Fermi Edge Polaritons Formed by Electron-Hole Pairs Interacting with Cavity-Confined PhotonsYossi Michaeli ii Fermi Edge Polaritons Formed by Electron-Hole Pairs Interacting with Cavity-Confined Photons Research Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science in Physics Yossi Michaeli Submitted to the Senate of the Technion-Israel Institute of Technology Adar Bet 5771 Haifa March 2011 ii . . .1 Second Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. . . . . . . . . . . . .2. . . 3. .1 Crystal Schrödinger Equation . . . . . . . . . . . . . . . . . . .1. . . . . . . . . . . . . . . . . . . . 2. . .2. . . . . . . . . . .1. . . . .1. . . . . . 2. 2. . . . . . . . . . . . . . . . . . . 3. 3. . . . . . . 3. . . . . . . .3 The Kinetic Term . . . . . . . .2 Two Particle Hamiltonian: Coulomb Interaction . . . . . . . . . . . . . . .3. . . . 2. . . . . . . . . . . . . . . . . . . . . . 3. . . .1. . 2. . . . . . . . . . . . .1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1. . . . . . . .1. . . . . . . .3 The k · p Method . . . . . . . . . . . . . . . . . .2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 The k · p Envelope Function Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 Particle – Electromagnetic Field Interaction Hamiltonian 3. . . . . . . . .1 Model Formulation . . . . . . . . . .3. . . . .3 Schrödinger-Poisson Model . . . . . . . . . . . . . . . . . . . . . . . . . .2. . . . . . .1. . . . . . . . . . . . .1 Introduction . . .2 Löwdin-Renormalization 2. . . 3. . . . . . . . . . . . . . . . . . . . . . . . . . .1. . . . . . . .3. . . . . . . . . . . . .3 The Zinc-Blende Models . . . . . . . .2 Two-Dimensional Electron Gas . . . . . . . . .1.1 Direct Interaction . . . . . . . . .3. . . . . . . . . . . . . 2. . . .1. . . . . .2. . . . . . . . . . . . . . . 2. . . . .2 Lattice-Cell Average . . . . . . . . . .1 Normalization . . . . . . 3. . . . . . .2. . . . . . . .2. . . . . . . . . . . . . . . . . .3. . . . . . . . . . . .4 The Interaction Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. . . . . . . . . . . . .1.3 Spin-Orbit Interaction . . . . . . . . . . . . . 2. . . iii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. . . . . . . . . . . . . .1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .1 Introduction . . . . . . . . . .3. . 2. .2.2 k · p Envelope Function Ansatz . . . . . . . . . . . . . . . . . . . . .1 Single Particle Hamiltonian . . . . . . . . . . 2. . . . . . .3. . . . . . . . . . .1. . . iii vi x 1 2 5 10 10 10 12 12 12 13 14 14 15 16 16 17 18 21 26 27 32 39 39 39 40 41 41 42 42 43 43 43 44 3 Free Carriers Optical Transitions 3. . . . .1. . . . . . . . . . . . . . . . . . . .2. . . . . . . .2. . . . 2. . . . . . . . . . . . . . . . . . . . . . . . .2 Remote Contributions . . .Contents Contents List of Figures List of Tables Abstract List of Symbols 1 Introduction 2 Electronic Properties of Semiconductors 2. . . . . . . . . . . . . . . 2. . . . . . . . . . . . . .1.2. . . . . . . . . . . . . . . . .1. . . .4 Two Band Model . . . . . . . . . . . .2 Bloch’s Theorem . .1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 Introduction of Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 Bloch States Formulation . . . . . . . . . . . . . . 2. . . . Dipole Matrix Element . . . . . . 4.4 Carrier Statistics . . . .5. . . . . . . . . . . . .1 Single Two-Level System . . .1 Bare Microcavity . . . . . . . . . . . . .3. . . . . . . .2. . . . . . .2. . . . . . . .3. . . . . . . . . . . . . . . . . . .4 Microcavity Polaritons . . . . . . . . .5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2. . . . . . . . . . . . . . . .3 3. . . . . . . . . . . . . .1 Microcavity Reflection Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. . . . .3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .iv CONTENTS 3. . . . . . . . . .2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 Coulomb Hole Self Energy 4. . . 3. . 5. . . . . . . . . . . . . . . . . . . . . .3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. . . . . . . . . . . . . . . . . . . . . . . . . . 5. . 4. . . . . 4. . . . . .4 Solving the Correlated Equation . . . . . . . . . . . .2. . . . . . . .3 Coupled Harmonic Oscillator .2 Plasma Screening . . . . . . . . . . . . . . . . . . . . . . . . .2 Multiple Two-Level Systems . . . . . . . . . . . . . . . . . . . . . . . .1 Background Screening . . .2 Transitions Calculation .2 Distributed Bragg Reflector . . . . 5. . .2 Quantum Microscopic Polarization . . . . . . .4. . . . . . . . . . . . . . . . . . . . . . . . . . . 45 45 45 47 47 49 50 51 55 55 55 57 57 58 58 59 61 61 61 63 63 63 64 69 70 75 95 95 97 98 101 101 103 103 104 104 106 107 107 107 108 110 110 112 116 132 135 138 143 4 Coulomb Correlated Optical Transitions 4. . . . . .4. . . . . . . . . . 3. . . . . . . . . . . . . . .2. . . . . . . . . . . . . . . . . . . . . . . . . 4. . . . . . . .1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .3. . . . . . . . . . .3 Heisenberg’s Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1. . . . . . . . . . . . . 4. . . . . . . . . . . . . . . . .2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. . . . . . . . . . . . . . 5. . . . . . 5. . .3 Introduction of Holes . . 5. . . . . . . . . . . . . . . . .2 Hartree-Fock Approximation . . . .3. . . . . . . . . . . . . . . . . . 5. . . . . . . . . . . . . . .5 Implementation Considerations . . . . . .1 Transfer Matrix Method (TMM) . . . . . . . . . . . . . . . . . . . . . . . .2 Strong Coupling in a Semiconductor Microcavity 5. . . . . . . . . . . . . . . . . . . . . . . .3 Embedded Doped QW . . . . 4. . . . . . . . . . . . . .4 Screened Exchange Energy 4. . . . . . .1. . . . . . . . . . . . . 4. . . . . .1. . . . . . . .5. . . . . . . . . . . . . . . . . . . 5.4 Solving the Equation for Free Carriers Spontaneous Emission . . . . . . . . . . .3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2. . . . . . . . . . . . . . . . . . . . .2 Diagonal Approximation . . . . . . . . . . . . . . . . . .3 Microcavity Optical Characteristics . . . . . . . . . . . . . . . . . . . . . . . Transitions Calculation . . . . . . . . . . . . . . . . .2 Linear Dispersion Model and Beyond . . . . . .1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2. . . . . . . . . 4. . . . . . . . . . . . .1 Introduction . . .2 Doped Quantum Well . .3 Many-Body Effects . . . . . . .5. . . . . . . . . . . . . . . . . . .4. . . . . .4 3. . . . 4. .2. . 5. . . .2 Microcavity Confined Photon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 Strong Absorber-Photon Coupling . . . . . . . . . . . . . . . . 5. . . . . . . . . .2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 3. . . . . . . . . . . . . 4. . . . 4. . . . . . . . . . . . . . . . . . . . . . . . 5. . . . . .1 Coupled Oscillator Model . . . 5. . .2. . . . . . . . . . . . . . . . . Summary and Outlook A Symmetry Properties of Wavefunctions B Two-Band Model Numerical Implementation C Self-Consistent Solution of Schrödinger -Poisson Model . . . . . . . . . . . . . . . . . . .1 Equations of Motion .2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2. . . . . . . . . . . . .2. . 4. .1 Second Quantization . . .2 Embedded Undoped QW . . . . . . . .1 Optical Resonator . . . . . .4. . . . . . . . . . . . . . . 3. . . . . . . . . . . . . . . . . . . . . . . . . 4. . . . . . . 5. .4. . . . . . . . . . . .1 Bare Quantum Well . . .3. . . .1.3. . . . . . . . . . . . . . . . . 5 Semiconductor Microcavities and Polaritonic Effect 5. . . . . . . . . .4. . . . . . . . . . . . . . . . . . . . . . . . . . .2. . . . . . . . . .4. .1. . . . . . . . . . . . . . . . . . . . . . . . . . . 149 F Refractive Index of Alx Ga1−x As Alloy Bibliography 152 155 . . . . . . . .1 Temperature dependence of the bandgap . 148 E. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .CONTENTS v D Lindhard Screening Model 145 E Bandgap Energy Modeling of Semiconductors 148 E. . . . . .2 Semiconductor Alloys . . . . . . . . . . . . . . . . . . . . . . .1 Ga0. . . . . . . . . . . . . . . . .19. . . . . Band structure of bulk GaAs around the Γ point at room temperature. . . . .1 Ga0. . . . . . . .9 As quantum well conduction subbands dispersion relation calculation and the appropriate density of states. . . . . . . . .List of Figures 2. . . . The band-edge potential and the self-consistent potential of a δ -doped single quantum well described in Fig. . . . . . . . . . . . . . . . . . . The calculated dispersion relations for the conduction and valence subbands. . . . . . . . . . calculated for an undoped 200 Å wide GaAs/Al0. . vi 12 14 19 23 24 24 25 26 27 28 29 29 30 31 31 32 33 33 33 34 34 36 37 38 53 54 . . . . . . Block diagram illustrating the process of self-consistent iteration. . . . . . . . . . . . . . . . . Areal charge density a for a 100 Å GaAs well. . The potential due to the ionized donor/electron charge distribution shown in Fig. . . . . 2. . . . . .9 As barriers. .4 2. . . . . .8 As 4 meV barriers. . . . . . . . Dependence of the transition strength |µk |on the angle between the electron k-vector and the electric field polarization vector E. . . . . . . . . . . . . . . . . .1 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The "in-plane" subband structure of the quantum well in the conduction band. . . . . .16 2. . . Electric field strength from an infinite plane of charge of volume density d(z ) and thickness δz .2 2. . . . The calculated wavefunction for the conduction. . . . . Dipole matrix elements calculated for 200 Å wide GaAs/Al0. . . . . Schematic band structure classification in the Löwdin perturbation theory. .9 As quantum well. . . . . . . . .19 2. 2. . . . . . .9 As undoped quantum well valence subbands dispersion relation calculation and the appropriate density of states. . . . . . .10 2. . . .24 3. valence heavy and light hole subbands for 200 Å wide GaAs/Al0. . . . . . . . . . . . .1 Ga0. . .9 As quantum well at T = 2 K with 1 × 1010 cm−2 2DEG concentration. . . . . . . . . . . . . . . . .3 2. . . . . . .11. . . . . . . . . . A schematic diagram of a GaAs/AlAs mixed type I . .18 2. . . . . . . . . . . The structure and band diagram of a modulation-doped heterojunction between GaAs and n − AlGaAs. .20 2.12 2. . . . . . . . . . . . . . . . . . surrounded by undoped Al0. . . . . The valence subband dispersion relations and boundary conditions at the interfaces of the well. .type II QWs structure. . . .1 3. . .1 Ga0. .7 2. . . . . . . . .8 2. . . . . . . . . . . . . . .11. . .2 Schematic representation of electronic functions in a crystal. . without additional doping. 2. . . . . .9 As quantum well at T = 2 K . . . . . . . . . . . . . . .1 Ga0. . Volume charge density a for a 200 Å GaAs well. .14 2. . . . . . . . . . . . . with various 2DEG concentrations in the well region. . . . . . . . . . .9 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5 2. . . . .23 2.1 Ga0. . . . . . . . . . . . . . . . .21 2. .15 2. . .9 As quantum well at T = 2 K . . . . . . . . . . . . . . . calculated using the 8 × 8 k · p for two crystal directions. .6 2. . . . . . . . . .22 2. . . . . . . . . . . . . . . . . n-type doped to 2 × 1018 cm−3 . . . . . . .9 As quantum well at T = 2 K with 1 × 1011 cm−2 2DEG concentration. . . . . . .17 2. . . . doped with two symmetrical δ -doping layers located 1000Å from the well. . . . . Contours of constant energy within any [100] plane of k -space for the heavy and light hole subbands in bulk GaAs. . . . . . . . . . . .2 Ga0. . The electric field strength E due to the charge distribution shown in Fig. . The calculated wavefunction for the conduction. The band-edge potential and the self-consistent potential of a modulation-doped single quantum well. . valence heavy and light hole subbands for 200 Å wide GaAs/Al0. . . . . . . . surrounded by Al0. . . . . . . Example to a 200 Å wide GaAs/Al0.13 2. . . . . . . . . . . . . . . . Schematic illustration of a semiconductor substrate and an epitaxial film containing a δ -doping layer. . . . . . . . . . . . . .1 Ga0. . . for a 200 Å wide GaAs/Al0. . . Wavefunctions of the conduction and valence subbands. . . . . . . . . Example to a 200 Å wide GaAs/Al0. The sum of the band-edge potential VCB and Poisson’s potential Vρ for single quantum well for the charge distribution shown in Fig. . . . . .1 Ga0. . . . .11. 2. .11 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The TE polarization absorption and spontaneous emission spectra calculated for the doped structure for various gas concentrations using the HF model at T = 2K . . . . A schematic depiction of the bandgap renormalization due to the introduction of 2DEG into the well region of the structure. . . . . . . . . . .27 4. . . . . . . . . . . . . . . . . . . . . . .1 As well with ne = 1 × 1011 cm−2 . . . . . . .2 4. . . . . . Calculated form factors for the electron-hole. . . . . . . . . . . . . . . . . . . .9 Al0. The bandgap renormalization dispersion for the various subband transitions calculated for the 200 Å wide GaAs/Ga0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The absorption spectra for both polarizations. . .28 Dipole matrix elements calculated for 200 Å wide GaAs/Al0. . . . . . . . . for the 200 Å wide GaAs/Ga0. . . . . . . . . . . . . . . . . . . . Spontaneous emission spectra calculated using the FCT and HF models for the considered undoped structure at T = 77K . . . . . . .20 4. . . . . . . .1 As well with various 2DEG concentrations for various conduction and valence subbands.10 4. . Infinite hierarchy of operator products for the equations of motion. . . The TM refractive index change spectra calculated for the doped structure for various gas concentrations using the FCT and HF models at T = 2K . . . . . . . . . . . . . . . . . . . . . . .9 As quantum well at T = 2 K . . . . . . . . . . . . .23 4.6 4. . . . . . . . . . . . . . . . . . . . . . . . . . . .8 4. . . . . . . . . . . containing 2DEG with various concentrations. . . . . . . . 54 59 59 62 65 66 66 67 68 70 71 72 73 74 75 76 77 78 79 79 80 80 82 83 84 85 86 87 88 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The complex elelctrical susceptibility calculated using the FCT model for the undoped well at T = 77K . . . . . . . . Outline of an approach that takes advantage of a phenomenological derivation of plasma screening. . . . . . . .14 4. . . . . . . . Schematic representation of the Hartree-Fock approximation operator product in the equation of motion.22 4. . . . . . . . . . . . . Calculated Coulomb-hole self energy and screened-exchange shift energy at the Brillouin zone center. . . . . . . . . . . . . . .26 4. . .11 4. . . . . . . . . . . . . . . . . . . . . . . . . .7 4. The complex elelctrical susceptibility calculated for the undoped structure using the HF model for the undoped well at T = 77K . . . . . . . . . . . . . . . . . . . . . . . . . . . both as a function of the 2DEG concentration. . . . .5 4. The total bandgap renormalization at the Brillouin zone center. . The absorption spectra for polarizations. . . . . . . . . . . . . . . . . . . . . The TE refractive index change spectra calculated for the doped structure for various gas concentrations using the FCT and HF models at T = 2K . . . . .18 4.9 4. . . . . . . . . . . .1 As well with ne = 1 × 109 ÷ 1 × 1011 cm−2 at T = 2K . . . . . . . . The TM polarization absorption spectra calculated for the doped structure for various gas concentrations using the FCT and HF models at T = 2K . . . .3 4. . . . .25 4. . . . . . . .17 4. . . . . . . . electron-electron and hole-hole combinations. . . . . . .4 4. .1 As well with various 2DEG concentrations. Schematic band edge absorption spectrum for a 2D semiconductor according to the Elliot formula. . . . . . . . . . . . . . . . . The complex elelctrical susceptibility calculated using the FCT model for the undoped well at T = 2K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 4. . . . . . . . . .LIST OF FIGURES vii 3. . . . .19 4. Schematic structure of the undoped QW examined in the numerical calculations. . . . . .24 4. . .1 Ga0.21 4. . . . . . . . . . . . . . . . . . . . . Spontaneous emission spectra calculated using the FCT and HF models for the considered undoped structure at T = 2K . . . The TE polarization absorption spectra calculated for the doped structure for various gas concentrations using the FCT and HF models at T = 2K . . . . . . . . . . . . . . . . .9 Al0. . Detailed view of the transition region in the absorption and spontaneous emission spectrum for the TE polarization using the HF model at T = 2K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The complex elelctrical susceptibility calculated using the HF model for the undoped well at T = 2K . . for the 200 Å wide GaAs/Ga0. . . calculated using the FCT and HF models for the undoped well at T = 2K .12 4.9 Al0. . which is incorporated directly into the many-body Hamiltonian to give the screened Hartree-Fock equations. The TM polarization absorption and spontaneous emission spectra calculated for the doped structure for various gas concentrations using the HF model at T = 2K . . . . . . . . . . . . . . . . Schematic structure of the doped QW examined in the numerical calculations. . . . . . . . . . . . . . . . . .9 Al0. . . . . . .13 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15 4. . . . . . . . . . . . . . . . . . . . . The energetic gap between the Fermi energy and the bottom of the conduction band and the Fermi wavevector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . The screening dielectric function (q ) calculated for the 200 Å wide GaAs/Ga0.16 4. . . . . . . . . . . .3 4. calculated using the FCT and HF models for the undoped well at T = 77K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Microcavity confined photon dispersion curves for various resonance modes. . . . . . . . . . . . . . . Calculation of the electric field amplitude of a microcavity for two DBR mirror configurations. . . . . . . The reflection spectrum minima energies and values. . . . . The TM reflection spectra as a function of the cavity mode energy. . . Amplitude and phase of the normal incidence reflection function for a microcavity with two DBR’s with 35 alternating high and low refractive index layers. . . . . . . . . . . . .24 5. 4. . at T = 2 K . . .14 5.33 The TM polarization absorption and spontaneous emission spectra calculated for the doped structure for various gas concentrations using the HF model at T = 77K .19 5. . . . . . . . . . . . . Schematic profiles of the simulated microcavity with embedded undoped QW in the cavity region. . . . . .18 5. . . . . .15 5. . . . . . . . . . . . . . . .8 5. . . . . . . . . . . . The TM reflection minima anticrossing curves extracted for each 2DEG concentration from the reflection curves . . . . . . obtained manually and from the coupled oscillator fit results for both polarizations. . . .7 5. . . . . . . . . . . . . . . . . . . . . . .27 Fabri-Perot etalon multiple reflections.9 5. . . . . . calculated for the simulated structure for various detunings. . . . . . . . . . . . .5 5. . . 5. . . . . . . . 4. .12 5. . . . . . . . . . . . . . . . . . .22 5. . . . . . . . . Schematic profiles of the simulated microcavity devoid of the embedded QW in the cavity region. . . . . . . . . . . . . . . . Reflection minima anticrossing curves for both polarizations of incident light. . .13 5. . . . . . . . . . . calculated for each of the considered 2DEG concentrations for various width detunings. . at T = 2 K . . . . . . . . . . . . . . . . . . . . . . . . . . . .20 5. . . . . . . . . . . . . . . . . . . . . . . 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . calculated at T = 2 K . . . . . . .16 5. . .10 5. . . . . . . . . . . . . .9 As quantum well. . . . . . . . . . . . Schematic description of the proposed computation model. . . . A microcavity schematic structure. . . . 89 90 91 93 94 96 96 97 99 100 101 102 103 104 105 109 110 111 111 112 113 114 114 115 116 117 118 119 120 121 122 123 . . . . . . . . . .2 5. . . . . . . . . . . . . . . . . . . . . . . . . . . calculated at T = 2 K . . . Schematic description of a typical sample wafer and the cross-sectional width profile of such a wafer. . . . . . . .4 5. . .29 Detailed view of the transition region in the absorption and spontaneous emission spectrum for the TM polarization using the HF model at T = 2K . . . . . . . . . . . .31 The TM polarization absorption spectra calculated for the doped structure for various gas concentrations using the FCT and HF models at T = 77K . . . . . . . . . calculated for each of the considered 2DEG concentrations for various width detunings. . . . . . The TE reflection minima anticrossing curves extracted for each 2DEG concentration from the reflection curves . . . . . . . . . . . . . The amplitude and phase of the normal incidence reflection calculation of a DBR structure with 35 layers with alternating high and low refraction indices. . . . . . . . . . . . . .11 5.17 5. Schematic illustration of a Distributed Bragg Reflector. . . . . . . . . . . . . . . . . . . . The TM polarization reflection minima anticrossing curves fitted using the coupled oscillator model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The reflection spectra for various width detuning values calculated for normal incident light both polarization polarizations. .32 The TE polarization absorption and spontaneous emission spectra calculated for the doped structure for various gas concentrations using the HF model at T = 77K . . . . . . . . . . .1 5. . . The total refractive index spectrum of a bare 200 Å wide GaAs/Al0. . . . .3 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The TE reflection spectra as a function of the cavity mode energy. . . . . . . . . . . . . . . . . . Resonator transmittance function for various values of mirror reflectance. . . . . . . . . .21 5. . . . . . . . . . .1 Ga0. . . . calculated using the transfer matrix method for width detuning δ = 0. . . The reflection spectrum of the simulated microcavity structure . . . . . . . . . Schematic profiles of the simulated microcavity with embedded doped QW in the cavity region. . . . . . . . at T = 2 K . . . . . . . . . . . . . . . . . Schematic description of light propagation through an interface between two adjacent dielectric layers and light propagation in a homogeneous medium. . . . . . . . . . . . . .94 at T = 2 K . . . . .26 5. . . .viii LIST OF FIGURES 4. . . . . . . . . . . . . . . . . . . Energy levels of the uncoupled absorber and field modes and the dressed energy levels of the coupled system. The reflection anti-crossing resonance energies as a function of 2DEG concentrations. . . . . . . . .25 5. . . . . . . The TM polarization reflection minima anticrossing curves together with respective polariton admixing coefficients. . . . . . . . . . . . . . . . . . with a 2DEG with concentration of ne = 1 × 1011 cm−2 in the well region. .23 5. . . . . . . . . . . . . . . . . . . . . . . . . .30 The TE polarization absorption spectra calculated for the doped structure for various gas concentrations using the FCT and HF models at T = 77K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 E. . . . . . . .3 Refractive indices for T = 77K . . . . . . . . . obtained for each resonance branch for both polarizations at ne = 3 × 1010 cm−2 . . . . . . . .35 The fitted amplitude and width of the transmition peaks as a function of cavity mode energy. . . . . . . 5. . . . . .95. . . . . . .1 The real and imaginary parts of the refractive various temperatures and x = 0 − 1. . of energy. . . . . . . . .1 Program flow for self-consistent solution of Schrödinger -Poisson under equilibrium condition with given donor concentration. . . . . . . . . . . . . . . . . . . . . . . . 5. . . . . . . . . . .4 Refractive indices for T = 300K . . . . . . . . . . . . . . . . .32 The resonance interaction linewidths extracted for various resonances using the coupled oscillator model fit of the anti-crossing curves as a function of the 2DEG concentration. . . . .29 The energetic distance between the anticrossing resonance lines as a function of 2DEG concentrations for both polarizations. . 144 D. . . . . . . 5. . . . 5. . . . . . . . . . 5. . . . . . . . obtained for each resonance branch for both polarizations at ne = 1 × 1011 cm−2 . .37 The fitted amplitude and width of the transmition peaks as a function of cavity mode energy. . 5. . . . . . . . . . 123 124 125 125 126 127 127 128 129 129 130 130 131 C. . . . . . . . . . .30 The coupling parameters extracted from the coupled oscillator fit of the reflection anti-crossing curves. . . . . . . . . . . . . . . . . . . . . . . . . . 5. . . as a . . . . . . . . . . . . . . . . . . . . . . 5. . . function . . . . index of Ga1−x Alx As . . . . . . . . . . . . . . F. . . . . . obtained for each resonance branch for both polarizations at ne = 3 × 109 cm−2 . . . . . . . . . . . . . . . . . . . . for both polarizations. . . . for both polarizations of the incident light. . for both polarizations. . . . extracted manually (via FWHM calculation) for each of the present resonances at width detuning of δ ≈ 0. F. . . . 146 E. . . . . . . .38 The polariton admixing coefficients extracted from the coupled oscillator model fit of the anti-crossing curves for ne = 3 × 109 cm−2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.LIST OF FIGURES ix 5. . . . . . . . . . . . . . . . . . . . . . .95 .34 The reflection spectra linewidths function of the 2DEG concentration. . . . . . . . . . . . . . . . .1 Change in the carrier distribution due to an electron at the origin. . . . . . . . . . . . . . . . . . . . . . . . . . . .2 Alx Ga1−x As alloy bandgap energy as a function of the Al composition for various temperatures. . . . . for both polarizations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40 The polariton admixing coefficients extracted from the coupled oscillator model fit of the anti-crossing curves for ne = 1 × 1011 cm−2 . . . . . .2 Refractive indices for T = 2K . . . .1 Temperature dependence of the Alx Ga1−x As alloy bandgap energy for various values of x. . for T = 2 K . . . . . . . . . . . . . . . . . . . . . . . . . 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31 The energetic distance between adjecent anti-crossing curves near resonance lines as a function of the 2DEG concentration for both polarizations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. . . . .36 The fitted amplitude and width of the transmition peaks as a function of cavity mode energy. . . . .33 Reflection spectra for various 2DEG concentrations calculated at width detuning of δ ≈ 0. . . 153 153 154 154 . . . . . . . . . . . . . . F. . . . . . . for . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. . . . . .39 The polariton admixing coefficients extracted from the coupled oscillator model fit of the anti-crossing curves for ne = 3 × 1010 cm−2 .28 Calculated first conduction and first three valence subbands dispersion relations for a bare doped QW with ne = 7 × 1010 cm−2 . . . . . . . . . . . . . . . 151 F. . . . . . . . . as a function of the 2DEG concentration. . . . . . . . . . . . . . . . 115 The calculated intersubband transition energies for a bare doped QW with ne = 7 × 1010 cm−2 and the difference between them. . . . . . . . . . . . . . .1 5. . . . The non-vanishing momentum matrix elements between the states corresponding to different irreducible representations of the tetrahedral symmetry group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7 E. . . . . . . . . . . . . .1 A. . . . . . . . . . . . . . . . . . . . Basis functions of the tetrahedral symmetry group Td . . . . . . . .3 A. . . . .9 The extracted parameters of the simulated structure for each of the seven resonances Xi . . . . . . . . . . . . . . .2 A. . . . . . . . . 136 136 137 137 149 149 149 150 150 x . . . . .1 E. . . . . . . . . . . Bandgap energies at room temperature compared to reported data (II). .5 E. Parameter values for modeling the bandgap energies (II). . . . . . . . . . . .4 E. . . . Parameter values for modeling the bandgap energies (I) . . .2 A. . . . . Bandgap energies at room temperature compared to reported data (I). . . . . . . . . . . . . . 118 Symmetry operations of the group Td using the Schönflies notation. . . . . . . . Direct products of the Γ15 representation with all representations of Td . . . . . . . . . .3 E. . . . . . . . . . . . . . . . . . . . .List of Tables 5. . . . . . . . . . . . . . . . . . . . . Parameter values for the bandgap of alloy materials. . . . In these structures the photonic mode is spatially confined inside the cavity layer. reflection and photoluminescence (PL) spectroscopy studies of such structures containing a twodimensional electron gas (2DEG) have been reported. where we incorporate both the Coulombic interactions between the charge carrier and also discuss the mutual screening effect. and thereby investigate the interaction of such electron-hole pairs with the light field excitations within these structures near the Fermi edge. and (b) the strong coupling regime. in the transfer matrix calculation of the entire MC reflection spectrum. We assume a dipole light-matter interaction. the calculation is performed with and without the presence of 2DEG of various concentrations. the light-matter interaction in the QW is modeled via a semi classical approach. the obtained electronic and optical properties of a bare QW are then used to calculate the reflection spectra of the full MC structure. and (b) the screened Coulomb-correlated model. where the 2DEG influence is incorporated through a self-consistent the Schrödinger-Poisson model. As for the bare QW. where the system eigenenergies are Rabi split modes termed cavity polaritons. We show that the strong interaction of the cavity-confined photon with the interband excitations of the twodimensional electron gas leads to the formation of cavity polaritons. Recently. In this study. Finally. The coherence between all these e-h pair excitations is induced by the strong interaction with the cavity mode. We start by considering the electronic properties of a bare QW via a two-level k · p method. the formulated approach is utilized to calculate the reflection anti-crossing curves of the investigated MC structures. In comparison. with QW embedded in their cavity region. which are embedded between two high reflectivity distributed Bragg reflectors (DBR). The interaction strength can be described in two regimes: (a) the weak coupling regime. Once the electronic properties of the QW are found. we present a theoretical framework for the modeling of MCs with embedded QWs. When the cavity layer width is comparable to the exciton wavelength. obtained from its electrical susceptibility calculation. This calculation is performed with and without the presence of the 2DEG in the well region of the QW. and discuss two approximated methods to model the optical properties of the considered QW: (a) the free-carrier model. This is achieved by using the refractive index spectrum of the QW. where the coherence stems from the Coulomb interaction between the electron and hole. Next. located at the Fermi edge of the 2DEG. yielding the complex electrical susceptibility. revealing sharp polaritonic lines. such as a quantum well (QW) or a heterojunction.Abstract Semiconductor microcavities (MC) are structures consisting of a heterostructure. our model calculations show that excitons are not formed in bare QWs having a 2DEG with such densities. composed of a cavity photon which is in strong interaction with unbound electron-hole (e-h) pair excitations. The most significant result is the demonstration that cavity polaritons are indeed formed for high 2DEG densities at the Fermi edge energies. which are analyzed manually and through a fitting procedure to the coupled-oscillator model. where this interaction can be described by perturbation theory. interaction between the exciton and the confined photon takes place. in contrast with the exciton-polaritons case. This allows us to investigate the influence of the 2DEG of variable concentrations on the coupling strength of the electron-hole resonances with the cavity photons and of the polaritonic linewidths. 1 . where the Coulombic interaction is absent and the dephasing is modeled phenomenologically. absorption and spontaneous emission spectra of the investigated of the structure. ne nh N2DEG .List of Symbols k En (k) EFM E B A φ H Ev Ec ∆ k φ e kt (k ) kz ψ nk unk (r) pnn hij Fnk. σy . ne f (E ) crystal momentum electron dispersion relation envelope function method electrical field magnetic field vector potential scalar potential Hamiltonian operator valence band-edge energy conduction band-edge energy spin-orbit splitting energy crystal momentum scalar potential elementary charge transverse/in-plane wavevector wavevector parallel to the growth direction particle wavefunction in a solid Bloch’s basis function momentum matrix elements Hamiltonian matrix entries slowly-varying wavefunction envelope optical matrix parameter nonzero interband momentum matrix element valence band-edge energy conduction band-edge energy Pauli operators Luttinger’s parameters spin-orbit splitting energy heavy-hole subband envelope function light-hole subband envelope function density of states density of states permitivity of the material volume charge density areal charge density two-dimensional electron gas electron carrier concentration hole carrier concentration 2DEG concentration Fermi-Dirac distribution function 2 .m (z ) Ep P Ev Ec σ (σx . σz ) γ1 . γ3 ∆ Fhh Flh ρ(E ) DOS ρ(z ) σ (z ) 2DEG N (z ). γ2 . mn (k) ω ˜ mn (k) Γn.n2 .m (k) N ( ω) rsp ( ω ) Isp ( ω ) KMS Rsp HF ϑq 1 . nr λc E+. ω ) BGR ∆ECH. t) (q.k.k P χ D k0 e ˆi χ and χ G α γ RPA ωn.and right-hand cladding material refractive indices the center wavelength of the high reflectevity region forward and backward propagating waves complex amplitudes the transfer matrix of the interface the transfer matrix of propagation in the same material the transfer matrix of a DBR structure reflection coefficient of the DBR structure transmission coefficient of the DBR structure .k Ωk (z.m k (ν ) R T F nL .k µcv.n4 Gn q.LIST OF TABLES 3 EF kB T kF a ˆ† ˆn n and a |0 FCT A L ˆ ˆ b† n and bn n ˆ ck n ˆ vk p ˆvc.n3 . nH nl .n ∆ESX. E− Mi Mp MDBR rDBR tDBR the Fermi level energy or the chemical potential Boltzman’s constant ambient temperature Fermi wavevector electron creation and annihilation operators vacuum ground state free-carrier theory volume of the translational invariant direction volume of the quantized direction hole creation and annihilation operators electron number operator in the conduction band hole number operator in the valence band microscopic polarization dipole matrix element macroscopic polarization optical susceptibility electric displacement photo wavenumber unit vector the real and imaginary parts of the optical susceptibility medium intensity gain medium optical absorption real part imaginary part phenomenological decay rate random phase approximation intersubband transition energy photon density of states spontaneous emission rate per unit volume per unit energy spontaneous emission intensity Kubo-Martin-Schwinger relation spontaneous emission rate per unit volume Hartree-Fock theory Fourier transform of the Coulomb potential form factor Rabi frequency screening dielectric function bandgap renormalization Coulomb-hole self energy shift screened exchange energy shift renormalized intersubband transition energy Coulomb enhancement factor etalon reflectance etalon transmittance etalon finesse coefficient the low and high refractive index of the DBR dielectric layers the left. 4 LIST OF TABLES RDBR TDBR MC MM C RM C TM C γc lef f Eph (kt ) mph HF f (r) HA HI Ω γcav . γabs Q γX EX (kt ) LDM FWHM reflectance of the DBR structure transmittance of the DBR structure microcavity the transfer matrix of the microcavity structure reflectance of the MC structure transmittance of the MC structure cavity mode linewidth the effective length of the caity spacer MC mode dispersion relation the cavity photon in-plane effective mass Fock Hamiltonian complex vector function of the polarization and relative amplitude of the cavity mode the Hamiltonian of a two-level absorber system the interaction Hamiltonian in the dipole approximation the atom-field coupling constant cavity and absorber damping factors the cavity quality factor exciton linewidth exciton energy linear dispersion model full width at half maximum . Thin dielectric structure growth technology has undergone a rapid development thougout the last two-decades. the ground state of the electronic system is determined by the carriers populating the conduction band resulting from the structure modulation doping. If the damping is weak enough. If the QW exhibits a pronounced exciton transition whose energy is nearly resonant with that of the confined photon. the photon-exciton interactions lead to crystal momentum conservation. the exciton-photon coupling strength is constant.Chapter 1 Introduction Semiconductor microcavities (MC). for a given semiconductor. The strong coupling of QW excitons with the MC trapped photons was first shown experimentally by Weisbuch [1]. resulting in a well defined polariton dispersion in this plane. The spectroscopic and dynamic properties of polaritons resulting from the basic resonances of the QW have been extensively studied. When the photon energy matches the energy difference between the excited and ground states of the atom. is about two-times larger for the 2D system compared the bulk semiconductor case. using the dipole and rotating wave approximations. which facilitates the easy attainment of the strong coupling regime in such a system. In quantum structures comprised of GaAs/Alx Ga1−x As QWs. in which the coupling strength is easily amplified and controlled. where a change in the emission has been detected for high finesse semiconductor MCs. much of the physics of light-matter interaction was first studied in the field of atomic physics. This 5 . The use of MCs with embedded QW enables to easily control and enlarge the exciton-photon coupling strength. who showed that momentum conservation in bulk semiconductors constrains an exciton to couple only to a photon with the same wavevector. This strong coupling is typically realized by embedding a quantum well (QW) in the center of the MC. 3]. the lowest excited states of the system are no longer degenerate. which typically consist of a GaAs layer with Bragg mirrors on the top and bottom. determined by the exciton oscillator strength. The so called Rabi splitting between these two modes is controlled by the cavity and the QW exciton properties [2. named exciton-polaritons. In this system the Rabi splitting is proportional to the excitonic oscillator strength. the strong coupling regime can be easily reached. where a Rabi splitting was manifest for a single atom confined in a metallic cavity [3]. facilitating the creation of ever more elaborate structures for the investigation of light-matter interactions. have been the subject of intense research more than a decade. The solid state analog was first discussed by Hopfield [5]. The polaritonic splitting. For planar MCs with embedded QWs. a strong exciton-photon coupling is manifest by the appearance of two lines on the reflection spectra. The new eigenstates of the system are thus superpositions of the uncoupled system states with an energy difference given by the Rabi frequency. The simplest model for the interaction of a radiation field with atoms is the Jaynes-Cummings model [4] which treats the case of a single two-level atom interacting with a single-mode radiation field. and does not depend on the exciton density. AlAs and Alx Ga1−x As. and when both bare exciton and empty cavity linewidths are small enough. As the lowest energy excitons in the QW (the heavy and light hole excitons) are those with the strongest oscillator strength. The first experiments that examined the spontaneous emission amplification in QW via light trapping were conducted by Yamamoto. the new eigenstates are mixed photon-exciton modes. Thus the spectroscopic and dynamic properties of such MC polaritons have been extensively investigated. In fact. which is in fact the first demonstration of a MC polariton. Such systems are typically comprised of commonly used alloy semiconductors such as GaAs. In contrast to the two-level atom case. since the first experimental evidence of strong coupling between the excitations of an electronic system located in the cavity and the confined photons [1]. where the system is devoid of excitons. introduced inside the QW through various doping mechanisms. we reformulate the problem using the two-band approximation. a quantum structure has to constructed where the energy of one of the two can be altered in a controllable and constant fashion. For the verification of this proposition. In order to describe the subband behavior of inhomogeneous quantum structures. it was conjectured that the source of the observed polaritonic lines are unbound electron-hole pairs with Fermi wavevector. the polariton energies equal the energies of the uncoupled exciton and photon. The introduction of the magnetic field facilitates in the separation of the interaction of the photonic mode with the 2DEG electrons and with the well excitations. there are still traces of strong interaction between the trapped photonic mode and various electronic resonances of the QW. The obtained experimental results were backed by theoretical calculations including the the dispersion relations of the studied QWs. The obtained experimental results were then fitted using the proposed experimental models in order to extract various experimental parameters. The achieved coupling is typically described by two models. An alternative method for the description of this coupled system is the linear dispersion model (LDM). such as optical spectra measurements and transform measurements based on the quantum Hall effect. Here. with the MC photon. the coupling Hamiltonian of the exciton and photon with a specific coupling strength can be diagonalized in order to obtain the polariton energy spectum. In this thesis we present a theoretical study of the electronic and optical properties of MCs with a single embedded QW in the cavity region and introduced 2DEG. Recently.6 Chapter 1. the refractive index of the QW is described by an effective refractive index where the exciton is modeled by an Lorentzian oscillator. of which we concentrate in the δ -doping scheme where a monoatomic layer of dopants is placed during the structure growth in the cladding regions of the QW. Here. The end result of this approach is that the exciton energy can be treated as constant compared to that of the MC photon mode. These studies have shown that in a doped QW placed inside a MC with high 2DEG concentrations (ne ≥ 0. In order to solve the appropriate Schrödinger equation. which plot the polariron energy as a function of the energetic difference between the excitons and the photon. The properties of such a gas can be probed through various experimental techniques. The first is the coupled oscillator model. Introduction doping can be attained through several growth techniques. and employ the Zinc-Blende model for the description of the bulk semiconductor. The first approach enables the investigation of the entire spectrum of the gas. In order to investigate the interactions between the electronic resonances of the QW and the MC photonic mode as a function of the their energetic difference. A convenient form for the representation of the polaritonic excitation mode are the anti-crossing diagrams. where the transfer matrix formalism is employed. we use the envelope function approximation for the description of the crystal Bloch function. For the computational complexity simplification. was assumed to originate from the strong interaction itself. in which each one of the interacting resonances is described by a simple quantum oscillator. dictated by the observed strong interaction. We start the analysis in chapter 2 by discussing the electronic properties of bulk semiconductors and quantum structures based on them. The coherence between these pairs. with and without the application of external magnetic field. In order to calculate the reflection spectra the transfer matrix method was employed. while all others are introduced as perturbations to the Hamiltonian. Here we propose a new modeling approach of such structures based on basic physical considerations. Away from the resonance region of the system (|δ | 0). 8] that examine the interaction of a 2DEG. the MC structure is constructed of layers with variable width along the diameter of the sample wafer. in contrast with the transport measurements which provide a glimpse only to the Fermi energy edge states of the electronic system. exhibit multi-particle phenomena stemming from electron-electron and electron-hole interactions in the presence of the 2DEG. To this end. we resort to the semi-empirical k · p approach where only part of the conduction and valence subbands of the crystal are contributing. As an explanation for this phenomenon. the optical susceptibility of these wells and finally the phenomenological dipole moments. the QW conduction band electrons can be described as a two-dimensional electron gas (2DEG). 7. δ . We start by presenting the most general formulation of the crystal Hamiltonian which is then simplified using the the Bloch formalism. Through illumination of a specific location along the sample the system can be brought to a interaction state where the photon mode energy equals that of the electronic resonance. a thorough experimental study was conducted including the measurement of the reflection and photoluminescence spectra such structures in the presence of 2DEG. In such a structure. The spectroscopic measurements. conducted at low temperatures and under applied magnetic field. .9 × 1011 cm−2 ). which stands in contrast to the accepted convention of fitting phenomenological models to experimental measurement results. while close to the resonance (δ ≈ 0) these energies deviate from those of the exciton and amount to the Rabi splitting of the system. several experimental studies have been conducted [6. This finding confronts the accepted view that the disappearance of well excitons leads to the destruction of the polaritonic modes. The addition of the Coulombic interaction to the model leads to the appearance of the bandgap renormalization effect. With the consolidation of the model for the calculation of the electronic properties of the considered structures. For the doped structures. we easily observe the clear influence of the Coulombic interaction on the absorption and spontaneous emission spectra. gradually dissipate with the rise of the carrier concentration as a direct consequence of the electronic screening effect. which plays a major role in all considered quantum structures throughout this thesis. together with assumed equilibrium conditions between them and simple phenomenological description of the carrier scattering process in the crystal. In order to consider the influence of external electrons on the electronic properties of the QW. performing the calculations for various ambient temperatures. The solutions of this system of equations is the main focus of theoretical discussion presented in these chapter. In chapter 3 we ignore to Coulombic interaction present between the various charge carriers in the system. where a parabolic dispersion relation is assumed. in particular of two variations of single bare GaAs/Alx Ga1−x As QWs. we confine ourselves to a semiclassical discussion of the semiconductor. we turn in chapters 3 and 4 to deal with their various optical properties. namely T = 2K and T = 77K . Identification of the observed interactions becomes possible by performing calculations of the electronic subbands of the considered structures using the results of chapter 2. To this end. thereby providing the selection rules for the energetic intersubband transitions of the structure. the magnitude of which is strongly dependent on the concentration of the introduced charge carriers in the system. In chapter 4 we reincorporate the Coulombic interaction. we are required to first obtain the microscopic polarization. an equilibrium conditions are assumed for the charge carriers and a naive phenomenological model for the carrier scattering is employed. To this end. through this quantitative presentation we introduce quantitatively the concept of the 2DEG. the conduction subbands are obtained using the effective mass approximation. For the attainment of the macroscopic optical parameters of the system such as the optical susceptibility.7 described by a 4 × 4 Hamiltonian matrix and the semi-empirical Luttinger parameters. which clearly appear in the first structures and also for low concentrations of the 2DEG in the second one. The EM field is subsequently added to the Hamiltonian under the dipole approximation. through the appearance of clear characteristic resonative lines suitable to excitonic coupling observed in such structures. calculated for a wide diapason of wavevector values. Using the introduced models. including the optical susceptibility. As in chapter 3. we next perform full calculation of the optical properties of quantum structures considered in this thesis. These line. These envelope functions are are subsequently used to calculate the matrix elements of the momentum in the dipole approximation. where the charge carriers are described quantum mechanically while the electromagnetic (EM) field description is classical in nature. In addition to these considerations. with the use of the full Hamiltonian of the system containing the light-matter interaction terms. For the sake of simplicity. we turn to the solution of the Heisenberg equation for the suitable quantum operator. In addition. The end result of this approach is a set of envelope functions and eigenenergies for each of the conduction and valence subbands. absorption. we add the proposed calculation scheme an electrostatic calculation in the form of an iterative Poisson problem solution together with the Schrödinger problem at hand. For these two structures we compare the absorption and spontaneous emission spectra calculated using the two theoretical models elaborated above and to the classical Elliot model. We start by performing a second quantization of the full system Hamiltonian without the presence of the EM field. which leads to the creation of system of three coupled equations typically termed the Bloch semiconductor equations. which leads to a self dependent series of equations treated in a approximate fashion using the Hartree-Fock approach with the Hamiltonian first order deduction. we also consider the influence of the 2DEG concentration on the spectral properties of the structure. the first devoid of external charge carriers in the well area and the latter with a 2DEG introduced through a δ -doping in the cladding area. refractive index and the spontaneous emission for a single bare QW. In . in this chapter we introduce the the carrier screening effect through the classical Lindhard model of the crystal dielectric function. we present in these chapters approximate models for the calculation of various optical parameters. From the calculation results for the two structures. absorption and the spontaneous emission of these structures. The microscopic polarization equation is coupled with similar equations for the charge carrier densities quantum operators. The solution the of the resulting coupled equations system for each value of the wave vector is obtained through the use of a transfer matrix formulation of the finite difference method computational scheme. Both approaches presented in these chapters. This iterative approach introduces the influence of the presence of the additional carriers through the subsequent alteration of the conduction and valence bands energetic profiles. thereby introducing the concept of the hole. In this model. which is further simplified by a diagonalization scheme which bring the problem down to a 2 × 2 Hamiltonian formulation. which is performed in several stages. provide us with the spectra of the electrical susceptibility. we also examine the influence of the carrier concentration on the properties of these interactions. Another classical effect. We first present the calculated reflection spectra for of bare MC structure. where the phenomenological description of the refractive index of the QW is replaced by the results of the models presented in earlier chapter. We first present the basic concepts of a classical optical resonator and discuss the implementation of such resonators through the growth of thin epitaxial layers composed of two or more types of semiconductor materials resulting in a creation of a couple of distributed Bragg reflector (DBR) mirrors with a cavity region of a target wavelength whole multiple width between the two. typical for such structures measured at low temperatures. After discussing the optical properties of these microcavities. and enables us to extract the optical properties of the entire structure from the optical properties of a bare QW obtained from basic physical principles. we elaborate on the quantum interaction theory of the optical and electronic excitation in microcavities. Introduction addition. and attests to the fact that the strong coupling between the EM field mode and the Fermi edge electron-hole resonance is cause to the appearance of the observed lines in the reflection spectra. we present a new approach based in the classical linear dispersion model (LDM). exhibited in the comparison of the emission spectra. we examine the coupling strength and the linewidths of discussed resonances. After the introduction of the computational procedure. Through the comparison of the anti-crossing curves attained for each of the concentrations we observe the existence of a strong coupling between the EM field mode and the electron-hole resonances up ne ≈ 1 × 1011 cm−2 . a way must be found to incorporate the two in a unified computational framework. describing the interaction between the electronic resonances created inside the QW placed in the cavity region and the EM field modes of this cavity. we introduce the two types of QWs. we first discuss the Jaynes-Cummings model of a two-level absorber developed in the framework of atomic physics and utilize it for the demonstration of the Rabi splitting of such a system when coupled with a single EM field mode. with the introduced 2DEG inside the well region. In chapter 5 we turn to the discussion of the main topic of this thesis. by extracting the admixing parameters of various resonances from the coupled oscillator fitting results. For the two resulting structures. We see that while the coupling strength diminishes with the 2DEG concentration. Equivalently to the atomic theory. Again. To this end. For the second structure. and thereby introduce the concept of the polariton as a coupling between the electron-hole resonance and the mode of the EM field. inside the cavity region of this MC. we show that they match the allowed transitions near the Fermi edge of the structure and not the Brillouin zone center. but the coupling stays well within the bounds of the strong coupling regime. This simple model can be generalized to a system with multiple such absorbers. To this end. This observation is compatible to the experimental observations presented earlier. which enable us to evaluate the properties of these light-matter interaction and their strength. and develop the concept of microcavity polariton. we present the calculation results of the reflection spectra obtained for various values of the broadening coefficient δ . Finally. Along these curves a typical splitting can be seen near the interaction region between the EM field mode of the MC and the electron-hole resonances of the QW. we apply it to a particular MC structure. We present numerical methods for the calculation of transmition and reflection spectra of such structures. discussed in chapter 4. This differs from the common approach of fitting the theoretical models to experimentally extracted data. In order to investigate the optical properties of a microcavity with an embedded QW in the cavity region. namely the microcavity structure. we show formulation of the coupled oscillator model for such a system. This differs from the results obtained in chapter 4 for the bare QW. and find that a linear change in the width of the epitaxially grown dielectric layers comprising the structure leads a proportional linear shift in the spectral position of the cavity mode of the trapped light.8 Chapter 1. By comparing the observed resonance energies of this MC reflection spectra to the calculation results of the intersubband transition energies of a bare single QW. this analysis affirms these interaction to be well in the strong coupling regime. By fitting these anti-crossing curves to the coupled oscillator model and manual extraction of various parameters. we observe the crucial part of the EM field mode of the MC in maintaining the strong coupling with . we examine the contents of the various anti-crossing branches in the attained curves. Next. For the structure with devoid of 2DEG in the well region. the excitonic lines undergo an energetic shift towards high energies as a result of the effective bandgap renormalization and the filling of the state space by the additional free charge carriers. The experimentally observed asymmetric spontaneous emission spectral lineshape is observed for high gas concentrations. and in particular the classical transfer matrix method (TMM). From these spectra we extract the energetic locations of the reflection for all values of δ and thereby attain the anti-crossing curves for each structure. where the excitonic resonances disappear at gas concentrations above ≈ 6 × 1010 cm−2 . By fitting these curves to a coupled oscillator mode a set of parameters can be extracted. which has also been previously investigated experimentally. is the Burstein-Moss splitting which leads to the separation of the maxima of the absorption and spontaneous emission spectra with the rise of the 2DEG concentration inside the QW. the linewidth rises considerably. .9 the electron-hole resonance even at high 2DEG concentrations. After introducing certain approximations to the crystal Schrödinger equation. with an emphasis on the two-band model . modifications of the electronic properties from their bulk properties. the lattice periodicity and its symmetry leads to the formation of electronic bands.2) (2. This model is then used to analyze the influence of a two-dimensional electron gas (2DEG).1 Crystal Schrödinger Equation Introduction 1 0 2 (pi − ezi A (ri )) + 2mi 2 The total Hamiltonian of all electrons. Finally.1. The term pi − ezi A (ri ) is the mechanical momentum of particle i. Note that the presented discusion doesn’t include the influence of strain effects on the electronic properties. which is invariant to our gauge choise (see below) for A and φ.1) In the second term E denotes the electric.1 2.1). We first present an extensive introduction to the Schrödinger equation within a semiconductor crystal.1.3) In the first term of (2. introduced by various methods into the quantum well region. E=− (2. The present chapter covers the theory of the calculation of the electronic states in nanostructures using the envelope function method (EFM). The combination of different materials within a nanostructure breaks the symmetry of the semiconductor crystal. The chapter is organized as follows. atoms and the electro-magnetic field of a solid state crystal is given by H= i 2 E 2 + c2 dr 0B (2. leading to quantization effects. which will later also serve as the basis for the analysis of optical effects covered in Chapter 3 and the many body effects in optical spectra in Chapter 4. on the electronic properties of the system. 2.1. i. we present the Schrödinger-Poisson model of the electronic bands structure where the electrostatics of the system are of relevance.1.and B the magnetic field.e.1. zi is the charge of particle i in units of e (-1 for electrons) and mi denotes the particle mass. continuously (within the band) relating the crystal momentum k of an electron to its energy En (k). The electrical field term A2 holds both transverse and longitudinal 10 . the response of electrons within the atomic lattice to external perturbations. the concept of solving the resulting single-particle Schrödinger equation using the bulk k · p method is presented. In bulk semiconductors. These modifications have a pronounced impact on the electronic and optical properties of nanostructures. e is the elementary charge.Chapter 2 Electronic Properties of Semiconductors The optical properties of nanostructures are intimately connected to the electronic states. a precise knowledge of the electronic properties is required for a proper device analysis. These fields are related to the vector potential A and the scalar potential φ via [9] ∂A − ∇φ ∂t B = ∇ × A. As such. The subsequent section derives the equations of the k · p envelope function method used in nanostructures and simultaneously introduces the k · p model for zinc-blende crystals. 1. The external contribution are Vext = i zi eφext (ri ).1. l dr = 0 Next. the Born-Oppenheimer approximation [10] can be used to separate the motion of electrons and atoms.5) with unit charges zi and zj . (2. j and atoms I.2. ∇ · A = 0 A 2 [9].6) The crystal Hamiltonian for electrons i. the vector potential A is restricted to external excitations. the effect of the external electro-magnetic field on the atoms has been excluded. the contributions to φ are distinguished between internal φint and external φext . This approximation is usually justified in the case of weak fields and heavy atomic masses. 4π 0 |ri − rj | (2. Therefore.8) 2 + i. As the atoms have a much higher mass than the electrons. the longitudinal contribution can be written in terms of the charge density ρ 2 (2. The electrons will react on movements of the atoms instantaneously. where the internal contributions are created by the crystal electrons and atomic cores. the interaction between ion cores and between electrons and ion cores can be concentrated into a potential U (x). where the local oscillations within the crystal lattice are averaged out.J e2 zJ +Vext 4π 0 |ri − rJ | Vea + 0 2 2 E2 dr.4) E2 (∇φ) dr = φ∇φ − φ∇2 φdr = φ dr.7) where T is the kinetic energy part of the Hamiltonian. i.j = e2 zi zj . J then reads H = i (pi + eA) + 2m0 Te 2 I (pI + ezI A) 1 + 2mI 2 Ta 2 i. The internal contributions are given by the longitudinal Coulomb interaction between the charged particles i and j Vi. where an electron experiences the mean-field potential U (r). t + c0 B HEM (2. and V the potential energy one. Using Poisson’s equation. and so we can split this term to the transverse and longitudinal 2 2 2 contributions E = El + Et . leading to the final crystal electron Hamiltonian H = i (pi + eA(ri )) + U (ri ) 2m0 1 2 e2 + Vext + HEM .j e2 4π 0 |ri − rj | Vee 1 + 2 I. Within this step.e. The equation reduces to a single electron equation. In the following discussion we assume a Coulomb gauge for the vector potential.1. In this gauge the term ∂ ∂t ∇φ in E vanishes.1. 4π 0 |ri − rj | (2. The interaction between electrons and atoms is restricted to a frozen lattice. This step is necessary as solving the equation for 1023 explicitly considered electrons is an impossible task.1. .J e2 zI zJ 1 − 4π 0 |rI − rJ | 2 Vaa i.and valence electrons can also be concentrated into the potential U (r).j The next step is to perform the single electron approximation or mean field approximation by assuming that the electron-electron interaction Vee with the core. Overall. The interaction between the movement of the atoms and the electrons can be reintroduced using phonons [10]. but this will be omitted throughout the thesis. Crystal Schrödinger Equation 11 contributions.1. The nk are the quantum numbers indexing the solutions. For any translation vector R mapping the infinite crystal lattice to itself.3.1: Schematic representation of electronic functions in a crystal (a) potential plotted along a row of atoms. The potential U (r) is periodic within the lattice (see Figure 2.11) Hk·p (k)unk (r) 2m0 m0 = En (k)unk (r). = − ∇2 + k·p+ k + U (r) unk (r) 2m0 (2.13). using symmetry arguments . the crystal is assumed to be homogeneous and perfect.r) (c) (d) ψ(k.12 Chapter 2.13) 2 2 An important consequence of the Bloch theorem is the fact that wavefunctions with different k values are not coupled together (due to the slowly varying plane wave) and therefore (2. 2.1. (c) amplitude factor of Bloch function having the periodicity of the lattice. One of the most frequently used is the semiempirical k · p method that serves to derive analytical expressions for the band structure.12) and the plane wave is the slowly modulating envelope.1.1).9) For the moment.1.1. (2.1.1.1 The k · p Method Introduction There is a vast number of methods to obtain solutions to (2. the time independent single electron Schrödinger equation to solve is given by − 2m0 ∇2 + U (r) ψ = Eψ.3 2.r) r Figure 2.1. (b) free electron wave function.13) has a parametric dependence on the crystal momentum k.2 Bloch’s Theorem 2 Assuming no external field. and (d) Bloch function ψ (after [11]) 2. Electronic Properties of Semiconductors (a) U(r) (b) exp(ikr) u(k.10) and the wavefunction can be expressed in the form known as Bloch function ψnk = unk (r)eik·r .1. the potential obeys U (r) = U (r + R) (2. Applying the differential operators to the plane wave and multiplying the equation on both sides from the left with e−ik·r gives the equation for the lattice periodic functions as 2 (2. where unk (r) denotes the lattice periodic path with unk (r) = unk (r + R) (2.1.1. 1. the matrix (2. the conduction and the valence bands lie at the Γ point energetically close together. Crystal Schrödinger Equation 13 and experimental observations.13). given by the free-electron dispersion. Hereafter a short introduction of the general concepts will be given. It is particularly useful to describe the band structure for direct semiconductors used in optoelectronic devices at the Γ point of the Brillouin zone with a high precision.1.2 Löwdin-Renormalization While the matrix hij (2.16) where pnn = u∗ n0 (r)pun 0 (r)dr (2.1.14) is not solved explicitly and no closed expression for un0 (r) is needed. we can point out that (2. one may express the lattice periodic functions unk (r) away from the Γ point in terms of the zone-center functions amk. (2.1. therefore a submatrix hij of the conduction.1. Therefore. As a remark. (2. 2. the assessment of light emission in optoelectronic devices usually requires only the knowledge of the lowest conduction and highest valence bands.1.15) is inserted into (2. so the interaction in-between these bands can be considered to dominate for the band structure of interest.1.1. Using these symmetry properties.13).18) and diagonalized.1.18) and a diagonalization of thek dependent. An extensive presentation can be found in [10] and [14]. In the model developed by Kane [15]. 2. Luttinger and Kohn [16] included the interaction of remote states onto the selected set of bands. Although the primary interest of Kane’s model was to include spinorbit interaction. multiplied from the left by un0 (r) and integrated over the crystal unit-cell to obtain En (0) + n k 2m0 2 2 δnn + m0 k · pnn amk. the expansion (2.e.3. For the relevant III-V semiconductors. similarities and equivalences for the momentum matrix elements pij can be deduced.1. These concepts will later reappear within the Envelope Function Methods (EFM). (2.13) reduces to 2 − 2m0 ∇2 + U (r) un0 (r) = En (0)un0 (r). usually the Γ point at k = 0.2. umk (r) = (2.13) for an extremal point with high symmetry of the band structure.18) is constructed by using group theory to derive symmetry properties of the zone-center functions un0 (r). where a matrix entry is given by hij = Ej (0) + k 2m0 2 2 δij + m0 k · pij (2.1. The idea of Löwdin perturbation theory is to divide the bands in two classes S and R (see Fig.n un 0 (r).and valence band at the Γ point is extracted out of (2. Other bands can be regarded as being remote. There.2).15) n To obtain the coefficients amk. A rough guideline illustrating the essence required by the k · p theory is presented in Appendix C. infinite matrix would lead to the exact coefficients and energies Em (k).1. the missing interaction with remote bands resulted in a heavy-hole band structure.1. i. the respective rows and columns of the matrix . using Löwdin’s perturbation theory [17]. The basic idea within the k · p theory [13] is to solve (2. The above equation can obviously be written in matrix form. This introduction is mainly based on [10].n . Bands in class S are considered explicitly. is is clear that the dispersion Em (k) will also be continuous.17) is used to express the momentum matrix element between two zone-center Bloch functions.n = Em (k)amk. Instead. As the matrix is continuous in k. A profound introduction into the group theory for semiconductors is beyond the current scope.1.18) is infinite.1.14) The solutions un0 (r) are denoted as zone-center functions and span the complete Hilbert space of all solutions to (2. leading to an infinite number of bands.1.n . The method was initially introduced by Bardeen and Seitz (see references in [12]) and applied by many researchers. bending for a hole into the wrong direction. one obtains the single band effective mass dispersion for band i as Ei (0) + k + 2m0 m0 2 2 R ν k · psν pνs · k Ei − hνν 2 = Ei (0) + 2 kT 1 k.14 Chapter 2. In nanostructures. m∗ (2. Hereby.2. the symmetry is broken by the atoms of the other species in one direction.1.j are a result of the combination of the free carrier dispersion and the (1) perturbation treatment of remote states in class R. as (2) (1) Hi. for a quantum wire. By keeping only one single band in class S .j stem from the direct treatment of the k · p interaction (and later from linear spin-orbit terms). (2.j =x. while zero order terms H(0) . which is fine close the zone-center. actually requiring self-consistency.1. 2. this assumption is no longer valid and the translational symmetry is broken in certain directions.z i=x.j ki kj + Hi.2: Schematic band structure classification in the Löwdin perturbation theory. the second order terms (2) Hi. The bands R are considered being remote and their effect on the bands in class S are included in the submatrix of hij using perturbation theory.1 The k · p Envelope Function Method Introduction In previous sections. the resulting matrix can be written. Electronic Properties of Semiconductors Class R Class S Class R Figure 2. the few remaining constants can be determined by comparing analytical dispersion expressions to experimentally determined effective masses. the energies Ei are approximated by the zone-center energy Ei (0). a carrier might be energetically confined within a lower-bandgap material.20) In the case of several bands within the class S. while first order terms Hi.y. the crystal was assumed to be homogeneous and infinitely extended.19) where i and j are in class S and ν is in R. ordered by the dependence on the wavenumber k. An important note here is that Ei denotes the energy of band i such that the renormalization of the matrix element depends on the result of the eigenvalue calculation.y. Group theory and symmetry arguments are then used to derive similarities and vanishing terms and reduce the perturbation expressions to a few constants. the renormalized matrix elements are given by R hij = hij + ν hiν hνj .2 2. it is broken in two directions and for a quantum dot in all three . hij are kept. given by the matrix element hii . As a consequence. By analytically diagonalizing the remaining matrix.z To summarize. Ei − hνν (2. In practice. contain zone-center energies (and possible terms if the zone-center basis is not orthogonal). but still free to propagate within the translational invariant direction.j ki + H(0) .1.21) i. In the case of a quantum well. The ordering is irrelevant within bulk materials.2 k · p Envelope Function Ansatz The traditional ansatz is to use the zone-center k · p lattice periodic functions un0 (r) for the expansion (2.5) .2. In the case of a multi-band equation. the plane wave term decouples the wavefunctions with different crystal momenta k.4) for all envelopes involved F(z ) = (F1 (z ).2.2. the Bloch function employing the plane wave ansatz has to be modified ϕmkt (r. FM (z ))T a system of coupled partial differential equations Hk·p (z )F(z ) = E F(z ). r denotes the coordinate of translational invariant direction(s). (2. but within the single-band effective mass theory.g. z is the coordinate of the direction(s) where the crystal symmetry is broken and un (r. the conduction band in a quantum well for kt = 0. Once the eigenvalues Em (0) and normalized eigenfunctions Fs. other orderings are suggested too (see [19] and references therein). the wavefunction is in each material expanded into the materials zone-center functions.m (z ). The k · p Envelope Function Method 15 directions. corresponding in the direct band gap semiconductors to the conduction band energy at Γ. F2 (z ).. z )eikt ·r Fnk. the states are now mixed together. (2.2. The result is that the effective mass like parameters from the perturbation interaction with remote states and the zone-center energies are position dependent.. the envelope equation is given by − ∂2 + Ec (z ) Fs (z ) = E (0)Fs (z ). one obtains in analogy to (2. depending on the transverse crystal momentum kt .2.4) The particular order of the coefficient and the differential operators is commonly referred to as operator ordering. where not only one. the in-plane dispersion is usually approximated using the dominant effective mass of the quantum well material Em (k) = Em (0) + k . In the bulk crystal. but plays a substantial role at a material interface: it is equivalent to the matching conditions for the bulk Bloch functions. the bands are split into subbands. In a nanostructure. justified by the requirement of a continuous probability flow.2. different materials are involved.2) are obtained.4) is denoted as Ben-Daniel and Duke ordering [18].1) and derive a proper equation to determine the envelopes Fnk. z ) = n un (r. The form of (2. together with a set of matching conditions. The equation for the envelopes is obtained by replacing the wave number ki by the corresponding operator −i∂i . how the lattice-periodic functions are mixed together.2.m (z ). 2. The expression Fnk. but several lattice periodic functions are included. (2. . Consequently.2. the number of translational invariant directions is reduced from two for a quantum well to zero for a quantum dot.2.m (z ) is referred to as slowly-varying envelope and denotes at every position in the symmetry broken direction z .2. The task of the envelope function method is now to select a suitable set of lattice periodic functions un (r. 2m∗ (z ) ∂z 2 2 (2. while in the symmetry broken direction. As a result. this decoupling is only true for the translational invariant direction.2..1). The crystal momentum k is only defined within the translational invariant direction. 2m∗ c 2 2 (2.m (z ) (indexed by the subband quantum number m) of (2. is to rewrite the second order differential operator [12] 2 2 − 2m∗ (z ) ∇2 → −∇ 2m∗ (z ) ∇. z ) for (2. it is in the presence of a material interface not Hermitian and therefore unexpected imaginary eigenvalues for the energy would result. As a result of the symmetry breaking.1) Here. As in the nanostructure. z ) is a lattice-periodic function. For the single-band effective mass approximation for e. The usual ad-hoc fix.3) The way the equation is written.2.2.2) Ec (z ) denotes the position dependent bulk band edge. which is called symmetrized operator ordering.7) It can be shown that this particular choice. The operator ordering is crucial at a material interface.5) to direct-bandgap semiconductor nanostructures.2. . 2 2 (2. which are required within the presence of material interfaces. the exact form of the k · p Hamiltonian at the Γ point is required. kt )∂j Hi. The traditional envelope equations for one and several bands are widely applied and used in a variety of numerical calculations of quantum wells (and superlattices) [20.16 Chapter 2.8) Here. The direct interaction terms (2.2. may lead to erroneous results.11) and P the nonzero interband momentum matrix element from the direct k · p interaction between the conductionand the valence band.R (r. corresponding to the Γ15 representation with p-type basis functions x.12) For the Γ15 states.6) Here the Hermitian operator ordering has already been introduced. where the material parameters change and therefore the ad-hoc operator ordering involves unknown approximations made to the effect of the interface.2. y and z are given by [13] ×4 H4 d |s = |x |y |z |s 2 Ec + 2m0 −ikx P −iky P −ikz P |x iP kx 2 Ev + 2m0 0 0 |y iP ky 0 2 Ev + 2m0 0 |z iP kz 0 0 2 Ev + 2m0 . In fact. y and z .3 2. Beside its successful application to some material systems.2.10) (2.2.2. The problem is that within the bulk k · p Hamiltonian. The momentum matrix element is often given as an energy parameter Ep . corresponding to Γ1 . the first order matrix elements within Γ15 vanish due to time reversal symmetry of the Hamiltonian [25].9) (2. (2. the top of the valence band at Γ is triply degenerate.2. Ec and Ev correspond to the zone-center energies Ec Ev P = = = s|H |s x|H |x i − s|px |s m0 (2.2.3. The usual fix is to split the contribution symmetrically N ki kj → ∂i N N ∂j − ∂j ∂i . Electronic Properties of Semiconductors where the k · p differential operator is given by Hk·p (z ) = − i. 24]. terms of the type N ki kj with i = j can appear. one would expect the direct interaction within the valence band k · x|p|y resulting into a linear term in k given explicitly by kz x|pz |y . 22. kt )∂i + ∂i Hi. 23. x.j (r. the optical matrix parameter. Ignoring spin.18) of the k · p matrix for s.1.1 The Zinc-Blende Models Direct Interaction To apply the envelope equations (2. 2. 21. (2. (2.L (r. while the lowest-lying conduction band is of s-type symmetry. kt ) i (1) (1) (2) + +H(0) (r.2. where no coordinate appears twice [10] (the crystal is invariant under a rotation of 180° around one of the axes). kt ). the traditional way of deriving the envelope equations contains several ad hoc fixes. the nonzero momentum matrix element is of the type x|py |z . related to P by 2m0 Ep = 2 P 2 . for which the operator ordering is not clear.2. Therefore.j ∂i Hi. The term N ˜+ operator ordering in the offdiagonal terms ky N ˜− contains the contributions from Γ15 (H1 ) and contains the contribution from Γ1 (F ) and Γ12 (G) bands. In contrast to the operator ordering within the valence band. the here chosen ordering is required to ensure the Hamiltonians Hermiticity. The ˜+ kx + kx N ˜− ky has been derived by Foreman [26]. (2. Only remote Γ15 type bands mix into the s-type conduction band.2. The term C is related to the common Kane parameter B = 2C . The k · p Envelope Function Method 17 For example. It is clear from table A. Hsx = iP kx and Hxs = −ikx P is an ad hoc fix. M 2 2m0 . Therefore. In a usual zinc-blende semiN 2 ˜ ˜ conductor such as GaAs.8). 2. L ¯ and M ¯ indicates symmetry arguments similar to the already used arguments within this chapter.2. the interaction is nonzero.f .e.14) Within the traditional k · p envelope function method applied to heterostructure.2. L and M . and therefore.7. The corresponding terms are given by the ×4 matrix H4 (2. In deriving these terms. given by the terms ky Ckz + kz Cky . (2. neglecting the influence of the Γ25 bands (H2 = 0) and using 2. e. N (2.19. is required to be symmetric.2. it is commonly neglected. The operator ordering in the first order terms. The derivation of the terms tese terms involves only ¯c .2. The bar in A ˜ ˜ ˜ the close relationship to the parameters A .2. while integration by parts gives xa |pz |y b = − y b |pz |xa . but can be estimated [27] using the following arguments: The term N− contains contributions of Γ15 and Γ25 bands. i. which include the free-electron dispersion c ˜c = A ¯c + A 2 2m0 ˜ =L ¯+ . If the states are from the same degenerate level.d-.2. the value for N given by ˜− = M ¯ =M ˜ − N 2 2mo . L 2 2m0 ˜ =M ¯ + .2.17) By crystal symmetry.M ˜ and N ˜ are usually determined by calculating analytical expressions for the dispersion and The parameters L ˜ into N ˜+ and N ˜− is not directly accessible relating them to measured effective masses.2. while N Γ25 (H2 ) bands.7) simply uses ˜ ˜+ = N ˜− = N . The Γ25 bands can only be formed by f -type and higher atomic orbitals. If the states are not from the same degenerate level.65 and N = −17.4 (in units of 2m0 ).2. the ordering in the off-diagonal terms between the conduction and valence bands. the non-vanishing elements from the perturbation are ky s | py | y r y r | pz | x kz + kz s | pz | y r z r | py | x ky . (2.19).18) and the coefficients (2.2.2 Remote Contributions The next step is to include the interaction between the remote states and keep track of the correct operator ordering.2. Although the approximation might be reasonable for very remote bands ν . C = 0.75 and therefore use N ˜ N− = −3. while the valence band states have contributions from all except the Γ2 states. a reflection in the (110) plane for xa |pz |y b results in y a |pz |xb . while Γ15 bands can be formed by p-.14) and the experimentally determined value for N ˜ . then a = b and the matrix elements vanish.g.15) N 2 ˜.13) whereas in this case the free electron dispersion is still contained in the direct interaction matrix (2. In the bulk k · p theory. these terms are equal.3.1 that the remote contributions to the conduction band stem from remote Γ15 states. Here a and b index the degenerate level. As B is usually small. M = −2. The symmetric operator ordering would ˜+/− = −8. while the Burt-Foreman ordering gives strongly asymmetric values N ˜+ = −13.65.2.and higher type orbitals [26]. If P is allowed to vary. (2. the ordering is irrelevant and the terms are summed together into the ˜ Kane’s parameter N ˜ =N ˜+ + N ˜− . it remains crude. the influence of f orbitals is ˜− is insignificant. the symmetrized operator ordering (2.16) ˜+ can be deduced using (2. . Within the usual semiconductors. The detailed splitting of N ˜ from experiment. the energy for the energy-dependent r renormalization has been replaced by the according band edge energies Ev and Ec .2. 22).2. |x ↓ . σz = 1 0 0 −1 .19) 2 Γ15 ν 2 Γ25 H2 = ν 2 Γ15 m2 0 ν | s | px | uν | .3. (Ec − Eν ) 2 2 G= m2 0 m2 0 ν | x | px | uν | . |y ↓ . if the point group of the crystal neglecting ˆ G .k = = 2 4m2 0c 2 4m2 0c (∇V × p) · σ . The spin-orbit energy leads to additional terms in the equation (2. Within the semiconductors involving heavier atoms.2. |s ↓ .2. (Ev − Eν ) 2 2 (2. The reason to this lies in the symmetry of the spin. (2. denoted here as E switches sign and is only invariant under the rotation of 4π. (2.21) is not diagonal in the basis (2. The spin variable is diagonal. which can be either up |↑ or down |↓ .18) .2. the Hamiltonian is diagonal in the spin variable.2.22) All operators in the zone-center Hamiltonian (2. the electron spin has been omitted. |z ↑ . M ˜− = H1 − H2 . (Ev − Eν ) | s | px | uν | .2.24) . the initial basis of four states describing the lowest conduction band and the top valence bands is doubled to eight states |s ↑ .2) for the zone-center functions. |z ↓ . (Ec − Eν ) 2 Γ15 C= m2 0 ν s | px | uν uν | px | z . L ˜+ = F − G.18 Chapter 2. then the point group including spin will be given by the elements of G and E ˜ definition of the double-group G of G defined as ˜ = {g.20) (2.2. (∇V × k) · σ . Therefore. σy = 0 i −i 0 . |y ↑ . g G ˜ = −g } ∀g ∈ G . the wavefunction including the spin any rotation around an angle of 2π. N 2 Γ1 ν ¯ = H1 + H2 . meaning that ↑|↑ = ↓|↓ = 1 and ↑|↓ = 0. the spin-orbit interaction has a large impact on the electron dispersions. 1 2 (Ec + Ev ) − Eν Spin-Orbit Interaction Up to this point.2. The crystal potential V (r) without spin terms is invariant under ˆ . Within the basis (2.20) and (2. The Bloch basis functions have to be extended to include the spin degree of freedom. Using the spin. namely HSO. leading to the spin is given by G .p HSO.2.3 m2 0 m2 0 | x | px | uν | . N 2 2 Γ12 F = H1 = ¯c = A 2. Electronic Properties of Semiconductors |s ×4 |x H4 = r |y |z |s ¯c k kA kz Cky + ky Ckz kz Ckx + kx Ckz kx Cky + ky Ckx |x ky Ckz + kz Cky ¯ x + ky M ¯ k y + kz M ¯ kz kx Lk ˜+ kx + kx N ˜− ky ky N ˜+ kx + kx N ˜− kz kz N |y kx Ckz + kz Ckx ˜ + ky + ky N ˜ − kx kx N ¯ y + kx M ¯ kx + kz M ¯ kz ky Lk ˜ ˜ k z N+ k y + k y N− k z |z ky Ckx + kx Cky ˜ + kz + kz N ˜ − kx kx N ˜+ kz + kz N ˜ − ky ky N ¯ z + kx M ¯ k x + ky M ¯ ky kz Lk (2. (2.14) do not act on the spin variable. (2.1. Therefore.2.22).21) where c is the vacuum speed of light and σ are the Pauli operators acting on the electron-spin variable σx = 0 1 1 0 . which is done by giving the spin z -component. ¯ = F + 2G.2. the Hamiltonian is given by ×8 H8 rd = ×4 ×4 H4 + H4 r d 0 0 ×4 ×4 H4 + H4 r d .23) In contrast to this Hamiltonian. (Ev − Eν ) | x | px | uν | . Under such a rotation.2. the spin-orbit interaction (2.2. |x ↑ . to [28] 0 0 0 0 0 0 −i 0 0 0 0 1 0 0 0 0 0 −i 0 0 0 −1 i 0 .p = 3 0 0 0 0 0 0 0 1 Here.p of (2. Further. 0 0 0 0 0 i ∆ 0 0 HSO.5 [111] SO −1 −0. For large band gap materials. where the band structure of GaAs calculated using the here presented k · p model is plotted around k = 0. The four-fold degenerate states corresponds to the irreducible representation Γ8 of the double group of Td .5 0 k (m−1) 0. the spin-orbit interaction HSO. The k · p Envelope Function Method 19 2 CB 1. The here elaborated model.k are terms linear in k . these states create the heavy-hole (HH) and the light-hole (LH) bands. the valence band edge Ev in (2. The effect of the spin-orbit interaction is to split the six-fold (threefold without spin) degeneracy of the valence band at Γ into a fourfold degeneracy with eigenvalue ∆ 3 and a two-fold degeneracy with an eigenvalue of − 2∆ 3 . HH.5 1 x 10 [100] 9 0 HH LH Figure 2. the states are separated by ∆. The experimental accessible bandgap is given by the difference between the Γ6 and the Γ8 states. 2.27) . Therefore.22). needs to be shifted by − ∆ 3. LH and SO band.2. including only the valence bands and a single band model for the spin degenerate conduction band. This division is achieved by perturbatively folding the conduction band onto the valence bands and vice versa. Away from Γ. The spin-orbit interaction for HSO.2. The result of HSO.3: Band structure of bulk GaAs around the Γ point at room temperature. while perturbative contributions from remote bands are in general neglected. The Γ7 band is referred to as spin-orbit split off band (SO).2. the coupling between the conduction and valence bands is weak and therefore the 8 × 8 k · p model can be divided into to a 6 × 6 model. The situation is illustrated in Fig. the spin-orbit energy ∆ is defined as ∆ = −3i 2 4m2 0c for the basis in (2.p is usually only included in the direct interaction. N+ = N ˜+ + P L =L Eg Eg (2.10).25) x | (∇V × p)y | z . is commonly referred to as the 8 × 8 k · p model.2.k of (2.2. The two-fold degenerate states correspond to the Γ7 representation and lie below the Γ8 states by −∆.2. In other words.21) is small and therefore commonly neglected [13]. 0 0 0 0 0 0 0 −1 0 0 i 0 0 −i 0 −i 0 0 i 0 0 0 0 0 (2. leading to an off-diagonal coupling between the conduction and the valence band. (2.5 Γ8 Γ7 −0. The lowest conduction band (CB) states finally correspond to Γ6 states.2. including the CB. calculated using the 8 × 8 k · p for two crystal directions.2.26) The k -dependent spin-orbit interaction HSO.2.p with modified parameters L and N given by 2 2 ˜ + P .3. The 6 × 6 k · p is therefore obtained by using only the valence band part of ×8 H8 rd + HSO.5 Γ6 1 Energy (eV) 0.20) leads. 2 2 3 1 .29) and neglecting the Γ7 rows and columns finally results in the 4 × 4 k · p model. using |J.2. Ac = A Eg (2. 16].2. remain unchanged.p . is given by uhh = ulh = u ¯lh = 3 3 .2. Electronic Properties of Semiconductors ˜ and N ˜+ . The reduction is performed by choosing a combination of basis functions (2.27).29) 3 1 .2. For some semiconductors.2.28) which is then equal to the common conduction band effective mass.30). The Luttinger parameters are usually those given in literature.31) [12. 2 2 1 = − √ (|x ↑ + i |y ↑ ) . The new basis is labelled according to its total angular momentum J and the angular momentum around the z -axis. Within the 4 × 4 k · p model. A possible choice. (2. Then. .30) 2m0 2 2m0 From (2. M and N . 2 1 = − √ (|x ↓ + i |y ↑ ) + 6 = = = = 1 √ (|x ↑ 6 1 √ (|x ↓ 2 1 √ (|x ↓ 3 1 √ (|x ↑ 3 − i |y ↑ ) + − i |y ↓ ) .2.31) 1 The full listing of these parameters is given in [12]. and consequently N ˜− .20 Chapter 2. 29. 2 2 1 1 u ¯SO = . 3 (2. Jz as notation. it is common to use the Luttinger parameters γ1 . In terms of the Luttinger parameters the full 8 × 8 Hamiltonian can be written explicitly as in (2. + i |y ↓ + |z ↑ ) . − 2 2 1 1 uSO = .22) effectively diagonalizing the spin-orbit interaction HSO. the correct operator instead of L ordering is conserved.2. the 6 × 6 k · p model can be further reduced to only include the HH and LH bands. − 2 2 The four states with total angular momentum of 3 2 belong to Γ8 and the two states with an angular momentum of 1 2 form the Γ7 (or SO) bands. The parameter M ˜ .− 2 2 3 3 u ¯hh = . the spin-orbit splitting is large and the SO band has only little influence on the top of the valence band. The parameters are related to each other via 2 L M N = − = − = − 2m0 2 ( γ1 + 4 γ2 ) ( γ1 − 2 γ2 ) 6γ3 . γ2 and γ3 1 [29] instead of the Kane’s parameters of the 6 × 6 model L . Jz . − i |y ↑ − |z ↓ ) .2.2. = Ec † P √ z † 2Pz √ † − 3P+ √0 † 2P− † P− 0 Pz P √+ ∆ 2Q √ −√ S/ 2 − 2 P+ 0 − √ 3/2S † − 2R † √ √ 2Pz 2Q† P −Q −S − P+ 3/2S † 0 R† √ − 3P + √ −S † / 2 −S † P +Q √0 † 2R R† 0 √0 † − 2P+ † −P+ 0 Ec † Pz √ † −√ 2Pz † − 3P− 2 P− 0 3/2S √ 2R Pz P √+ ∆ 2Q √ −S † / 2 √ P− − 3/2S 0 R √ − z √ 2P 2Q† P −Q S† 0 √ − 2R R √0 − 3√ P− −S/ 2 S P +Q H8×8 (2. The conduction band parameter is obtained using 2 ˜c + P . Transforming the Hamiltonian into the basis (2. Thereby. 3 2 |z ↓ . the parameters for 8 × 8 model can be calculated using the renormalization (2. 2 |z ↑ . 2.35) (2.2. use a simple effective mass model. 31]. Fortunately. 3 2 Eg Eg .2.2. The degeneracy of the light and heavy hole bands near the band edge generates a coupling term (as in the Luttinger Hamiltonian). as before.32) (2.2.e.2.45) 2. (2. The k · p Envelope Function Method 21 where 2 Ec P P± Pz = Eg + 2 2m0 2 2 2 kx + ky + kz .38) = = = 2m0 2 2 2 γ1 kx + ky + kz .2.37) (2. 12. this parameter is neglected. ∂z (2.2.40) (2.2. this yields a set of four coupled effective mass equations [30. Here . = γ1 + 2γ2 + λ.36) (2.2. In the case of the valence band. In most practical calculations.39) The Kane parameter B describes the inversion asymmetry. Only when we consider energy levels deep into the valence bands (close to the SO splitting energy. i. γ2 .43) where the dimensionless parameters λ and r are given by λ r = = 4m0 P 2 .41) (2.42) (2. Including spin degeneracy. 2m0 2 √ √ 2 2 − 3γ2 kx − ky + i2 3γ3 kx ky . Eg + ∆ (2. the strong interaction between the degenerate light and heavy hole bands (near the band edge) requires that these bands are taken into account explicitly.2.4 Two Band Model The conduction band can be modeled quite easily if we assume that the interaction with the other bands is weak enough for it to be treated perturbatively. 1 = γ1 + λr.2.2.5 eV splitting) can be introduced through the effective mass. 2m0 The parameter ∆ is.33) (2. 1 [iP (kx ± iky ) + B kz (ky ± ikx )] 6 1 (iP kz + B kx ky ) . the spin-orbit splitting energy. this set of coupled equations can be greatly simplified by a method described in [31].44) (2.2. The parameters γ1 .34) (2.2.2. however. 2 = γ1 − 2γ3 .2. about 300 meV in GaAs) do the coupling terms to the SO and conduction bands (1. The coupling between the Γ conduction band edge state |s and the Γ valence band edge state |z is given by 2 P=− m0 ϕs Vc ∂ ϕz . γ3 (kx − iky ) kz . 3 2 Q = R S = = 2m0 2 2 2 2 γ2 kx + ky − 2kz . γ3 and P can ne determined from effective masses at the Γ point of the bulk semiconductor [12] m0 mhh (001) m0 mlh (001) m0 mSO (001) m0 mhh (111) = γ1 − 2γ2 . (2. 51) W† P ∓Q where W = |R| − i |S |.22 Chapter 2. This allows us write the effective mass equation as ˆ hh + V H ˆ† W ˆ W ˆ Hlh + V Fhh Flh = E (k) Fhh Flh . We consider the case of a [100] plane.48) (2.2.57) 2 + k 2 )2 γ2 (kz t .29) into a new set uA . (2. showing the double degeneracy of the heavy and light hole bands. (2.55) where Fhh and Flh are the envelope functions corresponding to uA and uB .52) (2.46) (2. This expression can be rewritten to 2 k2 γ2 − γ2 kz t 2 2 E (k) = V0 + γ1 ± γ2 1 + 3 3 2 2 kz + kt . it can be diagonalized into two 2 × 2 block matrices.2. 2 (2.2. hole energies are taken to be positive. S +Q (2. 2 1 √ (ulh + u ¯lh ) . 2 1 √ (uhh + u ¯hh ) . is finding the solution in bulk material.2. respectively.2. (2. yielding the bulk energy dispersion relations for the heavy and light hole subbands. where we take V to be a constant V0 . which represents the (bulk) valence-band-edge offset with respect to an arbitrary reference energy.2.2.54) Finally.2. The first step in solving the quantum well problem. 2 1 √ (−ulh − u ¯lh ) . Electronic Properties of Semiconductors a unitary transformation of the four basis Bloch functions in (2. upper H U and lower H L . The solutions of the lower block can easily be determined from the latter.47) (2.2.2.2. We can identify P − Q and P + Q with the light hole energy (operator) H respectively.51) can be simplified into an effective-mass formalism with ˆ lh H ˆ hh H ˆ W = = = − (γ1 + 2γ2 ) ∂2 2 + (γ1 − γ2 ) kt . The upper and lower blocks are equivalent. the Schrödinger equation with the Hamiltonian (2. ∂z 2 ∂2 2 − (γ1 − 2γ2 ) 2 + (γ1 + γ2 ) kt . Note that in this formalism. given by P ±Q W Hσ = .56) where the plus sign refers to the light hole solution. The value of V0 will be different in well material and barriers reflecting the different valence band edge offsets. 4γ2 t z z t 3 2 (2.50) In terms of the the new base proposed above.53) (2. uD to decouple the set of four coupled equations into two coupled ones. ∂z √ ∂ for [100] √3kt γ2 kt − 2γ3 ∂z ∂ 3kt γ3 kt − 2γ3 ∂z for [110] (2.2. The index σ = U (L) refers to the upper (lower) ± signs. the 4 × 4 k · p Hamiltonian H4×4 = = = = 1 √ (uhh − u ¯hh ) . uB . Similarly to the conduction band case. we take into account the potential V (z ). uC. We can now easily solve for the eigenenergies E (k).2. ˆ lh and the heavy hole energy H ˆ hh .49) P − Q −S † 0 −S P +Q R = 0 R† P −Q † R 0 S† P R 0 .2. and the minus to the heavy hole one. It is therefore sufficient to solve the upper block and obtain its solutions. The Bloch functions ui are given by uA uB uC uD Consequently. writing the in-plane k component as kt 2 2 E (k) − V0 = γ1 kz + kt ± 2 (k 2 + k 2 ) + 12 (γ 2 − γ 2 ) k 2 k 2 . and y .directions.2.2.4: Contours of constant energy within any [100] plane of k -space for the heavy (right) and light (left) hole subbands in bulk GaAs. the effective masses along the z -axis [001] and x.64) . r) where the matrix notation implies ϕ = Fhh uA + Flh uB . = = Fhh.2 Flh.56) 2 2 E (k) = V0 + (γ1 ± γ2 ) kz + kt ±3 2 2 γ3 − γ2 2 kt .1 Flh. The effective HH mass is much larger along the [110] direction than along the [100] direction. we can expand the square root in (2. (2.2. as indicated by the larger contour spacing.63) (2. The effective LH mass is seen to be much more isotropic (after [32]).2.59) (2. resulting in a clearly anisotropic HH band and a quasi isotropic LH band. kt . r) ϕ2 (k. However. B± Flh. kt )e±ikhh z .axes [100] and [010] are identical (as expected). illustrating that γ3 can be related to the mass anisotropy along the [100] and [110] directions.2. In the plane of the well. γ2 (2. as the dispersion relation is given by E (k) = V0 + (γ1 ± 2γ2 ) k 2 .2. apart from a normalization constant ϕ1 (k. Constant energy contours are shown in Fig.56) with kt = 0 for [001]. kt )e±iklh z .2 (±kz2 . By taking a linear combination of the bulk solutions in each material.2. due to the lower energy of the HH bands the anisotropy term is relatively more important for HH than for LH.2.We can easily find this from (2. r).60).5 meV for the HH band and 3 meV for the LH band. a general solution can be constructed.2.60) (2. as Fhh Flh = eikt ·rt = eikt ·rt A± Fhh. 2.2. we see that in bulk material. We can construct a confined solution from the bulk plane wave solutions by imposing boundary conditions along the confinement axis. If kt small compared to kz . four plane wave solutions exist at a given energy.1 (±kz1 . 2.1 Fhh. and kz = 0 for the x. as in Fig. Still. Both ϕ1 and ϕ2 are two-component vectors as seen in (2.2 = eik·r = eik·r Hlh + V0 − Ehh −W † Hlh + V0 − Elh −W † . 2. A similar derivation can be formulated for the [110] crystal planes. r) + B± ϕ2 (±kz2 .1 (±kz1 .2.62) The four coefficients A± and B± are unknown constants. .4. We can write the components of Φ. kt )e±ikhh z + A± Flh.2. The xy -plane is the plane of the well. Fhh and Flh .2 (±kz2 .6. we choose the well growth direction (direction of confinement) along the z -axis. (2.2. yielding a general solution Φ of the form Φ= A± ϕ1 (±kz1 .2.5. The k · p Envelope Function Method 23 Figure 2.55) are found to be. The eigenvectors of (2. kt . there is no confinement and hence we retain the bulk plane wave solution. kt )e±iklh z + B± Fhh. (2. The energy spacing between each contour level is 0.58) The Energy term accounting for anisotropy for a given kt and kz is equal for the HH and LH subbands.and y .61) To solve the quantum well problem. As illustrated in Fig.59) and (2. to ±kz1 (light hole) and ±kz2 in (b) for the well layer of the quantum well (GaAs throughout this thesis). respectively. Within the plane of the well. . Electronic Properties of Semiconductors Figure 2. The boundary conditions at the interfaces then determine the energy eigenvalues and the coefficients. In the barriers (AlGaAs) a similar mechanism is employed.6: At any one energy on a bulk material. The amplitudes A± and B± in (a) correspond. we can find four wavevectors corresponding to the heavy and light hole bands. for each quantized level. the electron still behaves like a "free" electron.24 Chapter 2. Thus. An eigenstate of the Hamiltonian in a quantum well is then made of a linear combination of the bulk plane waves corresponding to those wave vectors. a parabolic energy subband exist (after [32]).5: The "in-plane" subband structure of the quantum well in the conduction band. LH HH Ev -kh -kl kl kh kz (a) (b) Figure 2. .7: (a) Valence subbands dispersion relations calculated for a 200 Å wide GaAs/Al0. As an illustration of the numerical method formulated in this section.2. see Appendix B). Particularly important is the density of states (DOS). for [100] (blue) and [110] (red) crystallographic directions at T = 2 K . The k · p Envelope Function Method 25 (a) 0 HH1 (b) −5 −10 LH1 HH2 E (meV) −15 HH3 −20 −25 −30 0 1 2 k|| (cm−1) 3 x 10 4 6 0 50 100 ρh/ρe1 Figure 2. calculated for the same structure for both crystal directions. giving rise to highly non-parabolic subbands.1 Ga0.65) (2.2. The boundary conditions at the interfaces between the regions and the demand that the solutions be confined in the quantum well provide the necessary relations to solve the problem. making a total of 12 unknowns over the three regions. (b) The ratio between the density of states of the valence subbands and the first conduction subband.2. The boundary conditions boil down to the continuity of the wave function and “generalized” continuity of its derivative. In order to symmetrize the problem the following quantities have to be matched across the interfaces Fhh and (γ1 − 2γ2 ) Flh dFhh √ + 3γ3 kt Flh .7(a) the valence subband structure of a 200 Å GaAs/Al0. π dE (2.9 As quantum well. 2. corresponding to current across the interface.9 As quantum well.2. Caution should be issued however that the above boundary conditions only apply when the Bloch functions in both well materials and are similar.2. The difference between the two crystallographic directions becomes clearly observable away from the zone center. as is the case for the GaAs − AlGaAs system used throughout this thesis.2. dz (2. The light and heavy holes are very heavily coupled.51).66) These boundary conditions were obtained by symmetrizing the Hamiltonian (2. The numerical implementation details of the two bands model are given in Appendix B. which can be found from ρ( E ) = 1 dk .1 Ga0. dz dFlh √ and (γ1 − 2γ2 ) − 3γ3 kt Fhh .67) assuming the dispersion relationship is isotropic (using the axial approximation. The subbands are named after their dominant character at the zone center (kt = 0). Thus we have four unknown constants in each region. As an illustration. we present in Fig. can be present in the conduction .9 As quantum wells considered in the examples above. 2. (2.2. 2. As stated at the beginning of the section. ↑ + |S.g.69) where |S.8(a) the dispersion relation for the first two conduction subbands calculated for the same structure as in Fig.1 Ga0. the conduction subbands are calculated using the effective mass approximation.8: (a) Conduction subbands dispersion relations calculated for a 200 Å wide GaAs/Al0. calculated for the same structure.9 As quantum well at T = 2 K . 2. 2.68) 2 dz m∗ (z ) dz We can write down the conduction band wave function as 1 Ψn (r) = √ eik⊥ ·r ϕn (z ) (|S. 2.26 Chapter 2.9 the wavefunctions for first two conduction and first four valence subbands calculated at the the Brillouin zone center. where the bands are assumed to be parabolic and the problem reduces to a one-dimensional Schrödinger equation of the form 2 d 1 d − + V (z ) ϕn (z ) = En ϕn (z ). electrons. we plot in Fig. A (2. (b) The ratio between the density of states of the conduction subbands and the first conduction subband. ↓ are the conduction band wave functions for the bulk case. 2. The spikes in the DOS are due to the band extrema away from the zone center.7(a) and the DOS of the first conduction subband. For the sake of completeness. Here we separate the heavy and light hole subband wavefunctions into figures (b) and (c) for the sake of clarity.1 Ga0. In many devices such models would be inadequate as large numbers of charge carriers.8(b) also presents the two-dimensional density of states for these subbands. Electronic Properties of Semiconductors (a) 1660 (b) 1640 1620 1600 E (meV) E2 1580 E1 1560 1540 1520 0 1 2 k|| (cm−1) 3 x 10 4 6 0 ρe/ρe1 5 Figure 2. Fig.2. ↑ and |S. we present in Fig. we plot in Fig.7 above. ↓ ) . e. for the 200 Å wide GaAs/Al0. As an illustration.7(b) the calculated ratio between the DOS of the valence subbands from 2.3 Schrödinger-Poisson Model In the discussion so far have concentrated solely on solving systems for a single charge carrier. The dotted lines mark the energetic locations of these wavefucntions and the quantum well energetic profile is outlined by solid black curves. 2.65 1.3. remembering that the quantum wells are assumed infinite in the x − y plane then any charge density ρ(z ) can be thought of as an infinite plane.e.06 0 2 z (m) 4 x 10 6 −8 Figure 2. or any other charge distribution ρ. Recalling that the potential then follow in the usual way [9] r ρ (2.3. it then becomes necessary to solve the electrostatics describing the system. i.1 Model Formulation When considering the case of an n-type material. Such an infinite plane of charge produces an electric field perpendicular .02 0.6 1. (b) valence heavy and (c) light hole subbands. i. then (although obvious) it is worth stating that the number of free electrons in the conduction band is equal to the number of positively charged ionized donors in the heterostructure.9: Wave functions for the (a) conduction.3. can be expressed by using Poisson’s equation ∇2 V ρ = − where is the permittivity of the material.02 0 0 −0.2. Schrödinger-Poisson Model 27 (a) 1. VCB (Z ) for example. a sheet.1) = r 0.04 −0.3. with areal charge density σ (z ) and thickness δz . The solution is generally obtained via the electric field (2.55 1.e. calculated at the Brillouin zone center (kt = 0) for a 200 Å wide GaAs/Al0.1 Ga0. are one-dimensional. (2. 2.3) Given that the potential profiles.3. we can point out the modulation doped system. band.02 −0.10(a). then they will also produce a onedimensional charge distribution.9 As quantum well at T = 2 K . As an example. In order to decide whether or not typical carrier densities would give rise to a significant additional potential on top of the usual band-edge potential terms (which will be labelled specifically as VCB or VV B ). The additional potential term Vρ (z ) arising from this. strength E. as shown in Fig.04 −0.5 E (eV) 0 1 2 3 z (m) 4 5 x 10 (c) 6 −8 (b) 0.06 0 2 z (m) 4 x 10 6 −8 −0.02 −0. In addition.2) E = −∇V V ρ (z ) = − −∞ E · dr. where the doping is located in a position where the free carriers it produces will become spatially separated from the ion. z =−∞ (2.3. as shown in Fig. where the planes are separated by the usual step length δz . Electronic Properties of Semiconductors Figure 2. 2.3.5) where the function sign is defined as sign (z ) = 1.4) Note that as the sheet is infinite in the plane. where the first would be the ionized impurities and the second the free charge carriers themselves. is then the sum of the individual contributions as follows ∞ E(z ) = z =−∞ σ (z ) sign (z − z ) 2 (2. the latter would be calculated from the probability distributions of the carriers in the heterostructure. (2. into an areal density.3. 2 (2.10(b).7) which ensures that the electric field. then for z > z . or expressed mathematically ∞ σ (z ) = 0. and hence the potential. then the field strength is constant for all distances from the plane. i. While the former would be known from the doping density in each semiconductor layer. E(z ) = +σ/2 . z ≤ 0 (2.28 Chapter 2.9) where q us the charge on the extrinsic carriers. introduced into the heterostructure is given by ∞ N= −∞ d(z )dz. The total electric field strength due to many of these planes of charge. E(z ) = −σ/2 . d(z ). then the total number of carriers.3. The step length δz selects the proportion of the carriers that are within that slab and converts the volume density of dopant. as defined at growth time. whereas for z < z .10: Electric field strength from an infinite plane of charge of volume density d(z ) and thickness δz (after [33]).6) and has been introduced to account for the vector nature of E. z ≥ 0 −1. there are as many ionized donors (or acceptors) in the system as there are electrons (or holes). Note further that it is only the charge neutrality. there would be two contributions to the charge density σ (z ). Thus if d(z ) defines the volume density of the dopants at position z . go to zero at large distances from the charge distribution. .3. For the case of a doped semiconductor.8) The net charge density in any of the planes follows as σ (z ) = q [N ψ ∗ (z )ψ (z ) − d(z )] δz (2. if a single sheet of charge is at a position z .e. to it.3. and with a strength E= σ . per unit cross-sectional area. n-type doped to 2 × 1018 cm−3 .5 −1 −1.2. Schrödinger-Poisson Model 29 1 0. the total number N of electrons in the quantum well is 100 × 2 × 1014 m−2 = 2 × 1012 cm−2 .2 0 −0. 2. The discontinuities in a occur at the edges of the doping profiles and are of magnitude 2 × 1014 m−2 .5 −2 E (1016 Vm−1) 0 100 200 300 z (Angstrom) 400 500 Figure 2.12. then the curve on top of the ionized impurity background clearly resembles −ψ ∗ ψ .11 shows the areal charge density along the growth axis for a 100 Å GaAs well.8 Al0. If the charge carriers are distributed over more than one subband. The ionized donors yield a constant contributions to σ within the well of d(z )δz = 2 × 1024 m−3 × 1Å = 2 × 1014 m−2 .2 As barriers. First.4 0.2 Ga0. then the contribution to the charge density σ (z ) would have to be summed over the relevant subbands n σ (z ) = q i=1 n ∗ Ni ψ i (z )ψi (z ) − d(z ) δz. as expected from the mathematics.5 0 −0. surrounded by undoped Al0. which plots the electric field strength E due to the charge distribution (as defined in equation (2.8 As 4 meV barriers. 2. (2.3. There are a number of points to note about Fig.2 −0.12: The electric field strength E due to the charge distribution shown in Fig. 2. 2 1. surrounded by undoped Ga0.11.10) where i=1 Ni = N . n-type doped to 2 × 1018 cm−3 . By assuming that the electrons introduced by such doping all occupy the ground state of the quantum well.8 0. Hence. again as expected.5 1 0. in each of the 1 Å thick slabs.11: Areal charge density a for a 100 Å GaAs well.6 σ (1014 cm−2) 0.3.4 0 100 200 300 z (Angstrom) 400 500 Figure 2. Fig.3. the field .5)) along the growth axis of the heterostructure. 2.14. Fig. even at this relatively high carrier density of 2 × 1012 cm−2 . The electric field strength itself is not an observable. The process is repeated until the energy eigenvalues converge. as displayed in Fig. at this point the wave functions are simultaneously solutions to both Schrodinger’s and Poisson’s equations—the solutions are described as self-consistent. (2. adding it to the original bandedge potential. the quantity which is significant is the potential due to this charge distribution. in this case for the potential. described in detail in appendix B.30 Chapter 2. for instance. Nonetheless it is important to calculate the effect of this perturbation on the electron energy levels by continuing with the iterative process and looking for convergence of the resulting energy solutions. The energy eigenvalues are calculated by considering the introduction of a further test electron into the system and incorporating the potential due to the carrier density already present into the standard Schrödinger equation. which is usually of the order of one or two hundred meV or more. merely an intermediate quantity which can be useful to plot from time to time. 2. so any test charge used to probe the potential is also an electron which would be repelled by the existing charge. Figure 2. represented by the function Vρ . In addition.55) becomes V (z ) → V (z ) + Vρ (z ). The numerical shooting method.3.11) where V represents the band edge potential at zero doping and the potential due to the non-zero number of carriers. Again.11.13 plots the potential as calculated from equation (2. While this is small compared to the conduction band offset. the symmetry of the original heterostructure and doping profiles are reflected in the symmetric potential. the zero field point at the center of the structure reflects the symmetry of the charge distribution.a process illustrated schematically in Fig.13: The potential due to the ionized donor/electron charge distribution shown in Fig. . 2. defining the origin. and so on . the charge density ρ. is rather small compared to the barrier height. solving Schrödinger’s equation again. Therefore. at the effective infinity at the left-hand edge of the barrier-well-barrier structure. it could still have a measurable effect on the energy eigenvalues of the quantum well. i.13. to the original band-edge potential for the single quantum well. The latter is an important point since the potential due to the charge distribution is itself dependent on the wave functions.e. The potential is positive at the center of the well since the system under consideration consists of electrons in the conduction band. The numerical implementation details of the self-consistent Schrödinger-Poisson model is presented in Appendix C. can be used without alteration to solve for this new potential. which implies charge neutrality.15 shows the result of adding the potential due to the charge distribution Vρ .3. it is necessary to form a closed loop solving Schrodinger’s equation.3). The carrier density in this single quantum well is reasonably high at 2 × 1012 cm−2 . 2.2. Electronic Properties of Semiconductors 5 4 3 Vρ (meV) 2 1 0 −1 0 100 200 300 z (Angstrom) 400 500 Figure 2. The perturbation. does reach zero at either end of the structure. and this produces a potential of up to 4 meV . calculating the potential due to the resulting charge distribution. which will thus yield new energies and wave functions. the potential term V (z ) in equations (2. 3. . Schrödinger-Poisson Model 31 Figure 2.2. 2.11. 180 160 140 120 100 V (meV) 80 60 40 20 0 −20 0 100 200 300 z (Angstrom) 400 500 Figure 2.14: Block diagram illustrating the process of self-consistent iteration.15: The sum of the band-edge potential VCB and Poisson’s potential Vρ for single quantum well for the charge distribution shown in Fig. Similarly to the modulation doped quantum well case considered earlier. the mobile charge in this case is often referred to as a two-dimensional electron gas (2DEG). This doping is termed the δ -doping and its concentration profile can be written to be N (z ) = N2D δ (z − zd ). Fig. 2.3. 100 Å Ga0. and the self-consistent potential. The total electron-gas concentration inside the well area is computed to be 9 × 1010 cm−2 . The δ -doping layers are introduced symmetrically in the barriers at a distance of 1000 Å from the well-barrier interfaces.16: The band-edge potential (solid blue) and the self-consistent potential (dotted red) of a modulation-doped single quantum well.18). indicated by the dotted horizontal line in Fig. for a system with an undoped single quantum well surrounded by doped barriers with the full layer definition being: 100 Å Ga0. 100 Å GaAs undoped. which collect in a quantum well. 2.21. The doping level narrow contributions are clearly observable.20. VCB . The charge distribution profile obtained from the self-consistent calculation is presented in Fig. one generates .17). N2D . so therefore instead of an ion/charge carrier plasma. 2. 2. VCB + Vρ . In this structure.3.12) where the 2D density. 2.2 Two-Dimensional Electron Gas Although mention has been made of quantum well systems in which doping in the barriers leads to a spatial separations of the ions and charge carriers. The appropriate band-edge potential profiles for the conduction and valence bands are presented in Fig. We consider a 200 Å GaAs quantum well with wide AlGaAs barriers.2 As doped n-type to 2 × 1017 cm−3 . illustrated in Fig. which consists of a wide QW cladded on both sides by narrow wells.8 Al0. An alternative method for creating the 2DEG is photoexcitation by utilizing a mixed type I .20). A similar method for introducing carrier plasma 2DEG at the material interface is to introduce dopings with high peak concentrations and narrow concentration width.19. under laser photoexcitation with energy below the narrow QW bandgap.type II quantum well (MTQW) structure. 2. 2.16 shows the band-edge potential. The electrons introduced into the system are physically separated from the ionized donors. Electronic Properties of Semiconductors 200 180 160 140 120 V (meV) 100 80 60 40 V 20 0 CB VCB +V ρ 0 50 100 150 z (Angstrom) 200 250 300 Figure 2. This can be achieved by confining the doping atoms to a single atomic layer of the host semiconductor (see Fig. us the number of doping atoms in the doping plane and z is the location of the dopand layer. quantitative calculations presented thus far have not considered these modulation-doped systems. we can introduce two symmetrical δ -doping layers into the two AlGaAs barrier layers and compute the influence of this doping on the potential profile. and the 2DEG accumulation at these areas is evident from the location of the Fermi level energy. which are effectively one-dimensional (1D) [34]. The physical separation leads to a reduction in the ionized impurity scattering and hence increased electron mobilities for in-plane (x − y ) transport.8 Al0. the electrons flow from the AlGaAs layer to GaAs in order to maintain a constant chemical potential throughout the two materials (see Fig. 2.32 Chapter 2. The substantial band-bending introduced by the doping levels is clearly seen at the well-barrier interfaces. As the Fermi level in the doped AlGaAs layer shifts from the middle of bandgap towards the donor levels.2 As doped n-type to 2 × 1017 cm−3 . (2. 2. surrounded by Ga0.3. Figure 2. . Schrödinger-Poisson Model 33 Ec EF 2DEG Ionized donors AlGaAs GaAs Figure 2.5 ρ (cm−3) 1 0. The achieved 2DEG inside the well is 9 × 1010 cm−2 . Also shown is a schematic lattice with the impurity atoms being confined to a single atomic plane (after [34]).2.5 x 10 18 2 1.5 0 0 500 1000 1500 z (Angstrom) 2000 2500 Figure 2.17: The structure and band diagram of a modulation-doped heterojunction between GaAs and n − AlGaAs. respectively. the conduction band edge and the Fermi level energy.18: Schematic illustration of a semiconductor substrate and an epitaxial film containing a δ -doping layer. Ec and EF represent.19: Volume charge density a for a 200 Å GaAs well.1 As barriers.9 Al0. doped with two symmetrical δ -doping layers located 1000Å from the well. 19. Figure 2.55 EF=1. Photoexcitation with EL1 creates electron-hole pairs in the wide QW.5 2 x 10 −7 Figure 2.21: A schematic diagram of a GaAs/AlAs mixed type I .1 0 0. while photoexcitation with EL2 creates a 2DEG in the wide QW and a 2DHG in the narrow QWs. 2.type II QWs structure.6 E [eV] 1.5325eV 1.5 0.34 Chapter 2.05 0 E [eV] −0. The locations of the δ -doping layers are indicated by the vertical dashed lines. Electronic Properties of Semiconductors 1. .5 1 z [m] 1.20: The band-edge potential (solid blue) and the self-consistent potential (dotted red) of a δ -doped single quantum well described in Fig.05 −0. 3. The photoexcitation created electron-hole bound pairs (excitons) in the wide QW have very short lifetimes (≈ ns) and thus their density is much smaller than the 2DEG concentration ne . At higher densities many-body effects such as bandgap renormalization and phase-space filling become important. we calculate using the Schrödinger-Poisson model the electronic properties of a 200 wide GaAs/Al0. 2.1 Ga0. We further discuss these effects in chapter 4.24 present. We formulate the statistical distribution of 2DEG electrons. kB is the Boltzman constant and T the temperature.9.22 we plot the dispersion relations obtained for the first conduction and first four valence subbands of this structure for various 2DEG concentrations.14) me where EE1 (0) denotes the bottom of the first conduction subband. Figures 2.3. the 2DHG in the narrow wells is free. The band-edge profile change can be clearly seen.13) where EF is the chemical potential of the electrons. the Fermi energy of the system equals the electron chemical potential π 2 EF = EE1 (0) + ∗ ne . Because of the large confinement energy in the narrow well its conduction band Γ minimum lies above the X-valley of the barrier. In contrast to the modulation doping.20. We concentrate on the low temperature regime (T → 0). but no noticeable change in the functional form of the function themselves can be seen compared to the undoped structure presented in Fig. the calculated wavefunctions for 2DEG concentrations of 1 × 1010 cm−2 and 1 × 1011 cm−2 . the 2D density of states should be multiplied by this statistical distribution and integrated over all possible states. and free-carrier screening [35]. thus electrons created in the narrow well can relax rapidly into the wide well.2. Introducing electrons into a QW results in profound changes in the optical spectrum.15) To complete the discussion. In contrast. The wavevector of the top most occupied state thus becomes (assuming a parabolic dispersion) kF = 2m∗ e 2 EF = √ 2πne . With the rise of the concentration the subbands shift to lower energies due to the induced bending of the band-edge potential profiles.3. 2. by changing the photoexcitation intensity. In order to calculate the Fermi energy. . (2. Selective excitation of the wide well can be used to create additional excitons in the wide wells. In Fig. thus the electrons in the wide well are not subjected to random static Coulomb fields and remain free even at low densities.9 As quantum well with δ -doped cladding layers presented in figures 2. The former effect appears as a decrease in the bandgap energy with increasing electron concentration (ne ) [36] while the latter results in a reduction of the exciton oscillator strength and binding energy [37]. as seen in Fig. and assuming that all available states are occupied in the first conduction subband. 2. The free electrons occupy the energy levels in the quantum well according to the Fermi-Dirac distribution f (E ) = 1 1+e E −EF kB T . respectively. the holes have to tunnel through a ≈ 0.23 and 2.3. As lower electron densities the effects of the electrons on the spectrum are related to their scattering with bound electron-hole pair (excitons) formed in the structure.19 and 2. (2. kB is the Boltzman constant and T the temperature.5 eV barrier and transfer very slowly. As a result. Schrödinger-Poisson Model 35 electron-hole pairs only in the wide QW.20. The 2DEG density ne obtained in this structure is controlled starting at very low values. It is obvious from these curves that the change in the gas concentration influences the location of the subbands energies but not the functional dependence. a spatial separation between electrons and holes is achieved and a stable 2DEG is formed in the wide QW while a two-dimensional hole gas (2DHG) remains in the narrow well. Photoexcitation with energy above the bandgap of the narrow well creates electron-hole pairs in both narrow and wide wells. (2. 4 0. 1 × 1010 cm−2 (red) and 1 × 1011 cm−2 (green).1 Ga0.6 1.9 As quantum well at T = 2 K .36 Chapter 2.22: The calculated dispersion relations for the (a) conduction and (b) valence subbands.6 0.8 x 10 2 6 Figure 2.2 0. Electronic Properties of Semiconductors (a) 1550 1545 1540 E1 1535 1530 E (meV) 1525 (b) 0 HH1 −5 HH2 LH1 −15 HH3 −10 −20 −25 0 0.2 1. with 2DEG concentrations of 1 × 106 cm−2 (blue). for a 200 Å wide GaAs/Al0.8 1 k|| (cm−1) 1. .4 1. 04 −0.56 1.54 1.5 E (eV) 0 1 2 3 4 5 x 10 6 −8 (b) 0.01 0 −0.9 As quantum well at T = 2 K with 1 × 1010 cm−2 2DEG concentration.52 1.01 −0.02 −0.05 0 2 z (m) 4 x 10 6 −8 Figure 2.62 1.02 −0.58 1.01 −0. .05 0 2 4 x 10 6 −8 (c) 0.6 1.23: The calculated wavefunction for the (a) conduction.04 −0. Schrödinger-Poisson Model 37 (a) 1.01 0 −0. (b) valence heavy and (c) light hole subbands for 200 Å wide GaAs/Al0.2.1 Ga0.03 −0.03 −0.3. 06 0 2 z (m) 4 x 10 6 −8 −0. .24: The calculated wavefunction for the (a) conduction.02 0.06 0 2 z (m) 4 x 10 6 −8 Figure 2.58 1.62 1.52 1.02 −0.04 −0.6 1.5 E (eV) 0 1 2 3 z (m) 4 5 x 10 (c) 6 −8 (b) 0.1 Ga0.04 −0.38 Chapter 2. (b) valence heavy and (c) light hole subbands for 200 Å wide GaAs/Al0.56 1. Electronic Properties of Semiconductors (a) 1.02 0 0 −0.02 −0.9 As quantum well at T = 2 K with 1 × 1011 cm−2 2DEG concentration.54 1. Coulomb-correlated transitions will be treated in Chapter 4. Such a treatment is beyond the scope of this thesis. But instead of using a complicated expression for the antisymmetric wavefunction. The resulting equations allow the calculation the optical susceptibility and therefrom of the absorption. A fully consistent treatment of the light-emission would in principle require to work in a fully quantized picture (see [39]). Then. The chapter is organized as follows.Chapter 3 Free Carriers Optical Transitions Absorption and emission of photons in semiconductors and semiconductor nanostructures is the result of the complex interaction between light and condensed matter. 40. we introduce the classical (monochromatic) light field and the self-consistent coupling to the electrons within the semiconductor nanostructure.1 Second Quantization Introduction The electrons are fermions and consequently their wavefunction must be antisymmetric to obey the Pauli exclusion principle. the representation is rewritten in terms of the Bloch states. 38. This theory review is based mainly on [12. We start by introducing the transformation of the crystal Hamiltonian (2. electromagnetism and quantum mechanics have to be combined. stimulated emission and refractive index change. This thesis focuses on the static properties of nanostructures. The derivation presented here serves to illustrate the imposed approximations and to clearly document the implemented equations for nanostructures considered in this thesis using the electronic structure obtained using the methods outlined in chapter 2. In order to capture the physics behind these processes. In two final sections. Starting with the Hamiltonian 39 . leading to a multitude of physical phenomena. with a particular focus on the details arising when electronic states obtained using the k · p envelope function method are used.1. the theory will be treated within the time-independent limit.8) into the second quantization. such as the constant emission or absorption of light. These equations have been used successfully over decades to describe optical properties within semiconductors. and ions. The resulting equations are famous and denoted as semiconductor-Bloch equations [38]. leading to the semiconductor-luminescence equations.1 3. the Coulomb interaction will be excluded and thereby.1. In the present chapter. The theory here is formulated within the classical limit of optical transitions. Therefore. 9. restricted to a constant number of particles.e. photons. 3. electrons. where the quantum-mechanically treated carriers couple to a classical electromagnetic field. we briefly explains how spontaneous emission can be obtained from optical susceptibility and cover the procedure to calculate required matrix elements from wavefunctions obtained using the k · p envelope equations. Then. only transitions between free carriers will be considered. the Hamiltonian can be formulated in the second quantization that allows to include a varying number of particles and maintain the antisymmetry of the wavefunction naturally. 41]. i. the corresponding operator in the Heisenberg picture is given by AH = eiHt/ Ae−iHt/ .8) and separating the interaction with the external electromagnetic field leads to H = i p2 i + U (ri ) + 2m0 H1 i e e2 2 Ai · pi + A m0 2m0 i He−EM + 1 2 i. a ˆn |0 = 0.1.1. Another detail is that here. 3.1) (or any other operator) into second quantization representation is given by the electron field operator ˆ r. .2) with resulting energies and wavefunctions.1. has been introduced.1) Here the pure electro-magnetic Hamiltonian HEM has been dropped as it constitutes within the present theory only an additional energy. The antisymmetry of the state is preserved by the fermion commutation rule a ˆ† ˆn n. . a particle occupying the eigenstates of the single particle Hamiltonian H1 . a and + =a ˆ† ˆn + a ˆn a ˆ† na n = δnn = 0.8) would not be diagonal in jk . a ˆn ]+ = a ˆ† ˆ† n.40 Chapter 3.7) where H is the Hamilton operator.1 Single Particle Hamiltonian ˆ r).1. the first quasi-particle. The representation in terms of second quantization could also be performed using another basis of the single-particle Hilbert space.1.8) = j a ˆ† ˆj Ej .1. These operators create and destroy a particle with ϕn and therefore act on n and a the state a ˆ† ˆn |1n = |0 . ja which is diagonal as ϕn is assumed to be an eigenfunction of H1 .k ˆ † (r)H1 Φ( ˆ r) drΦ a ˆ† ˆk ja drϕ∗ j H1 ϕk (3. 4π 0 |ri − rj | H2 2 (3.j e .1.3) n |0 = |1n . For an operator A in the Schrödinger picture. the representation of the single-particle Hamiltonian H1 in second quantization is Using the field operator Φ( given by ˆ1 H = = j. a n |1n = 0.1. . (3.1. The Fock space F can be defined in terms of the direct sum of N -particle Hilbert spaces HN F = H0 + H1 + H2 + . a n + A valuable tool to transfer the Hamiltonian (3. Let ϕn be the eigenstates of the single particle Hamiltonian H1 H1 ϕn = Eϕn (3. The state span an Hilbert space of varying number of particles termed as the Fock space. (3. but then (3. Next.6) n Here. As the Heisenberg picture will be generally used.4) (3. Free Carriers Optical Transitions (2. t) = Φ( ϕn (r)ˆ an (t). a ˆ† (3. (3.1.5) [ˆ an . we switch from the Schrödinger to the Heisenberg picture.1. the time-dependence of the operators is dropped from now on. the vacuum ground state |0 and creation and annihilation operators a ˆ† ˆn are introduced.1.1. 1.k · E − dj.3 Particle – Electromagnetic Field Interaction Hamiltonian The next Hamiltonian to quantize is the electron–EM field interaction Hamiltonian He−EM defined in (3. H1 ] = p. where the periodicity is much smaller than the photon wavelength of usual electromagnetic fields.10) Note that the m and the l are swapped in the integral compared to the annihilation and creation operators and that in jk | v | lm .1.1. but more complicated. while the full justification can be found in [41].k .k · E − + Ej − Ek A · dj. where d = −er is the dipole moment.9) ja ka 2 j.1. The first step is to relate the momentum matrix element p and the dipole transition matrix element r. ja (3. It is clear that He−EM is a one-particle Hamiltonian and therefore its representation in the second quantization is obtained by applying the field operators as in (3. with the difference that ϕn is diagonal in H1 but not in He−EM . it can e be treated as an additional energy constant and thus be neglected. the expression in the brackets vanished and the sipole approximation for the Hamiltonian is obtained. The result of the expansion is ˆ2 = 1 H jk | v | lm a ˆ† ˆ† ˆl a ˆm .1. the Coulomb term H2 .1.k e2 e j | A2 | k j |A·p|k + m0 2m0 a ˆ† ˆk . so jk | v | lm → δsj sm δsk sl jk | v | lm .3. H1 (r)] | k m0 i e = A· j | rH1 (r) − H1 (r)r | k (3.1. ja (3. The remaining term.2 Two Particle Hamiltonian: Coulomb Interaction The transformation of the two-particle interaction. The last step involved adding the term dj. which with the definition of E gives e ∂ j | A · p | k = −dj. m j | A · p | k is more 0 difficult to simplify and thus a simplified argument is brought here. 4π 0 |r − r | (3. the dipole approximation can be applied.12) m0 where H1 is the single particle Hamiltonian.k.1. As the crystal is a quasi-periodic structure.1.1).8). the second term in (3.11) ∂ A To simplify the Hamiltonian and in order to get rid of the vector potential A in favor of the electric field E = − ∂t (Coulomb gauge assumed as before). Second Quantization 41 3. is similar.1.1.14) m0 i ∂t If A is given by a plane wave A0 ei(ω0 t−k0 r) with center frequency ω0 and the transition energies Ek − Ej are ranged around ω0 .13) i e = A· j | r | k (Ek − Ej ) i i = − (Ek − Ej ) A · j | d | k . (3.1.k j |d·E|k a ˆ† ˆk . (3.k · E.1.15) . The relation is given by i [r. Thus.m The matrix element jk | v | lm is given by jk | v | lm = ∗ drdr ϕ∗ j (r)ϕk (r ) e2 ϕm (r)ϕl (r ). the electric field can be assumed to be constant within the range of a lattice cell. (3. This approximation removes the spatial dependence of A and reduces the second term j | A2 | k to a contribution diagonal in j. The resulting expression reads ˆ e−EM = H j. and is therefore proportional to the number of electrons and does not contribute to any interband transitions. ˆ e−EM = − H j.l.1.11) can transformed into e e j |A·p|k = A· j | [r. the spin variable must be included.1. 3. Therefore. k . 21) = 3. The envelope function F (z ) is then normalized over the quantized direction. (3.k. the coordinate z will be used for the symmetry broken direction while r will denote the translational invariant direction.22) The free direction is represented by the plane-wave and the symmetry broken part by F (z ). the conventions are given by w(r) w(q) = q w(q)eiq·r .19) in terms of the Bloch-functions.23) √ 1 m3−d 1 =√ A √1 1 =√ L md .42 Chapter 3. the Fourier transform of certain quantities (such as the electromagnetic field or the Coulomb interaction) in the translational invariant directions will be used to simplify the Hamiltonian.2.1. Within a bulk crystal. for a system quantized in d dimensions. z ) eik·r Fnk (z ) .n δk .1. the Bloch function is given by ϕnk (r) = unk (r)eik·r . ja (3.1.1. it is assumed that A is the volume of the translational invariant direction and L is the volume of the quantized direction. For the spatial Fourier transform.1. In the following. The indices n now include the subbands.k .1.20) (3.k j | H1 | k a ˆ† ˆk + ja 1 2 jk | v | lm a ˆ† ˆ† ˆl a ˆm − ja ka j. the units of the wavefunction parts are given by ϕnk (r. but the normalization over the translational invariant direction is distributed into the lattice periodic part unk (r. we have ignored the specific form of the wavefunction ϕn .16) 3.m j. √1 1 =√ Ω m3 (3.18) is still required to hold. z ).1. as presented in Chapter 2.1.l. which needs some clarifications and definitions. the Bloch function loses it’s plane wave dependence in the quantized direction and is therefore expressed as ϕnk (r. Free Carriers Optical Transitions The final Hamiltonian for the crystal electrons (3. Therefore.18) The goal now is to represent the Hamiltonian (3. z )eik·r Fnk (z ).17) where unk (r) is lattice periodic and the exponential term is slowly varying.k ϕnk a ˆnk (t) (3. As the Bloch-states are not localized. Here we consider the wavefunction to be a Bloch function. t) = Φ( n. (3.1.16) using the field operator ˆ r.1).1.1.16) into Bloch states. 1 A drw(r)e−iq·r .1.1 Normalization In the case of a quantum nanostructure. The Bloch-functions are required to be orthonormalized over the crystal domain Ω drϕ∗ n k ϕnk = δn . z ) = unk (r. Ω (3. z ) = |nk = unk (r. A (3. Ω = LA.1. Obviously.k j |d·E|k a ˆ† ˆk . The current section aims now to transform (3. (3.1. can therefore be written as ˆ = H j. Regarding the normalization.2 Bloch States Formulation Up to now. 3.1.1.28) As it is assumed that E(r. within the approximation that lattice-cell averaged quantities are used. t)eiq·r . the operator is constant over a lattice cell and the integral of the matrix element can be reduced using the orthogonality of the Bloch functions.2. z ) are not given in an explicit form.1. z )e (3. (3. t) is given by E(r.4. Second Quantization 43 3. the integration over it.2 Lattice-Cell Average A general problem of the approach using wavefunctions of the k · p envelope equation is that the lattice-periodic functions unk (r.electromagnetic field interaction Hamiltonian He−EM within the dipole approximation.24) Here the operator A is replaced with it’s lattice averaged quantity Vc−1 un k | A | unk Vc denotes the crystal cell and by Vc Vc .2. the kinetic term (3. k Bunk (r.1.3 The Kinetic Term To transform the kinetic part into the Bloch states.4 The Interaction Term The remaining step is to quantize the electron . t) is slowly-varying.k ∗ dzFn k (z )Fnk (z ). these properties are sufficient for building the Hamiltonian and calculating desired matrix elements. In that case. t) = q E(q. The other case is given by the long-range operator B .1.1. such as the Coulomb potential.22) can be simplified as n k | A | nk ≈ A un k | A | unk Vc Vc δk .1. Therefore to determine values for the short-range operator n k | A | nk .27) 3. the eigenfunctions ϕnk (r.k a ˆ† ˆnk En.1.25) Vc where u ˜i is the Bloch function normalized with respect to a single crystal cell.26) A further simplification depends on the actual form of the considered operator. Using that particular field operator. In the k · p method. (3.k . nk a (3.16) is given by ˆ1 = H n. Nevertheless. a matrix element between two wavefunctions (3.1. The matrix elements found in the literature are related A un k | A | unk Vc = u ˜n k | A | u ˜ nk (3. the fact that they are orthogonal to each other and some measurable quantities are available. Only their symmetry properties. We return to this relation further in section 3. The resulting . z )e ≈ Ω drdz 1 un k | unk Vc Vc ∗ ik·r e−ik ·r Fn Fnk (z ). The lattice-cell average reads n k | B | nk = Ω −ik ·r ∗ drdzu∗ Fn k (z )Bunk (r. z ) of the single particle Hamiltonian are inserted into the field operator.3.1.1. The Fourier representation of the electric field E(r. only the envelope function Fnk (z ) and the experimental value for u ˜n k | A | u ˜nk for the from bulk measurements are required. the envelopes and plane waves are assumed to vary little over a crystal cell. the relevant contributions to the sum will be around very small values of q. z )eik·r Fnk (z ) n k (r. This allows to pull the eiq·r factor out of the dipole integral in the calculation below.2. If an operator A mainly acts on the lattice-periodic part. 1.1.1.k = A ∗ dzu∗ n k Fn k dunk Fnk . if an electron with momentum −k is created (= a ˆ† −k ).32) Note that from now on.n. The term p ˆvc.29) L The delta function δk k results from the plane wave and ensures momentum conservation. while n ˆ vk = ˆ b† b−k bb−k ˆ is the number operator counting the holes in the valence band v .1. the Hamiltonian can finally be written as ˆ e−EM = −E(r.1. One important point is that in the Hamiltonian (3. giving the correlation between a particle in one band and an empty state in the other.k . some terms can already be identified: n ˆ ck = a ˆ† ˆck is ck a ˆ the number operator counting the number of electrons in the band c with crystal momentum k. Collecting all single particle interactions.k ˆ Ev (k)ˆ b† bk bbk −E(r. t) c.31) 3. In (3.n. p ˆ† vc.4) and (3. transitions between conduction subbands and transitions between valence subbands have been neglected.k µn . In order to introduce the concept of holes into the Hamiltonian. .32). n ka (3.k q δk k µn n.k p ˆvc.k a ˆ† ˆnk nk a drdzϕ∗ nk d· Ω q E(q. it is common to use for the valence band the absence of an electron. One obtains for the kinetic and dipole Hamiltonian ˆ H = c. as a quasi particle.k Ec (k)ˆ a† ˆck − ck a v.k · E(r.k (3.k ∗ ˆ ˆ† ˆ ck µcv. electrons in the valence band get excited into the conduction band and leave behind holes. Free Carriers Optical Transitions Hamiltonian then reads ˆ † He−EM Φ ˆ drdz Φ Ω = = − n . the valence band is fully occupied while the conduction band is empty. For the moment. where the dipole matrix element between two states n and n with same crystal momentum k is given by µn n. The sign is switched by convention. With increasing temperature.k bv −k a . In a simplistic view.3 Introduction of Holes In a semiconductor at T = 0K . the summation is distinguished between summation over conduction bands indicated with c and valence bands indicated using the index v .44 Chapter 3. a hole with momentum k is annihilated (= bk ) and vice versa.k a ˆ† ˆ nk .30) The transformation of the two particle Hamiltonian H2 is presented in the next chapter. ˆ Therefore.n. Consequently.1. the dipole matrix element µcv.k gives the coupling strength of such a correlation to an electric field.k .n. it is assumed that the carriers are uncorrelated and their coulomb interaction is included in the single particle Hamiltonians.n. only direct transitions are allowed.k a ˆ† ck bv −k +µcv.k a ˆ† ˆ nk . the Hamiltonian in the Bloch basis is given as ˆ = H n En (k)ˆ a† ˆnk −E(r. t) H n . In other words. (3. t)eiq·r .k µn .1.32).k = bv−k a ˆck is named microscopic polarization and is given by the off-diagonal density matrix element. the hole. n ka (3.5) are used to reestablish the normal ordering of creation and annihilation operators.n. t)eiq·r ϕnk − n . t) nk a n . the commutator rules (3.1. Using the dipole matrix element.v.1. EF. As the k dependence of the distributions for general band structures is quite evolved.1.1) leads to the more convenient form P( ω ) = b χ(ω )E(ω ).36) is not feasible. 3.2. These dipoles create a macroscopic polarization P. (3. (3.35) and (3. an analytical inversion of (3.36) for holes. (3.2. Therefore. (3. the electrons are free to move only in the plane of the active layer.4 Carrier Statistics The main interest of the current work is the continuous emission of light of a semiconductor.37) → (2π )d k the unknown Ω volume is removed and one ends up with known quantities (L = Ω/A). When the sum over k is transformed into an integral 2A dk (3.2 3.2. Note the factor 2 which stems from the implicit summation over the spin states.1. Transitions Calculation 45 3. the simplest way to calculate the quasi-Fermi levels is to perform a numerical Newton procedure to find the roots of N0 −N (EF ) = 0. The Fermi levels can be calculated from the 3D carrier density N N= for electrons and P P = 1 Ω 1 Ω fck ck (3.1) Taking the Fourier transform of (3. Their relation is defined in terms of a time-dependent susceptibility χ(t − t ) (in this case a scalar) t P(t) = b ∞ χ(t − t )E(t )dt .3.1.38) where we assume that the function summed over is cylindrically symmetric (which is a reasonable assumption for the subband structure near the zone center k = 0).1. so that the summation over k is restricted to two dimensions according to → k 2A (2π )2 dk 2πk. which is given in the steady state of the system.v −Ev (k))/kB T . given by the Fermi distributions for the electrons in the conduction band nc k = n ˆ ck = fck = and the holes in the valence band nv k = n ˆ vk = fvk = 1 1+ e(EF.2.33) Here.2) . kB is Boltzmann’s constant and T denotes the temperature. d denotes the dimensionality of the k-space. For an ideal 2D quantum well. an electric field E in a semiconductor induces dipoles.1.35) fv k vk (3. it can be assumed that the carriers relax into their quasi-equilibrium distributions.1.1 Transitions Calculation Introduction Classically.v denote the quasi-Fermi levels of the electrons and holes. which is usually unknown. Here Ω denotes the volume of the system. The induced polarization P(t) at time t depends on the electric field E(t ) at time t < t.c and EF.c )/kB T (3.1. In the limit of small light intensity and therefore absence of spectral-hole burning.1.1.34) 1 1 + e(Ec (k)−EF.2. As a first step.9) and neglect all terms containing ∂z E (z ).2. the term ∇ · E = 0 vanishes. ∂t2 (3.c.10) and (3. ∂t On the left hand side. φ(z ) us the real phase shift and E (z ) is the real field amplitude. Maxwell’s equations are rewritten to give a single relation between the electric field E and the macroscopic polarization P. ∂t (3. The next step is to assume a monochromatic electrical field E (traveling into z -direction) given by E(z. the self consistent equations are obtained as ∂z E (z ) ∂z φ(z ) = = ν k0 {P (z )} = − χ (z )E (z ). (3. 2k0 (3. varying little within an optical wavelength.4).8) which now allows to introduce the macroscopic polarization P using (3.2.c. ∂t (3.11) 2 2 2 The next step is to insert (3.12) Splitting into real and imaginary parts and using (3.2.10) where k0 = νn/c is the photo wavenumber and e ˆi is a unit vector orthogonal to e ˆz .2.6) (3.14) . (3. It is assumed that the spacial strong oscillatory part is properly described by the plane wave.4) (3.2. Free Carriers Optical Transitions The polarization influences the electric field via the electric displacement D= E= bE + P.46 Chapter 3.2.2. assuming charge neutrality and homogeneous media.2. 2 0 nc 2 1 ν k0 − {P (z )} = − χ (z ).2.5) ∂ (∇ × H) . On the right hand side. ∇ × H is replaced by the electric displacement D ∇ (∇ · E) − ∇2 E ≈ −∇2 E = −µ0 ∂2D . t) = e ˆi P (z )ei(k0 z−νt−φ(z)) + c. t) = 1 e ˆi E (z )ei(k0 z−νt−φ(z)) + c. the creation of dipoles and the amplification of an electric field needs to be treated self-consistently.2. Taking the curl of ∇×H− and using ∇×E=− ones obtains ∇×∇×E = ∇ (∇ · E) − ∇2 E = −∇ × −µ0 ∂µ0 H .7) ∂D =j ∂t ∂B . ∂z φ(z ) and ∂z E (z )∂z φ(z ).2). 0 ∂t2 ∂t2 (3.2. using (3.13) (3.11) into (3.2.2. ∂z E (z ) − iE (z )∂z φ(z ) = i µ0 ν 2 χ(z )E (z ).2. 2 (3.3) Therefore. The electric field induces a polarization 1 P(z.2.2.2.2. ν denotes the field frequency. The aim is therefore to calculate the polarization quantum-mechanically in the presence of an electric field and obtain a self-consistent formula for the steady state. E (z ) 2 0 nc 2 − (3.9) This equation is the inhomogeneous Helmholtz equation.3) − ∇2 E + µ0 b ∂2E ∂2P = − µ . This approach is denoted as the slowly varying envelope approximation and one ends up with an equation for the amplitudes E (z ) and P (z ). v.k a ˆ† ˆ nk = n ka n.2.v.k = H1 . Consequently.k (3. (3. the intensity gain (amplitude is half of it) is defined by G = −k 0 χ . one is left with the first term.k pvc.23) the second term containing p ˆvc.16) 3.k p† vc.k .15) and the change of refractive index (via a continuous phase change) is given by the real part of the susceptibility δn χ = .2.k H . p ˆnm.k (3.k The other term of the polarization containing is related to the complex conjugate part of (3.2. one obtains d i ˆ i ˆ p ˆnm.2. Therefore. O . vc.k = p ˆvc.22) (3. t) used for the optical properties is given by 1 P (z ) = 2e−i(k0 z−νt−φ(z)) µ∗ (3.2.k pvc. the expectation value pvc.2.k (3. The Heisenberg ˆ is given by [40] equation of motion for a time dependent operator O d ˆ i ˆ ˆ O= H. Transitions Calculation 47 where χ = χ + iχ is the electrical susceptibility of the system.18) from which the optical properties (3.k + µcv.21) (3.k has to be solved.2. Here the index m is used for the conduction band and n for the valence band of interest. (3. p ˆnm.v.k + µcv.k . The indices c and v will be used for the summation over remaining conduction and valence bands. Here.k vanishes because four anticommuting exchanges are required (leading to no sign ˆ ˆ = 0. n 2 The absorption can be obtained from the intensity gain through α = −G. where V is the volume of the system and P is defined as the macroscopic polarization density P= 1 V µn n.3 Heisenberg’s Equation of Motion In order to calculate the expectation value pnm.n .k c.k H − p ˆnm.16) can be obtained.2.19) cv.k .2.2 Quantum Microscopic Polarization The second step is to couple quantum mechanical observables to the solids properties. where operators are exchanged change) to obtain p ˆnm. (3.17) 1 V ∗ µcv.15) and (3.2.k .k + He−EM . the equation of motion of the microscopic polarization operator p ˆnm.3.2. dt Applying to the microscopic polarization operator.k p c.k 3. p† vc.2. The amplitude P (z ) of the macroscopic polarization P(z.20) 1 V ∗ µcv.k V c. The dipole interaction between electrons and the electric field is given by He−EM = −V P · E.2.2. dt The evaluation of the first commutator gives − For the second operator i (Ec (k) − Ev (k)) p ˆnm.k has been used.2.11) and is not required to determine the macroscopic amplitude P (z ).k p ˆ† ˆvc. 29) These equations constitute the semiconductor Bloch equations [38]. (3. the sums over c and v except forc = m and v = n will vanish when later the expectation value a ˆ† ˆmk is ck a taken.2.sometimes accompanied by exciton molecule (biexciton) resonances .28) dn ˆ mk dt ∂ n ˆ n. (3.25) Here. ∂t col.2. scattering between electron-hole excitations also becomes important. the screening of the Coulomb interaction by the optically excited carriers and the collective plasma oscillations are the relevant physical phenomena. Here.2. • the quasi-equilibrium regime which can be realized on relatively long time scales. The hand waving argument is that the expectation value a ˆ† ˆnk has a dominant time-dependence nk a a ˆ† ˆnk ∝ ei(ωn k nk a −ωnk )t (3. the equation of motion of the free carrier microscopic polarization is d p ˆnm.2.27) Similar equations can be formulated for the carriers density operators n ˆ mk and n ˆ nk dn ˆ nk dt = − + 2 E(z. 2 = − E(z.k i − E(z.k (ˆ nmk + n ˆ nk − 1) + ∂ p ˆnm.v δm.k dt i = − (Em (k) − En (k)) p ˆnm. one can distinguish several relevant regimes: • the low excitation regime in which the exciton resonances . Depending on the strength and time dynamics of the field. for the free-carrier case. v and k to a ˆ† ˆ mk + ck a c v ˆ bv−kˆ bn−k − 1. . Therefore. t) · µmn.26) and therefore rapidly oscillates for ωn k = ωnk and then averages out over time. (3.24) The delta functions of the remaining terms lead in the sum over c.k ∂t . the sums are neglected from now on. As the density increases gradually.k δn.k + ∂ n ˆ m.dominate the optical properties.48 ˆp ˆ to transform H ˆ into p ˆH ˆ† ˆ ˆmk a ˆ† ck bc−k bnk a Chapter 3.k ck a ˆ +ˆ b† v −k bn−k δc.k ˆ −δk .2. The main dissipative mechanism is the carrier-carrier Coulomb scattering.n δk .2.m δk . This approximation scheme is called the random phase approximation. Here. the excitations have relaxed into a quasi-equilibrium and can be described by thermal distributions. The interaction with phonons provides the most important relaxation and dephasing mechanism.k . (3.k (3. The terms with c = m and v = n are the density operators n ˆ mk and n ˆ nk . The relatively slow approach to equilibrium can be described by a semi-classical relaxation and dephasing kinetics. Free Carriers Optical Transitions → a ˆ† ˆmk δv.k ∂t . t) · µk p ˆ∗ nm.c + p ˆH. t) · µk p ˆ∗ nm. col. These semiconductor Bloch equations contain the basis for most of our understanding of the optical properties of semiconductors and semiconductor microstructures. • the high excitation regime in which an electron-hole plasma is excited. col. Collecting everything and taking the expectation values. A simpler way is to replace the oscillating microscopic polarization pnm. where ν is the frequency of the optical field.k . the high-excitation and quasi-equilibrium regimes dominate the theoretical analysis of the discussed quantum structures. of the single-particle density matrix elements to higher-order correlations. however..2.k | (nnk + nmk − 1) n.27) one could use the approach given in [40] and formally integrate the differential equation.g.2. and skipping all fast oscillating parts (as they should average out over time).e. in many cases one can identify specific physical mechanisms that dominate the scattering terms in some of the excitation regimes listed above. in the low excitation regime often the coupling of the excited carriers to phonons determines relaxation and dephasing. respectively [38].28) and (3. However. leads to i E (z ) nmk + nnk − 1 pnm.2. In all situations. the semiconductor Bloch equations are a very suitable theoretical framework which.2. This self consistent coupling of Maxwell and semiconductor Bloch equations (for shortness also called Maxwell-semiconductor-Bloch equations) is needed as soon as spatially extended structures are analyzed where light propagation effects become important. For example.2.4 Solving the Equation for Free Carriers To solve (3.k = pnm. the semiconductor Bloch equations have to be treated together with the Maxwell equations for the light field in order to determine the optical response.k = 0. Thoughout this thesis. the rate of change of the hole population (3.2. however. Under these conditions one can treat the semiconductor Bloch equations separately from Maxwell’s equations to calculate the optical response.k = − µmn. whereas at high carrier densities carrier-carrier scattering dominates.m. Inserting this equation into (3.m (k) − ν ) + γ 2 2 . The process of decoherence. In optically thin samples.k 2 (3.2.27).k e−i(k0 z−νt−φ(z)) . i. such as two-particle and phonon. Relevant examples are the polariton effects. as well as semiconductor lasers or the phenomenon of optical bistability [38].2. by a timedependent screening are governed by the quantum kinetics with memory structure of the carrier-carrier and carrier-phonon scattering.15) we obtain the expression for the absorption spectrum αF CT (ν ) = k0 χ = − ν 11 n 0 bc Ω |µnm.31) Inserting (3.30) 3.2.2.m (k) is defined by ωn.k by its slowly varying envelope snm. where propagation effects are unimportant.k ∂t ≈ col.28). Generally. (3. the scattering terms in the semiconductor Bloch equations describe all the couplings of the polarizations and populations.34) γ (ωn.2. The scattering terms require a more sophisticated treatment and some of these will be addressed in chapter 4.10) into (3.2. and the scattering terms can be described by Boltzmann-like scattering rates due to carrier-phonon or carrier-carrier scattering.31) and (3. The first term in (3. (3..or photon-assisted density matrices. −γ p ˆnm. In general. Transitions Calculation 49 • the ultrafast regime in which quantum coherence and the beginning dissipation determine the optical response.k e−i(k0 z−νt−φ(z)) .2. scattering is approximated using a a simple phenomenological decay rate model given by ∂ p ˆnm.29) is identical to the rate of change of the electron population (3. has to be supplemented with an appropriate treatment of the scattering terms in order to describe the various aspects of the rich physics which one encounters in pulse excited semiconductors.32) (3. dt (3. e.2.33) 2 i (ωn.m (k) − ν ) + γ where ωn.35) . and solve for the steady state of d snm.2.3.2. of the beginning relaxation and the build-up of correlations. Markov approximations for the scattering processes are often justified. For relatively long pulses.m (k) = Em (k) − En (k).29) describes the generation of electrons and holes pairs by the absorption of light. As long as the scattering terms are ignored. (3.2. the transmitted light field is proportional to the calculated polarization field. For the moment. i.e. In order to evaluate numerically the absorption spectrum we make the following assumptions: • The system under discussion is 2 dimensional and all k-vector dependencies are cylindrically symmetric.2. the spontaneous emission probability per second is given by Ge c/nb . The relation between emission and inversion gives 1 fck fvk = .3. the spontaneous emission probability per unit length is obtained.1) The first term on rhs denotes the probability of a photon emission.3. (3. which can be rewritten to nck + nvk − 1 = nck nvk − (1 − nck ) (1 − nvk ) . assuming that the conduction band and the valence band are two carrier reservoirs. 3. the complex part of the optical susceptibility χ .m (k) − ν ) + γ 2 2 .4) Note that the result can be related to the Kubo-Martin-Schwinger (KMS) [42] relation.35).5). G = Ge − Ga . leading to the KMS relation rsp ( ω ) = 2 n2 1 b ( ω) G( ω ) π 2 3 c2 1 − exp (( ω − (EF c − EF v )) /kB T ) (3. i. leads to nek = fek and nhk = fhk . The term nck + nvk − 1 is the inversion of the electron-hole population in the semiconductor.3. electron and hole is occupied. theµnm. the transition energy Ec (k) − Ev (k) is close to the photon energy.2) π c and therefore the spontaneous emission (s−1 m−3 eV −1 ) per second per unit volume per unit energy is given by rsp ( ω ) = 2 n3 b ( ω) Ge 2 3 2 π c (3. Using a phenomenological approach. which allows to pull the factor (3.35) can be divided into an emitting and an absorbing part.35) including only the emission probability term Ge ∼ nck nvk .m ∞ 0 n m dk k |µnm. while the second term denotes the probability of a photon absorption.5) out of the k -sum in (3. Neglecting the existence of a cavity.3. between the subbands n and m.38) transformation from summation to integration for the k-vector.36) where L is the length of the 2D quantum system. we can use (3. Free Carriers Optical Transitions Here.6) .2. Consequently. Therefore.e.s of (3. fck + fvk − 1 1 − exp ((Ec (k) − Ev (k) − (EF c − EF v )) /kB T ) (3. (3.3 Spontaneous Emission The spontaneous emission within a semiconductor nanostructure can be obtained from the amplification G of the photon field.k denotes the dipole matrix element.1.3.h.k .3. As the velocity of a photon in the semiconductor is given by c/nb . it may be replaced with ω on the r. (3.2.3. • The system undergoes a rapid equilibration of electrons and holes into Fermi-Dirac distributions. (3. by evaluating (3.50 Chapter 3. which. Under these assumptions. resulting from the scalar product between the monochromatic light field E and the dipole µnm.35). Thus. (3.3.5) Due to the Lorentzian in (3. the photon density of states (of photons with energy ω ) is given by n3 ( ω )2 N ( ω ) = b2 3 3 (3.2.2.k | (fek + fhk − 1) 2 γ (ωn. along the polarization of the light field.35) becomes αF CT (ν ) = − ν c 0 nb π L n.2.3) where as the spontaneous emission intensity (s−1 m−3 ) is given by Isp ( ω ) = ωrsp ( ω ) . k between the solutions of the k · p band structure calculation is not straight-forward. Using this expansion. (3.spontaneously. the average of all possible transitions is usually taken for the spontaneous emission µsp cv. the interband dipole moment is dominated by the admixture of uv0 into the conduction subbands and uc0 into the valence subbands.4) Assuming that n and m are eigenfunctions of the Hamiltonian H with corresponding eigenenergies En and Em .y. the spontaneous emission B coefficient (m3 s−1 ). Nevertheless. conduction and valence band. The root of this misbehavior is due to the fact that the quantization of the electro-magnetic field is not included. the last expression evaluates due to the orthogonality of uc0 and uv0 to zero1 .j Vc ∗ dzFm. Dipole Matrix Element 51 between gain and spontaneous emission. There is an obvious pathological feature when such a relation is used to obtain the spontaneous emission: if the electric field is zero.3) L But. conduction subbands is dominated by the envelope modulating uc0 .j k .i k dFn. 3. A simpler solution.g. given by ∂ i r = [H.5) fact also applies to 4 × 4 and 6 × 6 k · p valence band models. 44]. the dipole matrix element between two states is given by µn n. replacing the derivative of the position operator by the momentum operator and H |m by Em |m .j ∗ u∗ i0 Fm. In the general case.z cv. one obtains n|p|m = 1 This im (En − Em ) n | r | m .4. spontaneous emission would occur .4.k (3. r] . no transition between conduction and valence subbands in such models would exist. therefore the error of the classical treatment may be small enough for the present purpose.k = A dz L i.3.4. concluding that the expression is dominated by µn n. then no spontaneous emission would exist.4. The reason lies in the single-band approximation.ik duj 0 Fn. Another aspect to consider is the dependence of the dipole matrix element µcv.k = 1 µi . The issue has been addressed by Burt and others [43.ik ui0 .k ≈ A Vc ui0 | uj 0 i. (3. the spontaneous emission of a two level system treated using quantized electromagnetic fields does not differ from dipole radiation. the envelope functions are given by ϕmk = i Fm.3. Therefore.instead of the dipole matrix element. Including this. within a k · p calculation involving only single-band models for both. The relation can be derived using the Heisenberg equation of motion for the position operator r.4.k on the polarization of the light field. Working within the 8 × 8 model would therefore resolve the problem.2) The question to answer is to whether the dipole operator is acting on the lattice periodic part uj 0 or on the envelope part Fi . While the wavefunction of e. .j k .8) which defines another figure of merit. is to use the momentum. (3.7) The spontaneous emission rate per unit volume (s−1 m−3 ) is given by integrating rsp ( ω ) over the energy ∞ Rsp = 0 rsp ( ω )d ω = Bnp (3. (3.4 Dipole Matrix Element The evaluation of the interband dipole matrix element µmn.1) where ui0 are again zone-center Bloch functions. ∂t (3. Therefore. 3 i=x.4.3. allowing to use non-8 × 8 models without the pathological feature of vanishing transition probabilities. k ≈ A Vc ui0 | p | uj 0 i. However.10). the expression for |µk | can be much simplified.2. 3. the zone-center functions away from k include remote contributions.4. z ) |2 + 1 | ϕn (z )|g2. HH 2 2 |µk | = . effectively averaging out the polarization dependence. Therefore.7) In the approximation of slowly varying envelopes.4. z ) |2 . equation (3.10) 2 2 1 1 |µ0 | ˆ·e ˆ + k .4. When calculating µt .k = e pmn.m (k. (3. (3. the strength of interaction between each electron plane wave and the incident photon is highly polarization dependent. Consequently.4.e. As a general valence band wave function consists of both HH and LH components. (3.9) ˆ The factor 1 2 compensates for the spin degeneracy. for beam propagation in z -direction): 3 | ϕn (z )|g1. in bulk material this dependence doesn’t reveal itself as the incident field interacts with many electrons with different wave vectors. the approximation (3.4. transitions between remote contributions have to be included in a consistent definition of the momentum matrix elements [28]. We assume the envelope function overlap integral to be unity. the derivatives of the envelopes are small and the momentum matrix element is dominated by pmn. the energy difference En − Em can be replaced by the photon energy ω . beside the mixing of the explicitly considered states in the Hamiltonian. |µ0 | 3 | ϕn (z )|g3. the effect of remote bands is included using Löwdins perturbation theory.8) considers only transitions between zone-center functions ui0 included explicitly in the Hamiltonian. |µ0 | can be determined from measurements of the band curvature [12]. leading to the simple expression for µmn.k in terms of the momentum matrix element µmn. z. where i = x. (3.2.11) . we can no longer work with simple plane wave states. z ) |2 . Free Carriers Optical Transitions The electron dipole operator (electrons are negatively charged) is related to the position operator by d = er leading to n|p|m n|d|m =e .6) im (En − Em ) Within direct transitions. σ = U p | µ | 4 3 k σ Mnm = = .j k .i k Fn.m (k.4.4.u uv . y. We find for the normalized transition strength in bulk material 2 1 ˆ·e ˆ 1− k . all possible transitions between the involved conduction and valence subbands (HH or LH) have to be taken into consideration.k . 2 (3. It is customary to express |µk | as a function of |µ0 | = uc |e · p|ui . in (2. LH 2 3 ˆ is a unit vector pointing along the electron k-vector. | 3.m (k.m (k. σ = Down 4 3 (3.52 Chapter 3.8) L The momentum matrix elements are related via (3.25) to band structure parameters P .8) neglects the effect of remote bands.j Vc ∗ dzFm.4. the relative interaction strength is equal in all the three principal directions.1. yielding |µk | = 2 1 2u ˆ · p|uv | uc |e ¯c c . z ) |2 + 1 | ϕn (z )|g4. defined e. |µ0 |2 In quantum well structures.g. i.4.9) can be replaced by: 1. For the TE polarization (Ez = 0.1. but have to take into account the envelope functions. because of the moment we will be interested in is a transition between two bulk plane wave states. imω (3.u ¯v Fc |Fv | . Beside the neglected derivatives of the envelopes.4. In the k · p envelope equations. k As shown in Fig. This approximation is commonly applied and is also used in the remainder of the thesis. By expressing the valence 2 bands uv as linear combinations of the basis functions.. We can now formulate the transition matrix elements calculation for the special case of the 2 band model presented in section 2. In µk |2 =1 bulk. 4. σ = U p | ϕn (z )|g4.1: Dependence of the transition strength |µk |on the angle between the electron k-vector and the electric field polarization vector E. Dipole Matrix Element 53 Figure 3. we plot in Fig. For the TM polarization (Hz = 0): σ Mnm = |µk | = |µ0 | | ϕn (z )|g2. Here we use the results of the calculations from section 2. where the wavefucntions from Fig.12). As an illustration.4. while for kt ≈ 5 × 106 cm−1 the matrix elements start to diverge as the concentration rises. If we add 2DEG into the simulated quantum structure the matrix element are altered. 2.m (k.2 to plot the matrix element change with 2DEG concentration from 1 × 109 cm−2 to 1 × 1011 cm−1 .3. z ) | . the concentration dependence of the matrix element is negligible close to the Brillouin zone center. 2.3.3. The various conduction and valence subbands are indicated.1 Ga0. 3.4. 3 ω (3. 3. σ = Down (3.11) and (3. as shown in Fig.9 are used in the matrix element calculations. As can be seen for both polarities.2 the calculated dipole matrix elements (for both polarities) of a 200 Å wide GaAs/Al0. z ) | .4.13) where P is the dipole matrix element and ω is the transition wavelength.9 As quantum well at T = 2 K obtained using equations (3. where the curves are ordered according the rising value of the concentration.12) The term |µ0 | can be written as eP |µ0 | = √ .m (k. .4. 8 |µTM |2/|µk |2 k 0.2 0.6 |µTE |2/|µk |2 k 0 E1−HH1 N2DEG 0.9 As quantum well at T = 2 K .8 0. (a) 0.2 0 0 1 2 3 4 5 kt (cm−1) 6 7 8 9 10 x 10 6 Figure 3.5 0.1 Ga0.7 0. Free Carriers Optical Transitions (a) 0.6 0 0.8 0.2 0.1 Ga0.54 Chapter 3.4 E1−HH2 E2−HH3 E1−HH3 E2−HH2 0 0 1 2 3 kt (cm−1) 4 5 x 10 6 6 0 0.4 E1−HH2 0.2 Figure 3.9 As quantum well at T = 2 K . . without additional doping.6 E2−LH1 0.3 0.1 0 E2−HH2 E1−LH1 E2−LH1 E2−HH1 E1−HH2 E1−HH3 E2−HH3 (b) 1 E1−LH1 E1−HH1 |µTM |2/|µk |2 k 0.7 0. containing 2DEG with various concentrations in the the range of 1 × 109 cm−2 to 1 × 1011 cm−2 .5 0.3 0.8 0.2: (a) TE and (b) TM dipole matrix elements calculated for 200 Å wide GaAs/Al0.1 0 (b) 1 E1−LH1 E1−HH1 E1−LH1 E1−HH2 0.3: (a) TE and (b) TM dipole matrix elements calculated for 200 Å wide GaAs/Al0.4 0.4 0.6 |µTE |2/|µk |2 k 0 E1−HH1 0. Chapter 4 Coulomb Correlated Optical Transitions The preceding chapter only considered free carriers and omitted their Coulomb interaction. The Coulomb interaction leads to the formation of electron-hole pairs, also known as excitons. Due to their bound nature, the transition energy is lowered, leading to absorption below the fundamental band gap, while the correlated movement of the electron-hole pair increases the transition probability and consequently the transitions strengths. As in the previous chapter, the theory review is based mainly on [12, 9, 40, 38, 41]. The present chapter therefore focuses on the inclusion of such many body effects in the evaluation of optical properties. A large part of the theory presented in the preceding chapter can be recycled. In general, only the omitted two-particle interaction Hamiltonian H2 in (3.1.9) needs to be added to the equation of motion for the microscopic polarization (3.2.27). The chapter is therefore structured as follows: first, the Coulomb Hamiltonian is derived for Bloch functions and transformed into the electron-hole picture. Then the Coulomb Hamiltonian is included into the equation of motion, and the so called Hartree-Fock (HF) approximation is performed along with the introduction of screening. This leads to the formulation of the Coulomb enhancement of the absorption spectrum of the system. In order to utilize these observations, a numerical formulation of the HF model is outlined in detail facilitating a numerical calculation of the relevant optical characteristics. Finally, this formulation is applied to an actual quantum well structure, with and without the presence of 2DEG in the well region, and the results and discussed in detail. 4.1 4.1.1 Second Quantization Introduction If ni , ki denote the quantum number of a Bloch state, then the Coulomb Hamiltonian (3.1.9) can be written as ˆ2 = 1 H 2 n1 k1 n2 k2 | v | n3 k3 n4 k4 a ˆ† ˆ† ˆn3 k3 a ˆn4 k4 n1 k1 a n2 k2 a n1 k1 , n2 k2 , n3 k3 , n4 k4 (4.1.1) and the matrix element of the Coulomb operator reads n1 k 1 n2 k 2 | v | n3 k 3 n4 k 4 = e2 4π 0 · ∗ drdzdr dz ϕ∗ n1 k1 (r, z )ϕn2 k2 (r , z ) Ω Ω 1 ϕn k (r, z )ϕn3 k3 (r , z ). |(r, z ) − (r , z )| 4 4 (4.1.2) The Coulomb-potential is mainly long-range and therefore more or less constant within a lattice cell. Consequently, the lattice periodic part can be averaged over a single crystal cell. The integration is then performed over lattice averaged quantities, as suggested in (3.1.26). Therefore, rewriting (4.1.2) into a more suitable form including lattice 55 56 Chapter 4. Coulomb Correlated Optical Transitions cell averages, yields n1 k1 n2 k2 | v | n3 k3 n4 k4 = 1 i(k4 −k1 )·r i(k3 −k2 )·r e e A2 Ω Ω 1 ·gn1 k1 ,n4 k4 (z )gn2 k2 ,n3 k3 (z ) , |(r, z ) − (r , z )| drdzdr dz e2 4π 0 (4.1.3) ∗ where gn1 k1 ,n4 k4 (z ) ≈ i Fn (z )Fn4 k4 ,i (z ) and gn2 k2 ,n3 k3 (z ) is defined accordingly. The next step is to get rid 1 k1 ,i of the free directions r and r , where the slowly varying part of the Bloch-function is given by the plane wave. The 1 1 √ . Then, the integration over r is transformed into first step is to rewrite the term |(r,z)− 2 2 (r ,z )| as the integration over the distance between r and r , s = r − r . So, instead of performing for every point r the integration over r , one integrates for every point r over all possible distances s. This approach leads to several simplifications n1 k1 n2 k2 | v | n3 k3 n4 k4 = e2 4π 0 · A (r−r ) +(z −z ) 1 dzdz 2 gn1 k1 ,n4 k4 (z )gn2 k2 ,n3 k3 (z ) A L L 1 drds 2 2 A |s| + (z − z ) (4.1.4) ·ei(k4 +k3 −k1 −k2 )·r ei(k2 −k3 )·r . The integration over r and s is decoupled and therefore the plane-wave leads to a delta function drei(k4 +k3 −k1 −k2 )·r = Aδk4 +k3 ,k1 +k2 . A (4.1.5) The remaining integration over z, z and s depends on the dimensionality of the system. The Fourier transformation along the free direction A defined in (3.1.20) is used to expand ϑ(s, z, z ) = 1 |s| + (z − z ) 2 2 1 r = q ϑq (z, z )eiq·s (4.1.6) in terms of its Fourier representation. Taking the integral over s ds A q ϑq (z, z )eiq·s ei(k2 −k3 )·s = A q ϑq (z, z )δk3 −k2 ,q (4.1.7) leads to another delta function, which can together with (4.1.5) be used to simplify (4.1.1) into ˆ2 = 1 H 2 n1 ,n2 ,n3 ,n4 † Θq a ˆn1 k+q a ˆ† ˆn3 k a ˆ n4 k . ,k,k n2 k −q a (4.1.8) n1 , n2 n3 , n4 kk q n1 ,n2 ,n3 ,n4 Here, k4 has been relabeled to k and k3 to k . The term Θq contains all parts of (4.1.2) depending on ,k,k actual wavefunctions and on the dimensionality of the nanostructure. It is given by 1 ,n2 ,n3 ,n4 Θn = q,k,k e2 4π 0 dzdz gn1 k+q,n4 k (z )gn2 k −q,n3 k (z )ϑq (z, z ). L L (4.1.9) D In the case of a bulk crystal, there is no integration over z, z . Then, ϑ3 q is given by the 3D Fourier transform of the Coulomb potential 4π D ϑ3 . (4.1.10) q = Ωq 2 4.1. Second Quantization 57 The quantum well case has a 2D plane as the translational invariant directions and a 1D axis where symmetry is broken. The 2D Fourier transformation of (4.1.6) gives D ϑ2 q (z, z ) = 2π −q|z−z | e . Aq (4.1.11) 2π Hereby, the factor A q denotes the Fourier transform of an ideal quantum well with no extension in the symmetry broken direction, while the exponential factor comprises of the finite extension of the quantum well wavefunctions. For the sake of convenience, we define the form factor term to be 1 ,n2 ,n3 ,n4 Gn = q,k,k dzdz gn1 k+q,n4 k (z )gn2 k −q,n3 k (z )e−q|z−z | , L L (4.1.12) 1 ,n2 ,n3 ,n4 in terms of which Θn can be written as q,k,k n1 ,n2 ,n3 ,n4 n1 ,n2 ,n3 ,n4 Vq . = Gq Θq ,k,k ,k,k (4.1.13) For the 2D case we can write n1 ,n2 ,n3 ,n4 n1 ,n2 ,n3 ,n4 Θq = Gq ,k,k ,k,k e2 . 2 r qA (4.1.14) Inspecting the Coulomb terms, it is clear that these terms diverge in the limit of q = 0. The divergence can be resolved within the Jellium model [38], where the semiconductor is assumed to be intrinsically charge free. As a result, the diverging term of the electron-electron interaction at q = 0 is canceled by the diverging term of the interaction of electrons with the homogeneous positive charge background. Hence, terms q = 0 within the summation over q are discarded. 4.1.2 Diagonal Approximation The next approximation performed aims to remove two of the four subband indices in (4.1.8) by assuming that Coulomb interactions will not cause interband transitions. Then, n1 = n4 and n2 = n3 , which is called here diagonal Coulomb approximation. This assumption is for example justified at an approximate level in symmetrical quantum wells, as the diagonal Coulomb matrix elements are due to the symmetry of the wavefunctions, two orders of magnitude larger than the off-diagonal [45]. Also, for single-band effective mass wavefunctions, only the diagonal terms remain, while others vanish due to the orthogonality of the zone-center functions. Using this approximations, the Coulomb Hamiltonian reads ˆ2 = 1 H 2 m,n Θm,n,n,m a ˆ† ˆ† ˆ nk a ˆ mk . q,k,k mk+q a nk −q a kk q (4.1.15) Consequently, the notation Θm,n,n,m can be reduced to Θm,n q,k,k q,k,k . 4.1.3 Introduction of Holes The next step is to transform the Coulomb Hamiltonian into the electron-hole picture. In the diagonal approximation (4.1.15), one distinguishes between electron states in the conduction bands c, d and hole states in the valence bands v, w. Therefore, the following four combinations are possible n = c, m = d n = c, m = v n = v, m = c n = v, m = w. (4.1.16) As next, hole operators are inserted for electrons in the valence band. Further, a natural ordering is reestablished by propagating the creation operators to the left and the annihilation operators to the right and setting the hole creation operator before, the hole annihilation operator after the one of the electron. The terms involving the sum n=c,m=d are already in that ordering. Permuting the hole terms in the sum n=v,m=w introduces a new two-operator term while the sums n=c,m=v and n=v,m=c can be summed together. Hence, the final Hamiltonian 58 Chapter 4. Coulomb Correlated Optical Transitions which will be used to calculate the microscopic equation of motion is obtained as ˆ H = c,k Ec (k)ˆ a† ˆck − ck a v,k ˆ Ev (k)ˆ b† v k bv k (4.1.17) (4.1.18) (4.1.19) (4.1.20) (4.1.21) ˆ† ˆ ˆ ck . a ˆ† ck+q bv k −q bv k a (4.1.22) −E(r, t) · c,v,k ∗ ˆ† ˆ µcv,k a ˆ† ˆ ck ck bv −k + µcv,k bv −k a + + 1 2 1 2 Θc,d ˆ† ˆ† ˆdk a ˆ ck q,k,k a ck+q a dk −q a c,d kk q ˆ† ˆ† ˆ ˆ Θw,v −q,−k +q,−k−q bv k+q bvk −q bwk bv k v,w kk q ˆ† ˆ Θv,v −q,−k +q,−k bv k bv k v,k ,q − 1 2 c,v Θv,c −q,−k +q,k + Θq,k,q−k kk q The indices k and k have in some cases been swapped, shifted by q and occasionally changed sign. 4.2 4.2.1 Transitions Calculation Equations of Motion The next task is to evaluate the equation of motion (3.2.20) for p ˆnm,k , as it has already been done in previous chapter, excluding the Coulomb interaction. Therefore, the remaining task is to evaluate the commutator of the microscopic polarization and the Coulomb Hamiltonian, i ˆ d p ˆnm,k = H2 p ˆnm,k + rest dt (4.2.1) ˆ 2 are the Coulomb terms in (4.1.17). The evaluation is straight forward but lengthy: For each four-operator where H ˆ 2p ˆ 2 and then permutes p term, one writes H ˆnm,k − p ˆnm,k H ˆnm,k in the first term to the left to obtain the remaining terms which do not cancel finally. The remaining terms are resorted at the end into the natural ordering. The resulting equation of motion is given by d p ˆnm,k dt i = − Em (k) − En (k) − q Θn,n −q,−k+q,k p ˆnm,k (4.2.2) (4.2.3) (4.2.4) (4.2.5) (4.2.6) (4.2.7) (4.2.8) i − E(z, t) · µmn,k c a ˆ† ˆ mk + ck a v ˆ ˆ b† v −k bn−k − 1 + + − − + i 2 i 2 i 2 i 2 i 2 c,m ˆ Θm,c ˆ† ˆck a ˆmk+q −q,k+q,k + Θq,k ,k+q a ck +q bn−k a c,k ,q n,v ˆ† ˆ ˆ mk ˆ Θv,n bn−k−q −q,−k +q,k + Θq,k,−k +q bv k −q bv k a v,k ,q c,n ˆ Θn,c ˆ† ˆck a ˆmk −q,k,k + Θq,k ,k a ck +q bn−k+q a c,k ,q m,v ˆ† ˆ ˆ ˆmk−q Θv,m −q,−k +q,k−q + Θq,k−q,−k +q bv k −q bn−k bv k a v,k ,q m,n ˆ Θn,m ˆmk−q . −q,k,k−q + Θq,k−q,k bn−k+q a q k .20) and (4.22).27) reveals that in (4. Comparing this equation with the free-carrier result (3.2.2.2.9) The values usually inserted into Ev (k) are results of a preceding band structure calculation.8) are the results of the commutator with the conduction-valence terms (4. Therefore.2.6)-(4.8) is obtained at the end when (4.2.5) are the result of the equation of motion of the conduction-conduction and valencevalence terms (4. Figure 4.4.2: Schematic representation of the Hartree-Fock approximation operator product in the equation of motion. While permuting these operators. while it appears here explicitly. where the valence band is full [40]. all four-particle operators are factorized into all possible products of two-particle operators. [40].2. For this reason. the Ev (k) actually already holds the Coulomb energy of the full valence band. 4.26).9) of ref. (3. Therefore.1. the correct sign due to the anti-commutation relations has to be tracked. The free-carrier calculation implicitly included this energy within the band structure result. Due to the random-phase approximation (3. (4.n −q.2.2 Hartree-Fock Approximation In the next step.2).1.1.2. Taking the two-band approximation of this particular equation and ignoring the k dependence of the Coulomb matrix element results in Eq.2. Factorizing the four-particle operator into two two-particle operators yields the Hartree-Fock approximation. of which the expectation values are taken. particle densities nn.1.19). (4.k or to zero. Transitions Calculation 59 Figure 4. The term (4. one usually truncates the hierarchy at a certain order [40]. it would be necessary to calculate the equation of motion of the four-particle operator term which would couple to a six-particle operator term: one obtains an infinite hierarchy of equations.k obviously depends on the fourparticle operator terms. We summarize this approach for a two-operator expectation value . The obtained equation of motion for the two-particle operator term p ˆnm. Similar equations of motion can be obtained for the carrier population operators The terms (4.2.21). as demonstrated in Fig.4) and (4.2. the valence band energy is shifted to En (k) → En (k) − q Θn.7) is brought into proper order. all expectation values of two-particle operators reduce either to microscopic polarizations pnm. which is considered in the next section.k . The terms (4.1.1: Infinite hierarchy of operator products for the equations of motion.2. 4. where parameters are obtained from low excitation experiments.−k+q. k−q pnm. the factorization of (4.k + q Θn.k+q pnm.2. . Here. In practice.k . As an example.2.k + Θn. t) · µmn.n q. Inspecting (4.k E(z.k + q Θn.k+q The sum term in (4. Coulomb Correlated Optical Transitions d AB dt = = d AB dt d AB dt HF + + d d AB − AB dt dt d AB dt col. (4.2). ∂t and the renormalized transition energy is given as ω ˜ mn (k) = Em (k) − − q En (k) − q Θn. (4.4) and (4.2.12) and so on.2.k. we truncate the hierarchy at some point.k−q .2. t) = E(z.m −q. This coupling introduces the excitonic effects such as enhancing gain or absorption. the Heisenberg equation of motion gives the equation of motion for ABCD as d d ABCD = ABCD dt dt F + d d ABCD − ABCD dt dt F . We can continue by deriving the equation of motion for ABCDEF ≡ d d ABCD − ABCD dt dt F (4.5) can be added to the transition energy. Finally.k. The factorized terms of (4. The label F is used to indicate where dt the result from a Hartree-Fock factorization of the four and six operator expectation values. (4.15) . while the remaining terms are proportional to the inversion factor.k.4) leads to ˆ a ˆ† ˆck a ˆmk+q ck +q bn−k a HF = ˆ a ˆ† ck +q bn−k − a ˆ† ˆ ck ck +q a a ˆck a ˆmk+q ˆ bn−k a ˆmk+q ˆ bn−k a ˆ ck + a ˆ† ˆmk+q ck +q a = 0 − δq0 + δk k δcm nm. The term AB col. Other terms are factorized accordingly.k−q (nmk + nnk − 1) (4.1). t) · µmn.11) d ABCD contains expectation values of products of up to six operators. HF HF (4. 4. The expressions for the carrier distribution operators n ˆ nk and n ˆ mk can be obtained similarly.13) reveals that the microscopic polarization pnm. By introducing renormalized Rabi frequency Ωk (z.−k+q.2.k. the equation for the microscopic polarization in the Hartree-Fock approximation is obtained as d pnm.2. while in the free-carrier expression.10) where AB HF is the Hartree-Fock contribution to the equation if motion which will be treated in this chapter (see Fig.1.14) is called the exchange shift term.k dt = − − + i i ω ˜ mn (k)pnm.k+q .m q.14) Θm. The higher-order correlation (collisions) contributions are denoted similarly to previous chapter with the subscript col. where each succeeding equation describes a correlation contribution that is of higher order than the one before.2.n −q.m −q. The result is a hierarchy of equations.k at k is now coupled to the one at k by the Coulomb interaction.13) ∂ pnm.60 Chapter 4.2.k+q nm.k−q pnm.2. us a higher-order contribution containing four-operator expectation values and they will be discussed briefly later in this chapter.k |col.2. the microscopic polarization was entirely determined by the momentum matrix element and the inversion.k+q nn. we only mention that with the full many-body Hamiltonian (3. This implies ∂ ∂t pnm.2). Transitions Calculation 61 we can rewrite the semiconductor Bloch equations as d pnm. ω ) (4. .2.16)-(4.k that we can model with a relaxation time γ .3.2.16) (4. With the introduction of k · p states.29. The presence of unbound carriers of the electron-hole plasma leads to an adjustment of the carriers to a charge.k |col.1 Background Screening In the crystal electron Hamiltonian (2. Of course.2. t)p∗ nm.1. . only a restricted number of valence electrons are considered.8).k + ∗ i Ωk (z. the interaction between ion cores and the interaction between electrons and ion cores is merged into the potential U (r).k − iΩk (z. We can model this. i.k |col.2.k dt d n nk dt d nmk dt = = = −iω ˜ mn (k)pnm. ∂t (4.2 Plasma Screening A profound weakness of the Hartree-Fock approximation is the lack of screening of Coulomb interactions at elevated carrier densities. while core electrons are included implicitly. ∂t ∂ + nmk |col. effectively screening it and thereby reducing the Coulomb interaction energy.e. 4. ∂t The second term in (4.3.2. t)pnm. and they couple equations for different k states.2.2. nek = fek and nhk = fhk . The lack of screening in the Hartree-Fock equations is caused by the early truncation of the infinite hierarchy of equations of motion. .18) ∂ nnk |col.4. assuming that the conduction band and the valence band are two carrier reservoirs. t) (nmk + nnk − 1) + ∗ i Ωk (z. The semiconductor Bloch equations look like the two-level Bloch equations. the semiconductor Bloch equations reduce to the undamped inhomogeneously broadened two-level Bloch equations [40]. (q.2.18) in comparison to the corresponding free-carrier equations. the collisions between particles and the plasma screening. with the exceptions that the transition energy and the electric-dipole interaction are renormalized. 4.2.2. while the Coulomb energy between electrons has been retained and later transformed into the Bloch states.2.16)-(4. The main consequences of the collisions between particles are: • A decay of the microscopic polarization pnm. = −γpnm. The renormalizations are due to the many-body Coulomb interactions.2. The procedure is outlined in this subsection.1. The effect can be considered by replacing the dielectric permittivity of the vacuum 0 with the static semiconductor crystal background permittivity b in the Coulomb matrix element (4.k − Ωk (z.k − Ωk (z.2. t)p∗ nm. This coupling leads to significant complications in the evaluation of (4.2.3 Many-Body Effects There are two important many-body effects that do not appear in the Hartree-Fock limit of the equation of motion. and the carrier probabilities nnk enter instead of the probability difference between upper and lower levels. It is more difficult to include the plasma screening effect into (4. ω ) Vsq = Vq . t)pnm. (3.27)-3. If all Coulomb-potential contributions are dropped.k . These implicitly included core electrons and ion cores screen the Coulomb interaction between explicitly included electrons. They occur at high carrier densities. this limit is unacceptable for semiconductors. which are Fermi-distributed.15) is called the Coulomb field renormalization.17) (4.19) .2. One therefore introduces the screening effect phenomenologically using a dynamic dielectric function (q.k ∂ pnm. 4.18). • A rapid equilibration of electrons and holes into Fermi-Dirac distributions. including the factor due to the finite extend of the wavefunctions.k There are several far better models [48] for the dielectric screening but the Lindhard formula is still popular and commonly used in optoelectronic modeling [40] due to its simple structure. 38].n4 Θn = s.k−q − nn. where all have their strengths. Such problems can only be avoided within a systematic many-body approach. 1 Note that the Coulomb interaction Hamiltonian with the bare Coulomb potential already contains the mechanism for plasma screening. limitations and weaknesses.13). or more profound.3: Outline of an approach that takes advantage of a phenomenological derivation of plasma screening. one should be concerned that an ad hoc replacement of Vq with Vsq might count some screening effects twice. To use Lindhard’s formula in optical calculations. 0) is obtained. ω ) = 1 − Vq nk nn. q.n3 .q.21) Beside its physical importance.k (4. The outline of the screened Hatree-Fock approach is presented in Fig. Coulomb Correlated Optical Transitions Figure 4.2. The Lindhard formula can be derived using the self-consistent field (SCF) approach [46]. the screening introduces the desired effect of removing the divergence of the Coulomb potential with q → 0. the damped response of the screening is ignored. where Vsq denotes the screened potential and Vq the unscreened one1 .2.n4 .n2 .n3 . (4. the test charge potential Vq is replaced with the Coulomb 1 . For many practical applications. by neglecting vertex corrections in the equation of motion for the Green’s function [47] of the designated system and using the Kadanoff-Baym formalism [47]. therefore setting (ω + iδ) to zero.62 Chapter 4.k. 0) q.20) also denoted as randomphase approximation. (q. . There are several model dielectric functions.k .k 1 Θn1 .n3 .n2 . thereby facilitating the numerical evaluation.k. Therefore. One standard approach is the Lindhard’s formula [40.3).k.n4 potential Θn defined in (4. En (k − q) − En (k) + ω + iδ (4. one replaces the Coulomb potential within all equations with 1 . (q.n2 . Once (q. which is incorporated directly into the many-body Hamiltonian to give the screened Hartree-Fock equations (after [40]).1. The derivation using the SCF approach involves the reaction of a homogeneous electron gas to a test charge (see detailed derivation in Appendix D). 24 and 4. the Coulomb interaction is screened Θn. (4.24) The shift is denoted as Debye shift or Coulomb-hole self energy. 4.2. taking the time derivative. 4.k = .2. En (k) → En (k) − Θn. the valence band energy was shifted by a constant value. 0) and the difference between the unscreened and the screened Coulomb interaction leads to a density dependent shift ∆ECH.3 Coulomb Hole Self Energy By replacing the electron operators with hole operators in (4. Therefore.22) −q.2. the transition energy (4.m q.4 Solving the Correlated Equation In order to solve the equation of motion (4.k.n = (4.13).−k+q.k e−i(k0 z−νt−φ(z)) . compared to the free carrier result.26) Note that both contributions 4.m q.14) is. Therefore.k+q nm.k+q snm. ∆ESX. skipping the fast oscillating parts (as they should average out to zero) and solving for the steady state leads to snm. (4.23) s. In conclusion.2. the field independent solution variable is introduced.k+q .13) couples microscopic polarizations pmn.n −q.−k+q. (4.29) E (z ) .n − ∆ESX.−k+q. the equation system has to be solved self-consistently.3. within the screened Hartree-Fock limit.10).k.2.2.25) where the screened Coulomb potentials are used.2.27) into (4.k = − i 1 i (˜ ωmn (k) − ν ) + γ E (z ) · µmn.2. (4.3.2.n −q.k.k Θn.2. (q) (4.4.k .mn (k) = q n. reduced by the density dependent screened exchange shift. Transitions Calculation 63 4.27) Inserting (4. the equation is transformed using the same ansatz as in Chapter 3. (4.k (q. using the plane wave expression for the electrical field (3.k+q + Θq. given by 2snm.25 reduce the transition energy and can lead to a significant red shift.n = q Θn.2.n (4.2).2.−q.2.−k+q.2.k = pmn.4 Screened Exchange Energy Beside the Coulomb-hole self energy.k λnm.k . the transition energy is therefore renormalized ω ˜ mn (k) = ∆Emn (k) + ∆ECH.28) The polarization envelope still depends on the intensity of the considered field.2. As equation (4.13).2. In the high density limit. The fast oscillating pmn. the collision term is approximated using a decay rate −γpmn.k of different k values.k 1 −1 .2.k+q 2 · (nmk + nnk − 1) .k+q nn.2.2.2. q This shift is already implicitly included in the band structure.n Θm.k + Θn.mn (k).k is replaced with its slowly-varying envelope snm.2. n and the screened exchange shift energy ∆ESX. nb 0 E (z ) nb 0 V n. As an illustration. the lineshape and the inversion factor are combined into Λnm.n = q=0 Θn.2.k − ν ) + γ (4. Thus.k = −µmn.2. t) (fe (4. 4. surrounded by Ga0. we only need to solve the equation of motion for the polarization p ˆnm.64 Chapter 4. 4. and the chemical potentials. (4. using the trapezoidal method for integration.m q.k λnm.k .2.k q (q) = q − . In the next section we present the numerical implementation details of the solution.4 presents result of the calculation for the toy structure presented above with variable 2DEG concentration ne = 1 × 109 ÷ 1 × 1011 cm−2 .2.|k−q| − fn. are calculated as described in section 3.2.2. and illustrate the results for a 200 wide GaAs well.33) k + fhk − 1) − γpnm. • It is more convenient to evaluate q (q) than only (q). As can be seen. As the first step for the calculation.34) 2 rA En (k − q) − En (k) nk Converting the sum over the k-states into an integral.2. the elelctrical susceptibility is calculated using 1 1 1 P (z ) = 2 µ∗ (4.4.1 As barriers with δ -doped layers introduced at 1000 away from the barrier-well boundaries. as before. the electron and hole densities are given as input parameters.29) and (4.k . EF c and EF v .−k+q.30) This equation is the self-consistency equation for λnm.k − iΩk (z.1.37) = dq Gn. Using the decay model for the microscopic polarization we obtain d n m pnm.2.2. involved in the transition. Once solved.36) The integrals can be evaluated here.32) χ(ν ) = 2 mn.k Absorption.k−q − fn. we consider the conduction band and the valence band as two carriers reservoirs with Fermi-Dirac distributions for the electrons and holes. adding the sum over the spin.n q where the parameter n denotes the hole subband.mn (k) must be evaluated. Coulomb Correlated Optical Transitions In order to obtain a practicable expression. refractive index change and spontaneous emission can therefrom be calculated as already presented in the preceding Chapter 3.31) i (nmk + nnk − 1) . we have from (4. Here we elaborate on the practical numerics for the calculation of these terms (and others).k+q λnm.2.30) into (4.k . choosing θ as the angle between k and q and assuming no angular dependence for the subband structure.k.2.k e2 4π r ∞ 0+ 1 −1 (q) 1 −1 .k Λnm. With Fermi-Dirac electron and hole distributions.n −q.k Θn. we obtain q (q ) = q − with |k | = |k − q| = k 2 + q 2 − 2kq cos θ e2 4π 2 ∞ 2π dk k r n 0 0 dθ fn. • For the Coulomb-hole self energy with Fermi-Dirac distributions we can write ∆ECH. respectively.5 Implementation Considerations As a consequence of the collisions between particles in the assumed equilibrium state. .2.35) . yields Λnm.k+q + λnm.k .k = Inserting (4. As for the free carrier model.9 Al0. (q ) (4.m. En (|k − q|) − En (|k|) 1 2 (4. i (˜ ωmn.k .|k| .2. (4. influence which extends with higher gas concentrations. the Coulomb-hole self energy ∆ECH. (4.k = −iω ˜ mn (k)pnm. dt where n and m denote the conduction and valence subbands. Fig. the dielectric function (q).20) e2 fn.28). the screening effect mainly influential at low wavevector values. 1 As well with 2DEG concentration of ne = 1 × 1011 cm−2 .nm (k) = where |q| = |k − k | = k 2 + k 2 − 2kk cos θ e2 8π 2 ∞ 2π dk r 0 0 dθ k n m. 4.9 Al0.5 presents the calculated values of the Coulomb-hole self energy and screened-exchange energy shifts for various 2DEG concentrations.6 presents the dispersion relation of the total bandgap renormalization (BGR) energy. In this calculation we consider the transitions between the first conduction subband and first four valence subbands.2.k (z )| e−q|z−z | .k (z ) is the envelope function of the specific subband considered.41) where Fn. If we convert the sum over k into an integral with θ defining the angle between k and k and we assume a rotationally symmetric band structure. 4. In all the expressions above we use (4.q fe.n s. . we obtain ∆ESX.m m Θn.k . ∆ECH.k + Θs.nm (k).8 presents the calculated form factors used throughout the examples presented above. 4.m − ∆ESX. Fig.14) to describe the coulomb interaction for the 2D case. (4. both shifts are negative in nature and their values increase (in an absolute value) with the rise in the gas concentration.nm (k) = k =k n m.38) The band indices m and n are defined as before for the interacting subbands. calculated for the 200 Å wide GaAs/Ga0. 4. Transitions Calculation 65 200 180 160 140 120 εq 100 80 60 40 20 0 N2DEG 0 0.m m Gn. The form factor is explicitly calculated using ∞ ∞ Gn.5 x 10 2 8 Figure 4.2.2. Fig.40) Fig.39) .2. As predicted. The exponential fit of the calculated values is also presented for T = 2 K (red dashed curve in (a)). • The screened-exchange shift energy becomes ∆ESX.7 presents the total bandgap renormalization for the same structure for various 2DEG concentrations for ambient temperatures T = 2 K and T = 77 K (respectively in (a) and (b)).9 Al0.m = q −∞ dz −∞ dz |Fn. Fig.5 1 q (cm−1) 1.4.k + Gq q (q ) 1 2 (4.2.k q fe.n fh.4: The screening dielectric function (q ) calculated for the 200 Å wide GaAs/Ga0. 2 2 (4.1 As well with ne = 1 × 109 ÷ 1 × 1011 cm−2 at T = 2K .k (z )| |Fm. The renormalization value grown (in absolute value) with the concentration because of the state filling in the conduction subband. where the expected [49] ∼ 1/3 exponential dependence is clearly seen.1.q fh. (4. nm(k||=0) (meV) n.6 −4.4 −4. m=1 n=1.m=1.8 ∆ESX. −3.m=1.6 −3.4 −3.n (meV) −2 −3 −4 −5 (b) 0 n.6: The bandgap renormalization dispersion for the various subband transitions calculated for the 200 Å wide GaAs/Ga0.9 Al0. m=4 Figure 4. .5: Calculated (a) Coulomb-hole self energy and (b) screened-exchange shift energy at the Brillouin zone center (kt = 0). for the 200 Å wide GaAs/Ga0.1 As well with various 2DEG concentrations for various conduction and valence subbands.2 −4.4 0 1 2 3 k (cm−1) 4 5 x 10 6 6 n=1.2 −5.nm(k)+∆ECH. Coulomb Correlated Optical Transitions (a) 0 n=1 n=2 n=3 −1 ∆ECH.2 n.1 As well with ne = 1 × 1011 cm−2 .m (meV) −4 −4. m=3 n=1.m=1.66 Chapter 4. m=2 n=1.1 −1 ∆ESX.8 −5 −5.3 −2 −3 −4 −5 9 10 10 10 10 N2DEG (cm−2) 11 10 12 Figure 4.9 Al0. which leads to i wk µk E ( z ) 1 .2.9 Al0. q (q ) (4.48) .28) d snm. The last difficulty resides in Ωk (z.k µk . χ0 k (4.3649 − 0.m Γn. and the Coulomb enhancement factor Γn.k χ0 (ν ).k dt = − (i (˜ ωn.m (4.k has no angular dependence 1 µk where εk.m q .4. Substituting these two expressions into (4. k Γk (4.2. The blue dots denote the calculated values while the red dashed curve in (a) is the result of an exponential fit (were the fitted function is given in the legend box). (4.3528 10 10 (b) 0 −1 −2 −3 −4 −5 −6 −7 −8 −9 9 10 1 :HH 1 ∆Eg.k where defined in the previous section. t). To obtain susceptibility spectrum of a medium.k = 0 n. the k-dependent complex susceptibility function χ0 k = −Λnm.2.2.33).2.k = − + Θn.m dk εk.m s.k i (˜ ωn.k . for the 200 Å wide GaAs/Ga0.2.2.q snm.44) (4.2. k (ν ) = 1 + µk k=k We consider the last part of this equation and transfor the sum over k into an integral as before.m = k λnm.2.q χk Γk (ν ).2.2.45) where Λnm.m Θn. we assume that the microscopic polarization pnm. With the knowledge of these terms.k and λnm.k and to work with snm.1 As well with various 2DEG concentrations for the E1 − HH1 transition for (a) T = 2 K and (b) T = 77 K ambient temperatures. We define two new variables.m s.7: The total bandgap renormalization at the Brillouin zone center (kt = 0). The first step is to eliminate the rapidly varying phase factor from pnm.43) s.638×10−4 N0. E 10 N2DEG (cm ) −2 11 10 12 ∆Eg.k as in (4.m (k) − ν ) + γ 2 µk k=k i µk E (z ) wk 2 (4. E 1 :HH 1 10 10 10 N2DEG (cm−2) 11 10 12 Figure 4.42) is solved for the steady state (as in the previous section).m (k) − ν ) + γ ) snm. Furthermore. Transitions Calculation 67 (a) −1 −2 −3 (kt=0) (meV) (kt=0) (meV) −4 −5 −6 −7 −8 −9 9 10 Calculated −4. we can now concentrate on (4.2.k .46) s. snm.43) we obtain 1 0 n.q snm.m Θn.47) e2 k 2π 2 r 2π dθ 0 Gn.q χk Γk (ν ) = k=k 1 µk ∞ 0 n.k − i − wk k=k n m where we defined wk = fe k + fhk − 1.m (4.42) Θn. .8 Gee(q) 0. m=3 n=1.8: Calculated form factors for the (a) electron-hole.6 0.4 0. m=4 0 1 2 3 4 5 x 10 6 6 (b) 1 0.68 Chapter 4. m=1 n=1.8 Ghh(q) 0.2 0 n=1.4 0.8 Geh(q) 0. m=2 n=1.6 0. Coulomb Correlated Optical Transitions (a) 1 0.6 0. (b) electron-electron and (c) hole-hole combinations.2 0 m=1 m=2 m=3 m=4 0 1 2 3 q (cm−1) 4 5 x 10 6 6 Figure 4.4 0.2 0 0 1 2 3 4 5 x 10 6 6 (c) 1 0. m n.2. (4.m ˜ n.1 Ga0. The main advantage of this model is the elimination of the phenomenological modeling of the microscopic polarization decay through the use of γ parameter.2.2.k 1 1 k1 k1 . this model much more demanding computationally and thus ineffective for our purposes. Numerical Results 69 Inserting (4.2. ∆k − µk εkn .m (k) − ν ) + γ k n.32) and using the definitions in chapter 3. we investigate the optical properties of a 200 Å wide GaAs well.3. in-spite of its tendency to be rater resource consuming.m ( ν ) c 0 nb π L i (˜ ωn. .49) can thus be written in a matrix form M · Γn. ··· ∆k εk1 . Eq. .m = 1.50) can now be solved using standard mathematical methods to obtain Γn.2.k | Γn.2.2.2.55) where Γn.m Γn. Γkn T (4.53) into (4. 50]. . In this chapter we have discussed the screened Hartree-Fock approximation and its consequences for the absorption spectra.54) rsp.m k k1 . .m dk εk.m (ν ) = 1 + k 1 µk ∞ 0 n. absorption and spontaneous emission properties of this structure 2 An alternative choice for ∆k is to take the elements of the k vector as the support of a Gaussian quadrature [38].k 2 n kn k2 . ∆k 0 ε −µ kn . ω length if the 2D quantum structure. .k2 χ0 −µ k2 k1 ∆k 0 1− µ ε χ k . 4. The microscopic polarization can be obtain using 1 n.2.k | 0 2 wk Γn. refractive index.m (k) − ν ) + γ k (4.m (k) is the renormalized bandgap defined in (4.kn χ0 kn n Eq.52) M= ∆k εk1 .49) Various approaches exist in the literature to solve this self-consistent equation [38.15) and (3. .k χ0 (ν ). (4.53) k Γk 2 Inserting (4. · · · . We choose to use the most general matrix inversion method. We investigate elelctrical susceptibility. . we can write the expressions for the absorption (or the gain amplitude) and spontaneous emission spectra iν wk 2 αHF (ν ) = − |µnm.m = Γn. .m. 2 3 2 π c 1 − exp (( ν − (EF c − EF v )) /kB T ) (4.m ∞ dk k |µnm. surrounded by Al0.2. (3.k = e−i(k0 z−νt−φ(z)) E (z )χ0 . On the other hand.k 2 2 k2 k2 .6). To this end we approximate the integral over the k space by a sum of trapezoidal areas and discretize k and k with n values and constant step size ∆k 2 .2. It represents a simple model which accounts for the plasma screening but not for the particle collisions. as apposed to various approximate methods (such as Pade and the dominant momentum methods).50) (4.k2 χ0 k n 2 ··· ··· .3 Numerical Results In this section we present the results of the numerical modeling of sample structures using the models presented in this and previous chapters.9 As barriers with δ -doped layers introduced at 1000 Å away from the barrier-well boundaries.47) into (4. (4. ∆k 1 − µk εkn .m pnm. As a next level of complication.kn χ0 −µ kn k1 ∆k 0 −µ ε χ k .k = − and iν c 0 nb π L n.4.2. Γk2 .2.k1 χ0 1− µ k1 k1 ∆k 0 −µ ε χ k .46).HF (ν ) = − 2 1 n2 b ( ν) αHF (ν ). As before.26). (4. k with and n.2.51) .. next higher level of approximations can be computed referred to as Coulomb correlations.3.50).m k .m (ν ) i (˜ ωn.k1 χk k n 1 ∆k εk1 .2.2. and L is the k (ν ) are the solution of (4. we obtain Γn. k Γk (4. 13. In addition. 4. the materials are assumed to have a circularly symmetric crystallographic structure along the [100] orientation.14) and (3. (3. the results of both models are plotted in the same scale for comparison (using equations (3. presented in chapter 3. respectively). and the screened Hartree-Fock (HF) model. Coulomb Correlated Optical Transitions ݈ܣ.11 present the same calculation of χ performed using equations 3. the phenomenological broadening factor γ is taken to be 1 × 1011 sec−1 for all calculations.2. 4.12 presents the calculated absorption spectra for the two polarizations.2. This transition lines are the direct consequence of the incorporation of Coulombic interactions into the calculation. Finally. We perform the susceptibility calculation for this structure for two ambient temperatures. LH1 and HH2 ) are used throughout the calculations.70 Chapter 4. 4. only the first conduction subband (E1 ) and the first three valence subbands (HH1 . using the free-carrier (FCT) model. we investigate the influence of the various phenomenological parameters of the models on the optical properties. The interaction strength enhancement is clearly seen by comparing the absolute value of the absorption spectra calculated using the two models. 3. Calculation results for both TE and TM polarities are shown (figures (a) and (b).14 and 4. Along with the variation of the physical parameters of the discussed models. We can compare these results to the simplified excitonic spectral .13).2.9. presented earlier in this chapter.3.33) of the FCT model. while the insets are given to show a zoomed-in view of the high energy region of the spectra.2. As before.53 of the HF model.2. The most striking difference between the results of the two models is the appearance of strong transition lines bellow and above the bandgap for the HF case. Another prominent feature of the HF model results is the energetic shift towards the lower energies brought by the renormalization effect discussed earlier in this chapter. at various ambient temperatures and doping conditions. without a 2DEG presence in the well region. unless noted otherwise. Here. and for each of these the real and imaginary parts of the complex susceptibility are plotted on separate axis.2. We model this situation by introducing holes and electrons of equal and low concentrations (ne = nh = 1 × 106 cm−2 ) into the well region. unless noted otherwise. Fig. using equations (3.2. For all the considered calculations throughout this chapter and the entire thesis. The chosen value was obtained from a comparison of the calculated spectra to the experimental linewidths for similar structures. Fig.54 respectively for the FCT and HF models).35) and 4. The dashed vertical line marks the energetic bottom of the first conduction subband at this temperature.ଵ ܽܩ.e.10 presents the calculation of the complex susceptibility χ. The schematic depiction of the structure is given in fig. T = 2 K and T = 77 K . i.2. the dashed vertical lines mark the bottom of the first conduction subband. Fig. The legend for the coloring of the various layers is also given.9: Schematic structure of the undoped QW examined in the numerical calculations. 4. The relative energetic shift can be also clearly seen.1 Bare Quantum Well We start by considering the optical properties of an undoped structure.ଽ ݏܣ ݏܣ݈ܣ ݏܣܽܩ 1119Հ 200Հ Figure 4. 4. 524 1.528 1.4.3.01 0 0.532 1.005 0 −0.534 1.01 0.526 1.014 0.53 E (eV) 1.524 1. calculated using the FCT model for the undoped well at T = 2K . The two plots are the (a) TE and (b) TM polarization calculations.53 1.01 0.52 1.534 1.025 0.006 0.01 0.538 (b) 0.5323 eV 0.03 0.025 0.02 0.538 0 1.522 1.522 1.526 1.005 −0.005 −0. real (blue) and imaginary (red) parts. Numerical Results 71 (a) 0.52 1. .04 1.016 1.528 1.532 1.008 0.5364 eV 0.54 ℑ[χTE ] FCT ℑ[χTM ] FCT 0.54 0.536 1.01 1.10: The complex elelctrical susceptibility.012 0.5364 eV 0.02 ℜ[χTM ] FCT 0.004 0.015 ℜ[χTE ] FCT 0.015 Figure 4.03 1.01 1.02 0. χ.002 0 1.536 1. 72 Chapter 4.3 0.1 0 1.2 0.4 1.526 1.534 1. calculated using the HF model for the undoped well at T = 2K .5376 eV 1.2 1.7 0.6 ℜ[χHF] 0 0. Coulomb Correlated Optical Transitions (a) 0.4 0. The two plots are the (a) TE and (b) TM polarization calculations.4 1.528 1.5316 eV −0.4 0.2 −0.8 0.528 1. real (blue) and imaginary (red) parts.533 eV 1.5344 eV 1. χ.5275 eV −0.8 0.11: The complex elelctrical susceptibility.6 0.534 1.3 0.4 1.54 (b) 0. ℑ[χTE ] HF TE .2 0.2 0.4 1.536 1.5354 eV 1.532 1.3 −0.522 1.532 1.526 1.538 0 1.524 1.536 1.52 1.524 1.5 0.1 ℜ[χHF ] TM 0.1 −0.2 0.5242 eV 0.53 1.53 E (eV) 1.5306 eV 1.52 1.538 Figure 4.522 1.54 ℑ[χTM ] HF 0 −0. 522 1.8 x 10 5 (b) 4 x 10 5 1.528 1.538 1.534 1.538 1.538 Figure 4.2 0 1.6 1.534 1.536 1.3.524 1.54 3 2 1 0 1.6 0.522 1.532 1.526 1. Numerical Results 73 2 1.534 1.53 1.54 0.532 1.532 1.4 αTE (cm−1) 1.53 E (eV) 1.2 1 0.534 1.52 1.536 1.6 4 1.536 1. .4 αTM (cm−1) 1.4.5 1 4 x 10 1 0. calculated using the FCT (blue) and HF (red) models for the undoped well at T = 2K .12: The absorption spectra for the (a) TE and (b) TM polarizations.2 0 1.2 x 10 5 (a) FCT HF 2 1.536 1.528 1.4 0.52 1.8 0.53 1.5 0 1.6 0.8 0.8 1.524 1.532 1.4 0.538 2 1.53 1.526 1. nb c (2πa0 )2 eπ/ √ ∆ √ ∆ (4.12(a) can be easily seen.4) is the Coulomb enhancement factor. the electrical susceptibility can be written as [38] χ(ν ) = − + |µeh | Lπa2 0 E0 dx 2 ∞ n=0 (n π/x 2 + 1/2) 3 E0 (ν + iδ ) − Eg − En (4. Here the energetically lowest lying line at 1. The influence of the Coulomb enhancement term is clearly seen in the calculated plot.3. The curves presented thus far are all given for both light polarizations. The similarity between this result to our results in figure 4. the main difference is the appearance of low energy (near 1.10 and 4.3. which shows that the attractive Coulomb interaction not only creates the bound states but has also a pronounced influence on the ionization continuum compared to the free carrier case. where the most noticeable example is the difference between the sub-bandgap transition lines.13. The first exciton in the computed absorption spectra has been scaled by a factor of 0.1) xe E0 . For the TE polarization both E1 − HH1 and E1 − LH1 are dominant for the low kt values (which is region relevant for our calculations) compared to the single E1 − LH1 transition dominant for the TM case. theory presented extensively in literature.2) cosh eπ/ 1 4π 2 2 |µeh | A.74 Chapter 4.3.2. In the HF model results the differences between the polarizations are mainly exhibited in the transition strength of the various lines.12) compared to the TM case. Coulomb Correlated Optical Transitions Figure 4. Shown are the results obtained with and without including the Coulomb interaction. This additional transition is a result of the difference between allowed transitions for the two polarizations. (4.3. This effectcan be . TE and TM.1 (after [38]).532 eV ) transition peak in the TE absorption spectrum (see figure 4. cosh(π/x) (ν + iδ ) − Eg − E0 x2 and the resulting absorption spectrum is given by the Elliot formula ∞ 4 1 ν 2D δ ∆+ α(ν ) = α0 3 2 E0 n=0 (n + 1/2) (n + 1/2) where 2D α0 = + Θ(∆) eπ/ √ ∆ √ ∆ .13: Schematic (left) and calculated (right) band edge absorption spectrum for a 2D semiconductor.4843 eV (the E1 − LH1 transition) exchange their relative strength between the two polarities. as seen in figures 3. when comparing the dashed (free carrier) to the solid (elliot formula) curves.3) The term C (ν ) = cosh eπ/ (4. and we can easily detect the differences between the two cases. For the 2D case. 4. A schematic and calculated absorption spectra obtained using the results above are plotted in Fig. For the FC model case.4811 eV (attributed to the E1 − HH1 transition) and the next line at 1. occurring above the Fermi edge energy. for the TE and TM polarizations. To facilitate the comparison.1 in appendix E). The absorption. for both polarizations. As for the absorption spectra.54 Figure 4. The cause for these variations in the relative line strengths is. seen also in the higher energy lines. The results of the HF model reveal the stronger transitions compared to the FCT model results. As can be seen.535 1. Again. but for the considerable energetic shift due to the lowering of the bandgap with temperature (see figure E. revealed best for the HF model calculations.2 Doped Quantum Well In the previous section we have discussed in detail the calculated susceptibility and spontaneous emission spectra of a 200 Å wide quantum wells devoid of delta doping. This shift can be seen clearly from fig. i. We perform the simulation for two ambient temperatures. the curves are scaled to unity. and the scaling factor for each is indicated in the plots. the bandgap renormalization and transition peak difference between the two discussed models. reducing the Coulombic interaction between . which can be seen in figures 4.53 E (eV) 1. where we present the calculated and theoretical Fermi level energy as the function of the introduces electron concentration. Fig. for 2DEG concentrations of ne = 2 × 109 ÷ 4 × 1011 cm−2 . 4. 4.14: Spontaneous emission spectra calculated using the FCT (blue) and HF (red) models for the considered undoped structure at T = 2K . of two-dimensional gas.18. The spectra are presented in a waterfall chart style to effectively illustrate the influence of the 2DEG. most vividly in the insets of figure 4.5 2 rsp (s−1m−3J−1) 1.14 presents the calculated spontaneous emission spectra using the two models from the discussed structure. for both models in the introduction of electron gas into the well leads to an energetic shift of towards higher energies. Another major consequence of the rise the 2DEG concentration is the reduction of the absorption intensity.e. In this section we introduce a delta-doping into the cladding layers of the discussed bare quantum.3.525 1. the difference between the respective inter-subband transition probabilities seen in figures 3. Figures 4.23 present the absorption spectra from the structure for the various 2DEG concentrations.19. respectively. the energetic shift due to the large bandgap renormalization can be seen for the HF model spectrum. We investigate the influence of this gas on the optical characteristics of the structure for several concentrations of the gas using the same two theoretical models. the results of the two discussed models are quite different. thus moves according to this shift.5 1 0. We also observe here.2 and 3.5 0 1.17 and 4. causing the Fermi level to shift toward higher energies.52 1.16. 4. 4. 4. T = 2K and T = 77K .3. The reason of for this reduction in the strength of the resonances present in the structure is the enhanced screening effect of the 2DEG.12.15. Numerical Results 75 x 10 45 FCT HF 2.4. which results in the introduction of 2DEG into the well. We now present the results of a similar calculation. as for the T = 2 K case. together with the sharp transition peak. performed at higher ambient temperature of T = 77 K . This phenomenon can be attributed to the phase space filling due to the introduced electrons. calculated using the two discussed models. 4.21(a). FCT (in blue) and HF (in red).3. again. This structure is schematically described in fig.22 and 4. It can be easily seen that the results are similar to the T = 2 K case. 492 1.494 1.491 eV 0.04 1.01 0.15: The complex elelctrical susceptibility.496 1.482 1.482 1.484 1.04 ℜ[χTM ] FCT 0.49 E (eV) 1.02 ℜ[χTE ] FCT 1.02 0.488 1.4952 eV 0.498 0 1.03 0.005 0 1. χ.494 1.76 Chapter 4. Coulomb Correlated Optical Transitions (a) 0.496 1.484 1. calculated using the FCT model for the undoped well at T = 77K . ℑ[χTM ] FCT .015 1.48 1.4952 eV 0. The two plots are the (a) TE and (b) TM polarization calculations.488 1.486 1. real (blue) and imaginary (red) parts.01 ℑ[χTE ] FCT 0.492 1.02 0 1.486 1.5 (b) 0.49 1.48 1.498 0 1.5 Figure 4. 3 0.498 0 1.48 1.4932 eV 1.494 1.8 0.4903 eV 1.7 0.5 ℑ[χTM ] HF 0 −0.498 Figure 4.3 0.1 0 1. ℑ[χTE ] HF TE .5 1.484 1.4811 eV 1 ℜ[χHF] 0 0.49 1.6 0.492 1.486 1. real (blue) and imaginary (red) parts.488 1.5 0.492 eV 1.482 1.1 ℜ[χHF ] TM 0. calculated for the undoped structure using the HF mode for the undoped well at T = 77K .4957 eV −0.492 1.3.48 1.496 1.484 1.494 1.2 0.496 1.488 1.2 −0.1 −0.5 (b) 0. Numerical Results 77 (a) 0. χ.4 0.5 1.4843 eV 1.482 1.486 1.3 −0.4897 eV 1.16: The complex elelctrical susceptibility.4 0.49 E (eV) 1. The two plots are the (a) TE and (b) TM polarization calculations.2 0.5 1.4 1.4.4885 eV 1. 484 1.5 10000 1 5000 0 0.496 1.49 1.498 0.5 1. calculated using the FCT (blue) and HF (red) models for the undoped well at T = 77K .486 1.78 Chapter 4.496 1.5 x 10 5 (a) FCT HF 2 15000 αTE (cm−1) 1.492 1.484 1.48 1.492 1.494 1. Coulomb Correlated Optical Transitions 2.48 1.49 1.49 1.498 1.494 1.5 2.494 1.488 1.496 0 1.498 1.492 1.482 1.488 1.17: The absorption spectra for the (a) TE and (b) TM polarizations.486 1.492 1.5 0 1.5 3 2 1 1 αTM (cm−1) 1.488 1.488 1. .494 1.486 1.496 1.482 1.5 Figure 4.5 x 10 5 (b) x 10 5 4 2 4 1.49 E (eV) 1. 3.48 1.19: Schematic structure of the doped QW examined in the numerical calculations.485 1.49 E (eV) 1.495 1.ଽ ݏܣ ݏܣ݈ܣ ݏܣܽܩ ܵ݅ ߜͲĚŽƉŝŶŐ 1000Հ 1119Հ 200Հ ܵ݅ ߜͲĚŽƉŝŶŐ Figure 4. ݈ܣ.18: Spontaneous emission spectra calculated using the FCT (blue) and HF (red) models for the considered undoped structure at T = 77K . The δ -doping Si atom layers are marked with the red dashed lines on the cladding layers of the structure. Numerical Results 79 8 7 6 5 4 3 2 1 x 10 42 FCT HF rsp (s−1m−3J−1) 0 1.ଵ ܽܩ.4. .5 Figure 4. 3.2. The shaded blue area marks the energetic region occupied by the 2DEG.20: A schematic depiction of the bandgap renormalization due to the introduction of 2DEG into the well region of the structure. The band bending due to the internal electrical field is also shown. E1 (ne ) is the renormalized effective first conduction subband and HH1 is the first heavy-hole subband. EF stands for the Fermi level energy. Coulomb Correlated Optical Transitions EF E1 E '1(n ) e ω 'E −HH (ne ) 1 1 ωE −HH 1 1 HH 1 Figure 4. (a) 20 Calculated 15 Theory EF−Ec(kt=0) (meV) 10 5 0 5 20 x 10 (b) Calculated 15 kF (cm−1) Theory 10 5 0 10 9 10 10 10 N2DEG (cm−2) 11 10 12 Figure 4.21: (a) The energetic gap between the Fermi energy and the bottom of the conduction band and (b) the Fermi wavevector. E1 is the first conduction subband of the undoped structure. .80 Chapter 4. both as a function of the 2DEG concentration obtained using numerical calculation and the simple theoretical model presented in section 2. which losses its intensity because of the screening effect of the introduced electrons.26 and 4. As for the T = 2 K case discussed earlier. In figures 4. together with the asymmetric broadening.27 present a comparison between the absorption and spontaneous emission spectra. For both polarizations. which eventually disappears for high 2DEG concentrations. to the Fermi edge energy. For the HF model results. This can be best seen for the results of the FCT calculations for which this effect is most dominant because of the absence Coulombic interaction between carriers. Finally. In fig. For an undoped QW. 4.4.20. We have attributed the absorption peaks movement to the Fermi edge translation due to the phase space filling in the conduction band. both obtained using the HF model calculation. The lineshape profile of the peak together with the energetic shift pattern can be explained by considering two types of emission from the structure using fig. Similar analysis may be applied to the refractive index change spectra.22). In these figures two no ticable spontaneous emission peaks (marked 1 and 2) are present and the change in their position and shape should be addressed. This causes the spontaneous emission and the absorption spectra to diverge with the added carriers’ concentration.23) compared to the additional lower energy peak for the TE polarization (fig. It is a well known fact [6] that the emission due to the introduced external electrons (the 2DEG) stretches from the effective bandgap. though its influence is much weaker compared to the screening. 4. The differences between the absorption spectra for the two polarizations are to the description in the previous section.28. Along with the broadening. 4. leading to the shift of the effective bandgap towards low energies. retains its symmetric lineshape and also closely follows the lower most absorption spectrum peak. an effect proportional to the carrier concentrations (ne ). We can investigate the influence of the 2DEG concentration on the spontaneous emission and compare it to the absorption. respectively for both polarities. 4. This reduction in the strength of the absorption resonance peaks.21(a) we can see the shift of the Fermi edge towards high energies. indicating on the minor influence of the screening effect of the introduced electrons on the excitonic bonds in the . where the strong excitonic line attains its dominance up to ne = 5. we can note that as for the undoped case.7(a) shows an opposite trend of the bandgap renormalization. which can be attributed to excitonic transitions. Here.3. while fig. the FCT and HF differ greatly in the lineshapes and intensities of the resonances. as is observed for peak 2 in our simulations.31 present the absorption spectra calculation using the FCT and HF models at ambient temperature T = 77 K . while for th FCT model the main difference is the presence of a single peak in the TM spectrum (fig.20. is most noticeable for 2DEG concentrations above ∼ 6 × 1010 cm−2 . as the absorption.5 × 1010 cm−2 . an effect known as the Burstein-Moss shift (BMS) [49]. These plots also give a clear picture of the 2DEG emission asymmetric broadening. introduces in both models via the phenomenological factor γ . We can see that the spontaneous emission. the spontaneous emission peak location coincides with the energetic location of the first conduction subband [49]. denoted E1 (ne ).24 and 4. The relative intensity change can be clearly seen. These figures present a detailed view of the intercrossing region of the broad 2DEG induced peak and the excitonic line of the spontaneous emission. Peak 1. Another contributing factor to the observed diminishing interaction are the inter-carrier collisions. we may attribute this peak to the strong excitonic resonance present for lower 2DEG concentrations. the relative intensity of this emission peak intensified compared to peak 1. we can clearly see a milder reduction of the interaction strength compared to the HF results. for the entire diapason of the simulated gas concentrations.29 we give a detailed zoomed in view of the absorption and spontaneous emission spectra for 2DEG concentrations in range of ne = 3 × 1010 ÷ 8 × 1010 cm−2 . Numerical Results 81 the interacting conduction band electrons and valence band holes. Figures 4. an effect not seen in the presented plots because of the relative scaling. shifts this subband towards lower energies. marked by the dotted circle at the ne = 6 × 1010 cm−2 curve in figure 4.28 and 4. Along with the energetic shift in the peak location. the HF model calculation leads to the appearance of two strong spectral lines. 4. As we can see in fig. the Coulomb enhancement causes the spectra obtained from the HF model to be much stronger than the FCT ones. The combined influence of these two trends leads to an asymmetric broadening of the emission line. or the disappearance of the exciton transition lines. becoming larger with the 2DEG concentration.30 and 4. changes its energetic position with the rise of the doping concentration. after which the 2DEG emission retains the lead. the bandgap renormalization.25. Figures 4. the main differentiating feature is the relative strength of the two dominant peaks between the two polarizations. these spectra also attain a characteristic curve close to the Fermi edge energy. Along with the energetic shift. there is a noticeable alteration of the lineshape and the linewidth of the spontaneous emission spectra. These lines maintain their form and amplitude for much of the considered 2DEG concentration values. Thus. δn. 4. for both polarities. respectively for the TE and TM polarizations. shown for the same structure in figures 4. caused by the introduction of electrons inside the well. Here the overall shift to the lower energies can be clearly seen. EF . respectively for the TE and TM polarizations. 52 1. Coulomb Correlated Optical Transitions ×28 N2DEG=4x1011cm−2 ×24 N2DEG=3x10 cm 11 −2 ×2 ×19 N2DEG=2x1011cm−2 ×13 N2DEG=1x1011cm−2 ×13 αFCT (blue).22: The TE polarization absorption spectra calculated for the doped structure for various gas concentrations ne using the FCT (blue) and HF (red) models at T = 2K .55 Figure 4.545 1.53 E (eV) 1. .54 1.515 09 −2 ×1 1.525 1.82 Chapter 4. The scaling factor is indicated in the respective color.u.535 1. αHF (red) (a.) TE TE N2DEG=8x1010cm−2 ×3 N2DEG=6x1010cm−2 ×2 N2DEG=4x1010cm−2 ×1 N2DEG=2x1010cm−2 ×1 N2DEG=8x1009cm−2 ×1 N2DEG=6x1009cm−2 ×1 N2DEG=4x10 cm 09 −2 ×1 ×1 ×1 ×1 ×1 ×1 ×1 ×1 ×1 ×1 ×1 N2DEG=2x10 cm 1. αHF (red) (a.3.535 1.23: The TM polarization absorption spectra calculated for the doped structure for various gas concentrations ne using the FCT (blue) and HF (red) models at T = 2K .4.55 Figure 4.515 09 −2 1.u.52 1. Numerical Results 83 ×17 N2DEG=4x1011cm−2 ×18 N2DEG=3x10 cm 11 −2 ×18 N2DEG=2x1011cm−2 ×10 N2DEG=1x1011cm−2 ×9 αFCT (blue).54 1.) TM TM N2DEG=8x10 cm 10 −2 ×2 N2DEG=6x1010cm−2 ×2 N2DEG=4x10 cm 10 −2 ×1 N2DEG=2x1010cm−2 ×1 N2DEG=8x10 cm 09 −2 ×1 N2DEG=6x1009cm−2 ×1 N2DEG=4x10 cm 09 −2 ×1 N2DEG=2x10 cm 1.53 E (eV) 1.545 1.525 1. . 84 Chapter 4.52 1. δnHF (red) (a. .53 E (eV) 1.545 1.) TE TE ×1 N2DEG=5.525 1.0x1010cm−2 ×1 1.515 1.u.0x1010cm−2 ×1 N2DEG=3.5x10 cm 10 −2 ×1 ×1 ×1 N2DEG=3.0x1010cm−2 ×1 ×2 N2DEG=5.0x1010cm−2 ×5 N2DEG=6.55 Figure 4. Coulomb Correlated Optical Transitions ×5 N2DEG=7.5x10 cm 10 −2 ×1 ×1 ×2 N2DEG=6.24: The TE refractive index change spectra calculated for the doped structure for various gas concentrations ne using the FCT (blue) and HF (red) models at T = 2K .0x1010cm−2 ×1 N2DEG=4.54 1.535 1.5x1010cm−2 δnFCT (blue).5x10 cm 10 −2 ×1 ×1 ×1 ×1 N2DEG=4. 515 1.55 Figure 4.4.54 1. .5x1010cm−2 ×1 N2DEG=3.5x10 cm 10 −2 ×1 ×1 ×1 N2DEG=6.525 1.0x1010cm−2 ×3 N2DEG=6.535 1.0x1010cm−2 ×1 N2DEG=5.52 1. δnHF (red) (a.0x1010cm−2 ×1 N2DEG=4.) TM TM ×1 N2DEG=5.0x1010cm−2 ×1 ×1 ×1 1.25: The TM refractive index change spectra calculated for the doped structure for various gas concentrations ne using the FCT (blue) and HF (red) models at T = 2K .53 E (eV) 1. Numerical Results 85 ×3 N2DEG=7.5x1010cm−2 δnFCT (blue).5x10 cm 10 −2 ×1 ×1 ×1 ×1 ×1 N2DEG=4.u.545 1.3.0x1010cm−2 ×1 N2DEG=3. u.53 E (eV) 1.) TE sp N2DEG=8x1010cm−2 ×5 N2DEG=6x1010cm−2 ×1 ×13 ×13 ×3 ×2 N2DEG=4x1010cm−2 ×2 N2DEG=2x1010cm−2 ×4 N2DEG=8x1009cm−2 ×1 ×6 ×1 ×1 N2DEG=6x10 cm 09 −2 ×8 N2DEG=4x1009cm−2 ×17 N2DEG=2x1009cm−2 1. Coulomb Correlated Optical Transitions ×16 N2DEG=4x1011cm−2 ×16 N2DEG=3x10 cm 11 −2 ×28 ×24 ×15 N2DEG=2x10 cm 11 −2 ×19 ×12 N2DEG=1x1011cm−2 ×11 αHF (blue).54 1.52 1.545 1.55 Figure 4.86 Chapter 4. . rHF (red) (a.26: The TE polarization absorption (blue) and spontaneous emission (red) spectra calculated for the doped structure for various gas concentrations ne using the HF model at T = 2K .535 1.525 1 ×1 ×1 2 1.515 1. 545 1.) TM sp N 2DEG ×9 =8x10 cm 10 −2 ×5 N2DEG=6x1010cm−2 ×1 N2DEG=4x10 cm 10 −2 ×2 ×2 ×2 N2DEG=2x10 cm 10 −2 ×1 ×4 N2DEG=8x1009cm−2 ×1 N2DEG=6x10 cm 09 −2 ×1 ×6 ×8 N2DEG=4x10 cm 09 −2 ×1 ×17 N2DEG=2x1009cm−2 1.53 E (eV) 1.4. . rHF (red) (a.27: The TM polarization absorption (blue) and spontaneous emission (red) spectra calculated for the doped structure for various gas concentrations ne using the HF model at T = 2K .525 1 ×1 2 1.52 1.54 1.55 Figure 4.3.535 1. Numerical Results 87 ×16 N2DEG=4x1011cm−2 ×16 N2DEG=3x10 cm 11 −2 ×17 ×18 ×15 N2DEG=2x1011cm−2 ×12 N2DEG=1x10 cm 11 −2 ×18 ×10 ×11 αHF (blue).515 1.u. 5x1010cm−2 N2DEG=4.5x1010cm−2 N2DEG=3.u.) TE sp N2DEG=5.0x1010cm−2 N2DEG=4.0x1010cm−2 N2DEG=3. .525 1.5x1010cm−2 N2DEG=6.545 1. Coulomb Correlated Optical Transitions N2DEG=7. rHF (red) (a.52 1.0x1010cm−2 1.5x1010cm−2 N2DEG=5.55 Figure 4.28: Detailed view of the transition region in the absorption (blue) and spontaneous emission (red) spectrum for the TE polarization using the HF model at T = 2K .54 1.515 1.53 E (eV) 1.88 Chapter 4.0x1010cm−2 N2DEG=6.0x1010cm−2 αHF (blue).535 1. 0x1010cm−2 αHF (blue).29: Detailed view of the transition region in the absorption (blue) and spontaneous emission (red) spectrum for the TM polarization using the HF model at T = 2K .535 1. rHF (red) (a.515 1.5x10 cm 10 −2 N2DEG=4.0x1010cm−2 N2DEG=3.52 1.0x1010cm−2 N2DEG=4.5x1010cm−2 N2DEG=6.525 1.5x1010cm−2 N2DEG=5. .5x1010cm−2 N2DEG=3.0x1010cm−2 N2DEG=6.545 Figure 4. Numerical Results 89 N2DEG=7.53 E (eV) 1.0x1010cm−2 1.3.4.) TM sp N2DEG=5.u.54 1. .49 1.48 1.) TE TE ×2 N2DEG=4x1010cm−2 ×1 N2DEG=2x1010cm−2 ×1 N2DEG=8x10 cm 09 −2 ×1 N =6x1009cm−2 ×1 N2DEG=4x1009cm−2 ×1 N2DEG=2x1009cm−2 1.30: The TE polarization absorption spectra calculated for the doped structure for various gas concentrations ne using the FCT (blue) and HF (red) models at T = 77K .u. Coulomb Correlated Optical Transitions ×29 N2DEG=2x1011cm−2 ×5 N2DEG=1x1011cm−2 ×3 N2DEG=8x1010cm−2 ×2 N2DEG=6x1010cm−2 αFCT (blue).90 Chapter 4. αHF (red) (a.47 1.485 E (eV) 1.495 1.5 2DEG Figure 4.475 1. 495 1.47 1.485 E (eV) 1.3.5 Figure 4.49 ×1 1.31: The TM polarization absorption spectra calculated for the doped structure for various gas concentrations ne using the FCT (blue) and HF (red) models at T = 77K .475 1.4.u.48 1. . Numerical Results 91 ×3 ×11 N2DEG=2x1011cm−2 ×2 N2DEG=1x10 cm 11 −2 ×2 ×2 ×2 N2DEG=8x1010cm−2 ×1 N2DEG=6x1010cm−2 αFCT (blue).) TM TM ×1 N2DEG=4x1010cm−2 ×1 N2DEG=2x1010cm−2 ×1 N2DEG=8x1009cm−2 ×1 N 2DEG ×1 ×1 ×1 ×1 ×1 =6x1009cm−2 ×1 ×1 N2DEG=4x1009cm−2 ×1 N2DEG=2x1009cm−2 1. αHF (red) (a. These line amplitudes intensify considerably with 2DEG concentration.92 Chapter 4. which follow closely the excitonic absorption resonances. especially compared to the T = 2 K case discussed earlier. Coulomb Correlated Optical Transitions well. The spontaneous emission spectra.33 present the absorption and spontaneous emission spectra calculated using the HF model respectively for the TE and TM polarization at ambient temperature of T = 77 K . Figures 4.32 and 4. . indicating to a milder effect of introduced electron on the bandgap renormalization. like the absorption spectra. Here. two strong emission peaks are noticeable. Another striking difference of these curves from the T = 2 K case is the relative energetic immobility of the spectral lines with 2DEG concentration. reaching peak intensity for ne = 1 × 1011 cm−2 . indicating to a drastic reduction of the excitonic interaction. differ greatly from the results for T = 2 K discussed earlier in this section. maintaining approximately the same linewidth for this entire stretch of 2DEG concentrations. At ne = 2 × 1011 cm−2 both the absorption and spontaneous emission spectra change in both linewidth and lineshape. 485 E (eV) 1. . Numerical Results 93 ×3 N2DEG=2x1011cm−2 ×1 N2DEG=1x10 cm 11 −2 ×29 ×5 ×1 N2DEG=8x10 cm 10 −2 ×3 ×1 N2DEG=6x10 cm αHF (blue).3.) TE sp 10 −2 ×2 ×1 N2DEG=4x1010cm−2 ×2 N2DEG=2x1010cm−2 ×6 N2DEG=8x1009cm−2 ×7 N 2DEG ×2 ×1 ×1 ×1 =6x10 cm 09 −2 ×11 N2DEG=4x1009cm−2 ×22 N2DEG=2x10 cm 1. rHF (red) (a.u.4.47 09 −2 ×1 ×1 1.475 1.5 Figure 4.32: The TE polarization absorption (blue) and spontaneous emission (red) spectra calculated for the doped structure for various gas concentrations ne using the HF model at T = 77K .495 1.49 1.48 1. 485 E (eV) 1. .33: The TM polarization absorption (blue) and spontaneous emission (red) spectra calculated for the doped structure for various gas concentrations N2DEG using the HF model at T = 77K .u.) TM sp ×1 N2DEG=4x10 cm 10 −2 ×11 ×2 ×2 ×1 ×1 ×2 N2DEG=2x1010cm−2 ×6 N2DEG=8x1009cm−2 ×7 N =6x10 cm 09 −2 ×1 ×1 ×1 2DEG ×11 N2DEG=4x10 cm 09 −2 ×1 ×22 N2DEG=2x10 cm 1.5 Figure 4. Coulomb Correlated Optical Transitions ×3 N2DEG=2x1011cm−2 ×1 N2DEG=1x1011cm−2 ×1 N2DEG=8x1010cm−2 ×1 N2DEG=6x1010cm−2 αHF (blue).49 1.94 Chapter 4.47 09 −2 ×1 1.48 1.495 1. rHF (red) (a.475 1. 1 Optical Resonator An optical resonator is a device used for confining light at certain frequencies. a resonant interaction takes place. The method for the calculation of optical reflection and transmition of such structures is presented. The transmitions spectrum of this structure exhibits a pattern of repetitive peaks of large transmission corresponding to resonances of the etalon. These interaction modes are termed polaritons. can be embedded within the MC. 5. namely that of electromagnetic radiation. followed by a description of a distributed Bragg reflector (DBR) mirrors. The region between the two mirror is called the spacer layer and its refractive index is denoted by n. We also discuss the influence of the introduction of 2DEG into the well region of the embedded QWs and compare the results to the optical properties of bare QWs. presented in chapter 4. through the use of microcavity structures consisting of a cavity layer cladded by a distributed Bragg reflector on each side. the photon mode is spatially confined inside the cavity region.1) consists of two planar mirrors. which corresponds to a minimum in the 95 . together with the characterization of electromagnetic distribution inside the cavity region and the dispersion properties of the confined photonic mode. and the influence of the the introduced 2DEG inside the cavity region. In the last part of this chapter. these structures allow efficient confinement of charge carriers in a defined region of the structure. and the examination of their optical properties constitute the main objective of this chapter. obtained by combining the numerical approaches formulate in this and previous chapters. and then apply them to the case of semiconductor microcavities and the polaritonic modes. and finally the introduction of semiconductor resonator built of a cavity layer cladded with two DBR mirrors.Chapter 5 Semiconductor Microcavities and Polaritonic Effect In the previous two chapters. Next. We start our discussion by giving a brief introduction to the classical theory of microcavities. which allows us to easily probe their electronic and optical properties. As we have seen. This varying transmission function is caused by an interference between the multiple reflections of light between the two reflecting surfaces (see Fig. separated by a distance l. 5. A heterostructure. such as quantum well or heterojunctions. In this chapter we explore another form of confinement.1). 5. in order to introduce the interaction of electromagnetic field mode with a QW confine electron-hole resonance. by describing the optical properties of Fabri-Perot resonators. we have considered the optical properties of bare quantum wells. we present a unified framework for the calculation of the optical properties of MCs with embedded QWs in the cavity region. we present simple idealized models for the interaction of a two-level electronic system with a photon mode. Here we put a special emphasis on the coupled oscillator model to describe the polariton dispersion relations. Constructive interference occurs if the transmitted beams are in phase. Inside such a structure. We complicate the model by considering multiple two-level systems interacting with the photon field mode. The classical resonator (such a the Fabri-Perot etalon seen in Fig. which corresponds to a high-transmission peak. and when the cavity width is comparable to the wavelength of the QW resonant modes (excitons). for transmitted beams that are out-of-phase a destructive interference occurs. For now we assume that the medium outside the structure is plain air. 1) If both surfaces have reflectance R.9 0.1 0 −10 T R=0. (5. The phase between each successive reflection is [51] δ= 2π λ 2nl cos (θ) (5.1.7 0.3 0. the etalon thickness l and the refractive index of the spacer material n.e. The maximum transmission occurs when the optical path length difference 2nl cos(θ) between each transmitted beam is an integer multiple of the wavelength. for various values of R (or equivalently of F ).2 0. transmission spectrum.8 0. Fig. the transmittance function of the entire structure is given by T = 1 (1 − R)2 = 1 + R2 − 2R cos (δ ) 1 + F sin2 δ 2 . R + T = 1.85 R=0.1. the beam incidence angle θ.5 0.99 −8 −6 −4 −2 0 δ 2 4 6 8 10 Figure 5.1: Fabri-Perot etalon multiple reflections.3) 1+F which occurs when the path-length difference is equal to half an odd multiple of the wavelength. δλ is the full-width halfmaximum of the transmission band.75 R=0. Semiconductor Microcavities and Polaritonic Effect Figure 5. The maximum reflectivity is given by 1 Rmax = 1 − . the reflectance of the structure R is the complement of the transmittance. . 5.6 δλ 0.1.2 shows the dependence of the transmittance on the phase parameter δ .96 Chapter 5. (5. The multiple-reflected beams’ phase matching depends in the wavelength λ of the light. In the absence of absorption. i.4 0.2) 4R where F = (1− R)2 is the finesse coefficient.2: Resonator transmittance function for various values of mirror reflectance R. 1 0. 2. z represents the growth direction of the layered structure. Distributed Bragg Reflector 97 z n l nH n L Reϐlected Transmitted nH nr Incident lH lL lDBR Figure 5. The alternating high and low indices. The wavelength separation between adjacent transmission peaks or free spectral range (FSR) of the etalon. denoted respectively by nH and nL . the optical path length difference between reflections from subsequent interfaces is half the wavelength. nH − nL . The FSR is related to the full-width halfmaximum (FWHM) δλ of any one transmission band by a quantity known as the finesse F= ∆λ π √ = δλ 2 arcsin(1/ F ) (5. The reflection bandwidth is determined mainly by the index contrast. 5.2. For the design wavelength λc . . is given by λ2 0 ∆λ = .3). The reflectivity achieved is determined by the number of layer pairs and by the refractive index contrast between the layer materials. and the semiconductor layer thicknesses satisfy the following condition nL lL = nH lH = λc . we use the Transfer Matrix formalism [52]. which results in a strong reflection.1.and right-hand cladding material refractive indices. Therefore. Δλ. the reflection coefficients for the interfaces have alternating signs. The principle of operation can be understood as follows. the left.1) where λc is the center wavelength of the high reflectivity region of the structure.3: Schematic illustration of a Distributed Bragg Reflector (DBR).4) 2nl cos(θ) + λ0 where λ0 is the central wavelength of the nearest transmission peak. all reflected components from the interfaces interfere constructively. in addition.2 Distributed Bragg Reflector The reflecting surfaces (or mirrors) modeled in previous section by a reflectivity parameter R are realized in practice by a stack of multiple layers of alternating semiconductor materials with varying refractive indices called Distributed Bragg Reflector (DBR) (see Fig. as is described in the next section. 4 (5.5) 5.1. In order to analyze the reflection and transmittance of the DBR. The two indices nl and nr are. Each interface between the two materials contributes a Fresnel reflection. (5. respectively.5. (5.3) consists of plane waves along the in-plane direction. can be expressed as a 2 × 2 matrix transfer matrix. (5.2. The transfer matrix takes into account the propagation through the dielectric media and the boundary conditions at each interface between two adjacent layers. z1 and z2 .3) yields the following one-dimensional equation d2 Ekt .2) E(r)eiωt .e. dependent refraction index. The boundary conditions for the electric field become (see Fig. z ) + ω2 c2 (z ) E(ρ.2. assuming a frequency independent dielectric function. (5. For a given in-plane wave vector kt (parallel to the dielectric layer plane) we can make the following ansatz for the solution Ekt = Ekt . E − ). The relation between the two complex amplitudes between two point along the propagation direction. (5.2.2.ω (z ) = 0. c2 (5.5) This equation can be solved separately for each layer of the stack. The solution for a homogeneous layer with a constant refractive index is of the form Ekt . 5.2. the solution of Eq. A non+ evanescent wave solution exist only if ω > k . Therefore equation (5.9) .7) For simplicity we assume propagation only in the z ˆ direction and normal incidence of the light waves on the dielectric interfaces. n(r) = n(z ).4) Substituting into Eq. c0 being the speed on light in the vacuum and n(r) the location dependent refractive index.3) Because of the translational invariance along the in-plane. i.2. (5.8) − E1 − − E2 = + n1 E1 − (5. From here on we assume the non-unique basis for the transfer matrix is (E + . The boundary conditions resulting from Maxwell’s equations state that tangential components of the electric and magnetic fields are continuous across the interface.2. the speed of light in the dielectric c(r) = c0 /n(r).2.2.1 Transfer Matrix Method (TMM) A monochromatic electric field of frequency ω propagating in a dielectric medium.ω (z )eikt ·ρ .2) can be written as ∇2 E(ρ. can be shown to satisfy the following wave equation ∇2 E(r) + where E(r. z ) = 0.2. The complex amplitudes of the forward ( E ) and backward (E − ) t c propagating waves are determined from the boundary conditions at each interface between two adjacent layers of the stack.6) The solution is a linear combination of two traveling waves in opposite direction along the z ˆ axis.4(a)) + − E1 + E1 + n2 E 2 + − = E2 + E2 (5.ω (z ) kz = = E + eikz z + E − e−ikz z . A medium consisting of pairs of two different dielectric materials with different refraction indices has a growth axis. (5. z ˆ. t) = ω2 c2 ( r ) E(r) = 0. Semiconductor Microcavities and Polaritonic Effect 5. ω2 − k2 t. The boundary conditions for the field components can be written as D⊥1 = D⊥2 B⊥1 = B⊥2 E H 1 1 =E =H 2 2 (5.2.98 Chapter 5.2.ω (z ) + dz 2 ω2 c2 ( z ) 2 − kt Ekt . 5.12) where m is the dielectric layer index. (5.10) or a wave propagating from layer 1 to layer 2.2. so that the outermost layers are of high refractive index. and for the case of ki li = π 2 1 − |r| = 1+ nH nL nH nL 2N 2N n2 H nr nl n2 H nr nl 2 . 5.2. It can be shown that for a structure containing 2N + 1 layers of high and low refractive indices (as in Fig.15) The reflection coefficient can be expressed as rDBR (λ) = |r|eiαr .4(b)) + E2 − E2 = eikl 0 0 e −ikl + E1 − E1 ≡ Mp + E1 − E1 .11) where k = (ω/c0 ) n. The explicit relation between the amplitudes of the incident electric field in the DBR structure and the transmitted field on the other side can be written as + Eout 0 = M11 M21 M12 M22 + Ein − Ein .m Mp. for which the transfer matrix is simply the multiplication of Mi and Mp matrices for each boundary and layer MDBR = m Mi.m .14) (5.2.2. The in-layer propagation transfer matrix Mp .13) The reflection and transmission coefficients of the entire structure can thus be written as rDBR tDBR = − = M21 M22 det(MDBR ) M22 (5. can be written as + E2 − E2 = 1 2 n2 n2 + n1 n2 − n1 n2 − n1 n2 + n1 + E1 − E1 ≡ Mi + E1 − E1 (5. which relates different vectors at z1 and z2 in the same layer can be written as (Fig. (5. This amplitude propagation approach can be applied to the entire multilayer structure. nH . Distributed Bragg Reflector 99 Figure 5.16) .5. for both linear polarizations of the field.2.2.3).2.2. TE and TM.4: Schematic description of (a) light propagation through an interface between two adjacent dielectric layers and (b) light propagation in a homogeneous medium. The interface transfer matrix Mi . (5. (5. 5: Simulation of a DBR structure with 35 alternating high and low refraction index layers.51 φ/π 0. λ/λc . αr/π .52 0.4 0. where (a) is the refractive index profile as a function of the growth axis.04 1.55 Figure 5.9 0.47 0.1 −1 1 0.49 0.6 0.5 0 0 0.94 0.54 0. φ = 2πl λ . (b) the amplitude and phase of the normal incidence reflection function this structure as a function of normalized wavelength.48 0.8 |rDBR| 0.5 0. Semiconductor Microcavities and Polaritonic Effect (a) 3 2 1 0 n 0 1000 2000 3000 4000 z [nm] (b) 5000 6000 7000 1 1 |rDBR| 0.02 1.100 Chapter 5.96 0. and (c) the same amplitude as a i ni function of the phase acquired by the EM field at each layer.92 0.06 1.08 1.46 0.53 0.98 1 λ/λc (c) φ=π/2 1. as π can be clearly seen from these plots. whose mirrors are two DBRs and the spacer material is a semic conductor layer with refractive index nc and of length lc = 2λ nc m (m is a positive integer).3.5 an exact calculation of the reflection coefficient of a DBR containing 35 pairs of alternating high and low refractive index layers.3.6: A microcavity schematic structure. 5. the reflection and transmission coefficients of the left.1) yields ikl rl eiklef f −rr e−iklef f e ef f −R e−iklef f 2 nL T tl tr .and right-hand cladding material refractive indices.5) rl e−iklef f −rr eiklef f e−iklef f −Reiklef f nH tl tr T where T = tl tr and R = rl rr . In subplot (b) we also present the reflectivity phase αr .2 Ga0. tr ) are.98 (AlAs) and nr = 3. tl ) and (rr . nL = 2.1) where Mc is the transfer matrix of the spacer (cavity) layer. The refractive indices for the simulation are nl = 1 (air). Mc eiklc 0 0 e−iklc (5. The MC’s transfer matrix for a wave traveling from left to right can be written as r l MM C = MDBR Mc MDBR .3. nH = 3. 5.3 5.3) rl tl 1 tl . lef f is not equal to lc but rather to lef f = lc + lp . which corresponds to λc . where the zero-phase crossing is observed at λc as expected for the left-face reflection.3. where lp is the penetration depth . . λ/λc .59 (GaAs) (the structure refractive index profile is presented in (a)).8 As).1 Microcavity Optical Characteristics Microcavity Reflection Spectrum A microcavity (MC) is a Fabry-Perot resonator. As an illustration. and the phase acquired by the electromagnetic field at each layer. In subplots (b) and (c) we plot the reflection coefficient as a function of the normalized i ni wavelength.6) and nl and nr are the left. (5. (5. (5.2) (5. φ = 2πl λ . The microcavity can be analyzed similarly to the DBR using the transfer matrix formalism.3. 5. The stop-band.and right-hand DBR mirrors (see Fig. We can write the these three transfer matrices in their most general form as r MDBR = = = nc nr nl nc l MDBR 1 tr r −r tr 1 tl rl tl − tr 1 tr r r . respectively.3. we present in Fig.4) where (rl . MM C = (5. respectively.5.45 (Al0. Inserting these matrices into Eq.3. Microcavity Optical Characteristics 101 Figure 5. is centered at φ = 2 .3. 94 0.1 −1 1 0. As we increase the number of layers in the DBR mirrors the stop-band reflectivity approaches unity and the cavity field reflection line becomes narrower.9) RM C nc lef f γc is a homogeneous lifetime broadening of the confined cavity mode.55 Figure 5.6 0.96 0. we present in Fig.5 0 0 0. where (a) is the refreactive index profile as a function of the growth axis. This calculation was performed using the direct TMM calculation of the entire the structure.06 1. Semiconductor Microcavities and Polaritonic Effect (a) 3 n 2 1 0 0 2000 4000 6000 8000 z [nm] (b) 10000 12000 14000 16000 1 1 |rMC| 0.7 the simulation results of a sample microcavity with two DBR mirrors c with 35 alternating layers of high and low refractive index each and cavity length of lc = 2λ nc . similarly to the DBR simulation .46 0.9 0.5 0.8 |rMC| 0.3.5 0. The inset in (c) shows the π reflection profile in the vicinity of φ = 2 .54 0.5 0.8) It is clear that there are several modes which satisfy the condition kz lef f = mπ .08 1. the MC mode has a FWHM which can be shown to be [53] 1 − RM C c 2 γc = √ .3. (5.7: Simulation of a microcavity with two DBR’s with 35 alternating high and low refractive index layers each. φ = 2πl λ .52 0.3.7) (5.51 φ/π 0.53 0.49 1 0. (b) the amplitude and phase of the normal incidence reflection function this structure as a function of normalized wavelength. 5. of the cavity field into the DBR [53] and is given by lp = λc nL nH .98 1 λ/λc (c) φ=π/2 1.47 0.3.4 0. caused by the decay through the mirrors. 2 nc (nH − nL ) (5. Because of the finite transmission probability of the DBR.48 0.02 1.102 Chapter 5. As an illustration. λ/λc .04 1.92 0.6) The reflectance and transmittance of the MC can thus be written as TM C RM C = = 1 T = 2 22 2 det (MM C ) |MM C | (1 − R) + 4R sin2 (kz lef f ) 21 2 | MM (|rr | − |rl |) + 4R sin2 (kz lef f ) C| = 2 22 |2 | MM (1 − R) + 4R sin2 (kz lef f ) C 2 αr/π (5. and (c) the same i ni amplitude as a function of the phase acquired by the EM field at each layer.9 0.2 0.5 0. 8 As).45 (Al0. a MC is used to sort out a single mode of the electromagnetic field that is confined in one direction. the electric field buildup can be calculated inside the cavity. where kz = 2 this condition with the condition for a propagating wave indside the cavity. As we have seen in the previous section. 5. 5. Combining − kt The dispersion of the MC modes is defined as the kt dependence of the MC-mode energy. The DBR parameters are the same as in Fig.4 Microcavity Polaritons As we have seen in previous sections.9 and compare in to the free photon dispersion. for various c values of m.5. We plot the confined photon dispersion. For high kt values. it is obtained from the resonance condition kz lef f = mπ .11) The parabolic dispersion can be interpreted as a cavity photon that carries a very small in-plane effective mass nc mph = cl ≈ 10−5 m0 . kt < dispersion relation of the form [54] ω c2 2 . This is obtained by plugging the accumulated transfer matrix (5.3. Microcavity Polaritons 103 Figure 5.4. presented above.5. with the use of appropriate initial electric field amplitudes. where m0 is the free electron mass. (5.2 Microcavity Confined Photon ω2 c2 2 . The resonant transitions (excitons) in QWs behave like a two-level absorber system due to their enhanced oscillator strength and the large binding energy resulting from the quantum confinement [56]. + kt (5. 5.10) where m is a positive integer.13).8 the electric field profile inside a λ-wide microcavity for two configurations of the cavity DBR mirrors.3.2 Ga0. 5. the confined photon dispersion coincides with the free photon dispersion for all the modes. As an illustration we plot in Fig. For small kt we approximate the dispersion relation to a parabola Eph (kt ) ∼ = 2π c nc lc 1+ 2 2 kt 2mph . we can write the MC-mode Eph (kt ) = ck c = nc nc 2πm lc 2 2.1) for each layer of the structure into (5. Using the same procedure. 5. The spatial and spectral localization of the electric field intensity and its enhancement (from (a) to (b)) is clearly seen.3.3. The interaction of the MC-photon with .8: Simulation the electric field amplitude of a microcavity with DBR mirrors made of (a) 15 left and 25 right alternating layers and (b) 22 left and right alternating layers.2. while for the cavity we choose nc = nH = 3. Semiconductor microcavities with embedded QWs are of particular interest as they exhibit a one-dimensional confinement of both the electronic states and the electromagnetic field [55]. The cavity in this calculation is λ-wide and the layer materials are the same as in previous calculations. in Fig. 1 Strong Absorber-Photon Coupling Single Two-Level System To understand the interaction of a single photon mode with two-level absorber we employ the Jaynes-Cummings model [4.104 Chapter 5. The semi-classical model has the advantage of using the realistic MC parameters and it gives the reflectivity.4) .4.4. 58]. 61].4. the creation and annihilation operators that connect different photon Fock states. 1]. where a single photon mode is described by the Fock Hamiltonian HF = ω p† p + 1 2 .f (kt ) = n (dotted red). 1. A full quantum mechanical treatment developed in[62] is out of the scope of this thesis. and a quantum mechanical model representing the MC-photon and the QW exciton as two coupled harmonic oscillators [60. in which the active medium is represented by a Lorentzian oscillator [59. √ = n + 1 |n + 1 .9: Microcavity confined photon dispersion curves for various resonance modes (solid blue) compared to a ckt free photon dispersion curve Eph.1. respectively. The presented review in primarily based on [63]. as p |n p |n † √ n |n − 1 .3) p† and p obey the bosonic commutation relation p. defined in analogy to (3.1 5.2) (5. The field operator of the photon field E(r) is given by E(r) = ω pf (r) + p† f (r) .1) Here p† and p are. 5. by analogy with the exciton-polariton modes of the bulk material [5]. 2 0V (5. = (5.4. 54. transmission and absorption spectra.3). In this section we elaborate on the quantum mechanical model for the two-level absorber and extend the analog to a microcavity polariton. p† = 1. 64.1.4. Semiconductor Microcavities and Polaritonic Effect 10 9 8 7 6 E (eV) 5 4 3 m=2 2 1 m=1 0 0 2 4 6 8 k|| (m−1) 10 12 14 16 x 10 7 m=3 Figure 5. c the QW-exciton leads to several new effects that can generally be divided into two classes: (a) the enhancement or inhibition of the spontaneous emission rate in the weak coupling regime [57] and (b) the observation of new eigen-modes with a Rabi splitting (normal mode splitting) between them. The coupled exciton-photon eigenstates of the QW embedded in a MC are called cavity polaritons. The coupled oscillators model is more suitable for the description of the dynamics of the MC-polaritons. Various theoretical models have been proposed to describe the exciton-cavity polariton such as a semi-classical linear dispersion model. in the strong coupling regime [58. 65].4. (5. 11) The double degeneracy is lifted by the interaction and the separation between the dressed states within each group √ is 2 Ωf (r) n + 1.10(a)) |g . The effective cavity mode volume V is defined as V = |f (r)| d3 r. all the excited states of the entire system are grouped into doubly degenerate level groups (see Fig.3) and is given by HI = d · E = − Ωf (r) pσ+ + p† σ− .1.10) constituting symmetric and anti-symmetric combinations of the uncoupled states. d is the electric dipole matrix element between the ground and excited states of the atom and Ω. and without the atom-field interaction.1. 0 g.10(b). The atomic two-level system is described by a two-level Hamiltonian 1 HA = ωσ ˆz .1 + ω0 ω g . 0 Ω − g. the ground and excited states.4. the eigenstates of the system are 1 |±.e. At the resonance. The interaction then lifts the degeneracy within each group and new states are formed which are merely a linear combination of the doublets.1 e. We also define two additional operators σ+ |g σ− |e = = |e .8). .4. n . 0 ;ĂͿ ;ďͿ Figure 5. n + 1 ] . (5. σ− ] = 2σz . The atom-field interaction is once again the dipole interaction (as in 3. respectively. n ∓ |g.10: (a) Energy levels of the uncoupled absorber and field modes and (b) the dressed energy levels of the coupled system. n = √ [|e. seen in Fig.8) Here. 0 with the atom is the ground state and the optical cavity is devoid of photons.4. where f (r) is a complex vector function that describes the polarization and relative amplitude of the cavity mode 2 at the position r. termed the Rabi splitting.5) 2 where σz is a spin 1/2 like operator. the coupling term. Microcavity Polaritons 105 (ω + ω ) g .5. From (5. i. (5.4. |g . ω = ω0 .4. n + 1 and |e. 5. These new states are termed “dressed states” or the “dressed atom basis”.4.4.9) 2 0V The ground state of the entire system is |g . The atom-field interaction HI couples only the doublet states within each group and doesn’t couple states from different level groups. (5.7) where |g and |e represent. These operators obey the fermionic commutation relations [σ+ .6) (5.n = (n + 1) ω0 ± Ωf (r) n + 1. (5.4.4. 2 (5. 2 0 2 ω e. 5. is given by d ω Ω= . The eigenenergies are given by √ E±. (5. Since all the atoms are coupled to the field in a similar fashion.i .15) The atom-field interaction Hamiltonian is HI = Ω aσ+ + a† σ− . and the coupled atom-radiation system has N + 1 non-degenerate.4.9) and assuming f (r) is unity for all the atoms.4. equidistant energy levels. where we assume that the atoms interact only through their coupling to the radiation field. σ+ = i σ+. Semiconductor Microcavities and Polaritonic Effect We now assume the initial state of the system to be 1 |ψ (t = 0) = |e. The ground state of the coupled system of atoms and cavity atoms is |G. γabs . Due to the symmetrical form of the raising and lowering operators.i . (5. There is no irreversible decay in this model since there is only one photon mode. the excitation of the atomic system is equally shared among all atoms. which describes an atomic ground state with an empty cavity. The two lowest states of the atomic system are the ground state |G = |J.4. 1 = √ [|+ + |− ] . i. M with J = N/2 and −J ≤ M ≤ J . four in the third .106 Chapter 5. Ω > γcav .13) (5. 5.i . The atomic Hamiltonian and the atom-field interaction can be then expressed in terms of collective operators defined by σz = i σz. This description is valid also for QWs in resonant structures since typical exciton size is much smaller than the optical wavelength (a0 λ) [38]. 0 .2 Multiple Two-Level Systems An extension of the former system is a setting of multiple two level atoms interacting with a single photon mode. −J + 1 .4.12) The probability that the field mode is excited and the absorber is in its ground state is not stationary but rather of the form 1 2 | e. i. (5. since it involves a large ensemble of absorbers (or QW excitations). σ− = i σ−. −J and the first excited state |E = |J. For example. (5. (5. 0 ∓ |G. the atomic Hamiltonian as well as the atom-field interaction are sums of the individual atomic Hamiltonians.4.e.4. (5. 2 after a finite period of time t the system will be in the state 1 |ψ (t) = √ |+ e−iE+ t/ + |− e−iE− t/ 2 . the excitation is equally shared among the atoms.4.17) |±. 2 The eigenenergies of these states are given by √ E± = ω0 ± Ω N .16) where Ω is the single atom single photon Rabi frequency. In reality. In the strong coupling regime. This system is of relevance to the semiconductor MCs.4. If the absorber is in its excited states and it emits a photon. 1 ] . We consider a set of N identical two level atoms. there is an enhancement of the splitting because of the collective coupling of the atoms with the field. 1 and |E. the vacuum Rabi splitting must be greater than the cavity damping and the absorber damping. This discussion results hold only for weak light intensities. there are three equally spaced dressed levels in the second group. Thus. The structure of higher dressed state groups is quite different from lowest state groups and thus it is different from the single atom case. defined by (5. The atomic Hamiltonian has N + 1 non-degenerate levels |J. then it will subsequently reabsorb the photon.. when there is only one photon in the cavity. the cavity damping Q factor is finite leading to a cavity damping γcav .1.4.18) √ The splitting energy in this case becomes larger by the factor N than the isolated atom.. These two states are coupled by the atom-field interaction to form two dressed states exactly as for the single atom case 1 (5. The two degenerate states in the first group under resonant condition are |G. 0 .e.14) 2 termed the Rabi oscillations. n|ψ (t) | = (1 + cos Ωt) . 0 = √ [|E. √ As a result. (5. |1 a |1 b and |0 a |2 b . 5. An exciton in a semiconductor QW is in fact similar to such an assembly of N identical atoms.21) (5.4.3 Coupled Harmonic Oscillator When two harmonic oscillators interact resonantly. The m = 2 excitation state group has three degenerate bare states |2 a |0 b . The first is a coupled quantum oscillators and the second is the semiclassical linear dispersion model (LDM). namely each lattice site having a unit dipole moment corresponds to a single two level atom. 0 = 1 1 1 |2 a |0 b − √ |1 a |1 b + |0 2 2 2 1 1 1 = |2 |0 + √ |1 a |1 b + |0 2 a b 2 2 1 1 = − √ |2 a |0 b + √ |0 a |2 b .1 Coupled Oscillator Model In the coupled oscillator model.4. This system of many atoms where m N . thus in order to observe the differences a very high light intensity must be employed.4.4. and thus in analogy this modes are called dressed excitons. When this splitting is significantly greater then both the exciton and the cavity linewidths the system is said to be in the strong coupling regime. that split into three nondegenerate dressed states |2. where m is the photon number.4. The emission spectrum from the m = 2 to m = 1 state groups is. however. it was shown [64] that the dressed state ladder consist of levels remaining √ nearly equidistant over a large energy range. identical to that from m = 1 to m = 0 consisting of only two separate lines. The dressed bosons spectrum for m = 1 is identical to that of the corresponding dressed fermions. The coupling constant Ω is then not the single atom Rabi frequency Ωs . This is because of the six possible transitions.22) a with a frequency separation of 2Ω. − |2. This coupling mixes the exciton and the photon modes and the normal modes are similar to a dressed boson. (5.23) . In general. Microcavity Polaritons 107 and so on. 5. is similar to the dressed boson case described by the coupled oscillator model described in the next section. the above dressed boson is a valid concept. 2 2 a |2 b . A brief summary of these two models is given and a new approach to the cavity polariton modeling is presented.2 Strong Coupling in a Semiconductor Microcavity As mentioned in the previous sections. |2 b .4. the spectra of all the state groups will have only two distinct lines with an energy separation of 2 Ω N .4.5.1. the emission spectrum always has only two lines irrespective of the excitation state group. two are forbidden and two pairs are degenerate. 60]. in the case of N 1 identical atoms with m photons (m N ) that are resonantly coupled to a single mode. with a spacing between nearby levels being of the order of 2m Ω N . The dressed boson is then a perfectly linear system. The spectra corresponding to these excitations are therefore very complicated. excitons are similar to the two-level systems since only coupling between excitons and photon states of the same in-plane wave vector are allowed. Note that there is no difference between the fermion and boson cases for the first state group. 62. As we have mentioned before.2. we treat both the exciton transition and the photon mode as two quantum harmonic oscillators that are coupled together via their dipole interaction.4. + |2.20) (5. which are eigenstates of the total system Hamiltonian of the form H = ω0 a† a + 1 2 + ω0 b† b + 1 2 + Ω a† b + b† a . Two different models are usually used to describe such a system [59. 5.4.19) Here both field operators satisfy the boson commutation relation. The simplest Hamiltonian that describes this system can be written as H = (EX (kt ) + γX ) b† b + (EC (kt ) + γC ) a† a + Ω a† b + b† a . the coupled system can be respresented by new normal modes. but the enhanced √ Ωs N collective coupling of all the atoms. In the limit of very large N . (5. 28) where |C and |Xj are.3. 5.2. .27) where p† i and pi are.3. (5.108 Chapter 5. αC and αi. EC (kt ) is the photon in-plane dispersion inside the cavity given by eq. γX is the phenomenological linewidth of the exciton and γC is the cavity photon mode linewidth given by eq. Fig. (5. (5. 60].10). 5. 2DEG concentration ne and a phenomenological broadening parameter γ (set using a fit to experimental results.11 presents a schematic outline of the proposed numerical approach.i )p† i pi . the QW resonances (excitons) are treated as a resonant Lorentzian dispersive term in the dielectric response function of the QW material. where NX is the number of strongly interacting excitons present in the system.3). In this model.4. transmission and reflection spectra is that it takes into account the geometry of the MC including all the DBR’s. Semiconductor Microcavities and Polaritonic Effect where we assume a single MC mode interacting with one electron-hole (exciton) mode.25) We denote the eigenvalues of this This model can be extended to include larger number of exciton modes and the same single cavity mode. Ep. for E1 − HH1 and E1 − LH1 excitons the Hamiltonian becomes ΩXHH ΩXLH EC (kt ) + iγC 2 2 ΩXLH ∗ .2 Linear Dispersion Model and Beyond Another description for the optical response of QW excitons inside a MC structure within the linear regime can be given in the framework of the semiclassical linear dispersion model [62. 2 (5. we apply the Bogoliuvob’s transformation [5] to obtain HP ol = i (Ep. The use of this fully theoretical approach enables us to examine the optical properties of varieties of quantum structures.4. (5. In the typical formulation of the model. (5. respectively. This Hamiltonian can be written in a matrix form as H= EX (kt ) + iγX Ω 2 Ω 2 EC (kt ) + iγC .24) For this simple model. especially and specifically those of MC with embedded QWs in the cavity region. in which the results of the calculations presented in chapters 3 and 4 are used to model the dielectric response of the QW material. respectively.4. EXLH (kt ) − iγXLH 0 H= (5.i is its linewidth.4. For example.i is the eigenergy of this polariton branch and γp.4.i − iγp.j | |Xj .4. the i-th polariton branch creation and annihilation operators. b and b† are the exciton creation and annihilation operators.9).4. We propose an altered version of the LDM approach. the diagonalization of this Hamiltonian for each kt gives the polariton eigenenergies E± (kt ) = EX (kt ) + EC (kt ) ± 2 EX (kt ) − EC (kt ) + i γX − γC 2 + Ω 2 2 . the transfer matrix formalism is used to describe the light propagation inside the MC layers (as explained earlier in this chapter). in which case the Hamiltonian matrix to be diagonalized becomes (NX + 1) × (NX + 1). The computational starts with the definition of several inputs: • The physical parameters: ambient temperature T.26) 2 ΩXHH ∗ 0 EXHH (kt ) + iγXHH 2 In order to simplify the coupled oscillator Hamiltonian 5. the cavity mode and the j -th bare exciton wavefunctions.j are the admixing coefficients of the polariton. as discussed in section 4.23 and present a more particle-like picture of the polariton. The eigenstates of this Hamiltonian can be written as |Pi = |αC | |C + j 2 |αi. The advantage of the LDM approach in calculating the absorption. and uses the results of the first bare QW phase results as an input (marked as the MC+QW phase in figure 5. Once the band edges are calculated. • Various numerical calculation parameters.11). the energy diapason for the spectra calculations etc. The optical parameters of the bare QW are used to model the complex refractive indices of the cavity region of the MC.4. γ. To facilitate further analysis the minima of the reflection spectra are extracted and afterwards are fitted using the coupled oscillator model. Once the electronic properties of the QW are calculated the optical properties can be addressed. is used to calculate the conduction and valence subband dispersions up to kt values far away from the Brillouin zone center.5. To this end. which facilitates in the reflection spectrum calculation using the TMM described in section 5. the two models for the light-matter interaction presented in chapters 3 and 4 are used to calculate the optical susceptibility. the two-band k. DнYt dƌĂŶƐĨĞƌ DĂƚƌŝdž Yt ϮͲĂŶĚ Ŭ͘Ɖ .2. namely the material composition and width of the epitaxial layers comprising the investigated structures. The calculation is built of two separate phases. the cavity region of the full MC structure is considered as it contains the active section of the structure. is utilized to measure the influence of the introduced 2DEG on the band edges of the structure.11: Schematic description of the proposed computation model.p model. Once the dispersion relations are calculated. first the Schrödinger-Poisson model. First.1.2.4. such as the maximal value and quantization of transverse wave vector kt grid. formulated in section 2. As can be seen in the figure.4. as formulated in section 3. The reflection spectra represent the ultimate goal of this calculation and can analyzed much like experimental results. • The common physical parameters used throughout the calculations. and discuss the result and their analysis.3.. T ^ĐŚƌŽĚŝŶŐĞƌͲ WŽŝƐƐŽŶ Yt ďĂŶĚ ĞĚŐĞ ƉƌŽĨŝůĞƐ DĂƚĞƌŝĂů WĂƌĂŵĞƚĞƌƐ DŽŵĞŶƚƵŵ ŵĂƚƌŝdž ĞůĞŵĞŶƚƐ >ŝŐŚƚͲŵĂƚƚĞƌ ŝŶƚĞƌĂĐƚŝŽŶ ^ĐƌĞĞŶĞĚ ŽƵůŽŵď KƉƚŝĐĂů ƐƵƐĐĞƉƚŝďŝůŝƚLJ &ƌĞĞ ĂƌƌŝĞƌ &Ƶůů ƐƚƌƵĐƚƵƌĞ ƌĞĨůĞĐƚŝŽŶ ŽƵƉůĞĚ KƐĐŝůůĂƚŽƌ &ŝƚ Figure 5. absorption and spontaneous emission spectra of the structure at hand. the momentum (or dipole) matrix elements can be calculated for each subband transition combination. formulated in section 2. namely the QW and its cladding layers (marked as the QW region of the calculation in figure 5. In the next section we present an application of the second phase of the above elaborated approach.11). • The quantum structure definition. The second phase of the proposed approach considers the MC as a whole. Microcavity Polaritons 109 ^ƚƌƵĐƚƵƌĞ N2DEG. All samples for experimental investigation have a nonlinear width profile.12 presents a schematic cross-sectional picture of the epitaxial semiconductor layers.1 Ga0. which is uniform across all the epitaxial layers of the structure. respectively. i. followed by 25 alternating 602 Å and 710 Å wide AlAs and Al0. where the width detuning parameter δ is defined as the ratio between the width along the wafer radius W (r) and the width at the wafer center W (r = 0).12 changes by the same amount along the wafer radius. the widths along the radius diminish in a direct proportion to the detuning parameter profile. and thus is an ideal candidate for theoretical modeling. including their composition and physical width. The side view of the wafer presents the varying width profile along the radius. We start by presenting a sample structure of a bare MC used throughout this section. Figure 5. This collection of layers comprises the first of a pair of DBR mirrors of the MC.5.110 Chapter 5. atop of the entire MC a thin GaAs cladding is placed. i.13(b).13(a) presents the top and side views of a wafer from which a sample is cloven for actual experimental measurements. for both cases.9 As layer is deposited. we consider separately the doped and undoped QW structures.ଵ ܽܩ.ଽ ݏܣ ^ƵďƐƚƌĂƚĞ ݏܣ݈ܣ ݏܣܽܩ Figure 5. Semiconductor Microcavities and Polaritonic Effect ĂƉ >ĂLJĞƌ 710Հ 602Հ Z ϭϱ ƉĞƌŝŽĚƐ ĂǀŝƚLJ 2438Հ Z Ϯϱ ƉĞƌŝŽĚƐ ݈ܣ. being the second DBR mirror of the structure.9 As layers. Figure 5.1 Ga0. which is immediately followed by another structure of 15 alternating 602 Å and 710 Å wide AlAs and Al0. The reflection spectra.e.1 Ga0.12: Schematic profiles of the simulated microcavity devoid of the embedded QW in the cavity region. Figure 5.e. the thickness of each layer described in figure 5. The considered structure is deposited upon a GaAs thick substrate layer. 5.12 are in fact the wafer center values. presented in figure 5. all calculations in this thesis assume a linear δ − r dependence.1 Bare Microcavity We now present the structure of a sample used extensively in our experimental studies. As in the discussion of the optical parameters of bare QW in chapter 4. and calculate its reflection spectrum. Note that the widths of the epitaxial layers presented in figure 5. 5. Finally.9 As layers. In order to simplify result analysis.14 presents the reflection spectrum of the discussed bare MC structure calculated at T = 2 K using . respectively.5 Numerical Results In this section we present the results of the theoretical approach outlined in the previous section. plays the role of the cavity region. are analyzed using the coupled oscillator model and various extracted parameters are presented and discussed. Next we present the reflection spectra of the same MC strictures but with embedded QW structures in the cavity region. Atop of this structure a 2438 Å wide Al0. 5. Numerical Results 111 Figure 5.5 E (eV) 1.6 0.5.2 1.65 Figure 5.14: The reflection spectrum of the simulated microcavity structure (from figure 5. 1 0.45 1. the location of the cavity mode and the first low energy side-band.12).7 Reflection 0.4 0.94 at T = 2 K .35 1.13: Schematic description of (a) a typical sample wafer and (b) the cross-sectional width profile of such a wafer.6 1. The red and green dashed lines mark.4 1.5 0. .3 0. respectively.9 0.55 1.8 0. calculated using the transfer matrix method for width detuning δ = 0. 2. As an input to the calculation the layers and the refractive indices of the various epitaxial layers are used. calculated for T = 2 K .7 0. with the physical dimensions of the various epitaxial layers and their composition.9 0.45 0.5 0.2 Embedded Undoped QW As we have seen in the previous section. presented in section 5.12) for various detunings. for width detuning δ = 0. the refractive index can be linked to the complex . The refractive indices are obtained by using empirical results presented in chapter F for the Gax Al1−x As alloys. together with the width detuning parameter radial profile (shown in the inset). Figures 5.35 0. while the green dashed line marks the location of the first low energy side-band.9 0 0.5 1.2 1.95 δ 1 1.95 δ 1 Figure 5.112 Chapter 5. a phenomenon reflecting the linear change of the epitaxial layers widths mentioned above.16 presents such a structure. the reflection spectrum of the discussed microcavity structure has a characteristic high reflectivity wide window with a narrow deep trench at the cavity mode energy.7 1.4 0. While the cladding Al0. As we have seen in chapter 3. i.9 (b) 0. in red and green.6 Reflection minima 0. the refractive indices of the added QW layers have to be considered.5. calculated for the simulated structure (from figure 5.55 Emin (eV) 0.4 0. δ . The red dashed line marks the location of the low reflectivity cavity mode location. It can be easily seen that the cavity mode moves to higher energies as we probe the reflection away from the wafer center (δ = 1). the energetic locations of the reflection spectrum minima and the respective reflection values as a function of wafer width detuning δ .8 1.6 0. respectively. respectively. In this section we examine the consequences of introducing a single undoped QW into the cavity region of the MC on the reflection of the entire structure. 5.15: The reflection spectrum minima (a) energies and (b) values. This calculation can be performed for various values of width detuning parameter δ and ambient temperature T . Semiconductor Microcavities and Polaritonic Effect (a) 1.1 1. We have also seen that the energetic location of this mode follows linearly the change in the structure width detuning. Figure 5. the epitaxial layers widths shrink by 6% compared to the wafer center values.94.1 Ga0.1.15(a) and (b) present.9 As layers complex refractive index spectrum can be modeled using the semi-empirical results of appendix F.3 1. the TMM.e. the well layer index should be separately addressed being the active layer of the structure. The cavity mode and the side-band are again marked. In order to use the TMM method in the reflection spectrum calculation.65 0. namely the HF model. Numerical Results 113 ĂƉ >ĂLJĞƌ 710Հ 602Հ Z ϭϱ ƉĞƌŝŽĚƐ 1119Հ ĂǀŝƚLJ ߜ 1 200Հ Z Ϯϱ ƉĞƌŝŽĚƐ ݎ ݈ܣ.17 presents an example for such a calculation of n(ω ) for a 200 Å wide GaAs QW surrounded by Al0. The obtained dispersion plots are conventionally termed the reflection anti-crossing curves. The location of the noninteracting cavity mode is marked by the diagonal red dashed line.2 for this concentration are clear.ଵ ܽܩ. so that χ = χGaAs + χHF QW . Now that the refractive index spectra are known for each of the epitaxial layers of the structure. The linear width detuning parameter δ profile is given as an inset. the alteration of the detuning parameter δ leads to an energetic shift in the cavity mode location.ଽ ݏܣ ^ƵďƐƚƌĂƚĞ ݏܣ݈ܣ ݏܣܽܩ Figure 5. Moreover. Results of such a calculation at T = 2 K .19). As we have seen in the previous section. The inset in figure 5.94 ÷ 1. electrical susceptibility of the medium through n(ω ) = 1 + 4πχ(ω ). inducing a 2DEG with concentration of ne = 1 × 1011 cm−2 in the well region. where each set of points along the vertical represents a single δ value. the red diagonal dashed line marks the cavity mode dispersion. and each branch represents a polariton mode. The most striking feature of these representation of the data is the plethora of resonances present. and the results of the theoretical model presented in the previous chapter. Fig. Again. This interaction region splitting can be better seen by plotting the reflection spectra minima as a function of the cavity mode energy EC . thereby causing it to interact with the excitonic resonances present in the QW.18(a) shows the reflection spectrum at δ = 0.18 for the two light polarizations as a waterfall chart.5.5.16: Schematic profiles of the simulated microcavity with embedded undoped QW in the cavity region.19 (for both polarizations).1 Ga0.1) In our case.9 As barriers with δ -doped layers introduced at 1000 Å away from the barrier-well boundaries. reflecting the interaction region splittings in the dispersion curves (circled areas in figure 5. obtained using the empirical data of appendix F.5. this calculation can be performed for multiple values of the width detuning parameter δ of the structure in order to trace the influence of the layers width alteration of the reflection spectra of the entire structure. the complex electrical susceptibility spectrum is a superposition of the intrinsic GaAs susceptibility. 5. performed for δ = 0. most of which are hardly recognizable in the reflection spectra . The obvious similarities to the results of the susceptibility calculation of section 4. as shown in figure 5. These strong interactions lead to a characteristic splittings in the cavity mode location seen clearly in these figures for both polarizations. are presented in figure 5. (5. the reflection spectra can be obtained using a standard TMM calculation.3.98 as an illustration. 54 1.51 1.52 1.51 1.53 E (eV) ℑ[nTM ] HF 1.55 Reflection 1.525 1.505 1.54 (b) 3.94 ÷ 1) calculated for normal incident light of (a) TE and (b) TM polarizations.51 1.5 1.525 1.05 3. Semiconductor Microcavities and Polaritonic Effect (a) 3.545 1.505 1.6 1. and the red dashed diagonal line in (a) marks the location of the cavity mode.53 1.6 1.52 1.54 Figure 5. The inset in (a) shows a zoomed-in view on one of the reflection spectra comprising the waterfall plot (for δ = 0.7 0.525 1.515 1.52 E (eV) 1.05 3.97). (a) (b) Reflection Detuning 1.535 0 1. at T = 2 K .17: The real (blue) and imaginary (red) parts of the total refractive index spectrum of a bare 200 Å wide GaAs QW.51 1. .51 1.535 1. inducing a 2DEG with concentration of ne = 1 × 1011 cm−2 in the well region.8 0.7 0.53 1.53 E (eV) 1.1 Ga0. surrounded by Al0.55 E (eV) 1.5 1.54 1.515 1.9 As barriers with δ -doped layers introduced at 1000 Å away from the barrierwell boundaries.1 ℜ[nTM ] HF 3.1 ℜ[nTE ] HF 3.515 1.525 1.52 1. These results were obtained using the HF model for (a) TE and (b) TM polarizations for T = 2 K .535 0 1.114 Chapter 5.535 ℑ[nTE ] HF 1.55 Figure 5.52 1.54 1.515 1.18: The reflection spectra for various width detuning values (δ = 0.545 1.53 1.8 0. 535 1.545 Figure 5.54 1.5329 1.5.1 1. The fitting model energy parameter EX values are almost identical to the manually obtained values and the fitted coupling constant.11.53 1.52 1.54 1.1) with 7 excitons (denoted Xi ) corresponding to the seven resonances observed in the calculation results.626 3. The second column from the left lists the energetic location of the resonances extracted manually from the figure.52 1. the amount of splitting in the interaction regions and relative strengths of the resonance lines. for (a) TE and (b) TM polarizations of incident light. calculated at T = 2 K . i.545 (b) 1.5241 1. The lowest-most resonance in figure 5.5275 1.5328 1. The red dashed line represents the location of the bare MC cavity mode while the horizontal green dashed lines in (a) mark the location of the electronic resonances.1: The extracted parameters of the simulated structure for each of the seven resonances Xi marked in figure 5.5306 0. By comparing these curves to the bare QW complex susceptibility calculations in figure 4. The fit parameters EX .53 EC (eV) 1. .525 1. reflect the coupling strength in the interaction regions between the photon and the excitonic fields. Furthermore.19(a) exhibits the largest splitting. The fitting curves lie very close to the calculated anti-crossing data.52 1.5274 1.525 1.52 1.11 we can make several observations.535 E (eV) E (eV) 1. as in figure 5.1 together with a list of manually obtained energetic locations of these resonances.5343 1.377 2.2.5315 0. expressed mainly by the depth of the particular reflection spectrum minima. We thus state the direct connection between the amplitude of the electronic resonance of the bare QW to the amount of splitting in the reflection spectrum. in correlation with the largest absolute amplitude of this particular resonance for the TE and TM susceptibility spectra of 4. extracted from figure 5.5306 1.4 3.03 1.5243 0.20.7109 X1 X2 X3 X4 X5 X6 X7 CM in figure 5.5375 0.765 607 1.535 1. ΩX .20 presents the calculated TM polarization reflection anti-crossing curves (blue dots. The fitting was performed using the model presented is section (5. the interaction strength of light-matter interaction in the cavity.535 1.18.5242 1. the locations of the resonance lines in the reflection anti-crossing curves are unsurprisingly similar to the susceptibility resonances.545 1. Numerical Results 115 (a) 1. First.e. Table 5.545 1.28 1.19: Reflection minima anticrossing curves. and their size is in direct correlation to the observed splitting between adjacent branches of the anti-crossing dispersion curves. Using this data we can also convince ourselves that the strong interaction condition.4.19(b)) with a superimposed coupled oscillator model fit results (red curves).5315 1.54 1.5. 5. coincide with the relative amplitudes of the susceptibility spectra resonances.525 1.66 1.18.5376 Coupled oscillator EX (eV) ΩX (meV) γX (10−3 meV) 1.54 1.1 19. marked by green horizontal dashed lines. The black dashed circles in both figures mark the interactions areas of the cavity mode and the first three excitons.66 1. Manual EX (eV) 1. ΩX and γX are listed in table 5.53 1. Fig.23 1.8 3.53 EC (eV) 1.09 1. while the next three columns contain the coupled oscillator fit parameters.525 1. 23 and 5.52 1. First. As for the undoped stricture case.20. Figures 5. The blue dotted lines are the simulation results. is satisfied for each one of the polaritonic branches.21. where in the left panel we replot the anti-crossing polariton curves.54 1.e.53 EC (eV) 1.545 Figure 5. The strong cavity mode interaction with the bare excitonic modes leads to relatively large admixing of elementary electronic excitations that are very different in nature. the presence of polaritons is verified. The numbering on the left is of the polaritonic branches. already shown in figure 5. where i = 1 − 7. i. we perform a reflection spectrum calculation using the TMM approach. we use the results presented in section 4. For the sake of continuity. together with the physical dimensions of the various epitaxial layers and the width detuning profile (as an inset). respectively for the TE and TM polarizations.116 Chapter 5. i. Figure 5. the characteristic strong coupling splitting in the cavity mode and bare exciton interaction regions can be clearly seen.2 in the previous chapter for ambient temperature of T = 2 K . but now we use the calculated bare QW electrical susceptibility to obtain the well layer refractive index for each of the considered 2DEG concentrations.5. The presence of the cavity mode (curve 1) in each one of the polariton branches can be clearly seen. together with two dominant excitonic contributions. while the red solid lines represent the coupled oscillator fitting curves. The results of this procedure for the TM polarization are presented in figure 5. The various resonance energies are marked by the green dashed horizontal lines and are denoted as Xi . γ = 1 × 1011 sec−1 . namely that ΩX > γX .20: The TM polarization reflection minima anticrossing curves fitted using the coupled oscillator model. calculated at T = 2 K . We present such a structure in figure 5.26 show the appropriate reflection anti-crossing curves for each 2DEG concentration.52 1. γC . these polaritonic branches remain visible in the structure .545 1. shown in the |αi.54 1. marks the location of the interaction region and is omitted for the other seven subplots. The red dashed line.535 E (eV) X7 X6 X5 X4 X3 X2 X 1.22.25 and 5. These reflection spectra and the derivative anti-crossing dispersion curves present several dominant effects. and on the right panel we present the admixing coefficients spectra. while on the right we enumerate the bare cavity mode (1) 2 and the bare excitons (2-8).j | plot. the polariton admixing coefficients can be extracted by a simple diagonalization of the obtained Hamiltonian.535 1. respectively for the TE and TM normal incident light polarizations. extracted from the individual reflection spectra by registering the reflection dip minima. As before.525 1 1.14). where each subplot in both figures corresponds to a certain ne value.26 also presents the results of the coupled oscillator model fitting in the form of superimposed red curves. Moreover.24 present a collection of calculated reflection spectra in a waterfall plot style (similar to figure 5. Figures 5.53 1.e. the width detuning parameter δ values rise from the bottom up for each of the subplots. Semiconductor Microcavities and Polaritonic Effect 1.3 Embedded Doped QW Now we turn to the case of the doped QWs embedded inside the cavity region of the MC structure. 5.525 1. the calculation parameters are ne = 1 × 109 ÷ 5 × 1011 cm−2 . Now we have performed the data coupled oscillator fit to the calculated reflection spectra.3. An example of the interaction region is marked by the dashed red line and the cavity mode is marked by C . Numerical Results 117 Figure 5.j | (right panel). .5.21: The TM polarization reflection minima anticrossing curves (left panel) together with respective polariton 2 admixing coefficients |αi.5. 3 E3 − E2 (meV) 2.5396 1.4 2.2 for the bare doped QW.26. As we have already seen for the undoped well case in table 5.3.7 8. extracted manually (red empty circles) and obtained from the coupled oscillator fit (blue dots) of the reflection spectra presented previously as a function of the 2DEG concentrations (in a semi logarithmic scale). Fig.22: Schematic profiles of the simulated microcavity with embedded doped QW in the cavity region. Another noticeable effect present in these figures is the gradual shift in the location of the bare excitonic resonance line.1.ଽ ݏܣ ^ƵďƐƚƌĂƚĞ ݏܣ݈ܣ ݏܣܽܩ Figure 5. We present only the first conduction and three valence subbands.26.3.27 presents the energetic locations of the reflection anti-crossing resonances. EXi . kF and .2: The calculated intersubband transition energies for a bare doped QW with ne = 7 × 1010 cm−2 and the difference between them.5389 ∆EE1 −HH2 (E3 ) (eV) 1.2 we summarize the intersubband transition energies at kt = 0.5367 1. together with the Fermi edge energy for this particular 2DEG concentration. contrary to the excitonic lines disappearance for high 2DEG concentrations observed in section 4.5443 E2 − E1 (meV) 4.ଵ ܽܩ. The possible transitions at kF are explicitly marked. seen especially well in figure 5. Semiconductor Microcavities and Polaritonic Effect ĂƉ >ĂLJĞƌ 710Հ 602Հ Z ϭϱ ƉĞƌŝŽĚƐ ܵ݅ ߜͲĚŽƉŝŶŐ ĂǀŝƚLJ ܵ݅ ߜͲĚŽƉŝŶŐ 1000Հ ߜ 1 1119Հ 200Հ Z Ϯϱ ƉĞƌŝŽĚƐ ݎ ݈ܣ.536 ∆EE1 −LH1 (E2 ) (eV) 1. We can thus deduce that the strong interaction of the QW resonances with the cavity mode photon delays the effect of the introduced free electrons on these bound resonances. In table 5.5323 1.9 E3 − E1 (meV) 6. the distance between these line remains approximately costant along the whole investigated diapason of 2DEG concentrations.28 presents the results of a QW conduction and valence subbands dispersion relations calculation for this particular case using the k · p method outlined in chapter 2. marked by green dashed lines in figure 5. 5. Fig. 5. It can be also seen that while the shift is not linear for each of the three resonances. say ne = 7 × 1010 cm−2 . as was discussed in section 4. kt 0 kF ∆EE1 −HH1 (E1 ) (eV) 1. This shift can be attributed to the shift in the Fermi energy with the rise of the electron concentration. we examine the energetic transitions present in a bare doped QW for a specific 2DEG concentration.3 5. Table 5.2 for the bare doped QW. This can be seen clearly for both polarizations of the incident light. To that end. the coupled oscillator model fit gives very similar results to the manually extracted points.118 Chapter 5. Now we can identify each of the resonance lines in the reflection anti-crossing plots.4 along much of the discussed 2DEG concentrations diapason (up to ne = 2 × 1011 cm−2 ). 52 1.52 1.53 1.52 1.52 1.52 1.23: The TE reflection spectra as a function of the cavity mode energy.u.54 1.52 1.54 Figure 5.54 −2 1.53 1.54 −2 1.54 −2 1.53 1.54 −2 1.53 1.54 N2DEG=8x1009cm−2 N2DEG=1x1010cm−2 N2DEG=2x1010cm−2 N2DEG=3x1010cm−2 N2DEG=4x1010cm−2 Reflection (a.53 10 1.54 N2DEG=5x10 cm N2DEG=6x10 cm N2DEG=7x10 cm N2DEG=8x10 cm N2DEG=9x10 cm−2 1.54 N2DEG=1x10 cm N2DEG=2x10 cm N2DEG=3x10 cm N2DEG=4x10 cm N2DEG=5x10 cm−2 1.52 1.52 1.54 1.54 −2 1.52 1. at T = 2 K .53 11 1.54 1.53 1.53 11 1.52 1.52 1.53 11 1.52 1.54 −2 1. Numerical Results 119 N2DEG=3x1009cm−2 N2DEG=4x1009cm−2 N2DEG=5x1009cm−2 N2DEG=6x1009cm−2 N2DEG=7x1009cm−2 1.54 −2 1.52 1. calculated for each of the considered 2DEG concentrations for various width detunings.53 10 1.52 1. .54 1.52 1.53 11 1.53 10 1.54 1.54 1.52 1.53 1.54 1.52 1.53 1.52 1.53 1.53 10 1.) 1.54 1.53 1.5.53 EC (eV) 1.53 11 1.52 1.54 −2 1.53 10 1.5.52 1. 53 11 1.53 1.54 1. .54 1.54 −2 1.52 1. Semiconductor Microcavities and Polaritonic Effect N2DEG=3x1009cm−2 N2DEG=4x1009cm−2 N2DEG=5x1009cm−2 N2DEG=6x1009cm−2 N2DEG=7x1009cm−2 1.53 10 1.53 10 1.53 1.54 1.54 1.53 1.52 1.52 1.52 1.) 1.52 1.52 1.53 11 1.52 1.52 1.53 1.54 1.53 11 1.54 1.53 1. calculated for each of the considered 2DEG concentrations for various width detunings. at T = 2 K .52 1.54 N2DEG=1x10 cm N2DEG=2x10 cm N2DEG=3x10 cm N2DEG=4x10 cm N2DEG=5x10 cm−2 1.54 −2 1.54 −2 1.54 −2 1.53 11 1.53 1.54 1.52 1.53 1.52 1.54 −2 1.54 −2 1.52 1.52 1.54 N2DEG=8x1009cm−2 N2DEG=1x1010cm−2 N2DEG=2x1010cm−2 N2DEG=3x1010cm−2 N2DEG=4x1010cm−2 Reflection (a.54 −2 1.52 1.120 Chapter 5.54 Figure 5.u.53 1.52 1.53 10 1.53 1.24: The TM reflection spectra as a function of the cavity mode energy.52 1.54 N2DEG=5x10 cm N2DEG=6x10 cm N2DEG=7x10 cm N2DEG=8x10 cm N2DEG=9x10 cm−2 1.52 1.53 EC (eV) 1.54 1.52 1.53 11 1.53 10 1.54 −2 1.53 10 1.52 1.52 1. 54 1.53 1.53 1.52 1.52 1.52 E (eV) 1.53 1.53 1.54 1.52 1.52 1.54 Figure 5.54 1.52 1.53 1.53 1.525 N2DEG=7x1010cm−2 1.54 1.53 1.52 1.525 N2DEG=6x1010cm−2 1.54 1.52 1.52 1.54 1.54 1.54 1.54 1.535 1.52 1.25: The TE reflection minima anticrossing curves (dotted blue) extracted for each 2DEG concentration from the reflection curves in figure 5.53 09 1.5.53 EC (eV) 1.53 1.535 1.52 N2DEG=1x10 cm 1.525 N2DEG=5x1011cm−2 1.535 1.53 10 1. The cavity mode location is marked by the diagonal red line in each subplot.54 1.525 N2DEG=8x1010cm−2 1.52 1.53 1.52 1.54 1.53 1.535 1.54 1.535 1.52 1.53 1.54 N2DEG=3x10 cm 1.52 1.525 N2DEG=2x10 cm 1.54 1.54 1.54 1.53 1.52 1.535 1.52 1.54 1.54 1.53 1.52 1.53 1.53 1.535 1.52 1.53 1.54 1.53 1.525 1.535 1.54 1.525 1.54 1.535 1.535 1.52 −2 1.54 1.52 1.535 1.53 1.5.54 N2DEG=5x1010cm−2 1.525 N2DEG=2x1011cm−2 1.52 1.52 1.52 1.52 −2 1.535 1.54 N2DEG=1x1011cm−2 1.53 10 1.53 1.53 1.52 1.525 N2DEG=5x1009cm−2 1.535 1.53 1.52 1.52 1.525 N2DEG=4x1011cm−2 1.54 1.535 1.535 1.53 1. The suitable 2DEG concentration is shown above each subplot.525 N2DEG=6x1009cm−2 1.525 1.535 1.52 1.525 1.52 1.53 1.52 1.53 1.54 1.53 1.525 1.54 1.53 1.525 1.52 −2 1.54 1.54 1.52 1.54 1.54 1.53 10 1.54 1.53 1.53 10 1.535 1.53 1.52 1. .535 1.54 N2DEG=8x10 cm 1.53 1.525 N2DEG=4x1009cm−2 1.52 1.54 1.54 N2DEG=4x10 cm−2 1.535 1.53 1.54 1.23.535 1.525 N2DEG=3x1011cm−2 1.52 1.53 1.525 1.54 1.54 1.525 N2DEG=9x1010cm−2 1.54 1.52 1.53 1.52 1.52 1.52 −2 1.53 1.525 N2DEG=7x1009cm−2 1. Numerical Results 121 N2DEG=3x1009cm−2 1.53 1. 534 1.522 1.26: The TM reflection minima anticrossing curves (blue dots) extracted for each 2DEG concentration from the reflection curves in figure 5. Semiconductor Microcavities and Polaritonic Effect E (eV) X1 X2 X3 1.528 1.53 Figure 5.532 1.526 1.122 Chapter 5.535 EC (eV) 1. 3x10 4x10 5x10 6x10 7x10 8x10 1x10 2x10 3x10 4x10 5x10 6x10 7x10 8x10 9x10 1x10 2x10 3x10 4x10 5x10 09 09 09 09 09 09 10 10 10 10 10 10 10 10 10 11 11 11 11 11 .5251. i = 1 − 3. with the superimposed coupled oscillator fitted curves (solid red). The respective 2DEG concentration is shown above each subplot in units of cm−2 and the three visible resonances (green dashed lines) are marked as Xi .524 1.538 1.23.536 1. 525 10 10 10 N2DEG (cm−2) 11 Figure 5.8 1 kt (cm−1) 1. respectively red and black dashed lines„ together with intersubband transition energies.53 1.28: Calculated first conduction (E1 ) and first three valence (HH1 . for T = 2 K .54 X3 1.6 1.525 (b) 1.8 x 10 2 6 Figure 5. LH1 and HH2 ) subbands dispersion relations for a bare doped QW with ne = 7 × 1010 cm−2 . .63×105 cm−1 −25 0 0.531 eV X1 1.5.53 1. 1550 1545 1540 E (meV) 1535 1530 1525 1520 0 E1−HH1 (kt=kF) E1−LH1 (kt=kF) E1 EF=1.2 0.2 1.6 0. obtained manually (red circles) and from the coupled oscillator fit results (blue dots) for (a) TE and (b) TM polarizations.534 eV 1.535 ETM (eV) X i 1. Numerical Results 123 (a) 1.5.4 1.54 7×1010 cm−2 1.535 ETE (eV) X X2 i 1.4 0.27: The reflection anti-crossing resonance energies as a function of 2DEG concentrations.5319eV HH −5 E1−HH2 (kt=kF) 1 E (meV) −10 LH1 HH 2 −15 −20 kF=6. The Fermi edge energy EF and wavevector kF are marked. we can state that the observed X1 .5356 eV and 1. i.9 meV obtained from the intersubband transitions calculation in table 5.27 for this 2DEG concentration.29.5 meV for ne = 7 × 1010 cm−2 . The vertical dashed line marks the location of the ne = 7 × 1010 cm−2 concentration values.29: The energetic distance between the anticrossing resonance lines as a function of 2DEG concentrations for the (a) TE and (b) TM polarizations.2. For the considered 2DEG concentration. Granted that the estimate for this renormalization energy. while the situation is the other way around for the TM polarization. Second. 4. the cavity mode coupling to the E1 − HH1 resonance (or exciton) is stronger than to the E1 − LH1 resonance for the TE polarization. First. the energetic differences between X1 and X2 resonances is 3 meV . 5. respectively. where the for the TE polarization (fig.26 and 4. In order to verify this observation. 4. Now that we have identified the nature of the various resonances present in the reflection anti-crossing curves. E1 − LH1 and E1 − HH2 intersubband transitions at the Fermi edge energy. .1).j 7 6 5 4 3 2 10 11 ∆EX −∆EX 3 1 ∆EX −∆EX 3 2 ∆EX −∆EX 2 1 7×1010 cm−2 10 10 N2DEG (cm−2) Figure 5. the E1 − HH1 .27 are 1. we calculate the differences between the resonance energies in figure 5.j 7 6 5 4 3 2 (b) 10 9 8 ∆ETM (meV) i.27). and visa versa for the TM polarization (fig. the energetic difference between them. as these transition energies do not take into consideration the bandgap renormalization effect inherent to the electrical susceptibility calculation. We can draw several conclusions from these plots. It is not surprising that the calculated transition energies do not match the values marked in figure 5.5386 eV . X2 and X3 resonances in the reflection anti-crossing curves are. by comparing the absolute values of these coupling coefficients to those obtained earlier for the MC structure with the undoped QW (using the values listed in table 5.e. obtained from figure 4.7(a).124 Chapter 5. and plot them graphically in figure 5.30 presents the coupling coefficients obtained for each resonance from the coupled oscillator model as a function of the 2DEG concentration. the general trends of the coupling coefficients can be clearly seen (marked by the respective dashed lines). By performing the same procedure for the all other 2DEG concentration values. It is clear that the E1 − HH1 resonance is much more susceptible to the change in the 2DEG concentration than the E1 − LH1 resonance. a value very close to the 2. at kt = kF . This switch in the coupling strength can be traced back to the relative amplitudes of the lowest two resonance of the bare QW absorption (and thus of the complex susceptibility) spectra in figures 4. Fig. These values are very close to the values of the Fermi edge transition energies. we discuss their various other properties. is ≈ −4. respectively for X1 and X2 .26) the lowest resonance is the dominant one. the actual transition energy for the two examined resonances in figure 5.27 for various 2DEG concentrations. Finally. we can state that the introduction of 2DEG inside the cavity region doesn’t impact substantially the coupling strength of the cavity mode photon to the electronic resonances present in the QW.27. Semiconductor Microcavities and Polaritonic Effect (a) 10 9 8 ∆ETE (meV) i. (a) 2.5 1 0.5 2 TM ∆EDist (meV) 1.31: The energetic distance between adjecent anti-crossing curves near resonance lines as a function of the 2DEG concentration for (a) TE and (b) TM polarizations.5 2 1. The dashed lines mark the general trend of the various collections of data points. . Numerical Results 125 (a) 3 2.5 2 1. as a function of the 2DEG concentration.5 E1−LH1 1 0.5.30: The coupling parameters extracted from the coupled oscillator fit of the reflection anti-crossing curves. for the (a) TE and (b) TM polarizations.5 0 (b) 3 2.5 0 10 10 10 N2DEG (cm−2) 11 Figure 5.5 0 E1−HH2 E1−LH1 E1−HH1 ¯ hΩT M (meV) ¯ hΩT E (meV) 10 10 10 N2DEG (cm−2) 11 Figure 5.5 E1−HH2 0 (b) 2.5 1 0.5.5 E1−HH1 2 TE ∆EDist (meV) 1.5 1 0. dependence on the 2DEG concentration. Fig. namely E1 − HH1 and E1 − LH1 .30. For comparison. While E1 − HH1 linewidths rise almost monotonically with the 2DEG concentration. We can clearly see that the TM polarization fitting parameters in figure 5.32: The resonance interaction linewidths extracted for various resonances using the coupled oscillator model fit of the anti-crossing curves as a function of the 2DEG concentration. obtained from the fit to the reflection spectra presented earlier.33 plotted as a function of the 2DEG concentration. The first is the gradual disappearance of the excitonic resonances due to the lowering of the coupling strength. for a single value of the width detuning parameter δ .32 presents the coupled oscillator linewidth parameter. Now we turn to discuss the linewidth properties of the calculated reflection spectra. respectively for the TE and TM polarizations.5 0 (b) 2 1. We can claim this based on the similarity of the absolute values of the respective curves in the two figures. As expected.34 presents the results of the FWHM calculation of the three major resonances present in figure 5. Here.9528 for the full diapason of the 2DEG concentrations for both polarizations. By comparing the obtained plots to figure 5. for (a) TE and (b) TM polarizations. we plot the bare MC cavity mode linewidth. The TE polarization fitted parameters in figure 5. Fig. 5.3 1 E1−HH2 Bare MC E1−LH1 E1−HH1 0. reflecting the dephasing of the bound excitonic states due to the rising 2DEG electron concentrations.5 γTE (meV) X 1.5 0 10 10 10 N2DEG (cm−2) 11 Figure 5. Semiconductor Microcavities and Polaritonic Effect (a) 2 1.5.32(b) are unintelligible and thus should be ignored. Fig. γX . These figures reveal two additional effects. 5. the broadening of the resonance lines be clearly seen from the low concentrations up to the higher ones. the most dominant resonances. the there is a consistency between the results for both light polarizations. First of all. we perform a manual calculation of the FWHM of the reflection spectra along multiple values of the 2DEG concentration.32.3 eV .31 presents the Euclidean distances between the various polaritonic branches in the reflection anti-crossing curves in figures 5. we can convince ourselves that this.26.1 to be 1. exhibit quite different behavior. in fact. 5. can be an alternative method for the calculation of the cavity mode photon coupling to the QW excitons. discussed earlier in this thesis. where we first we observe a declining trend up .5 γTM (meV) X 1 0.33 presents the reflection spectra calculated at δ = 0. 5.32(a) exhibit a clear trends (marked by the dashed lines) towards rising linewidths of all three visible resonances. the obtained plots are somewhat similar to the coupled oscillator fitting linewidth parameters in figure 5. and the appearance of the discussed switch of the E1 − HH1 and E1 − LH1 coupling strength between the two polarizations. found to be during the calculations in section 5.25 and 5. Furthermore. Fig. and the second is the gradual shift of these fading resonance lines towards higher energies with the rising 2DEG concentrations. but there are also several differences.126 Chapter 5. In order to further validate these results. The dashed line mark the general trend of the appropriate resonance linewidth parameter. the E1 − LH1 linewidths have a somewhat more complex trajectory. 0x1009cm−2 (b) N2DEG=6.0x1010cm−2 N2DEG=5.0x10 cm 09 −2 N2DEG=2.0x1011cm−2 N2DEG=9.95.535 1.0x1011cm−2 N2DEG=2.0x1009cm−2 N2DEG=4.33: Reflection spectra for various 2DEG concentrations calculated at width detuning of δ ≈ 0.525 E (eV) 1.0x1009cm−2 N2DEG=5. for (a) TE and (b) TM polarizations.0x1011cm−2 N2DEG=4. (a) 2 1.5 0 10 10 10 N2DEG (cm−2) 11 Figure 5.0x1010cm−2 N2DEG=8.525 E (eV) 1.5 ETM (meV) FWHM 1 0. Numerical Results 127 N2DEG=5.0x1011cm−2 N2DEG=4.0x1011cm−2 N2DEG=9.0x1010cm−2 N2DEG=2.0x1010cm−2 N2DEG=1.0x1009cm−2 1.0x1010cm−2 N2DEG=2.0x1010cm−2 N2DEG=1.0x1010cm−2 N2DEG=5. extracted manually (via FWHM calculation) for each of the present resonances at width detuning of δ ≈ 0.0x1009cm−2 N2DEG=8.0x1010cm−2 N2DEG=6.0x1010cm−2 N2DEG=7.0x1011cm−2 N2DEG=3.52 1.0x1009cm−2 N2DEG=4.535 Figure 5.0x1010cm−2 N2DEG=8.0x1009cm−2 N2DEG=6.0x1010cm−2 N2DEG=3.0x1011cm−2 N2DEG=1.0x1009cm−2 N2DEG=7.95 .5 E1−HH1 0 (b) 2 1.0x1009cm−2 N2DEG=3.34: The reflection spectra linewidths function of the 2DEG concentration.5.0x1009cm−2 N2DEG=8.0x1010cm−2 N2DEG=7.0x1010cm−2 N2DEG=9.52 1.53 1.0x1011cm−2 N2DEG=2. .0x1009cm−2 N2DEG=6.0x1009cm−2 N2DEG=5.0x1010cm−2 Reflection N2DEG=4.53 1. for the (a) TE and (b) TM polarizations of the incident light.0x10 cm 10 −2 (a) N2DEG=5.0x1010cm−2 N2DEG=3.0x1010cm−2 Reflection N2DEG=4.0x1009cm−2 N2DEG=3.0x1009cm−2 N2DEG=2.0x1011cm−2 N2DEG=1.5.0x1010cm−2 N2DEG=9.0x1011cm−2 N2DEG=3.5 ETE (meV) FWHM E1−LH1 1 E1−HH2 0.0x10 cm 09 −2 N2DEG=7. Finally.53 1. a discussion which is outside the scope of this thesis. four of these branches are present. As the manual FWHM calculation of the various spectral dips for each reflection spectrum for every value of the width detuning parameter δ is a rather daunting task.40 the polariton admixing coefficients extracted through the coupled oscillator model fitting of the reflection anti-crossing spectra.5 1 0. For the low 2DEG concentration.6 0.52 1.525 1. Figures 5. we can easily see that the strong interaction condition that ΩXi > γXi is maintained for most of the discussed 2DEG concentrations diapason.38. Comparing the calculated coupling strengths between the cavity mode photon and the excitonic resonances. presented above in figures 5.35.4 0. while with the rise of concentration the high energy resonances phase and only strongest of these remain.52 1.2 0 TM 1.5 0 1.2 0 1 2 3 4 1 0. to the linewidths discussed above.52 1.54 EC (eV) Figure 5. are all determined through a least-squares fitting procedure. wi and Ei . The previous discussion considered the evolution of the polaritonic linewidths with the 2DEG concentration in the considered diapason. Semiconductor Microcavities and Polaritonic Effect TE 1 0.53 1.4 0. 3 × 1010 and 1 × 1011 cm−2 2DEG concentrations.54 2 1. 2 2π (E − Ei ) + (wi /2)2 (5. we opt to a more approximate approach.54 1.5 0 2 1. Here. respectively for ne = 3 × 109 .535 1.36 and 5.39 and 5. the strong presence of the cavity mode photon is clearly seen.8 0.535 1. In each of these figures we present the amplitudes of the transmitions peaks and their width for both light polarizations. The amplitudes Ai in each figures (and for each polarization) present a picture of the relative admixture of the various polaritonic branches in the resulting reflection spectrum picture. maintaining its strong coupling with the individual excitonic resonances and thus ensuring their survival up the very high 2DEG concentration values.31. Moreover.525 1.6 Ai 0. to ne ≈ 1 × 1010 cm−2 which then reverts and rises steadily up to the high 2DEG concentrations. 5.128 Chapter 5.525 1.53 1.535 1.525 1.37 present the results of such a fit. we convert the reflection spectrum to transmition and subsequently fit the obtained curve with a Lorentzian target function of the form T = 1 − |r | = i 2 Ai 1 wi .52 1. as for the undoped QW case.2) where the parameters Ai .5 wi (meV) 1 0. This complex behavior can be analyzed qualitatively by considering the lineshapes of the individual excitons interacting with the cavity mode electron [66].35: The fitted amplitude and width of the transmition peaks as a function of cavity mode energy. Now we turn to the calculation of the linewidths dependence on the cavity mode energy for a particular gas concentration.5.535 1. we present in figures 5. For each value of δ .8 0.53 1. a collective energetic shift can be seen with the rise of the 2DEG concentration in all of the present amplitude curves.54 EC (eV) 1. . similarly to procedure employed in the previous section. including the range of i. 5.30 and 5. but the overall trend of rising linewidth with the 2DEG concentration is still noticeable. The FWHM curves exhibit a somewhat more complicated behavior. obtained for each resonance branch (numbered 1-4) for the TE (left panel) and TM (right panel) at ne = 3 × 109 cm−2 . 53 1.36: The fitted amplitude and width of the transmition peaks as a function of cavity mode energy„ obtained for each resonance branch (numbered 1-3) for the TE (left panel) and TM (right panel) at ne = 3 × 1010 cm−2 .8 0.535 1.5.2 0 2 1 3 1 0.525 1.535 1.525 1.8 0.4 0.53 1.54 1.52 1.5 0 2 1.5.4 0.5 0 2 1.525 1.54 EC (eV) Figure 5.5 wi (meV) 1 0.4 0.52 1.6 0.2 0 TM 1.53 1. TE 1 0.54 1.54 2 1.54 2 1.5 1 0.53 1.2 0 TM 1. Numerical Results 129 TE 1 0.5 0 1.4 0. .52 1. obtained for each resonance (numbered 1-3) for the TE (left panel) and TM (right panel) at ne = 1 × 1011 cm−2 .525 1.8 0.52 1.5 wi (meV) 1 0.535 1.6 Ai 0.53 1.53 1.54 EC (eV) 1.52 1.6 Ai 0.52 1.525 1.5 0 1.52 1.2 0 2 1 3 1 0.535 1.525 1.53 1.525 1.52 1.525 1.535 1.535 1.53 1.54 EC (eV) Figure 5.54 EC (eV) 1.535 1.6 0.37: The fitted amplitude and width of the transmition peaks as a function of cavity mode energy.535 1.8 0.5 1 0. TE 1.525 1.545 | 1.54 1.535 E (eV) 1.53 1.5 0 1 α 0.39: The polariton admixing coefficients extracted from the coupled oscillator model fit of the anti-crossing curves for ne = 3 × 1010 cm−2 . Semiconductor Microcavities and Polaritonic Effect TE 1.53 1.5 0 1 0.535 E (eV) 1.i|2 | 2.52 1.53 E (eV) 1.i|2 1.i|2 | 1.5 0 1 α 0.5 0 1 0.i|2 α 0.5 0 1.52 1.5 0 1 0.535 E (eV) 1.52 1.130 Chapter 5.i|2 3 4 P2 P1 TM | 4.5 0 1 α 0.38: The polariton admixing coefficients extracted from the coupled oscillator model fit of the anti-crossing curves for ne = 3 × 109 cm−2 .5 0 1 0. .52 1 0.535 α α α α α α α α 1 1 2 P4 | 2.53 1.i|2 4 1 1.525 1.54 1.525 1.i|2 P3 2 3 1 | 3.53 E (eV) 1.545 | 1.54 Figure 5.5 0 1 α 0.i|2 1.5 0 1 1.i|2 | 3.5 0 1.535 E (eV) 1.5 0 1 0.54 1.535 Figure 5.5 0 1 0.54 1.53 1.52 1.525 1.i|2 TM | 1.i|2 | 3.525 1.i|2 | 2.535 EMC (eV) 1.5 0 1 0.535 EMC (eV) 1.525 1.54 α 0.53 1.525 1.i|2 | 2.525 1.52 1 | 4.i|2 | 3.53 1. 52 1.54 1.52 α 0.5. .53 1.53 1.5 0 1.i|2 | 2.52 1.535 EMC (eV) 1.525 1.525 1.525 1.i|2 1.53 E (eV) 1.i|2 TM | 1.535 Figure 5.5 0 1 α 0.53 1.40: The polariton admixing coefficients extracted from the coupled oscillator model fit of the anti-crossing curves for ne = 1 × 1011 cm−2 .5 0 1 α 0.525 1.5 0 1 α 0.5 0 1 1.5.54 α 0.i|2 | 2.i|2 | 3. Numerical Results 131 TE 1 | 1.5 0 1 α 0.i|2 | 3.535 E (eV) 1.54 1.535 E (eV) 1. the semiconductor. Second. such as bare quantum wells and semiconductor microcavities. Using this approach we compute the electrical susceptibility and spontaneous emission spectra of the structure. causing their effective disappearance for concentrations above ≈ 6 × 1010 cm−2 . we next consider microcavities and their optical properties in chapter 5. we concern ourselves with the two-band variation of this method. In chapter 3. we add to our quantum mechanical approach a complementary electrostatic part in the form of Poisson equation solution for the specific charge distribution. which are in good agreement with other theoretical models discussed in literature. Through this approach we attain doping dependent corrections to the band edges of the considered QW. Using this approach we calculate the dispersion relations of the conduction and valence subbands in a bulk semiconductor and single bare quantum well. and only simple dephasing mechanism is used. i. they enable us to compute various optical parameters of the bare QWs under the assumption of dipole interaction of the electronic system with the optical field. Our approach to this task is semiclassical in its core. mainly due to its implementation simplicity. where only the matter. After a general introduction of the approximate k · p approach. we reach the expected conclusion that the introduction of the Coulombic interaction changes dramatically the obtained results. This observation allows us to simplify our calculations by assuming full angular symmetry of the QW in the k -space without a substantial loss to the generality. through which the absorption and refractive indices can be easily computed. through the dynamics of the spontaneous emission and the absorption spectra with the 2DEG carriers concentration. We complicate this simple picture in chapter 4 by the introduction of Coulombic interaction between the electrons and holes in the structure. we observe a Burstein-Moss shift causing a split between the two types of spectra for high concentrations. the added 2DEG carriers lead to the appearance of a characteristic inhomogeneous broadened line in the spontaneous emission at low ambient temperature of 2 K . is treated quantum mechanically. we consider the introduction of doping inside the quantum well and explore its influence on the electronic properties. This computation is then performed for QWs with and without the presence of 2DEG carriers inside the well region. The observed bandgap renormalization for the Coulomb correlated model results also match the classical theoretical dependence of the 2DEG carriers concentration. and for a wide diapason of these carriers’ concentrations. The introduced Coulomb enhancement of reproduces strong bellow-bandgap resonance lines for the undoped QW absorption and spontaneous emission spectra. which introduce minor corrections to the conduction and valence subband energies in the middle of the first Brillouin region. The results of the doped QW spectra computation lead to several noticeable effects. Although these models are both approximate in nature. together with a simple screening scheme to model the influence of the introduced high concentration 2DEG. we turn to the investigation of the optical properties of these structures.Summary and Outlook In this work we presented a theoretical study of the various aspects of light-matter interactions taking place in variety of quantum structures. As a consequence of the attained results for the bare QW. The influence of the crystallographic orientation on the results is examined and found to be minor in the relevant portion of the k -space. First. the introduction of additional 2DEG carriers leads to a significant reduction in the strength of the observed absorption resonance lines. all these for various ambient temperatures. followed by a short discussion of the coupling of a quantum mechanical 132 . By comparing the computation results of these two models. Being in a position to model the electronic and optical properties of a bare QW. we first assume that the carriers present in the semiconductor are free and noninteracting. Next. we chose the Coulomb correlated model to be used for the modeling of such structures in the subsequent calculations. and the influence of the introduced 2DEG on their optical properties. We start our discussion in chapter 2 by presenting the electronic properties of quantum well structures.e. Once we have obtained the electronic properties of bare QWs with introduced 2DEG of various concentrations. Finally. First we consider the reflectivity properties of such structures and the numerical method for their calculation. To this end. further research should be performed to gain better understanding of the origin of the strong interaction between the confined cavity mode and the electron-hole pair resonances at the Fermi edge. we identify them to be unbound electron-hole resonances at the Fermi edge energy. especially when the linewidths of the various polaritonic branches are considered. • In addition to the improved modelling of the bare QW. Numerical Results 133 absorber with the light field mode inside the cavity region.5. extracted through the fitting procedure. a doped bare QW is introduced inside the cavity region of the MC and the reflection spectra are calculated for various values of 2DEG concentration. Although this work presents a complete theoretical framework for the calculation of the optical properties of MC with embedded QWs. . do not take into account higher order correlation effects and relaxation mechanisms. reaffirm this observation. Such extensions are farely easy to introduce into the current framework. Dispersion calculations in weak magnetic fields should provide a clue as to the nature of the various polaritonic branches. Here we present several possible pathways that may taken to address this issues: • As we have stated above. a cavity dampling mechanism can be introduced to the MC modeling in order to approximate the cavity mode behavior to the physical reality. the spontaneous emission spectra should also be calculated and compared. • Since excitons and unbound electrons and holes exhibit very different dependence on magnetic field (respectively diamagnetic and linear). • An angular calculation of the reflection spectra of the MC with embedded QW can be performed. in order to obtain the dispersion relation of the various polaritonic branches and thereby attain a better insight into their electronic nature. This indicates to a higher coupling of the found electron-hole resonances with the cavity mode than the bare QW case. in contrast to the observed disappearance of all such resonances for the bare QW. we find them to be generally larger from those obtained for the undoped QW case. the polariton admixing coefficients. As the next logical step. This observation is backed by the the parameters of a fit of the anti-crossing curves with the coupled oscillator model. and stands as the central finding of this work. By comparing the energetic locations of these resonances to the bare QW allowed energetic transitions.5. and thereby investigate the coupling of the cavity mode with the various bare QW resonances. seen through the characteristic splittings of the anti-crossing curves in the regions of interaction. This finding coincides with earlier experimental results [6] for similar structures. • In addition to the calculation of the reflection spectra of the entire MC. The reflection spectra of the MC with undoped QW inside the cavity region revealed a strong coupling of the cavity mode with the various QW resonances. effectively introducing a QW inside the cavity region of the MC. both the free carrier and Coulomb correlated models. • The temperature analysis performed in this work for the bare QW should be extended to the full MC calculations. Moreover. the electronic model of the bare QW should be extended to include the influence of external magnetic field. probably at the expense of computational complexity. used to calculate the bare QW electrical susceptibility. This procedure enables us to perform an investigation of the reflection spectra of the entire MC structure. When we compare the coupling strength parametes. despite the existence of Coulomb attraction between the electron and hole. as it may facilitate in differentiating between excitons and unbound electron-hole pair resonances. We conclude by stating that the main mechanism for the coherence between all the electron-hole pair excitations interacting with the confined cavity mode is the strong interaction itself. extracted using the coupled oscillator fit of the reflection anti-crossing curves for various 2DEG concentrations. The results of the bare QW optical properties calculations are incorporated into the full MC reflectivity computation through an altered form of the linear dispersion model. The most striking feature of the obtained reflection anti-crossing curves is the appearance of strong coupling between the cavity mode photon and certain QW resonances at relatively high 2DEG concentrations. Appendices 134 . . direct 1 A crystallographic point group is a set of symmetry operations. there are only a few distinct irreducible representations. the symmetry operations are listed in Table A. The basis functions for all irreducible representations are given in Table A.1) The kinetic operator T is naturally invariant under the action of the elements in the point group G. Γ2 . the point group G is a closed group. which is assumed to be of the dimension N .1. The elements are given by the 24 symmetry operations mapping a tetrahedron to itself. 135 . (A. leads to H (g −1 r)ψ (g −1 r) = H (r)ψ (g −1 r) = Eψ (g −1 r). leaving the wavefunction invariant. and let G be the point group1 of the crystal such that the symmetry operator g ∈ G. spanning a function space Iψ = ψ (r)|ψ (r) = ψ (g −1 r) ∀g ∈ G . An example of the Γ1 representation is given by the conduction band at the Γ point of zinc-blende. the function space Iψ is invariant under the action of G. (A. is left unchanged under the action of the elements of the group Td . Γ1 and Γ2 are one dimensional representations. Γ12 is a two dimensional and Γ15 and Γ25 are three dimensional. ψN } is the basis of the subspace Iψ . each point group corresponds to a crystal class. For an appropriate alignment and orientation. ∀g ∈ G. i. the tetrahedral group. . Therefore.14) with the kinetic operator T and crystal potential U (r). The wavefunction ψ (r) might be the same wavefunction as ψ (r) or might be linearly independent.3) Now. that leave a point fixed while moving each atom of the crystal to the position of an atom of the same kind. ψ1 . Let H (r) = T + U (r) be the Hamiltonian in (2. The basis is denoted as being the irreducible representation of the group G. . also denoted as trivial representation. In the following. Γ15 and Γ25 .e.4) Suppose that {ψ0 . The desirable property deduced from group theory is that for every symmetry group. i. Every irreducible representation describes an unique way how wavefunctions are transformed under the action of the elements of a symmetry group G. which are commonly labeled as Γ1 . The representation Γ1 is the identity representation. ψ (r) ∈ Iψ ⇒ ψ (g −1 r) ∈ Iψ . (A. That is. U (g −1 r) = U (r). f ∈ G : gf ∈ G. the crystal potential is invariant.Appendix A Symmetry Properties of Wavefunctions In this appendix we present a short overview of the symmetry properties of bulk semiconductor wavefunction. expressed in the mathematical formulation of the group theory.0. In the classification of crystals.2) Obviously ψ (r) = ψ (g −1 r) is also an eigenfunction of the Hamiltonian operator H (r) with eigenvalue E . like rotations or reflections. Td has in total five distinct irreducible representations.0. (A. for which the symmetry group of the Hamiltonian is Td . meaning that for g. Therefore. Assuming that ψ (r) is an eigenfunction of H with H (r)ψ (r) = Eψ (r) and applying the symmetry operator g . .1. An atomic wavefunction with s-like symmetry transforms accordingly.0. the eigenfunctions of the crystal Hamiltonian H can be classified according to the irreducible representation they form. the theory will be focused on the zinc-blende crystal. Γ12 . It is clear that each eigenfunction does only belong to one irreducible representation and it is clear that eigenfunctions corresponding to different irreducible representations are always orthogonal.e. Applying all symmetry operations g ∈ G therefore results in a set of wavefunctions with the same eigenvalue E .0.2. an infinite crystal would look exactly the same before and after any of the operations in its point group. Γi Γ1 Γ2 Γ12 Γ15 Γ25 Dimension 1 1 2 3 3 Basis functions xyz x4 y 2 − z 2 + y 4 z 2 − x2 + z 2 x2 − y 2 2 2 x2 − y 2 . Neglecting the later considered spin-orbit splitting. . (¯ zy ¯x) .) (xyz ) (¯ xy ¯z ) . (xz ¯y ¯) Table A. from which the vanishing momentum matrix elements (2. (A. (¯ xy z ¯) (zxy ) . the decomposition of the direct product into the direct sum is given in Table A. As an example. The p-type basis functions correspond to the representation Γ15 . y.5) Recall that functions not belonging to the same irreducible representation are orthogonal. The goal is now to use the introduced group theory to analyze the properties of the momentum matrix elements (2. the valence band is threefold degenerate. Another important irreducible representation is given by the top of the valence band at the Γ point. which can only be met if the matrix element is zero.17) can be calculated. but can be decomposed into a direct sum of irreducible representations Γ15 ⊗ Γj = u ⊕Γi .g. z and v = 1. (¯ xz ¯y ) . Therefore.3.1: Symmetry operations of the group Td using the Schönflies notation (notations after [10]). X denotes a non-vanishing and the 0 denotes a vanishing matrix element.17). (y x ¯z ¯) (¯ y xz ¯) . .2 and symmetry operations of G. (zyx) . 46). For the tetrahedral symmetry. as one has pu acting on the basis function ψv for u = x. Γ1 is represented by the function xyz and Γ15 by x. p.0. Type E 3C2 8C3 6C4 6σ Operation (xyz ) → (. (y z ¯x ¯) . The corresponding representation is given by the direct product Γ15 ⊗ Γj (see [10]. e.according to Table A. z x2 − y 2 bandgap III-V semiconductors.2: Basis functions of the tetrahedral symmetry group Td . with p-type basis functions x. N . (yxz ) . transforming under the action of Td according to the elements of a vector. y. They are given by the matrix in A. the only non-vanishing momentum matrix elements involve . (z y ¯x ¯) . (¯ z xy ¯) (¯ xz y ¯) . the momentum operator forms an irreducible representation of Γ15 . xyz | px | z . For the other. y and z . The direct product is not irreducible. .0.5) of the direct product Γ15 ⊗ Γj contains Γi . . The action of the momentum operator on wavefunctions of the irreducible representation Γj leads to a new expression. for the Γ1 type conduction band. y z 2 − x2 . (¯ y xz ¯) . equivalent matrix elements can be determined using the basis functions defined in Table A.1. (xy z ¯) . Symmetry Properties of Wavefunctions Table A. (z x ¯y ¯) .4. The conduction band at the Γ point is non-degenerate with a wavefunction obeying s-type symmetry. (¯ zyx ¯) . (¯ yzx ¯) . In order to further reduce the number of unknowns. (¯ zx ¯y ) . the matrix element ψ Γi | p | ψ Γj between two wavefunctions belonging to the irreducible representation Γi and Γj is nonzero only if the direct sum (A.. (xzy ) . z x y 2 − z 2 . pz obviously transforming like the elements of a vector. Within the matrix..4 . y and z . The momentum operator p is given by a vector of three operators px p = py . The only non-vanishing matrix element is of the type xyz | px | x . (¯ yz ¯x) . Therefore. a rotation of the crystal by 180° around the rotation axis [001] results in xyz | px | x = − xyz | px | z .136 Appendix A..only bands belonging to Γ15 . z 2 − 1 2 x +y x.1. (yzx) . p ψ Γ1 ψ Γ2 ψ Γ12 ψ Γ15 ψ Γ25 ψ Γ1 0 0 0 X 0 ψ Γ2 0 0 0 0 X ψ Γ12 0 0 0 X X ψ Γ15 X 0 X X X ψ Γ25 0 X X X X . A zero denotes a vanishing and X a non-vanishing element.3: Direct products of the Γ15 representation with all representations of Td (after [10]).137 Table A. Direct product Γ15 ⊗ Γ1 Γ15 ⊗ Γ2 Γ15 ⊗ Γ12 Γ15 ⊗ Γ15 Γ15 ⊗ Γ25 Direct sum Γ15 Γ25 Γ15 ⊕ Γ25 Γ15 ⊕ Γ25 ⊕ Γ12 ⊕ Γ1 Γ15 ⊕ Γ25 ⊕ Γ12 ⊕ Γ2 Table A.4: The non-vanishing momentum matrix elements between the states corresponding to different irreducible representations of the tetrahedral symmetry group. For this purpose. we can expand the first and second derivatives in terms of finite differences.6) − df dz z −δz 2δz f (z + 2δz ) − 2f (z ) + f (z − 2δz ) .3) (B.0.2. ∂z√ ∂z ∂ for [100] √3kt γ2 (z )kt − 2γ3 (z ) ∂z ∂ 3kt γ3 (z )kt − 2γ3 (z ) ∂z for [110] − (B. In order to allow for a location dependent effective mass. we consider the general one-dimensional form of the Hamiltonian equation (2.4) Fhh Flh = E (k) Fhh Flh . (B.0. and Fhh and Flh represent the hole wavefunction while under the effective mass and envelope function approximations.0. (B.7) . ∂z ∂z ∂ ∂ 2 − (γ1 (z ) − 2γ2 (z )) + (γ1 (z ) + γ2 (z )) kt .Appendix B Two-Band Model Numerical Implementation The method used to solve the Hamiltonian equations in the two-band model is based on the shooting method formulated in [33] for the simple case of the conduction band.5) The problem now is to find a numerical method for the the solution of both the energy eigenvalues E and the eigenfunctions F for any V (z ).2) (B. dz ∆z 2δz The second derivative follows as d2 f dz 2 ≈ = df dz z +δz (B.0.0. (2δz )2 138 (B. As a starting point.0. The first derivative of a function f (z ) can be approximated to df ∆f f (z + δz ) − f (z − δz ) ≈ = . We can rewrite the effective mass equations as ˆ hh + V (z ) − E H ˆ† W ˆ W ˆ Hlh + V (z ) − E Fhh Flh = 0.1) The potential V (z ) describes the valence band edge of the quantum well structure (in terms of hole energy).55). we rewrite this equation as ˆ hh + V (z ) ˆ H W † ˆ lh + V (z ) ˆ H W with ˆ lh H ˆ hh H ˆ W = = = ∂ ∂ 2 (γ1 (z ) + 2γ2 (z )) + (γ1 (z ) − γ2 (z )) kt .0. 139 As δz is an, as yet, undefined small step along the z -axis, and as it only appears in equation (B.0.7)with the factor 2, then we can simplify this expression by substituting δz for 2δz f (z + δz ) − 2f (z ) + f (z − δz ) d2 f ≈ . dz 2 (δz )2 (B.0.8) ˆ 0 = − ∂ (γ1 + 2γ2 ) ∂ in the light hole Hamiltonian, and express it in terms of finite Let us focus on the term H lh ∂z ∂z differences. We can rewrite this term as ∂ ∂Flh ∂ 2 Flh 0 ˆ lh . H =− (γ1 (z ) + γ2 (z )) + (γ1 (z ) + γ2 (z )) ∂z ∂z ∂z 2 (B.0.9) However, the shooting equations derived from this point by expanding the derivatives in terms of finite differences have led to significant computational inaccuracies in systems with a large discontinuous change in the effective mass (the Luttinger parameters), as occurs in the GaAs/AlGaAs system. The source of the inaccuracies is thought to arise from the δ -function nature of the effective mass derivative. ˆ 0 starting from the left-hand derivative A more robust scheme can be derived by expanding H lh 0 ˆ lh H ≈ lh (γ1 + γ2 ) ∂F ∂z z +δz lh − (γ1 + γ2 ) ∂F ∂z z −δz 2δz . (B.0.10) Recalling the centered finite difference expansion for the first derivative (B.0.6), we can write the numerator of the above expression as 0 ˆ lh 2δz H = (γ1 + 2γ2 )|z+δz (γ1 + 2γ2 )|z−δz Flh (z + 2δz ) − Flh (z ) 2δz Flh (z ) − Flh (z − 2δz ) , 2δz (B.0.11) or 0 ˆ lh (2δz )2 H = (γ1 + 2γ2 )|z+δz [Flh (z + 2δz ) − Flh (z )] (γ1 + 2γ2 )|z−δz [Flh (z ) − Flh (z − 2δz )] . (B.0.12) Making the transformation 2δz → δz then yields 0 ˆ lh H = 1 + (γ1 − 2γ2 ) Flh (z + δz ) (δz )2 − (γ1 − 2γ2 ) + (γ1 − 2γ2 ) + (γ1 − 2γ2 ) Flh (z − δz ) , − + − Flh (z ) (B.0.13) with (γ1 + 2γ2 ) ( γ1 + 2 γ2 ) + = = = = (γ1 + 2γ2 )|z+δz/2 , (γ1 + 2γ2 )|z−δz/2 , (γ1 − 2γ2 )|z+δz/2 , (γ1 − 2γ2 )|z−δz/2 . (B.0.14) (B.0.15) (B.0.16) (B.0.17) − + (γ1 − 2γ2 ) (γ1 − 2γ2 ) − ˆ 0 and a similar expression for the heavy-hole We now substitute the finite difference expressions for ∂/∂z , H lh 140 Appendix B. Two-Band Model Numerical Implementation counterpart into the effective mass equations, and obtain 0 = − − (γ1 − 2γ2 ) (γ1 − 2γ2 ) + (γ1 − 2γ2 ) Fhh (z + δz ) + Fhh (z ) 2 (δz ) (δz )2 − + + − 0 (γ1 − 2γ2 ) 2 Fhh (z − δz ) + (γ1 + γ2 )kt Fhh (z ) (δz )2 √ √ Flh (z + δz ) − Flh (z − δz ) 2 + (V (z ) − E ) Fhh (z ) + 3γ2 kt , Flh (z ) − 2 3γ3 kt 2δz √ √ Fhh (z + δz ) − Fhh (z − δz ) 2 = 3 γ2 kt Fhh (z ) + 2 3γ3 kt kt 2δz + − + (γ1 + 2γ2 ) + (γ1 + 2γ2 ) ( γ1 + 2 γ2 ) F ( z + δz ) + Flh (z ) − lh (δz )2 (δz )2 − ( γ1 + 2 γ2 ) 2 Flh (z − δz ) + (γ1 − γ2 )kt Flh (z ) + (V (z ) − E ) Flh (z ). (δz )2 − (B.0.18) (B.0.19) The Luttinger parameters γi can be found at the intermediary points z ± δz/2 by taking the mean of the two neighboring points z and z ± δz . It can be seen that we draw up a set of finite difference equations if we map the potential V (z ) and the Luttinger parameters γi to a grid along the z -axis. To solve these coupled equations and find the energies E and functions F we assume a equidistant grid zi , with a grid step δz , we can substitute z → zi , z − δz → zi−1 and z + δz → zi+1 . If we assume a given energy E , we are still left with 6 unknown parameters in the finite difference equations. However, we can rewrite these equations so that we are able to find Flh (zi+1 ) and Fhh (zi+1 ) from their values at the two previous nodes, zi−1 and zi Fhh (zi+1 ) 1 + 3 2 γ3 (γ1 + 2γ2 ) (γ1 − 2γ2 ) Fhh (zi−1 ) − + 2 2 − kt (δz ) − + = +3 γ3 2 γ3 + 2 2 + kt (δz ) − + (γ1 − 2γ2 ) √ (γ1 − 2γ2 ) 3 ( γ1 + 2 γ2 ) ( γ1 − 2 γ2 ) + kt δz − + Flh (zi−1 ) Fhh (zi ) + ( γ1 − 2 γ2 ) + + 1+ + ( γ1 + 2 γ2 ) γ1 + γ2 (γ1 + 2γ2 ) (γ1 − 2γ2 ) + (γ1 − 2γ2 ) (γ1 − 2γ2 ) 2 + (δz ) ( γ1 − 2 γ2 ) 2 2 + kt (δz ) V (zi ) − E ( γ1 − 2 γ2 ) (γ1 + 2γ2 ) (γ1 − 2γ2 ) √ γ2 2 2 + Flh (zi ) 3 + kt (δz ) (γ1 − 2γ2 ) γ3 + + + −3 γ3 γ2 + 3 3 + kt (δz ) − √ 3 kt δz (γ1 − 2γ2 ) ( γ1 + 2 γ2 ) √ γ3 (γ1 − γ2 ) 3 3 − 3 + + kt (δz ) (γ1 + 2γ2 ) (γ1 − 2γ2 ) √ γ3 3 − 3 , (B.0.20) + + (V (zi ) − E ) kt (δz ) (γ1 + 2γ2 ) (γ1 − 2γ2 ) (γ1 + 2γ2 ) + (γ1 + 2γ2 ) − 141 2 γ3 + Flh (zi+1 ) 1 + 3 (γ1 + 2γ2 ) (γ1 − 2γ2 ) Flh (zi−1 ) − 2 2 + kt (δz ) − + = +3 γ3 2 γ3 + 2 2 + kt (δz ) − + ( γ1 + 2 γ2 ) √ (γ1 + 2γ2 ) 3 (γ1 + 2γ2 ) (γ1 − 2γ2 ) + kt δz − + Fhh (zi−1 ) Flh (zi ) + ( γ1 + 2 γ2 ) + + 1+ + (γ1 − 2γ2 ) (γ1 − 2γ2 ) (γ1 + 2γ2 ) + (γ1 + 2γ2 ) (γ1 + 2γ2 ) 2 + (δz ) γ1 − γ2 (γ1 + 2γ2 ) 2 2 + kt (δz ) V (zi ) − E ( γ1 + 2 γ2 ) (γ1 + 2γ2 ) (γ1 − 2γ2 ) √ γ2 2 2 + Fhh (zi ) 3 + kt (δz ) (γ1 + 2γ2 ) γ3 + + + +3 γ3 γ2 + 3 3 + kt (δz ) + √ 3 kt δz (γ1 + 2γ2 ) (γ1 − 2γ2 ) √ γ3 (γ1 + γ2 ) 3 3 + 3 + + kt (δz ) (γ1 + 2γ2 ) (γ1 − 2γ2 ) √ γ3 3 . (B.0.21) + 3 + + (V (zi ) − E ) kt (δz ) (γ1 + 2γ2 ) (γ1 − 2γ2 ) (γ1 − 2γ2 ) + (γ1 − 2γ2 ) − These equations imply that, if the wavefunctions are known at the two points z − δz and z , then the value at z + δz can be determined for any energy E . This iterative equation forms the bias of a standard method of solving equations numerically, and is known as the shooting method. The equations can be rewritten in a matrix notation, which allows easy programmable implementation. Using a coefficient notation for these two equations of the form Fhh (zi+1 ) Flh (zi+1 ) the effective mass equations can be 0 0 a1 b1 = = a1 Fhh (zi−1 ) + a2 Flh (zi−1 ) + a3 Flh (zi ) + a4 Fhh (zi ), b1 Fhh (zi−1 ) + b2 Flh (zi−1 ) + b3 Fhh (zi ) + b4 Flh (zi ), (B.0.22) (B.0.23) written in a recursive transfer matrix expression Fhh (zi ) Fhh (zi−1 ) 0 1 0 0 0 1 Flh (zi−1 ) = Flh (zi ) . Fhh (zi+1 ) Fhh (zi ) a2 a3 a4 Flh (zi+1 ) Flh (zi ) b2 b3 b4 (B.0.24) Provided that we have initial values for the wave functions at the first and second nodes, we can determine the wavefunction values at any node by an iterative procedure. By multiplying matrices, it is possible to obtain an expression for the wavefunction values at any node (as a function of the initial values) Fhh (zn ) Fhh (z0 ) Flh (zn ) Flh (z0 ) (B.0.25) Fhh (zn+1 ) = Mn+1 Mn Mn−1 · · · M3 M2 Fhh (z1 ) . Flh (zn+1 ) Flh (z1 ) The questions that remain are what is the suitable choice for these initial values, and how to determine whether an energy is an eigenenergy of the system. Using four known values of the wavefunction components at z and z + δz , a fifth and sixth values can be predicted. Using the new point together with the known wavefunction components at z , we can subsequently find the wavefunctions at z + 2δz , and so on. Hence the complete wave function solution can be found for any particular energy. The solutions for steady states have wavefunctions that satisfy the standard boundary conditions F → 0 and ∂ F → 0, as z → ±∞. ∂z (B.0.26) as now there are two coupled wavefunction components. and the suitable choice is 0 for the first node. The function to be minimized can be found by generating the transfer matrix which propagates the wavefunction from the first two nodes to the last two nodes 0 m11 m12 m13 m14 Fhh (zN −1 ) Flh (zN −1 ) m21 m22 m23 m24 0 (B. and one cannot be scaled independently from the other. However. 2 2 (B. as changing it will only scale the wavefunction (the finite difference equations are linear) and this does not affect the eigenenergy. This guarantees a converging wavefunction. satisfying the boundary conditions. which can be found by substituting cmin into (B.142 Appendix B. we choose the initial values to be 0 and 1 for one subband.0.27) Fhh (zN ) = m31 m32 m33 m34 1 .29) Flh (z ) Flh (z ) 2 (z ) + F 2 (z )) dz (Fhh lh .0.28) and then the minimum of cmin (E ) is searched to obtain the energy. in many cases one of the wavefunction components exhibit a very sharp sign switching. The 1 can be any arbitrary number. A solution of the Hamiltonian equations is found when this minimum wavefunction amplitude is smaller than a certain threshold value.0. and 1 for the second. we minimize the amplitude of the wavefunction at the end of the grid. Here c is a parameter. c m41 m42 m43 m44 Flh (zN ) Minimizing the wavefunction amplitude at the final node leads to (m33 + m34 c) + (m43 + m44 c) → cmin . In order to work around these problems. Note that the wavefunctions obtained in this procedure are not properly normalized and should be transformed into 1 Fhh (z ) Fhh (z ) → (B. The valence band case is a bit more complicated. an additional problem arises from the parameter c defined above.25). The energy is varied systematically until both wavefunction components switch from diverging to ±∞ to ∓∞. often twice within a single energy search step. Two-Band Model Numerical Implementation As argued in [33]. which is to be determined when the equations are solved.0. in the one-band case of the conduction band only two initial values are required. On top of that. Therefore. and 0 and c for the other. so we set NA (z ) = p(z ) = 0. L is the length of the heterostructure and E is the eigenenergy. we have to solve for a given heterostructure with given donor concentration (C.i − ni ). which is equivalent to assume the device in equilibrium. it is necessary to consider not only the band-edge and external potential.0.0.0.1) and (C. we only consider the conduction-band.3. but also the carrier-carrier electrostatic potential. which is calculated from the envelope function ϕand the Fermi level EF n(z ) = m∗ (z )kB T πL 2 |ϕk (z )| ln 1 + e k 2 EF −Ek kB T . The second coupling term is the electron concentration n(z ). The boundary conditions for the envelope function are given by (B. The first coupling term between these two equations is the overall potential V (z ) = VCB (z ) − eVρ (z ) (C. whereas we set the electrostatic potential Vρ to zero at the boundaries [12].0. The Poisson equation (2. The summation over k represents the summation over all eigenstates (including spin).0. N.2) dz dz where (z ) is the position dependent dielectric constant. (C. As a consequence. we apply the shooting method from Appendix B to the Schrödinger equation and to the Poisson equation a finite-difference scheme. To solve (C.i +( i+ i+1 )Vρ.0. ND (z ) represents the ionized donor distribution and n(z ) is the electron distribution.4) where kB is the Boltzmann constant.3) where VCB (z ) represents the conduction-band profile given by the material composition. We restrict ourselves to model only the mesoscopic part and not the whole device.5) 143 . For − now. Vρ (z ) is the electrostatic potential and e denotes the + elementary charge.1) and (C. (C. so we have to define appropriate boundary conditions for the envelope function and the electrostatic potential.0.0. We start by considering the Schrödinger equation in the slowly varying envelope approximation in one dimension d 2 dz 2 d 1 m∗ (z ) dz ϕ(z ) + V (z )ϕ(z ) = Eϕ(z ) (C. T is the temperature.1) can be formulated as d d + − (z ) Vρ (z ) = −e[ND (z ) − n(z ) − NA (z ) + p(z )].i+1 ) + +e(ND.i−1 −( i−1 +2 i+ i+1 )Vρ.26).0.Appendix C Self-Consistent Solution of Schrödinger -Poisson Model For a quantitative discussion of carriers that are strongly confined to a small area.2) selfconsistently.1) where V (z ) is the overall potential and ϕ(z ) is the slowly varying envelope.0. which reads 0 = 1 2 (δz ) 2 (( i + i−1 )Vρ.2) numerically. (C. −∞ (C.new (z ) − Vρ (z )| is smaller than a pre-defined value δ.0. the electrostatic potential is updated Vρ (z )(z ) = Vρ (z ) + Γ(Vρ.new (z ). . First.4) with respect to (C.1 shows the program flow to obtain the self-consistent solution of the Schrödinger -Poisson system under equilibrium condition. Equilibrium conditions require us to choose the Fermi level appropriate to allow that ∞ ∞ −∞ + ND (z )dz = n(z )dz. Otherwise. where δz is the spatial discretization. we solve (C. we obtain the converged solution.6) To ensure this equilibrium condition. If |Vρ.1: Program flow for self-consistent solution of Schrödinger -Poisson under equilibrium condition with given donor concentration. Schrödinger ’s equation is solved assuming an initial potential V (z ) = VCB (z ). Then.4) with the constraint (C.0. Vρ (z ) = 0.6) to obtain the Fermi level EF . Self-Consistent Solution of Schrödinger -Poisson Model Figure C.new (z ) − Vρ (z )). where 0 < Γ < 1 is a damping parameter used to improve convergence. the Fermi level EF and the carrier distribution n(z ) are obtained by solving (C. the Poisson equation is solved resulting in a new electrostatic potential Vρ.0.0. Next.6). Figure C.0.144 Appendix C. To find out.k = q nq eiq·r . As such we neglect excitonic screening. we can note from (D. (D. we introduce a test charge. which is not a bad approximation for the elevated carrier densities present in conventional semiconductor heterostructures.e. (D.4) 1 Here.1) where nq = 1 V a ˆ† ˆk k−q a k (D.5) 145 . i.0. we first simplify the problem by assuming that the screening effects of an electron-hole plasma equal the sum of the effects resulting from the separate electron and hole plasmas. in turn changes the electrostatic potential.Appendix D Lindhard Screening Model To implement the screened Hartree-Fock approximation in chapter 4. (D. However. the electrostatic potential due to the electron is φ(r) = e/r.0.2) is the Fourier amplitude of the density distribution operator. We denote the carrier density distribution as an expectation value since we plan to calculate it quantum mechanically. The new carrier distribution. and V is the volume of the ka k−q a semiconductor medium. To derive the induced carrier distribution.k = q nsq eiq·r .1).3) k−q a V k. In vacuum. we can treat screening effects on the basis of an effective single-particle Hamiltonian ˆ ef f = H k Ee (k)ˆ a† ˆ k −V ka q Vsq ns.0. the Fourier transform of the density operator is given by nsq = V ˆ† ˆk . we need a screening model. in a semiconductor there is a background dielectric constant b which is due to everything in the semiconductor in the absence of the carriers themselves. there is the carrier distribution that is changed by the presence of the test electron at the origin (see Fig. D.−q .0. Starting with the electron plasma.1) that the corresponding quantum-mechanical operator for the screened electron charge distribution is en ˆ s (r) with 1 n ˆ s (r) = ei(k−k )·r a ˆ† ˆk (D. we wish to know what effect this electron has on its surroundings. At the level of a self-consistent Hartree-Fock approach. ns (r) . One approach is to use a self-consistent quantum theory of plasma screening involving arguments from classical electrodynamics and quantum mechanics [50]. Given an electron at the origin of our coordinate system. The density distribution operator can be defined as n ˆ (r) = ψ † (r)ψ (r) 1 ei(k−k )·r a = ˆ† ˆk k a V k..0. In a rigorous treatment we would use the electronic part of the many-body Hamiltonian to obtain an equation of motion for nsq . Furthermore.0. a charge sufficiently small as to cause negligible perturbation. 0.0. where Vsq = 1 V d3 rVs (r)e−iq·r . H k−q a (Ee (k) − Ee (k − q)) a ˆ† ˆk k−q a + p Vsp a ˆ† ˆk+p − a ˆ† ˆk .5). where the infinitesimal δ indicates that the perturbation k−q a has been switched on adiabatically.0.7) Taking the expectation value and keeping only slowly varying terms. (D. namely those with p = −q.11) . i..0. (D. k−q a k−q−p a (D.9) (D.1: Change in the carrier distribution due to an electron at the origin (after [50]). and φs (r) is the screened electrostatic potential.6) with Vs (r) = eφs (r). Lindhard Screening Model Figure D. we get i d a ˆ† a ˆk dt k−q = (Ee (k) − Ee (k − q)) a ˆ† ˆk k−q a +Vsq (nk−q − nq ) . (D.0. (D.0. (ω + iδ ) − Ee (k) + Ee (k − q) (D.8) We suppose that a ˆ† ˆk has solution of the form e(δ−iω)t . With the effective Hamiltonian (D.0.12) 4πe b [ne (r) + nsq ] . that we had a homogeneous plasma at t = −∞.0.e. This transforms (D. we get the equation of motion i d † a ˆ a ˆk dt k−q = = ˆ ef f a ˆ† ˆk .10) k The induced charge distribution is a source in Poisson’s equation ∇ 2 φs ( r ) = − The Fourier transform of this equation is φsq = 4πe 2 bq 1 + nsq V .146 Appendix D. We further suppose that the induced charge distribution follows this response.0.8) to a ˆ† ˆk = Vsq k−q a and nsq = Vsq V nk−q − nq (ω + iδ ) − Ee (k) + Ee (k − q) nk−q − nq . 15) = 1 − Vq nk nn. the poles are in the lower complex frequency plane. q (ω ) (D.14) where Vq is the unscreened Coulomb potential. (D. we find the screened Coulomb potential energy between carriers Vsq = where the longitudinal dielectric function is given by q (ω ) Vq . and adding the electron and hole contributions. .0.. and it includes spatial dispersion (q dependence) and spectral dispersion (ω dependence).q (ω + iδ ) − En (k) + En (k − q) (D.147 where for a point charge at the origin n eq = 1 V d3 rδ 3 (r) e−iq·r = 1 .12) and solve for Vsq to find Vsq = Vq 1 − Vq k nk−q − nq (ω + iδ ) − Ee (k) + Ee (k − q) −1 .10) into (D.0. substitute (D.16) This equation is the Lindhard formula. i.0.13) Using Vsq ≡ eφsq .e.0.k−q − nn. Repeating the derivation for the hole plasma. V (D. It describes a complex retarded dielectric function.0.0. The temperature dependence is calculated by Eg (T ) = Eg (0) − (E.Appendix E Bandgap Energy Modeling of Semiconductors The bandgap (or forbidden energy zone) is one of the most important semiconductor parameters. This is important to assure consistency between the values for alloy materials at x = 0 and x = 1 and the values for the respective basic materials. Various models define the temperature dependence of the bandgap energy in semiconductors (e.1 Temperature dependence of the bandgap αT 2 . In addition. For Si two additional models can be chosen which are based on polynomial fits of second and third order. E.1. the temperature-dependent bandgaps of the constituents (A and B) are calculated first.g.1. For an alloy A1−x Bx . Note. the resulting bandgaps at 300K. β+T The model of Varshni [67] is used for basic materials. Eg (300). The parameter values are summarized in Table E. 148 . For materials where the bandgap changes between direct and indirect the multiple valley conduction bands are considered. [67]). are given in Table E.1) where Eg (0) us the bandgap energy at 0K. for these materials always the lowest conduction band valley minimum is taken into account.3. The bandgap and the energy offset are then calculated depending on the material composition. 34-1.3: Bandgap energies at room temperature compared to reported data (I).2.278 2.2.2.350 2.67 1. 68.891 2.261 Reported value [eV ] 1.2) (E. 75] [76. the resulting bandgaps at 300K.2 Semiconductor Alloys A B In the case of alloy materials the temperature-dependent bandgaps of the constituents.Γ are summarized in Table E. Γ AB Eg = = = A B Eg. Table E.663 1. are included in Table (E. for materials where the bandgap changes between direct and indirect the multiple valley conduction bands are considered.1). Γ .37 1. Eg.163 0.78 × 10−4 5.X · x + Cg.338 α [eV /K ] 4. 77] [74. Semiconductor Alloys 149 Table E.X . A Eg. In addition. 72] [68.26-2.7437 1.2.0 × 10−4 β [K ] 204 332 83 327 300 Ref. The bowing parameters Cg.12-1. 68] E.424 2.1.2.X · (1 − x) + Eg.42-1.58 × 10−4 6.1242 0. [68.43 2. [76] [76] [78] [78] . For that purpose.1695 0.2.1: Parameter values for modeling the bandgap energies (I) Material Si Ge GaAs AlAs InAs InP GaP Minimum X L Γ X Γ Γ X Eg (0) [eV ] 1.14-2. 70] [75.88 α [eV /K ] 4. However. Eg and Eg .168 0.124 0.421 2. Material Si Ge GaAs AlAs InAs InP GaP Minimum X L Γ X Γ Γ X Eg (300) [eV ] 1.5: Parameter values for modeling the bandgap energies (II).354-0. 73. 69] [69. 68.771 × 10−4 β [K ] 636 235 220 408 75 162 372 Ref.272 Ref. 71.73 × 10−4 4.981 2.66 × 10−4 8. 69] [69. Material GaAs AlAs InAs InP GaP Minimum X Γ X X Γ Eg (0) [eV ] 1. Γ · x + Cg.E. [68.3) · (1 − x) + B Eg.0 × 10−4 2.3).774 × 10−4 5.Γ · (1 − x) · x. 72] [71. Γ (E.5 × 10−4 3.X AB Eg.360 1.66-0. 68.32 2. 74.239 0.1) (E. 69] [69] [70. additional model parameters are needed for the higher energy valleys in the respective III-V binary materials (Table (E. 75] [68. 76] [70. 73] Table E. AB AB min Eg.2)).78 × 10−4 7. The bandgap and the energy offset of an alloy A1−x Bx are calculated by AB Eg.X and Cg.X · (1 − x) · x. are calculated by (E.351 2.521 2.9. 73] [74.63 × 10−4 5.420 1.6 × 10−4 8. 32 Ref.2 presents the dependence of the bandgap energy on the Al composition for various temperatures.21-2.4 0. Material SiGe AlGaAs InGaAs InAlAs InAsP Cg. [76.143 0 -0. 68] [73.4) As an example of the procedure elaborated above.150 Appendix E. Bandgap Energy Modeling of Semiconductors Table E. Figure E. 79] [77] [80.9-1.Γ [eV ] 0 0 0 -0. In figure E. A direct-to-indirect (from Γ to X minimum) gap transition is observed at about x = 0. .4.7: Bandgap energies at room temperature compared to reported data (II). 68] [73.30 2.X [eV ] 0 -0.475 1.9: Parameter values for the bandgap of alloy materials.85 Ref. [76] [77] [73. 79] [78] Table E.2.899 2. 79] [80.142 2.73-2.2 -0.713 0 Cg [eV ] -0.766 2. we consider the properties of the Alx Ga1−x As alloy bandgap energy. Material GaAs AlAs InAs InP GaP Minimum X Γ X X Γ Eg (300) [eV ] 1.3 0 Cg.766 1.91 2. 68] Additional bowing parameters are given as a reference for the case when a one-valley bandgap fit is used AB A B Eg = Eg · (1 − x) + Eg · x + Cg · (1 − x) · x.76 Reported value [eV ] 1.21 2. (E.671-2.37-2.7 -0.14 2.1 we plot the dependence of the bandgap energy on temperature with Al (x) content as a parameter. 7 1.7 0.1: Temperature dependence of the Alx Ga1−x As alloy bandgap energy for various values of x.5 x=0.3 2.4 2 x=0.7 2. .5 1.6 1.E. 2.2 x=0.1 0.4 0.8 0.7 x=0.8 x=0.9 1.3 Eg (eV) 1.9 1.2: Alx Ga1−x As alloy bandgap energy as a function of the Al composition for various temperatures.2.6 1.2 2.8 1. 4K 2 300K Eg (eV) 1.3 0. Semiconductor Alloys 151 x=1 2.6 0.9 1 Figure E.1 2K.1 1.6 x=0.2 1.2 0.9 x=0.5 x 0.8 x=0.5 77K 0 0.4 x=0 0 50 100 150 T (K) 200 250 300 Figure E.1 x=0. This attribute has been of considerable technical importance. from which the refractive index can be calculated using n = (n) + i (n) = ( ) + i ( ). The second goal is achieved by a strait-forward interpolation of the experimental curves to the requested energy. This numerical data has been organized into a database of files which can be downloaded from [82]. which accounts for the direct bandgap shift of the alloy as a function of temperature.0 eV and for compositions x from 0 to 0. This appendix deals we the determination of the complex refractive index of this alloy.0. This methods are.1) With this experimental curve we can now go ahead and calculate the refractive index for any temperature and energy. and thus its properties are of paramount importance. In [81] the authors present a measurement of the room-temperature pseudo-dielectric function and related optical function data for Alx Ga1−x As alloys for energies E from 1. Of course. 300 K . As the theoretical modeling is two complex or ill-adjusted to our needs we resort to the empirical approach. 152 . This parameter is mainly important in determining accurately the optical properties of microcavity structures.3 and F.2. respectively.80 in steps of approximately 0. In figures F. These structures and their interfaces can be characterized conveniently and non destructively by visible-near UV optical reflection techniques.Appendix F Refractive Index of AlxGa1−xAs Alloy Alx Ga1−x As is the main semiconductor alloy of interest to us throughout this report. The materials in the binary Alx Ga1−x As are essentially lattice matched over the entire composition range. ( ) and ( ). discussed in chapter 5. this procedure is valid only in the spectral support region of the experimental data. because it has made possible the growth of artificial structures with electronic energy levels tailored to a wide variety of fundamental and applied purposes.5 to 6.1. F. 300 K ) and various values of x. carefully tailored to each material. (F. obviously. 77. but at present only incomplete dielectric function data are available. we present in figure F. calculated at T = 2. As a illustration. The temperature dependent extrapolation is performed by shifting the experimental spectra in energy by an amount of Eshif t (T ) = Eg (T ) − Eg (300K ). 77. These files include the real and the imaginary parts of the dielectric function.4 we present a zoomed in view of the refractive indices at energy intervals and x values relevant to this report. There are two main approaches in literature towards the determination of the refractive index: theoretical modeling of the dielectric function of the material at hand and semi-empirical approximations of this property [76]. However. an there is a considerable variation between then depending of various factor mainly the temperature.1 the results of the calculation of the refractive indices for temperatures values relevant to this report (T = 2. these techniques require an accurate data base. for any Al atoms molar composition x and temperature. 55 E (eV) 1.1 AlAs 0.8As Ga0.04 0.65 1.8 3.1Al0.2 3 2.5 5 x=1 2 2.02 0 1.6 3.5 4 4. .5 5 x=0 4.5 3 2.6 1.153 (a) 5.6 1.5 1.35 1.7 0.55 1.5 ℜ(n) 4 3.2Al0.7 Figure F.08 Ga0. T = 77K (red) and T = 300K (blue) and x = 0 − 1 (from left to right).9As GaAs 1.2: Refractive indices for T = 2K .4 ℜ(n) 3.5 5 4 3 ℑ(n) x=0 2 1 x=1 0 1.65 1.5 1.5 3 E (eV) 3.5 4 4.1: The (a) real and (b) imaginary parts of the refractive index of Ga1−x Alx As as a function of energy.4 0. 3.45 1.5 5 Figure F.5 2 2. for T = 2K (green).45 1.5 2 1.5 3 (b) 3.8 1.4 1.06 ℑ(n) 0. 02 0 1.02 0 1.4: Refractive indices for T = 300K .6 3.1Al0.55 1.2Al0.3: Refractive indices for T = 77K .1 0.4 Ga0.6 1.65 1.06 AlAs 0.8As Ga0.7 1.2 3 2.5 1.45 1.65 1.9As GaAs 1.8 3.154 Appendix F. Refractive Index of Alx Ga1−x As Alloy 3.4 0.55 1.8 3.2Al0.7 0.6 1.06 ℑ(n) 0.45 1.4 1.55 E (eV) 1.1Al0.7 Figure F.5 1.08 Ga0.04 0.55 E (eV) 1.65 1.6 1. .8As Ga0.45 1.8 1.7 Figure F.6 1.4 ℜ(n) 3.08 ℑ(n) 0.04 0.1 AlAs 0.45 1. 3.4 0.8 1.4 ℜ(n) 3.5 1.65 1.12 0.9As GaAs 1.2 3 2.5 1.6 3. O. p.2. 3. 1995.2.3. NATO Science Series B: Physics. 3. Lett. 5.1. 1.1. Cohen. B. Oct 2007. p. 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May 2001. על-אף השפעות הריכוז הגבוה של אלקטרוני הגז הדו-מימדי שמובילים להעלמות רזוננסים אקסיטוניים בבור הבודד.אך על-אף זאת הצימוד נשאר בתחום הצימוד החזק .באמצעות חילוץ פרמטרי העירוב של הרזוננסים השונים מתוך ההתאמה למודל האוסצילטורים המצומדים אנו בוחנים את הרכב כל אחד מהענפים בעקומי החצייה שלפנינו .אנו רואים כי בעוד עוצמת הצימוד יורדת במקצת כתלות בריכוז גז האלקטרונים . יט .Fermiולא במרכז אזור .הבור הבודד .רוחב הקו עולה בצורה משמעותית .אנו בוחנים את חוזק הצימוד ורוחב הקו של האינטראקציות בין הרזוננסים האלקטרוניים בבור ואופן השדה הא"מ .מתוך עקומים אלה אנו רואים שוב את החלק המשמעותי של אופן המהוד בשמירה על הצימוד החזק עם רזוננס האלקטרון-חור של הבור אף בריכוזי גז גבוהים.Brillouinתוצאה זו תואמת את התוצאות הניסיוניות ומהווה הוכחה לכך כי הצימוד החזק של רזוננס אלקטרון-חור של הבור בקצה אנרגיית Fermiעם אופן המהוד של השדה הא"מ הוא זה המשמר את העירורים בספקטרומי ההחזרה של המבנה .ניתן לומר כי הם מתאימים למעברים המותרים בין תתי-פסי הערכיות וההולכה בקצה אנרגיית .לבסוף . באמצעות התאמת עקומי החצייה הללו למודל האוסצילטורים המצומדים וחילוץ נתונים ידניים מתוך עקומים אלה . עבור ריכוזי גז גבוהים .הוא מבנה המיקרומהוד . עבור מבנה המיקורמהוד המכיל את הבור הקוונטי המסומם אנו בוחנים את השפעות ריכוז גז האלקטרונים הדו- מימדי על אינטראקציית הקרינה והחומר שבאזור המהוד .2קווים אלה .השונה במהותה מגישות מקובלות המעושות שימוש בהתאמה לתוצאות ניסיוניות .עבור שני המבנים המתקבלים . .אנו מציגים תחילה את מודל Jaynes- Cummingsהקלאסי לבולע בעל שתי הרמות שפותח במסגרת הפיסיקה האטומית .לאחר דיון זה על התכונות האופטיות של מיקרומהודים .ומוצאים כי שינוי ליניארי של עובי שכבות המבנה מוביל לתזוזה ליניארית מתאימה במיקום אופן המהוד של האור הכלוא בו .באופן אקוויולנטי למקרה האטומי .תחילה אנו מציגים את תוצאות חישוב ספקטרום הרפליקציה של מבנה ללא נוכחות בור באזור התהודה .שאף נחקר בצורה ניסיונית .אפקט קלאסי נוסף .עבור מבנה המיקרומהוד עם הבור הקוונטי הלא מסומם אנו רואים כי הצימוד עבור כלל הרזונסים המופיעים בעקומי החצייה הינו צימוד חזק.הולכים ונעלמים עם עליית ריכוז הגז במבנה כתוצאה מהסיכוך של האלקטרונים החופשיים .אנו מביאים את ניסוח מודל האוסצילטורים המתאים למערכת זו .המתאימה לתוצאות ניסויי פוטולומיניסנציה עבור מבנים דומים בטמפרטורות נמוכות.באמצעות השוואת עקומי החצייה עבור כל אחד מהריכוזים הללו אנו רואים כי קיים צימוד חזק בין רזוננסים אלקטרוניים של הבור ואופן המהוד של השדה הא"מ אף לערכים גבוהים מאוד של ריכוז הגז .ובכלל זה את שיטת מטריצת ההעברה הקלאסית . 6 1010 cm 2 -באמצעות השוואת אנרגיות הרזוננס הללו לחישוב אנרגיות המעבר עבור יח .האופיניים לאינטראקציות אקסיטוניות המופיעות במבנים אלה .ובאמצעותו אנו מתארים את האינטראקציה בין הרזוננסים האלקטרוניים הנוצרים בבור הקוונטי כאשר הוא מוצב באזור התהודה של מבנה המיקרומהוד לבין אופני התהודה של השדה הא"מ .אנו מציגים את התיאוריה הקוונטית של צימוד עירורים אלקטרוניים ואופטיים במהודים אלה . 1 1011 cm 2הדבר עומד בניגוד לתוצאות חישוב ספקטרומי הבליעה עבור הבור הבודד בהן ניתן לראות בוודאות כי הרזוננסים האקסיטוניים דועכים עבור ערכי ריכוז גז הגבוהים מ .תוצאות החישוב מביאות לשיחזור צורת קו אסימטרית של הפליטה הספונטנית .4-גישה זו . ע"מ לחקור את התכונות האופטיות של המיקומהוד המכיל את בור קוונטי המוצב באזור התהודה .מתוך הספקטרה הללו אנו מחלצים את מיקומי המינימה של ההחזרה עבור כלל ערכי ומקבלים את עקומי החצייה של המבנה .אנו מציגים את השיטות לחישוב ספקטרומי ההעברה וההחזרה .לצורך כך אנו מציגים גישה המתבססת על מודל הדיספרסיה הליניארית .הלא הם הפולריטונים האקסיטוניים .מתוך התאמת עקומים אלה למודל האוסצילטורים המצומדים ניתן להפיק מסקנות לגבי אופי האינטראקציה והחוזק שלהם .לצורך כך .זיהוי אינטראקציות אלה מתאפשר ע"י ביצוע חישובים אלקטרוניים באמצעות השיטות שפותחו בפרק .אנו תחילה מציגים את העקרונות המרכזיים של פעולת מהוד אופטי קלאסי ודנים במימוש מהודים כאלה באמצעות גידול שכבות אפיטקסיאליות דקות המורכבות משני מוליכים למחצה או יותר ליצירת מבנים עם זוג מראות Bragg מבוזרות שביניהן אזור ברוחב כפולה שלמה של אורך הגל .ובתוך כך מציגים את מושג הפולריטון כמייצג את הצימוד הנוצר בין רזוננס האלקטרון-חור ואופן השדה הא"מ.תוך החלפת המודל הפנומנולוגי של מקדם השבירה של הבור עם תוצאות המודלים שפותחו בפרקים 3ו .עבור מבנים אלה .לאורך עקומים אלה ניתן להבחין בפיצול אופייני באזורי הרוזוננס בין אופן המהוד של השדה הא"מ שבין הרזוננסים האלקטרוניים של הבור .ומציגים באמצעותו את תופעת פיצול Rabiברמות האנרגיה של המערכת המתרחשת כאשר היא מצומדת לאופן בודד של השדה הא"מ. בפרק .הקווים האקסיטוניים עוברים הזזה לכיוון האנרגיות הגבוהות בעקבות השפעת נירמול הפס האסור האפקטיבי ומילוי מרחב המצבים ע"י האלקטרונים החופשיים של הגז .אנו עוברים ליישומו על מבנה מיקרומהוד ספציפי . היא פיצול Burstein-Mossהמביא להיפרדות השיאים בשני סוגי הפליטה עבור ריכוזים עולים של גז האלקטרונים . מודל פשוט זה ניתן להכללה למערכת של מספר גדול של בולעים .לאחר מכן .לאחר הצגת המודל החישובי .נותנת לנו כלי לחישוב התכונות האופטיות של מיקרומהוד ע"ב תוצאות חישוב התכונות האופטיות של בור בודד הנובעות מעקרונות פיסיקליים ראשוניים .אנו מציגים את ספקטרומי ההחזרה עבור ערכים שונים של פרמטר שינוי הרוחב .אנו מציבים אל תוך אזור התהודה את שני הבורות הבודדים אותם חקרנו בפרקים הקודמים .אשר ניתן לראות בהשוואת ספקטרומי הבליעה והפליטה הספונטנית.המופיעים בצורה ברורה במבנה הראשון ואף בשני בריכוזים נמוכים של גז האלקטרונים הדו-מימדי .אף מעל .יש למצוא דרך לשלב את השניים במסגרת חישובים משותפת .5אנו עוברים לעסוק במבנה המרכזי אותו אנו חוקרים בעבודה זו .בנוסף לתופעה זו . האלקטרון והחור .אנו עושים שימוש בגישת מטריצות המעבר.באמצעות הופעת קווים רזונטיביים יז . הבליעה .מקדם השבירה והפליטה הספונטנית עבור בור קוונטי בודד.את פיתרון המערכת הנ"ל אנו מבצעים בשני שלבים .את המשוואה עבור הפולוריזציה המיקרוסקופית יש לצמד למשוואות דומות עבור אופרטורי הצפיפות של נושאי המטען .3 גם בניסוח זה אנו מניחים את הנחות שיווי המשקל עבור נושאי המטען ומודל נאיבי של תופעות הפיזור בגביש.אנו מוסיפים להשוואה בין המודלים גם את הדיון בהשפעה של גז האלקטרונים הדו-מימדי בתוך הבור על התכונות הספקטרליות שלו .3אנו מזניחים את האינטראקציה ה-Coulomb-ית בין נושאי המטען במערכת .Luttingerובעזרת ליכסון מתאים מביאים אותה לכדי פיתרון של משוואה מטריצית בגודל .בפרק 4אנו מציגים סיבוך אל תוך המודל באמצעות החזרת האינטראקציה ה-Coulomb-ית .בעקבות הוספת האינטראקציה ה-Coulomb-ית למודל .ו .ופישוט החישוב באמצעות השימוש בקירוב Hartree-Fockלאיברים מסויימים תוך ביצוע קירוב מסדר ראשון להמילטוניאן . באמצעות שני המודלים הללו . תוצאת החישוב היא סט של ערכי אנרגיה ופונקציות מעטפת עבור כל אחד מתת-פסי הערכיות וההולכה .אנו רואים בבירור את השפעת האינטראקציה ה-Coulomb-ית על ספקטרומי הבליעה והפליטה הספונטנית .בתחילה .מבנים קוונטיים שאינם הומוגניים .לצורך מציאת הפרמטרים האופטיים המאקרוסקופיים של המערכת כגון הסוסמפטיביליות החשמלית .המתואר באמצעות מטריצת המילטוניאן בגודל 4 4ופרמטרי .לצורך פיתרון מערכת המשוואות המצומדות הללו לכל ערכי וקטור הגל של הגביש .כגון הסוספטיביליות החשמלית .אנו מבצעים קוונטיזציה שנייה של המילטוניאן המערכת ללא נוכחות השדה האלקטרומגנטי )א"מ( .תוך הנחת שיווי-משקל עבור נושאי מטען אלה ומציגים מודל פנומנולוגי פשוט לתהליכי הפיזור בגביש . בנוסף להנחות אלה .ומציגים את מודל ה Zinc-Blende-לתיאור פסי האנרגיה במוליך למחצה הגושי .Elliotעבור המבנה עם הסימום .כמו בפרק .בעזרת גישה זו אנו מציגים את מושג גז האלקטרונים הדו-מימדי המשחק תפקיד מרכזי במבנים הנידונים בעבודה זו.אנו עוברים בפרקים 3ו4- לעסוק בתכונות האופטיות השונות של מבנים אלה .בכדי להתחשב בהשפעות נוכחות ריכוז גדול של נושאי מטען באזור הבור הקוונטי .ומבצעים את החישוב עבור שתי טמפרטורות סביבה שונות T 2K . אנו מנסחים את הבעיה בקירוב של שני פסים .ומוצאים את התלות של גודל נירמול זה בריכוז נושאי המטען במבנה .לצורך פישוט החישוב. 2 2במודל זה פס ההולכה מחושב בקירוב המסה האפקטיבית בה תתי-הפסים נלקחים כבעלי צורה פרבולית . עם גיבוש מודל לחישוב התכונות האלקטרוניות של המבנים הקוונטיים הנדונים .בפרט .אנו עושים שימוש בפונקציות המעטפת הללו לצורך חישוב אלמנטי המטריצה של התנע בקירוב הדיפולי .בפרק .עבור מבנים אלה אנו מבצעים השוואה של ספקטרומי הבליעה והפליטה הספונטנית המתקבלים משני המודלים החישוביים הללו ובינם לבין מודלים תיאורטיים אחרים כגון משוואת .בראשם שני בורות קוונטיים בודדים מסוג .Schrödingerחישוב איטרטיבי זה מציג את השפעות נוכחות נושאי המטען הנוספים דרך שינוי אנרגיות קצות פסי ההולכה והערכיות .מתוך שני מודלים אלה אנו מקבלים את ספקטרומי הסוספטיביליות החשמלית.בו נושאי המטען מתוארים בצורה קוונטית ואילו השדה האלקטרומגנטי מנוסח בצורה קלאסית .אנו מבצעים חישוב התכונות האופטיות של המבנים הקוונטיים הנדונים בעבודה זו .עבור טווח ערכים גדול של וקטור הגל של הגביש .לצורך הפשטות .מוצגים בשני פרקים אלה מודלים מקורבים המאפשרים חישוב פרמטרים אופטיים שונים .אנו מוסיפים למודל בפרק זה את השפעות אפקט הסיכוך של נושאי המטען באמצעות מודל Lindhardלפונקציה הדיאלקטרית של הגביש .GaAs / AlxGa1x Asהראשון ריק מנוכחות נושאי מטען חיצוניים באזור הבור ואילו השני בתוספת גז אלקטרונים דו-מימדי המוצג בתוכו באמצעות תוספת שכבות סימום בתוך מעטפת הבור .אנו מניחים את הנחת פונקציית המעטפת לתיאור פונקציית ה Bloch-של הגביש .אנו מציגים אל תוך החישוב המודל המוצע חישוב אלקטרוסטטי בצורת פיתרון מצומד של משוואת Poissonביחד עם משוואת .T 77K -מתוך תוצאות החישובים עבור שני המבנים .השדה הא"מ מתווסף להמילטוניאן זה באמצעות ניסוח במסגרת הקירוב הדיפולי .אנו מקבלים את תופעת נורמליזיית אנרגיית הפס האסור של המוליך למחצה .לצורך כך .אנו נדרשים למצוא את הפולריזציה המיקרוסקופית .דרכו אנו מציגים את מושג החור .וזאת בעזרת ההמילטוניאן המלא של המערכת הכולל את האינטראקציה של האור והחומר .המספקים לנו את החוקי הברירה עבור המעברים האנרגטיים בין תת-פסי ההולכה והערכיות במבנה .הבליעה והפליטה הספונטנית עבור מבנים אלה .אנו מגבילים עצמנו בדיון זה לטיפול סמי-קלאסי במוליך למחצה .לצורך מטרה זו אנו פונים לפיתרון משוואת Heisenbergעבור אופרטור זה .לקבלת מערכת משוואות המכונות משוואות Blochשל המוליך למחצה . לבסוף .הספקטרום של המבנה עם הגז . מראות תופעות רב-חלקיקיות הנובעות מאינטראקציות אלקטרון-אלקטרון ואינטראקציות של גז האלקטרונים עם החור המעורר.הראשון הינו מודל האוסצילטורים המצומדים .( ne 0.עם הפוטון הכלוא במיקרומהוד .ותוך הפעלת שדה מגנטי חיצוני על המערכת ובלעדיו .נהוג לתאר באמצעות שני מודלים .מדידות ספקטרוסקופיות .בניגוד למדידות התובלה הנותנות הצצה על המצבים שבמשטח קצה אנרגיית Fermiשל המערכת האלקטרונית .פותחה גישה חישובית הנובעת מעקרונות פיסיקליים ראשוניים בבואה לנתח את תכונות של מבנים קוונטיים .המודל השני המתאר מערכת מצודמת זו היא מודל הדיספרסיה הליניארית . .והשאר מוזנים אל תוך החישוב בתור הפרעות .חישוב הסוספטיביליות האופטית בהתבסס על הרמות הנ"ל ומומנטי דיפול פנומנולוגיים .9 1011 cm 2בה אין במערכת אקסיטונים .את הצימוד הנ"ל בין העירור האלקטרוני והאופן של השדה האלקטרומגנטי .ואילו קרוב לתהודה ) ( 0אנרגיות אלה שונות מאנרגיות האקסיטון .התוצאות הניסיוניות שהתקבלו במחקרים אלה גובו ע"י ביצוע חישובים תיאורטיים שכללו את חישוב רמות האנרגיה של האלקטרונים והחורים בבור כפונקציה של וקטור הגל . בעבודה זו אנו מציגים מחקר תיאורטי של התכונות האלקטרוניות והאופטיות של מיקרומהוד עם בור קוונטי בודד הנעוץ באזור המהוד .הגידול של מבנה המיקרומהוד נעשה כך שהעובי של כלל השכבות המרכיבות אותו משתנה לאורך הדגם .והן שוות בערכן לאנרגיית פיצול ה Rabi-של המערכת .התוצאה הסופית היא שניתן להתייחס לאנרגיית האקסיטון כקבועה ביחס לשינוי באנרגיית הפוטון הכלוא במהוד .לשם כך .לצורך מתן הסבר לתופעה זו .בו כל אחד מהעירורים מתואר באמצעות אוסצילטור פשוט .שבהם נעשה שימוש .המבוצעות בטמפרטורות נמוכות ותחת שדה מגנטי.(Anti-crossingהמציגות את אנרגיית הפולריטונים כפונקציה של ההפרש האנרגטי בין האקסיטון והפוטון .בו נעשה שימוש בפורמליזם מטריצת המעבר .באמצעות הארה במקום מתאים על הדגם ניתן להביא את המערכת למצב תהודה בו אנרגיית העירור האלקטרוני שווה לאנרגיית הפוטון .הפעלת השדה המגנטי נועדה לבצע הפרדה בין האינטראקציה של הפוטון הכלוא עם עירורים בגז האלקטרונים ובין האינטראקציה עם האקסיטונים של הבור .Fermiנטען כי הקוהרנטיות בין הזוגות אלקטרון-חור הללו.בנוכחות ריכוזי גז דו-מימדי . ע"מ לחקור את עוצמת האינטראקציה בין העירורים האלקטרוניים ובין הפוטון הכלוא במיקרומהוד כפונקציה של הפרש האנרגיה ביניהם .עדיין קיימת אינטראקציה חזקה בין הפוטון הכלוא לעירורים אלקטרוניים שונים בבור .צורת הצגה נוחה של אופני התהודה הפולריטוניים הן דיאגרמות החצייה ) .תחילה אנו מציגים את ההצגה הכללית של המילטוניאן הגביש ועושים שימוש בפורמליזם Blochבכדי להביאו לצורה פשוטה יותר .רחוק מאזור התהודה של המערכת ) . לאחרונה .לצורך פיתרון משוואת Schrödingerהמתאימה אנו פונים לגישה סמי- אמפירית הקרוייה שיטת ה k·p -במסגרתה נלקחים בחשבון רק חלק מתתי-פסי האנרגיה של פסי ההולכה והערכיות של הגביש .הותאמו בכלל המחקרים הללו אל התוצאות הניסיוניות ובוצע חילוץ של ערכי הפרמטרים הפיסיקליים של המודל. המודלים .בוצעו מספר מחקרים ניסיוניים הבוחנים את אינטראקציית גז האלקטרונים הדו-מימדי .ממצא זה נמצא בניגוד לדעה המקובלת כי העלמות אקסיטונים בשל נוכחות גז אלקטרונים בצפיפות גבוהה גוררת עימה בהכרח העלמות הפולריטונים .נובעת מהאינטראקציה החזקה עצמה . הדרושה לשם קיום האינטראקציה החזקה .יש צורך לגדל מבנה קווטני ביו ניתן לשנות באופן רציף את האנרגיה של אחד משני המרכיבים .במודל זה המילטוניאן הצימוד של אקסיטון והפוטון עם חוזק אינטראקציה מסויים מלוכסן לקבלת אנרגיות הפולריטונים .בוצעו חישובי רלפקציה בעזרת פורמליזם מטריצת המעבר.כאן מקדם השבירה של הבור הקוונטי מתואר ע"י מקדם שבירה אפקטיבי בו האקסיטון ממודל כאוסצילטור לורנציאני.בניגוד לגישות תיאורטיות קודמות .ובהם בורות קוונטיים מחוץ וכחלק ממיקרומהודים של מוליכים למחצה.המוצג במבנה הבור הקוונטי באמצעות מנגנוני סימום שונים .הועלתה סברה כי מקור אינטראקציה זו בזוגות אלקטון-חור שאינם קשורים בעלי וקטור גל .עם גז אלקטרונים המושרה בתוכו .כאן .בכדי לתאר את התנהגות פסי האנרגיה טז .מחקרים אלה הראו כי בבור קוונטי מסומם הנמצא בתוך מיקרומהוד ומכיל צפיפות גז גבוהה ) .תוך התחשבות באינטראקציה הקולונית שמקורה בגז האלקטרונים הדו-מימדי .לצורך הצדקת הסברה הנ"ל בוצע לימוד נסיוני מקיף של התופעה הכולל מדידת ספקטרומי הרפלקציה והפוטולומיניסנציה של מבנים מסוג זה .( 0אנרגיות הפולריטונים שוות לאנרגיות הפוטון והאקסיטון הלא-מצומדים . אנו מתחילים את הניתוח בפרק 2בבחינת התכונות האלקטורניות של מוליכים למחצה גושיים ומבנים קוונטיים המבוססים עליהם . מאז ההוכחה הניסיונית הראשונה לקיום של צימוד חזק בין העירורים של מערכת אלקטרונית הממוקמת במהוד לבין הפוטונים הכלואים בתוכו .שם פיצול Rabiמתקבל כאשר אטום בעל שתי רמות אנרגיה מוצב במהוד מתכתי .מערכות כגון אלה מורכבות מחומרים המשמשים לשימושים מעשיים שונים .התוכונות הספקטרוסקופיות והדינאמיות של פולריטונים במיקרומהוד שמקורם באקסיטונים אלה נחקרו בצורה רחבה.המודל הפשוט ביותר לתיאור האינטראקציה של שדה הקרינה עם אטומים כאלה הינו מודל Jaynes-Cummingsהמטפל במקרה של אטום בודד באינטראקצייה עם שדה בעל אופן תנודה בודד .עבור מיקרומהודים מישוריים עם בורות קוונטיים מובנים .המכונה פיצול .הניוון במצב המעורר הראשון מוסר .הצימוד החזק בין אקסיטונים של בורות קוונטיים לבין האור הכלוא במהוד הודגם לראשונה ע"י .בהשוואה למוליך למחצה גושי ועל-כן מושג הצימוד החזק בקלות יחסית .האלקטרונים בפס ההולכה של הבור הקוונטי ניתנים לתיאור כגז אלקטרונים דו-מימדי ) .השימוש במיקרומהודים עם בור קוונטי מובנה מאפשר שליטה והגדלה של צימוד הפוטון והאקסיטון .גודלו של פיצול זה .אינטראקציית הפוטון והאקסיטון מביאה לשימור תנע הגביש במישור הבור .Weisbuchובעצם מהווה את הדוגמא הראשונה ליצירת פולריטונים של מיקרומהוד . כאשר אנרגיית הפוטון תואמת את הפרש האנרגיות בין שתי הרמות האטומיות .GaAs / AlxGa1x Asמצב היסוד של המערכת האלקטרונית נקבע ע"י אלקטרונים המאכלסים את פס ההולכה כתוצאה מסימום מאופנן ) .(2DEGאת תכונות הגז הנ"ל ניתן לבחון באופנים ניסיוניים שונים .במערכת כזו פיצול Rabiפרופורציונלי לחוזק האוסצילטור האקסיטוני.דבר המאפשר יצירת מערכות חדשות למחקר האינטראקציה של אור וחומר.בוצעו ע"י Yamamotoבהם נצפה שינוי בפליטה זו במיקרומהודים של מוליכים למחצה בעלי finesseגבוה .( במבנים מסוג זה .הנקבע ע"י חוזק האוסצילטור האקסיטוני . שבהן מתקבלת הגברת הצימוד בין השניים ומתאפשרת שליטה בו .כגון בורות קוונטיים וצמתים ) .בעזרת מדידות אופטיות ומדידות תובלה המתבססים על תופעת Hallהקוונטית . תערובת של אקסיטון ופוטון והם קרויים פולריטונים אקסיטוניים .בעצם.(Heterojunctionsמהווים נושא מחקר פעיל בעשור האחרון . במבנים קוונטיים המורכבים מבורות קוונטיים .המצבים העצמיים החדשים של המערכת הם סופרפיזיציה של המצבים הלא מצומדים של המערכת.כאשר אנו מתרכזים בסימום המושג ע"י הכנסת שכבות מסמם דקות מאוד מחוץ לבור הקוונטי )סימום . חלק נרחב מהפיסיקה של אינטראקציות קרינה וחומר פותחו לראשונה בהקשר של הפיסיקה האטומית .כך שלפולריטונים ישנה דיספרסיה מוגדרת היטב במישור זה.GaAsוסגסוגת .המצבים העצמיים החדשים הינם .תקציר מורחב מיקרומהודים העשויים ממוליכים למחצה עם מבנים קוונטיים המוכלים בתוכם . והפרש האנרגיות הינו אנרגיית .צימוד אקסיטון-פוטון חזק מתקבל ע"י הופעת שני קווים בספקטרום ההחזרה .ניתן להגיע כאן בקלות לגבול הצימוד החזק . טכנולוגיות גידול של מבנים דיאלקטריים של מוליכים למחצה כמו מיקרומהודים ובמבנים פוטוניים אחרים התפתחה רבות בעשור האחרון . AlxGa1x Asהניסויים הראשונים שבחנו את הגברת הפליטה הספונטנית של בורות קוונטיים באמצעות כליאה של הקרינה .(Modulation dopingסימום זה יכול להיות מושג במספר אופנים . כיוון שהאקסיטונים בעלי האנרגיות הנמוכות ביותר בבור )אקסיטוני החור הכבד והקל( הינם בעלי חוזק אוסצילטור גדול .השיטה הראשונה מאפשרת לבחון את כלל טו .על-כן .כגון AlAs .אם מתקיימים רזוננסים אקסיטוניים ברורים באנרגיה הקרובה לאנרגיית הפוטון הכלוא וכאשר רוחבי הקווים של האקסיטון העירום ושל אופן התנודה הפוטוני של המהוד צרים דיים .Rabiבין שני אופני התנודה הללו נקבע ע"י תכונות המהוד והתכונות האקסיטוניות.גדל בשני סדרי גודל לערך במערכת הדו-מימדית .Rabi הראשון שדן באנלוגיה של המודל הנ"ל למוליכים למחצה היה .הפיצול הפולריטוני .Hopfieldכאשר הראה כי שימור התנע בחומר גושי מאפשר לאקסיטון להצמד אך ורק לפוטון בעל מספר גל זהה .התכונות הספקטרוסקופיות והדינאמיות של הפולריטונים הנוצרים מעירורי היסוד של הבור הקוונטי נחקרו רבות .ע"י שימוש בקירוב הדיפולי ובקירוב הגלים המסתובבים. 1 המצומדים עבור כל אחד מהרזוננסים X i 114 אנרגיית המעבר בין פסי הערכיות וההולכה עבור בור קוונטי ערום וההפרשים ביניהם ne 7 1010 cm עבור ריכוז גז אלקטרונים דו-מימדי 2 טבלא 5.9 פונקציות הבסיס של החבורה הטטרהדרלית Td מכפלה ישירה של ייצוג 15עם כלל הייצוגים של Td אלמנטי המטריצה הסופיים של התנע בין המצבים המתאימים לייצוגים שונים של חבורת הסימטריה הטטרהדרלית רשימת פרמטרים למידול אנרגיות הפס האסור )(1 אנרגיות הפס האסור בטמפרטורת החדר והשוואתם לערכים מהספרות )(1 רשימת פרמטרים למידול אנרגיות הפס האסור )(2 אנרגיות הפס האסור בטמפרטורת החדר והשוואתם לערכים מהספרות )(2 ערכי אנרגיות הפס האסור עבור סגסוגות שונות יד .5 טבלא E.4 טבלא E.2 132 132 133 133 145 145 145 146 146 אופרטורי סימטריה של החבורה Tdהמוצגים בעזרת סימון Schönflies טבלא A.2 טבלא A.3 טבלא E.רשימת טבלאות 111 הפרמטרים המחולצים עבור המבנה הנידון מתוך ההתאמה למודל האוסצילטורים טבלא 5.1 טבלא A.7 טבלא E.3 טבלא A.1 טבלא E. 2 איור F.4 מקדם השבירה של סגסוגת ה AlxGa1x As -כפונקציה של אנרגיית האור בT 77K - .149 150 150 מקדם השבירה של סגסוגת ה AlxGa1x As -כפונקציה של אנרגיית האור ב.עבור ערכים שונים של x מקדם השבירה של סגסוגת ה AlxGa1x As -כפונקציה של אנרגיית האור ב- . T 2K - עבור ערכים שונים של x איור F. T 300Kעבור ערכים שונים של x יג .3 איור F. 1 איור D.119 120 121 121 122 123 123 124 יחסי הדיספרסיה של תתי-פסי ההולכה והערכיות המחושבים עבור בור קוונטי עם גז ne 7 1010 cmב= T 2K אלקטרונים דו-מימדי בריכוז 2 הריחוק האנרגטי בין קווי הרזוננס כפונקציה של ריכוז גז האלקטרונים הדו-מימדי עבור שני הקיטובים פרמטרי הצימוד שהתקבלו מההתאמה למודל האוסצילטורים המצומדים של עקומי החצייה של מינימה ספקטרומי ההחזרה עבור שני הקיטובים כפונקציה של ריכוז גז האלקטרונים הדו-מימדי הריחוק האנרגטי בין עקומי חצייה קרובים של מינימה ההחזרה באזורי הרוזננס עבור שני הקיטובים כפונקציה של ריכוז גז האלקטרונים הדו-מימדי רוחבי הקו של האינטרקציה עבור הרזוננסים השונים שהתקבלו מההתאמה של עקומי החצייה של מינימה ספקטרומי ההחזרה למודל האוסצילטורים המצומדים כפונקציה של ריכוז גז האלקטרונים הדו-מימדי עבור שני הקיטובים ספקטרומי הרפלקציה עבור ריכוזים שונים של גז האלקטרונים הדו-מימדי בשינוי רוחב של 5%ביחס לקצה הדגם .שהתקבלו כתוצאה מהתאמת עקומי החצייה למודל ne 1 1011 cm האוסצילטורים המצומדים בריכוז גז דו-מימדי של 2 תרשים זרימה לפיתרון קונסיסטנטי של משוואות Schrödinger-Poissonתחת תנאי שיווי-משקל עבור ריכוז דונורים נתון תרשים זרימה לפיתרון קונסיסטנטי של משוואות Schrödinger-Poissonתחת תנאי שיווי-משקל עבור ריכוז דונורים נתון התלות בטמפרטורה של רוחב הפס האסור של סגסוגת ה AlxGa1x As -עבור ערכי x שונים איור 5.1 איור E.שחושבו באמצעות התאמה למודל לורנציאני עבור כל ענף רזונטיבי בשני ne 3 1010 cm הקיטובים בריכוז גז דו-מימדי של 2 האמפליטודות והרוחב של השיאים בספקטרום ההעברה כפונקציה של אנרגיית אופן המהוד .המחושב עבור שני הקיטובים של האור הפוגע האמפליטודות והרוחב של השיאים בספקטרום ההעברה כפונקציה של אנרגיית אופן המהוד .28 איור 5.31 איור 5.35 125 איור 5.34 איור 5.שהתקבלו כתוצאה מהתאמת עקומי החצייה למודל ne 3 109 cm האוסצילטורים המצומדים בריכוז גז דו-מימדי של 2 גורמי העירוב הפולריטוניים .37 126 126 127 140 140 147 147 149 איור 5.שהתקבלו כתוצאה מהתאמת עקומי החצייה למודל ne 3 1010 cm האוסצילטורים המצומדים בריכוז גז דו-מימדי של 2 גורמי העירוב הפולריטוניים .36 125 איור 5.1 רוחב הפס האסור של סגסוגת ה AlxGa1x As -כפונקציה של xעבור טמפרטורות סביבה שונות החלק הממשי והמדומה של מקדם השבירה עבור סגסוגת ה AlxGa1x As -כפונקציה של אנרגיית האור . שהתקבלו באמצעות חישוב ידני של FWHMעבור כל אחד מהרזוננסים בשינוי רוחב של 5%ביחס לערך בקצה הדגם .33 איור 5.1 איור E.38 איור 5.30 איור 5.שחושבו באמצעות התאמה למודל לורנציאני עבור כל ענף רזונטיבי בשני ne 3 109 cm הקיטובים בריכוז גז דו-מימדי של 2 האמפליטודות והרוחב של השיאים בספקטרום ההעברה כפונקציה של אנרגיית אופן המהוד .32 איור 5.שחושבו באמצעות התאמה למודל לורנציאני עבור כל ענף רזונטיבי בשני ne 1 1011 cm הקיטובים בריכוז גז דו-מימדי של 2 גורמי העירוב הפולריטוניים .40 איור C.עבור ערכים שונים של x יב .המחושבים עבור שני הקיטובים של האור הפוגע רוחבי הקו של ספקטרומי ההחזרה כפונקציה של ריכוז גז האלקטרונים הדו-מימדי.2 איור F.29 איור 5.39 איור 5. 17 110 איור 5.המחושב בעזרת שיטת מטריצת המעבר עבור שינוי רוחב השווה ל 0.22 איור 5.24 איור 5.14 איור 5.25 איור 5. 200 Å ne 1 1011 cmבאזור הבור עם גז אלקטרונים דו-מימדי בריכוז 2 ספקטרומי ההחזרה עבור ערכים שונים של שינוי רוחב המחושבים עבור פגיעה ניצבת של אור בשני הקיטובים .16 איור 5.26 איור 5.15 איור 5.94 -בT 2K - המיקום האנרגטי והערכים של המינימה של ספקטרום ההחזרה הניצבת של המבנה המתואר עבור ערכי שינוי רוחב שונים תמונת פרופיל סכמטית של מבנה מיקרומהוד עם בור קוונטי המוכל באזור המהוד של המבנה ספקטרום מקדם השבירה הכולל של בור קוונטי GaAs / Al 0.21 איור 5.11 איור 5. המחושבים עבור כל אחד מערכי ריכוז גז האלקטרונים הדו-מימדי בבור הקוונטי ועבור ערכים שונים של שינוי רוחב ב= T 2K עקומי החצייה של המינימה של ספקטרומי ההחזרה בקיטוב TEעבור כל אחד מערכי ריכוז גז האלקטרונים הדו-מימדי עקומי החצייה של המינימה של ספקטרומי ההחזרה בקיטוב TMעבור כל אחד מערכי ריכוז גז האלקטרונים הדו-מימדי אנרגיות הרזוננס של עקומי החצייה של מינימה ספקטרומי ההחזרה כפונקציה של ריכוז גז האלקטרונים הדו-מימדי .עבור T 2K עקומי החצייה של המינימה של ספקטרומי ההחזרה עבור שני הקיטובים של האור הפוגע המחושבים עבור T 2K עקומי החצייה של מינימה של ספקטרום ההחזרה של אור בקיטוב TMשהותאמו בעזרת מודל האוסצילטורים המצומדים ב= T 2K עקומי החצייה של מינימה של ספקטרום ההחזרה של אור בקיטוב TMביחד עם גורמי העירוב הפולריטוניים המתאימים תיאור סכמטי של מבנה המיקרומהוד הנחקר עם בור קוונטי מסומם המוצב באזור המהוד ספקטרומי ההחזרה עבור אור פוגע בקיטוב TEכפונקציה של אנרגיית אופן המהוד.19 איור 5. המחושבים עבור כל אחד מערכי ריכוז גז האלקטרונים הדו-מימדי בבור הקוונטי ועבור ערכים שונים של שינוי רוחב ב= T 2K ספקטרומי ההחזרה עבור אור פוגע בקיטוב TMכפונקציה של אנרגיית אופן המהוד.20 איור 5.שחושבו ידנית ובעזרת ההתאמה למודל האוסצילטורים המצומדים איור 5.18 איור 5.101 105 106 107 107 108 109 110 רמות האנרגיה של בולע לא מצומד ופיצול רמות האנרגיה עבור מערכת מצומדת תיאור סכמטי של מודל החישוב המוצע פרופיל סכמטי של המיקרומהוד המתואר ללא נוכחות בור קוונטי באזור המהוד תיאור סכמטי של דגם הבדיקה ותמונת החתך שלו ספקטרום ההחזרה של מבנה המיקרומהוד הנדון .9Asברוחב .12 איור 5.23 איור 5.13 איור 5.10 איור 5.1Ga 0.27 111 112 113 114 115 116 117 118 119 יא . 5 איור 5.3 איור 5.9 י .4 איור 5.7 איור 5.28 איור 4.30 איור 4.6 איור 5.26 איור 4.31 איור 4.2 89 90 92 92 93 95 96 97 98 99 100 איור 5.33 איור 5.1 איור 5.27 איור 4.8 איור 5.25 איור 4.81 ספקטרום שינוי מקדם השבירה בקיטוב TMהמחושב עבור המבנה המסומם עם ריכוזי גז שונים בעזרת מודל האלקטרונים החופשיים ומודל Hartree-FockבT 2K - ספקטרום הבליעה בקיטוב TEוספקטרום הפליטה הספונטנית המחושב עבור המבנה המסומם בריכוזי גז שונים בעזרת מודל Hartree-FockבT 2K - ספקטרום הבליעה בקיטוב TMוספקטרום הפליטה הספונטנית המחושב עבור המבנה המסומם בריכוזי גז שונים בעזרת מודל Hartree-FockבT 2K - תמונה ממוקדת של אזור המעבר של ספקטרומי הבליעה והפליטה הספונטנית עבור קיטוב TEהמחושבים בעזרת מודל Hartree-FockבT 2K - תמונה ממוקדת של אזור המעבר של ספקטרומי הבליעה והפליטה הספונטנית עבור קיטוב TMהמחושבים בעזרת מודל Hartree-FockבT 2K - ספקטרום הבליעה בקיטוב TEהמחושב עבור המבנה המסומם עבור ריכוזי גז שונים בעזרת מודל האלקטרונים החופשיים ומודל Hartree-FockבT 77K - ספקטרום הבליעה בקיטוב TMהמחושב עבור המבנה המסומם עבור ריכוזי גז שונים בעזרת מודל האלקטרונים החופשיים ומודל Hartree-FockבT 77K - ספקטרום הבליעה בקיטוב TEוספקטרום הפליטה הספונטנית המחושבים עבור המבנה המסומם עם ריכוזי גז שונים בעזרת מודל Hartree-FockבT 77K - ספקטרום הבליעה בקיטוב TMוספקטרום הפליטה הספונטנית המחושבים עבור המבנה המסומם עם ריכוזי גז שונים בעזרת מודל Hartree-FockבT 77K - ספקטרום ההחזרה של אטלון Fabri-Perot פונקציית ההעברה של מהוד עבור מספר ערכים של מקדמי החזרה של המראות תיאור סכמטי של מחזיר Braggמפולג תיאור סכמטי של התקדמות אור דרך הממשק בין שתי שכבות דיאלקטריות והתקדמות דרך תווך הומוגני אמפליטודה ופאזה של החזרה ניצבת המחושבים עבור מבנה DBRעם 35שכבות בעלי מקדם שבירה גבוה ונמוך תיאור סכמטי של מבנה מיקרומהוד אמפליטודה ופאזה של החזרה ניצבת המחושבת עבור מבנה מיקרומהוד עם שתי מראות DBRעם 35שכבות בעלי מקדם שבירה גבוה ונמוך חישוב אפליטודת השדה החשמלי בתוך מיקרומהוד עבור שתי קונפיגורציות מראות ה- DBR עקומי הדיספרסיה עבור פוטון הכלוא במיקורומהוד עבור אופני תנודה שונים איור 4.29 82 83 84 85 86 87 איור 4.32 איור 4. המחושבים בעזרת מודל האלקטרונים החופשיים ומודל Hartree-Fockעבור בור לא מסומם בT 2K - תיאור סכמטי של ספקטרום הבליעה בקצה הפס עבור מוליך למחצה דו-מימדי עפ"י מודל Elliot ספקטרומי הפליטה הספונטנית המחושבים בעזרת מודל האלקטרונים החופשיים ומודל Hartree-Fockעבור בור לא מסומם בT 2K - סוספטביליות חשמלית מרוכבת המחושבת בעזרת מודל האלקטרונים החופשיים עבור בור לא מסומם בT 77K - סוספטביליות חשמלית מרוכבת המחושבת בעזרת מודל Hartree-Fockעבור בור לא מסומם בT 77K - ספקטרומי הבליעה עבור שני הקיטובים של האור .1Ga 0.11 איור 4.9Asברוחב 200 Åהמכיל ריכוזים שונים של גז האלקטרונים איור 4.24 ט .8 איור 4.63 64 66 67 68 קוונטי נירמול הפס האסור הכולל במרכז אזור .14 איור 4.Brillouinעבור בור GaAs / Al 0.20 69 70 71 72 73 74 75 75 76 76 איור 4.אלקטרון-אלקטרון וחור-חור מבנה סכמטי של בור קוונטי לא מסומם הנבחן במסגרת החישובים הנומריים סוספטביליות חשמלית מרוכבת המחושבת בעזרת מודל האלקטרונים החופשיים עבור בור לא מסומם בT 2K - סוספטביליות חשמלית מרוכבת המחושבת בעזרת מודל Hartree-Fockעבור בור לא מסומם בT 2K - ספקטרומי הבליעה עבור שני הקיטובים של האור .7 גורמי הצורה המחושבים עבור שילובי אלקטרון-חור .18 איור 4.23 איור 4.Fermiשניהם כפונקציה של ריכוז גז האלקטרונים הדו-מימדי ספקטרום הבליעה בקיטוב TEשחושב עבור המבנה המסומם עבור ריכוזי גז שונים בעזרת מודל האלקטרונים החופשיים ומודל Hartree-FockבT 2K - ספקטרום הבליעה בקיטוב TMשחושב עבור המבנה המסומם עבור ריכוזי גז שונים בעזרת מודל האלקטרונים החופשיים ומודל Hartree-FockבT 2K - ספקטרום שינוי מקדם השבירה בקיטוב TEהמחושב עבור המבנה המסומם עם ריכוזי גז שונים בעזרת מודל האלקטרונים החופשיים ומודל Hartree-FockבT 2K - איור 4.19 איור 4.17 איור 4.21 איור 4.15 איור 4.13 איור 4.10 איור 4.9 איור 4.16 איור 4.המחושבים בעזרת מודל האלקטרונים החופשיים ומודל Hartree-Fockעבור בור לא מסומם בT 77K - ספקטרומי הפליטה הספונטנית המחושבים בעזרת מודל האלקטרונים החופשיים ומודל Hartree-Fockעבור בור לא מסומם בT 77K - מבנה סכמטי של בור קוונטי מסומם הנבדק באמצעות סימולציות נומריות איור סכמטי של תופעת נירמול רוחב הפס האסור בעקבות הכנסת גז אלקטרונים דו- מימדי לאזור הבור פער האנרגיה בין אנרגיית רמת Fermiותחתית פס ההולכה ווקטור גל .22 78 79 80 איור 4.12 איור 4. 6 .9Asברוחב 200 Å עבור .1Ga 0. המתואר באיור 2.1 איור 3.9Asברוחב 200 Åעם ריכוזי גז אלקטרונים דו-מימדי קוונטי בור עבור המחושבת 62 62 .T 2Kללא סימום נוסף אלמנט המטריצה של הדיפול עבור בור קוונטי GaAs / Al 0.3 איור 4.1Ga 0.9Asברוחב 200 Åעם ריכוז גז אלקטרונים דו-מימדי .9Asברוחב 200 Åעבור .T 2Kעם ריכוזי גז אלקטרונים דו- מימדים שונים באיזור הבור פונקציות הגל המחושבות עבור תתי-פסי ההולכה והערכיות עבור בור קוונטי GaAs / Al 0.1Ga 0.9Asברוחב 200 Å איור 2. 1 109 1 1011 cבטמפ' סביבה T 2K m 2 חישוב אנרגיית קולון-חור העצמית ואנרגיית ההחלפה המסוככת במרכז אזור .20 איור 2.T 2Kללא ערכים שונים של ריכוז גז אלקטרונים דו-מימדי באזור הבור היררכיה אינסופית של מכפלות אופרטורים עבור משוואות התנועה תיאור סכמטי של מכפלות האופרטורים של משוואות התנועה בקירוב Hartree-Fock תיאור הגישה העושה שימוש בניסוח הפנומנולוגי של סיכוך הפלזמה .2 איור 3.29 29 29 30 30 32 המבנה ודיאגרמת הפסים עבור אזור הממשק בין GaAsוAlGaAs - תיאור סכמטי של גוש חומר מוליך למחצה ושכבה אפליטסיאלית דקה עם סימום צפיפות מטען נפחית עבור הבור הקוונטי הנדון בעל סימום מסוג במרחק 1000 Å איור 2.1Ga 0.9Asברוחב 200 Åהמכיל 2 ח איור 4.1Ga 0.9Asברוחב 200 Åהמכיל ריכוזים שונים גז אלקטרונים דו-מימדי ועבור תת-פסי ההולכה וערכיות דיספרסיית נירמול הפס האסור עבור מעברי תתי-פסים שונים המחושבים עבור בור ne 1 1011 cm קוונטי GaAs / Al 0.2 איור 4.23 34 .19 תיאור סכמטי של בור קוונטי מעורב מסוג Iוסוג IIהמורכב משכבות של GaAs / AlAs קשר הדיספרסיה המחושב עבור תת-פסי ההולכה והערכיות עבור בור קוונטי GaAs / Al 0.19 איור 2. 1 1010 cבטמפ' סביבה T 2K m 2 פונקציות הגל המחושבות עבור תתי-פסי ההולכה והערכיות עבור בור קוונטי GaAs / Al 0.3 איור 4.1Ga 0.24 49 50 50 55 איור 3.1Ga 0.18 איור 2.Brillouin עבור בור קוונטי GaAs / Al 0.המשולבת ישירות להמילטוניאן של הגוף המרובה 55 58 61 הפונקציה הדיאלקטרית המסוככת ) (q GaAs / Al 0.21 איור 2.1 איור 4.22 מקצה הבור פוטנציאל קצה הפס עובר פסי ההולכה והערכיות עבור בור קוונטי בודד עם סימום מסוג .1Ga 0.17 איור 2.5 איור 4.4 עבור .9Asברוחב 200 Åעם ריכוז גז אלקטרונים דו-מימדי 33 איור 2. 1 1011 cבטמפ' סביבה T 2K m 2 התלות של עוצמת המעבר kבזווית שבין וקטור הגל של האלקטרון ובין וקטור הקיטוב של השדה החשמלי אלמנט המטריצה של הדיפול עבור בור קוונטי GaAs / Al 0. 100 Åעם 2 1018 cבשכבת הבור סימום מסוג nבצפיפות m 3 עוצמת השדה החשמלי הנוצר במבנה כתוצאה מהתפלגות המטען מאיור 2.11 איור 2.9Asללא סימום ברוחב 200 Å איור 2.5 איור 2.רשימת איורים 8 10 15 19 20 20 21 ייצוג סכמטי של פונקציות שונות של הגביש הצגה סכמטית של סיווג פסי האנרגיה בתורת ההפרעות של Lödwin מבנה הפסים של GaAsליד הנקודה בטמפרטורת החדר .14 איור 2.9 24 25 עוצמת השדה החשמלי הנובע ממישור אינסופי עם צפיפות מטען משטחית של ) d (z צפיפות מטען משטחית עבור בור קוונטי GaAs / Al 0.15 איור 2.10 איור 2.1Ga 0.16 25 26 27 27 28 ז .8 23 איור 2.2 איור 2.המחושב בעזרת מודל kp · 8 8 עבור שני כיוונים קריסטלוגרפיים שונים קווים שווי האנרגיה בכל מישור ] [100במרחב kעבור פסי החורים הקלים והכבדים בגוש GaAs מבנה תת-הפסים של פס הערכיות במישור הבור הקוונטי דיספרסיית תת-פסי הערכיות ותנאי השפה בממשקי החומרים בבור קוונטי קשר הדיספרסיה של תת-פסי הערכיות עבור בור קוונטי GaAs / Al 0.11 פוטנציאל קצה הפס ופוטנציאל העקביות העצמית עבור בור פוטנציאל עם סימום מאופנן איור 2.9Asברוחב 200 Åוצפיפויות המצבים המתאימות קשר הדיספרסיה של תת-פסי ההולכה עבור בור קוונטי GaAs / Al 0.7 22 איור 2.9Asברוחב 200 Åוצפיפויות המצבים המתאימות פונקציות הגל תתי-פסי ההולכה והערכיות עבור בור קוונטי GaAs / Al 0.1 איור 2.4 איור 2.12 איור 2.3 איור 2.1Ga 0.11 דיאגרמת בלוקים המתארת את תהליך העקביות העצמית סכום פוטנציאל קצה הפס והפוטנציאל הפואסוני עבור בור קוונטי בודד עם התפלגות המטען מאיור 2.6 איור 2.1Ga 0.11 הפוטנציאל החשמלי הנוצר במבנה כתוצאה מהתפלגות המטען מאיור 2.1Ga 0.9Asברוחב .13 איור 2. 3.5.5.1מהוד אופטי .4.1בורות קוונטיים ללא הוספת סימום .5.5.5.5.4פולריטונים במיקרומהוד .2.5.1צימוד חזק של כולא קוונטי ופוטון .4.2.5.4.5.4.4.5מיקרומהודים של מוליכים למחצה ואפקטים פולריטוניים .4.2קירוב Hartree-Fock .2מחזירור Braggמפולג .4.3.3.4פיתרון משוואות התנועה .4.5.2.3.3תוצאות החישובים הנומריים .2צימוד חזק במיקרומהוד מוליך למחצה .1מיקרומהוד ריק .3תכונות אופטיות של מיקרומהוד .3אפקטים של גוף מרובה .55 57 59 60 65 66 71 91 91 93 94 97 97 99 99 100 103 106 106 108 112 128 130 131 134 139 141 144 148 151 .5שיקולים במימוש נומרי .5.5.1שיטת מטריצת המעבר .1ספקטרום החזרה של מיקרומהוד .5.2מיקרומהוד עם בור קוונטי לא מסומם .5תוצאות החישובים הנומריים .5.2בורות קוונטיים בנוכחות סימום .2תכונות האור הכלוא במיקרומהוד .3מיקרומהוד עם בור קוונטי עם סימום סיכום נספחים נספח א' :תכונות סימטריה של פונקציות גל נספח ב' :מימוש נומרי של מודל שני הפסים נספח ג' :פיתרון נומרי של מודל Schrödinger-Poisson נספח ד' :מודל הסיכוך של Lindhard נספח ה' :שיטות לחישוב אנרגיית הפס האסור עבור חומרים מוליכים למחצה נספח ו' :חישוב מקדם השבירה של סגסוגות AlxGa1x As ביבליוגרפיה ו .2.2.5.4.5. תוכן עניינים v viii xii 1 2 5 6 6 6 8 8 10 10 11 12 17 22 23 28 35 35 35 38 40 41 41 41 43 43 45 46 47 51 51 51 53 53 54 54 ה תוכן עניינים רשימת איורים רשימת טבלאות תקציר רשימת סמלים וקיצורים .1הקדמה .2תכונות אלקטרוניות של מוליכים למחצה .2.1משוואת Schrödingerשל הגביש .2.1.1מבוא .2.1.2משפט Bloch .2.1.3מבוא לשיטת k.p .2.2שיטת פונקציית המעטפת של k.p .2.2.1מבוא .2.2.2פונקציית המוצא .2.2.3מודל Zinc-Blende .2.2.4מודל שני פסים .2.3מודל Schrödinger-Poisson .2.3.1הצגת המודל .2.3.2גז אלקטרונים דו-מימדי .3מעברים אופטיים בנוכחות אלקטרונים חופשיים .3.1שיטת הקוונטיזציה השנייה .3.1.1מבוא .3.1.2ניסוח מצבי Bloch .3.1.3הוספת חורים למודל .3.1.4סטטיסטיקות נושאי המטען .3.2חישוב המעברים .3.2.1מבוא .3.2.2ניסוח קוונטי של הקיטוב המיקרוסקופי .3.2.3משוואת התנועה של Heisenberg .3.2.4פיתרון משוואת התנועה עבור קירוב האלקטרונים החופשיים .3.3פליטה ספונטנית .3.4אלמנט מטריצה של התנע .4מעברים אופטיים בנוכחות אינטראקציה קולונית .4.1ניסוח הקוונטיזציה השנייה .4.1.1מבוא .4.1.2הקירוב האלכסוני .4.1.3הוספת חורים למודל .4.2חישוב המעברים .4.2.1ניסוח משוואות התנועה ד פולריטורנים הנובעים מהאינטראקציה בין זוגות אלקטרון-חור ופוטונים כלואים חיבור על מחקר לשם מילוי חלקי של הדרישות לקבלת תואר מגיסטר למדעים בפיסיקה יוסף מיכאלי הוגש לסנט הטכניון – מכון טכנולוגי לישראל מרץ 2011 חיפה הדר ב' תשע"ב . פולריטורנים הנובעים מהאינטראקציה בין זוגות אלקטרון-חור ופוטונים כלואים יוסף מיכאלי .