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Decoherence in Optically Excited Semiconductors: a Perspective from Non-Equilibrium Green Functions

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Decoherence in Optically Excited Semiconductors: a Perspective from Non-Equilibrium Green Functions Decoherence in Optically Excited Semiconductors: a perspective from non-equilibrium Green functions by Kuljit Singh Virk A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Physics University of Toronto Copyright c 2010 by Kuljit Singh Virk Abstract Decoherence in Optically Excited Semiconductors: a perspective from non-equilibrium Green functions Kuljit Singh Virk Doctor of Philosophy Graduate Department of Physics University of Toronto 2010 Decoherence is central to our understanding of the transition from the quantum to the classical world. It is also a way of probing the dynamics of interacting many-body systems. Photoexcited semiconductors are such systems in which the transient dynamics can be studied in considerable detail experimentally. Recent advances in spectroscopy of semiconductors provide powerful tools to explore many-body physics in new regimes. An appropriate theoretical framework is necessary to describe new physical effects now accessible for observation. We present a possible approach in this thesis, and discuss results of its application to an experimentally relevant scenario. The major portion of this thesis is devoted to a formalism for the multi-dimensional Fourier spectroscopy of semiconductors. A perturbative treatment of the electromagnetic field is used to derive a closed set of differential equations for the multi-particle correlation functions, which take into account the many-body effects up to third order in the field. A diagrammatic method is developed, in which we retain all features of the double-sided Feynman diagrams for bookkeeping the excitation scenario, and complement them by allowing for the description of interactions. We apply the formalism to study decoherence between the states of optically excited excitons embedded in an electron gas, and compare it with the decoherence between these states and the ground state. We derive a dynamical equation for the two-time ii correlation functions of excitons, and compare it with the corresponding equation for the interband polarization. It is argued, and verified by numerical calculation, that the decay of Raman coherence depends sensitively on how differently the superimposed exciton states interact with the electron gas, and that it can be much slower than the decay of interband polarization. We also present a new numerical approach based on the length gauge for modeling the time-dependent laser-semiconductor interaction. The interaction in the length gauge involves the position operator for electrons, as opposed to the momentum operator in the velocity gauge. The approach is free of the unphysical divergences that arise in the velocity gauge. It is invariant under local gauge symmetry of the Bloch functions, and can handle arbitrary electronic structure and temporal dependence of the fields. iii Acknowledgements I am indebted to the exceptional teachers and researchers from whom I learned through- out my university education. I am thankful to my supervisor John Sipe for suggesting many challenging problems for my thesis. John has always encouraged me to pursue my own ideas, and he provided much insight and feedback to guide me towards interesting and rich directions in research. My development as a physicist has been enriched by the broad range of topics of deep interest to him. His enthusiasm for research, and the time he devotes to his students, make him a great supervisor. I started my research path as an undergraduate in Prof. Jeff Young’s laboratory at UBC, and the remarkable balance of theory and experiment that exists there gave me a broad and rich perspective of physics. I have always drawn inspiration from Jeff’s creative approach to physics, and I learned from him how much fun physics is at many different levels. I thank him for his continuing support and encouragement in pursuing my own research. I have also benefited from numerous discussions and debates with Fred Nastos, Ali Na- jmaie, and Eugene Sherman. They happily gave me their time as I learned the basics while starting my life as a graduate student. I also thank Julien Rioux for providing the band structure plot shown in Chapter 1 of the thesis. I had the privilege to spend a period of two weeks in the research group of Prof. Steven Cundiff (JILA, University of Colorado at Boulder). I am grateful to him for his hos- pitality, and I thank Dr. Alan Bristow for his time and patience in explaining me the experimental side of two-dimensional Fourier spectroscopy of semiconductors. The constant love and support from my wife, Sumandeep, has been crucial in the com- pletion of this work. She always lifts my spirits, and keeps me going through the ups and downs of research. I also thank her for freeing me from various responsibilities so that I could pursue my research. Finally, I thank my parents for their love, encouragement, and unwavering support in all my pursuits. iv Contents 1 Introduction 1 1.1 Background .................................. 1 1.2 Perspective .................................. 2 1.2.1 Experiment .............................. 3 1.2.2 Theory................................. 6 1.3 ThesisOverview................................ 11 1.3.1 Brief introduction to non-equilibrium Green functions....... 11 1.3.2 Summaryandorganization. 15 2 Semiconductor Optics in Length Gauge 20 2.1 Background .................................. 21 2.2 NumericalMethod .............................. 24 2.2.1 Position matrix elements on a lattice in the discrete Brillouin zone 24 2.2.2 Implementationissues . 28 2.2.3 Numericaltime-evolution. 30 2.3 IllustrationandDiscussion . ... 34 2.3.1 Quantumwellbandstructure . 35 2.3.2 Comparison of length and velocity gauges . .. 36 2.4 Summary ................................... 40 v 3 Multidimensional Fourier Spectroscopy: Formalism 42 3.1 TheoreticalBackground . 43 3.1.1 Electrodynamics ........................... 43 3.1.2 Basis states and Hamiltonian in length gauge . ... 50 3.1.3 Greenfunctions............................ 52 3.1.4 The effective two-particle interaction . .... 56 3.2 ResponseFunctionsFramework . 59 3.2.1 The susceptibility expansion . 59 3.2.2 Hierarchy of correlation functions and its approximate termination 62 3.2.3 Equationsofmotion ......................... 68 3.3 Application .................................. 81 3.3.1 Externalandeffectivefields . 81 3.3.2 Analysisviadiagrams . 83 3.3.3 Signal and two-dimensional spectrum . .. 91 3.4 Summary ................................... 96 4 Multidimensional Fourier Spectroscopy: Exciton Decoherence 98 4.1 Background .................................. 101 4.2 RelationshiptoExperiment . 105 4.3 Equationsofmotion ............................. 108 4.4 Modeleffectiveinteractions . 110 4.5 Sources .................................... 119 4.6 Dynamics ................................... 126 4.7 NumericalMethod .............................. 136 4.7.1 Electrongas.............................. 136 4.7.2 Numerical time-stepping . 137 4.8 Results..................................... 139 4.9 Summary ................................... 150 vi 5 Conclusion 152 A Effective two-particle interaction: Details 159 B Derivation of Integral Bethe-Salpeter Equations 161 C Transforming between T (j) and X(j) 163 (2) D Derivation of EOM for Xn 165 E Diagram Rule 4 167 F Two-time Approximation 168 G Derivation of EOM 171 G.1 Derivationoftwo-particleEOM . 171 G.2 Derivationofsingle-particleEOM . .... 172 G.3 Expressions for interaction matrix components . ........ 176 Bibliography 177 List of Tables viii List of Figures 1.1 Schematic illustration of electron-hole excitations . ............ 3 1.2 Schematic illustration of a four-wave mixing (FWM) setup......... 6 1.3 Structure of dynamics controlled truncation . ....... 8 1.4 Modification of DCT in cluster expansion and Green function methods . 12 1.5 Keldyshcontour................................ 14 2.1 Discretized momentum space . 25 2.2 Interblock and intrablock matrix elements . ...... 26 2.3 Linkoperators................................. 27 2.4 Plaquetteoperator .............................. 27 2.5 Illustration of operators W x(k) and W y(k) ................ 33 2.6 Energy bands in the presence of Pöschl-Teller potential .......... 37 2.7 Comparison of density in the velocity and the length gauge calculations . 39 2.8 Conduction band populations for field frequencies far below the mid gap 40 3.1 Graphical symbols for constructing diagrams . ....... 56 3.2 TheBSEforfourpointfunction . 57 3.3 Diagrams for 4-point effective interaction I(2) ................ 57 3.4 Bethe-Salpeter equations for six and eight point functions......... 61 3.5 Illustration of the hierarchy problem in many body physics........ 64 3.6 Contributions of two-particle correlations to the two-particleEOM. 78 ix 3.7 Examples of pair evolution induced by three pulses . ....... 89 3.8 Example diagram of a term sensitive to the composite nature of excitons 89 3.9 Source diagrams for the generation of Raman coherence . ........ 91 3.10 Diagrams contributing to the biexciton amplitude . ......... 91 3.11 Diagrams that contribute to the third order signal . ......... 94 4.1 Diagrams of the interactions in 2-point Bethe-Salpeter equation . 111 4.2 Exchange process between an exciton and the electron gas due to vertex correctionstoself-energy.. 115 4.3 Source diagrams that transfer interband polarization to Raman coherence amongexcitonstates ............................. 122 4.4 Typical (τ) ................................ 135 Bnn 4.5 Two-time
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