um? flux“ . . . L..%a..,.a .1 mubfimewufl, atrnf, , inf , 1H5: It... a. f? .3 fury v . w, .\ i! 5 sq) 1V 9.... 1.. «4 . 5:1. .9 . : > Lighkrjrfivq. (new... ,afim. 95? Man a $~1¥ '1'] 1|. 13.3». t: . pl}; V I i z . v.4 \..42. .3... :r In. :9 V... '0 I , 11.2.: xccpzvragu , .§,_, .ll‘ 13": I. ’13:!!! :39 993' :9: E? P‘ [.2 «A {Lil LIBRARY Michigan State University This is to certify that the thesis entitled TRIONIC OPTICAL POTENTIALS FOR CHARGE CARRIERS IN SEMICONDUCTORS presented by Martin Johannes Alexander Schiitz has been accepted towards fulfillment of the requirements for the MS. degree in Physics and Astronomy Major Professor’s Signature 7 /ze/o°( 712-;{61‘4 Date MSU is an Affirmative Action/Equal Opportunity Employer A ———~.—_—.__.— PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 5108 KzlProj/AocanPresICIRCIDateDueindd TRIONIC OPTICAL POTENTIALS FOR CHARGE CARRIERS IN SEMICONDUCTORS By Martin Johannes Alexander Schiitz A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Physics and Astronomy 2009 ABSTRACT TRIONIC OPTICAL POTENTIALS FOR CHARGE CARRIERS IN SEMICONDUCTORS By Martin Johannes Alexander Schiitz Optical trapping of neutral particles has led to remarkable advances in precision measurement, quantum information, and addressing fundamental longstanding ques- tions in condensed matter physics. Despite recent advances in the optical and elec- tronic control in semiconductor systems, a similar laser-induced technique to trap and manipulate charged carriers in semiconductor devices has not yet been investigated. In this thesis, we will propose for the first time analogues optical trapping poten- tials for charge carriers embedded in a semiconductor quantum well by driving the trion resonance with intense, detuned laser light. A trion is a bound state between an exciton and an excess carrier, that modifies the exciton resonance frequency in the vicinity of a carrier. Accordingly, the Stark energy is modified in proportion to the the light intensity at the carrier location, which serves as a source of mechanical potential energy for the carrier. We show that this novel trion-mediated potential exhibits a non—local character, but can confine carriers at the lengthscale of optical wavelengths. It can be deep compared to the the single electron recoil energy and the carrier’s ef- fective temperature which benefits from the omnipresent cooling mechanism of the phonon bath. The model is extended to the new paradigm of a spin-selective carrier lattice in a true Solid State environment which is potentially much simpler to engineer and control than similar lattices in AMO physics. Our results suggest the possibility of integrating ultrafast optics and gate voltages in new single-carrier semiconductor devices with promising applications in quantum information processing, and explor- ing the physics of interacting electrons in the presence of a periodic potential readily controllable in space and time. DEDICATION Dedicated to my mum, my sister, Millhouse and Jaipur. iii ACKNOWLEDGMENT A few weeks before my year abroad at Michigan State University turns to its end, I am looking back on a year in my life, loaded with experiences. I have grown not only as a physics student, but as a person. Now, it is the right time to express my gratitude to all the people that stood by my side during the past year. First of all, I would like to thank my advisors, Prof. Michael G. Moore and Prof. Carlo Piermarocchi for the continuous guidance and support they have provided to me. After my undergraduate studies at the University of Karlsruhe, they gave me the opportunity to conduct research for the first time in my life as a physicist. They are great, passionate teachers with an astonishing ability to give precise explanations of complex phenomena and an seemingly inexhaustible patience. Our discussions have never been limited to the scope of this thesis and I have learned much more than I could have ever imagined. Most importantly, their uncomplicated, pleasant and humorous personality made it fun to work with them. Without any exaggeration, I cannot imagine a better team of advisors. Also, I would like to express my gratitude to Prof. Mark Dykman and Prof. Bhanu Mahanti for their intriguing style of teaching. In their classes I have learnt a lot of new physics, some of which already flowed into this thesis. Debbie Barratt has always been a very friendly, patient and helpful contact person. Thank you so much! My stay at MSU would not have been the same without Prof. Wolfgang Bauer and his golf league. I am very thankful that I got the opportunity to join this group. There is nothing better than going out on a warm Tuesday evening to hit some drives. The financial support by the golf league is gratefully acknowledged ;—). Although a Prussian by birth, Kai was more than a roomate that introduced me to the beauty of Ubuntu and Lyx, he has become a very good friend. V‘Ve will share many memories and I am looking forward to our trip to the “fest. iv I consider myself very fortunate to have met Kirstie: a friendship that is to be continued in Germany. Martin and Christine have become friends I had a lot of fun with. Then, I will never forget the glorious victories as a member of ’Team Michelle” together with Marshall and Kritsada. Michelle, I have played in your team, but unfortunately I never got to know you. Also, thanks to Philipp for selling a car that doesn’t have a radio, but features an on-board swimming pool! The National German Merit Foundation as well as the Fulbright program sup- ported my exchange year at Michigan State university with a generous scholarship. Also, they organized meetings at Princeton and Denver, where I shared experiences with many young, incredibly talented people that I will never forget. After a year abroad, far away from my home, I do appreciate even more what great friends I have back in my home territory. We always stayed in contact and I am looking forward to seeing you again! In particular, I would like to mention all of my golf team-mates, Enrico, Erik and Manuel who never gets tired of helping me out. Vor allem mochte ich aber meiner Mutter, meiner Schwester, meiner Oma und Karl danken. Ohne ihre standige Unterstiitzung, ihren Glauben an mich und ihren Zuspruch in personlich sehr schwierigen Zeiten ware die Anfertigung dieser Arbeit nicht moglich gewesen. Ich danke Euch von ganzem Herzen! TABLE OF CONTENTS List of Tables ................................. List of Figures ................................ Introduction .................................. Excitations in semiconductors 1.1 Excitons in quantum wells ........................ 1.1.1 Wannier-Mott theory ....................... 1.1.2 Confined Excitons ........................ 1.2 Trions in quantum wells ......................... 1.2.1 Hamiltonian ............................ 1.2.2 Variational solution ........................ 1.3 Light-matter interaction: Optical matrix elements ........... 1.3.1 Coupling to bound trions ..................... 1.3.2 Coupling to unbound trions ................... 1.4 Radiative lifetimes ............................ Optical Potentials: Atoms vs. semiconductors 2.1 Optical potentials for atoms ....................... 2.1.1 Optical dipole traps for atoms .................. 2.1.2 Dissipation in optical potentials ................. 2.2 Toy-model optical potentials in semiconductors ............ Effective Hamiltonian from second order perturbation theory 3.1 Perturbation theory ............................ 3.2 Orthogonalization of the excitonic continuum ............. 3.3 Diagrammatic approach ......................... 3.4 Second order energy correction ...................... 3.5 Effective Hamiltonian, Schrodinger equation .............. Optical potentials for carriers in semiconductor quantum wells 4.1 Influence of exciton level ......................... 4.2 Memory effect: Non-locality of the potential .............. 4.3 Potential depth ........... , ................... 4.4 Harmonic oscillator approximation ................... 4.4.1 Without memory: Local potential ................ 4.4.2 With memory: Non-local potential ............... vi viii ix 0000‘] 12 15 17 20 29 35 41 45 54 56 56 63 68 79 80 84 86 88 92 100 101 104 106 118 122 H0060??? CEO"? I-II Effective temperature for the electrons 5.1 Acoustic phonon scattering rate ................... 5.2 Equilibrium temperature ....................... Optical lattices for carriers in semiconductor quantum wells 6.1 Coulomb blocking: On-site repulsion ................ 6.2 Filling factor ............................. 6.3 Spin-selectivity ............................ Additional effects 7.1 Ponderomotive potential ....................... 7.2 Electron ionization .......................... 7.3 Restrictions on the electron density ................. Conclusion Material parameters Excitons in quantum wells Derivation of the volume element for Hylleraas coordinates Results of variational calculus Exchange effects E.1 Exchange effects in the coupling to unbound trions ........ E2 Exchange effects in the overlap ................... 'I‘rion radiative lifetime Three level system in rotating frame Excitonic second order energy shift Derivation of effective Hamiltonian I.1 Off-diagonal contribution H? I .................... 1.2 Diagonal contribution H} I ...................... 1.3 Trionic contributions in the effective Hamiltonian ......... Bibliography .................................................. vii 128 129 133 146 148 154 156 159 160 163 167 169 175 176 180 186 190 190 195 200 204 207 214 .. 214 .. 219 ...222 1.1 1.2 1.3 1.4 4.1 4.2 LIST OF TABLES Calculated trion binding energies for GaAs and CdTe. ........ Typical trion “size" compared to exciton “size” in GaAs and C dTe. . Calculated radiative exciton lifetimes, decay rates and level widths for GaAs and CdTe. ............................ Calculated radiative trion lifetimes, decay rates and level widths for GaAs and CdTe. ............................ Fitted parameters for g (x) for GaAs and CdTe in effective atomic units. ................................... Combinations of the time averaged probability for exciton creation and potential depth are presented as pairs in the form (Pm), V0 for two values of the saturation parameter c and the detuning from the trion resonance At. ........................... D.1 Variational parameters for GaAs and CdTe in atomic units. ..... viii 29 1.1 1.2 1.3 1.4 1.5 LIST OF FIGURES Images in this thesis are presented in color Schematic of optically-confined electron lattice. The 2DEG sample already has electrons confined to an :1: — y plane below the sample surface. Four incident lasers achieve further confinement using the AC Stark shift effect. For clarity, here the lasers are shown in the LIE-direction only; an identical pattern of light will be applied in the y- direction, too. The charge modulation can be probed based on a STM technique. ................................. Optical transition between the full valence band (VB) and the empty conduction band (CB) for a direct transition. ............. A schematic X - trion singlet. ...................... Ground state exciton binding energy and variationally optimized “rel- ative” trion energy in Hartree units. The trion is stable against disso- ciation for any possible massratio a. .................. Variationally optimized trion binding energy ET measured relative to the exciton binding energy EX. ..................... Schematic transition processes for a semiconductor QW, coupled to photons with momenta Q. Photonic lines, excitonic lines and trionic lines are marked by a red dot, green dot and blue dot respectively; free electrons and holes by black and grey dots, respectively. The upper half displays transitions of photons to excitons or unbound electron— hole pairs. The lower half describes the possible transitions to unbound and bound trions if the initial state contains also an excess electron. See also [32] ................................. ix 21 26 27 31 1.6 1.7 1.8 1.9 2.1 2.2 2.3 2.4 2.5 2.6 2.7 3.1 Coupling of a free electron to a bound trion singlet state ........ Plots of the functions It (k): They govern the trion oscillator strength for the interaction with a :I:Q mode. The variational parameters for GaAs were used. The value for Q corresponds to laser light tuned close to the trion resonance at an angle of 30° with respect to the 2—axis. Schematic diagrams of the exciton (left) and trion (right) in-plane dis- perion relations and the coupling to light. For excitons the radiative zone is restricted by the light cone effect (dashed lines) ......... A trion in its rest frame decaying into a photon and an electron. . . . Interaction of a laser with a single two-level atom with the chosen sign convention for the detuning A. ..................... Eigenvalues 6:}; of the Hamiltonian H in the rotating frame after the rotating wave approximation. ...................... The bare two-level system with eigenenergies E9 and E8 are shown in the center. The light shifts for red detuning are displayed on the right, creating an attractive potential. For blue detuning the potential is repulsive, as depicted on the left—hand side. ............. Random walk problem in momentum space. .............. An extract from the QW sample: It is discretized into small cells of size Am, the typical exciton size. If an electron is present, centered at re, excitons created within the area At, the typical trion size, give rise to the formation of a bound trion. In this model, the electron is coupled effectively to Nt = At/Ax trion states. ............ Level scheme with trion and exciton resonances and the chosen sign convention for the detuning. The ground state with an electron (blue dot) in the conduction band as well as th excited state after the creation of an electron hole (green dot) pair is depicted on the right. ..... Level shifts due to the optical Stark shift. The colouring of the level shifts indicates, if the laser frequency w is chosen below (red) or above (blue) the trion resonance. The presence of the trion shift causes a lower collective exciton shift, but an additional trion shift. ...... Feynman-like diagrams for diagonal contributions ............ 34 42 47 51 57 61 63 67 70 72 77 87 3.2 3.3 3.4 3.5 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 Feynman-like diagrams for off-diagonal contributions. ......... Plot of the function |I.. (k)|2: The variational parameters for 60/13 were used. The value for Q corresponds to laser light tuned close to the trion resonance at an angle of 30° with respect to the 2-axis. Memory functions mt (x) for GaAs and CdTe. Note that the natural length scale, the 3D donor Bohr radius (1.1), is slightly different for the two systems ................................. Memory functions mC (x) for GaAs and CdTe. Note that the natural length scale, the 3D donor Bohr radius 0.1), is slightly different for the two systems ................................. Predictions of the toy model and the effective Hamiltonian approach for the detuning dependent correction factors. ............. Memory function 7714(x) (solid lines) in the Q = 0 limit and fitted Gaussians g(.2:) (dashed lines): GaAs is displayed in blue, CdTe in orange. .................................. Memory function mt (x) for a finite value of the laser photon momen— tum Q projected onto the QW plane. We used the parameters for GaAs. ................................... Enhancement factor X as a function of the electron hole massratio 0'. Potential depth V0 for red (At > 0) and blue (At < 0) detuning in terms of the trion binding energy ET for various values of the saturation parameter c. ............................... Time averaged probability of formation of real excitons per excitorrcell as a function of the detuning from the trion resonance At for various values of the saturation parameter c. .................. The first three levels of the harmonic oscillator are compared to the potential depth (dashed black line) for CdTe and blue detuning: The blue branch shows the ground state level for the angles 6 = 20° (dark blue), 0 = 15° (blue), 6’ = 10° (lighter blue). Similarly, the first excited (green) and the second excited level (orange) are depicted. ...... Spatial spread AX of the harmonic oscillator ground state as a function of the detuning for a fixed saturation parameter 6 = 0.1 in CdTe and various values for the angle 6. ...................... xi 88 90 96 97 103 106 120 121 4.9 4.10 4.11 4.12 5.1 5.2 5.3 6.1 Trapping frequencies VT compared to the effective spontaneous emission rate for CdTe. 6 was chosen to be 30°. ................. Single photon recoil energy E R compared to the harmonic oscillator level spacing fiwho as a function of the detuning for a fixed saturation parameter c = 0.1 in CdTe and various values for the angle 6. Variational parameters (1* and 13* in the harmonic oscillator approxi- mation including the memory effect for e = 0.1 and red detuning. The 124 results for blue detuning (not presented here) are qualitatively the same. 126 am of the gaussian wave packet in the :E-direction as a function of the detuning At for 6 = 0.1 and CdTe. The results where the nonlocal effect of the potential was included are shown as solid lines: 6 = 10° (blue), 6 = 15° (green), 6 = 20° (orange). The corresponding values of the local limit where the non-locality was neglected are depicted as dashed lines. The non—locality causes a slight additional spread of the electron wavepacket. ........................... Acoustical phonon scattering at a lattice temperature T, = 0: Depen- dence of the intrasubband relaxation time 7'1_,1 on the quantum-well thickness L z. ............................... GaAs: The phonon cooling rate RP}, (E) is shown in blue. The heating rate Rheat is presented in orange for different values of the saturation parameter c: c = 0.1 (solid line), 6 = 0.05 (dashed line), 6 = 0.01 (dotted line). The thermal equilibrium energies can be found where the cooling rate intersects with the heating rate. ........... CdTe: The phonon cooling rate RP}, (E) is shown in blue. The heating rate Rheat is presented in orange for different values of the saturation parameter c: e = 0.1 (solid line), 6 = 0.05 (dashed line), 6 = 0.01 (dotted line). The thermal equilibrium energies can be found where the cooling rate intersects with the heating rate. ........... On-site repulsion U for CdTe for a square lattice with saturation pa— rameter c = 0.1. Singlet states (m = 0) and triplet states (m = 1) are shown in blue and orange, respectively, for angles 6 = 15° (lower branch) and 6 = 30° (upper branch). .................. xii 153 6.2 6.3 6.4 7.1 7.2 C.1 D.1 D.2 Coulomb interaction energies between electrons at different sites of the square lattice for the first six neighbours in CdTe. The angle 6 ranges from 7r/36 5 5° to 7r/6 E 30°. Note that, according to Tab. (4.2), the potential depth V0 can reach values ~ 1.6meV for blue detuning in CdTe, which is considerably larger than VC. ............. Spin selectivity of the optical potential: The trion system appears as a double two-level system. Electron and hole spins are depicted in blue and red, respectively. ........................... Cartoon for the effect of the spin-selective potentials Vi. They overlap for 45 = 0. As ab is increased adiabatically, the two potentials are pulled in two opposite directions. As a consequence, the two spin states of the electron move into opposite directions, and the electron wavepacket is split apart into its two contributions. .................. Bandstructures of GaAs and CdTe, taken from [29] ........... Scheme for a bilayer semiconductor QW system where electrons and holes are trapped, but short range dipole-dipole interactions are real- ized. Electrons are depicted as blue dots, holes as greens dots. Indirect 155 157 158 164 excitons X are composed of electrons and holes from different QW layers. 173 Possible set of coordinates to describe a three body problem in two dimensions. ................................ Numerical results for variational parameters a, ," , ’7’ in atomic units. . 187 Numerical results for variational parameters C1 and C2 in atomic units. 189 xiii IntrOduction Laser induced optical trapping and manipulation of neutral particles have played key roles to the advancement of modern atomic physics and biophysics [1, 2]. The realiza- tion of optical traps has enabled numerous fascinating experiments: in biology optical traps, also known as optical tweezers. have led to probing the mechanical properties of DNA [2]. In atomic physics. optical cooling and trapping has become a flourishing technology paving the way to remarkable advances in precision measurement, quan- tum information and addressing fundamental questions in condensed matter physics. In particular optical lattices, a regular array of microtraps for atoms generated by a standing wave laser field, have made important contributions to a deeper under- standing of fundamental Solid State questions and serve as a platform for various promising candidates in quantum information schemes. One exciting outgrowth of this general technique is for example the experimental implementation of a quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms [44] and, very recently, Anderson localization in a disordered optical lattice has been investigated experimentally [3, 4] A the same time, semiconductor nanostructures continued to attract significant interest due to the technological progress in their fabrication. Some of the most peculiar features of semiconductors become manifest in their optical properties. De- spite advances in the optical and electronic control in semiconductors [93. 94], a laser induced method to trap and manipulate charged carriers has not yet been explored. In this thesis, we will merge for the first time the expertise of these two blossom- ing fields, semiconductor heterostructure physics and cold atom physics, to propose the laser-induced trapping of a new system: an electron gas confined within a two- dimensional quantum well (QW). In analogy to optical potentials for atoms, an AC Stark shift is generated for the conduction electrons by driving the trion resonance with intense, detuned laser light. Trions are negatively (X‘) or positively charged (X +) excitons that have been observed as an important spectral signature of QW’s in which excess carriers are introduced by modulation doping [16, 17] or optical excita— tion [19]. In essence, this novel trion—mediated potential for carriers in semiconductors originates from the following mechanism: The trion resonance modifies the exciton resonance frequency in the vicinity of a carrier. Accordingly, the Stark energy of the complete system is modified in proportion to the light intensity at the carrier loca- tion. The dependence of the total energy on the carrier’s position serves as a source of mechanical potential energy for the carrier. The thesis is structured as follows: In the first Chapter we will review the core concepts of excitations in semiconductors. We will present a quantum mechanical treatment of excitons, the dominant excitations in undoped semiconductors. Readers familiar with the concept of excitons can skip this introductory part. In the next step, we will obtain a Hyllereaas—type wavefunction based on Ritz’s variational technique to accurately describe the trion state. We will verify that trions are stable against dissociation into an exciton and a free excess carrier for every possible electron-hole mass ratio. Our results for the trion binding energies are in a good agreement to experimental values and confirmed by previous theoretical investigations [28, 22]. In addition, we will underline the characteristics of the coupling to light for excitons and trions. While excitons display themselves as a coherent excitation over the whole sample, the trion oscillator strength shows a very different behaviour. In a last step, we will focus on the radiative properties of excitons and trions by calculating their 2 radiative transition rates and lifetimes, respectively. The second Chapter gives an introduction to the quantum optical foundation of this thesis. We will review the underlying mechanism of laser-induced dipole poten- tials for atoms, before we turn to our first theoretical treatment of the trion-mediated optical trapping potential for carriers. Here, we have modelled the QW sample as a collection of two—level systems. The collection consists of exciton-sized cells that can be excited by the photocreation of an exciton or of a trion state if a carrier is in the vicinity of this cell. We will find that the effective polarizibility of the background due to the exciton resonance marks a major distinguishing feature of the trionic po— tential. It leads to a correction factor to the pure trion contribution and gives a very different scaling behaviour of the potential depth with the detuning, compared to atomic optical potentials. Chapter 3 is devoted to a derivation of an effective Hamiltonian and the corre- sponding Schréidinger equation for a single electron that is coupled to virtually excited bound and unbound trion states. The features embodied in the effective Schrtidinger equation are discussed in the next chapter. In Chapter 4 we will report on the results of our investigations. In particular, the trion-mediated potential displays a non-local character that is a consequence of the light effective mass of the electron m: and the ’quivering’ of the electron motion once it has virtually mixed with the extended trion state. This non-locality occurs on a lengthscale of the order of the trion size (N 30-nm). We have been able to show that this effect can be neglected in the analysis of the electron trapping at the nodes or antinodes of the intensity pattern, because the trion size is small compared to the periodicity of the potential. In addition, we will show that the extended size of the excited trion level accounts for an effective enhancement factor X for the potential depth of almost two orders of magnitude with respect to typical excitonic Stark shifts. The quantity X has been intuitively linked to the integral over all possible bound trion 3 configurations having one electron and one hole ’on top of each other’. More simply, X can be viewed as the number of excitons that fit into the trion size without spatial overlap. Also, we will encounter and analyze the correction factor that first appeared in the framework of the previous cell model in Chapter 2. The results of the toy model and the effective Hamiltonian are in a very good agreement, qualitatively and quantitatively: the correction is smaller than one for red detuning and therefore accounts for a decrease in the potential depth. However, for blue detuning, it can give a strong enhancement to the potential depth. We will show that this novel type of potential can be deep compared to the single photon recoil energy E R and the equilibrium temperature of the electrons. It also benefits from an omnipresent cooling mechanism in the semiconductor environment: the phonon bath. Chapter 5 is devoted to an analysis of the phonon bath. We will calculate the effective equilibrium temperature of the electrons that arises from the competition of the interaction with the phonons and the heating due to spontaneous emission of photons. According to our results, it even seems feasible to obtain a potential depth that exceeds both the recoil energy and the thermal energy of the carriers by a factor of 10. Therefore, we predict the possibility of confining carriers embedded in a QW system on a lengthscale of optical wavelengths. In Chapter 6 we will extend our model to the idea of a spin-selective electron lattice which is much simpler to engineer and control than similar lattices for atoms. A possible setup for the experimental investigation of this system is depicted in Fig. (1). The strong Coulombic interactions between the trapped particles give rise to a very strong on-site repulsion and, in contrast. to the atomic counterpart, even the interaction between different sites of the lattice is not negligible. The combination of a periodic potential whose properties can be tuned in real time and which is strong enough to trap carriers at. a single site, but weak compared to the inter-particle interaction paves the way for intriguing new experiments not feasible in conventionz—il solid state physics. The level structure of our system allows for a directive mapping of the spin of the localized electron and the optical polarization of the photon that can couple to the trion state. Owing to this scheme. two different potentials for the two possible spin configurations might be created. This property may open up the way to important applications in quantum information processing. electron lattice /GaAs / AIGaAs 2DEG / Figure 1: Schematic of optically—confined electron lattice. The 2DEG sample already has electrons confined to an .r — y plane below the sample surface. Four incident lasers achieve further confinement using the AC Stark shift effect. For clarity, here the lasers are shown in the m-direction only: an identical pattern of light will be applied in the y-direction, too. The charge modulation can be probed based on a STM technique. We will focus on small side effects in Chapter 7. In summary, we will apply recently developed concepts from AMO physics to gen— erate new, interesting ideas for semiconductor systems. Owing to the specific proper- ties of these systems, like the background polarizability, the light particle masses, and the strong inter—particle interactions, semiconductor optical potentials are very differ- ent from the conventional optical trapping potentials in AMO systems. The outlook of optically manipulating the motion of electrons seems very promising for at least two purposes: we may expect a deeper understanding of fundamental models like the Hubbard model that are well established in the solid state community. The system proposed in this thesis stimulates further investigations, since this fermionic many- particle quantum state on a regular lattice possibly bridges the gap between current 5 ultracold atom systems and fundamental concepts in condensed matter physics. A unique dynamic control over all the relevant parameters of the optical lattice allows for experiments not feasible in condensed matter physics. In contrast to ultracold atom systems, strong long range interactions in a true solid state environment can be studied. In addition, the use of the trapped electrons as qubits in quantum in- formation systems may give rise to applications, well beyond the scope of this thesis. Owing to its spin-selectivity, electron spins may be addressed individually and used to create entanglement. In essence, our system might enable the implementation of the pioneering idea of real-time quantum simulation by dynamically simulating one complex quantum system with another. The thesis ends with a discussion of possible applications of this novel system and identifies possible future directions of research that go beyond the investigations presented in this thesis. Chapter 1 Excitations in semiconductors Typically, the optical spectra of undoped semiconductor QWs are dominated by exci- tons; these are optically excited bound electron-hole pairs that follow the absorption of photons in a semiconductor and result from the binding of the negatively charged electron of the conduction band with the positively charged hole left behind in the valence band. In the case of a small excess density of carriers, an exciton can capture an additional electron to form a charged exciton, which is the bound state of two electrons and a hole or two holes and one electron, depending on the type of dop- ing. The three—body structure of these charged excitons is the origin of its common name in the literature: trions [5]. The first theoretical prediction of the existence of trions by Lampert has celebrated its 50th anniversary in 2008 [6]. In best-quality samples, trions exist as free particles and thus carry original properties as mobile charged quasiparticles [7]. Just as excitons, trions can decay radiatively when one electron and one hole recombine emitting a photon. While the exciton problem can be mapped onto the hydrogen problem and is therefore analytically solvable, the anal- ysis of trions from a theoretical point of view bears at least two major difficulties: (i) As trions represent the eigenstates of two electrons and one hole - or viceversa - in the presence of Coulomb interaction, their energies and their wavefunctions are 7 not analytically known; (ii) since trions are bound objects, they cannot be described using from a (finite) perturbative approach [8]. In this introductory chapter, we will review excitons as the elementary excitations of undoped semiconductors and analytically derive their binding energies and wave- functions for the two dimensional limit. Based on a variational approach, we will do the same for trions and show that they are st able against dissociation for any possible electron-hole mass ratio. The binding energies we obtain are in a reasonable agree- ment with experimental values. We will present our numerical results for the two cases of GaAs/Ga.1-IAlIAs and CdTe/Cd1_IZnJ-Te QW systems. The material parameters we used are summarized in Appendix A. Using the so found wavefunc- tions, we will analyze the coupling of excess electrons to bound and unbound trions, induced by a standing electromagnetic wave. The optical matrix elements are a key ingredient for the work presented in this thesis and will be discussed in section 1.3. In the last section, we will focus on the radiative character of excitons and. trions by computing their decay rates and radiative lifetimes. 1.1 Excitons in quantum wells 1.1.1 Wannier-Mott theory The optical properties of semiconductors are strongly related to interband transitions between valence and conduction bands. Excitons are electronic crystal excitations that have become an indispensable concept in the understanding of these interband transitions. In principle, the term exciton has been coined to signify the modification of the absorption spectrum of photons because of the Coulomb interactions between the conduction band electron and the valence band hole. A very intuitive explanation for the exciton concept can be given based on the schematized configuration in Fig. (1.1). We assume a direct semiconductor, i.e. the conduction band minimum and the 8 valence band maximum are at the same point in k—space. The zero energy is chosen to be at the top of the valence band. Therefore, the bottom of the conduction band lies at the bandgap energy cc. Figure 1.1: Optical transition between the full valence band (VB) and the empty conduction band (CB) for a direct transition. Initially the system is assumed to be in its ground state, which means that the valence band is filled, and all conduction band states are empty. Optical transitions are induced by photons. For photons in the optical frequency range, the wavevector Q is small enough to be neglected, so that the photon is schematized as a vertical arrow. The photon promotes a valence band electron to an unoccupied state in the 9 conduction band. For simplicity we assume quadratic dispersion relations for both the valence and the conduction band, so that the single electron energies are h2k2 h2k2 613(k) 2' —--277;:-, CC (k) :- 6C + E, (1.1) where m]; and m; are the effective masses of the valence and conductions electrons, respectively. In 1937, Wannier was the first to interpret the interband transitions in semiconductors as a two-particle process [9]: as all the valence band states are completely filled, the removal of an electron from this band is accompanied by the creation of an excitation termed a. hole. The hole represents the absence of an electron in an otherwise filled band. It can be treated as a particle with an effective mass m2, positive charge 6 and energy 6,, (k). In the two-particle picture, the absorption of a photon of energy ha) in the semiconductor entails two excitations: one is the electron in the conduction band of wavevector k and energy cc (k) and the second one is the hole in the valence band of wavevector —k and energy 6h (k) = 712132 / 2771:. Accordingly, conservation of energy for this process implies h2k2 h2k2 2m; 2m; ' In the framework of this two-particle picture additional physics arise that are con- cealed in a one-particle approach. The essential idea is that the electron and the hole are particles with opposite charges. Therefore, there is a Coulomb attraction —e2/cr between them. Here, 6 is the static dielectric constant due to the effective screening of all not explicitly included charges (electrons and ions). The frequency dependence of e is neglected. The optical absorption rate for this transition was first calculated by Elliott in 1957 [10]. The final state of the system is described by a two—particle Schriidinger 10 equation 2V2 2 V2 2 h e h h e _ it 27718 [(1) (119,172) = Ed) (re,rh). (1.3) * _ 2mh 6 [re — rhl The problem can be transformed into relative r 2 re — rh and center of mass coor- dinates R in the standard way to obtain the reflective mass equation for excitons in real space representation —— — 9——“ — 7:] (I) (R, r) = E (R, r). (1.4) Here, [VI = m: + m]; is the total mass and pm 2 mZ‘mZ/ (in: + 771.71) is the reduced mass of the electron-hole pair. The center of mass motion is plane-wave like. In optical experiments the corresponding wavevector K equals the photon wave vector, which is, however, small. The center of mass motion can be separated as 4’ (R,r) = ‘1’ (R) 975(1‘) (1-5) and the Coulomb interaction essentially applies only to the envelope part of the wavefunction [_h2v¥_fi],,(,) = Erase) <1-6> 2pm er E : 60+ Err'l‘ —. (1.7) This is an intriguing and remarkable result: the exciton problem can be mapped in one-to-one correspondence to the well—known hydrogen problem. Excitons appear as quasiparticles in a solid that are the hydrogenic bound state solutions of an excited electron in the conduction band and the remaining hole in the valence band. Eqn. (1.6) is known as the Wannier equation [12]. At this point, we mention that the effective mass equation can be derived from 11 first principles, taking into account the full electron-electron interaction [11]. Based on a Green’s function formalism, Eqn. (1.4) can be obtained in the lowest order approximation to the effective electron-hole interaction. Moreover, Elliott showed that the optical transition rate A(w) depends on the relative wavefunction, evaluated at r = 0. The transition rate for the absorption of photons is 2 - At») = hipped: [463- (0)]25(m—ec—ej), (1.8) 2‘ where the summation over j includes both bound (cj < 0) as well as unbound (63- > 0) states of the relative motion of the electron-hole pair. The dependence on the relative wavefunction, taken at r = 0, can be understood by very intuitive arguments. Let us consider the situation of an emission process whereby the electron and the hole recombine to emit a photon of energy fun. Since the relative wavefunction governs the probability to find the electron and the hole at the same spot in the solid, the necessary condition for radiative recombination, it is evident that the emission rate should depend on the relative wavefunction at r = 0. Recombination occurs when the electron and the hole are “on top” of each other, consistent with physical intuition. Since the matrix elements for absorption and emission processes are identical by time reversal, the absorption rate must have the same dependence. Indeed, typical absorp- tion spectra of a direct gap semiconductor show sharp distinct exciton resonances well below the bandgap energy corresponding to the 13, 2s, . . . solutions of the Hydrogen problem. 1 .1.2 Confined Excitons Up to now, our review of excitonic excitations in semiconductors has been general. In the following we will focus on confined semiconductor structures, which attract significant interest due to their applications in submicrometer technology. The tech- 12 nological advances in crystal growth techniques have made possible the fabrication of various types of semiconductor heterostructures whose characteristic dimensions, typically a few nanometers, become comparable to the free carriers DeBroglie wave— length. In this regime the electronic and optical properties can deviate substantially from those of bulk materials, because the effects of spatial confinement become ap- preciable and restrict the electron and hole mobility to a reduced dimensionality. The energetically low lying electron and hole states are confined in one or more directions within a region of length LC. Since LC is still larger than the lattice constant but small enough to cause a quantization of the carriers envelope wave functions, structures of this size are called mesoscopic [12]. The most prominent examples of such. mesoscopic semiconductor structures are quantum wells (QW), where confinement along one spatial direction occurs due to the variation of the bandgap energy from one material to another. The translational motion in the plane perpendicular to the confinement direction is still unrestricted. Such a quantum well , e.g. for the III-IV compound semiconductor GaAs, can be realized by molecular beam epitaxy to sandwhich several GaAs layers in between layers of a material with a wider band gap, typically GamAl1_xAs, with 0 < :1: < 0.4. For substantially larger Al concentrations, the barrier becomes an indirect-gap semiconductor [12]. The barrier height, i.e. the strength of the confinement, is determined by the difference in the bandgap energies of the two materials involved. Quantum wells are currently the best studied quantum confined structure and they can serve as a paradigm for others. The system we study is also based on a quantum well structure, so that for the rest of this thesis we will refer to a quasi two dimensional or pure two dimensional system. For completeness, however, we mention the possibility to fabricate quantum wires, where the free electronic motion is confined in two dimensions and even quantum dots, where confinement exists in all three space dimensions. 13 Our next imminent goal is an accurate description of \N’annier excitons, confined in a QW structure. In QVV structures, excitons were first observed in an absorption experiment in 1974 [13], where up to eight resolved exciton transitions have been measured. For simplicity, we consider the pure two-dimensional limit, corresponding to a QW with a very thin QW width. In Appendix B we give a detailed treatment of excitons in two-dimensional structures. The two-dimensional exciton bound state energies are found to be 1 E, =—E ———., n=0,1,..., 1.9 n ()(71+I/2)2 ( ) where E0 serves as the energy unit ft? e2 82,111- E : ‘ Z Z —.—' 1.10 0 2.1110(2) 2600 262132 ( ) and a0 = 652 / [1:562 is the natural lengthscale. In principle, these are the bound state energies for the 2D hydrogen problem. Quite remarkably, the binding energy of the ground state exciton is 4E0, which is four times the bulk value. Intuitively, this increase in the binding energy due to the reduced dimensionality can be understood if one considers quantum well structures with decreasing width. Since the admixture of p-wave functions is energetically unfavourable, the wave function tries to conserve its spherical symmetry as much as possible [12]. Therefore, confinement parallel to the quantum wells induces a decrease in the Bohr radius perpendicular to the wells: the oppositely charged electron and hole approach each other, resulting in a higher binding energy. By reducing the dimensionality of the system, the binding energy is considerably increased, which relaxes the requirements for an experimental observation. This is one of the reasons for the remarkable theoretical and experimental interest in systems of reduced dimensionality. The lowest lying exciton state, the 13 exciton wavefunction will be of peculiar l4 interest, so that we explicitly state its wavefunction a. (r) = m (r) = i-gexm-Qr/aol- (1.11) 0 The exciton radius can be deduced from the exponential term of the relative wave— function. Indeed, in the 13 ground state of the two dimensional limit the exciton radius ax is only half the Bohr radius, [i.e. ax = a0 / 2, whilst the bulk value for the exciton radius is simply the Bohr radius do. The reduction in dimensionality from bulk to a purely two dimensional system causes the exciton radius to shrink to half of its original value. For a more accurate description of excitons in quantum well structures one has to take into account a finite well width and a subsequent finite extension of the exciton’s wavefunction in the direction perpendicular to the well width. The enhancement factor, which is simply four in the 2D limit, for real QWs is then a function of both the well width and the barrier height. At a first glance, one might be led to think that this enhancement factor of four is an upper threshold for excitons that are confined in real quantum well structures. However, accurate theories that have taken into account various effects as valence-band mixing, nonparabolicity of the bulk conduction band, the difference in the dielectric constants between well and barrier materials and the Coulomb coupling between excitons belonging to different subbands, predict very high binding energies, particularly for very narrow quantum wells, that can even exceed the two—dimensional limit of four times the bulk Rydberg [14]. 1.2 Trions in quantum wells In the quantum-mechanical description of excitons, we encountered a striking analogy between the exciton problem and the conventional hydrogen atom. Pushing this 15 analogy a step further, Lampert envisaged the existence of a class of mobile charged excitons, in analogy to the negative hydrogen ion H _ and the positively charged hydrogen molecule H2+ [6] . It can be regarded as a major breakthrough in our understanding of the optical response of doped semiconductor quantum wells. Indeed his theoretical predictions were demonstrated experimentally both in II-VI and III-V semiconductors [15, 16, 17]. Mobile charged excitations had been identified via their optical signature in quantum wells as the negatively X‘ and positively charged X + exciton, commonly referred to as trions. A trion is a charged bound three-particle complex, consisting of two electrons and one hole (eeh) or two holes and one electron (ehh), interacting via the Coulomb potential. When viewing the trion as a carrier bound to an exciton, it is immediately evident that, as the exciton is neutral, its attraction must be relatively weak which results in a comparatively small binding energy. This is why the unambiguous observation of a spectral line due to the charged excitons did not follow until the advent of high quality lightly doped QW structures where the binding energies are enhanced by approximately an order of magnitude relative to bulk [5]. The possibility of observing trions is tightly linked to the excess density of car- riers in the QW. The presence of free excess carriers in the sample is necessary to photocreate trions, whereas excitons can be created just by photoabsorption. This explains why trions appear in the absorption and emission spectra of QWs containing small excess carrier densities only. The most common way to realize an excess carrier density in the QVV is by modulation doping. The first observation of trions was also based on this technique [15]. It works as follows: The doping is introduced during the growth process of the sample, but only within a specific region of the barrier mate— rial. In order to hold the Fermi energy level constant throughout the doped sample, the donors tend to ionize. As a consequence, the freed carriers will migrate to the lower energy QW. Therefore. the two-dimensional carrier gas exists inside the QVV, 16 spatially separated from the donor layer. In contrast to bulk materials, the ionized donors do not affect the mobility of the electron gas. Subsequently, this technique allows for for high carrier mobilities [20]. The exciton and trion photoluminescence (PL) spectra in QWS have been investi- gated as a function of the doping [18]: starting from low doping, where the trion line is hardly seen, the excess carrier density has been progressively increased leading to a stronger signal of the trion line. The intensity of the exciton line decreases to the profit of the trion line by increasing the carrier density. The energy between these two lines defines the trion binding energy: it is the energy necessary to dissociate a trion into a neutral exciton and an excess carrier. It should be mentioned, however, that excitonic effects are quenched due to screening and phase space filling for carrier concentrations sufficiently high [21]. 1.2. 1 Hamiltonian We explicitly consider a negatively charged trion X _, consisting of two electrons and one hole, which we assume to be described by their effective isotropic masses m: and m2. We restrict ourselves to the limit of a purely two dimensional semiconductor, therefore omitting the form factors due to the finite extension of the electron’s and hole’s QW wavefunctions along the QW growth directionl. We neglect the electron- hole exchange interaction . Within the envelope—function approximation, the effective- mass Hamiltonian HT of the three-particle system reads Hr = T + Vc + 26c, (1.12) 1The growth direction is taken to be the é-direction throughout this thesis. 17 where cc is the energy at the bottom of the conduction band, while the top of the valence band was set to zero. T is the kinetic: energy operator A) 6226‘2’322 T: __ _ _ _ _——V , 1.13 2771: 1 2m; 2 2772.}: h ( ) The electrons and the hole carry a charge 6 and therefore the interactions are governed by mutual Coulombic potentials 9 e“ 1 1 1 "c = __( + -_ _), (1.14) 7‘1h T2h 7'12 where we introduced the mutual distances between the three particles r1 h, r2 h and fig in a self-explaining notation. To be accurate, ’HT should also include a contribution originating from image charges owing to the discontinuity in the dielectric constant at the interfaces of the well. Usually, however, the dielectric constants of the two materials involved are very similar and this effect can be simply modelized by taking into account an effective dielectric constant e, which is assumed to be the same in the two materials [22]. The total evelope energy is governed by the Hamiltonian H“)t : HT — 266. (1.15) we note that the Hamiltonian H W commutes with both the in-plane momentum operator P and the projection £z of the total angular momentum operator along the i-axis. Therefore the center of mass motion may be separated HtOt =HT+—. (1.16) Here, HT describes the ’relative’ envelope Hamiltonian of the trion complex. Since 18 this part of HT accounts for the binding energy, it is of peculiar interest. The total envelope energy is then given by 13%? . 1.17 2mt ' < ) EtOt : Eégfi + with E§9t being the ’relative’ envelope energy and K being the in-plane wavevector of the center of mass motion; mt = 2m; + m}: is the total effective trion mass. The stability of the trion complex X‘ against dissociation into an exciton and a free electron is ensured, if Eff” is smaller than EX, the exciton ground state energy: It has to be energetically favourable to bind the second electron to the exciton. Therefore, we can write the stability criterion for the trion as __ tot ET—EX-ET >0, (1.18) where ET is defined as the actual trion binding energy. In this notation it is defined as a positive quantity for a stable bound three particle object. In what follows, we restrict our considerations to the most stable state, i.e. a symmetric orbital envelope wave function with a zero angular momentum projection along the 2-axis. As a direct consequence of this simplification, the ’relative’ envelope wave function gob depends solely on the mutual interparticle distance 71h» r2}, and 7‘12. In this spirit, the total envelope wave function for the trion factorizes into a center of mass, the relative motion and the spin part 1 , 1 ‘Pr = — exp (KB) 3% (T1}z~r2lzaT12) X565 . (91:: 82:» Shzla (1-19) x/Z 27r h~ where R gives the center of mass position, A is the area of the sample and the factor 1 / \/ 271' accounts for the normalization of the overall angle degree of freedom. The fact that we only consider wavefunctions with a zero angular momentum projection 19 along the 2~axis m. = 0 gives rise to the simplification exp (im6) ——> 1, with 6 being the angle that does not affect the mutual distances between the particles. Se and S h z describe the spin state; the variables 81;, 32,: and Shz represent the spin components along the growth direction 2. we only consider heavy holes which are the the highest confined valence band in typical QW systems with a heavy—hole band angular momentum projection Shz E {i3/2}. The spins of the two electrons can either form an antisymmetric nondegenerate singlet state of zero total spin Se 2 0 or a symmetric triply degenerate state (triplet) with total spin Se 2 1. In total, heavy hole trions have eight spin states. To fulfill the Pauli exclusion principle for a symmetric relative motion wavefunction gob, the spin state XScShz has to be an anti- symmetric singlet state. Since we are interested in finding the trion ground state, we will only consider singlet trion states which are twofold degenerate because of the additional spin of the heavy hole. Bearing this in mind, the spin part of the trion wavefunction X5835)”, will be dropped in the following. The specific picture of a negatively charged singlet trion that we have in mind for the rest of this thesis is schematized in Fig. (1.2). This assumption is well justified since the X — triplet state is unbound in zero magnetic field and therefore it has been only observed at finite magnetic fields [17]. 1.2.2 Variational solution As stated earlier, .the theoretical treatment of trions is difficult. It is a few body problem with strong Coulombic interactions. However, we are interested only in the ground state, and we will use variational techniques. The Ritz procedure that has been successfully applied by Hylleraas to the similar problem of the He ground state in the early days of quantum mechanics [23]. In general, the starting point of this 20 Figure 1.2: A schematic X _ trion singlet. procedure is the variational principle ‘I’IHI‘I’l E[\II] = (H [II']) = ( (‘I’l‘yl ——> min. (1.20) Where H as usual denotes the Hamilton operator of the system of interest. One then has to make an “educated” guess for the general form of the eigenfunction \II. To begin with, its general form is assumed, but a number of parameters are left arbitrary. In the next step, the necessary integrals are carried out, so that E becomes a function of the parameters which have been introduced previously. By finding the absolute minimum of this function, the parameters, thereby the ground state eigenfunction, and above all, the energy of the system are determined. Subsequently, two major aspects are essential for a successful application and accurate results in this variational method: 21 first, a choice of the trial wave function suitable for the problem at hand and, second, a number of variational parameters commensurate with a reasonable handling of the calculation and optimization process [24]. Striving for a class of trial wavefunctions to handle two-electron problems that explicitly take into account correlation effects without having to deal with cumber- some angles, Hylleraas was led to introduce only metric distances that have a direct physical interpretation as variables. They are expressed in terms of the three distance coordinates, the two “elliptic coordinates” plus the inter-electron distance which are related to the mutual in—plane distances as S = 7‘1h. + T212,» t: 7‘11: — 72h, ‘u = 7‘12- (121) The variables 3 and u are positive by definition, whereas the variable t can take both positive and negative values. Owing to the Pauli exclusion principle, \11 can either be an even or an odd function of the variable t. Since H is necessarily an even function of t and since the integrals in the variational problem contain two factors, the contributions to the integral from -—t has to be the same as that from +t. Therefore, it is a legitimate simplification to restrict ourselves to positive values of t in the integrals and multiply the volume element by a factor of 2. The volume element that has to be respected when performing the necessary integrals is derived in detail in Appendix C. The transformation from Cartesian coordinates to the coordinate set {R, 6, s, t, u} it is given by 11(32 — t2) 2\/(s2 — 112) (112 — it?) 22 d2r1 (1er d217, __. d2RdQ as (It du (1.22) with the limits of integration for the Hylleraas coordinates .920. 031133, Ogtgu. (1.23) Ever since the introduction of this class of coordinates, their use has proved to be an efficient tool in a number of variational studies [25]. In general, the trial wavefunction used to analyzed the ground—state properties of two-electron problems is assumed to have the form \IJH (3, 15,11) = Ne—ks/2 Z chmlean'sltm-u", (1.24) where k and and the coefficients chm," are to be determined variationally. N is the normalization constant. The exponential decay factor ensures convergence and more accurate results are achieved by a systematic increase of the expansion lengths. This class of Hylleraas type wavefunctions haven proven successful for three body problems as they allow for a diffusive character of the wavefunction but also manage to take into account two—body correlations. Radial and internal angular correlations effects are reasonably modeled by the third Hylleraas coordinate u = r12. Therefore, a Hylleraas type wavefunction is expected to lead to an accurate calculation of the ground state of the charged exciton. The next step is a transformation of the “relative” Hamiltonian HT that governs the trion binding energy by using the Hylleraas coordinates s, t, and U. The notation can be simplified by switching to effective atomic units: We will e2 as a measure for length and Eh = mife‘1 #2172 use the atomic units a D 2 6712/71); as energy unit; a D is the effective 3D neutral donor Bohr radius and E, twice the effective 3D neutral donor Rydberg. l\«'Ioreover we introduce the effective mass ratio a = ing/”m; as an important parameter to characterize the specific QVV system. 23 The “relative” Hamiltonian in Hylleraas coordinates is given by 2 2 2 HT - ea_ - 5% — .212 (Se-ta» 512 - is 23 (u2 — t2) 2 2t (32 — 112) 02 11(82 t2) 5" 11(32 — t2) t“ _20 (32—212 02 11.2—t2_6_2_ s 8 t 8) 82—t2832 32—96132 92 2.9 s2—t2t _li'li-‘l ‘ Iii-’I +5 “'25) which corresponds to the strictly two dimensional limit of the Hamiltonian given in [22]. The first two lines govern the kinetic energy of the two electrons, the third line stands for the kinetic energy of the hole and the last line represents the Coulombic potential energy. In order to describe the binding of the trion complex, we use a trial wavefunction which looks strikingly simple in Hylleraas coordinates. It is the so called “Dritte N aherung” gob (s, t, u) = Ne_°‘3 (1 +1311. + 712) (1.26) that is equipped with the three variational parameters a, ,6 and r)»: Necessarily, tpb(s,t,u) is an even function of t to describe a trion singlet. Its normalization constant N is given by 1 7t (2560/1 + 240M313 + 405mm + 115272 + 256a2 (3,132 + 27)) _ ___ 8 . (1.27) N2 4096a The actual values of the parameters a, 1'3 and 7 are determined by varying these parameters until the mean value of ’relative’ Hamiltonian (HT), which gives the ’relative’ ground state energy E’Ot, reaches its minimum. As a consequence, the 24 variational parameters must satisfy the conditions = , = ,. :0. (1.28) However, no matter what the result for the variational parameters, the chosen trial wavefunction cpb (s, t, u.) only fulfills its assigned job if it can explain the experimen- tally confirmed trion binding energy reasonably well- When applying the dissociation criterion stated in Eqn. (1.18), we need to apply our variational found result to the exciton ground state energy EX. We obtain the ground state energy for the exciton from Eqn. (1.9) in effective atomic units as a simple function of the effective mass ratio a given by 2 EX =—4E0= -1+0. (1.29) Our results are presented in Fig. (1.3). We can confirm that the trion complex is indeed stable against dissociation for any possible effective mass ratio 0. Since ET is always positive, it is evident that the binding is stable in all cases. Indeed, the binding of an excess electron to a “neutral” exciton appears to be energetically favourable. Moreover we present the trion binding energy ET measured relative to the exciton ground state binding energy ET in Fig. (1.4). Intriguingly, this ratio is rather constant over the whole effective mass spectrum and is approximately 10%. Proceeding from this result, we can suggest the following useful rule of thumb: the exciton problem being by far simpler than the trion physics, accurate numbers for the exciton binding energies are very well known for all common QW structures. According to our result, a first valid approximation for the trion binding energy in this QW system can be simply achieved by taking 10% of the exciton binding energy. This result also supports the idea, that the trion originating from the electron-exciton interaction is rather loosely bound compared to the exciton. In this spirit, the trion binding stems from a dipole—charge interaction, whereas the exciton binding energy 25 —1.0,: _ 1 _ l—T—TW' “T _f ‘ l —1.2l- . .3 —1.4? 3 .. 353$ —1.6} >2 _ L“ _1'8; Exciton ‘2-0; — Trion f —2.2} L_1__._x_4u.______.__..1_.___1_ . ____ 1 . a . _ . M . . ‘ 0.0 0.2 0.4 0.6 0.8 1.0 0'=m;/m,"; Figure 1.3: Ground state exciton binding energy and variationally optimized “rela- tive” trion energy in Hartree units. The trion is stable against dissociation for any possible massratio 0. comes from a pure attractive charge-charge interaction, which is essentially stronger than the dipole-charge interaction. The result shown in Fig. (1.3) proves the stability of the trion with respect to the dissociation to an exciton and a free electron. The binding energy is given in terms of the effective Hartree energy E), = mge‘f/ezhz, which is twice the effective 3D donor Rydberg energy. This quantity, however, strongly depends on the properties of the QW material. A heavier effective electron mass and particularly a smaller dielectric constant lead to a bigger Hartree energy, favouring an increased trion binding energy. Intuitively, it is evident that a smaller dielectric constant 6, equivalent to less screening and therefore stronger Coulomb forces, enhances the binding energy. We specify our results for the two typical QW systems GaAs and CdTe. They are summarized in Tab. (1.1). 26 0.115: 0.110) l . l 1:; 0.105 Z is : 0.100: 0.095» 1: 0.0 0.2 0.4 0.6 0.8 1.0 0'=rn;/m;‘, Figure 1.4: Variationally optimized trion binding energy ET measured relative to the exciton binding energy E X- ] GaAs CdTe ET/EX 0.11 0.10 ET[meV] 2.05 3.61 Table 1.1: Calculated trion binding energies for GaAs and CdTe. Although the relative trion binding energy is comparable in both systems, a much stronger absolute trion binding energy can be expected for CdTe, evidently owing to the fact that E}, is more than two times bigger in CdTe than in GaAs. Our results for the absolute trion binding energies in these two systems are in an excellent agreement to previous theoretical studies and experimentally measured values. In real finite size GaAs QWs the trion binding energy varies with the well width, from~ 1meV in Wide 30 nm wells up to ~ 2meV in 10mm QW [5]. Our theoretical value is closer to the latter, which is no suprise at all, as our calculation refers to the purely two dimensional QW. As far as CdTe is concerned, previous theoretical studies have 27 obtained the value ET 2 3.7meV and experimentally values of ET z 3meV were reported [22, 26]. As a side note, we mention that, by taking the limit a ——> 0, we recover the two dimensional H ‘ binding energy [27]. We have obtained an analytic expression for (HT (a, ,3, 7)), which is presented in Appendix D along with the results for the variational parameters a, ,3 and y. We have specified the variationally optimized values for the QVV structures GaAs and CdTe. Despite its simple form and the fact that our trial wavefunction depends on three variational parameters only, it obviously captures the essential physics and produces very good results. For our purposes, it represents a very good trade-off between simplicity and accuracy. Now that we have obtained wavefunctions to describe both ground state excitons and ground state trions, we are in the position to investigate several properties of these two different semiconductor excitations. The first goal will be a comparison of their typical spatial extensions. We have already considered the “size” of the exciton in a purely two dimensional QW; we found that its characteristic size or is aw = 00/2 = aD(1+ o) /2, where up is the natural lengthscale that we are using. Naturally, we expect the trion “size” to be more extended compared to the exciton, as its binding energy is about one order of magnitude smaller. To approximate the extension of the three—particle complex of a trion, we calculate the expectation value (.9) for the Hylleraas coordinate .9, based on the variationally found wavefunction 99b (s, t, u), and compare this result to ax. Qualitatively, but very intuitively we can also understand the “size” of the trion at from the following picture: If we interpret at as the characteristic length associated to the weak binding of the trion’s second electron, we can express at in terms of the 28 trion binding energy ET 2 it” (mg—1 + (m; + n2;)_1)/2at2. (1.30) Similarly we define the size of an exciton or via EX 2 71” (mg—1 + 7712—1) /2a§,. (1.31) Both approaches lead to similar results; they are summarized in Tab. 1.2. In the case of GaAs we used EX = 20 meV, ET = 2meV and for CdTe EX 2 40 meV and ET = 3.6meV, which are the results we obtained for a strictly two dimensional QW with the corresponding material properties. As expected the trion has a more diffusive character compared to an exciton, its typical ”size” being about one order of magnitude bigger than the excitons typical spatial extension. ] GaAs ] CdTe ((59/61)2 11.6 12.6 (at/6,.)2 9.7 9.9 Table 1.2: Typical trion “size” compared to exciton “size” in GaAs and CdTe. 1.3 Light-matter interaction: Optical matrix ele- ments In this section we focus on the coupling of light with a semiconductor quantum well by calculating the optical matrix elements. These matrix elements express the transition amplitudes between different semiconductor eigenstates. Physically, they are based on the absorption and emission of photons. To be specific, we consider the interaction of the semiconductor QW with a classical monochromatic standing wave 29 E (r, t) = E0 cos (Qr) cos (tat), where E0 is the amplitude of the electric field, Q the in—plane wavevector and w the frequency of the light. Therefore, the spectrum of photonic modes contains only two contributions, namely photons with the in-plane momenta :le. Before going into the detailed calculations, let us understand the underlying physics first. Fig. (1.5) draws a scheme of the possible processes that we are about to investigate. Let us consider the case first where the conduction band is initially empty. This situation refers to the upper half of Fig. (1.5). Under this condition, we can only expect to photoexcite excitons, but not trions. For the creation of excitons the initial state [2) is characterized by a photon of momentum Q, which corresponds to a plane wave. It can either excite a bound exciton X or an unbound electron-hole pair. However, the creation of an exciton is much more likely to happen, because the center of mass of the exciton is a plane wave as well, and therefore the matching between the initial and the final state is larger. The matching is much poorer in the second case, because one initial plane wave photon Q has to split up into two plane waves, one for the free electron with momentum kg = k + aeQ and one for the free hole with momentum with momentum kc = —k + ahQ. The QW is translationally invariant and therefore the center of mass momentum is always conserved. Hence, in the first case the exciton’s center of mass momentum is just the momentum of the incoming photon Q, while in the second case it splits between the two particles according to their masses, 08 = 1 — a), = 7712/ (m: + mi); k describes the relative motion momentum of the electron hole pair. N ow, let us turn to the situation where the conduction band is not initially empty, i.e. excess electrons are assumed to be already present in the sample. As explained, this can be easily achieved by the method of modulation doping. This situation refers to the lower half of Fig. ( 1.5). The semiconductor-photon interaction then 30 (Q1 (le 7* 4 (f ] O f fe=lf+aeé 8h O ‘ Ehz—E+ahé r1 in the last term, we can see that the triplet part indeed vanishes and we are left with . . 1 - . - - ‘I’l (1‘1) ‘Pi (1‘1) = 5 {‘1’}; (1‘1) ‘I’j (1‘1) — ‘I’l (1‘)) ‘I’j (11)}- (142) The initial state lie) describes a free excess electron, which is simply taken to be 35 a plane wave with momentum kg 6:91 (1‘) = 31% (1-43) normalized to the area of the sample A. The final state is a bound trion state with total wavefunction \I't (r1, r1, rh), that separates into three pieces 1 ’ 7726 (PT + r1)+ mhrh 1 . \Ilt (rT,rl,rh) = Til—exp [7K( 771T )] —\/2_—7—;,9b(s,t,u), (1.44) namely a center of mass motion with wavevector K, a rotational part that is simply in its ground state, and the “relative” part 9% (s, t, u) that accounts for the binding of this three particle object and was determined by our variational approach. Electrons and holes being fermions, we can use the anticommutation relations for fermions {x11 (r) , \Ill,(r')} = 5.93/5 (r — r’) , (1.45) to write the transition amplitude for this process in the general form (fl Hint lie) = d0 fdzrdQI‘I ‘1’? (r,r',r) <15) (r') E(r). (1.46) Although this is not the ultimate result and will be subject to a further evaluation, it already bears some physical insight for the problem at hand. It shows the physically intriguing property that the photoexcited electron—hole pair is created exactly on top of each other at r, exactly at the spot where the photon is destroyed, as the dependence on E(r) in the integral shows. Owing to this property, we can circumvent a cumbersome evaluation of the remaining integrals due to the relative wavefunction formulated in Hylleraas coordinates: For one electron and one hole at the same spot - the situation we face now - the Hylleraas coordinates boil down to the simple case 36 where they all take on the same value, namely: 3 = t = 'u = |r -— r’ I. The evaluation of the remaining integrals essentially becomes a two body problem between an electron * . fi . * * . . .. of mass me and an excrton of maSs 77L; = me + mh, so that it IS appropriate for a solution in center of mass R = (ml-r + er’) /mt and relative coordinates r — r’. In this fashion, we find that the transition amplitude can be written in the compact form (fl Hint lie) = 32 l0K,ke+QI+ (ke) + 0K,ke—Q1— (1%)] 1 (1-47) where we have defined the quantities Ii (ke) as 1+ (kg) = % [frexpweowannabe,r) (1.48) I- (k6) = \/—1§: /d2r exp (2 (—,136Q — flyke) r) 5011 (717‘, 7“) . (1.49) The first term in Eqn. (1.47) stands for the interaction with the +0, mode of the standing wave, whereas the second term stems from the interaction with the —Q mode. The Kronecker deltas take care of the in-plane momentum conservation: They simply require that the trion center of mass momentum has to match the sum of the photon momentum :le and the initial momentum of the electron kg. The essential physics, however, are buried into the expressions Ii (ke). They cover the intrinsic coupling strength of a single electron to a bound trion. We note that expression (1.48) is identical to the optical matrix element found in previous theoretical studies for the trion-related absorption at T = OK [30, 31]. Before we we give a detailed analysis of the expressions Ii (k8), let us first show that we can derive the same expression using a formulation of the problem in k—space, based on the interaction Hamiltonian in Eqn. (1.37). Again, we fix the spin of the initial electron to be in the spin down state 1, whereby the polarization of the light has to be 0'_. The initial state we are to consider contains an electron of momentum 37 kc and a photon of in-plane momentum Q. Therefore, we can represent the initial state as lie) 2 k3,,Q) .—_ cLe’iaEr )0) '_ (1.50) The energy of this state E,- is simply the sum of the energy of the free electron in the conduction band and the photon 12kg ’ fiw. 1.51 2771' + ( ) 3k 8 Eizfc‘j‘ The final state If) of the transition is assumed to be a singlet bound ground state trion. By already incorporating momentum conservation into the notation, we express the final, state in short as If) 2 Ike +Q)t1 (1.52) where the subscript denotes a bound trion state. The center of mass momentum of the trion ke + Q appears in the bracket. No further quantum numbers are needed to specify the bound trion state, since we restrict our analysis to singlet trions described by the wavefunction given in (1.44). The energy of the final state E f contains a bound trion, made of two electrons in the conduction band and one hole in the valence band. The necessary energy has been provided by the initial photon, so that E f takes on the form ’12 (kc + Q)2 _ th In k-space the information about the wavefunction enters explicitly via its Fourier transform. In general, a singlet bound trion state is twofold degenerate with respect to the spin of the heavy hole; with a fixed center of mass momentum q can be 38 represented in second quantization as _ ” T .T T , IQ)t — 1;]; W1 (k11k21q — k1 - k2) CR1,TCkgaldq—kl—kgiwfl) |0), (1.54) 1: 2 where \Ilt is the Fourier transform of total trion wavefunction \Ilt, defined by ~ ‘1’t(k1,k2,P) = 43flf(12md2r2d2rhe"i(klrl+k2r2fl’rh)‘11);(r1,r2,rh), (1.55) where p = q — k1 — k2. Using the anti-commutation relations for electron creation and annihilation operators {61(33, Ck’,$’} : 0, {Ck"s’c1(’,s’} = 68,8,6k,k, (1.56) and similarly for the hole creation and destruction operators, we obtain that the transition amplitude becomes (kt. Q) Him Ike + Q). = Aco‘memu (kg). (1.57) This reasoning was based on a fixed mode +Q, because we only considered the final state Ike + Q>t- Taking into account the presence of both the +Q and the —Q mode of the standing wave, we find <3 Therefore, we have proven the equivalence of the two approaches (see Eqn. (1.47)), Hint |K), = )‘c [5K,ke+Q1+ (ke) + 5K,ke—Q1— (ke)] - (1-58) if the light matter coupling constant AC is chosen to be 292132120 AC 2 2 39 The factor of 1/2 arises from the expansion of the standing wave profile cos (Qr) into its two mode contributions according to 90 cos (Qr) = % (eiQr + e‘iQr) . (1.60) We have shown, that both approaches arrive at the same result. After all, the result should be independent of the basis. However, the analysis in terms of both basis provides a better insight on the physics of the process. Let us pause for a moment and give a physical interpretation to the expressions Ii (kg) in Eqn. (1.48) and (1.49). The trion transition amplitude, also called the trion oscillator strength, appears in terms of the Fourier transform 1+ (k) of the relative motion part of the full trion wave function; the Fourier transform taken for the situation of photocreation - photons create one electron and one hole ‘on top’ of each other - and k equal to the relative motion momentum p, of the e-X pair made with the initial electron ke and the photocreated exciton Q. On top of this physically intriguing result, we have obtained analytic expressions for the quantities Ii (k6). For the ‘relative’ trial wavefunction, given in Eqn. (1.26), they read _ ' 01 lifieQ — ark)“ + 02 lifieQ <11?ka2 + 03 _ «m 2 2 7,2 . (a + liBeQ "' 5.1-kl ) Ii (k) (1.61) Here, C1, Cg and Cg are simply constants, determined by the variational parameters (1,8 and 7 via the expressions C2 = a- (a (26 +13) — 95,) (1.63) C3 = a3 (a2 + 203 + 6,) . (1.64) 40 In order to gain a better physical insight, we present plots of the functions Ii (k) in k-space in Fig. (1.7). To be specific, the parameters for GaAs were taken, but no actual differences occur for CdTe. we can see that both are strongly peaked around k z 0. The different shifts of the centers are negligible, because the photon wavevector Q is very small on this scale. While in real space the wavelength A of the laser is much bigger than the trion size at, about one to two orders of magnitude, in momentum space the relation of the associated quantities is reversed: The momentum of the photon Q = 27r//\ is negligibly small compared to l/at, which gives the appropriate scaling behaviour of the Fourier transform of the relative trion wavefunction. This is why the functions Ii (k) rapidly go to zero on a scale of w a N 1/at. The fact that we have analytic expressions for the trion oscillator strength will turn out to be a helpful tool for further calculations. 1.3.2 Coupling to unbound trions Going back to Fig. (1.5), we see that an electron is not only coupled to the bound trion state, but also unbound trions, meaning free electron — exciton pairs. The bound trion state is energetically more favourable, but the photoexcited exciton can only capture the excess electron if it is created close enough to the electron. A intuitive picture we will refer to later is that the electron and the exciton have to be within the typical trion size ~ at2 to be able to generate a bound trion state. Otherwise the exciton and the electron are essentially independent of each other and coexist in the semiconductor Q\N. We are going to give a closer look at the transition rate for this process involv- ing unbound electron-exciton pairs. To characterize the unbound trion states, we restrict ourselves to 13 excitons, which typically give the strongest peak in absorption measurements and are well separated from the next higher exciton level by about 41 ky [l/aD] 5 0 kx [U00] 5 ky [l/aD] 5 , 0 _ _ kx [U00] 5’" Figure 1.7: Plots of the functions Ii (k): They govern the trion oscillator strength for the interaction with a iQ mode. The variational parameters for GaAs were used. The value for Q corresponds to laser light tuned close to the trion resonance at an angle of 30° with respect to the é-axis. 42 ~ 3.5E0. we start from a free electron state as our initial state , , exp (iker) (0(r) = (r1) = ————, (1.65) l 6 «E which is a plane wave with momentum kg. The final state, which shall be described by the wavefunction \IIC (r1,r2, rh), contains two plane waves, one for the free electron and one for the 13 exciton. The subscript 0 indicates the exciton states form a continuum. We can write \Ilc (r1,r2, rh) as a simple product of the electron and the exciton wavefunction exp (“(5 (aer1 + (111%)) exp (ikrz) WI x/Z 1 (913(1‘1 _ 171), (L66) \I’c (1‘111‘211'5) = (I‘lf> = where we introduced 016/}, = m: / h / mx; the electron has an wavevector k, while the exciton’s wavevector is km. Note that to describe this state we have to assign it the quantum numbers k and km, while for bound trions one wavevector was sufficient to characterize the state. We could equivalently assign it a center of mass momentum, but would still have to define a relative momentum. The nomenclature “exciton continuum” is based on this reasoning. It is important to emphasize that this ansatz to describe the continuum states \Ilc is not properly symmetrized. However, in Appendix E.1 we Show in detail that for a macroscopic sample the exchange effects that arise from a proper symmetrization of the wavefunction are negligible. To calculate the transition amplitude, we can simply refer to Eqn. (1.46) by re- placing \I't with ‘116. Expanding the cos (Qr) into its two modes, we find the transition amplitude to be (kel Hint lkx, k>c = 92251531 [5K—keo + 6K—ke,—Q] « (1-67) 43 Here we introduced the effective exciton Rabi frequency (2,, as o . or = 7051,. (0) «Z. (1.68) Although the bare coupling (20 being the same, the overall nature of this transition amplitude is strikingly different than from the one we have obtained before for bound trion states. Let us mention the obvious part first: The Kronecker deltas handle the impulse conservation, whereby the photo—excited exciton just takes the photon momentum, while the free electron is unaffected by the transition and simply stays in its initial momentum eigenstate. The fact that the amplitude is proportional to the relative exciton wavefunction taken at r = 0 goes along the line with the result found by Elliott in Eqn. (1.8). In this picture the corresponding expression for bound trions in Eqn. (1.48) can be regarded as a natural generalization of the excitonic transition rate, one relative coordinate being set to zero. The more interesting part is that the exciton oscillator strength appears as an extensive quantity that scales with the area of the sample according to the dependence ~ x/Z. Owing to the normalization constant of the 13 wavefunction (4513 (0) N 1/ (7mm)? (1.69) it can be visualized as proportional to the number of excitons that fit into the sample without spatial overlap. The bigger the sample is, the more excitons can be excited. Every single exciton transition at a particular spot is driven by the bare Rabi fre- quency {20, but the macroscopic size of the sample accounts for a huge collective enhancement factor for the exciton oscillator strength. This result sheds an interest- ing light on the nature of excitons: they appear as a coherent elementary excitation over the whole sample, resulting in a macroscopic transition dipole moment and sharp 44 absorption peaks in ideal cases. Since the presence of the electron effectively does not affect the process studied here, the unbound trion has essentially the same oscillator strength as the exciton. The oscillator strength decrease from ionized to bound trions is therefore of the order of ~ (lg/A, which is vanishingly small in the large sample limit. However, this is not suprising, because it. considers the somewhat artificial limit of a single electron in a huge macroscopic sample. One single electron cannot have a sizeable effect on the photon absorption. The exciton’s oscillator strength is approximately ~ A / (13,, while the trion oscillator strength scales as ~ (A / 0.3,) (0.3/14) :3 a? / (1%, which corresponds to a picture in which the trion oscillator strength is proportional to the number of excitons that fit into a single trion without spatial overlap [32]. We already estimated this ratio to be of the order of 10. 1 .4 Radiative lifetimes In this section we will present the radiative properties of excitons and trions. The intrinsic radiative decay of both free excitons and trions in low-dimensional systems is due to the coupling with a continuum of photon states. We will analyze the different characteristics of these two decay processes. The calculations will be based on the assumption of conservation of the in—plane wavevector, thereby disregarding possible effects of interface roughness and acoustic phonon scattering. The effect of the latter can be minimized in the low temperature regime. Therefore we will consider the T —+ 0 limit. The wavevector conservation is likely to be a plausible assumption, whenever the coherence length of the quasiparticles is longer than the wavelength of the light [33]. This condition can be satisfied in good quality samples at low temperatures. The proper framework to perform the actual calculations is Fermi’s Golden Rule: 45 The decay rate of a state [(1) or equivalently the probability to make a transition from the initial state [07) to the final state [3) per unit time to first order in perturbation theory is given by 2 11,43 = 7” [140,3]2 5 (12,3 — E0), (1.70) where M03 is the matrix element which couples the initial and the final states. To find the total decay rate F of the initial state [(1) we have to sum over all possible final state configurations, the 6-function appearing in Fermi’s Golden Rule effectively taking care of energy conservation in this process. We then find the general expression 27f 2 r = 7 2]./V103] 6 (E3 — Ea). (1.71) ,8 The interaction Hamiltonian H1 that couples the initial and final states can be ex- pressed in second quantized form as _ T T H1 — g 2 cQ+k.;(1/2)d—k,i(3/2)aQ~i + h.c. (1.72) Q,k Note that in-plane momentum conservation is already included, but the é-component of the photon is left arbitrary. The coupling constant 9 is the only difference from Eqn. (1.37), where we considered transitions induced by a classical electromagnetic standing wave. In order to describe the spontaneous decay process, we take 9 to be 9 : doég. (1.73) Here, (10 denotes, as before, the interband transition dipole moment, 6 is the polar- ization of the photon and the quantity 8 h... 5 = ”5.? (1.74) 46 E (k) E (k) Figure 1.8: Schematic diagrams of the exciton (left) and trion (right) in—plane dispe- rion relations and the coupling to light. For excitons the radiative zone is restricted by the light cone effect (dashed lines). can be visualized as the ‘electric field per photon‘ inside the material; 6 incorporates the refractive index of the material and V is the quantization volume. Let us first consider the case of an exciton decay. In a bulk crystal, the exciton can interact only with a photon that has the same wavevector in order to obey the translational invariance of the system, resulting in a hybrized exciton — photon mode, a polariton, which is the stationary state of an infinite dielectric medium. The situation is different for a QW system: the exciton with in-plane center of mass wave vector K interacts with all the photon modes that have the same in—plane wave vector, but no further restriction is imposed on the S-component of the photons wavevector. Compared to the bulk case, where radiative recombination occurs only by a leak of the polariton through the surface of the crystal or by (non-)radiative recombination at crystal imperfections, this leads to a large radiative recombination rate in the QW system [34]. In a two—dimensional QW system, an exciton combines the dual 47 merits of coherent nature on the two-dimensional QVV plane and superradiant decay in one direction along the growth direction, owing to its macroscopic transition dipole moment. Therefore, the breakdown in translational symmetry in the growth direction gives rise to profoundly modified coupling of excitons to photons. The radiative zone, however, is restricted by the light cone effect: Radiative recombination can only occur, if the exciton’s wavevector [Kl can be matched the photon’s wavevector: Consequently, it obeys the constraint [K] < 7162/ c. This is schematized on the left side in Fig. (1.8). Only excitons around K m 0 can couple efficiently to light; they are called bright states. States outside the light cone are called dark states and can be thermally populated for sufficiently high temperatures. We elegantly circumvent these problems by performing the calculation in the exciton’s restframe, i.e. we set K = 0, and assume the low temperature limit T —> 0. The initial state of the exciton decay process is characterized by the exciton’s wavefunction \le (re,r},). We will restrict our analysis to 13 excitons and neglect higher states as the interaction with the radiation field is typically dominated by this lowest state, particularly under near resonant excitation. The final state is given by the crystal ground state and the emitted photon. We fix the spin configuration of the initial state as composed of a spin down electron a = -1/2 and a heavy hole with spin Uhh = +3/ 2. As a consequence, the polarization of the photon is determined to be a 0+ photon. With this simplification, we will drop the spin indices for the moment and make the replacement 92 ——+ 9’2 : 92 / 3 to correct the dipole moment. we write the initial state [2) for an exciton with center of mass wavevector K as )7) = Z is. (k, K -— k) c]d]<__k )0), (1.75) k 48 where the Fourier transform of the exciton wavefunction ‘le (re, rh) is defined via ~ 1 . . \IIJE (k, K — k) = Z fdzredzrh \le (re, rh) exp (—27kr€) exp (—i (K — k) rh). (1.76) The energy of this initial state E) is given by h‘ZK'Q E) = + 6,3 — Ex. (1.77) 2771,: Similarly, we write the final state If) as If) = «1510) (1.78) with the photon energy .Ef::ne. (179) Using the (anti)-commutation relations for fermions (bosons), we obtain indeed the macroscopic superradiant transition amplitude for a QW exciton (fl H. II) = g’ f ,2, exp (-iQr) 111.». (r. r) ——— 9’7)... (0) 7.1—5m. (1.80) We can recognize that the in-plane photon momentum has to fulfill momentum con- servation, but no condition is imposed on Q2. In the subsequent step, the calculation of the radiative decay rate I‘m, the value of Q; will be determined by energy conser- vation. Inserting the transition amplitude into Fermi’s Golden rule (1.71), taking the large volume limit to replace the sums by integrals and evaluating the expression for simplicity in the exciton restframe, we find that the exciton‘s decay rate I}, and the exciton’s lifetime 7,;- can be obtained from 2 Mon —1 T_ : F : —— J: I 37ra.(2)h2ce c,—Ea. new 49 Here and in the following, 0 is the speed of sound in vacuum, 71 the refractive index of the QW material and (10 the effective exciton Bohr radius. The values obtained for the specific QW systems GaAs and CdTe are summarized in Tab (1.3). We also state the natural linewidths of the exciton level 75.1117, because we will refer to them in the course of the rest of this thesis. ] GaAs [ CdTe 71. [p5] 15.9 6.9 1‘, [10103-1] 6.3 14.5 51“,, [,ueV] 41 95 Table 1.3: Calculated radiative exciton lifetimes, decay rates and level widths for GaAs and CdTe. Indeed, we find very short lifetimes in the range of picoseconds: an evidence for the enhanced radiative recombination rate of excitons, caused by the breakdown of the translational symmetry of the system. Our results are in a good agreement with values from other theoretical and experimental investigations: For GaAs a radiative lifetime of 7'3; = 10 :t 4 ps has been measured in the absence of dephasing mechanisms [35]. Due to a smaller dielectric constant e and a bigger effective electron mass mg, the radiative lifetime in a CdTe semiconductor QW is found to be smaller. Esser et al. found T3 = 13 193 for GaAs and T1- : 4 p3 for CdTe in the T —> 0 limit. Let us turn to the radiative decay of a bound trion. The coupling of delocalized trions to light is very different from that of excitons, since not only a photon, but also an excess electron is involved. In an optical transition, where the photon momentum is negligible, the electron can absorb the trion center of mass momentum K. This fact opens up a radiative decay channel for trions with high K vectors, that doesn’t exist for excitons. In other words, there is no light cone effect for trions. A schema- tized version of this argument is presented in Fig. (1.8). All trion states can decay radiatively, since the mismatch in the wavevectors between the trion and the photon is absorbed to the excess electron during the optical transition. 50 The initial state [2) = TK>t contains a trion with center of mass momentum K. Again, we fix the spin configuration. The transition amplitude for this process can then be conveniently calculated from an adapted form of (1.47), the optical matrix element, where we replace the coupling constant 90 / 2 -—+ g’ to find Mif = = 9,5K.k€+Q1+(ke)- (182) We have calculated the trion decay rate F); in the trion restframe, where K = 0, as depicted in Fig. (1.9). In this frame, the in-plane momentum conservation takes on the form k6 = —Q. Figure 1.9: A trion in its rest frame decaying into a photon and an electron. We have obtained an approximative analytic expression for Ft, whose derivation is presented in detail in Appendix F. Our numerical results for GaAs and CdTe are summarized in Tab. (1.4). [ GaAs [ CdTe 'rt [ps] 18.9 18.4 P, [10103-1] 5.3 5.4 hr. [...eV] 34 35 Table 1.4: Calculated radiative trion lifetimes, decay rates and level widths for GaAs and CdTe. For both GaAs and CdTe we approximate the trion lifetime to be about ~ 20 ps. Experimentally very similar values have been measured in the range 45 — 60 ps [36]. Our theoretical prediction is in a good agreement to these experimental values, so that 51 our variational trial wavefunction again proves to be reliable and produces reasonable results. As a last remark, we want to address the question, why the lifetimes of excitons and trions are in a comparable range, although their oscillator strengths for a light induced transition are very different? While the exciton’s oscillator strength f); is an extensive quantity that scales as f I ~ A/AI, the trions oscillator strength ft is much smaller. since the presence of an excess carrier is required. It has the scaling behaviour of ~ Alt/Ax, that can be interpreted as the number of excitons that fit into a trion without spatial overlap. Here, A, At and A I are the sample, the typical trion and the typical exciton size respectively. In the case of an exciton’s radiative decay, the final state is marked by a photon with momentum Q = (Q,Qz). Since momentum conservation requires that the exciton’s momentum equals the photons in-plane momentum, Q is fixed. Moreover, the value of Qz is determined by energy conservation, so that in total there is no degree of freedom for the final state of this transition. The situation is different in the case of trions. When a trion decays radiatively, the final state is not only characterized by a photon, but also by an excess electron that is left in the two-dimensional sample. Consequently, in the case of a radiative decay of a trion the final state is completely determined by five momenta components, three for the photon and two for the free. electron. Since momentum and energy conservation again only fix three components, two degrees of freedom are left and need to be summed over to obtain the correct transition rate. This summation dies out for large k values and has the characteristic size of the trion‘s relative wavefunction’s Fourier transform, namely ~ 1/At. Thus we can get the scaling behaviour . 1 At A E — ~——~A A 1.83 it A) .4, A, / I ( ) which is exactly what we expect in the case of excitons. So, the fact that the trion’s final state of a radiative decay has two more degrees of freedom than the exciton’s exactly cancels the bigger oscillator strength the exciton has due to its macroscopic dipole moment. This gives an explanation for the fact that the radiative lifetimes of trions and excitons are not as different as one maybe expected at a first glance, when considering their very different decay channels and different oscillator strengths. Chapter 2 Optical Potentials: Atoms vs. semiconductors The possibility of trapping atoms at the nodes or antinodes of an intense laser standing wave was suggested first by Lethokhov in 1968 [37]. From 1986 on, when the first experimental observation of optically trapped atoms was reported, laser cooling and trapping has become a flourishing new field of modern physics [38]. Laser cooling and trapping rely on the fundamental interaction between laser light and atoms to exert controllable forces on the atoms. Today, the use of lasers to coherently control atoms is a well established technique and many sophisticated schemes have been developed based on special properties of the interaction. For cold atoms, a great variety of potentials can be designed via their interaction with the electromagnetic field, including static magnetic fields (Zeeman shifts), static electric fields (Stark shifts) and optical fields (light shifts). In this chapter, we will first review the mechanical effects of light on atoms, refer- ring to the concrete example of a two-level atom interacting with a single-frequency light field. Although this marks an idealized model rarely encountered in practice, it is pedagogically very valuable, because it shows many features that will be en- 54 countered in the rest of this thesis. We will restrict our attention to the problem of trapping atoms in a standing wave. A special property in the study of atoms confined in optical wavelength-size potential wells is that they can simulate many body physics previously accessible only in condensed matter systems. In these systems, the con- fining force is the so-called dipole force, which is proportional to the laser-intensity gradient, and which is based on a potential V (r) varying in space exactly as the laser intensity pattern. Technically, optical dipole traps rely 011 the coupling between an induced dipole moment in the atom d and an external electric field. The atoms then experience a spatially varying AC Stark shift which creates a trapping potential V (r) for the atoms v (r) = —a - E(r) o< (1(a)) |E(r)|2, (2.1) where a (w) gives the frequency-dependent polarizability of an atom and I (r) oc [E (r)|2 describes the intensity of the laser light field; E (r) is the electric field ampli- tude at the position r [39]. The sign of the polarizability a (w) is crucial in that it determines if the potential is attractive or repulsive. It is important to note that the light force is not fully conservative. Since the opti— cal potential derives from optical transitions between two atomic levels, spontaneous emission is inevitable and gives rise to an imaginary part of the polarizability [40]. Therefore, we will discuss below dissipation in optical potentials, an omnipresent, but readily controllable side effect of laser induced potentials. By reviewing the fundamentals of optical physics of two—level atoms, we will es- tablish notations and conventions and provide reference points we will further use in this thesis. Moreover, in the second part of this chapter, we will introduce our first theoretical approach to model an optical potential for carriers in semiconductors in the second part of this chapter. We will develop a simple picture, which essentially decomposes the quantum well into a collection of single two level systems. Based on C51 CI! this toy-model, we will be able to deduce similarities between the well established optical dipole trap for atoms and the novel scheme for carriers in a semiconductor QW we propose in this thesis. Besides the similarities, the existence of the extensive, coherent exciton excitations will give rise to striking and physically intriguing new effects that are not present in the case of optical potentials for atoms. 2.1 Optical potentials for atoms 2.1.1 Optical dipole traps for atoms Based on the dipole interaction between an atom and an electromagnetic standing wave, we will first derive the physics of optical trapping potentials for atoms from first principles. In the course of this derivation we will introduce the rotating reference frame and the rotating wave approximation. For simplicity, we consider a two-level system atom with an internal ground state [1) and an excited state [2) which is coupled to a monochromatic classical laser field with a detuning A. The system, including the chosen sign convention for the detuning A, is schematically shown in Fig. (2.1). The bare Hamiltonian H0 of the two level system can be written as H. = e1|1><1|+ e212) <2) , (2.2) with 61/2 being the energy eigenvalues of the eigenstates [1) and [2) for the unper— turbed problem. An optical field couples to the the dipole moment of the atom d. Explicitly, we consider the case of a monochromatic standing electromagnetic wave in the dipole approximation, i.e. we neglect the variation of the electromagnetic field E (r,t) across the atom and ignore the effects of the magnetic field B (r, t). Since a typical wavelength A used in optics is of the order of A ~ 400 71771, whereas the char- 56 |1> Figure 2.1: Interaction of a laser with a single two-level atom with the chosen sign convention for the detuning A. acteristic size of the atom is about a Bohr radius 0.0 ~ 0.1 run, this approximation is usually very good [41]. The interaction between the atom and the standing wave can be expressed as Hint = —d - E0 cos (Qr) cos (wt), (2.3) where E0 is the amplitude of the electric field, Q the wavevector of the laser and w the frequency of the light. The laser drives the transition between the two atomic states with a Rabi frequency (2 (r) which serves as a. measure of the interaction strength Q (r) = (1] d [2) E0 cos (Qr) = d12E0 cos (Qr) = {20 cos (Qr). (2.4) 57 It is proportional to the laser field and the dipole transition matrix element of the atom d12. The amplitude 00 is a constant that depends on the laser intensity as well as the properties of the atom. Without loss of generality, we can choose its phase such that it is a real-valued quantity (2 (r) = {2* (r); with this definition, we can express the coupling term of the Hamiltonian as Hm. = -9 (1)608 (wt) [[1) (2| + [2) (ill - (2-5) We now introduce a time-de endent unitary transformation U 75 to a frame rotating P - at the laser frequency 6.2 U (.) = exp (erg. (I2) <2I — |1> <19). (2.6) For the remainder of this section, we set 71. = 1. Defining the transformation as [\II (t)) = U (t) [\11 (t)>, the Schréidinger equation in the rotating frame reads . 5 ~ _ ~ ~ - ~ _ ”t . T - 25—. 12(0) —H[\Il(t)>, H—L HU—zU U. (2.7) To simplify the further analysis, we split up the Hamiltonian in the rotating frame H into two contributions H : f{0 + Hint (2-8) H0 = UTHoU — 11fo (2.9) H... = UTH...U (2.10) As far as H0 is concerned, we obtain the expression H. = e01) (1| + I2) <2I) + 5,3 (I?) <2) — 11) (1|). (2.11) where we defined the detuning A as the deviation of the laser frequency w from the transition frequency of the two level system A: (62—61)—w=w12—w. (2.12) In this definition, A is defined as a positive quantity for red detuning (see Fig. (2.1)). With E given by E = %(61 + 62), the first term in fig just gives a constant shift in energy, that we can drop in the following. Consequently, we can express H0 in the simplified form ~ A H. = 3 (I2) <2I — I1) (1|). (213) Let‘s focus on the evaluation of 1:1,") 2 U THz-mU 2: —Q (r) cos (wt) Ii (t), where we defined f1. (75) as (“2 (.) = UT (5 (I1) <2I + I2) (1|) U (.). (2.14) Taking the time derivative twice of h (t), we notice that Ii (t) obeys the differential equation of a simple harmonic oscillator with oscillation frequency w 11(t) + avg/3. (t) = 0. (2.15) Using that U (t = O) = U] (t = 0) = 1, the complete solution is given by fi(t) = li(0)cos(wt)+$fz(0)sin(wt) (2.16) 13(0) = I1><2I+I2><1I (2.17) 13(0) = w(I2><1I—I1)(2I). (2.18) In this fashion, we find Him to be ~ Hint = —Q (r) cos (wt) f7 (0) cos (wt) + g}? (0) sin (wt)] . (2.19) 59 Since in optics the frequency w is typically very high, of the order of 1.) ~ 1015 H 2, we will average the Hamiltonian if over one period of oscillation. This approximation is known as the rotating wave approximation, which helps us to get rid of the fast— oscillating terms. It is valid whenever the detuning A is small compared to the light frequency (.1. By introducing the time-avergaged Hamiltonian _ ,3 2.)... - H = — / dtH (t), (2.20) 27f 0 the Hamiltonian If in the rotating frame reads {2(r) H: (l2><2|-|1><1|)- 2 (|1><2|+|2><1|)- (221) A 2 We can see that by applying the rotating wave approximation we are left with a time— independent problem. This is indeed a very valuable simplification of the problem. The eigenvalues of [:1 show that the light—shifted energies are given by 1 I 1 2 9 . = — - 2.22 ,0 4,71) +12 (r) ( ) fir-:1: They are illustrated as a function, of the detuning A in Fig. (2.2). We see that the coupling Q (r) lifts up the degeneracy at A = 0. The level crossing is avoided as soon as a perturbation is applied to the system. The eigenstates of the Hamiltonian (2.21) are called dressed states; the light-induced perturbation mixes the states, so that the ground state is mixed with a component of excited state and vice versa. Compared to the unperturbed system, the levels of the dressed states are split farther apart, namely by the amount 52’. The coherent modification of the atomic spectrum in the electric field of a laser is called optical Stark effect. However, we can recognize another striking feature, namely that an inhomogeneous light field, as in a standing wave, produces a spatially dependent light shift. This is the physical origin, of an 60 €+/_ Figure 2.2: Eigenvalues (i of the Hamiltonian If in the rotating frame after the rotating wave approximation. optical dipole trap. The force. that stems from this gradient of energy is called the dipole force. It is simply based on the spatial variation of the internal energy of the atom. Proceeding from the Hamiltonian H it is straight forward to explain the presence of an optical potential for atoms. The energy shift to the ground state in second order perturbation theory due to the perturbation V (r) = —Q.];2 ([1) (2] + [2) (1]) is (2) _ |<2|V(r)|1>|2 _ _122 (r) _ _9_8 .. AE1 — BSD—Ego) — 4A — 4A cos (Qr), (2.23) which leads to the definition of an optical dipole potential for the specific, but im- portant case of a standing wave as V (r) = — , = —— cos2 (Qr) (2.24) The optical potential due to the optical Stark effect takes on the spatial pattern of the intensity profile with its characteristic spatial dependence ~ cos2 (Qr). However, the sign of the detuning will decide about the energetically favourable spots for atoms. For red detuning, corresponding to a positive detuning A > 0, the potential minima will be at the anti-nodes of the intensity profile, so that the atoms seek the spots of strong field intensities. In the case of blue detuning, where the detuning is negative A < O. the potential minima are at the dark spots of the intensity profile, i.e. the atoms are attracted towards the weak field spots. Similarly, we can derive the form of the optical potential starting from the eigen- values ci (r). Expanding them for a small perturbation Q (r), small compared to the detuning A, we get 2 2 r .. (5 = H) 1 (24(9) . 1H. 2 < >. 5.... Dropping the spatially homogeneous term, we obtain 2 43):] cos‘2 (Qr) . (2.26) 6:); (r) z i To show the equivalence to the optical potential V (r) we have to be careful about the choice for the sign in the expression (2.26). In the case of a large positive detuning A, we have to pick up the eigenvalue e_ to asymptotically approach the ground state of the unperturbed problem, whereas in the case of a large negative detuning it is vice versa. However, in both cases the spatial dependence of the perturbed ground state is then given by the same expression as derived for V (r) in Eqn. (2.24). In conclusion of this discussion. we may summarize that optical dipole potentials for atoms are based on the the light-induced Stark shift of the atomic energy levels. The essential findings are schematized in Fig. (2.3), where the different character of 62 A<0 A>0 7W 920") ~ I (F) Q2 7‘ V0?) 2 + 44X]? Figure 2.3: The bare two—level system with eigenenergies E9 and E6 are shown in the center. The light shifts for red detuning are displayed on the right, creating an attractive potential. For blue detuning the potential is repulsive, as depicted on the left-hand side. the potential for blue and red detuning is underlined. 2.1.2 Dissipation in optical potentials In the course of our introduction to light forces, we have swept one phenomenon under the rug: spontaneous emission. As a direct consequence of momentum conservation, the emission of a photon into the continuum of modes of the electromagnetic field must be accompanied by an atomic recoil. In quantum optics, the recoil aspect of spontaneous emission was ignored until the advent of ultracold atomic samples. The spontaneous emission events lead to random kicks in the atomic momentum, which has the ability to cause an unwanted heating of the atomic sample. We are going to address this dissipative character of the light—atom coupling in the following. 63 Let us first mention that the decay of the excited state is an irreversible process. In principle, the modes of the spontaneously emitted light also couple to the ground state of the system, but the number of modes is infinite in free space. To find the amplitude for the reverse process, all the contributions from the different modes have to be summed over. Since they add destructively, the total probability for the reverse process becomes zero. However, it is the competition between the excitation by an incident wave and the damping processes due to spontaneous emission that allows the atom to reach a steady state. To handle spontaneous emission, the standard approach is to introduce the density matrix p“ of the system and discuss the excitation of the atoms in terms of populations and coherences instead of ammitudes. The optical Bloch equations express that the rate of change of the atomic density matrix is a sum of two contributions describing the coupling with the incident wave and the coupling with the empty modes. To describe the effects of spontaneous emission, we add a phenomenological decay term P, which is the natural line—width of the excited state, equal to the inverse of the radiative lifetime of the excited state ’7'. The optical Bloch equations describing the two-level system subject to spontaneous emission read P22 = -FP22 — % (P12 — P21) (227) P11 = +FP22 + g; (P12 - P21) (228) P12 = (2A — g) P12 + 2% (P22 — P11) (229) P21 = - (2A + 3%) P21 - ig- (P22 — P11) . (230) where we recognize p11 and p22 as the population of the ground state [1) and the excited state [2) respectively as well as the coherence terms p12, p21. Note that the normalization of ,6 is conserved: 522 + .011 = 0. Throughout this thesis we will be 64 particularly interested in a weakly driven system, where the detuning is much bigger than the Rabi frequency. Therefore, the probability to be in the excited state p22 is small, resulting in a small loss rate out of the system and a quasi-closed system. In this regime we approximate the population of the upper state p22 with the steady state solution for a closed system given by Q2 2 4&2 + 202 + P2 ‘90 (2.31) 1 22 - - p 21+ 80 + (2A/F)2 Here, we defined the on-resonance saturation parameter so as 80 = --—. (2.32) In the regime where the detuning from the resonance A is much bigger than the driving Q and the natural linewidth I‘ of the excited state, the population of the excited state is approximately 92 p22 % 4—A—2. (2.33) Since the population in the excited state decays at a rate F, the total scattering rate of light from the laser field, or the effective spontaneous emission rate F58, is given by 80 F/‘Z . . 2.34 1 + 801+(2A/r’)2 ( ) Fse = P22F = For very high intensities, where 30 >> 1, F38 saturates at F/ 2, since an intense ex— citation equalizes the populations of the two levels. In the last step we defined the power-broadened linewidth of the transition I" = 1V1 + 30. (2.35) If the transition is “saturated”, the linewidth of the transition is effectively broadened 65 from its natural linewidth F to its power-broadened value F’. The absorption of an incident photon with momentum fiQ leads to a momentum transfer from the optical fields to the atom. The momentum is regained by the incident beam, if the photon is re-emitted by stimulated emission. However, if the atom returns to its ground state by spontaneous emission, the loss of momentum is zero on average, since the recoil associated with the spontaneous fluorescene occurs in a random direction. This picture leads to a dissipative force Fsp, often called “radiation pressure force” or “dissipation force”, that arises from absorption followed by spontaneous emission which is equal to the momentum transfer per photon liQ times the photons spon- taneously emitted per unit time. It relies on the scattering of photons out of the laser beam. The randomness of the spontaneous emission, being itself an irreversible process, gives rise to a heating mechanism, that needs to be taken into account. 11— lustratively it can be mapped onto a random walk problem in momentum space with. step size EQ and rate F 36. Fig. (2.4) depicts the situation of an atom starting out from the origin in momentum space at k = 0. The first kick takes it to a point somewhere on the first circle. The question is where the atom will be after the second kick. Clearly, it will be somewhere on the second circle. The red shaded area accounts for heating, since the energy of the atom has increased with respect to the energy it has gained from the first momentum recoil kick. The blue shaded area depicts an effective cooling process, because the atom’s energy has decreased again. However, the red shaded area is bigger than the blue one, so that the probability of a heating kick is bigger than the probability for a cooling kick; eventually, after several kicks, the electron will always have gained energy. We define the corresponding heating rate Rheat - the energy the atom absorbs because 66 Figure 2.4: Random walk problem in momentum space. of the dissipative light forces and measured in energy per unit time - as Rheat = PseER = P22FER (2.37) where we have introduced the single photon recoil energy E R- It is a figure of merit that sets a natural energy scale for the problem; physically, it is the kinetic energy of an atom of mass m with a momentum whose magnitude hQ equals that of the laser photons h2 Q2 E R 2m (2.33) Also, we have to consider this heating mechanism in the framework of optical dipole traps, since the light fields that form the basis for the traps can induce op- 67 tical transitions in atoms, which subsequently can result in spontaneous emission and a heating of the atomic sample. There is, however, a easy way to remedy this problem, resulting in an effectively dissipation-free, conservative potential. Assuming A >> 9, F, the potential depth of the optical trap is V0 = 92/4A, while the atoms spontaneously scatter photons from the field at a rate Fse = FQ2/4A2. If 92, which is proportional to the intensity, can be increased as A is increased, then the same potential well depth can be obtained with a reduced scattering rate. By choosing high laser intensities and large detunings, one can combine deep potentials with low scattering rates, because the excited level has been eliminated adiabatically. We can conclude that finite dissipation occurs in the optical dipole trap due to spontaneous photon scattering, but it can be sufficiently suppressed to an arbitrary degree in the far-detuned limit, provided that sufficient laser power is available to achieve the desired potential. 2.2 Toy-model optical potentials in semiconduc- tors Our fundamental review of optical dipole potentials for atoms has taught us that their basic mechanism relies on optical transitions between two internal atomic energy levels driven by intense off-resonant laser light. There, the electromagnetic field couples to to the dipole moment of the atom, inducing a shift in the atom’s ground state energy due to the AC Stark effect. For a non—uniform electromagnetic field the atom’s energy is spatially modified. Then, the atoms see an optical potential. Now, we draw an analog picture in a. semiconductor host environment: The elec- tron to trion optical transition effectively gives a second internal state to the spin- polarized electron, which is otherwise a pointlike, structureless particle in free space. In a simplified picture, the trion may be viewed as the excited state of an atomic 68 system, while the electron can be considered as the ground state. However, this com- parison has to be handled with care, since, in contrast to a single two—level atom, an optical excitation in a semiconductor material has to be described taking into account its spatially coherent nature in an ideal extended system. To cover this effect in our first theoretical approach, we propose the following idea: We envisage the quantum well as a collection of independent two level systems, each associated with a tiny region whose size is equivalent to exciton size AI = 7mg. Therefore, the system is discretized into small cells, as depicted in Fig. (2.5), and the total number of cells is J 17 = A/AI. First, let us think about the physics when no conduction electron is present in the sample. Under this assumption, of course, we expect excitations of excitons only. Every single cell is coupled to the laser light; we denote the Rabi frequency for this transition as the bare Rabi frequency {20 (r), since it refers to one specific cell and, . as a consequence, has no macroscopicly enhanced dipole moment. The ground state of one specific cell lg) is marked by the absence of an exciton, while the excited state of one cell Ie) is described by the presence of a photo-excited exciton. The ground state of the system does not contain a single exciton excitation, all the cells of the system are in the ground state which we express as |g, g, . . . , g). The elementary excitations of the total system are states in which one of the cells is “filled” with an exciton, such as for example the states |e,g, . . . , g), |g, e, . . . ,g) or |g,g, . . . e, ). Since excitons are plane waves that are coherent over the whole sample, in the language of the cell-model the creation of an exciton appears as an collective excitation over all the cells 1 . , (le,g,...,g)+|g,e....,g))+---+|g,g,...,e)). (2.39) «A? The light-matter interaction creates states that are a linear superposition of all the elementary excitations, whereby every single cell is addressed with the the bare Rabi 69 Figure 2.5: An extract from the QW sample: It is discretized into small cells of size AI, the typical exciton size. If an electron is present, centered at re, excitons created within the area At, the typical trion size, give rise to the formation of a bound trion. In this model, the electron is coupled effectively to Nt : fit/AI, trion states. 70 frequency 90 (r). Compared to the creation of an exciton in one specific cell, the collectivity of the exciton state gives rise to an enhanced, macroscopic dipole moment Now, let us examine the situation where conduction electrons are present in the system. If an exciton is photoexcited in one of the cells close to the electron, assumed to be centered at re, it is energetically favourable to form the three particle state of a bound trion; the exciton “captures” the excess electron. The coupling of the electron to the trion resonance is strongly enhanced, since the excited trion level is highly degenerate. Within this toy model the degeneracy factor is N = At/Ax, which is the number of excitons that fit into a trion. Literally, the exciton can be created anywhere within the trion size At around the initial electron, with the final result always being the same, namely a bound trion. Thus, the electron is coupled to M; degenerate trion states, as schematized in Fig. (2.5). It is well known, that the resulting effective Rabi frequency for the electron—trion transition Qt (r) is a (r) = mg, (r). (2.40) where we assumed 90(1‘) to be constant over At. Since the spatial extension of the bound trion is much smaller than the optical wavelength A, this is the equivalent to the approximation of evaluating the electric field at the atom’s center of mass position only. Still, we must not forget the excitation of excitons, even if conduction electrons are present in the system. We assume an unbound trion state Whenever an exciton is photoexcited outside the range At around the electron, because in this case the exciton is too far away from the single electron to capture it for a subsequent formation of a bound trion. Let us mention that we have not. restricted ourselves to the case of one single electron in the system. To be realistic, we assume a low density electron gas, but 71 f I 1 E‘ l *1 e A PS“ i ___): ET ?. ---------------- «—————---—-—— g 1A,, < O l _ _ __._ . __ Pt I At > 0 +_.___§s_ k E(k)“ Figure 2.6: Level scheme with trion and exciton resonances and the chosen sign convention for the detuning. The ground state with an electron (blue dot) in the conduction band as well as th excited state after the creation of an electron hole (green dot) pair is depicted on the right. the essential physics occur near a single electron. Therefore, when contemplating the effects on one electron, we can focus on the cells around that electron solely, disregarding the rest of the macroscopic sample. In the following we set up a Hamiltonian that takes into account both the trion as well as the exciton resonance. Thus, the system we investigate is drawn in Fig. (2.6). The detunings At and Ax from the trion and exciton resonances, respectively, are by definition positive for red detuning. They are separated by the trion binding energy ET, so that the following relation holds 72 Again, we will first analyze the situation in which only excitons may be excited, since initially the conduction band is assumed to be empty. We proceed from the many-body Hamiltonian H. given in the frame rotating at. the detuned laser frequency. We express H as H 2 HO + V (2.42) with the excitonic bare Hamiltonian H6" 2 AI/dgr I1‘,r) (2:,r| (2.43) and the light matter coupling V1; 2 /d2rp1. (r) H0) (.27, rl + Ir, 1‘) (0H (2.4-4) Here |0) denotes the crystal ground state, i.e. a completely filled valence band and an empty conduction band. The light matter coupling creates excitons, where we restrict ourselves to the case of one type of excitons, the 13 excitons. The state |:r., r) describes such a 13 exciton with its center of mass located at r. The strength of the coupling is governed by the quantity pl; (r), which is the spatially dependent Rabi frequency 90 (r) per exciton cell size _ Q0 (1‘) r pl? (1‘) _ m a (240) since the definition of the bare Rabi frequency 90 (r) is related to one cell of size AI. Even if this definition might seem cumbersome at a first glance, it is evident that the square of it over the detuning AI gives the energy shift density. This is the quantity we are interested in and which we will encounter below. 73 The orthonormalization relations are given by (0|0) = 1, (0|.r, r) = 0, (a:,r|.r,r’) = (5 (r — r’). (2.46) Along the lines of our analysis for atoms, we use second-order perturbation theory in order to calculate the ground state energy shift E0 as ‘ ' v, 0 2 44 2 I .I,‘ where we have taken the spatial dependence of the bare Rabi frequency as 90 (r) = {20 cos (Qr), corresponding to a electromagnetic standing wave. The square of this spatially dependent Rabi frequency is averaged over the area of the sample A, which accounts for the factor of 1 / 2. We discover that the coherent excitation of excitons results in a macroscopic background energy shift N, 9,2, (22 E = ___ = —N. — 2.48 0 2 A, la, ( ) that is proportional to NT. For notational convenience, we defined Q2 = $28 / 2. The consecutive step is to include the possibility of an excess electron, whose position we denote as re. The essential idea of the toy model is that the presence of the initial electron displaces all exciton states from an area given by the trion area At (centered at re) and replaces them with less energetic trion states. Furthermore, the assumption that the variation of the field intensity over At is negligible will explicitly enter our calculations. With the initial energy of the electron taken to be zero, the new bare Hamiltonian H6 in the rotating frame is H6 2 Ax/d2r |.r,r) (.r,r| — Ar/ dzr |r,r) (.r,r| + At/ d2r |t,r) (t,r| (2.49) A t At 74 which underlines that within the trion area At all exciton states are replaced by energetically more favourable trion states; we denote a bound trion with center of mass position r as lt, r). The light matter coupling can be written as v2 = / d2rpi: (r) [10> <13 rl + Ix,r> <01} — - re 2r :c,r .r r m )fAtd no>< . |+| , ><0|l +p. (re) [4 fr H0) (tarl + It,r> <0”, (2.50) ‘ t where the vacuum state ID) is now meant to include the presence of the initial electron, but no excitons or trions. In analogy to p33 (r), we introduced pt (r) as the effective trion Rabi frequency related to one trion cell of size At Pt (1‘) = Ni 03%) - (2.51) With the presence of the initial electron, the ground state of the system to second order perturbation theory is E0(re) = ——/d2r m, (r) — o, saws/it)? —— —/ er |9t(re)|2, (2.52) where 6 (r,At) is the two dimensional unit step function, i.e. unity for r 6 At and zero else. We can simplify this expression further to find prfi +Nth(re) N agape). E0(r€):——2—A.r A17 — t At (2.53) The total second order energy shift at the electrons position re consists of three terms: First of all, we discover again a macroscopic shift due to the coherent excitation of excitons. Moreover, the trion resonance causes a shift that is stimulated by the 75 degeneracy factor Nt. The exciton shift has to be corrected, since it is not allowed to count the exciton cells that are displaced around the electron. The number of these cells is exactly M, with the Stark shift of one single exciton cell inside At being (28 (re) /A3. The second term accounts for the excitonic shift that is replaced by a trionic shift. Therefore, the first two terms give the total excitonic contribution, while the third term represents the shift due to the bound trion state. We define the optical potential for the electron V (re) as the total optical Stark shift at re, measured with respect to the total optical Stark shift when the electron is placed at a node of the intensity pattern, where the electron sees no light and which therefore simply gives the excitonic background shift. In this way, the optical potential varies with the electron’s position as 1 1 _ 92 (re) V .= ’92 ——— =—N 0 A 2’4 (re) t 0(1‘e) [A13 At] t At ft( t): ( 0 l Where the factor ft (At) is defined as ET < 1 ,At > 0 (red) ft (At) = (2-55) ET + At 2 1 ,At 3 0 (blue) This is definitely an intriguing result with no analog in atomic systems. Because of its importance, we will reconstruct it based on Fig. (2.7). On the left, the macroscopic background shift is depicted. It is constant in space because of the coherent excitations of excitons. It gives the total shift of the system when no conduction electrons are present or, equivalently, when the conduction elec- tron is at the node of the laser light. When on electron is present (right side of Fig. (2.7)), the optical potential starts out from this macroscopic shift and shows a very different behaviour depending on the sign of the detuning from the trion resonance At. For red detuning the potential is attractive, whereas it is repulsive for blue de- 7'6 + NtQZQ (Fe) + —9—Nt92 (Fe) A3 _NEQ" _Ngfi2 V (Te) A3 A33 v A —o \ iV (T8) 4 Ntogm) Ntflcz) (Fe) Ax lAtl Figure 2.7: Level shifts due to the optical Stark shift. The colouring of the level shifts indicates, if the laser frequency w is chosen below (red) or above (blue) the trion resonance. The presence of the trion shift causes a lower collective exciton shift, but an additional trion shift. tuning. The electrons will be attracted either to the antinodes or the nodes of the intensity pattern. The optical trapping potential for electrons we study here shares this property with the optical potentials for atoms we analyzed earlier. However, new physics arises from the factor ft (At). This factor can be smaller than one (red detuning) or bigger than one (blue detuning), and consequently it causes a reduction or an enhancement to the potential depth, respectively. As it can be seen from Eqn. (2.54), the effects of the trion resonance and the exciton correction counteract, if the laser frequency w is tuned below the trion resonance. If the detuning from the trion resonance is in the blue regime, however, i.e. A; < O in our notation, the two effects actually add up in their contribution to the potential depth. Of course, there are subtleties that we have to take into account. As we approach the exciton resonance, 77 the potential depth increases, but. at the same time the probability of creating real excitons in the system goes up, in contrast to virtual excitations of the two level cells. Moreover, we have not addressed the issue of spontaneously emitted light and the subsequent heating processes so far. We will make up for that shortcoming later, when thoroughly discussing the properties of the optical potential in chapter 4. Still, we have introduced here the essential ideas that make this optical potential for carri- ers feasible, and encountered a novel feature that distinguishes this type of potential from the conventional optical trapping potentials for atoms. 78 Chapter 3 Effective Hamiltonian from second order perturbation theory This chapter will be essentially devoted to a derivation of an effective Hamiltonian Heff for a conduction electron embedded in a semiconductor QW coupled to an electromagnetic standing wave. The interaction with the light field is treated pertur- batively to second order, in both the eigenenergies and the eigenstates of the coupled system. This approach is valid as long as the Rabi frequency for a creation of a bound trion is small compared to the detuning from the trion resonance. The light field induces transitions to bound and unbound trion state with a small probability, according to our perturbative method. Therefore, the basic ingredients for this derivation will be the matrix elements for transitions to the bound trion states, as defined in Eqn. (1.47), and unbound trion states, as given in Eqn. (1.67). As we have observed in the discussion of the toy model, the exciton continuum with its macroscopic dipole moment is crucial to capture the physics of the optical dipole potential for the conduction electron. Since the overlap O (K,ke) = (K|K, kg) of a bound trion state |K) with the continuum states |K, Re) is non-zero, we have orthog- onalized the continuum states to the bound trion states, so that we can treat these 79 two different excitations as distinct levels in a three level system. In the next step, we will present Feynman like diagrams that illustrate the basic processes contributing to the second order Hamiltonian. The diagrams are meant to provide an intuitive understanding of the calculations to come that lead to the effective Hamiltonian and to the corresponding Schrodinger equation for a single electron. As it will turn out, this Schrodinger equation features some striking physical properties and will be the focus of the discussion in the subsequent chapters. In the course of our calculation we will often take the limit Q ——> 0 for the photon’s wavevector, since optical wavevectors are in general small compared to the the electron’s wavevector k: while Q scales as the inverse of the wavelength Q = 27r/A z 107m'1, the electron wavevector is of the order of the reciprocal lattice constant k = 27r/a x 10mm"1 and thus much bigger than Q. In the two dimensional QW system we study in this thesis the validity of this approximation can be further extended with respect to bulk systems, since translational invariance is given only in the QW. The in-plane wavevector Q depends on the angle between the laser and the QW, and by tuning this angle, we can easily control Q without changing the energy of the incident photons. In this spirit, it is easy to find a configuration in which the recoil energies can be neglected with respect to the detuning from the trion resonance. 3. 1 Perturbation theory Let us first explain the general scheme for the upcoming derivation of the effective Hamiltonian Heff of the coupled semiconductor - laser system. Following the stan- dard procedure of perturbation theory, the full Hamiltonian H of the system we are about to study can be split up into the unperturbed problem H0, which basically describes the conduction electron in the QW system, whose solutions are known and for simplicity taken to be two dimensional plane waves in the effective mass approxi- 80 mation, and into the perturbation V, induced by the light field, which gives rise to the formation of bound and unbound trions. Introducing the conventional perturbation parameter A, the full Hamiltonian H can be expressed as H: Hod-AV, (3.1) and the corresponding time—independent Schrodinger equation reads H |k) = E(k) lk). (3.2) In the next step, we expand both the eigenvalues as well as the eigenstates to sec- ond order perturbation theory. For the energy eigenvalues we introduce the notation E (k) = Em) (k) + AEm (k) + A2E(2) (k), (3.3) where the orders of perturbation theory in the energy eigenvalues are given by - 2:3: E”) (k) = (k(0>lv|k<0)> =0 (3.5) .. (0) V km) 2 E12) (k) n¢k ll (3.6) The zeroth order E (0) (k) simply represents the dispersion of a free electron of effective mass m2. We have already set the the first order energy correction E (1) (k) to zero, since it projects a single electron state |k(0)> onto a bound or unbound trion state, which is assumed to be orthogonal to the free electron state ’k(0)>. In the same 81 fashion, we analyze the new eigenstates of the system to second order k) = (140)) + A'k(1)>+ A2 lk(2-)>. (3.7) For the special case that the first. energy correction E (1) (k) vanishes - the case we con- sider here — the general solutions for the first and second order eigenstate corrections in terms of the zeroth order eigenstates are given by k(1)> : :15de 0)> (3.8) nsék ' V 1 V. V .191)- :ze—ggggln H140): <39) nyéklaék laék lk For convenience, we introduced the shorthand notation v,,,, = ' The starting point for the effective Hamiltonian Heff is Heff = Z (E(O) (k) + E(Q) (k)) [16”) (140)) + Z E(O) (k) (1140)) (162)] + he] , k k (3.12) where all orders higher than second order have been dropped as well as all terms involving the first order corrections to the eigenstates ’k(1)>, whose eventual projec- tion onto electron position eigenstates Ir) give zero because of orthogonality, when 82 computing the effective Schrodinger equation for the single free conduction electron. In addition, the effective second order theory should also include the first order cor- rections to the bound and unbound trion states, because these are coupled to zeroth order electron eigenstates. However, in the Appendix 1.3 it is shown that these terms are higher order corrections that are proportional to ~ 1 / Afi- Our model is set up for large detuning values and only respects contributions of the order ~ 1 /A1;,t- In order to keep track of the various contributions, we split up the effective Hamil— tonian ”Heff into three terms as follows Heff=H1+H},+H§I. (3.13) Note that in the following we will drop the superscript to indicate the perturbation theory order of the states, since all expressions will be given in terms of the zeroth order solutions, i.e. the free electron states. The first term simply describes the altered dispersion of the electron due to the mixing with the excited trion states HI = 2 (E(“) (k) + E(2) (k)) |k) (k| (3.14) k This term is diagonal in k—space; therefore, it cannot lead to the optical potential in r-space, because, when sandwiched between electron position eigenstates Ir) and |r'), it only depends on r —- r'. However, an optical potential requires terms that depend on r+r’ instead. The next two contributions, 71;] and 71%] originate from the second term in Eqn. (3.12), which we have expressed in terms of the zeroth order eigenstates solely and subsequently split up into a diagonal Hh and off—diagonal contribution 71%,. They are given by V ' 2 ‘ H11 = — 2 El“) 1k> (kIZ ' "' (3.15) k ’2 17a: EU: 83 and 713’, = 2 EU” (k) E cg, |q) (kl + 11.0, (3.16) k q;£k where we have defined the quantity ch as Vql Vlk . (3.17) Equlk qu:Z laék The term H31 will be of peculiar interest, because it is the only one that is not diagonal in k—space. As it will turn out, the optical trapping potential for the electron originates from the off-diagonal nature of H? 1. 3.2 Orthogonalization of the excitonic continuum The exciton continuum states |K,c) that we consider in the effective Hamiltonian Heff have to be orthogonal to the bound trion states which we denote as |K); therefore,|K,c) describes an electron - exciton pair with total crystal momentum K, properly orthogonalized to the bound trion states IK) according to (KlK’, c) = 0. (3.18) We can achieve the required orthogonality relation by defining the continuum states |K, c) in the following way lKacl = NC (lKvkel ‘“ 0(K1k6) HQ) 1 (3-19) where we introduced the overlap integral as o (K,ke) = (K|K,k,.). (3.20) 84 Since we assume both |K) and |K,ke) to be normalized to one, the normalization constant NC reads NC = 1 . (3.21) ([1 — I<9(K.1 = 92—“ [5K,k+Q1+ (k) + 6K,k_QI— (1)] (2) When computing Et (k), the sum over the intermediate states is de facto a sum over K, which, however, is readily simplified due to the in-plane momentum conservation. As a consequence, we obtain right away <2) _ 9% 11+12 11-0012 Et (k) __4— 521:2 52mm2 3212 vac-Q)? (324) 2771? — 277% _ At 2111: — mt — At This result is what we already expected, based on our discussion of the Feynman-like diagrams. We defined the detuning from the trion resonance At as AtzeC—Eéf’t—fiwzec—EX—ET—hw. (3.25) We can simplify expression (3.24) with the following approximations: as already argued, the limit Q —> 0, is a valid approximation. Moreover, we exploit the fact that the functions lIi (k)|2, physically the squared Fourier transforms of the relative trion wavefunction, are strongly peaked around k z 0, as illustrated in Fig. (3.3). Thus, we will keep only the value k z 0 in the denominator, meaning that the kinetic energy terms in the denominator can be neglected compared to the detuning At. 89 k.r [1/;1fl\\ Figure 3.3: Plot of the function |I_ (k)|2: The variational parameters for GaAs were used. The value for Q corresponds to laser light tuned close to the trion resonance at an angle of 30° with respect to the 2—axis. Within this approximation, we find (2) 96 2 E: (k) z —2— Ct lI+ (k)lQ=01 (326) Where we used the fact, that in the limit Q —> O we do not need to distinguish between the two modes. Of course, for Q = 0 the relation 1+ (k) = I. (k) holds. We already recognize that the strength of the electron’s energy shift is enhanced by the fact that the electron effectively couples to many trion states. The position where the additional electron-hole pair is photo—created can be literally anywhere within the trion size. In real space this degeneracy is described by the extension of relative trion wavefunction taken with one hole on top of one electron. In momentum space, this degeneracy is accounted for by the corresponding Fourier transform, namely Ii (k). 90 Unbound trion states: The next step is to find the second order energy shift E(2 ) (k) that arises from the unbound trion states. The total center of mass mo— . mentum K is not sufficient to specify these states; indeed, we can take the electron momentum k6 as an additional quantum number. Therefore, we aim at calculating __ZZ (lech) (KclVlk) (3.27) EK ,ke;k Since the evaluation of E?) (k) is rather lengthy and requires some approximations, a detailed derivation of EC?) (k) is presented in Appendix H. It is shown that E512) (k) can be written in the following form . 2 2 (2) _— QO 90 2 EC 0‘) — ”Arm— A: ‘21:;X l(Cm 41?) lI-l- (k)IQ=0 Q +2139... l0 (k + Q k) I+1Q=o (3.28) where we introduced the quantity N, = 12. (3.29) 7mm as the number of excitons that fit into the sample without spatial overlap. Therefore, we can recognize that the presence of the exciton resonance accounts for a constant background shift that is simply proportional to N33, which describes a macroscopic enhancement factor for the intrinsic shift Qg/Am. Physically, this macroscopic en- hancement factor arises from the fact that the exciton is a coherent excitation over the whole sample and thus carries a macroscopic dipole moment. Moreover, to shorten the notation, we introduced the quantity AI _ 11212 +A ' Re ‘37? I 91 where we at is the reduced mass of the trion 1712mm 2 ——'——— 3.31 M m; +7721 ( ) We will discuss the physical meaning of Xc (AI) in more detail later on. (2) . . . . Let us make some remarks on EC (k). Besrdes the macroscopic excitonic. back- ground shift, it contains competing terms with opposite signs which include infor- mation about the relative wavefunction of the bound trion and arise from the or- thogonalization of the diffusive electron - exciton pairs to the bound trion states. These physical results would not occur if we had not applied the orthogonalization procedure to obtain two distinct states. Full second order correction: we can combine the second order energy cor- rections, obtained for the bound trion states in Eqn. (3.26) and for the excitonic continuum states in Eqn. (3.28). to give the total second order energy shift E (2) (k). In total, we write it as 92 1 1 xx A, E12) (k) = wee—A: — 9811+(k>13=0 i (Z. + ___! 1)) S) 2A—UQ. 10 (k + an 1+ (1315:.) (3.32) This result for E (2) (k) represents a major piece of the puzzle in the derivation of the effective Hamiltonian Heff. The next section will be devoted to a discussion of the terms H}! and H; 11 as defined in the expressions (3.15) and (3.16), respectively. 3.5 Effective Hamiltonian, Schrfidinger equation In this section we will present an analytic form of the effective Hamiltonian Heff. Before achieving this ultimate goal, we will give the missing pieces, namely the con- 92 tributions H}! and H; 1’ which are by definition diagonal and off-diagonal in k-space, respectively. The intermediate steps that give rise to a more compact form for them are presented in detail in Appendix I. In the Appendix it is shown, that the diagonal part H}! can be approximated by 1 . 96 (0) » 1111 e :13: 2 E 1k11k> 0 )(k| — NT——:Z|k) k _123 + .. .2 0(5: + ‘ “(AA )) 2 11+ (1312):.) 1k) (k1 k 9 QB 20: 2k: '0 (k + Q1 1‘) 1+ (k)|Q=0 |k> (kl (3.34) +2 As far as the off-diagonal part Hg] is concerned. we have derived the following form 92 (- Alf Hf] “ “’19 (A: + X: )) Z lI+ (k — Qll2Q=0 (lk —‘ Q) 0‘ + Ql + h-C-l 1’ k c o, + 103:1: ‘11; 10 (k, k + Q) 1+ (k — Q)IQ=U (lk + Q) (k — £21 + h-c-)(3-35) These results mark a major result of this research: We have obtained an effective Hamiltonian that approximatively describes our system, an electron in a semiconduc— 93 tor QW coupled to standing wave that is tuned close to the trion resonance. Since we consider the limit Q = 0, we recognize that the diagonal and off-diagonal part of the effective Hamiltonian Heff contain similar contributions. As a result, we will be able to combine them; this simplification and a further analysis of the features of the effective Hamiltonian will be done in the framework of the corresponding position space Hamiltonian. Our ultimate goal is to derive an optical trapping potential in position space. Therefore, we will switch the basis from momentum space to position space. In order to translate our result from k-space to r-space, we sandwhich our expressions in momentum space with electron position eigenstates Ir) and |r’). Here, we use the relation 9 (Pk 3.38 g (271)“ f ( A 94 We find that the matrix elements are given by 112v? 112 I : _ 6 _ / _ NT __0 _ I (r|H11r) 2m; (r r) 1 IA$6 (r r) 122 122 -. +1—(im6 (r — r') — —29 (it + X2131») mt (r — r') (3.39) and 122 . (r| H?! 1r') 2 +21“ch (x) cos (2QR) 1‘ 9(2) 1 KC (AI) __ __ —‘ ‘ 1 ‘ 2 .40 2 (At + Am )mt (x)cos( QR) (3 ) where we introduced the functions mt (x) and me (x) in. the following way 2 . m (x) = f (:73, elk" 11+ (k— @1320 (3.41) and 2 . m.- (x) = +11... (0) (E f 7% elk" 10 (1.1. + Q) 1+ (k — Q)1Q=O 13.42) They give rise to a nonlocal character of the effective Hamiltonian, as they connect the electron’s position coordinates r and r’ in a way different from the local expres- sion (5 (r — r’). The notation underlines the picture in which we will identify them: They can be interpreted as memory functions. We will address their specific physical meaning in the next chapter. We have been able to find an analytic form of mt (x), since in the limit Q = 0 the problem naturally becomes isotropic. \Ne specifically evaluate mt (x) in the case of the QW systems GaAs and CdTe. The results are plotted in Fig. (3.4). Qualitatively, we see that both systems exhibit a. very similar character; for both 95 2.5: w ..1 gr I .A_L LasLl — GaAs — CdTe 1.0? L L L L41“; A I 1_L. L4- A 0.5} .1 p p b . A 1 A l a A L 4 A 1 A l = 4 L A A A l 4 6 8 10 X [(11)] CL N Figure 3.4: Memory functions 7m (x) for GaAs and CdTe. Note that the natural length scale, the 3D donor Bohr radius a D, is slightly different for the two systems. GaAs and CdTe, mt (x) falls off on a scale of approximately ~ 3a D- We will discuss this behaviour in more detail in the next chapter and give a physically intuitive explanation for this characteristic lengthscale of ~ 3a D- The second memory function me (x) that we encounter arises from the hybrid mixing of the diffusive electron-exciton pairs with the bound trion states, as the overlap function in its definition indicates. The exact value of me (x) depends on our choice of the parameter at that we introduced when simplifying the calculation of the overlap integral and, in principle, its best value should be obtained from a variational calculation. For consistency we approximate the actual value of at by tuning it to a reasonable trion binding energy according to the expression (1.30). Based on this approach, with ET 2 2.0meV and ET = 3.6 meV for GaAs and CdTe respectively, we estimate the values for (IT as 0T m 1.8 (1D and aT z 2.3aD. The corresponding 96 GaAS: (1,: 1 .8 CdTe: a,=2.3 X [01)] Figure 3.5: Memory functions me (x) for GaAs and CdTe. Note that the natural length scale, the 3D donor Bohr radius (11), is slightly different for the two systems. results are shown in Fig. (3.5). We see that the functions me (x) are very similar to the memory kernel mt (x) that arises from the trion resonance. We will exploit this fact for a further simplification by approximating the two types of memory kernels that we encountered in our calculation as equal, i.e. mt (x) z me (x) z m (x). Proceeding from the matrix elements given in expression (3.39) and (3.40), respec- tively, we can present the effective Hamiltonian of the system Heff in the following way 521’ h2v? _ . U 2 ”Heff — _NIE +/c:AA r |r) )<— 277% ) (r| I —Q% i +———-— Z:H/(Frdgr cos2 Qr + r mt (r — r’) |r) (r'l At 2 92 +250“ [d2rAi d2r cos 22(Qr + r ) 7m (r — r’) Ir) (r’l (3.43) I 97 where we used the trigonometric relation 1+cos (2:13) 2 2 cos2 (:1?) to express the result in a more compact form. The first term represents the constant background shift owing to the coherent excitation of excitons over the whole sample that we previously encountered in the toy model. The last two terms indeed carry the signature of the intensity profile and mark the optical potential for the conduction electron. Compared to the optical trapping potentials for atoms, however, there are important differences: The first one arises from the obvious non-locality of the potential, governed by the memory functions mt (x) and me (x). The second one stems from the presence of detuning dependent correction factors. We will discuss the details and the consequences of this result in the next chapter. We acknowledge the presence of the macroscopic excitonic background shift, but we will drop it in the following, since it only marks a constant shift in energy and has no influence on the electron’s kinetic or potential energy. Moreover, as announced above, we set the two memory functions equal for simplicity, because they show a similar behaviour. In this way, we find an effective Hamiltonian Heff that contains two terms, a conventional kinetic contribution as well as a term that arises from the interaction with the standing wave 112:7? Heff = /d2r Ir) (— 27,1,“ ) (r| 8 I r ) m (r — r’) Ir) (r'l (3.44) Here, we introduced the quantity fc (At). Since the detuning from the exciton res— onance Ax is uniquely related to the detuning from the trion resonance At via the trion binding energy ET, we can introduce the correction factor fc (At) as a function 98 of At only A A fc (At) = 1+ Xc (AI) —t — 2—‘ . AI = A, + ET (3.45) A1: A: The correction factor fc (At) pays tribute to the presence of the exciton continuum level in our system and will be subject to a further analysis. In particular, it will be compared to the correction factor ft (At), Eqn. (2.55), predicted by the toy model. For completeness, we state the effective Schrodinger equation for an electron with effective mass m: and wavefunction Q! (r) in the system we investigate: h2V2 r + r' * 2mg s22 . ‘11 (r) — ngc (At)/d2r' cos2 (Q ) m (r — r’) \I/ (r’) = E‘IJ (r). t (3.46) This equation underlines that a lot of intriguing physics is ahead of us, but, for now, it shall mark the end of this chapter. 99 Chapter 4 Optical potentials for carriers in semiconductor quantum wells We have predicted the existence of an optical trion—mediated trapping potential for carriers in a semiconductor QW system following two different approaches. First, based on an analogy to the well-established optical potentials for atoms, we have developed a toy model which stressed the importance of including the exciton level and its coherent character in our analysis. Second, we set up an effective Hamiltonian and extracted the corresponding Schrodinger equation that describes a conduction electron which virtually mixes with the extended bound and unbound trion states and, as a consequence, experiences light shifts. In this chapter we will cover the key points that feature this novel optical potential for carriers in a QW. Throughout this analysis, we will substantiate the results with simple, but plausible and physically intuitive explanations. In a first step, we will address the origin and the effects of the correction factor fc (At); we will highlight the stunningly comparable behaviour that our two different models predict in this respect. We will then turn to the non—local character of the potential embodied by the kernel m (x) and justify its presence with the properties of the trion wave 100 function and the light masses of the particles under study. Although the nonlocality is a striking feature of this potential, we will neglect it at first in order to predict the actual potential depth of this optically induced trion-mediated potential. Here, we will stress the key ingredients that support a deep potential with controllable dissipation. In the last section, we will characterize the potential using a harmonic oscillator approximation to show the quantization of the electron motion. Since this approximation is a well-established in the analysis of optical potentials for atoms, we will be able to draw analogies and emphasize the differences to optical dipole potentials for atoms. Finally, the fact that we neglected the non-locality within the harmonic oscillator approximation will be justified by a variational calculation which explicitly incorporates the memory function m (x) in the analysis. 4.1 Influence of exciton level In both theoretical approaches the toy model as well as the effective Hamiltonian method, we encountered detuning dependent factors which we called ft (At) and fc (At) respectively. They take into account the presence of the coherent, many- body excitation of the exciton level and its competition with. the formation of bound trions. The factors strongly affect the actual potential depth. In this section, we will Show that the predictions of our two distinct models are in an excellent agreement. Let us first summarize the expressions we have obtained. It is convenient to present ft (At) and fc (At) in terms of the dimensionless, system-independent quantity ET ET __ = ___— 4.1 g: 101 which obeys the conditions l'm =1. l'm :0. 4.2 aft—>05 , Athocé ( ) In terms of the quantity g, the toy model yields the compact result while the effective Hamiltonian approach leads to 262(6— 1—1n(§)> (5 — 1)2 fc (i) = (4.4) Although these two results might look very different at a first glance, it turns out that they agree not only on a qualitative, but even on a quantitative level. Fig. (4.1) displays the results of our two models in terms of the dimensionless parameter a: = At/ET, the detuning from the trion resonance At in units of the trion binding energy ET. It is positive for red detuning and negative for blue detuning, and related to 5 via the relation g = 1/(1+ :13). For red detuning (At > 0), both models predict that the correction factors and therefore the potential depth approach zero as the detuning from the trion resonance goes to “infinity”. On resonance, i.e. for At = 0, both correction factors become one: on resonance, the presence of the exciton level has no net effect on the potential for the electrons, so that there is no reduction nor enhancement in the potential depth. When we go to blue detuning (At < 0), where the potential swaps from attractive to repulsive, the correction factors start to increase non-linearly on the blue side, resulting in a drastic enhancement for the potential depth seen by the conduction electrons. For detunings closer to the exciton resonance than at = —0.9, the correction factors even cause an enhancement of more than a factor of 10, which directly maps 102 4- — fc(x) . — 1300 l 3- 3 i «s . 3 21 “3 t 1: i 0.- -__n-_-__.___ _. _._________ -____-____s . _ - . n: —1.0 —0.5 0.0 0.5 1.0 1.5 2.0 XzAt/ET Figure 4.1: Predictions of the toy model and the effective Hamiltonian approach for the detuning dependent correction factors. onto an increased potential depth by the same factor. As it will turn out later, this rather strong enhancement is crucial to obtain potential depths that dominate over competing effects such as the recoil energy or the thermal energy. In conclusion, we see that the predictions of our two models concerning the role of the coherent exciton level converge to a beautiful agreement. Therefore, in the following we will use only the correction factor ft (g) that we derived in the framework of the cell model in order to specify the potential due to its simpler, convenient algebraic form. 103 4.2 Memory effect: Non-locality of the potential Let us turn to another peculiar feature that is included in the effective Schrodinger equation (3.46), its non-locality. It is governed by the kernel m (x), that we identified and labeled as a memory function. According to its definition, it represents the interaction strength per unit area of a second order process in position space: the electron mixes with the extended trion state and “jumps” back gain. we have two physically intuitive ideas to explain the physics that cause this non- locality with its characteristic length of ~ 3a D: as seen in Fig. (3.4) and (3.5). The first idea is to associate the non—locality with the momentum kick that the virtual trion obtains from the photon absorption. After this kick, the virtual trion ‘travels’ for the period of its lifetime. The lifetime of the virtual excitation of the trion level is approximately ~ l/At. We work in the electron’s restframe, since the off-diagonality is independent of the thermal momentum of the electron ke. We can estimate the distance d the virtual trion travel as h|Q|_1_ _ 27th d. z _ _— Tllt At filtAHAt (4.5) where we expressed the magnitude of the in plane momentum Q in terms of the in plane wavelength A... Only close to the trion resonance, i.e. for small At, d can be comparable to a D, but for larger detunings it becomes smaller and falls short in explaining the nonlocality of the potential. However, we can give an even simpler explanation for the potentials non-local character that is essentially independent of the detuning. The optical potential is not diagonal in r-representation because the trion is a rather large object (N 20.1)) and the effective masses of the particles are relatively small. When the very light electron mixes with the diffusive extended trion state, it wiggles ‘inside’ the trion complex. Since the characteristic size of the trion matches perfectly with the char— 104 acteristic length of the kernel m (x), we believe that this simple picture explains the nonlocality of the optical trion-mediated potential. In principle, one can argue that optical potentials for atoms are nonlocal as well, but, due to the large proton mass and the small atomic size, the nonlocal effect is not observable. Let us mention that the form. of the memory function 772 (x) is well approximated by a Gaussian of the form 9 (x) z; (re-bf? (4.6) We have obtained the numerical values for the fitting parameters a and b based on a least-squares fit; the results are listed in Tab. (4.1). In Fig. (4.2) we present the memory function m (x) and the corresponding fitted Gaussians g (x) for both GaAs and CdTe. The very good agreement will allow us to simplify further calculations that explicitly include the kernel m (x) by using the fitted gaussian g (x). I GaAs I CdTe a 2.727 2.262 b 0.109 0.085 Table 4.1: Fitted parameters for g (x) for GaAs and CdTe in effective atomic units. So far, we have only considered the form of the nonlocal memory effect in the Q —+ 0 limit. For completeness, we present a numerical solution for a finite value of Q, corresponding to an angle of 70° between the laser and the QW plane. We can see that the Q ——> 0 limit is perfectly justified. The numerical solution is centered around k = 0 and shows a characteristic size of ~ 30. D, just as the approximative Q = 0 solution. It is displayed in Fig. (4.3). One might be led to think that the omnipresent quivering motion of the electrons due to their mixing with the excited trion state might impose a serious reduction on the localization effect of the potential. However, the characteristic length of this effect is one to two orders of magnitude smaller than the periodicity of the potential A“ / 2, depending on the chosen angle between the incoming laser and the QVV plane. 105 10 X IUD] Figure 4.2: Memory function mt (x) (solid lines) in the Q = 0 limit and fitted Gaus- sians g (1:) (dashed lines): GaAs is displayed in blue, CdTe in orange. Indeed, after this analysis of the nonlocality of the potential, our next step will be to neglect the nonlocality in order to prove the possibility to trap electrons and to estimate the actual strength of the potential. 4.3 Potential depth Although certainly an intriguing feature, we will neglect the non-locality of the optical trion-mediated potential in this section to present one of the main results of this thesis: the depth of the trapping potential. In order to achieve this goal we make the replacement m (r — r') —+ xd (r — r') , (4.7) which enforces a corresponding local potential for the electrons. The dimensionless quantity X takes into account the presence of the trion resonance and its electron-hole 106 y [do] 5 10 X [00] 10 Figure 4.3: Memory function mt (x) for a finite value of the laser photon momentum Q projected onto the QW plane. We used the parameters for GaAs. correlation effects. By definition, it is given by 2 . x= /d2xm(x) = /d2x/ (:52ezkxlr+(k)|5=0. (4.8) The integration is readily done and we obtain the result 2 X = Ifi/dzrcpbhwm) (4.9) The correlation factor x appears as the integral over all possible three particle trion configurations corresponding to one electron and the hole on top of each other. This is the natural extension of the two particle exciton picture, where at the moment of photocreation the electron and hole are at the same spot. The coefficient x takes into account that the exciton needs to be created somewhere inside a reasonable range to possibly bind to an electron and form a bound trion; the range is essentially governed by the relative trion wavefunction 4,01, (737", r): the arguments of 9% simply 107 95 9o 3 7 85 sd * I . 61T“‘_““0§77”“”7”OE’” "05 _7’ as 7 rd 0'=m;/m’,", Figure 4.4: Enhancement factor x as a function of the electron hole massratio a. underline that the wavefunction is evaluated with one electron and the hole on top each other, so that the Hylleraas coordinates we used in the formulation of gob boil down to s = t = u = r. The factor of 1/\/ 27r arises from the normalization of the overall angular degree of freedom. Inserting the relative wavefunction given in Eqn. (1.26), we obtain a compact analytic expression for X 2 o2 + 204/3 + 67 0,4 x=2mV2( aim that depends only on the values of the variational parameters (1,73 and y. Conse- quently, it can be expressed as a function of the electron hole mass ratio a = m: fin; solely. The result for X = X (a) is presented in Fig. (4.4). We recognize that X can take on rather large values: we obtain ~ 80 for GaAs and even ~ 90 for CdTe. At this point. we can close the circle between the cell model and the effective 108 Hamiltonian approach. In the toy model, the effective range around the electron in which the creation of an exciton leads to the formation of a bound trion was simply modelled by a step function around the electron of the trion size At. Based on this simplified picture, the fact that one electron is effectively coupled to many trion states gave rise to the definition of Nt = At/Ax, the number of excitons that fit into the bound trion state without spatial overlap. In principle, the effective Hamiltonian model yields the same result in terms of the relative wavefunction gob. Here, the step function is replaced by an integral over all the possible combinations in which the conduction band electron and the photoexcited exciton form a bound trion configuration. Therefore, based on their physical interpretation, the quantities Nt and X are equivalent. They express the degeneracy factor of the trion levels the free electron is coupled to, since the spot where the additional electron—hole pair is created can be literally al'iywhere within the trion size. While Nt was found to be ~ 10, the quantity X is in the range ~ 80 — 90, thus a factor of 8 —9 times bigger than the corresponding prediction of the toy-model. This is far away from being a perfect agreement, but still Nt and X are within the same order of magnitude. In addition, Nt only serves as an illustrative ingredient in the toy-model, but does not affect our results. For the numerical calculations, we will use the enhancement factor X, since it contains the actual information about the trion wavefunction. Based on the local limit in Eqn. (4.7), we recover a simplified Schrodinger equation for the electron with a local potential that obeys the profile of the laser intensity pattern 2 2 ’3' V m)—x:—(Zf..cos?(er)v(r> =E\I’- (“1) 4 :1: 27726 Consequently, the trapping potential V (r) varies with the electron position as V (r) = V0 cos? (Qr) = —6Atfc (At) cos2 (Qr), (4.12) 109 where we introduced the saturation parameter c for the electron trion transition 2 X (Bay. (4.13) The parameter 6 essentially describes two effects: Since the factor X08 can be viewed as an effective Rabi frequency for the electron trion transition, in the limit At >> Ft it gives the probability to be in the excited trion state. By choosing 6 sufficiently small, the dissipative aspects of the electron dynamics due to spontaneous emission of pho- tons are efficiently suppressed. Photon scattering occurs at the effective spontaneous emission rate I‘se = 61}, with I} N 10103—1 being the natural linewidth of the trion level. While I‘se scales as ~ 1/A2, the scaling behaviour of the trap depth V0 is very different for red and blue detuning, owing to the background-polarizability correction factor fc(At). While l/b scales as N l/Ai.r2 for red detuning, it scales as N 1/ At only for blue detuning. Therefore, intense blue—detuned light far enough from the trion-resonance provides strong confinement with minimal dissipation with respect to the trion-resonance. However, in this case one has to take into account the presence of the exciton resonance and the subsequent creation of real excitons in the system. we will do so later on by estimating the probability to excite one exciton per cell. Note that the degeneracy factor X gives rise to a reduction of almost two orders of magnitude in the required intensity to reach a given trap depth. In addition, a small 6 value prevents power broadening of the trion resonance and subsequent creation of real trions. Thus, we are interested in the large detuning, low saturation limit where the excess conduction band electrons are the only real particles in the system, whereas excitons and trions are excited only virtually. In the following, we will always associate the potential depth V0 = —€Atfc (At) (4-14) 110 with a fixed value of the saturation parameter 6 in order to control the dissipative effects. In this representation, the potential depth increases with the detuning from the trion resonance At, because simultaneously the Rabi frequency (20 is increased. For red detuning the only limitation is set by the maximum laser power available, while for blue detuning we will have to take into account the increasing probability of creating real excitons in the system as we approach the exciton resonance. Below, we will discuss both limitations. Before outlining the differences with respect to conventional optical dipole poten- tials for atoms. let us mention the similarities first. The potential V (r) given in (4.12) follows the spatial dependence of the intensity pattern of the laser and its depth is proportional to the applied laser intensity. For red detuning (At > 0) the potential is attractive, the electrons are attracted towards the bright spots, whereas for blue detuning (At < 0) the potential is repulsive, so that the electrons tend to accumulate at the nodes of the standing wave pattern, i.e they seek the dark spots of the intensity profile. Now, let us reiterate the specific properties of the trion-mediated optical poten- tial for electrons embedded in QW system that are distinctive and not present for conventional optical potentials for atoms. The major difference arises from the pres- ence of the factor fc (At). This is substantially underlined in Fig. (4.5), where the potential depth V0 is given for both red and blue detuning for various values of the control parameter c, which corresponds to a constant rate of spontaneous emission. The detuning At as well as the the potential depth l/b are represented in terms of the trion binding energy ET, which is of the order of ~ meV for typical semiconductor QVV systems. For red detuning the effects of the exciton level and the trion level on the shift of the electron’s ground state energy with respect to the excitonic background shift counteract. As a consequence, the potential depth I?) shows a saturation behaviour, 111 — e = 0.05 V// l 0.08- 1 0.06 Vo/Er 0.04% Vo/ET 0.0- . -- —l.0 -0.9 —0.8 —0.7 —0.6 A: / Er Figure 4.5: Potential depth V0 for red (At > 0) and blue (At < 0) detuning in terms of the trion binding energy ET for various values of the saturation parameter 6. 112 reaching the value eET at maximum, if sufficient laser power is provided. To reach the saturative regime we estimate the necessary laser intensity I to be I z 6 X 105VV/cm2 for both GaAs and CdTe QVV systems. The physics is different for blue detuning: The effects of the exciton level and the trion level on the shift of the electrons ground state energy with respect to the excitonic background shift add up, resulting in a potential depth that can reach several meV. For blue detuning and 6 held constant, the potential depth keeps on increasing when increasing the detuning from the trion resonance, thereby showing a highly non-linear behaviour. Of course, in doing so one approaches the exciton resonance whose natural linewidth I}; for both GaAs and CdTe is approximately I‘m/ET ~ 0.03. We can estimate the time-averaged probability for the excitation of real excitons (PI) using the two-level Rabi model. Referring to one exciton cell of size Am, the time-averaged probability that this “two-level” system is in the excited state reads > = f/X' (At/ET)2 4 (1 + At/ETl2 + 2e/x (At/ET)2 + (It/ET)” (P, (4.15) where we expressed all energies in terms of the trion binding energy ET and used the relation 0?, = 3A3. (4.16) In this way, (P55) essentially depends only on At, once the saturation parameter 6 is fixed. The results for (P1) are presented in (4.6). At this point, we have built up a powerful scheme which suggests that we can pick a (small) value for the saturation parameter c and tune the detuning from the trion resonance At. We will find a combination of the potential depth V0 and the time-averaged probability for exciton creation (PI). Four possible combinations are listed as examples in Tab. (4.2). Let us just describe one specific example explicitly: 113 (Px) Figure 4.6: Time averaged probability of formation of real excitons per exciton-cell as a function of the detuning from the trion resonance At for various values of the saturation parameter c. for e = 0.05 we find that a potential depth of V0 % 0.45ET is feasible, while the probability to excite an exciton per cell is still only about 1.2%. Here, laser intensities of I z 2.5 x 103W/cm2 and I z 7.7 x 103W/cm2 would be required in GaAs and CdTe respectively to achieve potential depths of VO :3 0.9 meV and V0 z 1.6meV. j A, = —0.9 ET] A, = —0.95 ET 6 = 0.05 1.2%, 0.45 4.7%, 0.95 e = 0.01 0.2%, 0.09 1.0%, 0.19 Table 4.2: Combinations of the time averaged probability for exciton creation and potential depth are presented as pairs in the form (Pm), V0 for two values of the saturation parameter 6 and the detuning from the trion resonance At. Lastly, we note that the excitons are created at the bright spots of the intensity pattern, whereas the electrons are located at the dark spots for blue detuning. This combination of a controllable, low excitation probability with significant spatial sep- 114 aration should strongly suppress the interaction of the electrons with real excitons in the system. The strength of Optical dipole traps for atoms is routinely measured in units of the single photon recoil energy E R = 71.2Q2/2m2. It simply sets a natural energy scale for the problem. Translational invariance along the QW implies conservation of the in- plane momentum only, which reduces the average recoil kick an electron experiences in the process of spontaneous photon emission. By averaging over the solid angle, the average recoil energy is (E R) = (2 / 3) E R. The small effective masses of the electrons, though, give rise to huge recoil energies, compared to atoms. The effective electron masses are about seven orders of magnitude smaller than the masses of rubidium atoms, one of the standard systems for atomic physics and quantum optics so that the recoil energies are correspondingly larger by a factor of ~ 107. For GaAs and CdTe, the average recoil energies amount to (E R) % 0.29meV and (E R) z 0.16meV respectively. Referring to the specific example just given above, the potential depth V0 exceeds the recoil energy E R. by a factor of three and even 10 in GaAs and CdTe respectively. Based on atomic optical potentials, many ground-breaking experiments have been performed. Some examples are the evidence for macroscopic Quantum interference [42], the observation of Bloch oscillations of atoms [43] and the investigation of the quantum phase transition from a superfluid to a Mott insulator [44] to name a few. In principle, they all relied on the power and versatility of optical potentials. The maximum potential depths in these experiments have been 2.1ER, 6E R and 22E R respectively. The fact that, despite the huge recoil energies involved in our system, we can still predict potential depths of several E R substantiates the strength of the trion-mediated trapping potential mechanism we investigate here. So the question that naturally arises in this context is: Why is the trion—mediated potential actually so strong? The bare interband dipole moment do in our system is N 66A, while the dipole 115 moment for an atom datom can be approximated based on the Bohr radius in vacuum as ~ 5 X 10‘16A. However, the effective dipole moment d for the electron-trion transition is enhanced by the degeneracy factor of the excited trion state. In total, we can estimate that the squared effective dipole moment in our system exceeds its atomic 2 atom ' counterpart by three to four orders of magnitude, i.e. d2 2 X118 as 103 — 104 x d Let us briefly summarize the parameters that are in favour of a deep potential depth: In general, a larger value for the saturation parameter yields a deeper po— tential, but at the same time the dissipative effects due to spontaneous emission of photons become more important. With 6 held constant, one can deepen the potential depth V0 by increasing the detuning from the trion resonance At, provided that suffi- cient laser power is available. Note, that when following this technique the behaviour is very different for blue and red detuning because of the correction factor fc (At). Fi- nally, host materials with higher trion binding energies allow for considerably stronger potentials. 4.4 Harmonic oscillator approximation In principle, the Bloch theorem states that the exact solutions to any periodic Schréidinger equation take on the form of non—localized Bloch wavefunctions 7.1-hug (r). They are labeled by a discrete band index n and a quasimomentum q within the first Brillouine zone of the reciprocal lattice. Upon translation by an arbitrary lattice vector R, Bloch functions are multiplied by a pure phase factor exp (z'qR). As a consequence, they are coherently extended over the whole lattice [40]. From this point of view, the next step we will take seems strikingly contradictive at a first glance and needs some clarification. In the framework of optical lattices of atoms, it has become well~established to approximate the atomic motion near the bottom of the wells by a simple harmonic oscillator that is thermally excited (see 116 for example [40, 58]). The basic idea is that for deep optical lattice potentials the atoms are tightly confined at a single lattice site which is approximately harmonic. The oscillation frequency who and the trapping frequency V1- : who/2w respectively which are easily obtained from this approximation have become figures of merit to characterize the optical potential. However, we note, that it. is only meaningful to talk about oscillatory motion if the trapped particle resides at one particular lattice site for a time at least comparable to the oscillation frequency [58]. This is precisely the regime where one has to describe the electrons motion quantum-mechanically. For atoms, this picture has already been justified experimentally: atoms have been successfully cooled down to the lowest bound state [45, 46, 47] entering a regime where their localized quantum wavepackets could be controlled in real time [48, 49]. As far as our system is concerned, it is easy to imagine mechanisms that lie beyond the scope of our model and do not appear in the Schréidinger equation formulated above: impurities due to defects in the lattice, phonons interactions, just to name a few. These effects can lead to decoherence which makes the picture of a single electron trapped at one lattice site more plausible by suppressing delocalization. Apart from that argument, an appropriate superposition of Bloch states, the energy eigenstates for single electrons, yields a set of Wannier functions which are well localized on the individual trapping sites. In this spirit, we will derive the essential quantities of the optical potential in the harmonic oscillator approximation to characterize its localization strength and to compare the results to typical values for atomic optical lattices. We will do so first in the limit of a local potential proceeding from Eqn. (4.12); in the second step we will explicitly include the nonlocality governed by the memory function 711 (x) and show that is is legitimate to neglect the nonlocality when describing the properties of an electron trapped at the bottom of the effective harmonic oscillator potential. 117 4.4.1 Without memory: Local potential Our considerations start out from (4.12). Without loss of generality, we align the in- plane photon wavevector along the sis-direction as Q = Qi‘. Expanding the cos2 (Qr) to second order around the local minimum at r = 0 (for red detuning At > 0) gives cos2 (Qr) z 1 — Q2$2. (4.17) The potential term in the Schrédinger equation becomes flat in the Q- direction and a simple harmonic oscillator in the fit-direction. In this approximation the Schrodinger equation consequently boils down to 52v? ‘ =1< 2771.8 4! (r) + V002 sin‘2 (6) 1:211; (r) : E\II (r), (4.18) where we expressed the momentum of the photon projected onto the QW plane in terms of the angle 6 between the 2~direction perpendicular to the sample area and Q according to Q = Qsin (6). By tuning the angle 6, one can easily alter the periodicity of the lattice a. = An / 2, while holding the potential depth of the lattice constant. The smallest periodicity is reached for two counterpropagating laser beams, which corresponds to the specific case where Q is in the plane of the sample and subsequently 6 takes on the value 7r/ 2, by definition the maximum value for 6. By choosing smaller values, one can control the validity of the harmonic approximation: a variation of the angle 6 has no impact on the depth of the potential since V0 is only affected by the total energy of the photons. However, the angle 6 does influence the periodicity of the lattice: making the angle 6 smaller leads to a bigger periodicity of the lattice, because the in-plane momentum Q decreases; equivalently, the in—plane wavelength increases. If the angle 6 is chosen sufficiently small, the harmonic oscillator approximation steadily improves, since the slope of the potential-well flattens out. 118 The Schrodinger equation describing our system has become the one for a simple harmonic oscillator in the indirection and we can readily identify m* fix/2:0 = ”.ng Sin? (6) 1 (4'19) where who is the oscillation frequency of the harmonic oscillator. Since the two dimen- sions are decoupled, we proceed by making the product ansatz ‘11 (r) = \le (:13) \I'y (y) for the electrons wavefunction and separate the energy eigenvalue E = Ex + E3, to obtain two decoupled one-dimensional Schrodinger equations. The solution to the Schréidinger equation in the g-direction h? a? ___—‘11. . = E 11;. ,_ 2771;05/2 y (y) y y (y) (4 20) is sim ly a lane wave with wavevector j: 2771’5E h. . C 9 Since we are about to study the localization effect of the optical potential, the fi-direction is of no further interest. Owing to the separability of the problem, we have simplified the problem to a one dimensional Schrédinger equation 51’ a? m; 2 9 ‘_“—_ 13(1‘) ‘1’ f—Cul} 1"”‘I/J; (.I‘) = E1:‘Px($), (4.21) 2771:0172 2 ’0 whose solutions are well known. The normalized ground state wavefunction of the one-dimensional harmonic oscillator in the Cir-direction is given by *, , 1/4 and KP“) (.r) = (BE—(751E) exp (—%i£9.r2) . (4.22) The level spacing of the harmonic oscillator is 2V - 2.‘ 2 6 V hay“) = h\/ GO in ( ) = 2\/V0ER sin2 (6) (4.23) me 119 In our context his)“, is the energy of local oscillations inside the trapping well. For a deep optical lattice with Vb >> (E3). the energy had/,0 of local oscillations in the well is large compared to the recoil energy and each well supports several quasibound st at es. I . | . .“ ] l 12) X f ‘x ‘\ g‘ 105 \ o [ \‘ s , N 03» \ \ i \ f: l [ \‘\ Afi 0 . 6 ' ‘\\ o 04) . 0.2 ‘2‘ 1] \‘ _ 0.0) " —35 —30 —25 —20 —1s —10 —05 00 A, [meV] Figure 4.7: The first three levels of the harmonic oscillator are compared to the potential depth (dashed black line) for CdTe and blue detuning: The blue branch shows the ground state level for the angles 6 : 200 (dark blue). 6 = 150 (blue), 6 2 100 (lighter blue). Similarly. the first excited (green) and the second excited level (orange) are depicted. Fig. (4.7) shows that for blue detuning in a CdTe system and appropriate val- ues for the angle 6, the first three energy levels of the harmonic oscillator can be resolved within the potential depth V0. As speculated above. the harmonic oscilla- tor approximation is favoured by deeper potentials V0 and bigger periodicities of the potential. In the next step, we examine the spatial spread AX of the ground state wave— 120 function AX .__ \/’,.,— : x/fi 1 4, (4.24) mewho (2772;14in sin2(9)l / The result for CdTe and various angels 6 is depicted in Fig. 4.8. For red detuning, the spatial spread saturates, simply because the potential depth follows a saturative behaviour. Typical values for the spatial spread AX are ~ 10a D, which is about five to ten times smaller than the corresponding lattice spacing a. 25~ — A X: 6=10° — A X: 6=15° .— 20- 52 —— A X: 6=20° ] >< <1 15- L.___. _i I _._ n_ _._._L __im+_1__.._ __ - - ___I- 0 5 10 15 20 A, [meV] Figure 4.8: Spatial spread AX of the harmonic oscillator ground state as a function of the detuning for a fixed saturation parameter e = 0.1 in CdTe and various values for the angle 6. For completeness we give the trapping frequency 11,. = who/271 according to 1 .- 2 V7. 2 EJVOER sm (6). (4.25) Typical trapping frequencies for atomic optical lattices are in the regime of ~ 100 kHz 121 [40]. On an absolute scale the potential depths and recoil energies in our system are several orders of magnitudes bigger which yield to trapping frequencies of the order of ~ 1010 — 1011112. As shown in Fig. (4.9), I/r can easily exceed the comparatively high effective spontaneous emission rate by two orders of magnitude. Accordingly, the vibrational level structure can be highly resolved, as the electron is forced to undergo many oscillations between the inevitable spontaneous emission events where it is randomly scattered due to a recoil kick. we note that in the limit E R << who the localization of the electron by the optical potential strongly suppresses the heating caused by spontaneous emission. This is because the single-photon recoil energy is not large enough to allow the electron to make a ’jump’ between the harmonic oscillator levels. In our system, as shown in Fig. (4.10), we have E R N fiwho, so while heating will not entirely suppressed, it may be significantly reduced. 4.4.2 With memory: Non-local potential Our next goal is to investigate the case of a single electron trapped at one site of a simple harmonic oscillator, but this time we will explicitly keep the non-local character of the potential in terms of the memory function. For simplicity. we approximate the memory function m (Ir — r’]) with the Gaussian fit presented in Fig. (4.2). we will compute the size of a Gaussian ground state wavepacket based on a variational approach. We will show the procedure for red detuning in detail. The calculation for blue detuning goes along the lines, but. one has to expand around a different local minimum as the character of the potential swaps from attractive to repulsive. In this way, we will be able to estimate the effect. the nonlocality has 011 the result obtained above in the local limit of the potential. Proceeding from the non—local Schrbdinger equation, i.e. including the memory 122 200: I — 6:0.10 1507 — 5:0.05 o —- €20.01 , Q” . . \L 100- « > ‘- i l l 50: 3 l l 0}. _ -_ re _ Addams“? 0 10 20 3O 4O A,[meV] 600i — e=010 500 — 6:005 o 400. — 6:0.01 LL” t S‘ 300: 200} l 100? i -3.5 —3.0 —2.5 -2.0 — 1.5 — 1.0 -O.5 0.0 A,[meV] Figure 4.9: Trapping frequencies Ur compared to the effective spontaneous emission rate for CdTe. 0 was chosen to be 300. 123 L 1‘ 0'5f -_--- ER 1 S. 2 a E GAE — it who: 9:100 l g3 0.31 o 9 a: ----- it who: 0: 15 “$3 0.2} LE P -------- h who: 9:200 0.1} ..................... w ; ........................ :.:::::-~.. onions __L a not in s bdrm” Hands 2 —3.5 —3.0 —2.5 —2.0 -—1.5 — 1.0 A, [meV] Figure 4.10: Single photon recoil energy E R compared to the harmonic oscillator level spacing hwho as a function of the detuning for a fixed saturation parameter 6 = 0.1 in CdTe and various values for the angle 6. function, 2 2 2 ’ _h V we) - fififcaa/CF“ C082 (Qfir ) m(Ir—r’l) \P (r’) = EM), t 2m; 2 (4.26) we will again expand the cos2 (Quill) to second order as I I 2 cos2 (Qr : r ) = cos2 (Q18 E :17 ) z 1— 94— sin2 ((9) (:17 + x’)2, (4.27) so that the non-kinetic part of the non—local Schrodinger equation explicitly becomes 92 2 __A_(:fc (At) /d2r' (1— Q7 sin2 (6) (a: + :c’)2> m (Ir — r'l) \I' (r') (4.28) 124 To find the ground state of this non-local Schrodinger equation, we can use a varia— tional approach with the following ansatz * 2 _,*,_, '2 \ll(r) =Ne‘a I e ’3 (y yo) (4.29) consisting of two gaussian wave packets. The normalization constant is given _._ : __ (4.30) (Ek) = 2772* (a +3 ) (4.31) and for the potential term it is given by .03 ., 1 (V) = —27ra—A—tfc(At) a*;’3* \/0‘* (2b + 01*)5’“ (2b + (3*) _ Q2 sin2 (6) 4\/a*3 (2b + a*) 15* (2]; + (3*)} (4.32) As we can see, both (Ek) as well as (V) do not depend on the parameter yo, since for the chosen direction for Q along the i-direction the problem is invariant under translations in the g-direction. By numerically minimizing the total energy (Ek)+(V), we obtain for example the values shown in Fig. (4.11) as a function of the (red) detuning At. While a* which describes the localization in the :E-direction is finite, the variational parameter [3* which is associated with the Q-direction is zero, independent of the chosen angle 6 and the detuning At. Thus, the electron’s wave function is flat in the 'Q-direction and shows no localization. Qualitatively, we obtain the same behaviour for blue detuning. 125 0.0030? 0.0025} 0.0020} " ................ E). : , """ . . 0.0015 0.0010; 0.00051 0.0000L 10 15 =20 O A, [meV] __. ‘ __ ‘ ___..___ .___. Figure 4.11: Variational parameters (1* and 3* in the harmonic oscillator approxima- tion including the memory effect for e = 0.1 and red detuning. The results for blue detuning (not presented here) are qualitatively the same. Horn the values for a“ we can deduce the spread of the gaussian as 1 0', = —, 'C )1 2a* which is depicted in F ig. (4.12). (4.33) Compared to the case where we neglected the memory term in the previous section by replacing it with a 6—function (see Fig. (4.8)), we recognize that it is well justified to neglect the nonlocal character of the potential. The spatial spread of the Gaussian wavepacket is effectively not affected by the presence of the non-local memory term m (x), since the additional spread caused by the non-local character of the actual trionic potential is negligibly small compared the spread that we obtained in the approximative local limit of the potential. 126 A X [610] 0’ A._._¥A ‘33“1T30TT—5215 if —i.5 —1.0 —0.5 0L0 A,[meV] A X [an] ---.---.........—.---- 5 10 15 20 A,[meV] J Figure 4.12: or of the gaussian wave packet in the ris-direction as a function of the detuning At for e = 0.1 and CdTe. The results where the nonlocal effect of the potential was included are shown as solid lines: 6 = 10° (blue). 6 = 15° (green), 0 = 20° (orange). The corresponding values of the local limit where the non—locality was neglected are depicted as dashed lines. The non-locality causes a slight additional spread of the electron wavepacket. 127 Chapter 5 Effective temperature for the electrons Up to now, we have focused on a proper description of the coupling of the conduction electrons to laser light, tuned close to the trion resonance. However, we should not forget that these electrons are embedded in a semiconductor host environment at the lattice temperature T), which also interacts with the conduction electrons. In this section, we will explore the electron interaction with phonons. More general, we will investigate the electron heating and cooling processes and calculate the strength of the different mechanisms. Due to the host environment, the electrons can emit phonons, whereby the electron energy is lowered; therefore, the emission of phonons serves as a natural cooling mechanism. In other words, the electrons are linked to a macroscopic refrigerator, the semiconductor lattice, and energy is exchanged in terms of its vibrational quanta, the phonons. Here, we will only consider acoustical phonons: Optical phonons will not contribute, because the exchanged energies we consider are always smaller than the characteristic optical phonon energy: In GaAs for example the energy of one optical phonon amounts to thO = 36 meV . Meanwhile, the electrons also experience effects that tend to increase the electron kinetic energy and 128 thus heat them up. In addition to cooling, we also need to consider heating effects: While phonons can not only be emitted but also absorbed, and in doing so cause a heating of the electron, such processes need to be activated thermally so that this effect goes to zero as the lattice is cooled down towards a low temperature regime. The primary heating effect that we need to take into account is then the intrinsic omnipresent heating mechanism of any optical potential, namely the spontaneous emission of photons. Once, the electron has entered the excited trion state after the absorption of a laser photon it is subject to spontaneous emission of photons with a rate that is governed by the trion radiative lifetime rt and the excitation probability of the trion state. The latter is controllable experimentally by the ratio of the Rabi frequency for the electron—trion transition to the detuning from the trion resonance. The essential goal of this section is a first-principle derivation of a time- dependent equation for the electron energy. Based on this equation, we will find the equilibrium solution at which the heating and cooling mechanisms balance each other. This equilibrium solution will culminate in the definition of an effective electron temperature T *, which is necessarily higher than the lattice temperature. We will have to compare the associated effective thermal energy kBT* to the depth of the optical potential, seen by the electrons, to determine its ability to effectively trap electrons. 5.1 Acoustic phonon scattering rate In a first approach to the problem, we will calculate the emission rate of acoustic phonons at T) = 0. In this way, we will get a feeling for the typical time scales of intrasubband relaxation processes, comparing our results to values obtained in pre- vious theoretical investigations [51]. Although these results are related to a different energy regime for the electrons and therefore not of interest for the further studies , 129 i ‘- .A_-:.“- ‘ .. . a good agreement will serve as a direct check of the validity of our approach. We restrict ourselves to the intrasubband contributions arising from the emission of acoustical phonons in order to estimate the relaxation time at which a warm carrier cools down to thermal equilibrium. Usually, it is a very good approximation to assume that intrasubband relaxations are faster than intersubband ones [51]. In the low temperature regime the absorption of phonons can be neglected. Moreover, optical-phonon scattering is negligible at low temperatures (3 40 K) [50]. Indeed, acoustic phonon scattering is the only way a carrier can relax down to the ground subband edge when its initial excess energy is smaller than the energy of an optical phonon th0- To compute the scattering rate Pph of an eigenstate W3) limited by transitions to all possible final states (21,115 f) induced by the electron-phonon interaction potential He—ph2 we exploit Fermi’s Golden Rule r,,,.— - 237:: ((222122. 22 I22> >I (22 + 22222- 22>, (52> where hid); is the energy of the emitted phonon and 6.2"]? give the energies of the initial and final electron’s states respectively. The computation of the scattering matrix element requires the knowledge of w,- and 11.7. As usual, we apply the envelope function formalism and restrict our considerations to a parabolic description of the host’s bands. This means that for a finite well—width in the fi-direction we write the envelope conduction states in the form 1 . - fi’i (r) = 33 8XI) (’lkl‘) X2? (2’5) (5'2) with the corresponding energy where A is the sample area. Furthermore, r = (.r, y) and k 2 (k1, Icy) give the in- plane components of the electron position vector and wave-vector, respectively, E,- the confinement energy of the 'i-th subband and x,- (z) the associated envelope function. Analogue expressions hold for the final states 22,012. For simplicity, we assume a very thin quantum well with one relevant mode in the growth direction. We do not take into account the dependence of the envelope function Xi/f (2:) on the energy level and take the form xii/f (Z) = 9% (0.4) for 0 g z s L z and zero elsewhere; with L z being the well width. This approximation is valid for very thin wells that support only a single bound state, so that intersubband relaxations are suppressed. The main scattering mechanism for electrons interacting with acoustic phonons is provided by the deformation potential interaction, and the assumed electron—phonon interaction Hamiltonian is given by HG-ph = Z [a (q) e-"'q'rb:i.+ he] (5.5) (f where bT is the creation operator for a phonon in the mode q"' and (Y D2 D2 2:0 |02 (2012 = 2,0ch V is the strength of the electron—acoustical-phonon interaction in the deformation po— tential approximation. For convenience, we buried most of the constants into the definition of the quantity co. The parameters that enter are the deformation poten- tial for electrons D, the density of the host material p and the longitudinal speed of sound cs. We neglect any possible change of the phonon spectra arising from the existence of the heterostructure and thus take isotropic acoustic branches with sound 131 1.1-£731“ =1 velocity cs. To apply Fermi’s Golden Rule, we will first evaluate the transition matrix element for the emission of an acoustic phonon with wavevector q and we obtain |<¢f| He—ph I212) (2 = Id (21)l2 IE (q.~.-)|2 6ki,kf+q (5.7) CO __ . = —5wo I: ((12) 2 (5.8) V where we introduced the function 2 (5.9) 3(q,) = f 222-2232.: (2)2; (2:) = / dze—‘iqzz 2.)) (z) and the phonons frequency wo that already incorporates the in—plane momentum conservation condition (.00 = Cs \/q3 + lki — kfl2' (5.10) Since the system is translationally invariant in the plane, the difference in the elec- tron’s in-plane wavevectors has to equal the in—plane component q of the phonons wavevector cj’ = (q, qz). For our particular choice of XIi/f (2:) being equally distributed on the interval 0 _<_ 2 S L; and zero elsewhere, we obtain _ 45in2 (QZLz/Q) 5.11 ((12123)? ( ) N ow, that the matrix element is evaluated, we have the necessary ingredients to turn back to our original goal, the calculation of the actual scattering rate I‘m due to the inelastic interaction with acoustical phonons. We obtain CO ‘ l f )2 277 k f ( ) f l P. — / dq; E qz / dp/ dk I: 0206 e: —e —fiw0 5.12 ph (27r)2 —oo 0 0 f f I f 132 To explore the non-zero temperature limit, this result would have to be multiplied by 1 (5.13) where f). is the Bose occupation function that in this context can be viewed as a temperature-dependent stimulation factor. We have solved the remaining integrals for the computation of I‘m numerically. In the case of GaAs, the calculations have been performed with the material parameters D = 8.6 eV, p = 5.39/cm3, and cs 2 3700 m/s [51]. Here, we give the results for the inverse intrasubband scattering rate 71.11 depending on the well-width Lz that can be directly compared to the results given in Fig. (6) of ref. [51], where the initial electron’s energy was chosen to be about ~ 0.25 eV. Our results are about a factor of 2 off from the ones stated in [51], where a more sophisticated approach for the electron’s wavefunction in the é-direction was used. For our purposes the simplified form catches the physics correctly and works fine for very thin well-widths, i.e. the quasi two-dimensional limit. In this limit, the intrasubband relaxation rate due to acoustic phonons 71.,1 is very fast, in the range of picoseconds. However, this refers specifically to the chosen electron’s energy which is different from the energy range we are to consider. Nevertheless, in this way we have confirmed the validity of our approach. 5.2 Equilibrium temperature In the next step, we will calculate one of the most important quantities in our analysis: the effective temperature of the conduction electrons T *. This effective temperature deviates from the temperature of the host environment Tl, because spontaneous emis- sion events render the optical potential dissipative, resulting in an effective heating 133 h- 72—]— f ff _1 T T ’T—‘I; ' ] 7.x10‘”; j l l 6.x10‘”[ 1 2—. 5.x10‘”2 ] 23 2 ; r _ .. : : 4.x10 “f ] 3.x10-“§ ] 2.x10‘“[ l ’ l 1%.. -.1_l 00 T‘EO”-“80 “T1059 ”_Tédff—TAG—TTGO 180“ ‘2 L: [A] Figure 5.1: Acoustical phonon scattering at a lattice temperature Tl = 0: Dependence of the intrasubband relaxation time Tl_,1 on the quantum-well thickness Lz. of the electron. The rate at which the electron experiences the spontaneous scatter- ing events is the effective spontaneous emission rate [‘38, which we estimate as the product of the time-averaged probability to be in the excited trion state times the ra— diative decay rate, i.e. the natural line-width, of the trion level: [‘38 = (P6) I} = d}; 6 is the saturation parameter. For our purposes, the associated energy gain rate or heating rate Rheat» measured in energy per unit time, is even more important than the scattering rate I‘se, because it carries the actual information about the strength of this mechanism to push the electron effective temperature away from the temperature of the refrigerator T). We approximate the corresponding energy gain rate as Rheat = (P6) Ft (ER) : GFt (ER) (5'14) 134 since, on average, every single spontaneous scattering heats the electron up by the average in-plane recoil energy (E 3). Experiments with Bose-Einstein condensates that are loaded into an optical lattice essentially take place in vacuum. The atoms are cooled first, before the lattice is ramped up. Thus, these systems lack a natural cooling mechanism and the heating process due to spontaneous emission of photons sets a boundary on the time-scale on which these experiments can be performed successfully. Once, the atoms have been cooled down to very low temperatures and the interaction with the photons is switched on, the recoil kicks heat the atoms up, which eventually knocks the atoms out of the traps. Therefore very small values for the saturation parameter 6 have to be chosen in order to control the heating rate. In general, when working with laser-cooled atoms the cooling mechanism stems from the same system as the optical potential, namely the laser, whereas in our system the cooling mechanism is completely independent of the laser-induced potential. In contrast to these limitations on the optical potentials for atoms, the system we study is equipped with the phonon bath which competes with the heating process described above. In equilibrium, the heating and cooling mechanisms cancel each other. We will derive the associated effective temperature of the electron at this equilibrium point. As a side note in advance of the later discussion we mention that phonons couple only very weakly to the spin—state of the electrons. so that spin-flip processes due to the phonon bath occur on rather long timescales. This fact has already been used to engineer quantum information systems based for example on quantum dot platforms. The interaction with the acoustic phonons is essentially a scattering problem. Here, we discuss the validity of Fermi’s Golden rule in the specific problem at hand. We will calculate not only a scattering rate, but also energy loss and gain rates because of the emission and absorption of acoustical phonons. 135 The Hamiltonian ’H for the problem of one single electron interacting with a bath of acoustical phonons can be written as H = H0+He_ph (5.15) h2g2 1 HO = 2m; + VC + ths [a] 0,705 (3.16) q He—ph 2 Ed (q) [e-i‘mbga— eff; b5] (5.17) (f where the unperturbed system contains the electron’s kinetic energy in the QVV plane, the confinement potential VC of the QW along the 2-direction and the energy of the phonon bath. Again, He—ph is the interaction of the electron with the phonon bath in the deformation potential approximation. Since we are only considering intrasubband scattering, the Hilbert space to describe the problem perpendicular to the QW plane is truncated to one mode only. Subsequently, the identity operator for the electron Hilbert space can be represented as 1 = Ix) (xl ® 1“, where we have decomposed the problem in the in-plane and the out-of—plane part; Ix) describes the wavefunction in the 2-direction. After this simplification, we can make the replacement eff”? .1 (XI 6222.222 IX) 6*qu = 33. (Qz) ef’iqR (518) where we introduced the Fourier transform of the squared electron’s wavefunction. in the E-direction 21(2) = (21 em“ 12> = / dzei’i‘m Ix (a)? (5.1a so that in the truncated problem which only allows for one mode in the quantized 2-direction the electron phonon interaction simplifies to ~ —= R —- - 2' R He-ph = 202121) [:._ (q;)e ”q 03+ 2+ ((1.2)22‘Ql 0,7] (5.20) (7 136 In this way the dependence on the 2-direction is governed as a matrix element in the interaction Hamiltonian which essentially goes to zero for large values of qz. Focussing initially on the emission of a single phonon, a convenient set of basis states to tackle the problem is (22> = l22>®lT> (5.21) _ 1 2 2 [(12+) — m[T(“Q)lhl®bq~lTl] (522) 1 A . Iq.—> = fi[r(+q>l22>®bqolT>] (5.23> which are tensor products of the electrons state and the state of the bath |T). We assume that the state [2) as well as the states [q, :t) are eigenstates of the unperturbed problem with the corresponding eigenvalues H0 [fl = Eu [17), 7'10 lqa i) = Ed: [(1, i) (524) The state [2) describes the initial state of the system. The electron is assumed to be initially in the state Ida") = [19) ® IX) (5135) which describes an electron free to move as a plane wave with wavevector R, in the QW and confined in the mode Ly) of the quantized é-direction. To adequately describe the physics of phonon emission and absorption we have introduced the in- plane momentum-shift operator T (iq) given by T (iq) = eiiqR (5.26) The phonon bath is initially at thermal equilibrium in the state |T). Thus, we 137 only require that it obeys the Bose-Einstein distribution . T _ — ~ 2' (TI bqbq'f IT) — 71902727 (0.27) which explains the chosen normalization for the states Iq, 21:), assuming IT) to be normalized. This assumption basically states that the phonon modes are uncorrelated. The given basis is sufficient to describe single phonon emission / absorption processes. The system starts out. in the state Ii.) and is then found to be in the state Iq, +), if an acoustic phonon of wavevector q has been emitted and in the state Iq, —), if an acoustic phonon of wavevector q has been absorbed by the electron. The momentum shift operator automatically takes care of the inplane momentum conservation. The orthogonalization conditions read (q,+lq’,—) = (q.ili)=0 (5.28) (q, ilq’, 2%) = ‘qar (529) O In its most general form the state of the system IIII (t)) can be written as I212 (2» = <2) (12> + 2 [22+ (2: 2) Iq, +> + c— (2: 0 Iq, —>1 (5.30) (7 where we introduced the time-dependent amplitudes c,- (t) and Ci (4’, t). Normaliza- tion of the state I‘II (t)) requires I2. <2>I‘2 + 2 [I22 (2: 012 + I2- (2,512] = 1 (5.31) q 138 The probability Pin,- (t) to find the system in state I2) at time t is 2,1,0) = I2.- (2: 5|? = 1— 2 [12+ (22012 + (2.02521 (5.32) q which is nothing but the condition of the conservation of probability. Hence we see that to calculate the transition probabilities to second order, it is only necessary to compute the corresponding amplitudes to first order. Later on, we will refer to the completeness relation for the whole system, i.e. the combination of the electrons Hilbert—space and the bath, 2: lk> (kl 2232 12> <21 22 12.22 = 1 (5.33) k The time dependence of the state’s system can be buried into the propagator U (1‘) defined as i U (2) = 23—727“ (.534) A similar definition holds for the free propagator U0 (t), where only the Hamiltonian of the unperturbed system H() enters. Expanding the full propagator U (t) to first order perturbation theory gives the result 2‘ 2 U (75) = Uo (t) — F/n dti Uo (15 — 1*1) V (t1) UO (I1) (535} which illustratively states that in first order the system can evolve in the time interval [0,t] with either no interaction or one interaction taking place somewhere in this interval. Before and after the interaction, the system evolves essentially as it. would in the unperturbed system. In first order perturbation theory, the time-dependent 139 r». amplitudes ci (q, t) to find the system in the state Iq, i) are given by .zj t Ci (it) = (q,il\1' (1)) = —E/0 dti (/r1q‘+13_(q,)e fiE+t/0a1‘f2‘(E E+>1 (5.37) -0212) = —%a r1..132(c1.~.>e‘7’2E—t f0 2212 ‘iIE E)’1<5.38> After the integration in time, we find that the probabilities [Ci (2i',1f)|2 to find the system in the state Iq, +) or Iq, —) respectively are , j. 2 E.'—E t 4s1n (41—312) 1220112))? = (22(2)? E- (21:22 (22.1221) (Ia-E1)? (5.39) 9 4sin2 #5 IC— (2212M2 = |O2(21)|‘|E+ (61:)l27'2q (E: E_)2 ) (5.40) At this stage, we can already recognize that the absorption of acoustical phonons becomes negligible in the limit T ——> 0, while there can still be spontaneous phonon emission. Only the stimulated processes that are proportional to rig vanish in the low temperature regime. The next step is well known from the general derivation of Fermi’s Golden Rule. It is a very crucial approximation that is based on the phenomenon of a separation of time-scales. we are confronted with the function (50) (Ef — E2) = “Sin ((gf—_1:))t/2h) (5-41) 140 which tends to the delta function 6 (E f — E,) for long times t —> 00. As a matter of fact, it is a diffraction pattern, whose maximal amplitude lies at t/27rfi for E f—Ez- = 0. Its width is on the order of 47rh/ t and its integral yields 1. Thus, it is an approximate delta function that expresses the conservation of energy with an uncertainty h/t because of the finite duration of the interaction [52]. In the Markovian approximation, the scattering events are independent of each other, the system is reset after each interaction: There is no memory. The separation of times scales appears as taking the limit t —> 00, whilst t has to be sufficiently small to justify the perturbative treatment of V, as we allowed for single scattering events only. In this limit, the transition probabilities become 1 27r ._ _ - 122101.512 = —,,—Ia(q>l21~—(qz>12(221+1)2>2 (5.42) _4 . 27r ,_ _ (2-21.222 = -;,j-l2:2(q)12|:+(222)1222215(E2--E—)t (5.43) from which we can readily deduce that the probability to remain in the initial state I2) decays at the rate 1,, = $212201»?[s_(q.>|2(22q+1)6 +121 (22> 2 22125 (1:, — 21.)] (5.44) The first term accounts for phonon emission, while the second term respects phonon absorption. However, this is not the end of the story. we will make use of the same framework to derive the energy gain and loss rates due to phonon emission and absorption. The strength of these mechanisms as compared to the photon recoil heating Rheat will then decide about the electrons equilibrium energy kBT*. In general, the time-dependent expectation value of the electrons energy (E (2‘)), 141 which is assumed to be purely kinetic, is (E (2» = Z ’2”. <222>111k><1212212><21® 21.111122» (5.25) k ’6 where we inserted the completeness relation (5.33). In the next step, we express the state I‘ll (t)) according to the general expression (5.30) and find that the deviation of the electrons energy from its initial value is given by 522? h2(k-——c02 522? E t — E = E E _ I it 2 h? (k,- + q)2 21.22,? .2 +- 1 - IC—-Uit)l 0146) 2Q: I 2m, 2111;; The difference in the kinetic energies is exactly the energy hug of the emitted or absorbed phonon. We define the change in the electron’s energy per unit time Rph (E2) according to 522? (E (15)) - E = Rph,(E1:)t (5247) . * 22728 When we plug in our results for the transition probabilities and use the energy con— servation, we find a very intuitive and simple result 27r _ _ 12,11. (13.) = 7 212(2112 1—_ (21.312 2222021 + 1) 21 (E.- — Ef — 2222,) 22' 27r _ _ +7 2 12 (2012 1:1 an2 122.2212 (E.- — Ef + 2222.) (5.13) (I The energy loss and gain rates due to acoustical-phonon emission and absorption respectively are obtained by weighting the scattering rates with the energy of the corresponding phonon. Thus, we just have to multiply by flow, before the summations are performed in (5.44). Now that we have understood the phonon mechanism and its influence on the 142 "": ‘L‘A‘A‘QUA .3 I electron energy we will find the equilibrium point for the electron effective temperature T*. To do so, we also have to take into account the recoil heating. Otherwise, without this heating process the result would be trivial: T * = T). A thermalized electron would automatically take on the temperature of its lattice environment. We can set up an equation for the electrons energy evolution in time, which respects both the contribution from the recoil heating processes as well as the phonon mechanisms. The electron’s energy evolves in time according to 20522» :12 T p12. (E) ”I" Rheat (5°49) In contrast to Rph (E), the heating rate Rheat depends neither on the lattice tem- perature Tl nor on the the electrons energy. Still, it can be tuned by changing the value of the saturation parameter 6. In equilibrium, the time-derivative vanishes and we find E* = kBT’E‘ as the point where RP), (E) and Rheat cancel each other, i.e. the equilibrium condition reads Rph (E*) + Rheat = 0 (5°50) We specify the results of this procedure in Fig. (5.2) and (5.3) for GaAs and CdTe, respectively. The lattice temperature was chosen to be T) = 300 mK. In this temperature regime the absorption of phonons is negligible and the dominant heating effect is clearly the spontaneous emission of photons. The trion radiative lifetime was taken to be rt = 100 p3 and the corresponding heating rate Rheat is plotted for the saturation parameter values 6 = 0.1, e = 0.05 and e = 0.01, respectively. Of course, a lower value for 6 results in smaller effective temperatures for the electrons, since the heating rate Rheat is linear in the parameter c. We find lower effective temperatures in CdTe than in GaAs, because the phonon cooling mechanism shows the same effec- 143 109% R [meV/s] 3:. 107g 106.. . . - . - A L 0.0 0.2 0.4 0.6 0.8 1.0 E [meV] Figure 5.2: 022.45: The phonon cooling rate Rph (E) is shown in blue. The heating rate Rheat is presented in orange for different values of the saturation parameter 6: 6 = 0.1 (solid line), 6 = 0.05 (dashed line), 6 = 0.01 (dotted line). The thermal equilibrium energies can be found where the cooling rate intersects with the heating rate. tiveness, while the recoil energy (E R) is noticeably smaller for CdTe. For saturation parameter values 6 z 0.01 — 0.1, we obtain equilibrium temperatures kBT* in the range 0.2 — 0.4meV and 0.1 — 0.3meV for GaAs and CdTe respectively. For red detuning, this energy range is of the order of or even bigger than the potential depth V0, even for laser powers that allow for the saturation regime. For blue detuning, however, we find that the potential depth can exceed kBT* considerably. In Tab. (4.2) we have specified for example the potential depth for 6 = 0.05. Compared to the corresponding electron equilibrium temperatures, the potential depth amounts to V0 % 3kBT* and V0 z 8kBT’E‘ for GaAs and CdTe respectively. Thus, by tuning the laser light above the trion resonance one can reach values for the potential depth V0 that are considerably larger than the effective thermal energies of the electrons in the system. 144 109: 3‘ . E 10% f a: ; / 107g 510MH- _‘TOTQ T "TE 6215*“ “TEST—f“ 578— T‘TM 176 E [meV] Figure 5.3: CdTe: The phonon cooling rate Rph (E) is shown in blue. The heating rate Rheat is presented in orange for different values of the saturation parameter 6: 6 = 0.1 (solid line), 6 = 0.05 (dashed line), 6 = 0.01 (dotted line). The thermal equilibrium energies can be found where the cooling rate intersects with the heating rate. Chapter 6 Optical lattices for carriers in semiconductor quantum wells For the sake of simplicity, so far we have focused on the case of two counter-propagating laser beams, which produces an intensity pattern in the form of periodic stripes along the QW. In this chapter we will extend this model to an intriguing novel phenomenon, namely a two-dimensional optical lattice for the conduction electrons in the QW: pe- riodic arrays of microtraps in one than more dimension are generated by a set of standing wave laser fields. A periodic potential in two dimensions can be easily formed by overlapping two optical standing waves along different, most commonly orthogonal, directions. To eliminate interference terms between the pairs of beams in the different directions, one can either choose orthogonal polarization vectors for the two laser fields [40] or slightly different optical frequencies [53]. In both cases, the resulting optical potential in the center of the trap is simply the sum of a purely sinusoidal potential in both directions. Superimposing two orthogonal standing waves in this way in combination with the confinement in the é-direction inside the QVV results in a two-dimensional array of artificial quantum dots for the electrons. This step essentially bridges the gap from the concept of a periodic optical po- 146 tential for electrons to the idea of an optical lattice for electrons. The concept of an lattice as a regular arrangement in space underlines a shift away from a picture of an disordered electron gas towards a possible new physical paradigm of an elec- tron gas whose density is spatially modulated and whose properties can be strongly determined by the periodic optical potential in a time-dependent way. To be pre- cise, we will specify our calculations for a two-dimensional square lattice with lattice periodicity a = A11/2; )1” is the in-plane wavelength of the laser. Thanks to recently developed quantum optical tools, a wide range of many body Hamiltonians have been realized with cold atoms in optical lattices. In particular, a whole toolbox to engineer various Hubbard type lattice models for 1D, 2D and 3D Bose- and Fermi systems has been provided whose properties can be controlled by varying external field parameters [54]. The original idea suggesting the possibility of studying non-trivial many-body systems with cold atoms in optical lattices was developed by Jaksch et al. [55]. The starting point in their analysis was the so-called Bose-Hubbard model. One of the key parameters that enters any Hubbard model is the on—site interaction energy U due to a repulsive interaction between the particles located at the same site of the lattice. In this chapter we will estimate the parameter U as an integral over a variational on-site wavefunction in a local harmonic oscillator potential. We will cover both spin singlet and triplet states. Compared to optical lattices for ultracold atoms, which interact via short-ranged s-wave interactions [40], we expect this parameter to be very big, because we deal with electrons that carry a charge 8 and thus interact via the strong, long—ranged Coulomb potential. For the same reason, the interaction between electrons that are located at different sites is not negligible and we will estimate the strength of these interactions as well. It is an interesting feature that the strength of the interaction can be tuned by the laser parameters in real time: For instance it is evident that the repulsion U increases with the potential depth due to a tighter squeezing of the on-site wavefunctions. 147 Another striking property that comes along with an optical lattice for the electrons is the possibility to engineer a spin-selective lattice: We will show how the underlying level structure of our system can be used to design differing traps for different internal spin states of the electrons. 6.1 Coulomb blocking: On—site repulsion Before considering into a more accurate approach to determine the on-site repulsion U, let us first get a feeling for the orders of magnitude with rather rough limiting, but simple approximations. First, as a lower limit to the actual value of U, we can easily compute the Coulomb energy between two classical pointlike electrons that are nearest neighbours on a square lattice and therefore separated by the lattice spacing a. The periodicity 0, depends only very weakly on the detuning from the trion resonance At, but can be tuned efficiently by altering the angle between the laser beam and the QW within about one order of magnitude. Still, a is typically of the order of ~ 10-7272 up to 22272 for both GaAs and CdTe. In this classical picture, we estimate the Coulomb energy Vc as VC- = 62/6a and find that the classical repulsive energy between the next neighbours is w 0.2 —- 0.6 meV for angles 6 in the range 6 = 10° — 30°. For red detuning and reasonable values for the saturation parameter 6 this possibly exceeds the potential depth V0. In the case of blue detuning, however, potential depths in the range of 1 —- 2meV are feasible, which is considerably larger than the next—neighbour interaction energy. Still, this reasoning underlines the strength of the Coulomb interaction in our framework and we will come back to these values when discussing the Coulomb energies between the various neighbours on the square lattice. To obtain a better understanding of the on—site interaction term U that goes beyond the classical approximation, we consider two electrons trapped at one lattice 148 site, neglecting the possibility of tunneling. As done before, we approximate the lattice potential by a harmonic oscillator of trapping frequency who, this time, though, in two dimensions: we remember that who depends on both the potential depth and the in-plane lattice periodicity. To describe the two electrons which shall both occupy the ground state I g) of the trapping potential, we simply take the product of two Gaussians 9 m*w’h 2 2 _ E ( ' (r1,r2Ig.g) = AF exp (—fi—’ (r1 + r2)), (6.1) . =1: , where N is the normalization constant for the single electron solution N = flffilm. This ansatz for the spatial wavefunction is symmetric with respect to particle ex- change, so that it only applies to a spin singlet state. Although the two electrons form a singlet state, both are assumed to couple individually to the trion resonance, since the trap size is much bigger than the trion size. Based on this ansatz, we cal- culate the expectation value for the Coulomb energy and obtain the compact result 2 :12. r 8 71'7712311,‘ . (2.21 112.10 = ?(/——2‘,,—”—0. (6.2) Of course, this straightforward calculation only sets a upper boundary on the actual value for U, since the wavefunctions were not allowed to change their form as a reaction to the inter-particle Coulomb interaction. With this ansatz we find Coulomb energies that can enter the range of several meV, depending on the combination of potential depth and lattice spacing. In total, we found a lower and a upper limit, from which we deduce that the actual value for U will be more or less around 1 —— 3meV. After these introductory estimates, let us apply a more sophisticated approach that covers both singlet and triplet. states on the same footing. The Hamiltonian for this two particle problem can be decoupled into a center of mass motion HR and the 149 relative motion H r 132 A] 2 2 2R2+P—+#v2r2+e— (6.3) H=H Hz— —' —a. 3+ r 211+ 2%“ 2n -2 ho e|r| where we introduced the total mass it! = 2m: and the reduced mass ,u, = mz/‘Z. While the center of mass of the system R moves in a harmonic potential of the trapping frequency who, the potential of the relative coordinate r is the sum of a harmonic trap of frequency who and the interaction potential. In addition, the Hamiltonian H is spin-independent, so that the total wavefunction can be factorized ‘1’ (1, 2) = u" (r) E (R) X (31, 82) (6-4) The solutions to the center of mass Hamiltonian HR are well known; therefore, we will focus on Hr in the following. In cylindrical coordinates r = (r, (b) the relative Hamiltonian H,- can be written as 6218+18212 1 1 = —— — —— 2d, — 6.5 Hr 2 8T2 + 7“ 8r+ r- —3¢2 + horz + ( ) where we switched to a unit system with h = ,u = 62/6 2 1. Starting from the Schrodinger equation for the relative Hamiltonian Hr-zi.’ (r) = 62,251 (r) (6.6) we introduce the replacement t (r) = to“) (6.7) so that we can express (6.6) in the following way a? 1 1 a? 1 2 2 1 5,3 + 17.5 + 3535] 9'9 (1‘) + (gw'hor + —) 99(1‘) = 699 (1') (6-8) Splitting 1,:- (r) up into a radial and an angular part ,9 (r) = R (r) emf (6.9) the radial wave equation to solve reads 1 r , ‘2 1 1 82 (m — 1) 1 2 2 2 (97“2 + 2722 + gwhor + ; R (T) " ER (7”) (6.10) and gives rise to the definition of the effective potential (7712—211) ,9 1 ————+ fr2+— (6.11) lirffff (r) = 27.2 2 1,0 7. If we introduce the scaled coordinate r 2: I/Vw, this differential equation becomes . 2 1 26 2 (7n “- 1) 2 - 1 , 2 . , . Who .I? who” R” (.1?) + R (.r) (6.12) Let us mention that for the case with no Coulomb interaction, i.e. if we drop the term 2 / Mr Eqn. (6.12) can be solved exactly by 12m = e—l’g/‘th'ctl/QLEI") (1:2) (6.13) with the quantized energy levels 6 2 who (2n + m + 1) (6.14) This short aside justifies our two parameter variational ansatz of the form 22471301 +13 .2 » ' is : —o-I' , ,3 ,ungt) 2 z 6.1.7 L, (r) Ne r c, , N ___27TF (5 +1) ( a) 151 m Ami to find a reasonable approximate solution to the problem including the Coulomb interaction between the electrons. The factor r5 is crucial in order to handle the divergence in the Coulomb potential. Under particle exchange the sign of the radial coordinate flips r —> —r, which corresponds to (7“. o) —> (r. o + 7r) in cylindrical coordinates. This gives eim(o+7r) _) ei'mrreimq') : ie'imo which shows that for even values of m = 0, i2, i4, . . . the system is in a spin singlet state, whereas for m = i1, i3, . . . the system has to be in a triplet state to establish an overall antisymmetric wavefunction. Therefore, our ansatz allows us to investigate singlet and triplet states. After performing the integrals, the expectation value for the “relative energy” is found to be [(7712 +3) 2_3___+1+\/2—5I‘(fi+1/2) .1 H 4,1,1 u,‘ . (Ll rllr>= + ho 4a I‘(,’3+l) (6.16) 11>le where F(z) is the Euler gamma function. To ensure convergence the variational parameters (1,6 have to be greater than zero; thus, we immediately see that the singlet state with m = 0 will always be the state with the lowest energy. We identify the on—site interaction matrix element U with the difference in energy caused by the Coulomb interaction U: [(7722 w) 26+1 M201" (,6+1/2) u, - — 6.17 + 110—— 40 11(3 +1) who . ( ) "Gal 9. Of course, our results confirm that the strong Coulomb interaction in our system results in a very strong on-site interaction U, as shown in Fig. (6.1). It is typically in the range of meV. We plot U for CdTe and a saturation parameter value c = 0.1. The qualitative behaviour is the same in GaAs, but the actual values for GaAs are 152 "0 " *5 “T i 10 * 5*”‘5’1’3—m‘m“ and] 20 A, [meV] Figure 6.1: On-site repulsion U for CdTe for a square lattice with saturation param— l eter e = 0.1. Singlet states (771 = 0) and triplet states (m = 1) are shown in blue and orange, respectively, for angles 6 = 150 (lower branch) and 0 = 30° (upper branch). considerably smaller due to a higher dielectric constant e and smaller potential depths. The intuitive picture that an increase in the lattice depth leads to higher barriers, therefore compressing the two particles and yielding higher values for U, is confirmed. In addition, we see that the singlet states lead to a smaller on-site interaction U than the triplet states. Furthermore, the strength of the interaction crucially depends on the angle 0 which essentially tunes the slope of the trap. A bigger angle 6 leads to a steeper potential with stronger squeezed wavefunctions and, as a consequence, higher values for the interaction parameter U. Based on these results, we are led to conclude that the the potential depth is too weak compared to the dominant on—site repulsive interaction U to favour more than one trapped electron per lattice site. 153 6 .2 Filling factor Dealing with ultracold atoms in optical lattices, one can usually restrict the analysis of the interactions between the atoms to the on—site contribution U. In other words, in the deep optical lattice regime the interaction between Wannier states localized at R and R’ reduces to a local form U 5R,R’ [40]. This regime, however, is different from our systems, where the 'r—1 Coulomb behaviour accounts for strong long-range interactions. Therefore, we will give a rough estimate for the interaction energies between electrons that are located at different sites of the square lattice in the next step. The estimate is given by the classical Coulomb energy V — (6.18) corresponding to the repulsive Coulomb interaction between two pointlike electrons with charge 6, separated by the distance (a. This approximation is allowed, if the electrons are far enough apart to neglect effects owing to the overlap of the wavefunc- tions. The parameter Q counts the separation between the two electrons in terms of the lattice periodicity a. The case C = 1 for example corresponds to next neighbours in the lattice. For the first six nearest neighbours in the two—dimensional square lattice, we have C =1. x/i, 2, x/E, x/é, 3 (6.19) For laser light tuned close to the trion resonance, the result of this approximation only depends on the angle (9, directly determining the lattice parameter a. Our results for CdTe are shown in Fig. (6.2). The values for GaAs are slightly smaller because of stronger screening effects in GaAs. The repulsive interactions between electrons trapped at different sites are still very strong and possibly in the range of the depth of the trapping potential. Only in the regime, where 6’ is chosen appreciably small, 154 l A .L. _ “my- - Vc [meV] Figure 6.2: Coulomb interaction energies between electrons at different sites of the square lattice for the first six neighbours in CdTe. The angle 6 ranges from 7r / 36 E 5° to 7r / 6 E 30°. Note that, according to Tab. (4.2), the potential depth V0 can reach values ~ 1.6meV for blue detuning in CdTe, which is considerably larger than Vc. resulting in larger lattice constans a, the neighbour interactions can become small compared to the potential depth. For both GaAs and CdTe QW systems, the lattice parameter a is N 1.5 pm for 6 2 5°. This lattice spacing is in the same range as the regime in which optical lattices for atoms are usually operated in the laboratory [56]. Last, but not least we would like to address the following question: What is the maximum electron density 711 to reach a commensurate filling of one electron per lattice site? With 71.1 = l/a2 = 4/A2 we find that 77.1 is typically in the range of II’ N 1096m’2. This is an electron gas in the low density regime. 155 ‘l 6 . 3 Spin-selectivity The strength of the optical potential crucially depends on the dipole moment between the two electron “internal" states. we can exploit the selection rules for optical transitions to present a remarkable feature of the optical potential we study, namely its spin selectivity: it depends on the spin state of the electron. Configurations are realizable where the two electron spin states are exposed to vastly different optical potentials. This situation is schematized in Fig. (6.3), where, depending on the spin state of the initial electron in the conduction band, two possibly different Rabi- frequencies couple the electron to the trion state, where the two electrons form a singlet state. A spin-down electron only couples to a" photons to form a trion singlet state, whereas a spin-up electron is coupled to a trion singlet via 0"“ photons only. Spin-flip Raman transitions are dipole forbidden: Consequently, the trion system at zero magnetic field appears as a double two—level system [57]; a remarkable feature that is not found in cold atom systems. For instance, spin polarized electron lattices can be created in a standing wave configuration made out of two counterpropagating laser beams with linear polariza- tion vectors enclosing a variable angle 63 [58]. This configuration is usually called lin-gb—lin configuration [59]. The result of this experimental setup is a standing wave light field which can be decomposed into a superposition of a 0+- and a a'-polarized standing wave laser field. In this way, two separate lattice potentials V+ (33,65) 2 V0c032(Q;1: + (9/2) and V. (513,95) 2 V0 cos? (Q3: — 65/2) have been real- ized. By changing the polarization angle to, one can control the relative separation between these two potentials Arr = (co/7r) Ax/‘Z. Both potentials can be shifted apart by increasing (1), until they overlap again for 65 2 mt, n being an integer. This theoret— ical scheme has already been applied to atoms: Based on such a configuration, atoms have been moved coherently across several lattice sites [60, 61, 62]. If the trapped par- 156 Figure 6.3: Spin selectivity of the optical potential: The trion system appears as a double two-level system. Electron and hole spins are depicted in blue and red, respectively. ticle is in a superposition of two internal spin states, state selective optical potentials can be used to split the wavefunction and transport the corresponding wavepackets in two opposite directions, as schematized in Fig. (6.4). This technique could be used to trap two sets of electrons: those in the spin-down state light shifted by the a- light, and those in the spin-up state light shifted by the 0+ light. We refer to these as the i—species. The ability to dynamically control the angle d) allows us to separate and move the electrons of the :lz—species relative to each other. In this way, two electrons of different species initially separated by As can be made to overlap by rotating (0 by to = 27TA1‘//\17. In this way, a spin~dependent lattice offers the intriguing possibility to tune the interactions between two electrons in different spin states by controlling the spatial separation and possibly the overlap of the on—site spatial wavefunction between zero and its maximum value, just by shifting the spin-dependent lattices to each other. 157 V+/_: ¢=0 V+: ¢=7r/4 V/Vo V_: ¢=zrl4 V+z ¢=37rl4 ‘ V_: ¢=37rl4 Figure 6.4: Cartoon for the effect of the spin-selective potentials Vi. They overlap for d) = 0. As qb is increased adiabatically, the two potentials are pulled in two opposite directions. As a consequence, the two spin states of the electron move into opposite directions, and the electron wavepacket is split apart into its two contributions. 158 Chapter 7 Additional effects In principle, our previous theoretical analysis has been based on three major corner— stones: We have thoroughly covered the coupling of the electrons to the standing laser light field and discovered the trion-assisted optical trapping potential. In addition, we have discussed the interaction of the conduction electrons with the phonon bath and we have determined the resulting effective temperature for the electrons. Then, we have extended this single-particle discussion by studying the interparticle Coulomb effects, which coined the picture of a possible electron lattice as a two dimensional reg- ular array of microtraps. In this final chapter of our investigations, we will complete our studies by considering a range of possible side-effects that we have not covered so far. The notation ’side effects’ suggests that these effects will be negligible compared to the three cornerstones identified above. To prove this statement will be the actual task of the present chapter. We will highlight the ponderomotive potential, which is an ubiquitous effect whenever an electron is placed in an intense electromagnetic field. Lastly, we will pay tribute to a possible ionization of the conduction electrons above the QW barrier and we will identify the range of electron densities in which an observation of the trapping potential is feasible. 159 7 .1 Ponderomotive potential \Vith ion traps, charged particles have been trapped in potential wells that are up to several eV deep [63]. In contrast to the trion-mediated potential we propose in this thesis, however, ion traps do not depend on the internal electronic state of the trapped particle. One of the most prominent forms of these traps have been developed by W. Paul [64], in which a spatially varying time-dependent field is used to confine charged particles in space: typically the so—called Paul traps are operated in the radio— frequency (rf) domain. Based on this technique, electrons have been successfully trapped for the first time in 1973 [65]. In this section we study the underlying concept of the “ponderomotive potential” generated by an electromagnetic standing wave. In principle, this is the interaction energy that results from the placement of a charged particle in a rapidly oscillating electric field. The well—known result of this problem states that the average kinetic energy associated with the quivering forced on an electron by a radiation field acts as an effective potential for the averaged motion of the electron [66]. In general, this ponderomotive potential is time dependent. Therefore, the total kinetic energy is not a constant of motion. As a consequence, the ponderomotive potential is, in general, not conservative. we introduce a time average defined by where u (t) is any dynamic variable and T is the period of the wave. In the following, we wish to analyze the average motion of an electron exposed to an electromagnetic standing wave along the i-axis, polarized along the Q-axis with wavevector Q = Qi‘ and angular frequency w. Thus the electric field can be written 160 in the form E(r) 2 E09 cos(Q:r) cos(wt) (7.2) For convenience, we choose a frame of reference in which near t = O the electron is, on average, at rest at the origin. The easiest approach to gain insight into the physics of this process is a classical description. With r (t) = (.r (t) , y (t) ,z (t)) and v (t) = (:i: (t) , y (t) , 2': (t)) denoting the position vector and the velocity of the electron, the classical equation of motion for the Q-direction reads y (.12, t) 2 —8E:) cos(Qr) cos(u2t) (7.3) mo which is simply the Newtonian equation for an electron of mass m; subject to the Lorentz force in the g—direction. Setting the integration constants to zero, the solution is readily found to be E y(.r, t) = 6* 02 cos(Q;r.) COS(th) (7.4) more which shows, that the motion in the g—direction is a function of the sit—position of the electron. The time—averaged kinetic energy Ek associated with this classical motion is then given by (Eklt = J <132>t + 1:2: t + 6.2.533 0082(Qgp) t 4m§w2 ~7 \J N 0* m-x- The average total kinetic energy splits into two parts. The last term acts as an effective potential for the average motion of the electron; it is the ponderomotive potential Up(:z:) defined by cos2(Q:r) = US cos2(Q.r) (7.5) 161 Forces arising from this interaction may be directly calculated as a spatial derivative of (7.5). The ponderomotive force simply turns out to be the time-averaged Lorentz force. Its effect is clearly to push the electron away from regions of high laser intensity, and it is completely independent of the light polarization. In this model, the total energy of the electron is the oscillatory energy Up (2:) plus the directed kinetic energy, the average energy of translation. Once a free electron travels from a. region of low intensity to a region of higher intensity, its original translational kinetic energy is converted into oscillatory energy. Thus, the highest intensity region which an electron can enter is one where the oscillatory energy equals the initial kinetic energy outside the beam [67]. Using the relation between the intensity of the laser I and the amplitude of the electric field E0 1 1 = EnccoEg (7.6) where 77.. gives the refractive index of the material, the potential depth U3 of the ponderomotive potential can be expressed in terms of the laser intensity I as r0 = ——e—2—I— (7.7) p 2772.: memo)? The ponderomotive potential is proportional to the intensity of the laser, but, at the same time, indirectly proportional to the squared frequency (.02 of the field. T here- fore, we can clearly see that the associated Paul traps can possibly represent a strong confinement mechanism for rf frequencies. Using typical values for our system, how- ever, with frequencies in the optical domain, the ponderomotive potential is strongly suppressed: it is of the order of ~ 116V at maximum for rather large laser intensities of I % 107W/ c7712. Despite the small effective masses of the electrons in the semicon- ductor QW, it is still a negligible effect compared to the strength of the trion-based optical potential; indeed, it is at. least three orders of magnitude smaller than the 162 trion—mediated optical potential which can enter the meV range for blue detuning for relatively small intensities of I ~ 10311707712. 7 .2 Electron ionization Up to now, we have analyzed the interaction of the laser photons with the semi- conductor QW’ only in terms of photoexcitations of excitons or, if possible, bound trions. This section will be devoted to the possibility of exciting the conduction band electrons above the barrier of the QW system, which we shall call electron ionization. Of course, the occurrence of this effect is objectable to our goal of trapping electrons that are confined inside the QW. The ionization process essentially depends on the band structure of the QVV system at hand: The incoming photons are tuned close to the trion resonance, a few meV below the conduction band. Since the trion and exciton binding energies are much smaller than the bandgap cc the photon energies ha) are approximately fiw z 6c- Both GaAs and CdTe belong to the class of direct semiconductors and their bandgap cc is shown in Fig. (7.1) between the symmetry points F 8 and F5. In principle we also have to consider ionization processes into the barrier material as AlgGalflrAs, but the changes in the bandstructure are negligibly small on the energy scales for this specific process. The ionization process goes along the following mechanism: In principle, a con- duction electron initially close to the F6 point in the conduction band may be excited for example to the F7 point in a higher conduction band. Since photon wavevectors are very small, this process appears as a vertical line in the bandstructure diagram; in (7.1) we can see with the naked eye that the gap Eg that needs to be overcome is much bigger than the bandgap 62, the photons of the laser are close to in energy. Since we work in the low electron density regime, only this ionization process of con- 163 1'5 /\ 7 7 T6 1" 7 40 — H la .12 __ E/ \E Z 1" L A I" A X U,K It i; / ‘0 I/\‘ ,\ 0 l'L‘S n I; :f\;’ r ls -2 /\ x5 / ___—4 ' -1, l‘ X, I. A I" A X U.K 2 I" Figure 7.1: Bandstructures of GaAs and CdTe, taken from [29]. 164 duction electrons needs to be considered. Therefore, as far as the QVV systems we have specialized on are concerned, electron ionization can safely be neglected. For completeness and to keep the discussion as general as possible, though, we underline this reasoning with a calculation based on Fermi’s Golden rule, assuming a QW system where the gap Eg for the ionization is comparable to the bandgap cc. The initial state Ii) 2 |k, Q) of the process we are about to consider is characterized by a photon of energy hw and momentum Q = (Q, Q2) and a initial excess electron confined in the QVV of width L2 in the growth direction 2. In fi-space, the initial elec- tron is assumed to be in the lowest conduction band close to the F6 point. Assuming a plane wave inside the QVV, we write the wavefunction of the initial electron \112- (7") as __ exp (ikr) m with 65 (2.) being the electron’s wavefunction along the i-direction and uc (k, r) being ‘Pt (‘7‘) Q5 (3) no (k, 1‘) (7-8) the periodic Bloch function in the conduction band. Modelling the confinement as a infinite square well problem in the ground state, (I) (2) is given by ac.) = g sin (Liz) . (7.9) Consequently, the energy of the initial state E2; is — EC, (7.10) where the electron energy consists of two contributions: the kinetic energy of a two— dimensional plane wave and the confinement energy E; due to quantized é—direction. For an infinite square well EC amounts to EC: fig (”)2. (7.11) The final state | f) = [Ef> consists of an ionized electron which, as a consequence of the photon absorption. is supposed to have left the quantum well. Essentially, it is a free electron which is not subject to the two—dimensional confinement anymore. Consequently, its wavefunction \I'f (F) is modelled as a three-dimensional plane wave exp (if?) of (F) = W at, (Eff) , (7.12) where V is the quantization volume and up (16f, F) is the Bloch function in the higher conduction band which we label as c’. The energy of the final state is Wk} 7127;? E = E ——: 7.13 f g + 2772:: + 2171; ( ) where we have separated the z-component explicitly. We calculate the electron’s ionization rate I‘2oo based on Fermi’s Golden rule. The corresponding interaction term Hint is the light-matter coupling for this specific ionization process; we write it in the general form Him = 9c Z CL,k+Qcc~kaQ + h.c. (7.14) k The light—matter coupling constant for this process is called 92. A photon of in— plane momentum Q is annihilated, while an electron of in—plane wavevector k in the conduction band c is promoted to the upper conduction band c’. The strength of the interaction 92 is essentially governed by the overlap between the two wavefunctions of the initial and final state which describe electrons in different bands one being confined in the QW and the other one being free to move in all three directions. Neglecting a mismatch in the Bloch part of the wavefunctions the coupling constant 166 gc for this transition is ~ exp —il::fF oikr‘ ~ (A: 96 : 9c/d3r (\/F ) fld) (z) 2 9c V0 (kfz) 6kf,k+Qv (7.15) where _EJC governs the bare dipolemoment for this transition, (,I') (kf‘z) represents the Fourier transform of o (z) and the photon momentum was neglected. For the ioniza— tion rate Iiio." we obtain F . = 891.2 1 1+cos (nj/Acl/Ede a 6m (1 _a,,/E,)2 (Ad), (7.16) where AC] 2 fw — (E9 + EC) is the detuning of the photon from the energy that has to be overcome: The bandgap energy Eg plus the confinement energy EC. This result confirms our original expectations: Due to the big energy mismatch for this transition, the ionization rate is zero for the QW systems GaAs and CdTe. In principle, the conduction electrons could be ionized by a phonon-assisted pro- cess, i.e. a combination of photon plus phonon excitation, as well. Since, this is a second—order process, it is less likely to occur than the first-order photon excitation and can therefore be neglected. In conclusion, we find that the conduction electrons can be assumed to be perfectly confined inside the QW, since ionization processes do not play a great role in our setup. 7.3 Restrictions on the electron density The density of the excess carriers inside the QW can be easily controlled via modu- lation doping and, in the course of our studies, we have only touched on the electron density twice: We have determined the density for a commensurate filling of one 2 electron per site 77.1 to be ~ 109cm— . We have also mentioned before that exci- 167 tonic effects are quenched for sufficiently high carrier densities. We will specify this restriction first: We have presented the trion-mediated potential in the framework of a single-particle effect. For this regime to be valid, we need to require that the maximum number of trions that fit into the sample without spatial overlap, given by N?” = A/(7rag), is much smaller than the number of conduction electrons in the sample N2 = 712A; here, no denotes the electron density. In a sense, we restrict our considerations to the regime where the trion state can be assigned to one single electron only. Consequently, our results are valid in the regime l 716 << —-—2 (7.17) 7rat Based on a typical trion size of approximately at N 2 — 3a D, the requirement boils down to 712 << 8 x 1010c-m—2. Since this density is higher than the one-electron—per- site density m, the condition, stated in Eqn. (7.17), is a very weak restriction on the electron density n2. Let us mention another restriction on the electron density 722 we have to be aware of: Electrons obey the Fermi—Dirac distribution and therefore acquire energy up to the Fermi energy simply by phase-space filling. I In a two—dimensional system, the Fermi energy 61? is related linearly to the electron density: 6 F = (77712 / m2) Ane. For an electron density of ne 2 1010cm‘2, the Fermi energy amounts to 0.25meV and ‘ 0.36 meV for CdTe and GaAs respectively. This energy range is comparable to the the depth of the optical potential for the electrons. If the electron density 772 2 is modulated to be in the lower 10907727 regime, however, the Fermi energy 6 F is appreciably smaller than the depth of the trapping potential. 168 Conclusion We have merged concepts from semiconductor physics and cold-atom physics to prove the existence of a novel laser—induced optical dipole potential for electrons in a semi- conductor QW. This potential, generated by driving the trion-resonance with intense, detuned laser light is proportional to the light intensity. To properly describe the trion resonance, a bound state between an exciton and a charge carrier, we have obtained a Hylleraas-type wavefunction using Ritz’s variational technique. We have investigated the optical potential following two approaches: In Chapter 2.2 we modeled the QVV sample as a collection of two-level systems. The trion modifies the exciton resonance frequency in the vicinity of a carrier. Accordingly, the Stark energy is modified in proportion to the light intensity at the carrier location, which serves as a source of mechanical potential energy for the carrier. Chapter 3 was devoted to the deriva- tion of an effective Hamiltonian and the corresponding Schrodinger equation for an electron that is coupled to virtually excited trion and exciton continuum states. We have found that this trion-mediated potential exhibits several remarkable fea- tures. In particular, it displays a non-local character that is a consequence of the light effective masses of the electrons m: and of the “quivering” of the electron motion once it has virtually mixed with the extended diffusive trion state. This non-local effect takes place on a lengthscale of the order of the “trion size”, which is approximately ~ 30mm. Since this lengthscale is small compared to the periodicity of the potential, we have been able to show that the non—locality can be neglected in the analysis of 169 the electron trapping at the nodes or anti-nodes of the intensity pattern. The diffusive character of the excited trion level accounts for an effective enhance- ment factor x for the potential depth of almost two orders of magnitude with respect to typical excitonic Stark effects; the quantity x has been intuitively explained as an integral over all possible bound trion configurations having one electron and one hole “on top of each other”. Simply put, X can be seen as the number of excitons that fit into the trion size without spatial overlap. The most striking and distinguishing property of this potential relative to its atomic brother, though, stems from the strong effective background polarizability of the semiconductor medium, an effect not present for conventional atomic optical potentials. The coherent exciton resonance gives rise to a very important correction factor fc (At) to the pure trion contribution: for red detuning, this correction is smaller than one and thus leads to a decrease in the potential depth. However, for blue detuning, it can give a strong enhancement to the potential depth. We have shown that this novel type of potential can be deep compared to the single photon recoil energy ER and to the effective temperature of the electrons. The latter strongly benefits from an omnipresent natural cooling mechanism in the semiconductor environment, the phonon bath. A regime where the potential depth exceeds both the recoil energy and the effective thermal energy of the carriers by a factor of 10 seems feasible, according to our calculations. In addition, we have extended our model to the idea of a spin-selective electron lattice which is much simpler to engineer and control than similar lattices in AMO systems. There is a direct mapping of the spin of a localized electron and the optical polarization of the photon that can couple the electron to the trion state. Owing to this level scheme, two different potentials for the two spin configurations can be created. In summary, we have applied concepts from the AMO community to generate new 170 ideas for semiconductor systems. Owing to the specific properties of these systems like the background polarizability, the light particle masses, the strong inter-particle interactions, and the electron-phonon cooling, semiconductor Optical potentials are very different from the conventional optical potentials in AMO systems. From a fundamental point of view our results can open up a new paradigm in solid state physics: a gas of charged particles in a periodic potential that is possibly strong enough to trap the carriers at a single site, but weak compared to the inter-particle interactions. This system deserves further investigations, since this fermionic many- particle quantum state on a regular lattice possibly bridges the gap between current ultracold atom systems and fundamental concepts in condensed matter physics. A unique dynamic control over not only all the relevant parameters of the optical lattice allows for experiments not feasible in conventional solid state systems. A wide range of properties can be adjusted through the laser beam geometry, the polarization, the intensity and the frequency of the light. Contributions to a fundamental understand- ing of the quantum behaviour of a many electron system in a solid might be expected from these systems. In contrast to ultracold atoms in an optical lattice, strong long range interactions occur, as in a true solid. The system we propose might even touch on highly complex questions as high-Tc superconductivity or Mott-insulating phases, since strongly interacting fermionic atoms in optical lattices have previously been sug- gested for investigations on these ongoing problems [68, 69]. To quote Richard Feyn- man in 1965: “I think it is safe to say that nobody understands quantum mechanics.” Undoubtedly, experiments where the motion of electrons is optically manipulated in real time would continue to push the bizarre features of quantum mechanics to the forefront, realizing the thought experiments once envisaged by the founding fathers of quantum theory. Today, about 45 years later, with the new language of quantum information emerg- ing we might hope to deepen our understanding of the principles of quantum physics. 171 But to get there, we need to master one of the great modern scientific challenges: the development of tools to prepare, manipulate and measure the quantum mechanical state of a system. Experimental and theoretical progress has been made based on many different platforms. Examples in which quantum control is sought or has been accomplished include quantum optical systems, such as those utilizing trapped ions [70, 77] or neutral atoms [60, 78, 79], cavity quantum electrodynamics [80, 81, 82] and nuclear magnetic resonance [83, 84]. as well as solid state systems, based on nuclear spins [85, 86], quantum dots [87] and Josephson junctions [88]. All this work con- stitutes important steps towards the realization of a “quantum computer”, in which algorithms are implemented as unitary transformations on a many—body quantum state. To come back to Feynman, our model, in essence, might lead one day to the implementation of the pioneering idea due to Feynman of simulating one quantum system with another [89]. Owing to the long spin decoherence times [90] and the potential for scalability of a semiconductor based system, electron spins in semicon- ductors have been identified as one of the most promising candidates for quantum information science. Cutting-edge technologies for solid state quantum electronics might be merged with those from quantum optics by using light to localize, manipu- late and probe the entanglement of electron spins in semiconductors. Our findings are based on a very general technique so that a wide range of vari— ous systems with different applications could be investigated. Obviously, our scheme can be extended for further research including one dimensional systems (quantum wires) with an increased trion binding energy and/or holes as the excess carriers with larger effective masses. More strikingly, though, we can envisage for example a comparable semiconductor based system where the trapped particles interact via shorter-ranged dipole-dipole interactions, instead of the strong long-ranged Coulom- bic coupling present in the current system. How could we achieve this? Indirect excitons are formed from electrons and holes, that are confined to two different par- 172 allel QW layers by a potential barrier; they behave as dipoles perpendicular to the plane [93, 94]. The repulsive character of the their interaction has been evidenced in experiments as a positive and monotonic line shift with increasing density [95]. Because of the spatial separation between the electron and hole layers in this coupled QW structure, the intrinsic radiative lifetimes of optically active indirect excitons ex- ceeds that of their direct counterparts by orders of magnitude and can be in the range of several microseconds [94]. The system which we propose as a strikingly promising candidate for interesting future research projects is schematized in Fig. (7.2). Figure 7.2: Scheme for a bilayer semiconductor QW system where electrons and holes are trapped, but short range dipole—dipole interactions are realized. Electrons are depicted as blue dots, holes as greens dots. Indirect excitons X are composed of electrons and holes from different QW layers. We start out from an indirect exciton present in the coupled QVV system. By coupling the electron or the hole in one of the layers to the trion resonance via a laser standing wave as proposed in this thesis the carrier’s position can be controlled; at the same time, its excitonic partner in the other layer will follow due to the strong exciton binding energy. Since the exciton complex is overall neutral, this system has the potential to localize and control carriers which are effectively subject to a short—ranged dipole-dipole interaction because of their excitonic oppositely charged 173 companion in the second layer. Based on the spin-selectivity of the trion—mediated potential for the qubit carriers, two carriers well separated from each other first can be addressed and subsequently a differential energy shift can be induced due to their dipole-dipole interaction, conditional on the state of the two qubits. Eventually, this could implement a two—qubit phase gate, required. for universal quantum computation. In conclusion, optical trapping and cooling techniques have had a profound impact on the fields of quantum many-body physics and information processing [91, 92]: the flexibility they naturally impart could have a similar impact on solid state physics, vastly beyond the scope of this thesis. 174 Appendix A Material parameters Quantity [Units] GaAs [ CdTe ] Fundamental gap (2 K) Electron effective mass Heavy hole effective mass Effective mass ratio 3D donor Bohr radius Hartree dielectric constant Kane optical matrix element Dipole moment 6c * 771. e * 772 h. eV 777.0 7710 nm me V eV eA 1.5192 0.0665 0.34 0.2 9.95 11.58 12.74 23 6.2 1.6063 0.096 0.19 0.5 5.4 27.6 9.7 21 r 0.6 [29] [22] [22} [22] [22] [22] [29] [29] Appendix B Excitons in quantum wells In this Appendix we give a quantum—mechanical description of excitons that are confined in QW systems. For simplicity, weconsider the pure two-dimensional limit, corresponding to a QW with a very thin QW width. Introducing the scaled radius 0 = <73 (31) we can express the \Vannier equation, Eqn. (1.6), as A 2prr . _V2 _ _ = 8.2 [ p p] W) W W) < > 2115562 2 = = _— B3 0326 cao’ ( ) where a0 is given by 6H2 a0 = 2. (13.4) ”I8 In analogy to the hydrogen—problem, the energy Er is negative for bound states and positive for the ionization continuum. We define Here, E0 serves as the energy unit 7,2 2 2 . 2/‘1‘0‘0 26(10 26“}1“ A 1 [ p pjéflp) 4éhfi, ( ) where we introduced A as 2 6 11.1- : _ _ , B. he 2E7- ( 8) With our choice of <2, the parameter A will be real for bound states and imaginary for the ionized continuum. We are interested in a solution for two-dimensional semi— conductors. For this purpose, it is convenient to write the Laplace operator in polar coordinates 2 1 (9 0 5? =__n__,g 39 where £3 is the operator for the 2 component of the total angular momentum 2 82 It obeys the eigenvalue equation Cz—l—ei'71‘¢ = m——1—eim*9 m = 0 21:1 i2 (B 11) m 277' 7 ‘ a a 7 ' ' ' ‘ The ansatz _ 1 .y. (p (,0) 2 fm (P) 7:61,,” (8.12) I 177 yields the following equation for the radial part of the wave function 169 a A 1 772.2 , [ +——f‘“? fm(p)=0' (B'13) In order to solve this differential equation, we will first examine the asymptotic be- haviour of the radial wavefunctions for large radii. In the limit p —-> 00, the leading order equation reads [ d2 1 Z132- - E] foo (P) = 0- (314) Therefore, the convergent solution takes on the form foo (p) = e‘P/Q. (13.15) In the next step we investigate the asymptotic form of the wavefunctions for small p. Thus we express f (p) as f (p) 2: f0 (,0) foo (p) and see that f0 (p) should vary as plml. Factorizing the asymptotic solutions, we make the ansatz for the total radial wave function fm. (p) = plm’le—p/2R (p). (B.16) Inserting this ansatz into Eqn. (B.13) leads to 8212 BR 1 p555+(2|ml+1—p)_0—,0-+ (A— |m| —§) RZO. (Bl?) To shorten the notation we introduce the quantities l p=2|m|, q=A— |m| —5. (B18) 178 We can obtain the solution to Eqn. (B.17) by a power series expansion R (p) = 2 mp“. (319) V20 By inserting this expansion into Eqn. (B.17) and comparing the coefficients of the different powers of p we get a recursive relation for the coefficients V—q (u+1)(u+p+1)° [Bl/+1 = {331/ (8.20) Of course, only a result that can be normalized has a physical meaning. Therefore, the series must terminate for some value 1/ = Vma-x, so that all coefficients above this value vanish, i.e. 131/2111” al. 2 0. The condition um“;- — q 2: 0 gives 1 1 117710.17 + ITnl + a = A E n + 5, (13.21) where the main quantum number n can assume the values 77. = 0, 1, 2, . . . , respectively, for |m| ——- um“, = 0, Iml = 1 and 12mm 2 0 or |m| = 0 and umax = 1, etc. As a matter of fact, we have derived the bound state energies: Combining Eqs. (BS) and (B21) yields the two-dimensional exciton bound state energies, as given in Eqn. (1.9). The solution to the differential Eqn. (B.17) with integer numbers p and q can be expressed in terms of the associate Laguerre polynominals L3 (,0). Using these orthogonal Laguerre polynominals, the properly normalized two dimensional exciton wave functions can be written in general as 1 (n — |m|)! . _ 2, , . 3 l 3plmle p/2L-n.l-l-n|’l’rt| (’0) 8mm, (B22) flag (n + 312—) [(72 + l7nl)l ¢‘?7.,T71 (r) = where p = 2r/ [(n +1/2)a0]. 179 Appendix C Derivation of the volume element for Hylleraas coordinates In this appendix we will give a detailed derivation of the volume element in Hylleraas coordinates for a two dimensional problem. At the expense of mathematical elegance, but for reasons of intuitivity we choose a multi-step derivation. The ultimate goal is to find the volume element when one makes a variable transformation from Cartesian coordinates {r1,r2,rh} to a set of coordinates {R,Q, .9,t,u}, where R is the center of mass of the three-body system, 9 marks the overall angle degree of freedom that doesn’t affect the mutual distance between the particles and s, t, u are the Hylleraas coordinates. They are related to the three mutual in-plane distances between the particles 71h: T2h, r12 as 8=”‘1h+7‘2ha t=7‘1h—7‘2h» 'U-=7"12- (Cl) Our starting point for this derivation is depicted in Fig. C.1. In the first step we express the positions of the three particles in terms of the coordinates {R, 6’, p, 1, a5}, where p is the distance between the hole and the center of 180 :1: r > Figure C.1: Possible set of coordinates to describe a three body problem in two dimensions. The factor 11. is given by ,u 2 7722/2772; 181 mass of the three body system, 1 is the distance between one electron and the center of mass of the two electron system and Q5 is the angle around the center of mass of the two electrons, measured with respect to the line connecting the center of mass of the two electron and the holes position. We can express the Cartesian coordinates in terms of this set of coordinates. The holes position is given by 1*}, = X+pcos(6) y}, = Y + p sin (6), while the positions of the two electrons are 771* .171 = X -— h pcos(6) +lcos((,b — 6) 2772.; m}: y1 = Y — psin (6) — lsin (gb — 6) . * 27716 a: mh 2mzpcos (6) — [cos (qb — 6) y2 = Y—27:17:;psin(6)+lsin(¢—6). 8 The J acobian for this variable transformation d2r1 (1% (1%,, .9 |M (R, 9,1), 1, an dQRdQ dp all da is found to be |M(R,6,p.z.¢)l = 9112. 0119 ex z o 182 (C8) 777* The next step is rather trivial: We introduce the coordinate q = (I + 277%) p, so ’8 that we obtain d2r1 (1‘ng d217, —+ 4(11 dQRdB dq (11 da ((3.10) Now, that we have already separated the center of mass R and the overall angle 6, we are left with expressing the coordinates {(1, 1,09} in terms of the Hylleraas coordinates {3, t, 11}. Before achieving this ultimate goal, we take one more intermediate step by writing {(1,l,q>} in terms of the interparticle distances {r1h, rgh, u} Using the law of cosines, we find the following relations u = 21 (CH) 7‘1}, 2 12 + q2 — 2lq cos (Q5) (C.12) 7'2}, 2 \/l2 + q‘2 + 21g cos ((p), (C.13) where the identity cos (7r — (:5) = — cos ((9) was used in the last line. Inverting these equations to solve for l, q and 6 gives l = u/2 (CM) 1 r) r) U2 q = \/§(7‘ih+7§h) ‘1— (015) 7.2 —’f‘2 95 = arccos 2" 1h . (C.16) u\/2 (Tfh + r22h.) — {”2 Making use of the relations 7%,, + r3}, = % (32 + t2) and T31: — rfi = —st, the coordi— nates {q, l, 0} can be expressed in terms of the Hylleraas coordinates {3, t, u} solely 183 z = u/2 ((3.17) 1 . . . q = -2-\/.92 + t2 — u2 (C.18) c5 = arccos (— 8t ) (C.19) -'u\/.92 + t2 — u?- VVe are now in a position to take the final step of this derivation d2r1d2r2 d2rh —> 4.] (.9, t, u) q (s, t, u) l (s, t, u) d2R d6 d3 dt du. (C20) The .lacobian for this transformation is 01 (91 01 6.?- 27? M 9 t2 , . _ , a z) a _ 3" — , 4J(s,t, u) — 4 273 0% D6 — 2 2 , (C21) ,, '(2+t2—“2) 1_ 8t 84*) 06) 099 u S u 2 2 2 2 m ‘0?- m u (S +t —u ) so that we can readily deduce 'u (.92 — t2) 4J (s, t, 'u) q (9. t, u.) l (s, t,u) = — (C22) 1 4 \/(32 -112) (112 — t2) If the integrand is symmetric in t, as in our case, we can restrict ourselves to positive values of if only, multiplying the volume element by a factor of two. Finally, we state the result d2r1 d2r2 (1%,, ——> (1211 (16 ds dt du. ((3.23) u. (s2 — t2) 2\/(s2 .212) 1.2 _ t2) with the limits of integration for the Hylleraas coordinates 320,03ugs,03t3u (0%) 184 Our result for the volume element d7 agrees with the one given in [22], where the specific case of an integrand independent of 6 was assumed. In this case, the integra- tion over 6 can be performed right. away yielding a factor of 27r. We can confirm their result (13 dt du. (C25) Appendix D Results of variational calculus In this appendix we give a detailed presentation of our results for the variational calculation of the trion binding energy. Based on the trial wave function, given by Eqn. (1.26), the expectation value for the ’relative’ Hamiltonian (HT (0,117)) has been solved analytically. We have found that it obeys the following analytic expression (HT) = {a [9(557r — 204.8% + 102405 (1 + a) + +1024 ((2 — 37r) ,3 + 2,32 (1 + 0) + 27 (1 + 0)) + +128an ((16 — 4.577) ,3 + 681.» (1 + 0)) +4a2 (8,32 (157r — 256) + 47 (2l7r — 512) + 3mm (127 +12oa)) + 64a4 (37r (2 + 33 + 430) — 64)] } / (6402 (16a2 + 157rcr,8 + 4832) + 407 (512a + 4057(3) + 460872) , where the effective mass ratio 0 enters as an external parameter. In a subsequent step we have minimized this function with respect to the variables a, ,3, "y. The results are presented in Fig. (D.1). hv'Ioreover, let us mention a trick commonly referred to in the literature in the context of this type of trial wavefunction to simplify the minimization process [25, 28]. 1.86 0.5~ 0.0 0.2 0.4 0.6 0.8 1.0 Usz/m; Figure D.1: Numerical results for variational parameters a, ,6, 7 in atomic units. Omitting the normalization for the moment, one can rescale all the arguments in the general form of 99b (.9, Lu) as I f 8‘03 (1+ (311 + ";’t2) —> 63—8 fl (1+ 012/2 + C262) 2 95b (9’, t’, 'u’) (D.1) with the scaling factor k = 2a. The parameters C1 and C2 are linked to the param- eters a, (3 and 7' by the relations (3 (3 ’7' . C1 2a k 3 C2 402 k2 ( ) i.e. they absorb the scaling factor k. This scaling factor is often referred to as “effective charge”. If one substitutes 95b (.9, t, 11.) into the original expectation value 187 (HT), one obtains the (ompact form (QM — kL ET 2 — with the abbreviations 2—t2 , L =/0DC (1.9/S (111/0112 dt (8 ) [24t2—1]@§(D.4) 0 :—u2) (112—t2) s —t u (.9 :2—t:)2 62 82 M = «'3 —— — — [000 ds/S du/Ou dt 2\/a(:2u_ t2) vb] (9.92 (91,2 2 1 ___—328 _ t2 (8 05— tat) — —a: "‘ $011 2.9 (11.2 — t2) ,2 2t (.92 — 212) 2 ~ r — 11(32 — t2) ‘5” — 'u (.92 — 11:2) m b (Dd) N = [000 d9 / du /u2 dt ,5; (D.6) 0 — (.112) (112 — t2) The expressions L, M and N are quadratic in the coefficients Cl and Cg only. Due to Eqn. (D.3), ET also depends on k. The essential idea is to minimize ET with respect to k in the first place, which is immediately satisfied by L k = 2M’ (13.7) which yields the minimum ET to solely depend on the parameters C1 and C2 as L2 E = — . D- 41)! 1V ( 8) Our results for the scaled variational parameters C1 and C2 are shown in Fig. (D2). They are in perfect agreement with previous theoretical investigations [28]. The optimized variational parameters for the specific QVV systems GaAs and 188 0.18~~~—~——- 1 __ . , j 016* —— C1 .l 0.14; — C2 6 0.12[ , 2.3" 0.10] , l 0.08] f f 0.06: , +// r . 0.0“" 7*" 0‘2"“ ““7 0.4“" "”06 "““”‘“0.”8—”“‘” "1.0 0' = m; /m’,", Figure D.2: Numerical results for variational parameters Cl and Cg in atomic units. CdTe are summarized in Tab (D.1). Variational parameters ] GaAs {CdTe a 1.293 1.008 0 0.396 0.260 7 0.347 0.233 C, 0.153 0129 C2 0.052 0.057 Table D.1: Variational parameters for 00/15 and CdTe in atomic units. 189 Appendix E Exchange effects In this section we will highlight the exchange effects that arise from a proper sym- metrization of the wavefunctions describing the bound and unbound trion states. In the first part, we will show that exchange can be neglected when computing the transition matrix element from a free electron to a the unbound exciton continuum, whenever the sample size A is macroscopic. In the second part, we give a detailed derivation for the overlap between the bound and unbound trion states, which is a key ingredient for the orthonormalization procedure. In the course of the derivation, we present in detail the approximations we made to neglect the exchange effects. E.1 Exchange effects in the coupling to unbound trions The properly symmetrized version of the continuum wavefunction for an exciton with momentum km and a free electron with momentum k is given by \Pc (r1. r2. 17.) = \/2A {elki<‘*er1+0I2r/z)e‘kr2a1,, (r1 — rh) ie-ikr(aer2+ohrh)eikr10518 (r2 _ rh) (El) 190 where Ore/h = 771://h/771._1- are the mass ratios of the effective mass of the electron / hole to the exciton mass. The upper sign renders an even orbital wavefunction, corresponding to a spin singlet state, while the lower sign represents an odd orbital wavefunction and is therefore only valid in combination with a spin triplet state. Note that, in contrast to the standard textbook examples, this symmetrized ansatz is not built upon a set of separable orthonormal basis wavefunctions. This circumstance leads to the occurrence of the normalization constant 00, which we will discuss below. Proceeding from our ansatz, the wavefunction at the moment of photocreation is ikl‘ ((lerl‘i'flhr) lkl‘ I ie 6 (013 (r — r) (E2) ‘31 . - I - .- I. = [ezkl‘rezkr 0513 (0) It e‘l(Qhkx+k)l‘elaek1§r $13 (r, _ r):kE3) flx‘q Inserting this expression into Eqn. (1.46) gives the transition amplitude 00m . - (k8l Hlnt lkLE') k>c : 7% {Q18 (0) fiékeak [OkISQ + 6kx’_Q] :l: Ali/2 [dzr (121‘, e‘1'(9hk.r+k)re—'i0’ek.rl‘l¢’fs (r’ — r) xeikcl‘, [eiQr + e—z‘Qr] } (E4) The first line gives the direct term with the exciton wavefunction taken at zero. Note that it scales extensively with the sample size. To calculate the second line, we introduce the variable transformation r’—»p=r—r’ (13.5) 191 which leads to 14:72 /(121,d‘zpe—i(0hk1+k)rei(ke-Oekr)(r—p)¢:{s (,0) [(3,er + e—iQI‘] (E6) so that the two integrals can be separated l .- 43] /d2r 62(ke—kx—kiQ)r/d2pez(aekI-ke)p@={8 (p) (E7) The first one simply imposes conservation of total momentum 1 - .- _ ) ,* '—~"‘5keiQ,kl~.+k d2P€l(aekI kaparsfpl (38) /11/2 Then, we are left with evaluating the Fourier transform of the relative 1.9 wavefunction. W’ith q = aekr — k5, we obtain _ oo 27r .- _ [d2pe-qu,,=;3(p) = [0 dpp¢78> a. 1;. The importance of exchange is therefore limited to a small, finite range. At the same time, we have seen that the contribution from the direct term in Eqn. (E4) scales extensively as L / (1;. Consequently, the direct term outnumbers the exchange term by a factor of ~ A/ag. Let us address the proper normalization of 01 of our ansatz in Eqn. (E1). For convenience, we change variables according to {r17r23 I'h} -) {r,p,rh} (E'14) r = r1 — rh (E.15) p = r2 — rh (E.16) Note that under exchange of the two indistinguishable electrons the coordinates r and p flip: r <—> p. Expressed in terms of these new coordinates, the square of We (r1, r2, rh) becomes |‘1’c(r,1m‘h)l2 = ,7; {I013 (0V +|<213(p)|2 1.2-(/d2pe-lqp¢i.961.- (r2 — n.) 90(1‘1 — 17.)} (E23) The residual factor N takes into account that this symmetrized ansatz is not built upon the Slater determinant of a set of orthonormal basis wavefunctions. We will give an explicit expression for N later on. The calculation of the overlap is simplified in the coordinates {R, r, p} with the three particle center of mass R : (361.1 + )361‘2 +1.13hrh (E.24) in which the wavefunctions for the continuum and the bound trions respectively read We (RAND) = W€l(k‘r+k€)R {elprelqpfibls (7‘) i elqrelpp¢913(l))} (E25) 195 and ‘11: (R, r, p) = 72—36iKR {6.51.9 (7‘) <9 (p) i 6.61.9 (p) 90 (7‘)} (E26) Here, for the sake of a more compact notation we introduced the quantities q : flmke — 38km = ke — 88K (E28) where we already used in the last step that the total momenta of the continuum and bound trion states have to be the same to have a nonzero overlap, i.e. K 2 km + ke. Now, let us turn to the calculation of the overlap between We and ‘IJt. As far as the bound trion states are concerned, we only consider spin singlet states. Consequently, \Ilt has to be even under particle exchange, i.e. we pick up the upper plus sign. To have a nonzero overlap, the spin functions have to match, so that also for the continuum states We we pick up the plus sign, corresponding to a singlet state of the two electrons. In the first step, the integration over the center of mass coordinate R gives the condition on the total momentum mentioned above. Assuming g0 (p) to be real valued, we have ‘flN , mm = max,k.+k./d2rd2p [951mm+¢1s1 X [eipreiqpqfils (T) + e’iqre’ippéls (10)] (13.29) By relabeling the integration variables, we can simplify this expression to one direct 196 and one exchange term (QR-I‘M) = %6K,kr+ke { / dQPt‘Bi‘”lc“)1s(7‘)l2 / 612103qu (p) +/d~reprm (w) M #2“...qu on} (12.30) In the following, the momentum conservation is assumed to be fulfilled implicitly and we label the overlap by the total momentum K and the electron momentum kg as O (K, k3). To better distinguish between the direct and the exchange contributions, we split 0 (K,ke) as with 0.1mm = 3% ereiP’Imm )l )Q/deequwp) (12.32) mN . Gamma) = fi ere'Pmar)-/d2pe'qpo1(p> (E33) To proceed with our analysis, we have to specify the form of the wavefunction «,9 (7“). For the sake of simplicity we assume a 13 wavefunction up (7") = —2—e-T/at (E34) 7rat with at characterizing the "trion size”; $1.9 (r) was assumed to be the same for the continuum and the bound trions . 2 _ c.9130) = ,/—m.2e W (192.35) .17 197 Carrying out the integrals, we find the following analytic expressions for the overlap 2 {RN 8 2‘/27rat 0d (K, ke) = (E36) V A (4 + (1%122)3/2 (1 + a.%q2)3/2 and ‘flN 8 V27ra§a$at (ax + at) ‘ 3 2 fl (1 + (1525(12) / (( 061: (K, k6) = (13.37) a]; + (102 + agagifi)3 Let us compare the overlap 0d (K, kg) to (96$ (K, k3). Both decrease with increasing momenta p and q. Very interestingly, the magnitudes of both p and q are propor- tional to [363K - Re, which, due to K = Q + ke, can be equivalently expressed as fleQ — flmke. This is exactly the relative motion momentum that determined the momentum dependence of the optical matrix elements (see (1.48) and (1.49)). At maximum, if the relative motion momentum p,- = ,Bxke — BeQ is zero, (9d (K, kg) and (9m (K, ke) show the following scaling behaviours . a. 0m.- = 0) ~ It (E38) and O ( ._ 0) (f2)? 1 at (E 39) w pz _ L (1+ax/at)2—E ' which shows that the exchange contribution is considerably smaller, by more than a factor of (2.3/A: this is nothing but the microscopic exciton size over the macroscopic sample size and therefore a very small number. In the spirit of a1, / L —> 0, we can focus our analysis on the direct term. We have obtained 2 7m Nix/5 Od(K,ke) = am“ A‘ 2 3/2 2 3/2 (4 + ago-g I/BQK — kel ) (1+ (1% l/BeK — kel ) (123.40) 198 We can make further simplification by using the following approximations: First of all, a8 = m: /mI is relatively small, since the effective masses of the holes are usually much bigger than the effective masses of the electrons. Moreover, we know that the typical trion size at is considerably bigger than the exciton size ax due to the weaker binding. In this limit, we find 2 (9d (K,ke) x mm/ 72‘ N5 3/2 (E41) (1+ at? lfieK — kel2) Let us address the normalization constants in front: We have already shown in Eqn. (E20) that in tends to one in the macroscopic limit. However, we have not specified N so far. We are going to make up for that shortcoming now. In analogy to our normalization procedure to find 91, for the bound trion states we obtain / (1%erde lxrt (R,r, p)|2 = N2 1+ f/dgrolmwr) 2 ] (E42) which leads us to 2 2 / d‘ZRerde |th (R, r, p)|2 = N2 1+ 163—31ng (E43) (“I + at) The trion is spatially more diffusive than the exciton and in the limit at >> am the correction term is much smaller than one and, subsequently, N is approximately one. Using these simplifications, we find that the overlap between the bound trion states and the continuum states is approximately given by 2 0(1(K,kc) z (/ Tit 2J2— 3/2 (E44) ‘ (1+ (1%],I’36K — kel2) It is intrinsically small and depends on the relative motion momentum Pi- 199 Appendix F Trion radiative lifetime In this appendix we present a calculation that gives an approximative analytical form for the radiative decay rate F t of a bound trion. First of all we fix the spin configuration of the initial trion singlet state. We specifically refer to a trion consisting of a heavy hole with spin 0h}: = —3 / 2 and an electron singlet. Subsequently, it decays radiatively to a a- photon and an free electron with spin down i. Let us analyze the decay process in the usual lab frame first: The initial state consists of a singlet trion with total momentum K, which we denote as Ii) = lKlt' (F1) The corresponding initial energy is h'2K2 t t El 2 2771). + 266 — ETO , (F.2) where the ’relative’ energy E?” is measured w.r.t to the bottom of the conduction band. We can express the initial trion state |i) :2 |K>t with a fixed center of mass K 200 in terms of its Fourier transform of the trion wavefunction as 1 i |0), (F23) _ ~ T M — Z ‘11 (k13k2,K - k1 — k2) Ckl,TCk2,1dK—k1—k2,—(3/2) lqflkg where the Fourier transform for a trion of center-of—mass momentum K is defined in qul- (1.55). We write the final state as If) = ké. Q) = 61,, ,4, I0) (F4) with a corresponding final energy of h2k2 Ef — 2m; + cc + ha) (F5) which gives us for the energy-conserving 6-function in Fermi’s Golden Rule th2 hQA'Z ~ E—E.- = e hw— —E. F6 6( f ') 6(2m;+ 2mt ) ( ) where the quantity E was introduced as E 2 cc — Em; it simply expresses the energy gap between the top of the valence band and the bottom of the bound trion dispersion curve. In Eqn. (1.82), the transition amplitude was given as (fl “I I?) = 9,5K,ke+QI+ (Re) Taking the large volume limit for the photon momentum (Q,Qz)we write the sum over the final states as Ea: : VB/d3QZ (R7) 6 C2 (2”) kc 201 Consequently the decay rate can be calculated in the trion’s resframe, where the trions center of mass momentum is set to zero, as ’52 .. r: . (13’212— (hQ: hay—E) F.8 t (270%)] Q9 +( (2)6 2m;+ ( ) where 13 (—Q) is explicitly given by C 4 C 2 C" 2 13(_Q)=27rN2( 1Q + 2Q 4; 3) . (F.9) (aha?) The approximation we will make is to neglect the recoil energy of the electron 52g? . . . . . Emmi] = 2m... , which drastically Simplifies the evaluation of the energy conserv- 6 ing 6-function. This simplification is allowed, since all the values Q can take on in the integral are effectively restricted to ~ 1/at owing to the effect of the function I: (—Q). Because of 0’) d << E (F.10) f) . .2): .. 27"th we neglect the recoil term; in other words, the photon energy takes on the gap E and we neglect small changes due to the dispersion relation of the electron. Since we consider a process starting from K = 0 and because of the smallness of the optical photon wavevector Q, the electron will always end up close to to the bottom of its dispersion relation, so that the photon energy is approximated as simply hw z E. In this approximation, we can solve the remaining integrals simply in spherical coordinates ~ ~ 2 1 CQ4 1—1122+CQ21——u2 +0 Ft=3d—(—:N2Q3/0 du( 1 ( ) ~ 2 ( 7 ) 3) , (FM) 6 (a2 + Q2 (1 — (12)) where in the last step the quantity Q was introduced as Q = E/ (he/n). The remaining 202 integral over u can be solved analytically. 203 Appendix G Three level system in rotating frame In this appendix, we will derive the general form of a driven three level system Hamil- tonian in a frame rotating at the detuned driving frequency a). We set it ~—- 1 and follow the steps taken at the derivation of the optical potential for the two—level atom. Setting the energy of the ground state I1) to zero, the bare Hamiltonian of the three-level system can be written as Ho = we I?) (2| +w3 l3) <3l- (G.1) For the unitary transformation of our system, we define the operator U (0 : e—ztwt(12)(2)+I3) (3|) ((2.2) which gives the Hamiltonian H0 in the rotating frame 1% = H0 — my = A2 I2) <2I + A3 l3) (3|, (G3) 204 where we defined the detunings A2 and A3 as positive for red detuning in both cases, i.e. A2 = tug —- w, A3 = (4J3 - w. ((3.4) The ground state I1) is coupled to the excited states [2) and I3) with the spatially dependent Rabi frequencies {22 (r) and 93 (r), respectively. Assuming a harmonic time dependence. we can then write the interaction Hamiltonian in a semiclassical picture as Hint = -92 (r)COS(wt) H1) (2| + I?) <1|l - Q3 (1‘) COS (wt) [11) (3| + I3) (11] (G5) which becomes in the rotating frame - I I3. 0 ' Hint = —Qg(r)cos(wt) l12(0)cos(wt)+—2f(—sin(a’t) “’ 1 ‘ I. If 0' l —§23 (r) cos (wt) fig (0) cos (wt)+ if )sin(wt) . (G6) Here, we introduced the quantities (32(0) = |1><2| +|2><1| 132(0) = 3wII2)<1I — I1)(3II (G7) and similarly for I33 (0) and (2.3 (0). We now apply the rotating wave approxima- tion by averaging over one period of oscillation T = 27r/w. After the rotating wave approximation, the full Hamiltonian H in the rotating frame reads H = A2 I2) <2I+A3 I3) (3I — 92;—>II1>(2I+I2>(1I— 91,9 Ill) (3I + I3) <1|]- (G3) 205 This is the general form we used extended to the framework of the trion and exciton resonances in a. semiconductor system. 206 Appendix H Excitonic second order energy shift In this appendix we give a detailed derivation of the second order energy shift E?) (k) coming from the diffusive unbound trion states. The first approximation we make when calculating (2) __ (k|V|K,) c )(K ,cl VIk) EC (kW-Z: Em. k (H3) K k8 87 is to identify the energies of the orthogonalized states with the corresponding energies of the unorthogonalized states. This approximation is based on the fact that the overlap with the bound trion states O(K,ke) is rather small; it even vanishes as the sample size A goes to infinity. In this approximation, we express the energy denominators as n? (K — k6)2 + h? (I? — k2) 2m;- 2m; —FAm, (H2) EK,k8;kg where we introduced the detuning from the exciton resonance A]; as A$ = 66 — EX — fiw. (H3) 207 Next, we write the orthogonalized states |K,c) in terms of the free continuum |K, kg) and the bound trion states IK), respectively. Accordingly, we arrive at the equation [(k| V |K, kel (K, kel V lkl ——::g; +0 (K, ke)2 (kl V IK) (Kl V Ikl —20 1. (H4) Here, we used that the matrix elements (kI V IK) as well as (K,k€I V Ik) are real- valued quantities. Clearly, we recognize that the intermediate states can be free continuum states |K,ke), but also bound trion states IK), which gives rise to a contribution proportional to the squared overlap O (K, kg)2 or even a hybrid mixture that is proportional to the overlap (9 (K, kc). We will approximate the squared normalization constant, given by 1 N2 = _ O(K.ke)2 m1 (H.5) as one, again owing to the fact that the overlap integral (9 (K, k8) is assumed to be small, which is always true for a large enough sample. In this fashion, we have simplified the expression for E52) (k) to lelK ke> (K,kelV|kI “Egg: (2') +9 (K. kc)2 , (H6) 208 As indicated by the underlying subscripts, we will evaluate the different contributions ('27), (2'2') and (2'22) piece by piece. In order to evaluate the first term (2) we exploit the fact that (kl V |K, kc) (K’ kel V lkl = 9.351238 (Gk—keg + 5K—ke,—Q) - (H7) The presence of the Kronecker deltas eases the evaluation of the first term drastically. We obtain kyf —:ZEKke;l(< 1 1 K,k€) (K, keI V Ik) = 42}; + . Ek+Q,k;k Ek-Q,k;k - (as) Neglecting the exciton's recoil energy compared to the detuning from the exciton resonance, i.e. h2Q2 EkiQJck = 2m + Ax % A177 (HQ) ’1' we obtain (2.2 ‘23:? (kl VIK,ke) (K,ke| V|k>= — —‘?— (H.10) hl(kek 13x In order to retrieve a physically very intuitive compact form for the contribution of the term (2), we can rewrite the effective exciton Rabi frequency QT in terms of the number of excitons NJ; that fit into the sample without spatial overlap s22 8 92 A 92 92 = __Q__A : _fl___ -_— AI. _0 H.11 .r. 4 W012) 2 7mg I 2 , ( ) where we used that the exciton radius is half the Bohr radius a1: = a0 / 2. To sum it up, the term (2') gives the contribution 92 kV K2,.) (K,ke| VIk) = —N,._L). (H.12) EK ,ke;k Ad? -ZZ-E-—— 209 The presence of the exciton resonance accounts for a constant background shift that is simply proportional to NE, which describes a macroscopic enhancement factor for the intrinsic shift {lg/AI. Physically, this macroscopic enhancement factor arises from the fact that the exciton is a coherent excitation over the whole sample and thus carries a macroscopic dipole moment. Next, we evaluate the term (22') Proceeding from the electron—trion transition matrix element '30 H (KlVlk>— — g- I6K,kk+QI+< >+6K,k_QI_ (kII, (H13) we find 2 Q - 02 = flax,k+q0< I II “732 BMW; +() mic—(aka2 I- of] . (H.15) Ek—Q,ke;k Again, we will approximate this expression by taking the limit Q —+ 0, but also k —-> 0, because big values for k are effectively suppressed by the functions Ii (k)2. However, there is no restriction on ke, so that we keep this dependence. In this way, 210 the denominators simplify to 2‘2 (k i Q — k2)2 2‘22 (k3 — k2) E . = A kiQ-Ikek 2m, + 22228 + 5” h2k2 z ‘ e + Ax, (H.16) 2,221; where we introduced the reduced mass of the trion M as m*ml~ pt 2 —£——— (H.17) * ‘ _ me + my, Within the approximation of both Q ——> 0 and k —> 0, we find O(k+Q,ke) == (9 (k — Q, kg) and 1+ (k) = I _ (k). Finally, we write the term (22') in a rather compact form as C’),(Kk) Q? X A, -22— e (lelK>z—70|I+(k)|?2=0 (I), (H.18) K k E,Kke;k I where we introduced the quantity Xc (Ax) as x145): Z 3520 kelz (H.19) ke ‘Q—f'l’AJ: Since the squared overlap (9 (0, kg)2 boils down to 2 7m. 8 O(0,ke)2 = At 2 2 3 (H20) we can give an explicit solution to find Xc (AI). Taking the large sample limit in order to replace the sum by an integral and expressing the integral in polar coordinates, we 211 obtain Xc (Ar) 2 00 = 4. 222; A2, (”/0 "'( ke 2 2 3 £sz + 23,) (1+ 22,2223) 3‘7 I. 0 (E72122 ‘1' AI) (1+ (C?) where we identified in the last step the trion binding energy 52 ET = 2, (H22) This is certainly an approximation; at has not been specified so far, but was left as a free parameter in the approximative ansatz for the trion wavefunction in (??). Appre- ciating this ansatz as an educated guess, we should be in the right order of magnitude with this approximation for the trion binding energy. Based on this approximation, we have obtained an analytic expression for Xc (Ax), namely Z2521n(£)+(4—3€)£—1 H23 (5 -— 1)3 ( ) Xc (6) which is expressed in terms of the quantity g = ET/Ax, the trion binding energy over the detuning from the exciton resonance. As far as the evaluation of the (22') term is concerned, we shall halt here. Therefore, we are left with an evaluation of the (222') term, which has an hybrid character as it contains as an intermediate state the bound trion part as well as the unbound diffusive continuum states; this mixed character is governed by the overlap O (K, kg). The product of matrix elements that occur here are given by S2 - - (kI v IK) (K, kel v M = 709261,)... [ox,k+QI+ (k) + 0122-221.— (k)I . (H24) 212 Again, neglecting the optical wavevector Q, we find O(,K k) (2 222—17—— 6 (kI V IK) (Kakel V M a 2—“92 I0 (k + Q,k)1+ (k)IQ=o- Mth Ax K ke (H25) In conclusion of this appendix, we have shown that the second order energy shift Egg) (k) due to the orthogonalized continuum states can be approximately expressed as n2 ()2 J , L( )(k) = _wrfi— 2a,, XE (AE) II+ (1<)I"Q_0 no +2E—Qr l0 (k ‘1' Q) l") 1+ (kllQ=0- (H26) 33 213 Appendix I Derivation of effective Hamiltonian In this appendix we close the gap in our analysis of the effective Hamiltonian Heff by deriving compact expressions for H} I and Hg]. We will put emphasis on the off-diagonal term Hg], which is of a much greater importance for the understanding of the optical trapping potential for the electron. Therefore we will first cover the derivation of Hg], before turning to the diagonal part 71h. I.1 Off-diagonal contribution Hg] The starting point for an analysis of the off—diagonal term H; I is the quantity ch, as defined in Eqn. (3.17). The intermediate states can be bound trion states |K) or continuum states |K, c), properly orthogonalized to the bound states. Separating qu into these two pieces we have ' qVK)()KVk (qVKc)(,Kch We emphasize, that, according to the general definition in Eqn. (3.16), the case q = k is ruled out right away. The energies EKke;k are approximately given by the 214 expression (H2). In the first step we express the orthogonalized continuum states in terms of the bound trion states IK) and the free diffusive continuum states |K, kg) to obtain (QI V IK) (Kl VIk) C- . Z q,“ E: EquK;k +Zm +ZNEO (K,1(,.)2< ke EquKke;k — ZN20(K k.) (‘1' VlK) (Kkel V|k) k8 c . ' EquKke;k V K k K V k _ZNEWK’RMQI I ,a< I II ke . Eqkake;k (1.2) We can drop the second line right away, because it imposes the conditions 5k,ke as well as 6,1,1“: which results in 5k,q and is therefore not allowed. In the consecutive step, we use that the overlap O (K, kc) is a small quantity to approximate . <9 0 and k ——> O, that we have already made before, we find h2K2 71%? E . = A 2A, I.“ K“ 2m, 21772; + " t ( 0) where we used that, owing to iii-plane momentum conservation, K has to be a com- bination of k and Q, which we both neglect. Proceeding from these approximations, 215 we express ch in the following way (a l (b) ch— ch +ch, (1.6) (a) (b) where ch and ch have been defined as ZK|V|1< 1 fi§_(__0(K,)2ke (1.7) q" — EKke;k and =—Z 2: E 1.11;; )1. | v IK) (K, kel v |k> + (k k): egg/(q ) )(kl +h c. (1.10) qaék If we plug in 6513:) from expression (1.9) into 71%“) and let k —> k + Q in the first line 216 and k —> k — Q in the second line, we arrive at 2mI_ 25 (n _ u1k—Qw;w+1v H” — :§;&3 0, and exploiting the relation Ema—QI,HWW+QI,HWw—Qr,flww+Q) —— = -—1 1.13 Ek+Q,k—Q Ek—Q,k+Q Ek+Q,k—Q Ek+Q,k—Q ( ) the simplified version of 71??) is given by 201 X' Ar. H13“) s— jQZ|I+( (—k Q)|Q_0(Z—t + C; )) (|k—Q) (k+Ql +h.c.). (1.14) (b) 2(5) Now, let us turn to ch and its contribution to the effective Hamiltonian HI I . 217 It implies products of mixed matrix elements which we can write as Q (ql v IK, ka O, we get 2(b) Q092: H11 % T 2 :l0 (R, k + Q) 1+ (k “ Q)|Q=0 (lk + Q) (R '— Ql + h-C) (1-21) (11' _ k 2(a) 2(1)) In conclusion we have shown that the off-diagonal part H I I =71! + H I I takes on the form 1 A 2 ~ Xc ( :1") H11 ~ Qj(2)::|1+(k Q)|2Q=O (— +—— As ___)(lk - Q) (R + Ql + h-C) Q Q: + 2, 2k: |O(k,k+ Q) 1+ (k — Q)|Q=o (lk+ Q) qIVIKc) , sex—fl: +223“ - 024) K ke - 2 2 lyék Elk K EK; k EKke; k Plugging in the results for the transition matrix elements, for the contribution from the bound trion states we obtain 2 O, we can further simplify the expression above to (QIVIK)(K|V|1<)%9I2) 2 2K: I (k)| _ . (1.26) 2 2 l + —0 13Kk 22:, Q Following the same analysis and approximations as in the derivation of the second order energy shift E572) (k), we obtain for Hh only terms that are proportional to 1 / A? or 1 / Ag; this arises from the squared denominator in the general equation for 7'61. We then find , 52(2) (28 H}, = —ZE“’)(k)|k)>(K(1),| (1.29) where IK(I)> is the first-order correction of the bound or unbound trion states and (0) EK is the corresponding zeroth-order energy eigenvalue. The states ‘K(1)> are coupled to the zeroth-order electron eigenstates lk(0)> via the relation |K(1)> = Z 73%; |k(”)>, (1.30) k so that 71sz can be rewritten as Heff __ _ZEl'O) Z— V__kK _q_VK |k(0)>