Justus-Liebig-Universität Gießen I. Physikalisches Institut Control of Emission Properties of Semiconductors through Functionalization Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften - Dr. rer. nat. - dem Fachbereich 07 Mathematik und Informatik, Physik, Geographie vorgelegt von Florian Dobener aus Herborn Gießen, im September 2019 1. Gutachter: Prof. Dr. Sangam Chatterjee 2. Gutachter: Prof. Dr. Wolfram Heimbrodt Conten t s 1 Introduction 1 2 Light Matter Interaction 5 2.1 Absorptive Optics in Semiconductors . . . . . . . . . . . . . . . . . . . . 6 2.1.1 Band Structure and Density of States . . . . . . . . . . . . . . . . 7 2.1.2 Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.3 Photoluminescence . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.1.4 Influence of Strain and Localized States . . . . . . . . . . . . . . . 19 2.1.5 Nonradiative Recombination . . . . . . . . . . . . . . . . . . . . . 22 2.1.6 Influence of Defects and Disorder . . . . . . . . . . . . . . . . . . 23 2.2 Nonlinear Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2.1 Anharmonic Oscillator Model . . . . . . . . . . . . . . . . . . . . . 29 2.2.2 Sum-Frequency Generation . . . . . . . . . . . . . . . . . . . . . . 30 2.2.3 Phase Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.2.4 Second-Harmonic Generation . . . . . . . . . . . . . . . . . . . . . 34 2.2.5 Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.2.6 Supercontinuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.2.7 Molecular Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3 Heterostructures 45 3.1 Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 Interfaces and Band Alignment . . . . . . . . . . . . . . . . . . . . . . . . 49 3.3 Light-Matter-Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4 Electron Matter Interaction 55 4.1 Three-Step Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.2 Background and Line Shape . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.3 Valence-Band-Edge Spectroscopy . . . . . . . . . . . . . . . . . . . . . . 60 I Determination of the Ga(N,As,P)/GaP Band-Offset 5 Ga(N,As,P) Heterostructures 65 6 Methods 71 6.1 Photoluminescence Excitation Spectroscopy . . . . . . . . . . . . . . . . 71 6.2 X-Ray Photoeelectron Spectroscopy . . . . . . . . . . . . . . . . . . . . . 73 7 Hetero-Offsets in Ga(N,As,P) Structures 81 7.1 Indirect Measurements by Photoluminescence Excitation Spectroscopy 81 7.2 Direct Measurements by Photoelectron Excitation . . . . . . . . . . . . . 86 7.2.1 GaP/Si Offset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 7.2.2 (BGa)(AsP)/GaP Offset . . . . . . . . . . . . . . . . . . . . . . . . . 91 7.2.3 Ga(N,As,P)/GaP Offset . . . . . . . . . . . . . . . . . . . . . . . . . 92 7.3 Review of the Offsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 II Nonlinear Effects 8 Materials 105 8.1 KNbO3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 iii iv Contents 8.2 Quartz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 8.3 Organotin Sulfide Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 9 Methods 109 9.1 Absolute Second Harmonic Generation . . . . . . . . . . . . . . . . . . . 109 9.2 Steady-State White-Light . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 10 Results 115 10.1 Absolute Second-Order Nonlinear Coefficient of KNbO3 . . . . . . . . . 115 10.2 White-light Generation in Organotin Sulfide Clusters . . . . . . . . . . . 120 11 Conclusion and Outlook 123 A Appendix 139 B Statement of Authorship 141 C Acknowledgements (in German) 143 D Abstract (in German) 145 L i s t o f Abbrev i a t i ons APB Anti phase boundary BAC Band anti-crossing BEC Bound exciton complex CB Bonduction band CCD Charge-coupled-device CHA Concentric hemispherical analyzer CL Core level CMOS Complementary metal-oxid-semiconductor cw Continious wave DAP Donor-acceptor-pair DFT Density functional theory DOS Density of states erf Error function ESCA Electron spectroscopy for chemical analysis FAT Fixed analzyer transmission FRR Fixed retardation ratio FWHM Full-width-half-maximum GVD Group velocity dispersion hh Heavy-hole HHG High harmonic generation HOMO Highest occupied molecular orbital IMFP Inelastic mean free path LCAO Linear combination of atomic orbitals LED Light emitting device lh Light-hole LUMO Lowest unoccupied molecular orbital v vi Contents MOVPE Metal-organic vapour-phase epitaxy MQW Multi quantum well MST Model solid theory NA Numerical aperture Nd:YLF Neodymium doped yttrium-lithium-fluorine PCF Photonic crystal fiber PES Photoelectron spectroscopy PL Photoluminescence PLE Photoluminescence excitation spectroscopy QW Quantum well SBE Semiconductor bloch equation SFG Sum frequency generation SHG Second harmonic generation SPM Self phase modulation TEM Transmission electron microscopy UHV Ultra high vacuum UPS UV photoelectron spectroscopy UV Ultra violet VB Valence band VBM Valence band maximum VCA Virtual crystal approximation XPS X-ray photoelectron spectroscopy XRD X-ray diffraction 1 In t roduct ion Today’s world is governed by information. Life revolves around the accessibility and availability of communication channels and the speed of digital information interchange has reached a prior unimaginable pace. It feels natural to us to communicate with people anywhere in the world in a fraction of seconds, with devices so small, that they fit in our pockets. Hence, wirelessness and seamlessness are the terms of modern society’s communication, shaped by big technology companies. The consequence is a gradual expansion of the boundaries of technological progress and possibilities. Recent advances towards an always-reachable pool of information and data, known as the cloud, stresses again the timelessness and importance of information availability at any time in the modern world. Some influential companies even reduce the amount of ports of their devices, stating that the future is solely wireless. However, these developments required significant technological efforts. The first landmark and technological backbone dates back to 1947 when Shockley, Bardeen and Brattain invented the world’s first transistor1. The device was introduced with the words “[The transistor] may be employed as an amplifier, oscillator, and for other purposes for which vacuum tubes are ordinarily used”1, which hardly captures today’s importance of the transistor. Eleven years later, Jack Kilby solved the tyranny of numbers2, which is the ever-growing demand for more wiring and components in complex electronic designs. His simple, yet ingenious idea was to combine transistors on a single piece of semiconductor, creating the first integrated circuit3. This device was simple and just generated a plain sine wave4, but what followed is a success story of semiconductor technology, leading to devices of billions of transistors in handheld devices, super- computers able to simulate our physical world, globe-spanning systems of connections and thus, our modern world in all. Modern transistors consist of silicon and technological advances in silicon growth led to the availability of single crystals with a pureness of 99.9999%5, most likely rendering silicon the purest material on earth. It was a prospecting candidate to redefine the unit of weight due to the precise control of the material, too6. Current silicon based devices have a transistor size of 14nm, only a factor 70 bigger than the atomic radius of silicon. This strong miniaturization raises the question: where is the limit? In fact, the past pace of roughly doubling the amount of transistors per area every two years, known as Moore’s law, is already stagnating. Therefore, recent research point towards the control of single quantum states (e.g. single atomic levels) as a building block of quantum computers7, which enable a completely new way of computation and bring consequences for society yet unclear. Most likely, the yet to come technology will call for further technological advances, such as the progression of semiconductor technology fueled the demand for information exchange, culminating in the publication of the hypertext protocol by Tim Berners Lee and Robert Cailiau in 19908. It was the birth of the internet, which further powered the effort, money and research put into semiconductor technology. Certainly, the emerging possibility of inter- device communication called for fast communication capabilities. As light is the fastest possible way of information interchange, its use in this context seems inevitable. Indeed, semiconductor light emitters and detectors cover most of the communication devices 1 2 Introduction today, arisen from breakthrough inventions, such as the light emitting device (LED) or the laser. Semiconductor light-emitter development took place parallel to the birth of integrated circuits. Theodor Maiman first demonstrated lasing in a ruby crystal in 19609. Within two years, Hall et al.10 transferred the laser concept to semiconductors by demonstrating laser emission of GaAs heterostructures. The history of light emission from semiconductors actually outdates electronic diodes - over 100 years ago an electroluminescence device was reported by Round11. In the 1920’s, Oleg Vladimirovich Losev carefully characterized the electrical and optical properties of LEDs in 16 publications12, describing it as the reversal of Einstein’s photoelectrical effect. After 40 years of little research on the topic, the impact of semiconductor technology and the possibility of lasing in heterostructures took over the development of LEDs, too. Ga(As,P) structures were the first LEDs emitting in the visible part (red) of the spectrum13, quickly followed by green (GaP, 1970)14 and blue (GaN, 1980/90)15 light emitting diodes. Today, except for a small region in the green spectral region, LEDs exist for emission wavelengths throughout the complete visible spectrum, making them cost and energy effective devices for many applications. Therefore, LEDs are the basis for most illumination devices, such as screens or home lightning16, where it completely replaced the ineffective tungsten light bulb17. However, white-light emission from semiconductor devices remains challenging due to the inherent restriction to the band gap of the material. For lasers the situation is even worse, since they require direct-band-gap materials, thus, even less wavelengths are available. This thesis contributes to the study of materials for functionalization of semiconductors, to broaden the spectral regions of wavelengths accessible by light emitting semiconductor devices. Three main topics are covered: the engineering of novel structures for lasers directly integrated on silicon and two investigations related to the transformation of laser light in the scope of non-linear optical processes. The non- linear optical processes are investigated in the framework of second harmonic emission and broadening of the incident laser light to the emission of visible to near-infrared white-light. The lack of efficient electrically driven lasers in direct connection with silicon tran- sistors circumvents the realization of on-chip photonics. Silicon photonics could solve several bottlenecks in state-of-the-art computer technology. In particular information transfer between building blocks of a modern computing devices like volatile memory access or fast inter-device communication. A direct realization of bulk or quantum well silicon lasers is impossible, since the indirect band structure of the material renders the material incapable in providing an efficient laser action18. Albeit, mass-communication links operate on laser driven optical fiber systems19, their connection to complementary metal-oxid-semiconductor (CMOS) structures is complex and costly. Currently, the employed lasers are (Ga,In)As on InP substrate, which are externally wired to computers for information interchange. The interlink is not only comparably slow, but the com- plete material system is expensive due to the high costs of In, too. Therefore, laser manufacturers and scientists are searching for a material directly grown on a CMOS structure. A promising material is Ga(N,As,P)20. While already the ternary Ga(As,P) allows for nearly lattice matched growth on silicion, the quaternary composition enables for pseudomorphically strained growth on silicon. Moreover, the additional degree of freedom in the composition of Ga(N,As,P) enables tuning of the emission wavelength in dependence of the composition. Common heterostructures for laser applications consist of several layers in addition to the active Ga(N,As,P) layer21. Lasing of such multilayer Introduction 3 structure has already been demonstrated at cryogenic temperatures22. However, room temperature lasing has not been reported to date. The first part of this thesis aims to investigate the internal band offsets between the several layers, which are crucial in device design and further optimization of Ga(N,As,P) structures. Photoluminescence excitation spectroscopy (PLE) and X-ray photoelectron spectroscopy (XPS) experiments provide the means to determine all of the materials band offsets and to contribute to a general understanding of the structure. The second part of this work deals with conversion of laser light into other wave- lengths. Frequency conversion is a powerful tool for reaching new wavelength regimes, either for scientific purposes or for consumer devices. Frequency doubling is a common mechanism in green light emitting laser pointers. In spectroscopy applications, various devices make use of frequency conversion: frequency doublers or triplers, optical para- metric amplifiers and lasing systems. Materials exhibiting non-linear properties are a necessity for these applications. Potassium niobate (KNbO3) is a perovskite showing a strong second order non-linearity and is thus an appropriate material for frequency doubling. Albeit the fact that it is already in use for some devices, it has so far only been characterized at the standard wavelength of 1064nm and its performance as frequency doubler in conjunction with tunable systems is unclear. The participation of this work centers on themeasurement of the absolute second order coefficient over a broad energy range. The second part is devoted to the characterization of potassium niobate as well as the investigations of another non-linear optical material: recently synthetized amorphous material show very strong non-linear responses leading to the generation of white-light23. The molecular structures studied here are tin-sulfide cluster cores in an adamantine- like arrangements. Aromatic ligands are attached at the four tin sites. A prototypical system is the [(PhSn)4S6] structure, but many different compounds exist allowing for a investigation of the role of different cluster atoms or ligands. The white-light generation process itself seems to be of a different nature than what is assumed in established white-light emitters such as photonic crystal fibers (PCFs)24. The optical nonlinearity is very strong, such that even continious wave (cw) laser irradiation triggers white-light emission, which is not possible for existing non-linear white-light converters. Yet, the underlying mechanisms of the process remains partly unknown. Recent investigations on the material suggests that its π-ligand system and its habitus play a major role in the generation process25. The second part of this work tries to progress the understanding of the structure-property relation by investigating a series of compounds, in which a coinage metal complex replaces one of the ligands and the implications of this replacement for white-light generation. In all this thesis is structured as follows: first, the theoretical framework is discussed, namely a brief introduction to the theory of semiconductors and their interaction with light is given. It is followed by a discussion of non-linear processes and a description of the interaction of electrons with matter interaction. The following two parts describe the methods and discusses results of the hetero-offsets inGa(N,As,P)multilayer systems and the non-linear properties on potassium niobate as well as of SnS clusters, respectively. Finally, the results are summarized and an outlook on the various topics concludes this dissertation. 2 L ight Mat te r In te rac t ion Generally, Maxwell’s equations govern all the properties ofmacroscopic light propagation. The four formulas ∇ · D = ρ (2.1) ∇ · B = 0 (2.2) ∇× E = − ∂ ∂tB (2.3) ∇× H = j+ ∂ ∂tD (2.4) with the electric displacement D = ε0E+ P (P is the polarization density) and magnetic induction B = µ0H+M (M is the magnetization density), constitute the propagation in vacuum (with M = 0 and P = 0) as well as in any material (M 6= 0 and P 6= 0). E and H denote the electric and magnetic field strength, respectively. j is the electrical current density. For non magnetizable materials and vanishing currents, inserting Eq. 2.4 into Eq. 2.3 yields the wave equation( ∇2 − n2 c2 ∂2 ∂t2 ) E = µ0 ∂2 ∂t2 P. (2.5) The interaction with a material is split into two parts: into a resonant part represented by the polarizability P and a general and constant material response represented by the dielectric function ε. In the fully coherent regime, the dielectric function can be assumed to be constant; for weak incoherent carrier occupations it is safe to assume macroscopic constancy of this value as well. In vacuum, the wave equation becomes homogeneous and the dielectric constant ε = 1, yielding an analytical solvable equation. One of themost used andmost convenient solutions to this problems are monochromatic plane waves given by E(r, t) = E0 exp [i (k · r− ωt)] (2.6) Basically, any spectroscopic problem can be broken down to the propagation of such plane waves and, therefore, they are a most versatile tool in the description of wave prop- agation optics. It becomes obvious from the source term in Eq. 2.5 that the interaction with a material is solely due to the polarization of the atomic orbitals. Actually, to antic- ipate quantum-mechanical dynamics described later in more detail, the light creates a quantum mechanical state as superposition between excited and ground states in matter, at least in a simple two-level scheme. This superpositional state dephases if the driving wave vanishes leading to the possible creation of carriers in the excited state. With the creation of carriers the system enters the incoherent regime, i.e., the excitation lost its phase relative to the driving wave. In semiconductors, it is necessary to have sufficient energy to surpass the energy gap of the material to create a significant carrier population. In this regime, the quantummechanical nature of the incident light becomes important, leading to absorption of the incident photons through the photoelectric effect. However, in a more detailed view the states of the incident photon and the materials excitation mix as well forming new quasi-particles called polaritons. Polaritons generally 5 6 Light Matter Interaction describe photon-excitatonmixingwaves insidematerials, therefore, several types of them exist, such as exciton-polaritons, plasmon-polaritons, phonon-polaritons and many more. The regime in which such signatures arise in spectroscopic experiments is commonly termed the strong-coupling regime26. Polaritonic effects aremore andmore exploited for technological uses, such as the polariton laser27,28, polariton cooler29 andmanymore30,31. Albeit, its use in common functionalized materials for mass-market applications is rather limited. Already the roughness of interfaces or the surface of a sample can destroy the narrow spectral features arising from polaritonic resonances. Not surprisingly, such resonances do mostly not show in heterostructure devices, since they often exhibit internal interface roughness, strain or alloy fluctuations to a certain degree. If the photon finally is absorbed in the material, an electron hole pair is created in the material. This pair, a so-called exciton, is attracted by the Coulomb force lowering the fundamental band gap energy by the exciton binding energy. Therefore, the first exciton resonance is below the actual band gap energy of the material, which is visible in both, emission and absorption of a material. Generally, the binding energy of excitons scale with the size of the band gap and its resonances are similar to the orbital state resonances of a hydrogen atom altered by the dielectric environment. This similarity makes it possible to measure Rydberg series of excitons in reflection or absorption measurements32,33. However, for III-V semiconductors with band gap energies in the near infrared regime, the exciton binding energy is often below the inhomogeneous broadening and therefore not visible in most experiments, e.g.,GaAs has a exciton binding energy of 4.2meV34. Nevertheless, it is important to note that all emission and absorption resonances are mostly arising from the 1s state of the exciton even if there is no excitonic population inside the sample. This behaviour and a further discussion of influences of dimensionality, defects or interfaces will be given in Sec. 2.1 of this chapter. If the incident photon energies are too low to excite carriers according to the pho- toelectric effect the wave simply propagates along the material interacting with the induced polarization. In certain, non-centrosymmetric materials this polarization is non-centrosymmetric as well, creating frequency mixing phenomena according to an anharmonic oscillator model. This phenomenon is often described by the term non- linear optics, since the electrical susceptibility of the material has to be expanded by a Taylor series of the electric field to accurately describe the physics around this effect. Accordingly, these effects depend on the incident electric field strength. Since such effects are rather weak, they became of importance with the invention of the laser in 1960 by Maiman9, providing sufficiently strong fields to significantly exploit such ef- fects. Nowadays, non-linear effects are a standard tool in each spectroscopic laboratory, e.g., for reaching unreachable wavelengths through frequencymixing or second harmonic generation (SHG) as well as high harmonic generation (HHG). A detailed theoretical description of such effects will take place in Sec. 2.2 of this chapter. 2.1 Absorptive Optics in Semiconductors Crystalline semiconductors exhibit periodic crystal lattices with certain symmetric prop- erties. The complex interplay of the mixing of the outer shell single atomic orbitals of the constituents determine their material properties. The challenging part of the light interaction in crystals is: the transition from the interaction of light with the orbitals of a single atom to the interaction of simultaneous interaction with 1023 atoms and 2.1 Absorptive Optics in Semiconductors 7 their corresponding orbitals. Accordingly, each light-induced change in the material has to be treated not only in the scheme of this single interaction event but also in the response of the surrounding system, leading to a vast correlation of interactions. Light-matter interaction is mostly governed by the change of electronic motion and occupation or lattice vibrations, therefore, Coulomb interaction between electrons and atomic binding strengths between the atoms become important. It is convenient to deal with such transitions in reciprocal space when the dispersion relation, i.e., the wave vector (k) dependence of the energy is known. This dependence is termed the band structure of a material and first treatments of it started in the 1920s35,36 and lead to rigorous calculations in the 1950’s. 2.1.1 Band Structure and Density of States Theoretically, the transition from single atoms to a crystal lattice can be treated in two ways: as the free dispersion of an electron perturbed by the surrounding atoms and as mixing of the discrete lines of the crystals atoms. Electron perturbations are the method of choice for conduction-band calculations. Contrary, the valence band can be accurately described by the mixing of atomic levels. This mixing smears the single atomic energies to broadened bands with many electronic levels giving rise to the band structure. The transition from the free electron energy given by E(k) ∝ k2 to a reduced band structure scheme is given in Fig. 2.1. The periodic potential of the atoms and electrons is the foundation of a band structure treatment, i.e., the electronic potential is assumed to fulfill V (r+ R) = V (r), (2.7) where r is the space coordinate and R is the periodicity of the potential. With this periodicity there exists a solution of the Schrödinger equation of the form ψk,i(r) = eikruk,i(r) (2.8) with uk,i(r) = uk,i(r+ R). k represents the wave vector and i is the index of the band. This is known as Bloch’s theorem, established by its eponym Felix Bloch in 192935. Similarly the eigenenergies are periodic in k-space: Ei(k) = Ei(k+ G). (2.9) Accordingly, Bloch’s theorem exists in k-space, too: ψk+G,i(r) = ψk,i(r). (2.10) This relation enables to reduction of the band structure zone schemes to the first Brillouin zone, which is bound by k = ±π/a. A basic approach of deducing the zone scheme is to introduce the Bloch wave function (2.8) into the Schrödinger equation and to cancel the exponential functions out, yielding[ ~2 2m ( −i∇+ k)2 ) + V (r) ] uk,i(r) = Ek,iuk,i(r). (2.11) The simplest solution can be found in the empty lattice approximation37, where V (r) = 0, i.e., the electrons see no potential and are in free space. A solution is the constancy 8 Light Matter Interaction of the Bloch function (uk = const.), thus the gradient vanishes leading directly to the dispersion relation E(k) = ~2 2m k2. (2.12) This is as expected for a vanishing surrounding potential the dispersion relation for an electron in vacuum. Since it contains the free electron mass m, it can be used to generalize the concept of the particles mass to the effective mass, present in the vicinity of a potential. Accordingly, E(k) is derived two times to strip the wave vector from the equation, yielding after a rearrangement of Eq. 2.12 (m∗)−1 = 1 ~2 ∂2E ∂k2 . (2.13) The termm∗ is then called the effective mass and is only dependent on the dispersion relation of a particle, i.e., yielding the free electronmassm in a potential free environment. With it is possible to describe the carrier mobility inside a material and, therefore, it is an important parameter for optical and electrical interaction. However, the aim is to describes electrons inside amaterial rather than in vacuum. Accordingly, a non-vanishing potential is present inside of Eq. 2.11, which is generally given by periodic lattice potential of the crystal under study. For a simple theoretical treatment the Kronig-Penney-model is often used38, which employs a periodic rectangular potential, i.e., a superlattice or periodic quantum well structure. With this simplification Eq. 2.11 is solved analytically, leading to an equation of wave vector and energy k(E) ∝ arccosL(E). The right-hand side L(E) is part of the solution and is not restricted between 0 and 1 (for a full form of the function see Seeger39). Therefore, for some values of E the relation may not have a solution: a band gap exists, i.e., no states exist at these energies. This is due to the mixing of inter-potential wave functions, where standing waves are created to fulfil the boundary conditions in the periodic lattice. The outcome can be interpreted as crude approximation to an atomic lattice, where the atomic orbitals are assumed to be rectangular. Whilst changing the parameters of the model, i.e., the relation between potential thickness and distance, the mixing behaviour of the wave functions can be studied. In the extreme case, where the potential thickness is zero the model resembles the empty lattice approximation, yielding the free electron dispersion (grey in Fig. 2.1). As the potential thickness is increased the free electron dispersion slowly is shifted away (thus creating a gap) at the boundaries of the reduced zone scheme (black in Fig. 2.1), which are tied to π/a due to the periodicity of the arccos function. For some bands the curvature becomes negative (k = 0 in the middle band in Fig. 2.1), therefore the effective mass is negative as well. Since such a negative mass is not physically meaningful, this behaviour is attributed to hole quasi-particles, i.e., an electron vacancy in the corresponding band. This explanation scheme is very convenient for the treatment in semiconductors, since the most of the behaviour can be modeled much easier (e.g., recombination is the interaction of an electron with a hole). However, the Kronig-Penney model is of course over-simplified to explain the band structure of a real atomic lattice. The structure of such lattices are not only inherently three dimensional, but also exhibit more symmetric properties and the atoms have many energy levels they provide for wave function mixing. Therefore, the deduction of a band structure of a common semiconductor is less easy and often computationally expensive, i.e., the problem can only be solved numerically. Two very common approaches of band 2.1 Absorptive Optics in Semiconductors 9 k E Band gap d a ...... Free e- a/d = 0 a/d = 0.1 Figure 2.1: Band structure of the empty lattice approximation (dashed grey lines) and of the Kronig-Penney model (solid black line). The left side depicts the potential and the right side their electronic dispersions. structure calculations are the k · p-method or the Korringa–Kohn–Rostoker-method. Nevertheless, they both still need many material parameters, which have to be deduced experimentally and therefore there exists the attempt to use ab-initio methods, such as density functional theory (DFT). DFT mostly is used in the calculation of molecular orbit theory to deduce the orbitals of a single molecule or simple complexes. Unfortu- nately, the method is weak in the deduction of excited states as well as absolute energy scales, therefore the derivation of band gap energies is very faulty. Nowadays, various approaches overcoming these shortcomings exist, e.g., by using better functionals40 or combining DFT and k · p calculations41. For a typical direct-gap zinc-blende structure III/V semiconductor (such as GaAs) the valence band is mainly created from s and p orbitals of the constituting atoms. This process is called sp3 hybridization. A simplified band structure of a such amaterial shows mainly a parabolic conduction band (band above the energy gap in Fig. 2.2) and three valence bands. In unstrained materials, the two upper valence bands are degenerate and exhibit two different slopes (two black bands below the energy gap in Fig. 2.2). According to the effective-mass formula (Eq. 2.13) they can be attributed to heavy-holes and light-holes, respectively. The lower valence band (grey in Fig. 2.2) is the split-off band due to the spin-orbit interaction of the electrons. Under equilibrium conditions, the bands are fully filled up to the Fermi energy EF , which lies in the energy gap for semiconductors, and as electrons are Fermions they fulfil the Fermi-Dirac-Distribution f(E) = 1 exp{(E − EF )/kbT}+ 1 , (2.14) yielding the occupation probability f(E) of a state with energy E. kb is the Boltzman constant and T is the temperature of the material. The distribution is shown on the right side in Fig. 2.2. Folded with the density of states (DOS) (left side in Fig. 2.2) it yields the total occupation at temperature T . The DOS itself describes the number of energy 10 Light Matter Interaction D(E) E 0 1 f(E) Fermi energy Γ-pointBand gap k heavy-holes light-holes split-off band co nd uc tio n b an d Figure 2.2: Typical band structure scheme of a III/V semiconductor (middle) with the conduc- tion band (upper parabola) and the three valence bands (lower three parabolas). The left and right sides show the materials density of states and its Fermi-Dirac statistic, respectively. levels in an infinitesimal energy interval E + dE. It can be derived for a d dimensional Fermi gas (where k ∝ √ E according to Eq. 2.12) very easily. The number of states can be found by the integral D(E)dE ∝ ∫ ddkδ (E − E(k)) . (2.15) The k-space infinitesimal element can be converted to the wavenumber according to ddk ∝ kd−1dk, carrying out the integral leads toD(k)dk ∝ kd−1. Substituting k for E in this relation yields D(E)dE = D[k(E)] dk dE dE. (2.16) Together with free electron dispersion (Eq. 2.12) we get the energy dependent DOS: D(E)dE ∝ Ed/2−1. (2.17) Accordingly, the three dimensional DOS has a root shape (as in Fig. 2.2), the two dimen- sional one is constant (i.e., a step function when higher states are present) and the one dimensional case shows an inverse root behaviour. Therefore, the one dimensional case shows a singularity at the edge and decays to zero the higher the energy. For the full understanding and completeness the lattice temperature, i.e., the atomic distance variation with temperature, has to be taken into account. The semi-phenomeno- logical Varshni formula42 Eg(T ) = E0 − α T 2 T + β (2.18) with the material parameters α, β and the zero temperature gap E0, describes this shift of the electronic band gap with temperature. This leads to an increase of the 2.1 Absorptive Optics in Semiconductors 11 e- e- Excitation Relaxation Recombination h h Figure 2.3: Simplified schematic of the photoluminescence process. A photon (wavy arrow on the left) excites an electron, creating an hole and an electron. The (quasi-)particles relax to the band minimum, where they recombine and emit a photon with a higher wavelength (wavy arrow to the right). band-gap energy with decreasing temperature, altering the total occupations under equilibrium conditions. Regarding emission of light, the energy gap shift leads to a shift of the emission to higher energies with decreasing temperature. Nevertheless, there exist various approaches connecting the band gap change of temperature to intrinsic parameters of the semiconductor and is mentioned here for completeness. A generally accepted simple formula is ∆Eg(T ) = αΘp 2 [ p √ 1 + 2T Θp − 1 ] . (2.19) Here, Θp is the average phonon temperature, α is related to the maximum (high tem- perature) entropy of the system and p is a material parameter related to the degree of phonon dispersion43. Accordingly, the formula enables to deduce microscopic quantities (e.g., phonon temperature) for a measured temperature dependence, yielding a more descriptive approach of the physics behind this shift. All considerations so far are for equilibrium conditions. If light interacts with the system the material the first approach is that the light is absorbed and the energy is transferred to an electron, lifting it from the valence band to the conduction band (Fig. 2.3). This action can take place whenever the energy of a photon itself is greater than the band gap energy. In this case, the electron populates the conduction band for a certain time T2, which is in the order of 0.1 to 1 ps for common semiconductors (e.g., Si or GaAs) at room temperature. This radiative lifetime broadens the linewidth of the spontaneous emission which takes place after the decay time T2 through the energy time uncertainty ∆E ·∆t ≥ ~/2. (2.20) 12 Light Matter Interaction n = 1 n = 2 n = 3 K E continuum Figure 2.4: Excitonic levels (n = 1− 3) below the conduction band. K is the center-of-mass momentum of the exciton. However, the simple picture of photons lifting electrons in the band structure quickly fails. Since the band structure is a single particle picture in a periodic potential it does not take Coulomb interaction of the electron and its hole it leaves behind into account. The attraction between those two (quasi-)particles lead to a lowering of the fundamental gap energy, which is accurately described by the Wannier equation44: ~2k2 2µ φn(k)− ∑ k′ Vk−k′φn(k′) = Enφn(k). (2.21) This equation is very similar to the Schrödinger equation of the hydrogen atom with the eigenfunctions and -energies φn and En. The system is simply altered by the dielectric environment through the dielectric constant ε contained in the Coulomb potential Vk−k′ and the reduced effective mass µ of electron and hole. This combined electron hole system bound by the Coulomb force is called an exciton and is per definition a quasi- particle, as it only exists in the vicinity of a dielectric environment. For parabolic bands and a direct gap semiconductor, the system can be transformed in the center-of-mass system, yielding the dispersion relation of excitons: E(n,K) = Eg −Ry∗ 1 n2 + ~2K2 2M (2.22) with the scaled Rydberg energy Ry∗ = 13.6eVµ/m0ε 2, translational massM and exci- ton wave vector K. The radius of the exciton is scaled relative to the one of the hydrogen atom by ε2m0/µ. In inorganic semiconductors, excitons typically have a rather large radius (low binding energy), more specifically they are then called Mott-Wannier exci- tons44. For GaAs, the binding energy of the 1s exciton state is roughly 5meV below the conduction band edge34 and the exciton Bohr radius is around 10nm. Contrary, in organic semiconductors the exciton is typically confined to a single molecule with a 2.1 Absorptive Optics in Semiconductors 13 comparable small binding radius (large binding energy), these are called Frenkel exci- tons45. Another species of excitons are charge-transfer or type-II excitons. They exist at the interface between two materials, i.e., electron and hole exist in another material. Generally, the exciton binding energy increases with increasing band gap. In second quantization, the excitons can be described by creation and annihilation operators, such as XK = ∑ k φλ(k)a † v,k−Kh ac,k+Ke (2.23) as an example for the exciton annihilation operator. a(†)v/k are the creation and annihilation operators for electrons in the conduction and holes in the valence band. φλ(k) is the wavelength dependent exciton wave function With the similar constructed annihilation operator BK the excitons fulfil the permutation relation〈[ XK, X † K ]−〉 = 1− 4 3 π · na3B, (2.24) i.e., for low densities n excitons behave bosonic, becoming more and more fermionic for higher densities until they end up in an uncorrelated electron-hole plasma which can only be described by Fermi-Dirac-statistics. a3B describes the exciton volume in a bulk crystal, which can be easily transferred to lower dimensional systems, where the term is simply ∝ adB with dimension d. 2.1.2 Absorption The optical interaction presented in this work is mainly governed by exciton resonances and contributions from the electron hole plasma. To accurately determine optical absorp- tion or emission properties of an ideal semiconductor, the system has to be treated fully quantum mechanically or at least semi-classical (for absorption). Neglecting vibrational effects, the system Hamiltonian in the semi-classical case is of the form H = He +He−e +He−EM . (2.25) The first part,He, describes the occupation and dispersion of the electrons in the ma- terial, He−e is the electron-electron Coulomb scattering term and He−EM describes the interaction of electrons with the electric field. The electron occupation is deter- mined by using the counting operator constructed from the ladder operators a(†)λ,k for the conduction (λ = c) and valence band (λ = v), respectively. This yields He = ∑ k Ec ka † c,kac,k + ∑ k Ev ka † v,kav,k (2.26) for the free electron Hamiltonian, including the conduction band and valence band energies Eλ k . The scattering contribution reads He−e = ∑ k,k′,q6=0 ( 1 2 Vqa † c,k+qa † c,k′−qac,k′ac,k + 1 2 Vqa † v,k+qa † v,k′+qav,k′av,k − Vqa † c,k+qav,k′a † v,k′−qac,k ) . (2.27) 14 Light Matter Interaction In this scattering Hamiltonian, the first two terms on the right-hand side of the equation represent the electron-electron (1st line) and hole-hole (2nd line) scattering, i.e., two particles with wave vectors k and k′ get annihilated and create two particles with new wave vectors k+ q and k′ − q. The last term includes interband effects between the electrons and holes, i.e., electron hole scattering. The last part of the systems Hamiltonian is also the most important for light matter interaction, the coupling to the electromagnetic field: He−EM = −E(t) ∑ k ( d∗cva † v,kac,k + dcva † c,kav,k ) . (2.28) Here, E(t) is the classical electromagnetic field strength, the d’s represent the transition dipole moments for the creation of an electron from an hole (a†c,kav,k) or vice versa (a†v,kac,k). The transition dipole moment is represented by expansion of the macroscopic polarization P into the Bloch basis P = ∑ k dcvPk + c.c. (2.29) With the Heisenberg equation i~ ∂ ∂t 〈O〉 = 〈[O,H]〉 (2.30) and the full Hamiltonian (Eq. 2.25) the dynamics of the polarization Pk(t) can be cal- culated. To reach this goal, the commutators of the Heisenberg equation have to be carried out. Unfortunately, one not only gets a single particle density matrix through the free electron/hole Hamiltonian and the interaction term, but four operator terms from the Coulomb interaction, too. Accordingly, the equation expands further and further and has to be truncated at some point. A method to do this in a self-consistent fashion is the cluster expansion approach, which is presented in Kira and Koch46. In this approach, the expectancy value ofN interacting particles are expressed asN particle interaction with an additional full correlation term containing theN + 1 interaction alone. Therefore, the interaction is said to be clustered into singlet, doublet etc. interactions (up toN ). The additionalN + 1 correlation term is truncated, stopping the expansion of the equation and closing it to a N particle problem for an expectancy value for N particles, thus making the equation analytically solvable. With this approach, the equation is reduced to a closed form: the two band semiconductor bloch equations (SBEs): They read as[ i~ ∂ ∂t − E′ k e(t)− E′ k h(t) ] Pk(t) = − [ 1− fek (t)− fhk (t) ] Ωk(t) + ∂ ∂t Pk(t) (2.31) ∂ ∂t fλk (t) = −2 ~ Im [Ωk(t)P ∗ k (t)] + ∂ ∂t fλk (t). (2.32) The fλk represent the occupation probabilities for electrons (λ = e) and holes (λ = h), respectively. Ωk = dcvE(t) + ∑ k6=k V|k−k′|Pk(t) (2.33) 2.1 Absorptive Optics in Semiconductors 15 is the Rabi energy which is renormalized due to Coulomb screening. The energies E′ k λ(t) = Eλ k − ∑ k′ 6=k V|k−k′|f λ k′(t) (2.34) are renormalized as well. A description of polarization dephasing, screening of the interaction potential or relaxation of carrier distribution is beyond the scope of the generic SBEs, since such effects are not included. To include such effects, time dependent correction factors for the polarization and occupation probabilities are introduced at the end of Eq. 2.31 and Eq. 2.32. Since they are not of major interest for the absorptive behaviour studied here, their full form is only given in the Appendix or in Koch and Kira47 or Haug and Koch48. However, in the case of weak electric field strengths acting upon the system the generic SBEs are still able to accurately describe the effects taking place. If static conditions and vanishing (or at least low) carrier occupations are assumed, i.e., the system is in equilibrium and not under illumination, Eq. 2.31 is homogeneous and breaks down to the Wannier equation (2.21). Since the wave functions of the Wannier equation are well known due to the hydrogen problem, they can be used to deduce a solution for the inhomogeneous part of the SBE (the system is illuminated: E(t) 6= 0), yielding χ(ω) = 2|dcv|2 ∑ λ |φλ|2 Eλ − ~ω − iγ , (2.35) with a phenomenological dephasing factor γ. This is the Elliot formula49 for the linear semiconductor susceptibility, thus describing the absorptive behaviour of a semicon- ductor. Taking the imaginary part of Eq. 2.35 yields the absorption α(ω) for a material. Viewing the resulting absorption spectrum reveals that the absorption is strongly governed by the Lorentzian shaped 1s exciton resonance, followed by less pronounced peaks for the 2s, 3s etc. resonances (Fig. 2.5 shows the behaviour for GaAs). Obviously, no p-type or higher orbital contributions are visible since their transition dipole moment is zero and excitons from this bindings are dark. Around the band-gap energy (Eg) the absorption coefficient becomes quasi-constant (zero in Fig. 2.5) and is enhanced due to Coulomb renormalization compared to the exciton free absorption edge. For higher photon energies the difference between the absorption with excitons and the exciton-free absorption vanishes due to the less importance of the Coulomb force in the electron-hole plasma. Generally, the Coulomb interaction strength is given by the excitonic Rydberg constant (Ry∗), therefore the smaller Ry∗ the more the Coulomb free absorption is resembled (Fig. 2.6a). Furthermore, the strength of the 1s peak gets weaker with decreasing Ry∗ and higher resonances vanish in the absorption edge, i.e., the edge is red shifted due to the higher states and Coulomb enhancement. This is important that even for inorganic semiconductors with small band gap, where exciton resonances may not be visible, the effects still play a major role as a shift of absorption. For example the GaAs absorption edge gets shifted by about 25meV, even if the broadening is so strong that no clear 1s resonance is visible. Generally, increasing broadening (introduced as phenomenological γ in Eq. 2.35) shifts the 1s peak maximum only weakly to higher energies (Fig. 2.6b). This mainly bleaches the resonance and broadens its Lorentzian line shape, yielding a strong red shift and less steep rise of the absorption edge. Figure 2.6b shows the change of the absorption for ten broadenings in between γ~ = 10meV . . . 50meV. 16 Light Matter Interaction - 1 0 - 5 0 5 1 0 - 5 0 0 0 2 0 0 0 Ab so rp tio n E - E g ( m e V ) c o n t i n u u m1 s r e s o n a n c e 2 s , 3 s . . . . R y * = 4 . 6 m e V h γ = 0 . 4 m e V Figure 2.5: Absorption of GaAs (solid line) with a low broadening (hΓ = 0.4meV). The exci- tonic 1s and higher states are visible at the Rydberg energy and right before the continuum edge (dashed vertical line). For comparison the dashed line (scaled for visibility) depicts the absorption without excitonic effects. The inset shows a broader range of the absorption, visualizing the Coulomb enhancement in the continuum. The x-axis in both graphs depict the detuning, i.e. the difference to the band gap energy. 2.1.3 Photoluminescence The derivation of photoluminescence can be done in the same framework as for the SBEs. However, spontaneous emission is an intrinsically quantum-mechanical effect, therefore, the problem has to be treated fully microscopic. Accordingly, the light field has to be quantized as well, changing the light field interaction Hamiltonian of Eq. 2.25 to He−EM = ∑ k,q ( iFqbqa † c,k+qav,k + iFqbqa † v,k+qac,k ) + h.c. (2.36) and replacing the electric field strength with its expectancy value 〈E〉 = ∑ q Equq 〈bq〉+ c.c.. (2.37) Thereby, Fk = dcvEquq denotes the interaction strength, with the vacuum field ampli- tude Eq and the mode strength uq, i.e., it represents the wavefunction overlap between electron and hole states. b(†)q denotes the creation and annihilation operators for photons with momentum q, which is very small compared to the quasi-momentum of the carriers k. The intensity of light is given by 〈EE〉, which contains the single particle photon 2.1 Absorptive Optics in Semiconductors 17 - 2 0 0 - 1 0 0 0 1 0 0 2 0 0 Ab so rp tio n 3 0 m e V 1 0 m e V 0 m e V 5 0 m e V D e c r e a s i n g E x c i t o n R y d b e r g E n e r g y R y * = 5 0 . . . 0 m e V Ab so rp tio n E - E g ( m e V ) I n c r e a s i n g B r o a d e n i n g h Γ = 1 0 . . . 5 0 m e V Figure 2.6: Influence of the Rydberg energy (upper graph) and the broadening (lower graph) on the absorption strength. The x-axis shows the difference to the band gap energy of the material. expectancy values 〈 b†b 〉 and the change of photon quanta∆ 〈 b†b 〉 = 〈 b†b 〉 − 〈 b† 〉 〈b〉. If pulsed excitation is assumed, the light field and polarization is decayed and the remain- ing radiative contribution stems solely from the quantum-optical intensity correlation ∆ 〈 b†b 〉 . Using the Heisenberg equation-of-motion with the cluster expansion scheme again, yields for the temporal change of this intensity correlation i~ ∂ ∂t ∆ 〈 b†qzbq′z 〉 = ~ ( ωq′z − ωqz ) ∆ 〈 b†qzbq′z 〉 + i ∑ k|| ( Fqz∆ 〈 bq′za † c,k|| av,k|| 〉 + F∗ q′z ∆ 〈 b†qza † v,k|| ac,k|| 〉) . (2.38) Thus, the equation describes the emission of light from a carrier distribution in the material. For a more convenient treatment, the material is assumed to be confined with the wave vector of the incident light parallel to the z-axis. The wave vector is therefore split in components in the area normal to it and one part parallel to it (k = (k||, kz)). The three parts of Eq. 2.38 resemble the existing photonic field, the absorption process, where a photon and a hole are destroyed and a conduction-band electron is created. 18 Light Matter Interaction The last part contains the photon induced polarization term ∆ 〈 b†qza † v,k|| ac,k|| 〉 which describes the creation of a photon by the annihilation of an electron and a hole. Thus, it contains the most part of the materials emission and is given by i~ ∂ ∂t ∆ 〈 b†qza † v,k|| ac,k|| 〉 = ( E′ k c − E′ k h − ~ωqz ) ∆ 〈 b†qza † v,k|| ac,k|| 〉 − ( 1− fek − fhk ) ΩST k||,qz + iFqz ( fekf h k + ∑ k′ ∆ 〈 a† c,k′|| a†v,k||ac,k||av,k ′ || 〉) + i~ ∂ ∂t ∆ 〈 b†qza † v,k|| ac,k|| 〉 ∣∣∣∣ scattering . (2.39) The first two terms of the equation represent the behaviour known from the SBEs in the fully quantum mechanical picture with the generalized Rabi energy ΩST k||,qz = ∑ q′z iFq′z∆ 〈 b†qzBq′z 〉 + ∑ k′|| Vk||−k′|| ∆ 〈 b†qza † v,k′|| ac,k′|| 〉 . (2.40) The last line represents scattering terms introduced from third order correlation terms, i.e. it includes triplet contributions from the cluster expansion approach. Furthermore, the third term describes the source for light emission, coming either from an electron hole overlap (fekf h k ) or the exciton occupation sum. If slow varying densities are assumed the carrier populations fλ and∆ 〈 a† c,k′|| a†v,k||ac,k||av,k ′ || 〉 can be taken as constant. Further, neglecting higher-order scattering terms the equation can be solved analytically48, yielding for the steady-state photon flux IPL(ωq) = ∂ ∂t ∆ 〈 b†qbq 〉 = 2|Fq|2 ~ Im [∑ λ |φλ|2Nλ Eλ − ~ωq − iγ ] . (2.41) This is the photoluminescence Elliot formula, strongly resembling Eq. 2.35, but with an additional source term Nλ = ∑ k |φλ,k|2fekfhk + ∑ φ∗λ,kφλ,k∆ 〈 a c,k†|| av,k′|| ac,k′av,k 〉 . (2.42) This term yields the actual carrier population in the material, either from the electron- hole-plasma (first sum) or excitonic occupation (second sum). A simple approach to this formula is to solve the density dependent Wannier equation (2.21), where the effective Coulomb potential Vk,eff = (1− fe − fh) · Vk is used, numerically, yielding the exciton wave functions φλ. The photoluminescence intensity can then by calculated from these wave functions directly. From the obvious similarities of the Elliot formulas for absorption (Eq. 2.35) and photoluminescence (Eq. 2.41) it is clear that the emission of a semiconductor is centered around the 1s exciton peak as well. In fact, even if no exciton population is present in the system the strongest emission stems from this resonance. Therefore, the character of the population cannot be determined by photoluminescence spectroscopy alone. Figure 2.7 shows a comparison of photoluminescence for no and for weak exciton populations in an ideal semiconductor. The electron-hole plasma is only visible as a weak side peak in the 2.1 Absorptive Optics in Semiconductors 19 - 5 0 5 1 0 1 5 0 . 0 1 0 . 1 1 In te ns ity E - E 1 s ( m e V ) N o e x c i t o n s 1 0 % e x c i t o n s Figure 2.7: Theoretical photoluminescence for no (solid line) and 10% (dashed line) of relative excitonic population. The energy axis is scaled with respect to the exciton 1s energy. photoluminescence signal, barely visible in a non-logarithmic plot. Moreover, for small Ry∗ or typical broadening this sideband vanishes completely, but it is still important to know that the emission is governed by the 1s peak and shifted accordingly, even for completely uncorrelated electron-hole pairs. 2.1.4 Influence of Strain and Localized States There are various extrinsic and intrinsic parameters influencing the band gap of a semi- conductor. The main dependencies are temperature, strain, composition, and localized states, which will be explained for nitrogen states in a semiconductor. The influence of composition, i.e., band gap change in binary, ternary or quaternary alloys is treated in the virtual crystal approximation (VCA). Imagine a III/V ternary material of the atomic species A, B and C, where A and B have the same amount of valence electrons (i.e., are in the same periodic element table group) and C is their covalent counterpart. Therefore, the A, B to C ratio has to be 50 % and the ratio of A and B can be controlled during growth, yielding the compositional formula AxB1−xC, with the alloy fraction x. This yields for the energy gap of the material Eg(AxB1−xC) = Eg(BC) + x [Eg(AC)− Eg(BC)]− bx(1− x), (2.43) where b is the phenomenological bowing parameter. This has to be introduced since the behaviour is often found to be non-linear and is therefore corrected to this slight changes 20 Light Matter Interaction 4.4 5.4 5.6 5.8 1.0 1.5 2.0 2.5 3.0 3.5 Ba nd ga p( eV ) Lattice constant (A) GaAs GaP GaN Si Ga(NAs) Ga(NAsP) Ga(NP) Figure 2.8: Gallium based materials band gap and lattice constant shift with respect to their composition. The arrows indicate the strong bowing introduced by the incorpora- tion of nitrogen. Filled circles (solid lines) depict direct materials and hollow circles (dashed lines) indirect ones. of the linearity. The VCA model can be easily expanded to quaternary materials, with a second alloy fraction y. Figure 2.8 depicts the change of band-gap energies and lattice constants depending on the composition for gallium-based systems. Albeit most of the changes can be determined accurately, the change with nitrogen incorporation shows a rather strong bowing. Contrary to the excepted behaviour, a monotonic increase of the band gap from GaAs to GaP a huge bowing is observed, even lowering the band gap when small amounts of nitrogen are introduced into GaAs. This behaviour cannot be understood in terms of the VCA, which is due to the fact that nitrogen is partly diluted in the lattice of the material. Therefore, nitrogen acts as an isovalent local impurity level, leading to a dispersion less band in the structure of the material. For small doping levels of nitrogen in GaP, the local impurities lie energetically inside the band gap, giving rise to defect related photoluminescence (PL) and it further does not affect the band structure50. Contrary, in the case of GaAs the local impurity band lies within the conduction band of the material, leading to quantum mechanical anti-crossing effects. Generally, bands never cross each other in a band structure scheme, because a crossing always leads to a wave function mixing, which in turn creates new state bands. This behaviour may be described by the band anti-crossing (BAC) model. 2.1 Absorptive Optics in Semiconductors 21 In a simple two level description, the band gap splitting reduces to determinant equation∣∣∣∣∣E − Ec V V E − EN ∣∣∣∣∣ = 0, (2.44) where Ec is the conduction-band-edge energy (relative to the vacuum level), EN is the impurity level energy and V is an empirical coupling constant. Solving of the determinant leads to the two new energy levels 2E± = Ec + EN ± √ (Ec − EN )2 + 4V 2. (2.45) This behaviour was studied in great detail for Ga(N,As)51,52, (Ga,In)(N,As)53 or other (qua)ternary materials54. With this model it is possible to accurately predict the band gap bowing in diluted nitrogen alloys. Another effect on the band edges is due to strain inherent in the material. The strain tensor 2εij = ∂uj ∂dxi + ∂ui ∂xj (2.46) describes the infinitesimal length change of the material due to extrinsic strain. u is the displacement vector, pointing from the strain free position to the strained one. Thus, in one dimension the stress can be imagined to reflect an infinitesimal length change over the total length ε = δd/d. It is convenient to split the strain into hydrostatic strain and shear strain. The hydrostatic strain is simply given by the trace of the tensor: εh = Tr ε = εxx + εyy + εzz . For a direct zincblende material the shift of conduction (∆Ec) and valence band (∆Ev) is only affected by the hydrostatic strain and given by ∆Ec/v = Ξc/v Tr ε, (2.47) with the hydrostatic deformation potential for either the conduction (Ξc < 0) or the valence band (Ξv > 0). These are material specific properties, relating the energy change of a material to its length change: Ξ = a dE dx , (2.48) where a is the lattice constant of thematerial. These potentials aremostly experimentally known and the deformation potentials for binary, ternary or quaternary systems can be calculated by VCA as well. With the strain induced changes for the bands (Eq. 2.47) the total energy gap change is calculated to ∆Eg = ΞTr ε, (2.49) with Ξ = Ξc − Ξv . Contrary, shear strain does not affect the edge energies but leads to a degeneracy lifting of the heavy-hole (hh) and light-hole (lh) valence bands ∆Ev,hh−lh = E0 v ± Eεε (2.50) with the shear strain energy shift E2 εε = Ξ2 b 2 [ (εxx − εyy) 2 + (εyy − εzz) 2 + (εxx − εzz) 2 ] + Ξ2 d [ ε2xy + ε2yz + ε2xz ] . (2.51) 22 Light Matter Interaction k|| E No strain Tensil strain Compressive strain Figure 2.9: Shift of the hh and lh bands under tensile (middle) and compressive (right) strain. The shift of the band gap energy is not shown for clarity. Here, Ξb/d are the shear strain deformation potentials. Accordingly, there are two forms of strain possible: compressive or tensile strain. They lead to a different behaviour, for compressive strain the hh is lifted above the lh and for tensile strain it is the other way round (Fig. 2.9). For tensile strain, this leads to a complex behaviour due to the crossing and therefore mixing of the valence bands (right hand side in Fig. 2.9). The influence of hydrostatic and shear strain acts on valence band and the effective masses of electrons and holes, as well. The behaviour is described with formulas very similar to Eq. 2.47 and a complete listing is found in Aspnes and Cardona55. 2.1.5 Nonradiative Recombination Various competitive processes to radiative recombination exist in ideal semiconductors. For instance, a third entity may be involved in the process and the energy difference may be dissipated thereby. This process is called Auger recombination after Pierre Victor Auger who described this process in 192356, although the effect was first discovered by Lise Meitner in 192257. The basic scheme for Auger recombination is that the transition through the band gap is possible by intra-band lifting of either an electron in the conduc- tion band or an hole in one of the valence bands (Fig. 2.10). Accordingly, the processes are termed eeh (two electrons) or hhe (two holes) Auger recombination, respectively. The probability for each scales with the carrier density, i.e.,∝ n2p for eeh and ∝ np2 for hhe processes. A three-particle process it is statistically less favourable in most cases, however, it becomes more prominent for elevated carrier densities, e.g., for high excitation densities, high doping or temperatures, or for small band gaps (which in some terms also enriches the carrier density in the conduction band through Fermi-Dirac statistics). The process may also occur phonon-assisted58. In that case, a k translation is possible during the process, enabling more carriers to participate in the process. Thus, the statistical probability is further enhanced. The Auger effect can be used as tool for material analysis, i.e., here, the lifted electron is emitted to the vacuum level where it is detected. The energy of the emitted electron sheds light on the internal structure of the material. However, as spectroscopic tool the Auger effect is mostly restricted to lower material orbitals and thus not suitable to investigate the behaviour around the band gap. The counterpart of the Auger effect is 2.1 Absorptive Optics in Semiconductors 23 e- h e- eeh process h h e- hhe process Phonon Phonon x E Figure 2.10: Auger recombination in the energy over space picture. The process can either happen with two electrons (left) or two holes (right). The recombination energy is transferred to the up(down)-lifting of an electron (hole). The graph shows the process in a phonon-assisted form, but up-lifting in higher bands is possible, too. the impact ionization, which can result in an avalanche breakdown in the material. This breakdown is the basis of avalanche photodiodes, which are able to detect very low light intensities through this effect. 2.1.6 Influence of Defects and Disorder Despite ideal semiconductors presented in the theoretical treatments in this chapter, several defects and disorder effects commonly determine semiconductor’s response. Both are inevitable in the growth of such materials for thermodynamic reasons26. There are several defects, such as dislocations, stacking faults, grain boundaries or voids only to name a few. Nevertheless, the most influential defect regarding optical interaction is the point defect, i.e., an additional atom or atomic replacement in the crystal lattice. Such defects are present in a concentration around 1013−1017cm−3 in typical materials. Commonly, point defects are characterized by their position in the band gap. They can either act as an electron donor, typically these are defects energetically near the conduction band edge, or as an electron acceptor, which are near the valence band edge (Fig. 2.11). Both defect types can be separated into neutral and ionized traps, i.e., for donors the exchange is through adding or removing an electron: D0 ↔ D+ + e−. (2.52) Similarly, an acceptor can be described by adding or removing an hole: A0 ↔ A− + h. (2.53) 24 Light Matter Interaction D+ e- D0 A- h A0 e- h e- Donors h Acceptors Donor-Acceptor Pair e- h Recombination center he - Bound exciton complex Figure 2.11: Energy vs. space diagram for possible point defects in a semiconductor. Shallow donors (acceptors) interchange from charged and neutral by emitting or collecting a particle. Recombination with the localized particles is possible (vertical lines), resulting in an emission of lower energy photons. A recombination center (middle) gives rise to an even lower emission wavelength. Each shallow donor or acceptor can bind an exciton, forming a bound exciton complex. If the wave functions of donor and acceptor overlap they form a donor-acceptor-pair (right). Typically, these defects are two fold spin degenerate, but due to the spatial confinement one of the electrons is strongly blue shifted due to Coulomb repulsion59. These types of defects are generally called shallow traps. Whenever shallow traps are spatially near to each other and their wave functions overlap, they may form a donor acceptor pair, enabling an optical recombination channel with lowered transition energy compared to the band transition. Contrary, so-called deep traps occupy energies in the vicinity of the band gap. These are termed recombination centers if they are able to capture an electron or a hole. Such recombination centers show distinct optical features at energies considerably lower than the actual band gap energy (for examples see Klingshirn26). In the band structure defects appear as dispersion less energy bands. Therefore, shallow traps represent a positively (negatively) charged center to which an electron (hole) can be bound. Accordingly, a localized exciton forms in the vicinity of an ionized shallow trap. Thus, the point defect is able to recombine, giving rise to emission bands below the energy gap transition of a material. Moreover, trap levels can attract excitons themselves, a bound exciton complex (BEC) forms. This is typically denoted by an addition of a X to the defect symbol, e.g.,D0X for an exciton bound to a neutral donor. Generally, excitons can bind to any type of shallow trap, except for A+X . In the ionized acceptor the binding to the free electron is energetically more favourable in most materials and therefore the exciton is less likely to bind to it. The binding energy of such BECs following in ascending order D+X , D0X and A0X . Since the bound complex offers no translational freedom, the peaks in absorption or emission or usually very narrow in bulk structures. The visibility of these narrow lines depend on the concentration of the point defects. Otherwise, the peaks broaden due to BEC-BEC interaction, e.g., their statistical spatial distance is to small too regard them as real point defects. However, the peaks generally merge with the 2.1 Absorptive Optics in Semiconductors 25 Figure 2.12: Various absorption regimes of the absorption coefficient in disordered semicon- ductors. Reproduced from Wood and Tauc60. free luminescence of the material for lower dimensional structures, such as quantum wells (QWs), and are therefore only visible as tailing structure. Three regimes (I, II and III in Fig. 2.12) are visible at the band edge of an absorption spectrum. The first one (I) is due to general alloy impurity or amourphousness in the material. The second (II) is the Urbach tail26 which accounts for an exponential tail of the DOS due to localized excitons. This tail is often approximated with the formula N(E) = N0 εloc exp ( − E εloc ) , (2.54) for E below Eg . εloc describes the density of localized states below the gap. For a thermal energy greater than the localization energy (kbT > εloc), the thermal reexcitation into extended states is possible. This is often observed in temperature-dependent PL experiments, where the change of emission energy deviates from the expected Varshni-like behaviour (Eq. 2.18) around a certain temperature (Fig. 2.13). This deviation is known as S-shape and is mainly divided in three regimes: at very low temperatures the carriers are frozen not able to move and hence not able to reach the traps. At slightly higher temperatures the carriers are able to diffuse and find the most favourable trap, therefore, the emission energy drops. Further increase of the temperature leads to thermal reexcitation, returning to the expected Varshni behaviour of temperature shift. This effect can be used to characterize the energetic position and amount (at least to a certain degree) in a material61,62. Drawing back to the absorption spectrum and the third regime (III): this is the so called Tauc regime and the behaviour in this regime can be approximated by the formula (α~ω)n = A (~ω − Eg) , (2.55) 26 Light Matter Interaction 0 1 0 0 2 0 0 3 0 0 E ( R T ) E ( 0 ) Ba nd g ap T ( K ) i d e a l V a r s h n i d e f e c t + d i s o r d e r r e l a t e d Figure 2.13: Example of the Varshni shift of band gap energy in dependence of temeparature (solid line). The dashed line shows a defect and disorder related curve with a change in a certain temperature range due to freezing into localized states. where A is a fitting factor and n is an exponent to scale for the type of transition. It constitutes the following values: • direct allowed: n = 2 • direct forbidden: n = 2/3 • indirect allowed: n = 1/2 • indirect forbidden: n = 1/3. In comparison to the direct allowed transition described by the Elliot formula (Eq. 2.35) it is obvious that no excitonic effects are taken into account. Accordingly, the formula is a rather crude approximation as it only takes the root shaped DOS edge (for direct allowed) into account. Nevertheless, it is a valuable tool to investigate the underlying type of transition and to get a first estimate of the band gap energy for materials with low exciton binding energies and moderate broadening. Additionally, in most cases the exciton binding energy can be estimated at least to correct the Tauc fitted band gap energy. Commonly, disorder has an effect on the optical properties aswell. Mostly it arises from alloy fluctuations or geometrical defects. This is why such effects become important for lower dimensional structures, because the interface fluctuations are much more relevant in the materials response. To estimate the effects of disorder, two variations of the previous presented Kronig-Penney model exists: diagonal and off-diagonal disorder. The first describes a variation of the well depths in a certain interval, whereas the latter accounts for well-width fluctuations. Generally, a combination of both effects will occur in reality. Regarding diagonal disorder, the case is rather simple: the depth is smeared out around a mean well depth, leading to Gaussian decaying number of states for higher or lower energies around this mean value. Accordingly, the DOS is extended by a tail 2.2 Nonlinear Optics 27 ideal disordered E DOS Stokes shift E I I E Absorption PL Figure 2.14: Influence of disorder in the Kronig-Penney model on the DOS (middle), PL and Absorption (right). for these energies around the non-disordered genuine DOS (Fig. 2.14). Contrary, the off- diagonal disorder rather influences the form of the DOS, since the boundary conditions of the standing waves of the electronic wave functions change. All in all, the influence is mainly the existence of a tail at the onset of the DOS. Due to the smearing the absorption of a material is generally broadened (as shown for different broadening scales in Fig. 2.6). This leads to a bleaching of the exciton 1s peak and general broadening of the edge. Another indicator of disorder is the shift between PL and absorption. This is due to the continuous nature of disorder, i.e., the carriers are able to diffuse to a lower energy and recombine afterwards. Accordingly, besides inhomogeneous broadening and the formation of a lower energy tail the PL shifts red, too. Since the absorption is assumed to not excite any carriers and is created solely from the induced polarization the spectrum reflects a statistical averaging across the disorder scale. As a consequence, there is a shift between absorption and emission of disordered samples, known as the Stokes shift63. This shift can be used to estimate the general disorder strength of a material. 2.2 Nonlinear Optics The previous section treated the effects attributed to the absorption of photons in solids, mainly semiconductors. Absorption in this sense means that an electron transitions from one energy level to another. This transition is always enabled through the previous buildup of a polarization. This section describes the interaction of lightwith non-centrosymmetric solids in the polarization regime, i.e., the photon energy is typically lower than the band gap energy. Through the symmetry breaking in a material, the conversion of light into another energy becomes possible. This is due to the interaction of the induced electron motion (i.e. polarization) and the incident field, which happens at sufficient strong electric fields. Typical processes include the generation of optical harmonics, sum- or difference frequency generation if two beams are incident on sample. A prominent example is SHG, which is widely used in everyday devices (i.e., in green laser pointers) or in scientific research to reach other wavelength regimes. Franken et al. first described the effect of SHG, only one year after the invention of the laser byMaiman. Since then, the field of 28 Light Matter Interaction Table 2.1: Transformation into the contracted notation of the non-linear coefficient d. jk 11 22 33 23,32 31,13 12,21 l 1 2 3 4 5 6 non-linear optics emerged rapidly and nowadays there is a huge range of devices from parametric amplifiers to high-harmonic generation to reach deep ultra violet (UV) light. Time domain spectroscopy became possible over a wide energy range as ultra-short laser systems became available. These sources supply much larger field strengths than continuous wave lasers which further enhance the non-linear processes. The description of non-linear interaction is mainly through the relation of polarization and electric field, which typically reads P (t) = ε0χE(t). (2.56) However, this treatment is not sufficient when E(t) becomes strong, since the polarization itself emits light due to acceleration and deceleration of electrons. The solution to this problem is to expand the polarization in the electric field, yielding P (t) = ε0 ( χ(1)E(t) + χ(2)E2(t) + χ(3)E3(t) + . . . ) (2.57) Therefore, the polarization generation becomes field dependent with proportionality constants of different order. Typically, the second and third order expansions are treated for most materials. Their strength is generally relatively weak, i.e.,χ(2) ≈ 1pm/V and χ(3) ≈ 1pm2/V 2. The non-linear polarizations are generally due to spatial asymmetry; hence a non- centrosymmetric material is necessary to create a bulk non-linear response. Accordingly, Eq. 2.57 is a tensor equation in reality. For the second order term, this reads as Pi(ω3) = 2ε0 ∑ jk χ (2) ijk(ω3, ω1, ω2)Ej(ω1)Ek(ω2). (2.58) Thus, the interaction of field elements Ej and Ek give rise to a polarization Pi in the i axis. For the third order term, the χ tensor has order four and so forth for higher orders. Hence, the interaction in higher orders become a complex interaction of many fields. It is often convenient to stick to a contracted notation where dijk = 1 2 χ (2) ijk. (2.59) Further, symmetry properties of the materials play a role if the material is lossless (which is always assumed in this theoretical treatment) and the dispersion of the susceptibility is weak. This is often the case in red or near-infrared wavelength regimes, since the resonant part of the spectrum lies around the band gap of the material in the blue or ultraviolet regions. The so-called Kleinman symmetry65 states, in this case, that the enumeration variables j and k are interchangable, hence the χ tensor is symmetric in this quantities. Accordingly, the quantity from Eq. 2.59 is further contracted to dil with the substitutions given in Tab. 2.1. If the factor is measured such that multiple matrix elements contribute to the emission an effective coefficient (deff) is given. 2.2 Nonlinear Optics 29 2.2.1 Anharmonic Oscillator Model A simple treatment of the second-order non-linearity is by a simple anharmonic oscillator model, which attributes for the symmetry breaking in a material. The one dimensional anharmonic oscillator differential equation reads as ∂2x ∂t2 + 2γ ∂x ∂t + ω2 0x+ ax2 = −eλE(t)/m. (2.60) Here, ω0 is the resonance frequency, γ is the damping and a is a scaling factor for the anharmonic potential. The right hand side describes the driving by the electric field and x is the space coordinate. λ is a strength scaling factor and acts as a perturbative term in the equation. The potential of the restoring force equates to V (x) = 1 2 mω2 0x 2 −max2. (2.61) A modification of this potential allows for the treatment of centrosymmetric material, i.e., V (x) = 1 2 mω2 0x 2 − 1 4 mbx4. (2.62) This yields a symmetric potential with interaction anharmonic bending b. To solve Eq. 2.60 x is expanded in λ: x = λx(1) + λ2x(2) + λ3x(3) + . . . . This leads to the equation system ∂2x(1) ∂t2 + 2γ ∂x(1) ∂t + ω2 0x (1) = −eE(t)/m (2.63) ∂2x(2) ∂t2 + 2γ ∂x(2) ∂t + ω2 0x (2) + a ( x(1) )2 = 0 (2.64) and so forth for higher orders in x. Therefore, the higher orders depend solely on the lower order contributions in the anharmonic potential. For the symmetric potential (Eq. 2.62) the equations odd and even in x become decoupled. The consequence of the decoupling will be analysed later. The first-order solution of Eq. 2.63 yields x(1)(ωj) = − e m Ej ω2 0 − ω2 j − 2iωjγ ≡ − e m Ej D(ωj) . (2.65) Introducing this equation into Eq. 2.64 yields a possible second-order solution x(2)(ω1 + ω2) = −a(e/m)2E2 1 D(ω1 + ω2)D2(ω1) . (2.66) Here, not only the case ω1 + ω2 is possible but also the difference frequency or equal incident frequencies. For simplicity, only the SHG case will be studied here. However, the solutions of the electronic motion in a potential has to be transferred to an expression for the susceptibility. If one assumes that the anharmonic potential represents the potential of an atomic species in the material the polarization can be expressed asN times this contribution, hence P (1)(ωj) = ε0χ (1)(ωj)E(ωj) = −Nex(1)(ωj), (2.67) 30 Light Matter Interaction which yields analogously for higher orders the susceptibilities χ(1)(ωj) = Ne2/m ε0D(ωj) (2.68) χ(2)(ω1 + ω2, ω1, ω2) = N(e3/m2)a ε0D(ω1 + ω2)D(ω1)D(ω2) . (2.69) Thus, it is obvious that an anharmonic potential gives rise to a second order non-linear response. Contrary, for the symmetric case of Eq. 2.62 the decoupling even and odd expansions of the equation system leads to a non-interaction between first and second order terms. Hence, for symmetric potentials there is no possibility to create an even non-linear response, i.e., SHG or fourth harmonic generation (or higher) are not possible in centrosymmetric materials as volume response. Although, virtually any material emits surface SHG, due to the inherent symmetry breaking of an interface (see Sec. 2.2.5). Reviewing Eq. 2.68 and Eq. 2.69 shows that the second order susceptibility can be expressed in terms of the first order. Thus, χ(2)(ω1 + ω2, ω1, ω2) = ε20ma N2e3 χ(1)(ω1ω2)χ (1)(ω1)χ (1)(ω2). (2.70) If the prefactor is constant this leads to the Millers rule66, which reads δ = χ(2)(ω1 + ω2, ω1, ω2) χ(1)(ω1 + ω2)χ(1)(ω1)χ(1)(ω2) . (2.71) The constant δ is called the Millers delta and is measured for many materials. Accord- ingly, it is possible to predict the SHG or other second-order interactions by the linear dispersion of the material. However, it is stressed here that the Millers rule is a crude approximation from a simple anharmonic potential and often fails for real materials67. It should therefore be treated with caution if used in any experiment. 2.2.2 Sum-Frequency Generation In this section, the derivation of sum-frequency generation, which is a second order contribution, will be treated. The general approach is by using the polarization driven wave equation (Eq. 2.5), where the electric field represents the field generated from the polarization. Accordingly, the polarization represents the interaction of the incident elec- tric field. To account for linear and non-linear parts of the polarization the driving factor is split, such that P = P (1)+PNL represents the linear and non-linear parts, respectively. Further, monochromatic plane waves along the z direction and a lossless medium is assumed. Hence, the generated electric field and polarization read as (exemplary for the third vector component) E3(z, t) = A3 exp [i (k3z − ω3t)] + c.c. (2.72) P3(z, t) = 4ε3deffA1A2 exp [i(k1 + k2)z] exp [−iω3t] + c.c. = P3 exp [−iω3t] + c.c. (2.73) The complex conjugates can be omitted without changing the underlying physics. Intro- duction in the wave equation (Eq. 2.5) and rearranging leads to ∂2A3 ∂z2 + 2ik3A3z = −4deffω 2 3 c2 A1A2 exp [i(k1 + k2 − k3)z] , (2.74) 2.2 Nonlinear Optics 31 Figure 2.15: Schematic of sum-frequency generation. Dashed lines depict virtual levels and solid lines are electronic levels. with the wave vector mismatch∆k = k1 + k2 − k3. Typically, the second derivative of A3 is much smaller than k3 ∂A3 ∂z , hence it can be dropped. This is known as the slowly varying amplitude approximation. This simplifies the equation to ∂A3 ∂z = iω3 2ε0n3c P3 exp [i∆kz] = β exp [i∆kz] . (2.75) In the assumption of undepleted pump, i.e.,P3 is constant, the equation can be readily integrated for an interaction length L: A3(L) = β ∫ L 0 exp (i∆kz) dz = β exp (i∆kz)− 1 i∆k (2.76) With this result, the emitted intensity given by Ii = 2niε0c|Ai|2 equates to I3 = 2n3ε0cβ 2L2sinc2 (∆kL/2) . (2.77) 2.2.3 Phase Matching It is apparent in Eq. 2.77 that the emitted intensity is crucially dependent on the wave vector mismatch ∆k. Figure 2.16 shows the change of intensity related to the wave vector mismatch for a constantL. A perfect matching, thus maximum output intensity is achieved by the phase matching condition∆k = 0. In this case, the total intensity scales with L2, since the generated and incident wave stay in phase leading to constructively enhancing of the signal. Contrary, the generated wave gets out of phase with the driving field if there is a mismatch in the wave vectors of the generated waves (∆k 6= 0). This leads to back conversion from the non-linear field to the fundamental wave, thus the light intensity changes periodically throughout the interaction region (Fig. 2.17). Since the prefactor ∆k/2 in the sinc function of Eq. 2.77 determines the periodicity of the oscillation it is convenient to define it as the coherent length of non-linear interaction: Lcoh = 2 ∆k . (2.78) 32 Light Matter Interaction - 1 0 - 5 0 5 1 0 0 . 0 0 . 5 1 . 0 No rm al ize d in te ns ity ∆k L Figure 2.16: SHG intensity in dependence of the wave vector mismatch∆k. However, phase matching is not possible for many materials. To illustrate why this is the case it is convenient to rewrite the phase matching condition in terms of the refractive indices, yielding n1ω1 + n2ω2 = n3ω3. (2.79) For SHG (ω1 = ω2 = ω3/2) the equation breaks down to n(ω) = n(2ω). Typically, the dispersion of a material is a monotonically increasing function to higher refractive for higher energies. This is often described by the Sellmeier formula68, but it is only an approximation far away from the resonance taking place in the blue spectral region. For even higher energies the dispersion of the material will lower rapidly, theoretically allowing for phase matching conditions in this region. The resonance is due to the band gap of the material, thus if the condition is chosen such that the SHG energy is above this resonance, it will be absorbed in the material. Hence, phase matching is not achievable under these circumstances. Yet, it is still possible to generate optical harmonics with high efficiency. In essence, this can be done in two ways: critical and non-critical phase matching. Both of the methods require a birefringent material, i.e., the material shows different refractive indices depending on the light polarization. Luckily, this is the case for many materials exhibiting non-linear properties due to their asymmetric nature, which often gives rise to birefringence, too. The refractive indices of fundamental and SHG are then able tomatch if the material is rotated and tilted to match the criteria. This angle-tuning matching procedure is often termed critical phase matching. It is critical in the sense that the angles have to be oriented with great precision. Additionally, temperature plays a major role, since it affects the refractive indices of the material. Thus, for critical matching conditions, the environmental influence has to be very controlled. The angle-based condition for phase matching69 is 1 ne(Θ)2 = sin2(Θ) n2e + cos2(Θ) n2o , (2.80) 2.2 Nonlinear Optics 33 0 5 1 08 0 . 0 0 . 5 1 . 0 No rm al ize d in te ns ity L � k = 0 ∆k = 0 . 1 ∆k = 0 . 2 5 L c o h ( ∆k = 0 . 2 5 ) Figure 2.17: SHG intensity along an interaction thickness L for different phase matching factors (∆k). The line below depicts the coherence length for a phase mismatch of∆k = 0.25. with the angle of incidence Θ and ordinary no and extraordinary ne refractive indices, respectively. Moreover, this leads the non-normal angle of incident to a spatial walk-off due to amismatch between the Poynting vector and thewave vector of thewave70. This is only the case for extraordinary rays, leading to a mismatch between the fundamental and harmonic generation, thus limiting the interaction region of the two waves. Additionally, for ultra-short pulses also temporal walk-off has to be accounted for which is a generally independent phenomenon in harmonic generation. As a consequence of the energy time uncertainty (Eq. 2.20), the bandwidth of a pulse is broader than its cw counterpart. Each of the frequencies contained in the pulse see a different refractive index in the material, thus they travel at a different speed through the material. This leads to pulse broadening in general, which is quantized by the group velocity dispersion (GVD) around the pulses central frequency ω0: GVD(ω0) = ∂ ∂ω ( 1 vg(ω) ) ω=ω0 . (2.81) vg(ω) is the group velocity of thematerial. Certain materials exhibit negative GVD, hence pulse compression occurs. In the regime of harmonic generation, this effect appears as well: the harmonic travels at another speed than the fundamental. This leads to a walk-off between the two waves, while the induced polarization still produces harmonic waves. Accordingly, the generated pulse broadens strongly, while the peak shifts away from the driving wave. The phenomenon is quantized by the group velocity mismatch, i.e., GVM = v−1 g1 − v−1 g2 . Rarely, the group velocity mismatch is negligible for ultra-short 34 Light Matter Interaction pulses below the ps regime. Thus, thin crystals in combination with strong focusing is used for the generation of ultra-short harmonic pulses. In contrast to angle tuning, the non-critical approach is to use temperature tuning to change the refractive indices of a material. It is only possible for materials with a strong influence of temperature71,72 and is limited by the temperature intervals applicable. Technically, temperature gradients in and around the crystal can lead to disturbances of the harmonic generation. This technique is often employed in commercial systems using non-linear effects due to its better reproducibility and resilience to disturbances. Furthermore, the type of the phase matching is further divided into two cases, de- pending on the polarizations of the incident beams relative to each other73,74. Collinear polarization is called type-I phase matching and type-II for orthogonal polarizations of the incident waves. The emitted harmonic polarization depends on the crystal properties, e.g., if its positive (ne > n0) or negative (ne < n0) uniaxial (two optical axes) or biaxial (three optical axis). Generally, the conditions can be derived from the tensor components of χijk . An example for the elements of such tensors for the mm2 (orthorombic) and 32 (trigonal) crystal class will be given in Ch. 8. Notably, even for circumstances where the above criteria are not able to achieve phase matching, it is possible to efficiently convert light. The basic idea is to alternate the prop- erties throughout the interaction length. Armstrong et al.75 proposed to rotate thin slices of crystals by 180◦ to achieve the condition for the so called quasi-phase matching75. However, the slices have to be too thin to be of practical use to achieve good efficiencies, but Yamada et al. showed the possibility of the process by an alternating static electric field to switch ferroelectric domains. The intensity increase in the interaction region is then a modulated parabola, leading to a slightly lower total intensity than for perfect phase matching. Nevertheless, good efficiencies are possible with different approaches to this idea77. The angle dependence is employed in a measurement technique for SHG. By rotating a sample one obtains an angle dependent oscillatory intensity function. These are Maker fringes first observed by Maker et al.78. In the wave picture, the oscillatory and envelope parts are known, allowing for measurements with less statistical error. The relevant equations are outlined in Sec. 9.1 as correction terms for the measurement. 2.2.4 Second-Harmonic Generation In addition to the derivation of sum-frequency generation outlined in Sec. 2.2.2, which contains the SHG in the case ω1 = ω2, an outline of a solution with depleted pump inten- sity will be given here. This is necessary, since for short pulses and good experimental conditions the efficiency of SHG can reach several tens of percent of the fundamental intensity. Notably, the total intensity of the three interacting waves I = I1 + I2 + I3 is still conserved. Furthermore, the spatial photons per unit area and time obey the Manley- Rowe relations79,80: ∂ ∂z ( I1 ω1 ) = ∂ ∂z ( I2 ω2 ) = − ∂ ∂z ( I3 ω3 ) . (2.82) This means that the spatial change of intensity in the driving waves leads to the same intensity change of the generated wave. 2.2 Nonlinear Optics 35 Figure 2.18: Schematic of second harmonic generation. Dashed (solid) lines depict virtual (real) levels. With these relations it is possible to solve for the SHG intensity with a depleted pump as will be shown here schematically, only. For a full treatment, the reader is referred to Armstrong et al.75 or Boyd69. In the following, the index one and two will refer to the fundamental and second harmonic wave, respectively. First, the amplitudes of the waves are divided into their complex amplitude and phase: Ai = √ I 2niε0c ui exp (iφ1) . (2.83) According to the Manley-Rowe relations, it follows that u21(z) + u22(z) = 1. Introducing each of the monochromatic plane waves with amplitudes from Eq. 2.83 into the wave equation (Eq. 2.5) leads to three coupled differential equations: ∂u1 ∂zn = u1u2 sinΘ (2.84) ∂u2 ∂zn = −u21 sinΘ (2.85) ∂Θ ∂zn = ∆s+ cosΘ sinΘ ∂ ∂zn ln ( u21u2 ) (2.86) with the reduced space coordinate zn = √ ε0cI 2n21n2 2ω1deff c z ≡ z/l (2.87) and phase relation Θ = 2φ1 − φ2 +∆kz. ∆s = ∆k is the wave vector mismatch. For perfectmatching conditions (∆s = 0) Eq. 2.85 can be integrated and it can be shown that the quantity Γ = u21u2 cosΘ is conserved75. It follows that Γ = 0, when assuming that the incident second-harmonic field is zero at the boundary of the interaction region (u2 = 36 Light Matter Interaction 0 2 4 0 . 0 0 . 5 1 . 0 No rm al ize d am pl itu de z / l F u n d a m e n t a l S H G Figure 2.19: Normalized SHG (dashed) and fundamental (solid) amplitude in dependence of the normalized interaction length z/l. 0). Accordingly, cosΘ has to be zero throughout the crystal to keep the conservation law. If we arbitrarily choose sinΘ = −1 Eq. 2.84 and Eq. 2.85 can be solved directly, yielding u1(zn) = sech(zn) (2.88) u2(zn) = tanh(zn), (2.89) for the intensity increase along the interaction region. Figure 2.19 shows that the increase of SHG is now limited by the availability of pump intensity. Further, it is obvious that the complete incident power can be possibly converted into SHG power under perfect circumstances. In reality, mainly spatial and temporal walk-off effects and decoherence will lead to a lower output efficiency. Notably, modulation between fundamental and harmonic wave shows up if the SHG intensity (u2 6= 0) is not zero at the input or the two waves are out of phase75. Moreover, for∆s 6= 0 Eq. 2.84 to Eq. 2.86 can be solved, too. This leads to the oscillatory behaviour already known from Sec. 2.2.2. The major difference is that envelope of the oscillations now follow the tanh function from Eq. 2.89 (Fig. 2.20). However, for focused Gaussian beams there a still some peculiarities: the π phase jump during the focus transition81 rises complex interaction patterns for not perfectly matched waves. Even in the phase-matched case, the generated harmonic shows a phase jump equal to the order of the non-linearity times π. Therefore, the waves dephase across the focus. Since the field strengths in the focus region are the biggest, this effect is very pronounced. Accordingly, the wave vector mismatch has to be slightly detuned to a higher value to account for the phase jump in the focus to restore the full output efficiency82. Further, the harmonic beam shows a non-symmetric pattern and fringe structure for birefringent materials83. The most important consequence for 2.2 Nonlinear Optics 37 0 1 2 0 . 0 0 . 5 1 . 0 SH G am pl itu de z / l ∆ s = 0 ∆ s = 2 Figure 2.20: Normalized SHG amplitude for two different wave vector mismatches in depen- dence of the normalized interaction length (z/l). measurements of SHG is the replacement of the effective interaction region, which is dependent on the Rayleigh length of the Gaussian beam83. For non-focused beams the interaction region was simply given by the length of the crystal. This has major impact on the generation of light in the non-matched case, since the oscillations inside the Rayleigh length are more pronounced in the output intensity than anywhere else in the crystal. 2.2.5 Interfaces As already stated previously, even-order non-linear effects are created mainly from asymmetries along the polarization of the electric field. Despite the fact that this creates harmonics or mixing of the incident light for non-centrosymmetric bulk materials, it implies that an interface emits photons of a different energy, too. This implication simply follows from the symmetry breaking inherent to an interface. Since the coefficients for higher orders are relatively weak for most materials, only SHG will be treated here. In essence, surface SHG follows from the continuity of the tangential component of the electric and magnetic fields at the interface. In conjunction with the wave equation (Eq. 2.5) the surface SHG intensity can be calculated84. Similarly to simple reflection and transmission, the surface harmonic generation is dependent on the polarization relative to the plane of incidence. The harmonic light is generated in a thin slab of thickness l = λ/4π, for a material exhibiting bulk non-linearity’s. Further, the intensity vanishes for phase matching conditions in the surface slab. If the slab is thin compared to the wavelength, the transverse electric field emitted from the surface equates to ET = ER = i4πPNLd nT cosΘT + nR cosΘR , (2.90) for transmission (T) and reflection (R), respectively. Any other case of incidence and different slab thickness can be treated as well. Since the formulas are rather lengthy the 38 Light Matter Interaction reader is referred to Bloembergen and Pershan84 for a full treatment of the conditions. A more detailed derivation of surface emission for a particular polarization will be given in Sec. 9.1 in the correction terms for the measurement. The wave amplitudes in this cases can be approximated as AR ≈ PNL 4ε0c2εr (2.91) AT ≈ πAR (1− 2ikT (ωs)z) . (2.92) Contrary, in a centrosymmetric material the emitted light depends critically on the surface properties andmolecular orientations of thematerial. Thus, a rigorous theoretical treatment is complicated, but typically the emitted light amplitude is three orders of magnitude weaker compared to a non-linear material. 2.2.6 Supercontinuum In contrast to harmonic generation and mixing processes there exists the possibility of white-light generation of a material through non-linear interactions. The process cannot be understood in terms of a simple picture, since many processes are involved in the generation of the light and the underlying contributions are not often clear. Therefore, nu- merical simulations are used to model such behaviours24. This mechanism is employed in PCFs85, where certain asymmetries are introduced and simulated to broaden the fundamental pulse to an extent of two or three octaves in the visible and near-infrared spectrum. Here, some basic principles and the foundation of these processes will be given. Typically, supercontinuum generation takes only place for light pulses, since they deliver the necessary spectral width for the processes involved. The simplest effect the incident pulse undergoes is its modification through the Kerr effect86 or intensity dependent refractive index, which reads n = n0 + n2I(t), (2.93) where the non-linear refractive index is related to the third order susceptibility n2 = (2π/n0) 2χ3. The effect has two major consequences: a focusing of intense light and self phase modulation (SPM) of a short laser pulse. The first process is known from ultra-short laser systems, such as the Ti:sapphire laser87, where the focusing of the pulsed light is used to favor it over the cw light, thus enabling continues pulsing of the Ti:sapphire laser88. According to Eq. 2.93 the non-linear phase change is given as89,90 φ(t) = ω0/cn2I(T )L, (2.94) for a material of thickness L and pulse central frequency ω0. Since the phase change induces a temporal change of the incident pulse this leads to a symmetric broadening of the pulse and a frequency deviation to a maximum value of ∆ω = ω0/cn2LI0/τ, (2.95) where I0 and τ are the intensity and width of the pulse. Moreover, the pulse is affected by a second mechanism leading to a broadening of the pulse, the GVD, i.e., the change of dispersion in the group velocity. However, Kerr lensing and GVD effects are not significant enough to generate white-light of multiple octaves bandwidth. For a further 2.2 Nonlinear Optics 39 broadening mainly Raman, self-steepening, and four-wave mixing effects take place. Raman scattering typically leads to a broadening of the pulse around the center frequency, as the bandwidth further increases the scattering has more cross-sections, thus the effect gets stronger, the broader the pulse. Self-steepening occurs due to temporal intensity changes of the pulse, hence, the pulse sees different refractive indices if the Kerr effect is strong enough. This leads to a slower propagation of the peak pulse and thus gets shifted to the temporal tail of the pulse, introducing spectral asymmetry. Four-wave mixing is a non-linear third-order process converting two (or three) waves into two (one) waves of different frequency. Hence, for a pulse with a great bandwidth four-wave mixing can lead to strong transformation of the light by mixing of its inherent frequencies. In essence, a working white-light emitter counter-balances each of these non-linear effects and the dispersion of the material, leading to a form-stable wave: a soliton91. A soliton itself can undergo fission which further broadens the pulse. In summary, white-light generation is a complex process involving multiple optical non-linearity’s, which counterbalance themselves to generate a stable travelling pulse with a very high bandwid