Excitonic Analysis of Many-Body Effects on the 1s 2p. Intraband Transition in Semiconductor Systems. Andrew M. Parks. A thesis submitted to the

Transcription

1 Excitonic Analysis of Many-Body Effects on the 1s 2p Intraband Transition in Semiconductor Systems by Andrew M. Parks A thesis submitted to the Department of Physics, Engineering Physics and Astronomy in conformity with the requirements for the degree of Master of Science Queen's University Kingston, Ontario, Canada May 211 Copyright c Andrew M. Parks, 211

2 Abstract I present a detailed study of many-body eects associated with the interband 1s transition and intraband 1s-2p transition in two- and three-dimensional photo-excited semiconductors. I employ a previously developed excitonic model to treat eects of exchange and phase space lling. I extend the scope of the model to include static free-carrier screening. I also develop a factorization scheme to obtain a consistent set of excitonic dynamical equations. The exciton transition energies are renormalized by many-body interactions, and the excitonic dynamical equations provide simple expressions for the individual contributions of screening, phase space lling and exchange. The eects of exchange and phase space lling are quantied by a set of excitonic coecients. I rst calculate these coecients analytically by omitting screening eects. In contrast, the screened coecients involve multi-dimensional integrals which must be evaluated numerically. I present a detailed discussion of the numerical methods used to evaluate these integrals, which include a novel algorithm for segmenting multi-dimensional integration regions. The excitonic model correctly predicts the blue shift and bleaching of the 1s exciton resonance due to exchange and phase space lling. Free-carrier screening is found to enhance these eects by lowering the exciton binding energy. In contrast, the eects of free-carrier screening on the 1s-2p transition energy are more subtle. In the absence of free-carrier screening, exchange and phase space lling lead to a blue shift of the transition energy. However, screening decreases the 1s binding energy faster than the 2p binding energy, which in turn decreases the transition energy. Thus, screening eects oppose exchange i

3 and phase space lling, and the overall magnitude and sign of the 1s-2p transition energy shift depends on the free-carrier density. Specically, for low-moderate excitation densities exchange and phase space lling can be dominated by screening, leading to a net red shift of the transition energy. The results for two- and three-dimensional systems are qualitatively similar, although the magnitudes of the shifts are much smaller in three dimensions. ii

4 Acknowledgements I would like to express my foremost gratitude to my supervisor, Prof. Marc M. Dignam, for his wise and friendly guidance during the course of my studies. Our stimulating and fruitful discussions inspired me to write this thesis. I beneted immensely from his patience and foresight, which helped me optimize the quality and scope of my work. I am thankful for the opportunity to attend a conference and present our work, and I enjoyed the time we spent together. I am grateful to Dr. Dawei Wang, whose impressive work laid the foundation for my thesis. He was a continuing source of helpful advice and discussion, even after he had left Queen's University. I thoroughly enjoyed the courses taught by Prof. Stott, Prof. Zaremba, and Prof. Dignam. I learned a great deal from them, and I appreciate the extensive time and consideration they dedicated to their lectures and problem sets. I would like to thank the physics department for nancial support, and for giving me the opportunity to pursue graduate studies. I am especially obliged to Loanne Meldrum, who was always available to answer questions, and provided invaluable assistance during the nal stages of my thesis. Special thanks are due to Matthew Sharp, who is never afraid to question me, and understands me better than anyone. Lastly, but most importantly, I am very grateful to my parents. They have always loved and supported me, and I dedicate this thesis to them. iii

5 Contents Abstract Acknowledgements Contents List of Tables List of Figures Table of Abbrevations and Notations i iii iv vi vii ix Chapter 1: Introduction Semiconductor systems Ultrafast spectroscopy in semiconductors Thesis overview Chapter 2: The excitonic dynamical equations Carrier dynamics in an excitonic basis Population-dependent nonlinear phenomena Chapter 3: Evaluation of the R coecients in the absence of screening Momentum space wavefunctions R coecients for a two-dimensional system R coecients for a three-dimensional system Chapter 4: Evaluation of the R coecients in the presence of free-carrier screening Model for free-carrier screening Screened wavefunctions Expressions for the R coecients Numerical evaluation of the R coecients iv

6 Chapter 5: Analysis of nonlinear optical phenomena Excitons in a two-dimensional GaAs quantum well Excitons in bulk GaAs Discussion of the energy shifts Dimensionless analysis of nonlinear phenomena Chapter 6: Concluding Remarks Bibliography Appendix A: The second order coherence function Appendix B: Fourier transform of exciton wavefunctions B.1 Two-dimensional wavefunctions B.2 Three-dimensional wavefunctions Appendix C: Analytic treatment of the expansion integrals C.1 Indenite integrals in terms of elementary functions C.2 Denite integrals in terms of hypergeometric functions Appendix D: Evaluation of the C coecients D.1 Two-dimensional coecients D.2 Three-dimensional coecients v

7 List of Tables 4.1 The number of points in equivalent product and sparse grids for L = Comparison of numerical and analytic results for the R coecients vi

8 List of Figures 1.1 Illustration of an electron-hole excitation between parabolic energy bands Absorption spectrum for a quantum well Exciton energy spectrum Microwave-ultraviolet region of the electromagnetic spectrum Geometric depiction of the inverse stereographic projection Plots of the exterior integrand a (1) l (u) for l = 1, 2, Convergence of the R1s,1s,1s coecient Convergence of the R 2p 2p,1s,1s coecient Convergence of the R 2p 1s,2p,1s coecient Eect of screening on the 2D variational parameters Eect of screening on the 2D binding energies Eect of screening on the 3D variational parameters Eect of screening on the 3D binding energies Surface plot of the r1 1s 1s,1s,1s (k, 1, φ) level set Surface plot of the r1 1s 1s,1s,1s (k, p, ) level set Surface plot of the r1 1s 1s,1s,1s (x +, x,, ) level set Surface plot of the r1 1s 1s,1s,1s (, x,, y ) level set Process ow for calculating screened R coecients Illustration of the trapezoid integration rule Plot of a test function for one-dimensional quadrature Comparison of quadrature results for the test function Comparison of two dimensional grids for Clenshaw-Curtis quadrature Eects of screening on the 2D R coecients Eects of screening on the 3D R coecients Eects of screening on the R1s1s1s coecient in 2D and 3D Temperature- and density-dependence of the 2D screening wavenumber D 1s exciton energy (constant screening) D 1s-2p transition energy (constant screening) D 1s exciton energy in the low density regime (proportional screening) D 1s exciton energy in a higher density regime (proportional screening) D 1s-2p transition energy in the low density regime (proportional screening) D 1s-2p transition energy in a higher density regime (proportional screening) D 1s oscillator strength (constant screening) vii

9 5.9 2D 1s oscillator strength (proportional screening) Relationship between fermion density and the chemical potential in 3D Temperature- and density-dependence of the 3D screening wavenumber D 1s exciton energy (constant screening) D 1s-2p transition energy (constant screening) D 1s exciton energy in the low density regime (proportional screening) D 1s exciton energy in a higher density regime (proportional screening) D 1s-2p transition energy in the low density regime (proportional screening) D 1s-2p transition energy in a higher density regime (proportional screening) D 1s oscillator strength (constant screening) D 1s oscillator strength (proportional screening) D shift ratio (constant screening) D shift ratio (proportional screening) D shift ratio (constant screening) D shift ratio (proportional screening) viii

10 Table of Abbreviations and Notations Symbol COM EXE PSF RPA H L T V α k β k B k Meaning centre of mass excitonic dynamical equation phase space lling random phase approximation Hamiltonian operator orbital angular momentum operator kinetic energy operator interaction energy operator creates a conduction-band electron with crystal momentum k creates a valence-band hole with crystal momentum k creates a qboson pair with relative momentum k B µ creates an exciton in the state µ ϕ µ (k) ψ µ (r) x κ m exciton wavefunction in momentum space exciton wavefunction in conguration space scaled (dimensionless) representation of x screening wavenumber exciton reduced mass continued on following page... ix

11 ...continued from previous page Γ(x) J ν (x) N ν (x) K ν (x) H ν (x) T ν (x) P µ ν (x) Y µ ν (θ, φ) Gamma function Bessel function of the rst kind Bessel function of the second kind modied Bessel function of the second kind Struve function Chebyshev polynomial of the rst kind associated Legendre polynomial of the rst kind spherical harmonic function pf q generalized hypergeometric function (see page 134) x

12 Chapter 1 Introduction 1.1 Semiconductor systems Some of the most signicant technological achievements in recent history exploit the remarkable physics of semiconductor materials. By controlling their growth and composition, early researchers discovered how to engineer the electronic properties of semiconductors. This led to miniaturized integrated circuits in the early 195s, followed two decades later by the rst microprocessors. Semiconductors also exhibit useful optical properties which have given rise to devices such as light-emitting diodes and semiconductor lasers. Some modern devices, such as optical disc drives and ber-optic communications equipment, often exploit the electronic and optical properties of semiconductors simultaneously. Semiconductors also provide practical systems for studying interesting many-body phenomena, such as the quantum Hall eect. Theory is inexorably coupled to technological and experimental research, and a useful theoretical model must account for the electro-optical properties of semiconductors, as well as interactions with external elds Energy bands in semiconductors A solid material contains a very large number of nuclei and electrons. Most of the electrons are bound to individual nuclei, forming positively charged ions. The remaining - 1 -

13 1.1. SEMICONDUCTOR SYSTEMS valence electrons are delocalized and can form covalent bonds between ions. When a very large number of nuclei are bonded together the energy dierence between individual electronic orbitals becomes vanishingly small, and the allowed quantum states for valence electrons form continuous energy bands. In the absence of excitations, electrons will occupy states with the lowest energies. When considering the electronic and optical properties of semiconductors and metals, the most important bands are the highest energy occupied bands and the rst unoccupied bands. In metals the entire energy spectrum is spanned by a set of overlapping energy bands, and the higher energy occupied bands are only partially lled. Conversely, in insulators and semiconductors the occupied and unoccupied bands are separated by an energy gap of forbidden states. At absolute zero, the highest energy band beneath the gap is fully occupied and is called the valence band, while the rst unoccupied band above the gap is called the conduction band. An electric current in a solid results from the changing momenta of charge carriers, and this can only occur in partially occupied energy bands. Thus, metals can conduct current in their ground state, but insulators and semiconductors exhibit vanishing conductivity. However, a semiconductor is characterized by an energy gap which is small enough that valence band electrons can be readily excited into the conduction band. When this occurs, the valence and conduction bands are only partially occupied, and the semiconductor can then conduct a current. Thus, thermal excitations can actually cause the conductivity of a semiconductor to increase with temperature. Semiconductors derive many of their unique electronic properties (as well as their name) from their variable conductivity. When an electron is excited to the conduction band it leaves an unoccupied state in the valence band called a hole (see Fig. 1.1). The conduction band electrons and valence band holes are both charge carriers that contribute to the conductivity of a semiconductor. Moreover, electrons and holes assume a natural role as fundamental particles of a manybody description of semiconductor phenomena [1, 2]. Thus, it is important to obtain accurate representations of the quantum states for electrons and holes. The full electron and hole band structures can be quite complicated and are dicult to calculate in general. However, most - 2 -

14 1.1. SEMICONDUCTOR SYSTEMS interesting electrical and optical phenomena only involve charge carriers near the maximum of the valence band and the minimum of the conduction band [3]. In this neighbourhood the energy bands are approximately parabolic, and they can be accurately represented using the eective mass approximation. In this approach, the energy dispersion for non-interacting carriers is written as E(k) = 2 2 i,j k i k j M ij, (1.1) where M ij is the eective mass tensor and i, j label degrees of freedom in momentum space. The components of the eective mass can then be determined experimentally. In this thesis we focus on semiconductors in which the dispersion is approximately rotationally invariant near the gap, such that the eective mass can be treated as a scalar quantity [1]. To describe optoelectronic processes in semiconductors we also require information about the carrier wavefunctions. For bulk and mesoscopic structures, the characteristic length of the system is much larger than the lattice constant of the semiconductor material. Thus, it is often reasonable to disregard microscopic contributions to the wavefunctions. To this end it is customary to employ the ansatz Ψ λ,n (r) = χ n (r)u λ (r, k ), (1.2) E Egap h! opt k Fig. 1.1: Illustration of an electron-hole excitation between parabolic energy bands

15 1.1. SEMICONDUCTOR SYSTEMS known as the envelope function approximation [1], which factorizes the wavefunction into two components. The microscopic component, u λ, is the part of the Bloch function [4] satisfying the periodicity of the crystal lattice. In contrast, mesoscopic eects are described by the slowly-varying envelope function, χ n, which is essentially constant within a unit cell. The quantum number λ labels the energy band, n labels the subband energy levels, and k is the crystal momentum 1. The spectrum of subband states (bound and unbound) may arise from Coulomb interactions between electrons and holes, or from spatial connement of the system, as in quantum wells, wires, and dots [5]. The essential features of the wavefunction relevant to optoelectronic phenomena on the mesoscopic scale are embodied by the envelope function. Thus, the wavefunctions considered in the remainder of this thesis are simply envelope functions. Specically, we consider eigenstates of the single-particle Hamiltonian H = T + V e, (1.3) where T is the non-interacting energy (whose momentum space representation is given by Eq. 1.1), and V e is an eective one-body potential Optical interactions and excitons Optical interactions provide an important mechanism by which electrons can be excited to the conduction band. Fig. 1.1 shows a schematic illustration of a semiconductor absorbing a photon with energy ω opt E gap. This process excites an electron from the valence band to the conduction band, and is referred to as an interband transition. If the valence band maximum and conduction band minimum occur at the same position in momentum space (k = in Fig. 1.1), then there is no net transfer of momentum associated with this transition. Thus, interband transitions can be mediated solely by photons tuned to (or above) the energy gap, which impart negligible momentum. Materials with this property are known as direct gap semiconductors. Conversely, if the band extrema do not coincide in k-space, 1 The k limit corresponds to the semiconductor band edge

16 1.2. ULTRAFAST SPECTROSCOPY IN SEMICONDUCTORS then interband transitions must also involve interactions with phonons to satisfy momentum conservation. This decreases the transition probability and makes optical processes slower in so-called indirect gap semiconductors. In this thesis we focus on direct gap semiconductors (e.g. GaAs), which are ideal for studying optical phenomena. Unless a semiconductor is subjected to very strong excitations, charge carriers do not behave as free particles. Rather, Coulomb interactions between carriers can have a profound eect on the optical response of a semiconductor. Most importantly, the attractive interaction between electrons and holes can lead to the formation of a bound state which has less energy than an unbound electron-hole pair. This allows a semiconductor to absorb a photon with less energy than E gap, and the resulting quasiparticle is called an exciton. Excitons are composite particles of two oppositely charged fermions, so they are electrically neutral and have integral spin. The lack of net charge makes it impossible to detect excitons directly. However, exciton population can be inferred from experiments using optical spectroscopy. 1.2 Ultrafast spectroscopy in semiconductors Optical spectroscopy is a powerful technique used to investigate the physical properties of condensed matter systems. Applications of optical spectroscopy in semiconductor materials include the study of band structure, excitation spectra, charge-carrier distributions, defects, phonons, and plasmas [6]. A wealth of information is encoded in the polarization eld (P ), which describes the response of a system to an applied electric eld (E). For low intensities, the relationship between P and E is well approximated by the linear expression [7] P (r, ω) = ɛ χ (1) (ω)e(r, ω), (1.4) where ɛ is the vacuum permittivity, ω is the angular frequency and χ (1) (ω) is the rst-order susceptibility tensor. In linear spectroscopy, the primary object of study is the absorption coecient, α(ω), which is proportional to the imaginary component of χ (1) (ω) [1]. One - 5 -

17 1.2. ULTRAFAST SPECTROSCOPY IN SEMICONDUCTORS way to calculate the linear absorption coecient is to employ a single-particle quantum mechanical model, which leads to the Elliot formula, [1] α(ω) = E gap F n 2 πδ(e n ω), (1.5) cηɛ n where c is the speed of light and η is the material refractive index. The quantum number n labels single exciton states, with corresponding energy E n and oscillator strength F n 2. The oscillator strength quanties the interband transition probabilities, and may be written as F n 2 = µ 2 ψ n (r = ) 2, (1.6) where µ is the transition matrix element in a two band model, and ψ n (r) is the part of the exciton wavefunction describing the relative motion of the electron and hole. From Eqs. (1.5) and (1.6), we see that linear spectroscopy is directly related to the single particle eigenstates of a semiconductor system. At very high intensities, the optical properties of a medium can be aected by an applied electric eld. For example, the optical Kerr eect describes an intensity-dependent change in the refractive index of the propagation medium [8]. Such material changes are called nonlinear optical phenomena, and typically occur when the applied eld is comparable in strength to the inter-atomic elds ( 1 9 V/cm). In this regime, the relationship between P and E becomes nonlinear, and Eq. (1.4) is no longer valid. One way to model the nonlinear polarization is to generalize Eq. (1.4) using a perturbative expansion, [7] P α (r, ω) = ɛ n=1 β 1...β n χ (n) αβ 1...β n (ω)e β1 (r, ω)...e βn (r, ω), (1.7) where χ (n) αβ 1...β n are the components of the n th -order susceptibility tensor, and α, β j {1, 2, 3} label spatial degrees of freedom. Many interesting nonlinear phenomena, such as secondharmonic generation and the Kerr eect, are adequately described by the second- and thirdorder susceptibility tensor, respectively [8]. However, the susceptibility tensors are calculated - 6 -

18 1.2. ULTRAFAST SPECTROSCOPY IN SEMICONDUCTORS using single-particle wavefunctions, which are not generally the eigenstates a many-body system. This has important implications for the optical spectroscopy of semiconductor systems. Even for moderate exciting eld strengths, Coulomb interactions between carriers can lead to nonlinear phenomena which are best modeled non-perturbatively. Thus, nonlinear spectroscopy can be used to detect signatures of many-body interactions in photoexcited semiconductors Nonlinear spectroscopy of semiconductor systems Excitation of a semiconductor with an intense, ultrashort ( 1 fs) laser pulse creates excited carriers which have a coherent, well-dened phase in relation to the exciting eld [6]. In the coherent regime, which lasts 2 fs after the pulse, scattering processes have not yet destroyed coherences between carriers, and the system exhibits interesting many-body behaviour. Later, between 2 fs and 2 ps after the pulse, carrier relaxation and dephasing occur due to a buildup of carrier-carrier scattering, optical phonon scattering, intervalley scattering, and intersubband scattering [6]. After dephasing destroys coherences, the distribution of carriers is typically non-thermal. Optical spectroscopy can be used to probe the system in this nonthermal regime, in order to study the scattering processes which lead to the formation of a thermalized distribution. Eventually, 1 ps after the exciting pulse, acoustic phonon scattering and carrier recombination bring the excited carriers into thermal equilibrium. Depending on the timescales of interest, as well as the interactions targeted for study, there are a variety of possible techniques for ultrafast spectroscopy. For example, luminescence spectroscopy measures the linear properties of a system following photo-excitation, whereas four-wave-mixing techniques can be used to study carrier dynamics and decay in the nonlinear, coherent regime [6]. However, the most common method is pump-probe spectroscopy, whereby a sample is rst excited by a strong (pump) pulse, which induces material changes that are probed by a second pulse (probe) after a time delay. The wavelengths of the two pulses may be similar (as in optical pump, optical probe spectroscopy), or quite dierent (as in optical pump, terahertz probe spectroscopy), - 7 -

19 1.2. ULTRAFAST SPECTROSCOPY IN SEMICONDUCTORS depending on the application. In this thesis, we focus on pump-probe spectroscopy and analyze the eects of many-body interactions on the absorption spectrum for the probe pulse. We restrict our attention to the coherent regime, in which the many-body eects depend solely on the photo-excitation density, and regard the system as being independent of time [6]. At low temperatures 2, excitonic signatures dominate the optical absorption spectrum below the band gap energy. Fig. 1.2 shows a typical absorption spectrum for a GaAs/AlGaAs quantum well. The absorption peaks indicate population of excitonic states with dierent binding energies. 3 Fig. 1.2 shows that the absorption peaks become less pronounced for higher lying states, and the optical response approaches a continuum absorption around the band gap due to free electrons and holes. Clearly, the dominant feature of the absorption spectrum is the 1s exciton resonance. Nonlinear eects arising from many-body interactions inuence the shape and location of the excitonic absorption peaks, and the 1s resonance in particular has been a focus of intense study for several decades [1, 11, 12, 13, 14]. 2 Specically, when kt is smaller than the exciton binding energy. 3 In a quantum well there are two non-degenerate hole bands which have dierent eective masses; the so-called light-hole (lh) and heavy-hole (hh) bands. In this thesis we consider only heavy-hole excitons, and ignore the spin degree of freedom [9]. 1s (hh) absorption 2s (hh) 1s (lh) photon energy Fig. 1.2: Illustration of a typical absorption spectrum for a semiconductor quantum well. The labeled absorption peaks correspond to the 1s and 2s heavy-hole exciton, as well as the 1s light-hole exciton

20 1.2. ULTRAFAST SPECTROSCOPY IN SEMICONDUCTORS The population of excitons depends on the characteristics of the exciting optical pulse. If a spectrally narrow pulse is tuned to a specic excitonic resonance, the ensuing exciton population will be dominated by a single state. Naturally, the total density of opticallygenerated electron-hole pairs (bound or unbound) increases with the intensity of the pulse. However, the behaviour of the exciton density is more subtle. Since an exciton is a composite particle of fermions, it occupies a region of space which cannot accommodate additional excitons without violating the exclusion principle. Thus, as the spatial density of excitons increases, the number of unoccupied excitonic states decreases. This is referred to as phase space lling (PSF), and it can strongly inuence the optical response of a semiconductor. As the number of available states decreases, fewer photons will be absorbed at excitonic resonances. This so-called bleaching of the resonances manifests as a attening of the absorption peaks. For suciently high excitation densities the absorption peaks vanish, and the population is dominated by free electrons and holes [13, 1]. Since the total carrier density depends on the intensity of the optical pulse, the bleaching of excitonic resonances is a nonlinear optical eect due to many-body interactions. Another nonlinear optical eect results from exciton-exciton interactions. Specically, the exchange and PSF components of the Coulomb interaction, in addition to eects of free-carrier screening, lead to a density-dependent renormalization of the exciton energies. This results in spectral shifts of the excitonic resonances in the optical absorption spectrum. The majority of previous experimental studies have focused on nonlinearities associated with the interband 1s transition [1, 13, 14, 12, 15], but the energy shift of the intraband 1s-2p transition has attracted some recent attention [16, 17, 18]. Interband transitions to optically active exciton states (s states) are driven by optical elds at near infrared (NIR) frequencies. For example, the energy of the 1s exciton resonance in GaAs is 1.5 ev (Fig. 1.3), which is in the NIR part of the electromagnetic spectrum (Fig. 1.4). Quantum mechanical selection rules prohibit interband transitions involving states with non-zero orbital angular momentum (e.g. p states). However, these optically forbidden exciton states can be populated by intraband transitions (i.e. transitions between - 9 -

21 energy (ev) 1.2. ULTRAFAST SPECTROSCOPY IN SEMICONDUCTORS 2p ω 4 mev 1s ω 1.5 ev Fig. 1.3: Approximate exciton energy levels for GaAs (not to scale) ultraviolet near infrared mid infrared far infrared terahertz microwave frequency (Hz) Fig. 1.4: Microwave-ultraviolet region of the electromagnetic spectrum. exciton subbands). Figs. 1.3 and 1.4 show that the 1s-2p transition energy corresponds to the terahertz (THz) part of the electromagnetic spectrum. In an optical-pump terahertz-probe experiment in GaAs, absorption of THz radiation at 4 mev would indicate transitions of photo-excited 1s excitons to 2p states. Thus, pump-probe spectroscopy can be used to study nonlinearities associated with the intraband 1s-2p transition. Moreover, since unbound electron-hole pairs inuence the NIR absorption spectrum, optical-pump terahertz-probe spectroscopy can provide unambiguous measurements of exciton populations. However, the technical challenges of producing coherent THz radiation have limited the amount of experimental study in this eld [16, 17]. In this thesis I present a detailed study of manybody induced nonlinear eects associated with the interband 1s transition and intraband 1s-2p transition in photo-excited semiconductors Theoretical approaches A number of approaches have been developed to analyze the nonlinear dynamics of photoexcited semiconductor systems. Since this is ultimately a many-body problem, the most general approach is to employ quantum kinetic equations using non-equilibrium Green's functions [1, 19]. However, such calculations can be cumbersome, and they do not provide an intuitive description of the interactions. On the other hand, density matrix approaches can provide simple dynamical equations for combinations of creation and annihilation - 1 -

22 1.2. ULTRAFAST SPECTROSCOPY IN SEMICONDUCTORS operators representing carrier distributions and polarizations. However, this approach leads to an innite hierarchy of density matrix equations, which must be truncated by imposing an approximation. For example, factorizing expectation values using the random phase approximation leads to the well-known semiconductor Bloch equations (SBEs) [2, 1]. The SBEs have long been used to model interband dynamics in photo-excited semiconductors [1, 6], however, they generally fail to adequately describe intraband coherences [9, 21, 22]. More general truncation schemes, such as cluster expansions [23] or the dynamic truncation scheme [24] can account for coherences up to arbitrary order, but the resulting dynamical equations are numerically intractable. The SBEs and their generalizations represent interactions using a basis of fermionic (electron and hole) operators. Since exciton-exciton correlations are fundamental to the nonlinear response of semiconductors [6, 2], it is natural to wonder if an alternative dynamical model can be formulated using a basis of excitons. Some authors have proposed phenomenological models employing bosonic or quasibosonic representations of many exciton systems. For example, the authors of Ref. [25] formulate a mean-eld Hamiltonian for an N exciton system that can be treated as a multilevel system, similar to the N-level atom. This approach is founded on the observation that these dynamical systems have very similar symmetry groups (specically, SU(N, N) and SU(N) for the N exciton system and the N-level atom, respectively). Although the authors ultimately obtain an intuitive and insightful model, they do not account for the underlying fermionic nature of excitons at a fundamental level. Rather, they start from a bosonic mean-eld Hamiltonian, which is augmented with fermionic corrections to ensure their result for the interband polarization agrees with semiconductor Bloch equations. Since excitons are not true bosons, an alternative approach is to represent the Hamiltonian using operators which explicitly account for the fermionic nature of a many exciton system Excitonic model The excitonic model developed in Ref. [9] provides an accurate description of fermionic eects. Starting from the usual electron-hole semiconductor Hamiltonian [1], the Usui

23 1.2. ULTRAFAST SPECTROSCOPY IN SEMICONDUCTORS transformation [26, 9] is used to represent the Hamiltonian in a basis of quasi-bosonic pair operators. The commutation relations for the pair operators contain a population dependent term which represents the eects of phase space lling (PSF). Using the Heisenberg equation of motion, one can obtain a hierarchy of dynamical equations for expectation values of pair operators. Finally, these equations can be transformed to a basis of excitons, yielding the excitonic dynamical equations (EXEs). The excitonic basis oers important advantages over the SBEs, which use a basis of fermion operators in momentum space. Interactions with THz elds break the rotational symmetry of momentum space by creating excitons with nonzero orbital angular momentum [23]. This drastically increases the complexity of the SBEs by introducing additional degrees of freedom. However, the EXEs are immune to this complication, since they employ a basis of exciton states. Furthermore, the EXEs treat intraband coherences in an unfactorized form [9], which is crucial for the 1s-2p transition. Importantly, the EXEs and SBEs yield equivalent results for interband coherences. However, for low to moderate excitation densities, a numerical simulation of the excitonic dynamical equations is typically more ecient than the SBEs, even in the absence of THz elds [9, 21]. Lastly, the excitonic model provides a clear and intuitive description of interactions involving excitons. This leads to simple, analytic expressions quantifying nonlinear eects (such as the bleaching and energy shifts of exciton resonances), without having to perform tedious self-energy calculations [11]. In this thesis, I incorporate free-carrier screening into the excitonic model to study nonlinear eects associated with excitonic transitions in two- and three-dimensional semiconductor systems. As discussed in Ref. [9], the excitonic model correctly predicts the blue shift and bleaching of the 1s exciton resonance due to exchange and PSF. In this thesis, free-carrier screening is found to enhance these eects by reducing the exciton binding energy. In contrast, the eect of screening on intraband transitions is more subtle. I nd that free-carrier screening opposes exchange and PSF, and the direction of the net shift of the 1s-2p transition energy depends on the free-carrier density

24 1.3. THESIS OVERVIEW 1.3 Thesis overview In chapter 2 I discuss the model developed in Ref. [9] which employs an excitonic representation to treat carrier dynamics. Using this model, the resonance shifts and bleaching of the absorption peaks due to many-body interactions are quantied by excitonic coecients. Chapters 3 and 4 focus on the excitonic coecients associated with the interband 1s transition, as well as the intraband 1s-2p transition. In chapter 3, I use the bare (unscreened) Coulomb interaction to obtain analytic results for the excitonic coecients. I consider strictly two-dimensional (i.e. zero thickness) quantum wells, as well as threedimensional bulk semiconductor structures. In chapter 4, I incorporate static free-carrier screening into the model, and use a simple variational ansatz for the exciton wavefunctions. This leads to expressions for the excitonic coecients involving multi-dimensional integrals, and I discuss important aspects of the numerical method used for their evaluation. In chapter 5, I combine the results of chapters 2-4 to analyze the population dependence of nonlinear phenomena associated with excitonic transitions. Specically, I calculate the 1s exciton energy and oscillator strength, as well as the 1s-2p transition energy, for two- and threedimensional GaAs semiconductor systems. Finally, in chapter 6 I summarize the analysis presented in this thesis, and discuss future topics of study

25 Chapter 2 The excitonic dynamical equations 2.1 Carrier dynamics in an excitonic basis To model the dynamics of charge carriers (electrons and holes) in a photo-excited semiconductor, one typically starts with the Hamiltonian [1, 9] H s.c. =H + V e e + V h h + V e h + H I, (2.1) where H represents the one-body energies of electrons and holes, V e e, V h h, and V e h are electron-electron, hole-hole, and electron-hole Coulomb interactions, and H I is the energy of interactions between carriers and electric elds. Beyond this point, all dynamical models may be separated into two broad categories; fermionic and bosonic. In the former, one works directly with the semiconductor Hamiltonian in the electron-hole representation, whose noninteracting component is [1, 9] H = k (E gap + E e k )α k α k + k E h k β k β k, (2.2) where the operator α k (α k) creates (annihilates) an electron in the conduction band with crystal momentum k, β k (β k) creates (annihilates) a valence-band hole, E gap is the energy gap between the conduction and valence bands, and E e k ( E h k ) is the kinetic energy of an

26 2.1. CARRIER DYNAMICS IN AN EXCITONIC BASIS electron (hole). Fermionic models are quite general, and they have long been used to treat semiconductor systems over a wide range of carrier densities [24, 2, 1, 23]. The fermionic approach also provides a simple framework to develop a microscopic model of semiconductor dynamics, such as the semiconductor Bloch equations [1]. However, these models often fail to provide a simple, intuitive picture for the phenomena they describe. This is particularly true when interactions between excitons (as opposed to free-carrier interactions) dominate the response of the system [2]. In such cases the bosonic approach, which employs composite bosons (i.e. excitons) as the fundamental particles of the system, can provide an accurate and insightful description of an optically-excited semiconductor. Although excitons have integral spin, they cannot be considered true bosons because they do not exactly satisfy the canonical commutation relations. The deviation from true bosons depends on the total exciton density (see Eq. (2.11)), and this tends to complicate the development of a microscopic model within the bosonic approach. In this thesis, we employ the quasi-bosonic method developed in Ref. ([9]), which describes the ultrafast optical response of a semiconductor using an excitonic representation. The usual starting point of a bosonic model is to derive a bosonized version of the semiconductor Hamiltonian, which is generally a dicult task. One method is to project the Hamiltonian onto a basis of pair operators using the Usui transformation [9, 21, 26, 27]. Consider the projector U F F exp B k e,k h α ke β kh B B, (2.3) k e,k h where B k e,k h is the creation operator for an electron-hole pair, F is the vacuum state in the (fermionic) electron-hole representation, and B is the vacuum in the pair representation. Applying the operator U to a Fock state of N electrons and holes, N ψ = α k e,j β k h,j F, (2.4) j=1-15 -

27 2.1. CARRIER DYNAMICS IN AN EXCITONIC BASIS yields a superposition of Fock states in the pair representation [9, 26, 2]. Each state in the superposition U ψ corresponds to a specic pairing of the momenta k e,j, k h,j. Thus, the transformation UH s.c. U (2.5) is not unitary, but it can be made unitary by imposing a unique pairing of the electrons and holes. To this end, the Usui projector is augmented with a pairing operator P, whose action is to pick out the term in the superposition which satises a specic pairing. The transformed Hamiltonian is then given by H s.c. = PUH s.c. U P. (2.6) The simplest pairing scheme matches electrons and holes with opposite momentum, so that α k β k B k, k B k (2.7) creates an electron-hole pair with zero centre of mass (COM) momentum. This prescription is especially suitable for optically excited direct-gap semiconductors, which are the focus of this thesis. Applying the Usui transformation according to Eq. (2.6), one may establish the correspondence [9] α k α k B k B k; (2.8) β k β k B k B k, (2.9) as well as the commutation relations, [B k, B k ] = [B k, B k ] = ; (2.1) ( ) [B k, B k ] = δ k,k 1 2B k B k. (2.11) The second term on the RHS of Eq. (2.11) represents the deviation from true bosons, and

28 2.1. CARRIER DYNAMICS IN AN EXCITONIC BASIS electron-hole pairs created by B k shall henceforth be referred to as quasi-bosons (or qbosons). To satisfy the exclusion principle, we also impose the constraint B k B k =, (2.12) which prevents the creation of identical qbosons. Note that when k k, B k and B k commute like bosons, but when k = k Eq. (2.11) gives [B k, B k ] = 1 2B k B k {B k, B k } = 1, (2.13) which is consistent with the canonical anti-commutation relation for fermions. Thus, the bosonized representation does not destroy the fermionic nature of the system. Using the Usui transformation, the qboson semiconductor Hamiltonian may be written as H s.c. = H Q + V X + H I, (2.14) where H Q is the energy of a non-interacting qboson pair, V X is the Coulomb interaction between qbosons, and H I accounts for interactions between qbosons and electric elds. The non-interacting component, H Q = k Ek B k B k V k k B k B k, (2.15) k,k includes the binding energy (Ek ), as well as the internal Coulomb interaction, of a qboson pair; and the matrix elements of the Coulomb interaction are denoted by V k k. The twobody component of the Hamiltonian, V X = k,k V k k B k B k B k B k, (2.16) includes only the exchange interaction, since direct Coulomb scattering interactions are

29 2.1. CARRIER DYNAMICS IN AN EXCITONIC BASIS eliminated by the zero COM momentum pairing scheme [9]. Interactions with external electric elds are treated semi-classically according to H I = E opt P inter E THz P intra, (2.17) where E is an electric eld, P inter is the interband polarization, and P intra is the intraband polarization. Here we have made the explicit distinction that interband transitions are driven by electric elds at optical frequencies (E opt ), whereas intraband transitions are driven by terahertz elds (E THz ) [9, 21, 28]. The expression for the interband polarization is P inter = k (d cv B k + d cvb k ), (2.18) where d cv is the interband dipole moment [1, 28], while the intraband polarization is P intra = G k,k B k B k, (2.19) k,k where G k,k = e k r r k (2.2) is the intraband transition matrix element [28]. To describe the dynamics of a semiconductor in the qboson representation, we use the Heisenberg equation of motion together with Eqs. (2.11) and (2.12) to obtain the dynamical equation [9, 21] i d dt B p = E pb p + k V p k B k + 2 k V p k ( B pb k B k B k B pb p ) + E opt d cv ( 1 2B pb p ) + k E THz G kp B k (1 2B pb p ). (2.21)

30 2.1. CARRIER DYNAMICS IN AN EXCITONIC BASIS The Coulomb terms in the second line of the RHS represent exchange and phase-space-lling (PSF) eects, respectively. The population dependent terms in the third and fourth lines are also due to PSF eects. The PSF terms arise from the population dependent component of the qboson commutation relation (Eq. (2.11)), which accounts for the underlying fermionic composition of the qbosons. Clearly, the dynamics of a single qboson operator is coupled to products of two and three qboson operators through the optical and Coulomb terms on the RHS of Eq. (2.21). One may readily determine that the dynamical equations for these higher order terms depend on even larger products of operators. Thus, the evolution of qboson operators are governed by an innite hierarchy of dynamical equations Truncation of the dynamical hierarchy In order to use the qboson dynamical equation, we must rst nd a suitable approximation to truncate the dynamical hierarchy. Since observable phenomena are ultimately related to the expectation value of Eq. (2.21), a natural approximation is to factorize expectation values of a large number of operators into products of expectation values of smaller number of operators. For example, the familiar semiconductor Bloch equations are obtained by imposing the factorization α pβ p β qα q α pβ p β qα q, (2.22) which is often referred to as the random phase approximation (RPA) [1, 9], and is essentially equivalent to the coherent limit [23, 29, 2]. To develop a factorization for qbosons in this limit, we consider expectation values with respect to the coherent state [23, 3, 31] ψ coh = k (u k + v k B k ) B, (2.23)

31 2.1. CARRIER DYNAMICS IN AN EXCITONIC BASIS which is clearly normalized if u k 2 + v k 2 = 1. Consider the single qboson expectation value B p ψ coh B p ψ coh = k,k B (u k + v k B k)b p(u k + v k B k ) B = B (u p + v pb p )B p(u p + v p B p) k p(u k + v k B k)(u k + v k B k ) B, (2.24) where we have isolated the k = p term from the product using the qboson commutation relations. This ensures that each term in the product commutes with the operators to the left of the product. So, to evaluate the product it suces to consider the expectation value ( B (u k + v k B k)(u k + v k B k ) B = B = B u k 2 + u k vk B k + u k v kb k + v k 2 B k B k ( ) u k 2 + v k 2 v k 2 B k B k B ) B = 1, (2.25) where terms which annihilate the vacuum have been discarded. We thus have B p = B (u p + vpb p )B p(u p + v p B p) B ( ) = B u p 2 B p + u pv p B pb p + u p vpb p B p + v p 2 B p B pb p B ( ) = B u p vp(1 B pb p ) + v p 2 (1 B pb p )B p B ) = B (u p vp + v p 2 B pb pb p B = u p v p. (2.26) Now, to evaluate the expectation value of two qboson operators, B pb q = k,k B (u k + v k B k)b pb q (u k + v k B k ) B, (2.27) - 2 -

32 2.1. CARRIER DYNAMICS IN AN EXCITONIC BASIS we must consider two distinct cases. If p q then B pb q = B (u p + v pb p )B p(u p + v p B p)(u q + v qb q )B q (u q + v q B q) B =u p v pu qv q, (2.28) whereas B pb p = B (u p + vpb p )B pb p (u p + v p B p) B ( ) = B u p 2 B pb p + u pv p B pb p B p + u p vpb p B pb p + v p 2 B p B pb p B p B ( = B u pv p B p(1 B pb p ) + u p vp(1 B pb p )B p + v p 2 (1 B pb p ) 2) B = v p 2. (2.29) Comparing Eqs. (2.26), (2.28) and (2.29) reveals the coherent limit factorization B pb q = B p B q (1 δ p,q ) + B pb p δ p,q. (2.3) For the three operator expectation value, B pb qb q, (2.31) we know that p q due to the exclusion constraint (Eq. (2.12)). Thus, [B p, B q ] =, and the expectation value with respect to the coherent state is ψ coh B pb qb q ψ coh = B (u p + vpb p )B p(u p + v p B p)(u q + vqb q )B qb q (u q + v q B q) (u k + v k B k)(u k + v k B k ) B k p,q = B (u p + vpb p )B p(u p + v p B p)(u q + vqb q )B qb q (u q + v q B q) B. (2.32) Since the p and q terms commute with one another, we can immediately apply the results

33 2.1. CARRIER DYNAMICS IN AN EXCITONIC BASIS of Eqs. (2.26) and (2.29) to obtain ψ coh B pb qb q ψ coh = u p v p v q 2. (2.33) The coherent factorization for the expectation value of three qboson operators is thus B pb qb q B p B qb q. (2.34) Applying this factorization to the expectation value of Eq. (2.21), we are still left with expectation values of two qbosons. It can be shown [9] that the dynamical equation for B pb q couples to higher order expectation values of the form B pb k B kb q. To determine the coherent factorization for this four operator expectation value, we rst note that B pb k B kb q = B pb k B kb q (1 δ p,k )(1 δ q,k ), (2.35) due to the exclusion constraint. The expectation value with respect to the coherent state is thus ψ coh B pb k B kb q ψ coh = B (u p + v pb p )B p(u p + v p B p)(u k + v k B k)b k B k(u k + v k B k ) (u q + v qb q )B q (u q + v q B q) B =u p v p v k 2 u qv q. (2.36) Comparing this result with Eqs. (2.26) and (2.29), we see that the dynamical hierarchy can be truncated using B pb k B kb q B pb q B k B k, (2.37) together with the factorization of Eq. (2.34). However, this factorization scheme leads to inconsistent expressions for physical observables which are renormalized by many-body interactions (see App. A). Indeed, it has been remarked in the literature that the coherent state argument alone is not sucient to provide a consistent factorization [29]. To extend

34 2.1. CARRIER DYNAMICS IN AN EXCITONIC BASIS this factorization we also consider the dilute limit, that is, when the total exciton density is small. For the coherent state of Eq. (2.23) this corresponds to v p 1 [31]. Now, using the normalization of the coherent state we may write B pb p = v p 2 = v p 2 ( u p 2 + v p 2) = u p 2 v p 2 + v p 4 = B p B p + B pb p 2, (2.38) where the last equality follows from Eqs. (2.26) and (2.29). If v p 1 we may neglect the second term, and hence approximate B pb p B p B p. (2.39) Thus, in this limit we factorize an N-operator expectation value into N single-operator expectation values. This leads to the symmetrized factorizations 2 B pb qb q 2 B p B q B q = B p B qb q + B q B pb q, (2.4) and 2 B pb k B kb q 2 B p B q B k B k = B pb q B k B k + B pb k B k B q. (2.41) By choosing this factorization scheme we have restricted the applicability of our model to the coherent and dilute regime. However, by applying the symmetrized factorization scheme to either the interband or intraband dynamical equations (Eqs and A.1), we obtain identical expressions for the intraband transition energy (Eqs. (2.49) and (A.12)). This is sucient motivation for the symmetrized factorization scheme in this thesis, since the description of nonlinear phenomena associated with the 1s-2p excitonic transition is our primary goal

35 2.1. CARRIER DYNAMICS IN AN EXCITONIC BASIS The exciton basis The qboson dynamical equations are not immediately useful because the qbosons are not eigenstates of the system. The dynamical equations can be cast in a more insightful (and computationally ecient) form by applying the unitary transformation B µ = k ϕ µ (k)b k ; B k = µ ϕ µ(k)b µ, (2.42) where ϕ µ (k) is an eigenfunction of an electron and hole with relative momentum k, coupled by the Coulomb interaction. This shall serve as the denition of an exciton in this thesis, so that ϕ µ (k) is the momentum space wavefunction for an exciton in the state µ. Applying this transformation to Eq. (2.21) and taking the expectation value, we obtain the excitonic dynamical equation: i d dt B µ =Γ µ B µ E µ B µ + {γ j } ( R µ γ1,γ 2,γ 3 + R µ γ 2,γ 1,γ 3 ) B γ1 B γ 2 B γ3 + E opt (t) Mcv C µ 2 C µ,γ1,γ 2 B γ 1 B γ2 + {γj} γ E THz (t) G µ,γ B γ + 1 ( E THz (t) G γ4,γ 2 5 T µ γ1,γ 2,γ 3,γ 4,γ 5 + T γ µ ) 2,γ 1,γ 3,γ 4,γ 5 B γ1 B γ 2 B γ3. (2.43) {γ j } The quantity B µ is known as the interband coherence function [21], E µ is the noninteracting exciton energy, Γ µ is a phenomenological interband dephasing term, and the excitonic coecients

36 2.2. POPULATION-DEPENDENT NONLINEAR PHENOMENA R µ γ 1 γ 2 γ 3 k,p V p k [ ϕµ (p) ϕ γ 1 (p) ϕ γ 2 (k) ϕ γ3 (k) ϕ µ (p) ϕ γ 1 (k) ϕ γ 2 (p) ϕ γ3 (p) ] ; (2.44) C µ k ϕ µ (k); (2.45) C µ,γ1,γ 2 k ϕ µ (k) ϕ γ 1 (k) ϕ γ2 (k); (2.46) T µ γ 1 γ 2 γ 3 γ 4 γ 5 k,p ϕ µ (p) ϕ γ 1 (k) ϕ γ 2 (p) ϕ γ3 (p) ϕ γ4 (k) ϕ γ 5 (p) (1 δ k,p ), (2.47) account for Coulomb, optical, and THz interactions. The two components of the excitonic R coecient account for exchange and PSF eects, respectively. The excitonic dynamical equation for the intraband coherence function B µb ν is shown in App. A. 2.2 Population-dependent nonlinear phenomena One especially useful feature of the excitonic dynamical equations is that they provide simple expressions for the population-induced shifts of the exciton energy and oscillator strength. The Coulomb part of Eq. (2.43) may be written as i d [ dt B µ = E µ ] ( R µ µ,γ1,γ 2 + R µ ) γ 1,µ,γ 2 B γ1 B γ2 B µ Coulomb γ 1,γ 2 + ( R µ γ3,γ 1,γ 2 + R µ ) γ 1,γ 3,γ 2 B γ1 B γ2 B γ 3. (2.48) γ 1,γ 2 γ 3 µ The rst line on the RHS gives the renormalized exciton energy, ε µ E µ γ 1,γ 2 ( R µ µ,γ1,γ 2 + R µ γ 1,µ,γ 2 ) B γ1 B γ2, (2.49) while the second line represents couplings to other exciton states. If the system is excited by an optical pulse at the 1s exciton resonance, then the initial pair population will be dominated by 1s excitons. To model this excitation condition, we set γ 1 = γ 2 = 1s in

37 2.2. POPULATION-DEPENDENT NONLINEAR PHENOMENA Eq. (2.49) to obtain ( ) ε µ E µ R µ µ,1s,1s + Rµ 1s,µ,1s N 1s. (2.5) This shows that the magnitude of the energy shift depends on the initial 1s exciton population. Moreover, this population is related to the intensity of the optical pulse, so the energy shift is a nonlinear optical eect. Experimentally this manifests as a shift of the exciton absorption peak with respect to the intensity of the optical pulse. It should be noted that the sign of the energy shift is not solely determined by the excitonic R coecients, since free-carrier screening leads to an additional change of the exciton binding energy. In this thesis, we consider the energy shift of the 1s exciton ε 1s = E 1s 2R 1s 1s,1s,1sN 1s, (2.51) which corresponds to the initial interband (optical) excitation, as well as the shift of the 2p exciton ( ) ε 2p = E 2p R 2p 2p,1s,1s + R2p 1s,2p,1s N 1s, (2.52) which allows us to analyze the 1s 2p intraband transition. Another optical nonlinearity is the so-called bleaching of the exciton resonance. This refers to a attening of the optical absorption peak due to a reduction of the oscillator strength. From Eq. (2.43) we see that the renormalized oscillator strength is given by 2 f µ = C µ 2 C µ,γ1,γ 2 B γ 1 B γ2 C 2 µ 4C µ C µ,1s,1s N 1s, (2.53) {γj} where we have again assumed resonant excitation of the 1s exciton, and have discarded the O ( N 2 ) 1s term. The excitonic C coecients are positive-denite (see App. D), so the second term in Eq. (2.53) accounts for a population-dependent bleaching of the optical oscillator strength. In order to evaluate the nonlinear shifts, we require values for the C and R coecients

38 2.2. POPULATION-DEPENDENT NONLINEAR PHENOMENA In the next chapter we will determine the momentum space wavefunctions for the 1s and 2p states, and hence calculate the R coecients related to the 1s-2p transition. The calculation of the C coecients is quite trivial, as shown in App. D

39 Chapter 3 Evaluation of the R coecients in the absence of screening To determine the renormalized energies in Eqs. (2.51) and (2.52) we must evaluate the excitonic R coecients dened in Eq. (2.44). It is convenient to express the R coecient in terms of two components R µ γ 1 γ 2 γ 3 = R1 µ γ 1 γ 2 γ 3 R2 µ γ 1 γ 2 γ 3, (3.1) where R1 µ γ 1 γ 2 γ 3 k,p V p k ϕ µ (p) ϕ γ 1 (p) ϕ γ 2 (k) ϕ γ3 (k) (3.2) and R2 µ γ 1 γ 2 γ 3 k,p V p k ϕ µ (p) ϕ γ 1 (k) ϕ γ 2 (p) ϕ γ3 (p). (3.3) In this section, we use the rotational symmetry of the system to obtain analytic results for these coecients. First, the momentum space wavefunctions are derived by exploiting the dynamical symmetry of the unscreened exciton Hamiltonian. We then calculate the R

40 3.1. MOMENTUM SPACE WAVEFUNCTIONS coecients using two- and three-dimensional eigenfunction 1 expansions for the Coulomb matrix elements. In chapter 5 these will be compared to numerical results for the R coecients calculated in the presence of free-carrier screening. 3.1 Momentum space wavefunctions Within the zero COM momentum pairing scheme, the envelope function for a qboson is e ik r χ k (r) =, (3.4) LD/2 where k(-k) is the crystal momentum of the electron (hole), r r e r h is the separation between the electron and hole, and L is the characteristic length for the D dimensional system. The conguration space (ψ µ (r)) and momentum space (ϕ µ (k)) representations of the wavefunction for an exciton in the state µ are thus connected by ψ µ (r) = k ϕ µ (k)χ k (r). (3.5) The eigenvalue equation for Wannier excitons is [1, 9] (T + V) µ = E µ µ. (3.6) where T is the kinetic energy and V is the Coulomb interaction energy. Since we consider only the relative motion of the electron and hole, the system is rotationally invariant and it is a straightforward exercise to derive two- and three-dimensional solutions of this equation in conguration space [1]. However, this approach does not provide much insight for the remarkable result that the energy spectrum depends only on the principal quantum number. This hidden degeneracy cannot be foreseen by considering the rotational symmetry of the N-dimensional Wannier equation [32, p.224]. This is because the (special orthogonal) group SO (N) is merely a subgroup of the full symmetry group of the N-dimensional Wannier 1 specically, eigenfunctions of L 2, where L is the total angular momentum operator

41 3.1. MOMENTUM SPACE WAVEFUNCTIONS equation, which is actually SO(N +1). This symmetry manifests naturally when the Wannier equation is considered in momentum space. The solubility of Eq. (3.6) in momentum space was rst demonstrated by Fock for the three-dimensional hydrogen atom [33, 34]. In this section we use Fock's stereographic projection technique [35, 36] to solve the twodimensional Wannier equation. Not only is this approach insightful, it also leads directly to the momentum space wavefunctions that we require. The momentum space representation of Eq. (3.6) is k T + V µ = E µ k µ E µ ϕ µ (k). (3.7) where k is a two-dimensional wave-vector. Using the Fourier transform convention dened in Eq. (B.1), the matrix elements of the two-dimensional attractive Coulomb interaction are (in SI units) [9, 1] k V k = e2 1 2ɛL 2 k k V k k, (3.8) where e is the elementary charge and ɛ is the dielectric constant. Inserting this expression into Eq. (3.7) we obtain the integral form of the Wannier equation ( 2 k 2 2m E µ ) ϕ µ (k) = ( ) L 2 k V k k e 2 ˆ µ 2π 2ɛL 2 k dk ϕ (k ) k k, (3.9) where m is the reduced mass of the electron and hole, and the vector sum has been converted to an integral (assuming L is suciently large). It is convenient to introduce dimensionless quantities Ē and k dened by E = ĒR y; k = k a, (3.1) where R y = 2 2m a 2 ; a = 4πɛ 2 m e 2 (3.11) are the three-dimensional Rydberg energy and Bohr radius [1, 9]. The Wannier equation - 3 -

42 3.1. MOMENTUM SPACE WAVEFUNCTIONS Fig. 3.1: A geometric depiction of the inverse stereographic projection used to map twodimensional momentum space to the unit sphere. may thus be written as ( k2 + η 2) ϕ ( k) ˆ 1 = π d k ϕ ( k ) k k, (3.12) where η 2 Ē. To solve this integral equation, consider the coordinate transformation u = η2 k 2 η 2 + k 2 ẑ + 2η η 2 + k k, 2 (3.13) where ẑ is the unit vector perpendicular to the k x k y plane. The magnitude of the vector u is { [η 2 k 2 ] 2 [ ] } 2ηk 2 1/2 [( η 2 k 2) ] 2 1/2 + 4η 2 k 2 u = η 2 + k 2 + η 2 + k 2 = (η 2 + k 2 ) 2 = 1, (3.14) which means u is restricted to the surface of a unit sphere embedded in three-dimensional Euclidean space, as shown in Fig This conformal mapping of the plane to a sphere is an example of an inverse stereographic projection [37, p.136]. If θ and φ are the usual polar and azimuthal components of spherical coordinates charting the sphere, then the elemental solid angle is ( η 2 k 2 ) ( ) 2η 2 ( ) 2η 2 dω u = d (cos θ) dφ = d η 2 + k 2 dφ = η 2 + k 2 kdkdφ = η 2 + k 2 dk. (3.15)