological classes
of the bulk and the vacuum occurs in the topmost vdW layer. For this reason, the
Sb adlayer can be considered topologised in that it quasi belongs to the non-trivial
substrate and produces TSSs at its interface with the trivial vacuum [P7].
9.3 The topological phase transition
The chapter concludes with a deeper analysis of the electronic interplay between the
Bi2Se3 substrate and the Sb adlayer and confirms that latter indeed topologises. For
this purpose, the distances between the Sb BLs and the substrate are varied between
long (free-standing sheets plus clean substrate) and short (heterostructure). By cal-
culating the bands for each configuration, the electronic properties can be tracked
along a simulated desorption-and-readsorption process.
According to pp. 126ff, the fundamental behaviour of surface states follows from the
topological class of the underlying bulk. The combined band structure of the clean
substrate and a free-standing, thin Sb sheet is expected to be a superposition of two
uncoupled entities: the non-trivial surface bands of the former (analogous to Fig. 7.5
(c)) and the trivial 2D bands of the latter (analogous to Fig. 7.5 (b)). In the adsorp-
tion process, which finally results in the bands of the heterostructure (Fig. 9.4), the
non-trivial Dirac cone of BS(0001) and the trivial 2D bands of the Sb sheet obviously
vanish and, instead, a joint, non-trivial band structure appears. Thus, at a certain
point, the Kramers pairs of the Sb sheet have to change partners so that they integ-
rate themselves into the non-trivial surface states of the substrate. Such a partner
change is a topological phase transition. The topological class of surface states can be
checked by means of the connections between the surface Kramers pairs (SKPs) at
the four surface TRIMs. Fig. 9.9 shows the hexagonal surface Brillouin zone and the
TRIM rhombus (bold). One TRIM is Γ0, the Γ-point of the zeroth Brillouin zone (black).
Two further TRIMs are the M-points M0 and M010 01. The last TRIM is M011, which equals
156
2BL-Sb
1BL-Sb
Clean
9.3 The topological phase transition
M11̄1 of the next Brillouin zone (grey) and is thus identical to M01̄1. The original k-path
of Fig. 9.4 connects the non-TRIM K0 0 011 and the TRIMs Γ and M1̄1. It is prolonged in
the x-direction so that it comprises the line segments Γ0M0 0 01̄1 and Γ M11 (dashed lines
in Fig. 9.9). The symmetries of BS(0001) imply that all M-points yield the same 𝛿𝑖
product2. Thus, the SKP connections between Γ0 and any of the M-points character-
ises the entire surface band structure. Nevertheless, the complete k-path in Fig. 9.9 is
employed for the band structure calculations as it permits tracking the spin texture
at the same time.
The details of the desorption-and-readsorption procedure
are as follows: Starting from the stable 2BL-Sb hetero- M01̄1 M001
structure (𝑑1 = 2.3 Å and 𝑑2 = 2.5 Å), BL2 is rigidly lif- K0 M
0
11 11
0
ted by 6.0 Å in the first stage so that Γ𝑑2 = 8.5 Å. This rep-
0
resents the 1BL-Sb heterostructure plus an isolated 1 BL M10 Γ1
sheet. In the second stage, BL1 is lifted by 6.0 Å as well so y
that 𝑑1 = 8.3 Å and 𝑑2 = 2.5 Å. This represents the clean x
substrate plus an isolated 2 BL sheet. In the third and last
stage, the 2 BL sheet is lowered by −6 0 Å so that the ori- Figure 9.9: Hexagonal Bril-. louin zone with TRIMs.
ginal 2BL-Sb heterostructure is restored. Each stage is lin-
early sampled by 20 structures, the images, so that the BLs move vertically by 0.3 Å
from image to image. For each image, the band structure is calculated and plotted
according to four different modes:
1) Line width → localisation in the upper half slab; RGB colour → localisation in
BL2, BL1 and tQL, respectively. This extracts the surface bands and indicates
whether they belong to BL2 (red), BL1 (green) and/or tQL (blue).
2) Line width → tQL; blue/red colour → magnetisation of tQL. This extracts the
bands of tQL and shows their spin texture. The direction of the plotted magnet-
isation is perpendicular to the k-path (spin-momentum locking, vide supra).
3) Same as 2) for BL1.
4) Same as 2) for BL2.
Fig. 9.10 shows the band structures of 16 out of the total 56 images.
The energies are reset to the VBM of bulk Bi2Se3. The Fermi energy of the slab is in-
serted as a black horizontal line. Four special cases provide important subsystems
(yellow background, blue labels): (a) the 2BL-Sb heterostructure, (e) the 1BL-Sb het-
erostructure plus an isolated 1 BL sheet, (i) the quasi clean substrate plus two quasi
isolated 1 BL sheets, (m) the clean substrate plus an isolated 2 BL sheet. For these im-
ages, all four plotting modes are shown. For all other images, only mode 1) is shown.
The respective structure model with the interlayer distances is displayed next to the
band structures. Red arrows indicate the order of the images. The full set of images
is available as an mp4 file in the supplementary material of [P9].
2This restriction of the combinatorics is the central reason why it is easier to realise TIs in trigonal
systems, than in any other symmetry class.
157
M0 K0 Γ0 M0 M0 K0 Γ0 M0 M0 K0 Γ0 M0 M0 0 0 011 11 1̄1 11 11 1̄1 11 11 1̄1 11 K11 Γ M1̄1
(a)
1
D*
tQL BL1 BL2
0.5
R my mx my mx my mx
0 P
-0.5
-1.5 -1 -0.5 0 0.5 -1.5 -1 -0.5 0 0.5 -1.5 -1 -0.5 0 0.5 -1.5 -1 -0.5 0 0.5
tQL BL1 BL2 k‖ (Å−1) m (a.u.): − +
(b) (c) (d)
gap
closing
(e)
D* 1BL-Sb 1BL-Sb isolated BL
R
P
(f) (g) (h)
partner
change
(i)
P1 quasi quasi
R clean BS isolated BL isolated BL
P2
D'
(j) (k) (l)
gap
closing
(m)
R clean BS isolated 2BL isolated 2BL
P1'
D'
P2'
(n) (o) (p)
gap gap
opening opening
Figure 9.10: Topological phase transitions in the Sb@BS heterostructure (details in text).
Reproduced from [P9].
E (eV)
9.3 The topological phase transition
In the first image (a), the surface band structure of 2BL-Sb is recognisable as well as
the SKPs D∗, RBS (abbreviated to R) and PSb (abbreviated to P). The SKP connections
are as follows: Both branches of P are connected to the VBs (denoted v-trivial) and
both branches of D∗ are connected to the CBs (denoted c-trivial). Only the branches of
R are connected to the VBs and CBs (denoted non-trivial) wherefore this SKP reflects
the topological non-triviality of the substrate. That the lower branch going from R
into the VBs changes its character from tQL to Sb indicates a strong mixing between
the substrate and the adlayer.
Lifting BL2 deforms the bands. The lower branch of D∗ bends down from (a) to (d)
and closes a gap along Γ0M011, while the character changes from BL2 (red) to BL1
(green). The gap closing does not entail any partner changes, though, as R is still non-
trivially connected along Γ0M01̄1. At the same time, P rises towards R and two further
v-trivial peaks belonging to BL2 (red) emerge from the bulk VBs. From (d) to (e), the
band structure changes only slightly when it transforms into a superposition of the
surface band structure of 1BL-Sb and the (red) 2D band structure of an isolated 1 BL
sheet (Fig. 8.4 (c)). The Kramers pairs of the latter are v-trivial. The SKPs of 1BL-Sb
show the same connections as those of 2BL-Sb, even though P and R are very close
together. The PAW projections shift so that TSSSb has a mixed BL1+tQL character
(cyan). Since no partner change occurs from (a) to (e), the desorption of BL2 does not
involve topological phase transitions. In the reverse direction of adsorbing a second
BL to 1BL-Sb, the transport of the TSSs across BL2 towards the surface of the 2BL-
Sb heterostructure consists in a continuos shifting of the PAW projections, while the
valence bands of BL2 disappear in the bulk VBs.
Lifting BL1 drops D∗ into the bulk CBs (f). At 𝑑1 = 3.6 Å (g), D∗ touches P and R
whereupon a partner changing occurs: The upper branch of D∗ connects with the up-
per branch of R, which produces a c-trivial SKP, again denoted R. The lower branch
of R connects with the upper branch of P, which produces a v-trivial SKP, denoted P1.
The lower branch of P connects with the lower branch of D∗, which produces a non-
trivial SKP, denoted P2. The partner change is a topological phase transition which
detaches the topological non-triviality from R and stores it into a new point P2. It fur-
thermore unravels the zigzag manner in which PSb is connected in the 1BL-Sb bands
(the B band is connected to the same TRIM as TSSSb, while the lower branch goes
elsewhere). As a consequence, the upper Sb peak it is allowed to form the twofold
degenerated peak of the isolated 1 BL sheet. In the course of the topological phase
transition, the projections of TSSSb shift from tQL+BL1 (cyan) to BL1 (green). In ad-
dition to the three new SKPs from mixing the old ones, an additional, v-trivial SKP
denoted D′ emerges from the bulk VBs. The middle image of the second stage (i) rep-
resents a system of two quasi isolated 1 BL sheets and the quasi clean substrate. The
SKPs are clearly visible, as well as the almost complete formation of the valence band
peaks of the isolated BL1.
Lifting BL1 further towards BL2 closes a gap (j) which restores the Dirac cone of the
substrate. By this means P2 swaps SKP connections with D′ whereupon P2 is v-trivial
and D′ is non-trivial. This proves that D′ is indeed the original D. From (i) to (m), the
bands of the two isolated 1 BL sheets transform into the bands of an isolated 2 BL
sheet (Fig. 8.4 (d)). At 𝑑2 = 4.0 Å (k), the upper peak of BL2 and the two peaks of BL1
159
9 The antimony on bismuth selenide heterostructure
(P1 and P2) merge whereupon the topmost band with its characteristic furrow comes
off. After the merging, the two topmost 2D Kramers pairs are denoted P′1 and P
′
2. As
no gap closing occurs in the bands of the Sb BLs, the topological class of the 2 BL sheet
is trivial, as already confirmed in Sec. 8.2.2.
In the last stage, the 2 BL sheet is lowered towards the substrate. At 𝑑1 = 4.8 Å (n),
the topmost (twofold degenerate) band of the sheet (the 2 BL band) arranges with
the bulk CBs so that P′1 touches R. At the same time, a gap opens between the 2 BL
band, the Dirac cone of the substrate and the lowermost bulk CB. As a result, D′, P′1
and R change the SKP connections: The inner upper branch of D′ connects to the
outer lower branch of the 2 BL band, which produces the B band and renders D′ is v-
trivial. The inner upper branch of P′1 connects to the outer upper branch of the Dirac
cone and the inner lower branch of P′1 connects to the bulk CBs, so P
′
1 c-trivial. The
inner lower branch of R connects to the outer upper branch of the 2 BL band, which
produces TSSSb and renders R non-trivial. The new connections are better visible
along Γ0M01̄1, while along Γ0M011, R is apparently c-trivial and P′1 non-trivial due to the
vanishing gap. This gap opens at 𝑑1 = 3.1 Å (p) as well and, after a reordering between
D’ and the lower Sb peak, the final band structure of 2BL-Sb is restored (a).
The magnetisations show that the twofold degenerate bands of the Sb sheets polar-
ise when they approach the substrate. The spin texture of the Dirac cone picks the
band with the same magnetisation and hybridises with it, while the other band re-
mains unaffected. This explains why D* inherits the spin texture of D. If a Sb sheet
approaches the substrate, the first topological phase transition happens in the top-
most VB, if the sheet is 2 BL thick, and in the second top VB (lower Sb peak) if the
sheet is 1 BL thick. The latter is quite surprising at first glance, but can be explained:
The bands of the isolated sheets in Fig. 8.4 (c) and (d) show that the topological phase
transition happens in those VBs which provide p𝑧 contributions. On the contrary, the
topmost VB of a 1 BL thick sheet provides only (p𝑥 + p𝑦) contributions. This leads
to the conclusion that the p𝑧 orbitals play the decisive role in the surface topological
phase transitions, similarly to the band inversion in the bulk TIs.
9.4 Discussion and summary
The analysis of the 1BL-Sb@BS and 2BL-Sb@BS heterostructures showed that the
structure models in which the BLs continue the natural stacking sequence of the sub-
strate are indeed favourable. However, structure models with inverted BLs are less
stable by only ∼10 meV per Sb atom, less than 𝑘B𝑇 at room temperature. Hence, it is
likely that inverted structures are thermodynamically relevant. Similarly to the case
of bulk Bi2Se3, vdW corrections and SOC reduce the interlayer distance between the
substrate and the Sb adlayer. SOC is even more important in the heterostructures
than in bulk Bi2Se3. The favourable natural stacking minimises the interlayer dis-
tances for geometrical reasons and thus the Sb chemical potential.
The band structures were calculated for the naturally stacked heterostructures as
well as for the inverted ones. The bands of the latter have never been published be-
160
9.4 Discussion and summary
fore, while the bands of the former are unprecedented in detail and clarity. The theor-
etical band structures of the natural structure models compare very well with ARPES
measurements. Unfortunately, the band features which distinguish the natural struc-
ture models from those with inverted BLs, D* and PSb, are invisible in ARPES due
to negative binding energies and matrix-element effects. Thus, ARPES is limited in
identifying BL inversions. The STM simulation of the natural structure models agrees
very well with experimental STM images. In particular, the differences in the sym-
metries of the image and the atomic geometry, hexagonal vs. trigonal for 1BL-Sb and
inversion for 2BL-Sb, were proven to be due to deviations between the charge dens-
ities at the Fermi surface and the atomic geometries.
The topological phase transitions during a simulated desorption-and-readsorption
process illuminated how the Dirac cone of the clean BS(0001) surface merges with
the bands of a thin Sb sheet to produce the joint bands of the heterostructure. When
a 1 BL or a 2 BL thick Sb sheet approaches the substrate, a complicated series of to-
pological phase transitions occurs. It begins in the p𝑧 bands of the Sb sheet and trivi-
alises the Dirac cone, whose Dirac point D is buried in the bulk valence bands. The
Dirac point D* is shown to be a remainder of the Sb peaks and topologically trivi-
ally connected to the conduction bands. The only non-trivial surface Kramers pair
is RBS, the crossing of the Rashba split conduction bands. Its lower branch follows
an electron-like parabola near the Γ-point and bends down away from the Γ-point to
form TSSSb, which is visible in ARPES. The topological phase transitions occur only
when the first (set of) BLs approach the BS surface. The addition of a second BL to
1BL-Sb is continuous in that there are no partner changes.
The literature on DFT band structures of the Sb@BS system is spare. Two works with
intersecting author lists exist, containing band structures for 1BL-Sb (Jin et al. [168])
and 2BL-Sb (Kim et al. [166]). The band structures look rather different from those
presented here. The 1BL-Sb band structure of Jin et al. shows a strong misalignment
of D∗ and PSb, the former being in the bulk CBs and the latter being 0.2 eV below D∗,
not 0.3 eV. The desorption-and-readsorption process shows that the misalignment is
due to a too large interlayer distance 𝑑1 (Jin et al.: 2.71 Å vs. this work: 2.38 Å). Jin
et al. obtained the structure by DFT relaxations, but they do not mention whether
they employed vdW corrections and/or SOC. The large 𝑑1 suggests that they omitted
at least one of the two. The 2BL-Sb band structure of Kim et al. shows the major
qualitative difference that there is no gap in the ΓK direction, contrarily to the bands
presented here. Unfortunately, Kim et al. give neither details on whether they em-
ployed vdW corrections or SOC during the DFT relaxations, nor structure parameters
like the interlayer distances. In both [168] and [166], the band structures of the ΓM
path are missing, despite the pronounced gaps in TSSSb which show that D∗ does
not reflect the topological non-triviality of the substrate. Jin et al. calculated also the
band structures under variation of the interlayer distance between a 1 BL sheet and
a Bi2Te2Se substrate. Although this reminds of the desorption-and-readsorption pro-
cess of this work, they did not analyse the details and did not extract the topological
phase transitions. In particular, they did not notice that RBS is non-trivially connected
while D∗ is trivially connected to the CBs. Consequently, the SKP method developed in
this thesis is indeed a novel tool for extracting surface topological phase transitions
in a systematic manner.
161
10 Conclusions
In this PhD thesis, two classes of nano-scaled systems were successfully investigated
by means of DFT: rare-earth silicide nanowires on silicon surfaces and thin antimony
layers on bismuth selenide. A close inspection of the configuration spaces of the
nanostructures resulted in different structure models, which were optimised with
respect to the atomic positions and whose stability was evaluated by means of ab
initio thermodynamics. The detailed analysis of the structural and electronic prop-
erties lead to several conclusions about the underlying chemistry and provided a
deep insight into the physics of the nanostructures. All results agree well with exper-
imental reference on lattice parameters as well as ARPES measurements, Fermi sur-
faces and STM images. Finally, the systematic approach of this work and the detailed
cross-linking between the structural and electronic properties permitted the identi-
fication of fascinating novel phenomena: the dimensional crossover in the nanowire-
Tb@Si(557) system and the complex series of multiple topological phase transitions
in the Sb@Bi2Se3 heterostructure.
The heavy sixth-row elements which occur in both systems were shown to have prop-
erties which are unique in the periodic table. The lanthanoids Tb and Er have a
stable trivalent state in the bulk RESi2 – x phases, despite the incompleteness of the 4f
shell. Also the lanthanoid contraction was demonstrated for the silicides from TbSi2
to ErSi2. The element Bi is even more peculiar as the relativistic effects give rise
to the inert-pair effect of the 6s electron, as explicitly demonstrated in bulk Bi2Se3.
Furthermore, the relativistic effects are so strong that Bi2Se3 transitions into a topo-
logical insulator. Similarly, bulk 𝛽-Sb turned out to be a topological semimetal. The
heavy elements necessitated methods beyond DFT: the LDA+U method, vdW correc-
tions and spin-orbit coupling, which all proved to be crucial for obtaining correct
results for the respective systems. The ℤ2 invariant was elaborated in order to ex-
plicitly calculate the topological classes of Bi2Se3 and 𝛽-Sb. Furthermore, it is the
basis of the novel SKP method, which was developed in this thesis in order to track
and understand the topological phase transitions in the Sb@Bi2Se3 heterostructure.
This method can be applied to all surfaces which establish a connection between a
substrate and an adsorbate of different topological classes.
It proved to be indispensable to analyse the related higher-dimensional systems be-
fore treating the actual nanostructures. These were the bulk RESi2 – x phases and the
monolayer-Tb@Si(111) system in the case of the TbSi2 nanowires, and bulk Bi2Se3
and 𝛽-Sb in the case of the Sb@Bi2Se3 heterostructures. Though, the connection
between the bulk and the respective nanostructure is not as obvious as one may
assume. For example, the popular argument of surface-induced strain, which can
describe the epitaxy of bulk-like layers on substrates, was proven to be inapplicable
162
for the herein investigated nanostructures. On the contrary, surface strain is neg-
ligible in both the monolayer-Tb@Si(111) and the Sb@Bi2Se3 systems, while a large
strain would follow from the lattice constants of the respective bulk phases. Hence,
the strain in nanostructures has to be evaluated with special caution in order to avoid
wrong conclusions. The bulk structures are useful because they permit to understand
fundamental structural mechanisms, which apply to the nanostructures as well, e.g.
the silicon cages in RESi2 – x or the layer-layer interaction in Bi2Se3 and 𝛽-Sb. Fur-
thermore, the electronic properties of the higher-dimensional systems can help to
identify the origins of the band structures of the nano-scaled systems. In this work,
the bands of the nanowire-Tb@Si(557) system were shown to be derivable from the
bands of the monolayer-Tb@Si(111) system, which in turn can be derived from the
bands of (hypothetical) CaSi2-TbSi2 by zone projection. In the case of the Sb@Bi2Se3
system, the topological classes of Bi2Se3 and 𝛽-Sb were absolutely crucial for under-
standing the topological phase transitions in the heterostructure.
Outlook: Rare-earth silicide nanowires on silicon surfaces
The investigations of the RESi2 nanowires give rise to at least two further scientific
questions. Firstly, the bulk phases of RESi2 – x were shown to be a complex system,
despite the small unit cells of the structure prototypes. The results for the supercells
with ordered Si vacancies give only an idea about the underlying physics. In order
to fully understand the RESi2 – x phases, it is necessary to introduce probability into
the distribution of the Si vacancies and to calculate the temperature-concentration
phase diagrams. This may finally explain why the stoichiometry of AlB2-RESi2 – b is
equal for all REs (𝑏 ≈ 13 ), while the stoichiometry of ThSi2-RESi2 – a depends on the
atomic number of the employed RE (𝑎 < 13 ).
The second scientific question concerns the RESi2 nanowires on Si(001). Although
not included in this thesis, investigations of this system were published in [P3, P6].
Despite a systematic inspection of the configuration space of reasonable structure
models, the stability of the nanowires could not be proven by means of straightfor-
ward grand-potential-chemical-potential phase diagrams. On the contrary, the width
of the nanowires was found to be unlimited in that they would aggregate to a 2D film.
The contradictory results were attributed to kinetic effects. Understanding the peri-
tectic transition in the bulk phase diagrams might support the further elaboration of
this issue. A further consequence of this work for the nanowire-RE@Si(001) system
results from the finding that the stable structure models of the monolayer-Tb@Si(111)
and nanowire-Tb@Si(557) systems obey the ZKB concept in a strict manner. Thus, it
is likely that the nanowires on Si(001) obey the ZKB concept as well. Therefore, the
structure models already found should be evaluated regarding this issue and further
structure models could be set up based on this.
Outlook: Thin antimony layers on bismuth selenide
The findings on the Sb@Bi2Se3 heterostructure raise two further scientific questions.
The first concerns the layer inversions, which introduce only small penalties into the
163
10 Conclusions
Sb chemical potentials. Since these are below 𝑘B𝑇 at room temperature, the omis-
sion of the lattice dynamics is certainly questionable. This point should be further
investigated, e.g. by the frozen-phonon method.
The second question is about GW-corrected band structures. Since Bi2Se3 is a topo-
logical insulator, GW exerts a strong effect on the region around the Γ-point, which
reduces the band gap and alters the Dirac cones of the clean surfaces of Bi2Se3, Bi2Te3
and Sb2Te3 [163, 164, 170]. For this reason, GW effects on the bands of the Sb@Bi2Se3
heterostructures would be very interesting from a general point of view. Moreover,
a corrected band structure would permit a quantitative comparison with ARPES,
which could be very helpful for identifying layer inversions, e.g. by the slopes of
TSSSb.
164
Appendix A
A.1 Mathematical identities
The Dirac notation
Dirac∑︁nota ti〉o〈n (discrete basis):
〈 𝑛〉

 𝑛〉 𝑛
 = 𝐼 Completeness
 𝑛 𝑛′∑︁= 𝛿 𝑛𝑛〉′〈  〉 ∑︁  〉 〈  〉 OrthogonalityΨ = 〈 𝑛  𝑛〉Ψ = Ψ𝑛 𝑛 , Ψ𝑛 = 𝑛 Ψ Series expansion𝑛 𝑛𝑂𝑛𝑛′ = 𝑛 𝑂 𝑛′ Matrix representation
Dira∫c notat〉i〈on (continuous basis):〈 d𝑥 〉𝑥 𝑥 = 𝐼 Completeness 𝑥〉𝑥′ ∫= 𝛿(𝑥〈 
−
〉〈
𝑥′
 
)
〉 〉
∫  〉 〈  〉 Orthogonality
Ψ = d𝑥 𝑥 𝑥 Ψ = d𝑥 Ψ(𝑥)𝑥 , Ψ(𝑥) = 𝑥 Ψ Series expansion
𝑂(𝑥, 𝑥′) = 𝑥 𝑂  𝑥′ Matrix representation
The translation operator
The translation operator𝑈𝑡 is defined by its action on the real space distribution of a
state, i.e. 𝑈𝑡 shifts the expectation value of the position operator 𝑥 for every |Ψ⟩ ∈ H
by a constant 𝑡. Since translations do not change the scalar product, the translation
opera〈to〉r ha〈s to
𝑥 = Ψ 
be u〉nitary〈. 〉 〈   〉 〈   〉 〈   〉𝑥 Ψ ⇔ 𝑥 + 𝑡 = Ψ̃  𝑥  Ψ̃ = Ψ 𝑈−1𝑥𝑈 Ψ = Ψ  𝑥 + 𝑡 𝑡 𝑡 Ψ
As this is, in particular, valid for the vectors of any basis, the condition can be sum-
marised as a commutator equation.
𝑈−1𝑡 𝑥𝑈𝑡 = 𝑥 + 𝑡 ⇔ [𝑈𝑡, 𝑥] = −𝑡𝑈𝑡 (∗)
A1
Appendix A
This can be solved by the following ansatz:
𝑈𝑡 = exp(i𝑡 · 𝑠) with [𝑠, 𝑥] = 𝑐 ∈ ℂ
⇒ [𝑈𝑡, 𝑥] = [exp(i𝑡 · 𝑠), 𝑥]
(A.93)
= i𝑡 exp(i𝑡 · 𝑠) [𝑠, 𝑥] = i𝑡 𝑈𝑡 [𝑠, 𝑥]
⇔(∗) [𝑠, 𝑥] = i
As the position ope[rator 𝑥]and the momentum operator 𝑝 hold the canonical com-
mutation relation ( − 1ℏ 𝑝, 𝑥 = i), the solution for the translation operator𝑈𝑡 is:
𝑈𝑡 = exp(−i 1ℏ 𝑡 · 𝑝) (A.91)
The commutator of operator functions
Let 𝐴 and 𝐵 be linear operators acting on a Hilbert space H . The commutator of 𝐴
and 𝐵 shall be a constant 𝑐 [∈ ℂ. It] then holds for 𝑛 ∈ ℕ \ 0:
𝐴𝑛, 𝐵 = 𝐴𝑛𝐵 − 𝐵𝐴𝑛 = 𝑛𝐴𝑛−1𝑐 (A.92)
Proof. (A.92) will be prov[en by]mathematical induction.
∀𝑛 ∈ ℕ \ 0 : [𝐴𝑛, 𝐵] = 𝐴𝑛𝐵 − 𝐵𝐴𝑛 = 𝑛𝐴𝑛−1𝑐 (IV)
𝑛 = 1 : [ 𝐴1, 𝐵] = 𝐴𝐵 − 𝐵𝐴 = 𝑐 ⇔ 𝐵𝐴 = 𝐴𝐵 − 𝑐 (IA)
𝑛→ 𝑛 + 1 : 𝐴𝑛+1, 𝐵 = 𝐴𝑛+1𝐵 − (𝐵𝐴)𝐴𝑛 = 𝐴𝑛+1𝐵 − (𝐴𝐵 − 𝑐)𝐴𝑛 (IS)
= 𝐴(𝐴𝑛𝐵 − 𝐵𝐴𝑛 + 𝑐𝐴𝑛−1) IV= 𝐴(𝑛𝐴𝑛−1𝑐 + 𝐴𝑛−1𝑐)
= (𝑛 + 1)𝐴𝑛𝑐
□
A corollary is the generalisation to functions of operators which per definitionem
have a converging p∑︁ower series:∞
𝑓 (𝐴) = 𝑓 𝑘𝑘𝐴 ⇒ [ 𝑓 (𝐴), 𝐵] = 𝑓 (𝐴)𝐵 − 𝐵 𝑓 (𝐴) = 𝑓 ′(𝐴)𝑐 (A.93)
𝑘=0
The Fourier series in multiple dimensions
Let 𝑓 : 𝒙 → ℂ be a periodic function on a latticeR with unit cell C
∑︁ R
, reciprocal lattice
G, as defined in Sec. 2.2. Then 𝑓 has the following Fourier series:
𝑓 (𝒙) ∼ e∫xp(i𝒙 · 𝑮) 𝑓𝑮 (A.94a)𝑮∈G
1
𝑓𝑮 = |C | d𝑥 exp(−i𝒙 · 𝑮) 𝑓 (𝒙) (A.94b)R CR
A2
A.2 The principle of minimal coupling
If, in addition, 𝑓 ∈ H = 𝐿2(C∑︁R), the Parseva∫l theorem ho 2 1 
lds [174, pp. 145f]:
𝑓𝑮 = |C | d𝑥
 2𝑓 (𝒙) (A.95)
𝑮∈G R CR
If, in addition, 𝑓 is co∑︁ntinuous and its d∫erivative is piecewise continuous, then:
𝑮 2 −i𝑓𝑮 = |C | d𝑥 𝑓 ∗(𝒙)∇ ′ 𝑓 (𝒙′𝑥 ) (A.96)′
𝑮∈G R CR 𝒙 =𝒙
Eq. (A.96) follows from the Parseval theorem applied on 𝑓 and its gradient ∇ 𝑓 ex-
pressed as a Fourier series [174, pp. 148ff].
(∇ 𝑓 )𝑮 = i𝑮 𝑓𝑮
A.2 The principle of minimal coupling
Let 𝐻0(𝒑, 𝒙) be the Hamilton operator ofa f〉ree par〉ticle with charge 𝑞:𝐻0(𝒑, 𝒙) Ψ = 𝐸 Ψ
Then, the introduction of a vector potential 𝑨(𝒙, 𝑡) and a scalar potential 𝜑(𝒙, 𝑡) mod-
ifies the Hamilton ia〉n as follows: 〉  〉  〉𝐻0(𝒑 − 𝑞𝑨, 𝒙) Ψ = (𝐸 − 𝑞𝜑)Ψ ⇔ [𝐻0(𝒑 − 𝑞𝑨, 𝒙) + 𝑞𝜑] Ψ = 𝐸Ψ (A.97)
Eq. (A.97) can be motivated by the Hamiltonian for the classical problem. For this
purpose, let 𝐻c0 be the Hamiltonian for a classical charged particle moving in an elec-
tromagnetic field. The coupling of the electromagnetic potentials to the momentum
and the energy yields the correct classical Lorentz force:
c (𝒑 − 𝑞𝑨)2𝐻0 = + 𝑞𝜑2𝑚
𝒙¤ ∇ c 1= 𝑝𝐻0 = (𝒑 − 𝑞𝑨)𝑚
1
𝒑¤ = −∇ 𝐻{c𝑥 0 = − } ∇𝑥 (𝒑 − 𝑞𝑨)2 − 𝑞∇𝑥𝜑 = 𝑞 [(𝒙¤ · ∇𝑥) 𝑨 + 𝒙¤ × (∇𝑥 × 𝑨) − ∇ 𝜑]2 𝑥𝑚d𝑨 𝜕𝑨 + c 𝜕𝑨= 𝑨, 𝐻0 = + (𝒙¤ · ∇𝑥[) 𝑨d𝑡 𝜕𝑡 𝜕𝑡 ]
⇒ d d𝑨𝑚𝒙¥ = (𝒑 − 𝑞𝑨) = 𝒑¤ − 𝑞 = 𝑞 −∇ − 𝜕𝑨𝑥𝜑 + 𝑞𝒙¤ × (∇d d 𝑥 × 𝑨) = 𝑞𝐸 + 𝑞𝒗 × 𝑩𝑡 𝑡 𝜕𝑡
Eq. (A.97) follows from the correspondence principle by replacing the canonical vari-
ables 𝒑 and 𝒒 in 𝐻c0 with the respective operators.
A3
Appendix A
The proof for the meaningfulne ss〉of Eq. (A.97) consists in a test for gauge invariance.For this purpose, the solution Ψ is subjected to a 𝑈 (1) unitary transformation, i.e.
for an arbitrary function 𝜒 (𝒙〉, 𝑡), the t〉ransformed states ar〉e:Ψ̃ = 𝑈𝜒 Ψ = exp(i 𝑞ℏ 𝜒(𝒙, 𝑡))Ψ (A.98)
Consequently, the operators transform as follows:
𝒙 = 𝑈 −1𝜒 𝒙𝑈𝜒 = 𝒙 (A.99a)
?̃? = 𝒑 −1 = exp(i 𝑞 ) (−i ∇ ) exp(−i 𝑞𝑈𝜒 𝑈𝜒 ℏ𝜒 ℏ 𝑥 ℏ𝜒) == 𝒑 − 𝑞∇𝑥𝜒 (A.99b)
𝑞 𝜕 𝑞 𝜕𝜒
𝐸 = 𝑈𝜒 𝐸𝑈
−1
𝜒 = exp(i ℏ𝜒) (iℏ ) exp(−i𝜕𝑡 ℏ𝜒) = 𝐸 + 𝑞 (A.99c)𝜕𝑡
With these relations, the full transformation of the
holds:  〉
differential equation Eq. (A.97)
0 = [𝐻0(𝒑 − 𝑞𝑨, 𝒙) − (𝐸 − 𝑞𝜑)] Ψ
= 𝑈[ [𝐻 −1  〉𝜒 0(𝒑 − 𝑞𝑨, 𝒙) − (𝐸 − 𝑞𝜑)]𝑈 𝑈 𝜒 𝜒 Ψ ]
[  〉
(A.100)
= 𝐻0(𝒑 − 𝑞∇𝑥𝜒 − 𝑞𝑨, 𝒙) − (𝐸] 𝜕𝜒
= 𝐻0(𝒑 − 𝑞?̃?, 𝒙) − (𝐸 − 𝑞?̃?) 
+ 𝑞〉 − 𝑞𝜑) Ψ̃𝜕𝑡
Ψ̃
with
𝜕𝜒
?̃? = 𝑨 + ∇𝑥𝜒, and ?̃? = 𝜑 − (A.101)
𝜕𝑡
The transformed differential equation in Eq. (A.100) has the same structure as be-
fore, but different electromagnetic potentials. These are linked to the original ones
by a gauge transformation according to Eq. (A.101). Therefore, the electromagnetic
fields and, thus, the physics remain the same. The consistency between the phase
shift applied to |Ψ⟩ (Eq. (A.98)) and the gauge-transformed electromagnetic poten-
tials (Eq. (A.101)) proves that the way of introducing electromagnetic potentials into
the Hamilton operator is gauge invariant.
Eq. (A.97) is called the principle of minimal coupling since the electromagnetic poten-
tials couple only via the charge 𝑞 to the physics of the quantum particle. In other
words, the effect of electromagnetic fields on the physics of a quantum particle is
determined only by one scalar parameter and nothing else (like dipoles etc.). The
principle of minimal coupling fulfils the physical prerequisite of gauge invariance.
Moreover, it is a proof that the properties of electromagnetic fields arise naturally
from the phase degree of freedom of the wavefunction |Ψ⟩. The principle of min-
imal coupling can introduce electromagnetic fields into any differential equation of
a free particle depending on energy and momentum operators, in particular the non-
relativistic Schrödinger equation and the relativistic Dirac equation.
A4
Appendix B
B.1 The terms and ionisation energies of the lanthanoids
Hund’s second rule can be illustrated by the terms of the isolated, neutral Ln atoms.
The divalent VEC of the first lanthanoid La (4f15d06s2) is not stabilised by any ex-
change energy. Therefore, the electron leaves the 4f shell and occupies the 5d shell
instead (actual VEC: 4f05d16s2, trivalent). For the second lanthanoid Ce, the exchange
energy of two 4f electrons is too small to stabilise the divalent VEC wherefore, as in
the case of La, one 4f electron is lifted to the 5d shell (actual VEC: 4f15d16s2, trival-
ent). For all subsequent lanthanoids of the first half-series, the divalent VEC is more
stable than the trivalent one, ensured by the increasing exchange-related penalty on
removing an electron from the 4f shell. The exchange penalty peaks in Eu, whose 4f
shell is half-filled (4f75d06s2). The first element of the second half-series Gd assumes
the trivalent VEC (4f75d16s2), because it would otherwise begin to fill the other spin
component of the 4f shell without stabilising exchange energy. The next element Tb
assumes the divalent VEC, unlike Ce, though, the trivalent VEC is less stable by only
0.04 eV. All subsequent lanthanoids of the second half-series assume the divalent
VEC, analogously to the first half-series. The last lanthanoid, Lu, is trivalent, the only
possible VEC due to exhaustion of the 4f shell [75].
Hund’s rules manifest themselves also in the third ionisation energies. All Ln2+ ions
assume the divalent VEC (4f𝑛5d06s0), except La2+, Gd2+ and Lu2+, whose VECs are
trivalent (4f05d16s0, 4f75d16s0 and 4f145d06s1, respectively). However, the energy
difference to the divalent state for La2+ and Gd2+ is small compared the ionisation
energies. The peculiar dependence between the third ionisation energies and the
atomic numbers can completely be explained by three contributions:
1) The exchange penalty, which increases the ionisation energy proportional to
the number of 4f electrons with same spin. It is the most effective contribution
and gives rise to the zigzag pattern with maxima at Eu and Yb.
2) The paring energy, which destabilises the second half shell and thus reduces the
ionisation energies of the second half-series by a constant offset.
3) The orbital energy from the change in total angular momentum, which mod-
ulates the zigzag patterns by wavy slopes with inflexion points at quarter and
tree-quarters fillings of the 4f shell.
If all three contributions are subtracted, the residual of the third ionisation energies
would follow a quite smooth curve with a positive slope, which accords with the
increasing charge of the nucleus [75].
A5
Appendix B
B.2 The RESi2 – x bulk phases
𝑎 𝑏 𝑐 𝑐/𝑎 𝑉/𝑁Er 𝜇Er Δ𝜇Er
(Å) (Å) (Å) (Å3) (eV) (meV)
PBE
ort-AlB2-ErSi2 4.24 3.84 3.91 0.923 56.8 −1.484 266
hex-AlB2-ErSi2 – b 3.76 4.14 1.101 50.8 −1.750 0
tet-ThSi2-ErSi2 3.89 3.89 14.99 3.852 56.7 −1.500 251
ort-ThSi2-ErSi2 – b (1) 3.92 3.98 13.25 3.376 51.8 −1.708 43
ort-ThSi2-ErSi2 – b (2) 3.89 4.02 13.31 3.422 52.0 −1.658 92
PBEsol
ort-AlB2-ErSi2 4.18 3.85 3.83 0.917 54.7 −1.739 259
hex-AlB2-ErSi2 – b 3.72 4.10 1.102 49.0 −1.998 0
tet-ThSi2-ErSi2 3.85 3.85 14.79 3.838 54.9 −1.750 247
ort-ThSi2-ErSi2 – b (1) 3.86 3.95 13.09 3.393 49.8 −1.942 56
ort-ThSi2-ErSi2 – b (2) 3.83 3.98 13.18 3.444 50.1 −1.888 110
Exp. reference
hex-AlB2-ErSi2 – x 3.78 4.09 1.082 50.6
ort-ThSi2-ErSi2 – x — — — — —
Table B.1: Lattice parameters and relative Er chemical potentials of ErSi2 – x in the stoi-
chiometric and the vacancy-populated AlB2 and ThSi2 phases. The unit cells were optim-
ised by code-level relaxations with different xc-functionals. Experimental reference: [85].
TbSi2 – x: Tab. 4.6.
𝑑a(Å) 𝑑b(Å) 𝑑c(Å)
PBE
hex-AlB2-TbSi2 2.37 (mm)
ort-AlB2-TbSi2 2.36/2.42 (mm)
hex-AlB2-TbSi2 – b 2.43 (mv)
tet-ThSi2-TbSi2 2.36 (mm) 2.36 (mm) 2.43 (mm)
ort-ThSi2-TbSi2 – b (1) 2.49 (vv) 2.44 (mv) 2.36 (mm) 2.39 (mv)
ort-ThSi2-TbSi2 – b (2) 2.50 (vv) 2.47 (vv) 2.37 (mm) 2.37 (mv)
PBEsol
ort-AlB2-TbSi2 2.34/2.39 (mm)
hex-AlB2-TbSi2 – b 2.42 (mv)
tet-ThSi2-TbSi2 2.33 (mm) 2.33 (mm) 2.38 (mm)
ort-ThSi2-TbSi2 – b (1) 2.48 (vv) 2.44 (mv) 2.35 (mm) 2.38 (mv)
ort-ThSi2-TbSi2 – b (2) 2.49 (vv) 2.46 (vv) 2.36 (mm) 2.37 (mv)
Table B.2: Si–Si nearest neighbour distances in TbSi2 – x optimised with different xc-function-
als. Complement to Tab. 4.7.
A6
B.2 The RESi2 – x bulk phases
𝑑a(Å) 𝑑b(Å) 𝑑c(Å)
PBE
hex-AlB2-ErSi2 2.36 (mm)
ort-AlB2-ErSi2 2.35/2.42 (mm)
hex-AlB2-ErSi2 – b 2.43 (mv)
tet-ThSi2-ErSi2 2.35 (mm) 2.35 (mm) 2.42 (mm)
ort-ThSi2-ErSi2 – b (1) 2.51 (vv) 2.45 (mv) 2.36 (mm) 2.39 (mv)
ort-ThSi2-ErSi2 – b (2) 2.52 (vv) 2.48 (vv) 2.36 (mm) 2.37 (mv)
PBEsol
ort-AlB2-ErSi2 2.39/2.34 (mm)
hex-AlB2-ErSi2 – b 2.42 (mv)
tet-ThSi2-ErSi2 2.33 (mm) 2.33 (mm) 2.39 (mm)
ort-ThSi2-ErSi2 – b (1) 2.51 (vv) 2.44 (mv) 2.35 (mm) 2.38 (mv)
ort-ThSi2-ErSi2 – b (2) 2.51 (vv) 2.48 (vv) 2.35 (mm) 2.37 (mv)
Table B.3: Si–Si nearest neighbour distances in ErSi2 – x optimised with different xc-
functionals. Complement to Tab. 4.7.
𝑎 𝑐 𝑉/𝑁RE 𝛾 𝑑a Δ𝜇RE
(Å) (Å) (Å3) (Å) (meV)
TbSi2, PBE
ort-AlB 4.23 (3.88)𝑎 4.00 58.3 124.7◦ (110.6◦)𝑎 2.36 (2.42)𝑎2 0
tr3-CaSi2 3.78 5.56 63.4 103.6◦ 2.41 −30
tr6-CaSi2 3.94 9.34 60.9 104.9◦, 118.9◦ 2.49, 2.29 −67
TbSi2, PBEsol
ort-AlB2 4.17 (3.88)𝑎 3.92 56.2 124.0◦ (112.0◦)𝑎 2.34 (2.39)𝑎 0
tr3-CaSi2 3.75 5.50 61.6 103.2◦ 2.39 +54
tr6-CaSi2 3.91 9.22 59.1 104.6◦, 118.9◦ 2.47, 2.27 −34
ErSi2, PBE
ort-AlB2 4.24 (3.84)𝑎 3.91 56.8 125.4◦ (109.3◦)𝑎 2.35 (2.42)𝑎 0
tr3-CaSi2 3.75 5.48 61.4 103.0◦ 2.40 −140
tr6-CaSi ◦2 3.92 9.17 59.1 104.6 , 118.8◦ 2.48, 2.28 −100
ErSi2, PBEsol
ort-AlB 4.18 (3.85)𝑎 3.83 54.7 124.6◦ (110.8◦)𝑎 2.34 (2.39)𝑎2 0
tr3-CaSi2 3.72 5.42 59.7 102.6◦ 2.38 −63
tr6-CaSi2 3.89 9.05 57.4 104.3◦, 118.8◦ 2.46, 2.26 −70
Table B.4: Lattice parameters and relative RE chemical potentials of TbSi2 and ErSi2 in the
stoichiometric AlB2 and CaSi2 phases. The unit cells were optimised by code-level relaxations
with different xc-functionals. tr6-CaSi2-RESi2 has two independent M-star angles 𝛾 and Si–Si
bond lengths 𝑑 . 𝑎a orthorhombic distortion. Complement to Tab. 4.8.
A7
Appendix B
(i) s-orbitals (ii) py-orbitals (iii) pz-orbitals (iv) px-orbitals
(a)
0
-2
-4
-6
-8
-10
-12 hex-AlB2-TbSi2
(b)
0
-2
-4
-6
-8
-10
-12 hex-AlB2-Si2
(c)
0
-2
-4
-6
-8
-10
-12 ort-AlB2-TbSi2
(d)
0
-2
-4
-6
-8
-10
-12 ort-AlB2-Si2
M Γ K|Γ A M Γ K|Γ A M Γ K|Γ A M Γ K|Γ A
Figure B.1: Band structures of (a) hex-AlB2-TbSi2, (b) hex-AlB2-Si2, (c) ort-AlB2-TbSi2 and (d)
ort-AlB2-Si2. The Si2 structures are the respective TbSi2 structures without the Tb atoms. The
line width corresponds to the PAW projections of (i) the s orbitals, (ii) the p𝑦 orbitals, (iii) the
p𝑧 orbitals and (iv) the p𝑥 orbitals of the Si atoms.
(i) tet-ThSi2-TbSi2 (trivalent) (ii) AMO (iii) FMO (up) (iv) FMO (down)
0
-2
-4
-6
-8
-10
-12
Γ XY Σ ΓZ Y1Σ1 Z Γ XY Σ ΓZ Y1Σ1 Z Γ XY Σ ΓZ Y1Σ1 Z Γ XY Σ ΓZ Y1Σ1 Z
Figure B.2: Band structures of (i) trivalent and (ii) – (iv) f-valent tet-ThSi2-TbSi2. The plotting
mode is analogous to Fig. 4.9 (b,i). Colour: yellow→ Tb, blue→ Si. AlB2 structures: Fig. 4.11.
A8
E − EF  (eV)
E − EF  (eV) E − EF  (eV) E − EF  (eV) E − EF  (eV)
Appendix C
C.1 The geometric interpretation of the Ahanorov-Bohm
effect
As already mentioned, geometric phases become manifest in the Aharonov-Bohm
effect, although not always recognised as such. Berry transferred his considerations
to the ABE by a gedankenexperiment consisting in a thin solenoid like that in Sec.
7.1 and a particle with charge 𝑞 confined to a box outside the magnetic field [127].
The origin of the box defines the three-dimensional parameter 𝝃, so a variation of 𝝃
corresponds to a movement of the box. Unlike that in [127], the approach presented
here works with the operator algebra, independently of the representation.
Let Ω be the space outside the solenoid and 𝑊 (𝝃) ⊂ Ω the box domain, a simply
connected neighbourhood of 𝝃. Let 𝐻off(𝒑, 𝒙) be the Hamilton operator on𝑊 for the
solenoid-off setup and 𝝃 = 0. Let |Ψoff(0)⟩ be the eigenstate of the corresponding
stationary Schrödinger equation.  〉  〉
𝐻off(𝒑, 𝒙)Ψoff(0) = 𝐸 Ψoff(0) (C.102)
Without loss of generality, the expectation values of 𝒙 and 𝒑may be 0, i.e. the particle
is located at the b〈ox origin and do〉es not move〈on average. 〉
Ψoff(0) |𝒙 |Ψoff(0) = 0 ∧ Ψoff(0) |𝒑|Ψoff(0) = 0 (C.103)
Now, two gauge transformations are being applied to the system: switching on the
solenoid and moving the box. Since the box is simply connected and field-free, the
solenoid field alters the wavefunction by a unitary transformation 𝑈D, also called
Dirac factor [127]. The movement of the box transforms implies the unitary trans-
formation linked to the translation group𝑈T (vide appendicem, A.1).
𝑈T = exp(−i
1
ℏ 𝝃 · 𝒑) (Translation) (C.104)
𝑈D = exp(i
𝑞
ℏ 𝜒(𝒙 | 𝝃)) (Dirac factor)
with ∫ 𝒙
𝜒(𝒙 | 𝝃) B d𝒙′ · 𝑨(𝒙′) = 𝜒(𝒙) − 𝜒(𝝃) ⇔ 𝑨(𝒙) = ∇𝑥𝜒(𝒙 | 𝝃)
𝝃
A9
Appendix C
Nota bene, the Dirac factor depends on 𝝃, which is chosen to be the origin of the box.
The entire integration path has to stay within the box so as to ensure the integrability
of 𝑨. The 𝝃-dependent term of 𝜒 must not be omitted, as it accommodates the pre-
condition that 𝜒 is defined only locally. Omitting it would correspond to trying ti find
a global scalar potential for 𝑨, which, of course, does not exist.
By means of the explicit representations of both unitary transformations, the station-
ary Schrödinger equation for the solenoid-on, boxed-moved setup, described by the
Hamilton operator 𝐻on(𝒑, 𝒙 | 𝝃), can be solved.
𝑈 −1D 𝒙𝑈D = 𝒙, 𝑈T 𝒙𝑈
−1
T = 𝒙 + 𝝃
− (C.105a)𝑈D 𝒑𝑈
1
D = 𝒑 − 𝑞𝑨(𝒙), 𝑈 −1T 𝒑𝑈T = 𝒑
⇒  𝐻on(𝒑〉, 𝒙 | 𝝃) =𝑈T𝑈D 𝐻〉off(𝒑, 𝒙)𝑈−1D 𝑈−1T = 𝐻off(𝒑 − 𝑞𝑨(𝒙+ 𝝃), 𝒙 +〉 𝝃) (C.105b)Ψon(𝝃) = 1 𝑞𝑈T𝑈D Ψoff(0) = exp(−i ℏ 𝝃 · 𝒑) exp(i ℏ 𝜒(𝒙 | 𝝃)) Ψoff(0) (C.105c)
Inserting the transformed wavefunctions from Eq. (C.105c) into Eq. (7.55) yields the
Berry connection A. In the last step, Eq. (C.103) is used.
𝑈〈 −1D i∇𝜉𝑈 D = i∇𝜉 +〉
𝑞
ℏ𝑨〈(𝝃), 𝑈−1T i∇𝜉𝑈T = i∇𝜉 + 1ℏ𝒑⇒ A(𝝃) = i〈  〉Ψon(𝝃) ∇𝜉 Ψon(𝝃) = Ψ (0) 𝑈−1𝑈−1off D T i∇ 𝜉𝑈T〉𝑈D Ψoff(0)= Ψoff(0) (i∇ 𝑞𝜉 + 𝑨(𝝃) + 1 (𝒑 + 𝑞𝑨(𝒙))) Ψoff(0) (C.106)ℏ ℏ
= 𝑞ℏ𝑨(𝝃) +
𝑞
ℏ𝑨(0)
The Berry connection A indeed equals the magnetic vector potential 𝑨 up to prefact-
ors. Integrating it over a cl∮osed loop 𝛾 : 𝑡 ∮∈ [0, 1] → 𝝃(𝑡) ∈ Ω returns the observableBerry phase 𝜒B:
𝜒B = A(𝝃) · d𝝃 = 𝑞ℏ 𝑨(𝝃) · d𝝃 =
𝑞
𝑁 ℏΦ0 (C.107)
𝛾 𝛾
where 𝑁 is the winding number around the solenoid.
In conclusion, the treatment of the ABE within the Berry approach results in the same
findings as those obtained in Sec. 7.1. The Berry phase in Eq. (C.107) equals the
ABE phase in Eq. (7.52), both being proportional to the flux through the solenoid.
The Berry approach confirms that the central origin of the ABE consists in the non-
existence of a global gauge for 𝑨 onΩ. Like Berry phases from closed loops in general,
the ABE is observable, although the box containing the particle has never penetrated
the magnetic field inside the solenoid.
Moreover, Eq. (C.107) reveals an important property of geometric phases. Because
the phase function exp(i𝜒B) is periodic, it remains unchanged after the addition of
integer multiples of 2π to 𝜒B. Consequently, a magnetic flux Φ0 being integer mul-
tiples of the quantum flux ℎ𝑞 results in the same phase factor as the zero-flux con-
figuration and, thus, in the same interference patterns. Therefore, the presence of
a magnetic field permits the prediction of interference patterns, but the reverse, i.e.
A10
C.2 Proofs concerning the ℤ2 insulator
the unambiguous deduction of the magnetic field from the interference patters, is
impossible. Only continuous flux changes allow the observer to count the number of
circles passed and thus to determine the absolute magnetic flux difference from the
interference patterns.
C.2 Proofs concerning the ℤ2 insulator
Gauge invariance of the electric polarisation under𝑈 (?̄?) transformations
The electric polarisation in terms of the Berry phase in Eq. (7.64) is invariant under
𝑈 (?̄?) transformations, which mix the states of the occupied Bloch space.
Proof. Let 𝑆(𝝃) be a 𝝃-dependent, unitaryℂ?̄?×?̄?-matrix rotating the occupied subspace
spanned by the Bloch factors {|𝑢𝜆𝛼(𝒌)⟩}. Then the Berry connection A(𝝃) transforms
as follows: ( )ᵀ ( )
𝑢(𝝃) | (𝜆 )(𝒔)⟩ (𝜆 ) (𝜆 ) (𝜆 )B 𝑢1 , . . . , |𝑢 (𝒔)⟩ , 𝑢
∗(𝝃) B ⟨𝑢1 (𝒔) |, . . . , ⟨𝑢 (𝒔) |?̄? ?̄?
?̃?(𝝃) B∏𝑆(𝝃[)∫𝑢(𝝃)1 ] ∏ [∫ 1 ]
Ã𝑖 (𝝃) = d𝑠 ?̃?∗(𝝃) i∇ ?̃?(𝝃) = d𝑠 𝑢∗(𝝃) 𝑆∗(𝝃) i∇ 𝑆(𝝃) 𝑢(𝝃)∏ 𝑗𝑗≠𝑖 [∫0 1 ] ∑︁ 𝜉 𝑗 𝜉𝑗≠𝑖 0
= ( [∫ d𝑠
∗
𝑗 ] 𝑢𝑘 (𝝃) 𝑆
∗
𝑘𝑙 (𝝃)) i∇𝜉 𝑆∏ ∑︁𝑙𝑚
(𝝃) 𝑢𝑚(𝝃)
𝑗≠𝑖 0 𝑘,𝑙,𝑚
1
= d𝑠 𝑢∗(𝑗 (𝝃) i∇
∗
𝜉 𝑢)(𝝃) + 𝑆𝑘𝑙 (𝝃) i∇𝜉 𝑆𝑙𝑘 (𝝃)𝑗≠𝑖 0 𝑘,𝑙
= A𝑖 (𝝃) + i tr 𝑆∗(𝝃) ∇ ( )
[128]
𝜉 𝑆 𝝃 = A𝑖 (𝝃) + i∇𝜉 ln(det(𝑆(𝝃)))
= A𝑖 (𝝃) − ∇𝜉𝜃(𝝃)
(C.108)
where 𝜃(𝝃) is the phase of the determinant of 𝑆(𝝃). Thus, also a more general 𝑈 (?̄?)
transformation basically reproduces Eq. (7.59), so band reordering within the occu-
pied subspace during the parametric variation does not affect the results. □
Nota bene, while band reordering within the occupied manifold is allowed, the con-
servation of the band gap separating the occupied subspace from its unoccupied com-
plement is still obligatory for all values of 𝝃. In other words, the system strictly has
to remain a band insulator for all 𝜆.
A11
Appendix C
Kramers link between the Berry connections of the components of a
T-symmetric system
Eqs. (7.79) state that the Berry connections of the components of a T-symmetric Bloch
system are linked by the Kramers symmetry. For the proof, it is crucial to keep track
of the chain rule when dealing with the gradient of a function like 𝑓 (−𝒌). This is
facilitated by the following notation: ∇𝑘 𝑓 (−𝒌) means that the ∇-operator has not yet
been executed and acts on the exterior 𝑓 and its argument −𝒌, the latter giving a
minus sign. On the other hand, (∇𝑘 𝑓 ) (−𝒌) means that the gradient of 𝑓 is evaluated
at −𝒌. The difference between the two definitions is thus a minus sign.
By using the Kramers symmetry (Eq. (7.74)), the above remarks and the relations
⟨𝑎 | ∇𝑎⟩ = −⟨∇𝑎 | 𝑎〈⟩ and ⟨Θ𝑎 | Θ𝑏⟩ = ⟨𝑏〉| 𝑎⟩, the Berry connection of component Ievaluated at −𝒌 can be rewritten as follows:AI𝛼(−𝒌) = i 𝑢〈I𝛼(−𝒌) (〈 
∇ I𝑘𝑢𝛼) (−𝒌) 〉
= −i 𝑢I (−𝒌) 𝛼 ∇ 𝑘 𝑢I𝛼(−𝒌)
= −i Θ𝑢II
 〉
𝛼 (𝒌)  exp〈(−i𝜒𝛼(𝒌))∇𝑘 exp(i𝜒 (𝒌)) Θ𝑢II𝛼 𝛼 (𝒌)= +(∇ 𝜒 ) (𝒌) − i〈Θ𝑢II(𝒌) Θ(∇ 𝑢II 〉𝑘 𝛼 𝛼 𝑘 𝛼)〉(𝒌) (C.109a)= +(∇ 𝜒 ) (𝒌) − i〈 (∇ 𝑢II𝑘 𝛼 𝑘 𝛼) (𝒌) 𝑢II𝛼 (𝒌)= +(∇ 𝜒 II II 〉𝑘 𝛼) (𝒌) + i 𝑢𝛼 (𝒌) (∇𝑘𝑢𝛼) (𝒌)
= +(∇ 𝜒 ) (𝒌) + AII𝑘 𝛼 𝛼 (𝒌)
For the second com〈 ponent, a similar expressiII 〈II  II 〉 〈
on holds:  〉
A𝛼 (−𝒌) = i 𝑢𝛼 (−𝒌) (∇𝑘𝑢𝛼) (−𝒌) = −i 𝑢II(−𝒌) ∇ 𝑢II𝛼 𝑘 𝛼 (−𝒌) 〉
= −i Θ𝑢I𝛼(𝒌) exp(〈−i𝜒𝛼(−𝒌))∇𝑘 exp(i𝜒𝛼(−𝒌)) Θ𝑢I (𝒌)
= −(∇ 𝜒 ) (−𝒌) − i〈 𝛼Θ𝑢I (𝒌)  〉𝑘 𝛼 𝛼 Θ(∇ I𝑘𝑢𝛼) (𝒌) (C.109b)= −(∇ 𝜒 ) (−𝒌) + i 𝑢I (𝒌) (∇ 𝑢I 〉𝑘 𝛼 𝛼 𝑘 𝛼) (𝒌)
= −(∇𝑘𝜒𝛼) (−𝒌) + AI𝛼(𝒌)
The sum and the difference between the partial connections of component I and II,
defined asA+(𝒌) andA− (𝒌) in Eqs. (7.80a) and (7.81a), can be transformed into each
other as follows:
A− (𝒌) + A− (−𝒌)
AI(𝒌) − AII(𝒌) + AI(−𝒌) − AII(−𝒌)
Eqs. (C.109) ∑︁
= AI(𝒌) + AII(𝒌) − A∑︁I(𝒌) − AII(𝒌) + (∇𝑘𝜒𝛼) (𝒌) + (∇𝑘𝜒𝛼) (−𝒌) (C.110)𝛼
A+(𝒌) − A+(−𝒌) + (∇𝑘𝜒𝛼) (𝒌) + (∇𝑘𝜒𝛼) (−𝒌)
𝛼
A12
C.3 The antimony on bismuth selenide heterostructure
C.3 The antimony on bismuth selenide heterostructure
1.0K Γ M K Γ M K Γ M(a) P01 (b) P0102 (c) P0120
0.5 D*
TSS PSb
0.0 TSS
D*
R
B BS B PSb
-0.5 RBS
-1.0
1.0 (d) P10 (e) P1012 (f) P1021
0.5
0.0
-0.5
-1.0
-1.0 -0.5 0.0 0.5 -1.0 -0.5 0.0 0.5 -1.0 -0.5 0.0 0.5
BS Sb k  (Å−1) k  (Å−1ǁ ǁ ) kǁ (Å−1)
Figure C.1: Band structures of 1BL-Sb and 2BL-Sb structure models with different stacking
optimised with DFT-D2+SOC. (a) P01, (b) P0102, (c) P0120, (d) P10, (e) P1012, (f) 1021. The line
width, colouring and Fermi levels are analogous to Fig. 9.4. Arrows highlight special features.
Stable structure models are highlighted bold. Reproduced from [P7].
A13
E − EF (eV) E − EF (eV)

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A23
Publication list
[P1] S. Sanna, U. Gerstmann, E. Rauls, Y. Li, M. Landmann, A. Riefer, M. Rohrmüller, N. Vollmers,
M. Witte, R. Hölscher, A. Lücke, C. Braun, S. Neufeld, K. Holtgrewe and W. G. Schmidt, Sur-
face Charge of Clean LiNbO3 Z-Cut Surfaces, in High perform. comput. sci. eng. ‘14 (Springer
International Publishing, Cham, 2015), pp. 163–178, doi: 10.1007/978-3-319-10810-0_12.
[P2] A. Lücke, U. Gerstmann, S. Sanna, M. Landmann, A. Riefer, M. Rohrmüller, N. J. Vollmers, M.
Witte, E. Rauls, R. Hölscher, C. Braun, S. Neufeld, K. Holtgrewe and W. G. Schmidt, Solving the
Scattering Problem for the P3HT On-Chain Charge Transport, in High perform. comput. sci. eng.
´15 (Springer International Publishing, Cham, 2016), pp. 155–170, doi: 10.1007/978-3-319-
24633-8_10.
[P3] I. Miccoli, F. Edler, H. Pfnür, S. Appelfeller, M. Dähne, K. Holtgrewe, S. Sanna, W. G. Schmidt
and C. Tegenkamp, Atomic size effects studied by transport in single silicide nanowires, Phys.
Rev. B 93, 125412 (2016), doi: 10.1103/PhysRevB.93.125412.
[P4] S. Sanna, C. Dues, U. Gerstmann, E. Rauls, D. Nozaki, A. Riefer, M. Landmann, M. Rohrmüller,
N. J. Vollmers, R. Hölscher, A. Lücke, C. Braun, S. Neufeld, K. Holtgrewe and W. G. Schmidt,
Submonolayer Rare Earth Silicide Thin Films on the Si(111) Surface, in High perform. comput.
sci. eng. ´16 (Springer International Publishing, Cham, 2017), pp. 163–175, doi: 10.1007/978-
3-319-47066-5_12.
[P5] C. Hogan, K. Holtgrewe, F. Ronci, S. Colonna, S. Sanna, P. Moras, P. M. Sheverdyaeva, S. Ma-
hatha, M. Papagno, Z. S. Aliev, M. Babanly, E. V. Chulkov, C. Carbone and R. Flammini, Tem-
perature Driven Phase Transition at the Antimonene/Bi2Se3 van der Waals Heterostructure, ACS
Nano 13, 10481–10489 (2019), doi: 10.1021/acsnano.9b04377.
[P6] K. Holtgrewe, S. Appelfeller, M. Franz, M. Dähne and S. Sanna, Structure and one-dimensional
metallicity of rare-earth silicide nanowires on Si(001), Phys. Rev. B 99, 214104 (2019), doi: 10.
1103/PhysRevB.99.214104.
[P7] K. Holtgrewe, S. K. Mahatha, P. M. Sheverdyaeva, P. Moras, R. Flammini, S. Colonna, F. Ronci,
M. Papagno, A. Barla, L. Petaccia, Z. S. Aliev, M. B. Babanly, E. V. Chulkov, S. Sanna, C. Hogan
and C. Carbone, Topologization of 𝛽-antimonene on Bi2Se3 via proximity effects, Sci. Rep. 10,
14619 (2020), doi: 10.1038/s41598-020-71624-4.
[P8] S. Appelfeller, K. Holtgrewe, M. Franz, L. Freter, C. Hassenstein, H.-F. Jirschik, S. Sanna and M.
Dähne, Continuous crossover from two-dimensional to one-dimensional electronic properties
for metallic silicide nanowires, Phys. Rev. B 102, 115433 (2020), doi: 10.1103/PhysRevB.102.
115433.
[P9] K. Holtgrewe, C. Hogan and S. Sanna, Evolution of Topological Surface States Following Sb Layer
Adsorption on Bi2Se3, Materials (Basel). 14, 1763 (2021), doi: 10.3390/ma14071763.
[P10] C. Hogan, P. Lechifflart, S. Brozzesi, S. Voronovich-Solonevich, A. Melnikov, R. Flammini, S.
Sanna and K. Holtgrewe, Theoretical study of stability, epitaxial formation, and phase trans-
formations of two-dimensional pnictogen allotropes, Phys. Rev. B 104, 245421 (2021), doi: 10.
1103/PhysRevB.104.245421.
[P11] K. Holtgrewe, F. Ziese, J. Bilk, M. N. Pionteck, K. Eberheim, F. Bernhardt, C. Dues and S. Sanna,
Tuning the Conductivity of Metallic Nanowires by Hydrogen Adsorption, in High perform. com-
put. sci. eng. ’20 (Springer International Publishing, Cham, 2021), pp. 133–146, doi: 10.1007/
978-3-030-80602-6_9.
[P12] S. Sanna, J. Plaickner, K. Holtgrewe, V. M. Wettig, E. Speiser, S. Chandola and N. Esser, Spectro-
scopic Analysis of Rare-Earth Silicide Structures on the Si(111) Surface, Materials (Basel). 14,
4104 (2021), doi: 10.3390/ma14154104.
[P13] Z. Mamiyev, C. Fink, K. Holtgrewe, H. Pfnür and S. Sanna, Enforced Long-Range Order in 1D
Wires by Coupling to Higher Dimensions, Phys. Rev. Lett. 126, 106101 (2021), doi: 10.1103/
PhysRevLett.126.106101.
[P14] C. Fink, F. A. Pfeiffer, K. Holtgrewe and S. Sanna, Au adsorption on stepped Si(ℎℎ𝑘)-Au surfaces,
Surf. Sci. 718, 122010 (2022), doi: 10.1016/j.susc.2021.122010.
A24
Danksagung
Ich möchte mich hiermit bei all denjenigen bedanken, die mich auf meinem Wege
bis zum Abschluss der Promotion begleitet haben.
Zuallererst gilt ein großer Dank meinem Doktorvater Prof. Simone Sanna, der mich
immer kräftig unterstützt hat. Er hat sich nie davor gescheut, mir auch schwierige
Aufgaben zu übertragen, und mir gleichzeitig den Freiraum gegeben, sie entspre-
chend gründlich zu bearbeiten. Insbesondere die Tatsache, dass er zum Beginn mei-
ner Promotion den Ruf an die JLU bekommen und folglich mit mir und Dr. Christof
Dues zusammen die Arbeitsgruppe gegründet hat, war eine Herausforderung, auch
aus organisatorischer Sicht, die sich im Nachhinein als wertvolle Erfahrung erwie-
sen hat. Bei Christof möchte ich mich in diesem Zuge auch sehr herzlich bedanken,
ebenso bei allen anderen Mitgliedern der AG Sanna, für gute fachliche Gespräche
und ein hervorragendes Arbeitsklima, auch extra laborem.
Ich möchte mich sehr herzlich bei Prof. Wolf Gero Schmidt bedanken, der während
meines Studiums an der Universität Paderborn mein Interesse an der theoretischen
Materialphysik geweckt hat und mich bis zum Masterabschluss und darüber hinaus
bestens unterstützt hat.
I want to thank Dr. Conor Hogan, who was my supervisor during my ERASMUS+ re-
search internship at the university “Tor Vergata”, Rome, Italy. The collaboration with
him was very precious and fruitful. Furthermore, I want to thank all other colla-
borators with whom I worked on the RESi2 nanowires and the antimony on Bi2Se3
heterostructures.
Ein ganz großer Dank gilt meinen Eltern, die schon früh mein Interesse an der Na-
turwissenschaft geweckt und mich in meinen Vorhaben immer unterstützt haben.
Des Weiteren gilt mein Dank auch dem Rest meiner Familie und meinen Freunden
dafür, dass sie mir immer Halt gegeben haben.
Ich möchte mich bei der Deutschen Forschungsgemeinschaft (DFG) für die finanzielle
Unterstützung bedanken. Ferner möchte ich mich bei den HPC Einrichtungen bedan-
ken, deren Rechenzeiten die Ergebnisse dieser Arbeit ermöglicht haben: das Hoch-
schulrechenzentrum der JLU Gießen, das Hochschulrechenzentrum der TU Darm-
stadt und das Höchstleistungsrechenzentrum Stuttgart.
A25