qcl lecture long

8/12/2019 QCL Lecture Long

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Intersubband Optoelectronics

Jerome Faist

ETH Zurich

Zurich, September 17, 2009


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Contents

List of figures xiv

List of tables xv

List of variables and symbols xv

1 Introduction: intersubband optoelectronics as an example of quantum en-gineering 11.1 Quantum engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Basic requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Tools of the mid-infrared . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3.1 Transparent materials . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.2 Reflectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.3 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.4 Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.5 Polarizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.6 Spectrometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Notes and acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Technology 52.1 Molecular Beam epitaxy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Metal Organic Vapor Phase epitaxy . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Reactive Ion etching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Quantum Cascade laser processing . . . . . . . . . . . . . . . . . . . . . . . 6

3 Electronic states in semiconductor quantum wells 93.1 Band structure of semiconductors in the kp approximation: origin of the

effective mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.1.1 Basic approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.1.2 Beyond the perturbation expansion . . . . . . . . . . . . . . . . . . . 123.1.3 Example: a two-band Kane model . . . . . . . . . . . . . . . . . . . . 12

3.2 Envelope function approximation . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.1 Multiband case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

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3.2.2 One Band model: the Ben-Daniel Duke . . . . . . . . . . . . . . . . . 15

3.2.3 Two band model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.4 Formal derivation of the 2x2 model . . . . . . . . . . . . . . . . . . . 16

3.3 Hartree potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.4 Building blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.4.1 The single quantum well . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.4.2 The coupled well system . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4.3 The superlattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.5 In-plane dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.6 Full model: the valence band . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Intersubband absorption 294.1 Interaction Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2 Intersubband and interband transition . . . . . . . . . . . . . . . . . . . . . 30

4.3 Selection rules and absorption geometries . . . . . . . . . . . . . . . . . . . . 30

4.4 Absorption strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.4.1 Absorption for a 2D system . . . . . . . . . . . . . . . . . . . . . . . 32

4.4.2 Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.4.3 Gain and loss cross sections . . . . . . . . . . . . . . . . . . . . . . . 33

4.5 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.6 Sum rule in absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.7 Absorption in a quantum well: a two-band model . . . . . . . . . . . . . . . 374.8 Absorption linewidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.8.1 Non-parabolicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.8.2 Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.9 Stark-tuning of intersubband absorption . . . . . . . . . . . . . . . . . . . . 42

5 Intersubband scattering processes 45

5.1 Spontaneous emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.2 Scattering rate by bulk optical phonons . . . . . . . . . . . . . . . . . . . . . 47

5.2.1 Temperature dependence . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.3 Quasi-elastic intersubband process . . . . . . . . . . . . . . . . . . . . . . . . 50

5.3.1 Acoustic phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.3.2 Electron-electron scattering . . . . . . . . . . . . . . . . . . . . . . . 52

5.3.3 Impurity scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.4 Comparison with experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.4.1 Interband pump and probe . . . . . . . . . . . . . . . . . . . . . . . . 56

5.4.2 Intersubband pump probe . . . . . . . . . . . . . . . . . . . . . . . . 57

5.4.3 Intersubband saturation experiments . . . . . . . . . . . . . . . . . . 57

5.4.4 Intersubband electroluminescence . . . . . . . . . . . . . . . . . . . . 57

5.4.5 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . 58


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6 Detectors 61

6.1 Fundamentals of infrared detection: noise, BLIB.. . . . . . . . . . . . . . . . 61

6.2 Quantum Well Infrared Photoconductor (QWIP) . . . . . . . . . . . . . . . 61

7 Mid-infrared waveguides 63

7.1 Dielectric waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

7.2 Dispersion of doped layers, Drude approximation . . . . . . . . . . . . . . . 63

7.3 Metal-based waveguides for the Terahertz . . . . . . . . . . . . . . . . . . . 63

7.4 Far-field computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

8 Quantum Cascade lasers I: fundamentals 65

8.1 Historical perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658.2 Active region: fundamental concepts . . . . . . . . . . . . . . . . . . . . . . 66

8.2.1 Injection/relaxation region . . . . . . . . . . . . . . . . . . . . . . . . 66

8.2.2 Cascading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

8.3 Intersubband versus interband lasers . . . . . . . . . . . . . . . . . . . . . . 69

8.4 Rate equation analysis, threshold condition, slope efficiency . . . . . . . . . 70

8.5 Optimization of the active region: intersubband toolbox . . . . . . . . . . . 71

8.5.1 Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

8.5.2 Optical phonon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

8.5.3 Phase space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

8.5.4 Escape time, Bragg reflection and upper level confinement . . . . . . 78

8.5.5 Injection efficiencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

8.6 Optimization of the active region: different designs . . . . . . . . . . . . . . 81

8.6.1 Designs: general trends . . . . . . . . . . . . . . . . . . . . . . . . . . 81

8.6.2 Three quantum well active region . . . . . . . . . . . . . . . . . . . . 81

8.6.3 Double phonon resonance . . . . . . . . . . . . . . . . . . . . . . . . 83

8.6.4 Boundtocontinuum active regions . . . . . . . . . . . . . . . . . . . 85

8.7 Cascading: scaling with the number of periods . . . . . . . . . . . . . . . . 87

8.7.1 Threshold current density . . . . . . . . . . . . . . . . . . . . . . . . 87

8.7.2 slope efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 888.8 Temperature dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

8.8.1 intersubband physics . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

8.8.2 Backfilling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

8.8.3 Self-heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

8.9 Doping of the active region . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

8.10 Fundamental limits of quantum cascade lasers . . . . . . . . . . . . . . . . . 95

8.11 Short wavelengths: discontinuity and strain compensation . . . . . . . . . . 98

8.11.1 Conduction band discontinuity and performance . . . . . . . . . . . . 98

8.11.2 Strain-compensated InxGa1xAs/AlyIn1yAs/InP material system . . 99


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iv CONTENTS

9 Quantum Cascade laser II: Mode control 1079.1 Fabry Perot cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1079.2 Distributed feedback cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

9.2.1 Multilayer approach, Bragg reflection condition . . . . . . . . . . . . 1099.2.2 coupled mode analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 1129.2.3 Fabrication geometries . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Surface grating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114Buried grating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

9.2.4 Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119Dynamical behavior and linewidth . . . . . . . . . . . . . . . . . . . 120Cavity pulling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

9.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1239.4 External cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

10 Quantum Cascade lasers III: applications 12510.1 Energy deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

10.1.1 Wallplug efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12610.1.2 High power operation . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

10.2 Telecommunications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12610.2.1 high frequency modulation . . . . . . . . . . . . . . . . . . . . . . . . 126

10.2.2 Telecommunications experiments . . . . . . . . . . . . . . . . . . . . 12610.3 G as sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

10.3.1 Gas absorption lines: pressure and temperature depdendence . . . . . 12610.3.2 General system consideration . . . . . . . . . . . . . . . . . . . . . . 12610.3.3 multipass cells and fringe noise . . . . . . . . . . . . . . . . . . . . . 12610.3.4 interpulse modulation . . . . . . . . . . . . . . . . . . . . . . . . . . 12610.3.5 intrapulse modulation . . . . . . . . . . . . . . . . . . . . . . . . . . 12610.3.6 Wavelength modulation . . . . . . . . . . . . . . . . . . . . . . . . . 12610.3.7 Photoacoustic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

10.4 Liquid sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

11 Quantum Cascade lasers IV: Transport models 12711.1 Classical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

11.1.1 Drude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12711.1.2 Esaki-Tsu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

11.2 Transport as a result of intersubband transitions . . . . . . . . . . . . . . . . 12711.2.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12711.2.2 Diagonal transition lasers . . . . . . . . . . . . . . . . . . . . . . . . 12711.2.3 With charge conservation and the optical field . . . . . . . . . . . . . 12711.2.4 Computing the electron distribution in the subband . . . . . . . . . . 127

11.3 Resonant tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129


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11.3.1 Problem of coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . 12911.3.2 Density matrix approach: Kazarinov, Willenberg . . . . . . . . . . . 12911.3.3 second-order current . . . . . . . . . . . . . . . . . . . . . . . . . . . 12911.3.4 Full model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

11.4 Non-equilibrium Greens function theory . . . . . . . . . . . . . . . . . . . . 129

12 Quantum Cascade Lasers V: Device properties and characterization 13112.1 Electrical charateristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13112.2 Gain and loss measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 13112.3 Electron temperature measurements: hot electrons and hot phonons. . . . . 13112.4 Active region non-linearities . . . . . . . . . . . . . . . . . . . . . . . . . . . 13112.5 Confined phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

13 Intersubband processes II (collective excitations) 13313.1 Depolarization shift. (Dielectric model and connection to the oscillator model)13313.2 Intersubband plasmon, linewidth versus charge density . . . . . . . . . . . . 13313.3 Raman spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13313.4 Optical microcavities and cavity polaritons (Alessandro, Carlo, ..) . . . . . . 133

14 Interlevel transition in Quantum dots 13514.1 Fabrication: self-assembled Stransky Krastanov growth . . . . . . . . . . . . 13514.2 Interband spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13514.3 Intraband spectroscopy techniques . . . . . . . . . . . . . . . . . . . . . . . 135

14.4 The intersubband polariton in quantum dots. (Ferreira, Luke Wilson, ...) . . 135

15 Intersubband non-linearities and non-linear devices 13715.1 Non-linear intersubband polarization (Jacob Kurghin, Ghilad Almoghi, Carlo

Sirtori,) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13715.2 Second harmonic generation structures (step well, coupled wells) . . . . . . 13715.3 Trade-off between power and non-linear power; phase matching condition . . 13715.4 Non-linear generation of THz (Carlo) . . . . . . . . . . . . . . . . . . . . . . 13715.5 Quantum Cascade lasers with non-linear elements: mid-IR (Claire) and THz

(Belkin) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

15.6 NIR-THz mixing in QCL (Carlo) . . . . . . . . . . . . . . . . . . . . . . . . 137

16 Appendix: Designs 13916.1 Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

16.1.1 3QW 10.3um diagonal . . . . . . . . . . . . . . . . . . . . . . . . . . 13916.1.2 3QW 5.3um vertical . . . . . . . . . . . . . . . . . . . . . . . . . . . 13916.1.3 Two-phonon resonance at 9um . . . . . . . . . . . . . . . . . . . . . . 13916.1.4 Bound-to-continuum at 9um . . . . . . . . . . . . . . . . . . . . . . . 13916.1.5 Bound to continuum at 16um . . . . . . . . . . . . . . . . . . . . . . 13916.1.6 Broad bound-to-continuum . . . . . . . . . . . . . . . . . . . . . . . . 140


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List of Figures

2.1 TEM image of a QCL active region . . . . . . . . . . . . . . . . . . . . . . . 52.2 STM image of the active region of graded and abrupt interface design . . . . 6

2.3 Processing steps for the fabrication of a Fabry-Perot cavity quantum cascadelaser. a) MBE or MOCVD growth of the active region. b) Defining the ridgeby photolithography ( spin resist, expose with UV light and develop). c) Theridge is etched in a wet etching solution. d) Si3N4 is deposited by PECVD. e)Open the insulating oxide on top of the ridges. f) Top and bottom metalization. 7

2.4 Schematic drawing of the facet of a processed device. . . . . . . . . . . . . . 8

3.1 GaAs band structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Comparison for various model for the energy-dependent effective mass . . . . 173.3 Computed confinement energy of electrons in the conduction band as a func-

tion of well width. Bound states are shown with full lines, resonances withdotted lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.4 Energy states of a quantum well computed with a two-band model, and com-pared with a one-band model (dashed lines). The growing importance ofnon-parabolicity as one moves away from the gap is clearly apparent. . . . . 21

3.5 A coupled quantum well system. . . . . . . . . . . . . . . . . . . . . . . . . . 213.6 A coupled quantum well system. . . . . . . . . . . . . . . . . . . . . . . . . . 223.7 Schematic band structure of a superlattice. . . . . . . . . . . . . . . . . . . 233.8 Computed dispersion of a 100A InGaAs well and 30A AlInAs barrier super-

lattice. A conduction band discontinuity of Ec = 0.52eVas well as a Kaneenergy of 18.3eVwas assumed in an effective two-band model. For clarity, thezero of the energy scale has been set to the bottom of the InGaAs quantumwells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.9 Dispersion of the first miniband (green), compared to a sinus function (red).The result of the finite superlattice are also plotted as full squares . . . . . . 24

3.10 Square of the wavefunctions for a finite superlattice formed by eight 100AInGaAs wells separated by 30A AlInAs barriers. . . . . . . . . . . . . . . . 25

3.11 Dispersion of the state of a quantum well in the valence band. . . . . . . . . 263.12 Schematic description of the origin of the valence band dispersion in the quan-

tum well showing schematically the effects of confinement and interactions. 27

4.1 Experimental geometries allowing the measurements of intersubband transitions 31

vii


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viii LIST OF FIGURES

4.2 Photocurrent in a quantum well photoconductor as a function of the polar-ization. The residual responsivity for the TE polarization is less than 0.2%than for the electric field normal to the layers. (From H.C.Liu et al, APL 1998) 31

4.3 Intersubband absorption in an 65A thick quantum well. (From West andEglash, APL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.4 Intersubband absorption between two bound states . . . . . . . . . . . . . . 35

4.5 Intersubband absorption in a multiquantum well designed for triply resonantnon-linear susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.6 Intersubband absorption from a bound state to a continuum . . . . . . . . . 36

4.7 Comparison of the intersubband absorption for various structures, with energylevels schematically drawn close to the curves . . . . . . . . . . . . . . . . . 37

4.8 Lineshape functions in various limiting cases. . . . . . . . . . . . . . . . . . . 404.9 Transmission through a 15nm thick InAs/AlSb mulitquantum well system.

Because of the non-parabolicity of this system, a broad (about 20meV wide)peak would be expected, in contrast with the narrow experimental line ob-served. This narrowing was interpreted by Warburton et al. as a result ofdepolarization shift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.10 Linewidth of the absorption of quantum wells of various widths. The dottedline shows the compute value of the broadening caused by the optical phononscattering. The decrease in linewidth as a function of quantum well width is a 42

4.11 Experimental measurement of the broadening of the intersubband transition

and mobility as a function of temperature, compared with the model proposedby Unuma et al. [1]. Optical and acoustic phonon intra- and inter-subbandprocesses are also included in the model, as indicated. . . . . . . . . . . . . . 43

5.1 Schematic description of the relevant scattering mechanism in the two relevantcases. The simplest case is the situation occuring in the Mid-Infrared (de-picted on the left) where the LO phonon scattering is dominant. In contrast,in the terahertz, at low temperature, a number of non-radiative processesshould be taken into account including acoustic phonons, electron scatteringand other elastic processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.2 Radiative lifetime of a square quantum well as a function of the transitiionenergy. A constant value of the oscillator strenght of 23, corresponding to anInGaAs/AlInAs quantum well, has been assumed . . . . . . . . . . . . . . . 46

5.3 Scattering by an optical phonon between two subbands. An electron withinitial energy Ei and initial wavevector ki is scattered to the lower subbandat kf, therefore losing (or gaining) an energy equal to LO in the process. . 47

5.4 Computed electron lifetime (left vertical axis) due to optical phonon scat-tering as a function of transition energy in a square quantum well, whosewidth is indicated in the rigth vertical axis. Zero initial kinetic energy in theupper subband, and zero temperature was assumed. The filled squares are

experimental measurements, as indicated. . . . . . . . . . . . . . . . . . . . . 48


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LIST OF FIGURES ix

5.5 Computed electron lifetime due to optical phonon scattering as a functionof the initial kinetic energy. A square, 220Athick InGaAs/AlInAs quantumwell was considered. The equivalent electron temperature is shown in the tophorizontal axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.6 Computed temperature dependence of the intersubband lifetime for a 100Athick InGaAs/AlInAs quantum well . . . . . . . . . . . . . . . . . . . . . . . 50

5.7 Intersubband relaxations processes in a quantum well with an energy separa-tion smaller than the optical phonon energy E21 < LO. In this configuration,various processes may play a role: optical phonon emission (op), electron-electron scattering (EE), ionized-impurity scattering or interface roughnessscattering (II), acoustic-phonon emission (AC). . . . . . . . . . . . . . . . . 51

5.8 Various electron-electron scattering processes. The 2, 2

1, 1 is the domi-nant electron-electron scattering term, as the Auger-like term 2, 2 2, 1 wasshown to vanish in symmetric quantum wells. The 2, 1 2, 1 will lead tothermalization between subband while the 2, 2 2, 2 or 1, 1 1, 1 will tendto thermalize the subband themselves. . . . . . . . . . . . . . . . . . . . . . 52

5.9 The electron-electron scattering time for the 2, 2 1, 1 process presentedas a function of well width for an infinite quantum well at zero temperaturefor different screening models: no screening, constant screening length, staticsingle subband screening for and ideal 2D system with no z-dependence (i.e.|A2,21,1| = 1), static single subband screening includingq- dependence formfactor. The Fermi energy of the excited state Ef2 is equal to 10meV and the

initial electron state wave vector ki is equal to zero. Adapted from Ref [2],with the correction suggested by [?], with permission . . . . . . . . . . . . . 54

5.10 Techniques for the measurement of the intersubband lifetime. . . . . . . . . . 56

5.11 Lifetime as a function of pumping density. . . . . . . . . . . . . . . . . . . . 60

8.1 Schematic band structure of the device considered by R. Kazarinov and R.Suris. A superlattice is under a sufficiently strong electric field such that theground state of a well is above the first excited state of the following well . . 66

8.2 a) Schematic conduction band diagram of a quantum cascade laser. Each stageof the structure consists of an active region and a relaxation/injection region.

Electrons can emit up to one photon per stage. b) General philosophy of thedesign. The active region is a three-level system. The lifetime of the 32transition has to be longer than the lifetime of level 2 to obtain populationinversion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

8.3 Highly simplified picture of the effective potential in a quantum cascade laser.At zero field, the current is blocked by the injection regions that prevent thecurrent from flowing. With the application of a strong electric field, theseinjection regions are flattened by the electric field and the current flowsalong the electronic cascade, generating the photons. . . . . . . . . . . . . . 68

8.4 Population inversion between the two upper levels of a ladder of three states.

The relevant non-radiative process are also indicated. . . . . . . . . . . . . . 70


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x LIST OF FIGURES

8.5 Schematic illustration of the behavior of the population inversion and thephoton flux with injected current. . . . . . . . . . . . . . . . . . . . . . . . . 72

8.6 Two states coupled by a tunnel barrier. The lifetime of the upper state iscontrolled by the thickness of the coupling barrier . . . . . . . . . . . . . . . 73

8.7 Photon-assisted tunneling transition. Plotted are the oscillator strengthf =2m02

z2Eij and optical phonon scattering rate as a function of barrier width forsystem consisting of a 4nm and 1.5nm InGaAs well coupled by an AlInAs bar-rier. The additional points are the two values for a 6nm thick single quantumwell exhibiting the same transition energy . . . . . . . . . . . . . . . . . . . 74

8.8 Product of the oscillator strength and the lifetime as a function of tunnelbarrier thickness. This number is a figure of merit of the gain cross section.The value forL

b= 0 is the one computed for a single quantum well 6nm thick,

as in Fig. ??. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

8.9 Engineering a population inversion using an optical phonon resonance. Ina ladder of three states, where the two lower ones are spaced by an opticalphonon energy, the n=3 lifetime will naturally be longer than the n=2 becauseof the smaller exchanged wavevector for the optical phonon emission betweenthe n=2 and n=1 states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

8.10 Engineering the lifetimes. In a superlattice, a population inversion builds upat the edges of the minigap because of the phase space for scattering out fromupper state of the lower miniband is much larger than the phase space toscatter in the same state. a) real-space and (b) reciprocal space picture of theenergy bands. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

8.11 Product of the oscillator strength and the upper state lifetime (left) andbranching ratio3/32 (right) for a 60/5nm GaInAs/AlInAs superlattice withan number of period N. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

8.12 Computed transmission of a injector designed to provide maximum reflectionat the energy of the excited state, as shown. . . . . . . . . . . . . . . . . . . 79

8.13 Comparison of the electroluminescence efficiency of a structure designed withand without a Bragg reflector facing the upper state. . . . . . . . . . . . . . 80

8.14 Schematic description of the relevant levels and injection efficiencies in a cas-

cade laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818.15 Schematic diagram of the first generation of quantum cascade lasers. . . . . 82

8.16 Schematic of the parameters optimized in the active regions. . . . . . . . . . 82

8.17 Schematic potential profile and design of the active region of a quantum cas-cade laser based on three quantum wells. a) Two quantum well before theapplication of the electric field. b) The electric field bring the two groundstates in resonance and yields a splitting equal to the optical phonon energy.A third thinner well is added upstream. If the state of this well is resonantwith the excited state of the coupled well, the resulting transition is diagonal(c), if the well is thinner and the resonance is above, the transition is vertical

(d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83


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LIST OF FIGURES xi

8.18 Examples of active regions based on a three quantum wells, using lattice-matched InGaAs/AlInAs material. Structure a) is designed for a wavelengthof = 10.3m using the concept shown in Fig. 8.17 c), that leads to adiagonal transition with a longer lifetime. Structure b) is designed for =5.3m and exhibits a vertical transition, as explained in Fig. 8.17 d) [3]. . . 84

8.19 Schematic band diagram of a two-phonon resonance gain region designed foroperation at 9m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

8.20 Low-temperature (T= 80K) luminescence spectra of the active region basedon a two-phonon resonance. Calculated transition energies are shown on thetop horizontal axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

8.21 Oscillator strength (right axis) and product of oscillator strength and upperstate lifetime (left axis) as a function of applied electric field for a superlatticeconsisting of 5 period of a 59AInGaAs quantum well and a 9AAlInAs barrier 87

8.22 Three first miniband edges as a function of position for a chirped superlattice(a) and for a bound-to-continuum structure c). . . . . . . . . . . . . . . . . 88

8.23 a) Schematic conduction band diagram of one stage of a bound-to-continuumactive region under an applied electric field of 3.5 104V/cm. The modulisquared of the relevant wavefunctions are shown. b) luminescence spectrumof the active region at 300 and 80K, as indicated. The applied bias is 9V.Lower curve: computed oscillator strength of the various transitions from the

upper state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

8.24 Overlap as a function of the number of period . . . . . . . . . . . . . . . . 90

8.25 Slope efficiency as a function of the number of periods in a series of other-wise identical quantum cascade lasers. Inset: slope efficiency per stage.(Fromref. [4]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

8.26 Temperature dependence of the light-versus current characteristics of a quan-tum cascade laser based on a bound-to-continuum transition operating at = 9m.(From ref. [5]). Inset: Threshold current density as a function oftemperature, fitted by the expression 8.8.23 with a T0 of 190K . . . . . . . . 92

8.27 a) Electrons thermally excited from the injector populate the lower state. Thecritical energy is the difference between the Fermi energy of the injectorand the lower state. b) Threshold current density versus temperature fora series of quantum cascade lasers with an identical active region based ona vertical transition operating at = 4.6m but with injectors of variouslengths showing different values of inj . . . . . . . . . . . . . . . . . . . . . 93

8.28 Comparison of the electroluminescence from a two otherwise identical sampleswith different doping profiles. In the sample with the setback, only the injectorregion is doped. In the reference sample, the area marked in green in the

schematic band structure in the inset is also doped. . . . . . . . . . . . . . . 94


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xii LIST OF FIGURES

8.29 The threshold current densityJTHin open squares and the maximum currentdensityJNDR in solid squares as a function of sheet densities correspondingto the doping study (n67, n72, n69, n71). In dashed lines, fitted curves corre-sponding to an extrapolated doping offset ofnoffset = 1.0 1011 cm2 addedto the active region doping. Open and solid circles represent the thresholdand the maximum current densities respectively of a laser grown after a longerMBE running time (n120). . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

8.30 Wallplug efficiency as a function of photon energy. Filled squares, experimen-tal values in continuous wave, from references [6, 7, 8, 9, 10, 11, 12]. Soliddisk: pulsed values from references [13, 14, 10, 15, 16]. The solid line rep-resents the prediction from the simplified formula with the key parametersderived from [15]. The influence of a shorter or longer in-plane dephasingtime // is shown by the two dotted curves. Long dashed lines represent theprediction from the model scaled by a factor 0.4. . . . . . . . . . . . . . . . 97

8.31 Characteristic temperatureT0 for various quantum cascade lasers, as functionof the photon energy and for various material systems. A clear relationshipbetween discontinuity and temperature behavior is is apparent. . . . . . . . . 99

8.32 Characteristic temperatureT0 for various quantum cascade lasers, as functionof the photon energy and for various material systems. A clear relationshipbetween discontinuity and temperature behavior is is apparent.(Redrawn fromdata from [17] ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

8.33 Band gap versus lattice constant for the main semiconductor materials, tothe exception of group III nitrides. Of particular interest for quantum cas-cade lasers are the GaAs/AlxGa(1x)As, the InxGa1xAs/AlyIn1yAs/InP, theInAs/AlSb heterostructure material system . . . . . . . . . . . . . . . . . . 100

8.34 Conduction band discontinuity as a function of In content in the quantumwell material, for a strain-compensated material with the well thickness rep-resenting 40% of the total stack thickness. Full line, result of the computa-tion, dashed line: value adjusted to the measured discontinuity of the latticematched material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

8.35 Electron effective mass of In1x

Gax

As material as a function of the Ga contentx (in contrast to the text where x labels the Indium content). Full line:unstrained material. Dotted, dashed lines: in-plane, respectively out of planemass for material grown on InP. From Suwagara et al., . . . . . . . . . . . . 104

8.36 Conduction band structure of a strain-compensated quantum cascade laserdesigned for short wavelength operation. The predicted emission wavelengthwas= 3.16musing the effective mass of the unstrained material. Measuredwavelength were between 3.4mand 3.6m, compatible with a much heaviermass as predicted by theory. . . . . . . . . . . . . . . . . . . . . . . . . . . 105

8.37 Emission spectrum of a short wavelength quantum cascade laser at various

temperatures, as indicated. . . . . . . . . . . . . . . . . . . . . . . . . . . . 105


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LIST OF FIGURES xiii

8.38 Threshold current density versus temperature for the device operating at =3.6m. The relatively lowT0 observed (85K) is an indication of leakage pathsabove the barriers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

9.1 Subthreshold luminescence of a short wavelength quantum cascade laser asa function of increasing current, from 100mA to 800mA by steps of 100mA.The temperature is 20K and the threshold current is at 950mA. The spectraare shown with two different horizontal scales a) and b). The shift of theindividual Fabry-Perot resonances is caused by the device heating. . . . . . 108

9.2 Bragg reflection condition for gratings of various order N a) First order grating,only backscattering is possible. b) For N=2, either backscattering or verticalemission is possible. c) In the third order grating (N=3), emission is only

possible inside the substrated because of the large refractive index differencebetween the guided mode (neff= 3.2) and the vacuum. . . . . . . . . . . . 110

9.3 Computed transmission for a finite quarter wave stack (640 pairs) with aperiodicity of 1.56m of average refractive index 3.2 and a refractive indexstep of n= 1.5 102. A stop band of width about 4cm1 appears in thetransmission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

9.4 Computed gain for a 1mm long device, with the same parameters as the oneof Fig. 9.3. Symmetric facets have been assumed . . . . . . . . . . . . . . . . 112

9.5 Distribution of the optical field in the cavity of the laser as a function of theproductL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

9.6 Distribution of the optical field in the cavity of the laser for an overcoupledlaser with a L = 4.7. This overcoupling enables very low mirror losses butat the cost of a lower efficiency. . . . . . . . . . . . . . . . . . . . . . . . . . 114

9.7 a) Scanning electron micrograph of a distributed feedback quantum cascadelaser with a grating on the surface. b) Computed effective index, losses andmode profile of the waveguide mode for three values of the etch depth: 0, 0.3and 0.5m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

9.8 a) Schematic cut through a waveguide design that uses air as the top claddingand a lateral injection through a doped InGaAs layer. b) Scanning electronmicrograph of a finished device. . . . . . . . . . . . . . . . . . . . . . . . . 115

9.9 a)Light and bias versus current characteristic of distributed feedback quantumcascade laser with a air/semiconductor top cladding and a lateral injection.b) Representative spectra of the device at various operation temperatures. . 116

9.10 a) Atomic force microscopy of a buried grating prior to the regrowth step.b) Scanning electron micrograph of the active region of a device after theregrowth was performed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

9.11 a) Computed mode for a 10mwavelength with (black line) and without (blueline) 200nm thick grating. The computed change of refractive index is n=1.5 102 while the losses are not affected. c) Real part of the mode effectiveindex as a function of etched depth. As expected from equation 9.2.32, the

relationship is linear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117


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xiv LIST OF FIGURES

9.12 Single mode optical power versus injected current for a distributed feedbackquantum cascade laser combining a buried grating and a buried heterostruc-ture waveguide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

9.13 Temperature tuning of the a buried grating device operating in continuouswave above room temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . 119

9.14 Mode position of a distributed feedback quantum cascade laser as a func-tion of both the temperature of the submount and the electrical power dissi-pated.(From??) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

9.15 Single mode spectra of a distributed feedback quantum cascade laser as a func-tion of time during a current pulse. The chirp is thermally driven. (From [18]) 121

9.16 Measured linewidth as a function of pulse length. The point labeled Aero-dyne is the result of a spectral fit of the absorption line of NH3 at 967cm

1

that did yield a linewidth of 0.012cm1 . Adapted from [19] . . . . . . . . . 1229.17 Measured half width at half maximum of the emission spectra of a quantum

cascade laser as a function of pulse length. The linewidth is obtained by fittinga vibrational molecular line, and the pulse length is a increasing function ofthe voltage on the pulser, indicated in the horizontal axis. Adapted from [ ?] 123


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List of Tables

3.1 Fundamental parameters for various III-V semiconductors . . . . . . . . . . 11

5.1 Summary of lifetime measurements for transition energies above the opticalphonon energy in GaAs quantum wells. Early measurements were inaccurate

due to the strong pumping needed. ISB: intersubband, IB: interband. Onlylow temperature measurements (T=10K) are reported . . . . . . . . . . . . . 59

5.2 Summary of some lifetime measurements for transition energies below theoptical phonon energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

8.1 Some elastic constants, energies and deformation potentials relevant for In-GaAs/AlInAs material system. (from ??). . . . . . . . . . . . . . . . . . . . 102

9.1 Measured temperature tuning coefficients of distributed feedback quantumcascade lasers, as reported in various references. The tuning coefficient issensitive to temperature but in a very good approximation independent onthe wavelength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

xv


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xvi LIST OF TABLES

variables used

Physical constants:0 vaccuum permettivityq electron chargek Boltzmann constantm0 Electron massP Kanes matrix elementEP Kane energyVariables:

Absorption coefficient (conventionally in cm1)2D Absorbed fraction by a 2D system (uniteless)E Energy EnergyE Electric field Dielectric constant Chemical potential (energy units)|, (x) Electron wavefunction Lifetimenrefr Refractive index

ni Sheet density in subband iNi Volume density in subband i


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Chapter 1

Introduction: intersubbandoptoelectronics as an example ofquantum engineering

1.1 Quantum engineering

Intersubband transitions in quantum wells are transitions between states created by quan-tum confinement in ultra-thin layers of semiconductors. Their unique physical properties,such as a atomic-like density of states, as well as the fact that they can be manufactured

in the technology mature III-V semiconductors makes them an attractive building blockfor mid-infrared optoelectronics. In fact, modulators, detectors and lasers were all demon-strated using this technology, and are becoming mainstream devices for a new generation ofoptoelectronics for sensing and telecommunication applications.

The developement of intersubband optoelectronics is a good example of the power of quan-tum engineering: the same way the understanding of Maxwells equations lead first toexplanation of the electrical phenomena, and then to their engineering; quantum mechanicsis moving from the realm of explanation to the one of engineering.

1.2 Basic requirementsA basic knowledge of solid-state physics and of quantum electronics. Level: master/graduate.

1.3 Tools of the mid-infrared

The intersubband optoelectronics has developed the mid-infrared portion of the spectrum.As compared to the visible or the near-infrared, this portion of the spectrum is based onrather different tools and techniques. In this chapter, we briefly review the different aspects

of the mid-infrared concerning materials, detector and sources.

1


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2

CHAPTER 1. INTRODUCTION: INTERSUBBAND OPTOELECTRONICS AS ANEXAMPLE OF QUANTUM ENGINEERING

1.3.1 Transparent materials

Traditional insulators: Glass (SiO2) and Saphire (Al2O3). Salts (NaCl, KCl, .) II-VI semiconductors: ZnSe, ZnS III-V semiconductors: GaAs IV semiconductors: C, Si, Ge

1.3.2 Reflectors

Dielectric constant of metals. subsectionDetectors

Thermal detectors: Pyroelectric, thermoelectric, bolometers, Golay cells Photoconductors: Mercury Cadmium Telluride photoconductors, InSb, PbS, ..

1.3.3 Sources

Blackbody source

CO2 laser Interband lasers

1.3.4 Fibers

In general, the optical fibers in the mid-infrared have general performance levels much belowwhat is achieved in the near-IR by the glass fibers.

Fluorides (seem the most advanced). Limited to the short wavelength portion of themid-IR

Chalcogenides Silver Halides. Very wide transparency range, low loss, but high chemical reactivity. Holllow fibers. High loss, limited bandwidth.

1.3.5 Polarizers

Brewster plates

Grid polarizers


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1.4. BIBLIOGRAPHY 3

1.3.6 Spectrometers

Grating spectrometer Fourier Transform Infrared Spectrometer

1.4 Bibliography

At this stage there is not real textbook covering the topic. Some textbooks covers aspectsof the material presented in this lecture:

Power point presentations They will be made available in pdf format before the lecture.

E. Rosencher and B. Winter Optoelectronics . Publisher: Cambridge UniversityPress (1998). Good coverage of the fundamental properties. Include the basic descrip-tion of the intersubband transitions.

G. Bastard Wave mechanics applied to semiconductor heterostructures (Les edi-tions de physique). Very good description of the computation of electronic state insemiconductor heterostructures (sections 3-5).

H. Schneider and H.C. Liu Quantum Well Infrared Photodetectors (Springer se-ries in optical sciences) The reference book on quantum well infrared detectors by theleaders of the field

Lecture notes

1.5 Notes and acknowledgements

Under this form, this script is meant for internal use since it does not always give propercredit to the figures, especially for the ones taken from the books cited above. The authorwould like to acknowledge Giacomo Scalari for his help in preparing this lecture.


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4

CHAPTER 1. INTRODUCTION: INTERSUBBAND OPTOELECTRONICS AS ANEXAMPLE OF QUANTUM ENGINEERING


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Chapter 2

Technology

2.1 Molecular Beam epitaxy

Active region

Injector

50 nm

MBEGrowth

MO

CVDGrowth

Figure 2.1: a) TEM image of a of the cleaved cross section of a quantum cascade laser active region(sample S1840) composed by a part of a active region (grown by MBE) and an upper cladding layer

(grown by MOCVD). b) Enlargement of 3 periods of the active region

5


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6 CHAPTER 2. TECHNOLOGY

Figure 2.2: STM image of the active region of (a) the structure with graded interfaces and(b) the structure with abrupt interfaces. (c) shows the averaged line profiles of the activeregion. With courtesy of P.Offermans and P. M. Koenraad.

2.2 Metal Organic Vapor Phase epitaxy

2.3 Reactive Ion etching

2.4 Quantum Cascade laser processing

A laser consists of a gain medium inserted in a optical resonator cavity. In a quantumcascade laser, as in most semiconductor lasers, the optical cavity is formed by the epitaxial

layers themselves. In the growth direction, optical confinement is achieved by total inter-


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2.4. QUANTUM CASCADE LASER PROCESSING 7

nal reflection between the high refractive index gain region and the lower refractive indexsubstrate and cladding layers. Laterally, confinement is achieved by defining a narrow (1030wide) stripe by removing the semiconductor material. Processing of the laser structure

InP Substrate

Gain regionCladding/contact

mask

MBE/MOCVD growth

a) b)

c) d)

e) f)

Spin resist / expose

Wet etch S3N4CVD deposition

S3N4Dry etch Top/substrate metalisation

Figure 2.3: Processing steps for the fabrication of a Fabry-Perot cavity quantum cascade

laser. a) MBE or MOCVD growth of the active region. b) Defining the ridge by pho-tolithography ( spin resist, expose with UV light and develop). c) The ridge is etched in awet etching solution. d) Si3N4 is deposited by PECVD. e) Open the insulating oxide on topof the ridges. f) Top and bottom metalization.

proceeds through the steps outlined in Fig. 2.4. Thirty-five periods of the active regionand relaxation region pairs are grown on a low-doped (n=131017cm3) InP substrate byMolecular Beam Epitaxy (MBE). The top cladding layer may be grown by MBE or MetalOrganic Chemical Vapor Deposition (MOCVD). Ridges are defined by photolithography andwet chemical etching. An insulating layer of Silicon Nitride is then deposited by Plasma En-hanced Chemical Vapor Deposition (PECVD). This Insulating layer is then opened on the

top of the ridges by a dry etching process. Contacts are then evaporated on the top andbottom of the device for contacting.


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8 CHAPTER 2. TECHNOLOGY

Figure 2.4: Schematic drawing of the facet of a processed device.


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Chapter 3

Electronic states in semiconductorquantum wells

3.1 Band structure of semiconductors in the kp ap-proximation: origin of the effective mass

Band structure of the GaAs

Figure 3.1: GaAs band structure

9


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10 CHAPTER 3. ELECTRONIC STATES IN SEMICONDUCTOR QUANTUM WELLS

3.1.1 Basic approximations

Behind the effective mass approximation lies a very powerful approach to the computation ofthe band structure. It relies on the knowledge of the band structure at k = 0 and expandingthe wavefunctions in this basis. The Schroedinger equation for a crystal writes:

p22m0

+ V(r) + 2

4m20c2

( V) p

(r) =E(r) (3.1.1)

For simplicity, we drop the spin-orbit coupling term. The latter arises as a relativistic term:the motion of the electon in the field of the ion, the latter sees an equivalent magnetic fieldthat operates on the angular momentum variable. Let us first compute the action ofp on, written in terms of Bloch wavefunctions so that

nk(r) =eikrunk(r), (3.1.2)

we obtain:

p (eikrunk(r)) = i (eikrunk(r)) (3.1.3)

= keikrunk(r) + e

ikrp unk(r) (3.1.4)=ei

kr(p + k)unk(r). (3.1.5)

Using the above relation, the Schrodinger equation is obtained for unk(r):

p22m0

+

mk p +

2k2

2m0+ V(r)

unk(r) =Enkunk(r) (3.1.6)

The HamiltonianH=H0+ W(k) may be splitted into a k-independent

H0 = p2

2m0+ V(r) (3.1.7)

and k-dependent part

W(k) = 2k2

2m0

+

mk p. (3.1.8)

The solution of the the equation

H0un0(r) =En0un0(r) (3.1.9)

are the energies of the band structure at the point k = 0. The fundamental idea of thek papproximation is to use the un0(r) as a basis for the expansion of the wavefunction andenergies at finite k value. In the simplest cases, taking an interband transition across thegap and looking at the conduction band, taking the second-order perturbation expansion:

Ec(k) =Ec(0) +2k2

2m0

+2k2

m0m=c

|uc,0|p|um,0|2

Ec Em(3.1.10)


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3.1. BAND STRUCTURE OF SEMICONDUCTORS IN THEKPAPPROXIMATION:ORIGIN OF THE EFFECTIVE MASS 11

GaN GaAs InP GaSb InAs InSb

Eg(eV) 3.4 1.519 1.424 0.811 0.418 0.235mc/m0 0.17 0.0665 0.079 0.0405 0.023 0.0139Ep(eV) 20.2 22.71 17 22.88 21.11 22.49rvc(A) 2.55 6.14 5.67 11.5 21.5 39.50(eV) 0.017 0.341 0.11 0.75 0.38 0.81

Table 3.1: Fundamental parameters for various III-V semiconductors

In the lowest order approximation, all other bands except for the valence band may beneglected, in which case the dispersion can be written as (taking the zero energy at the top

of the valence band):

Ec(k) =EG+2k2

2m0+

2k2

m0

p2cvEG

(3.1.11)

Defining the Kane energy EP = 2m0P2 such that

EP = 2

m0|uc,0|p|uv,0|2 (3.1.12)

the dispersion of the conduction band can be written as:

Ec(k) =Ec+

2k2

2m0

1 +

EP

EG

(3.1.13)

We then obtain the effective mass as:

(m)1 = (m0)1

1 +EPEG

(3.1.14)

The Kane energy is much larger than the gap EP >> EG and is rather constant across theIII-V semiconductors. As a result, the effective mass in inversly proportional to the bandgap.One should be careful that some authors use the definition

EP = 2m0P2 (3.1.15)

(Bastard, for example), while others (Rosencher) use

EP =P2. (3.1.16)

At least they all use EP having the dimension of energy!. In the above expressions, thespin-orbit term can be included by replacing the operator pby

=p +

4m0c2

(

V). (3.1.17)


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3.1.2 Beyond the perturbation expansion

A very powerful procedure is to expand formally the solutions of Equ. 3.1.6 in the solutionsat k = 0, writing formally:

unk(r) =m

c(n)m (k)um,0(r) (3.1.18)

and restricting the sum to a limited relevant subset of bands. Improved accuracy can beachieved by introducing more bands. In this basis, and projecting the equation onto thestateuM,0, the Hamilton equation is written as:

EM0+

2k2

2m0 En(k)

c(n)M(k) +

m=M

HkpMmc(n)m (k) = 0 (3.1.19)

where the kp Hamiltonian is

HkpMm=

m0k uM,0|p|um,0 (3.1.20)

To the extend the matrix elements are known, equation 3.1.19 can be solved.

3.1.3 Example: a two-band Kane model

Let us write explicitely a two-band Kane model. The latter can be a fairly realistic model ifone is only interested at the effect of the valence band onto the conduction one, replacing the

three spin-degenerate valence bands (heavy hole, light hole, split-off) by an effective valenceband. The unk can then be expressed as:

unk =acuc0+ avuv0 (3.1.21)

Replacing this expansion into Eq. 3.1.19, we obtain the following matrix equation: Ec+

2k2

2m0

m0k pcv

m0k pcv Ev+

2k2

2m0

acav

= E

acav

. (3.1.22)

AssumingEc = 0, Ev = EG, the solution of the matrix equation satisfies:2k2

2m0 E)(EG+ 2k2

2m0 E) 2

m20|k pcv|2 = 0. (3.1.23)

This second order equation in E can be of course solved directly; it is however more instructiveto write it under the form a pseudo effective mass equation (remember EP =

2m0

p2cv:

E(k) = 2k2

2m0

EP+ EG+ 2E

E+ EG. (3.1.24)

For k 0 the above expression reduces itself to the result of the perturbation expansion:

(m)1

= (m0)1

1 +

EP

EG

. (3.1.25)


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3.2. ENVELOPE FUNCTION APPROXIMATION 13

Equation 3.1.25 can be expressed in a somewhat simplified form:

m(E) =m(0)

1 + EEG

(3.1.26)

that is commonly used in the literature.

3.2 Envelope function approximation

The problem we will try and solve now is the one of a heterosructure, in which two materialsA and B are sandwiched together. Of course such a material could be seen as a new materialby itself, and its band solved by ab initio techniques, but such a computation is very heavy,

time consuming and moreover does not give much physical insight into the result. Theenvelope function approximation solves this problem in a very efficient and elegant manner.It is widely used to predict the optical, electrical properties of semiconductor nanostructures.

3.2.1 Multiband case

At the core of the envelope function approximation is a generalization of the k p approx-imation: it is postulated that the wavefunction can be written as a sum of slowly varyingenvelop functions fA,Bl (r) that will modulate the Bloch function of the material, namely:

(r) =l

fA,B

l uA,B

l,k0 (r). (3.2.27)

Behind the equation 3.2.27 is the idea that at each point, the wavefunction is described byak pdecomposition and that this decomposition depends on the position. Furthermore, itis assumed that

1. the envelop functionfA,Bl (r) is slowly varying compared to the Bloch wavefunction, iffA,Bl (r) is written in a Fourier decomposition, the wavevectors are close to the centerof the Brillouin zone

2. the Bloch functions are identical in both materials, i.e. uA

n,k0(r) = uB

n,k0(r). This also

implies that the interband matrix elementS|px|X is equal in both materials.It allows us to write te wavefunction as

(r) =l

fA,Bl ul,k0(r). (3.2.28)

Let us assume first a quantum well, in which a layer of material A is sandwiched into abarrier material B. Because of the in-plane translational invariance, the wavfuction may bewritten as plane waves:

fl(r

, z) = 1

Seikr

l(z) (3.2.29)


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where z is choosen as the growth direction and k = (kx, ky) is the in-plane wavevector.Note that this convention there is confusion in the litterature as the sign

may either mean

perpendicular to the plane of the layers or to the growth axis. The Hamiltonian is then

H= p2

2m0+ VA(r)YA+ VB(r)YB (3.2.30)

where the functions YA(z) and YB(z) turn on the potential in the respective layers. Wewill develop our system close to k=0. To solve the system, we must:

1. Let H act upon (r)

2. multiply on the left byum0(r)eikrm(z)

3. integrate over space

We have to use the following relations. As the envelop function is slowly varying, we maywrite

cell

d3rflfmul um=f

lfm

cell

d3rul um= flfmlm (3.2.31)

and take advantage of the fact that the band edge are eigen function of the Hamiltonian at(k=0): p2

2m0+ VA,B(r)

um,0(r) =

A,Bm,0 um,0(r). (3.2.32)

The derivation is rather tedious, but one should note the similarity with the normal k

ptechnique by considering the action of the operator pon the wavefunction:

p(ekrl(z)ul(r)) = (k iz

+p)eikrl(z)ul(r) (3.2.33)

and we then may consider the substitution

p (k iz

+p) (3.2.34)

where it is understood that p then acts only on the Bloch part of the wavefunction. Usingthe above substitution into the Hamiltonian, one finally get the following set of diferential

equation written in a matrix form:

D(z, iz

)= (3.2.35)

where the elements of the matrix D are given by the equation

Dlm =

Al YA+ Bl YB+

2k22m0

2

2m0

2

z2

l,m

+km0

l|p|m im0

l|pz|m z

(3.2.36)

and the matrix elementl|p|m = cell ulpumd3r.


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3.2.2 One Band model: the Ben-Daniel Duke

As an example, let us consider first a pure one band model in the effective mass approxima-tion. Setting

Al YA+ Bl YB =V(z) (3.2.37)

we obtain the Schrodinger equation:

2

2m2

z2+ V(z)

(z) =(z) (3.2.38)

In this equation, the band is assumed to be parabolic with a curvature given by an effectivemassm. For an isolated band such as the heavy hole band, it is a rather good approximation.

It can also be used successfully for thick quantum well structures in the conduction band,when the confinement energies are much smaller than the material energy gap. One firstformal difficulty arises because the mass of the barrier material is in general different thanthe one of the quantum well. For this reason the effective mass should be considered as aposition-dependent quantitym =m(z). In that case, one shows that the proper boundaryconditions that will match the solutions are the interfaces are notby matching the envelopefunction and its derivative, but by matching the envelope function

(z) (3.2.39)

and the quantity

(z)mz

. (3.2.40)

The latter condition can be shown to be equivalent to the conservation of the probabilitycurrent.The one band model can be formally derived in the k p framework by adding the effect ofthe remote bands in the matrix 3.2.36, the result of which is given (see Bastard) for a 8 bandmodel with the remote bands as

8

m=1Am+ Vm(z) +

2k22m0

2

2m0

2

z2 l,m+k

m0l|p|m i

m0l|pz|m

z

2

2

z

1

Mzzlm

z i

2

2

=x,y

k

1

Mzlm

z+

z

1

Mzlmk

+2

2

,=x,y

k1

Mlmk

m=l

(3.2.41)where effective mass parameters Mlm are defined as

m0

Mlm=

2

m0

l|p| 1 0 V(z)m|p| (3.2.42)

One sees that for a one band model, neglecting the in-plane dispersion, the term 2

2z

1Mzz

lm

z

is the only one remaining and will then change the effective mass.


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3.2.3 Two band model

For the conduction band, a very nice model is the one in which one keeps one valence band,creating a two-band model. For simplicity, let us look at the states at k = 0 and neglectthe free electron term. We obtain the system of equation given by:

Vc(z)fc im0

pcv

zfv =fc (3.2.43)

im0

pcv

zfc+ Vv(z)fv =fv (3.2.44)

Extracting fv from the second equation yields:

fv = 1

Vv(z) i

m0pcv

zfc (3.2.45)

replacing into the first equation, after substitution, the following reslult is obtained.

2|pcv|2m0

z

1

Vv(z)

zfc+ Vc(z)fc = fc. (3.2.46)

Recalling the definition of the Kane energy EP = 2m0

p2cv and defining an energy-dependenteffective mass:

1

m(, z)=

1

m0

EP

Vv(z)

(3.2.47)

we obtain finally a Schrodinger-like equation

2

2

z

1

m(, z)

zfc+ Vc(z)fc=fc. (3.2.48)

This model is very useful to model the electronic states in the conduction band with theinclusion of the non-parabolicity. It is very widely used in the study of intersubband transi-tions.

3.2.4 Formal derivation of the 2x2 model

The 2x2 model can be formally derived from the 8x8 Kane Hamiltonian close to the point.In this approximation, the heavy hole is decoupled from the other valence bands and one isleft with two equivalent 3x3 Hamiltonian (one for each spin direction):

H=

Ec(z)

23pcvm0

pz

13pcvm0

pz

23pcvm0

pz Elh(z) 013pcvm0

pz 0 Eso(z)

(3.2.49)

acting on the three-dimentional vector of the envelop function = (c, lh, so), where c, lh

and so label the conduction, light-hole and split-off band edges, respectively. Notice that in


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the Hamiltonian 3.2.49 we have also neglected the diagonal free electron termsp2z/m0 thatcan be shown to contribute of terms of the order ofE

g/E

p1. Moreover, as we are only

interested in the energy levels located above the edge of the conduction band, the problemcan be solved by using the second and third rows of

H =E (3.2.50)

to express the equation in the first row as

pz1

2m(E, z)pzc+ Ec(z)c =Ec (3.2.51)

but with now the energy- and position dependent effective mass given by

1m(E, z)

= 1m0

23

EpE Elh +

13

EpE Eso

(3.2.52)

The solution of the differential equation 3.2.51 give the conduction component c and theenergy of the stationary states. We must recall, however, that the totally stationary wave-functions is given by the three componentsc, lh,and so, weighted by their correspondingBloch functions, so that the only knowledge ofc is insufficient for the complete physicaldescription of the stationary state. The 2x2 Hamiltonian can be recovered by a unitarytransformation, defining an effective valence band by Ev = (2Elh+ Eso)/3 and thereforethe gap is now an effective gap Eg = Eg + 0/3, which has also a contribution comingsplit-off edge 7. In doing so, we neglect contributions on the wavefunctions and energies of

the order of (/|Ec Ev|)2 = (/Eg)2, with = 23 0. For typical III-V semiconductorsused for QC lasers this factor is quite small: in GaAs (/Eg)

2GaAs 0.01 and in InGaAs

(lattice matched on InP) (/Eg)2GaInAs 0.04 and therefore the corrections never exceeds

few percent. For devices based on InAs, however, this approximation is likely to fail as Eg in this material.A feeling for these approximations can be obtained by comparing the various expressions forthe energy-dependent effective mass. This is done in Fig. ??

Figure 3.2: Comparison for various model for the energy-dependent effective mass


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3.3 Hartree potential

For the carrier densities used (typically about 1016cm3), electron-electron interaction is nota dominant term but must be taken into account if the electronic states are a good accuracyis to be aimed at. The effects are especially strong in devices where the electrons are remotefrom the ionized donors. It is usually treated in a mean-field approximation by adding aHartree potential VH(z) to the Hamiltonian:

H= 2

2m2 + VQC(z) + VH(z) , (3.3.53)

This term is computed from the local charge density

(z) =q0ND(z) q0i ni|i(z)|2

, (3.3.54)

where ND is the doping profile of ionized dopants, i(z) the probability density andniis thenumber of electrons per unit area in the i-subband. The potentialVH(z) is computed from(z) using Poissons equation:

2VH(z)

z2 =

(z)

0. (3.3.55)

The electronic densities ni on the subbands are not known a priori, and in the general casedepend on the transport in the device. However, a good starting approximation is to assumethat the electron distribution is thermal, charaterized by Fermi distribution with a common

chemical potential in each period: and that charge neutrality is achieved in each period,i.e.: i

ni=i

Di(E)f(E)dE (electrons) (3.3.56)

=

ND(z)dz= ns (impurities) (3.3.57)

where

Di(E) = mi (E)

2 (E Ei)

is the density of states of the i-subband, (E

Ei) is the Heavyside function, f(E) =

11+exp(EkT )

is the Fermi distribution function, andnsis the total areal electron concentration,

which is equal to the impurities concentration. The Hartree potential depends on the solutionof Schrodingers equation; therefore Schrodingers and Poissons equations must be solvediteratively until convergence is achieved.

3.4 Building blocks

When designing intersubband structures, it is important to acquire an intuitive grasp ofsimple building block. This is done most easily in a simple one-band model. Solving for such

structure usually follows the following methodology:


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3.4. BUILDING BLOCKS 19

1. The wavefunctions are found for the regions in which the potential is constant (orassumed constant). They will be plane waves or decaying exponentials

2. Those solutions are then matched at each interfaces, using the boundary conditionscompatible with the Hamiltonian. For one band model, it means matching the valueof the wavefunction and of the derivative, divided by the mass, at the interfaces.

3. Boundary conditions are then imposed on the edge of the sample: decaying exponentialfor bound states, or periodicity for a periodic structure

We will then review some of these results rapidely.

3.4.1 The single quantum wellThe simplest case is of course the quantum well with infinite barrier height. The wavefunc-tions are simply sinus and cosinus functions (if the well is taken symmetric around zero) orsinus only if the quantum well is lying in the interval [0,L]. In the latter case, the conditionon the value of k

kn=n

L (3.4.58)

with n integer enables immediatly to write the energy as:

En= 2k2n

2m =

n222

2m . (3.4.59)

Of course, in real structures one should taken into account the finite barrier height. FollowingBastard, we assume a potential with the form:

Vb(z) =Vb if |z| L/2

0 if |z| > L/2

the bound energy solutions (


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For positive energies, one does not obtain real bound states anymore, but resonances forwhich the transmission function of the quantum well (corresponding to the transmissionprobability of an incident electron at energy ) has a maximum has a maximum. Thetransmission function can be computed and is given by

T() = |t()|2 =cos(kwL) i

2

+

1

sin(kwL)

12 (3.4.64)where is given by the weighted ratio of the wavevectors:

=mbkwmwkb

(3.4.65)

The following figures show the computation of such energy levels in various cases:

Figure 3.3: Computed confinement energy of electrons in the conduction band as a functionof well width. Bound states are shown with full lines, resonances with dotted lines

A very attractive feature of the two-band model is that it can be implemented numericallythe same way as the one-band one by just introducing the energy-depedent effective mass.This is of course also valid in the formula given above. The importance of introducing thenon-parabolicity via the interaction of the conduction and valence band is shown clearly inthe folloginw figure, where the energy states of a single quantum well are compared in thecase where the non-parabolicity is neglected (dotted line) and where it is taken into account(solid lines). The discrepancy for the position of the highest level is 90meV, much larger

than any experimental uncertainty.


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Figure 3.4: Energy states of a quantum well computed with a two-band model, and comparedwith a one-band model (dashed lines). The growing importance of non-parabolicity as onemoves away from the gap is clearly apparent.

3.4.2 The coupled well system

The system to be considered here is two quantum well coupled through a tunnel barrier.Of course this system can be modeled directly by solving the Hamiltonian equation of the

whole structure. We however wish to study this system in a thight-binding model in whichwe use a base formed by the solution of the individual wells. Again, following Bastard, theHamiltonian to solve is:

H= p2z2m

+ Vb(z z1) + Vb(z z2) (3.4.66)

Figure 3.5: A coupled quantum well system.

We solve the problem using the basis wavefunction of the isolated wells, that satisfy:

p2z2m

+ Vb(z z1)(z z1) =E1(z z1). (3.4.67)

Neglecting higher excited states, the complete wavefunction can be expanded in terms of thebasis function using:

(z) =(z z1) + (z z2). (3.4.68)


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Introducing the expansion in the Hamiltonian yields the following matrix equation:

E1+ s (E1 )r+ t(E1 )r+ t E1+ s

= 0 (3.4.69)

Solving for E1 yields the equation:

= E1 t1 r +

s

1 r (3.4.70)

where r is the overlap,

r=

1(z

z1)

|1(z

z2)

s is the shifts= 1(z z1)|Vb(z z1)|1(z z2)

and t the transfer integrals

t= 1(z z1)|Vb(z z1)|1(z z2).

This result can be expressed graphically, as shown in the following figure. The most impor-tant term is then the transfer integral because it is responsible for the splitting between thetwo states. The larger the transfert integral, the wider the spacing between the two states.

Figure 3.6: A coupled quantum well system.

3.4.3 The superlattice

In a superlattice, the potentialVb(z) is defined as a sum of potential of the individual wells:

Vb(z) =

n=Vb(z nd)

with each potential well being

Vb(z

nd) =

Vb if

|z

nd

| L/2

0 if |z nd| > L/2


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Figure 3.7: Schematic band structure of a superlattice.

The wavefunctions are then given, for the quantum wells, by:

(z) = exp(ikw(z nd)) + exp(ikw(z nd)); |z nd| L/2

For the barriers, for positive energies ( >0), the wavefunction is then given by:

(z) =exp(ikb(z nd d2

)) + exp(ikb(z nd d2

)); |z nd d2| h/2.

In addition to the boundary conditions at both interfaces of the quantum well, we should alsoimpose the periodicity of the structure. As it is well-known, it will force the wavefunctionto follow Bloch theorem, that is

q(z+ nd) = exp(iqnd)q(z) (3.4.71)

for a periodicity d.The resulting transcendental equation is given by:

cos(qd) = cos(kwL)cos(kbh)

1

2

1

sin(kwL) sin(kbh) (3.4.72)

where has the same meaning as for the single quantum well case. For negative energies, iand the trenscendental equation becomes:

cos(qd) = cos(kwL) cosh(bh) 12

1

sin(kwL) sinh(bh) (3.4.73)

The above equation defines a dispersion relation between the energy (appearing in kw andkb) and q. An example of such a dispersion, showing clearly the minibands and minigaps, isshown in Fig. 3.8. In particular, the increase in the miniband width with energy is clearly

apparent.


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Figure 3.8: Computed dispersion of a 100A InGaAs well and 30A AlInAs barrier superlattice.A conduction band discontinuity of Ec = 0.52eVas well as a Kane energy of 18.3eV wasassumed in an effective two-band model. For clarity, the zero of the energy scale has beenset to the bottom of the InGaAs quantum wells

The superlattice dispersion of the first miniband is compared to a sinusoidal fit, showing thethat the latter is a very good approximation for narrow minibands.

0.005 0.01 0.015 0.02 0.025

Wavevector A1

0.048

0.049

0.05

0.051

0.052

0.053

0.054

Energy

eV

100A30A Superlattice dispersion

Figure 3.9: Dispersion of the first miniband (green), compared to a sinus function (red).The result of the finite superlattice are also plotted as full squares

The equivalent real space picture is shown in Fig. 3.10 where a the energies and wavefunctions

of a finite eight quantum well long superlattice


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3.5. IN-PLANE DISPERSION 25

Figure 3.10: Square of the wavefunctions for a finite superlattice formed by eight 100AInGaAs wells separated by 30A AlInAs barriers.

3.5 In-plane dispersion

For electronic states, the in-plane dispersion can be computed with various levels of accuracy.The simplest approach is to assume that each subband has an effective mass given by itsmass at k=0. This approximation holds relatively well in the conduction band. However, itcompletely fails in the valence band because the in-plane wavevector couple states belong-ing to different bands and a complicated spectrum arises. For this reason, the analysis of

intersubband devices operating in the valence band is much more complicated.


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Figure 3.11: Dispersion of the state of a quantum well in the valence band.

3.6 Full model: the valence band

To be accurate, the full kp envelope function model must be solved, that is the set of 8differential equations defined by the equation 3.2.35. Solving the full model is essential inthe valence band because no simplified model can easily be used if some degree of accuracyis sought. It is observed that:

It is usualy convienient to use the growth direction as the quantization direction forthe angular momentum.

The confinement potential lifts the degeneracy between the heavy and light hole bands,because their different mass induce a different confinement energy

The in-plane dispersion is highly non-parabolic because of the coupling between thebands induced by the in-plane momentum. In particular, this coupling prevents anycrossing between the light-hole derived and the heavy hole-derived band (see dahsedlines)

In some cases, the mass is inverted: the bottom of the LH1 band has a electron-like character over some portion of reciprocal space because of the repulsion and itsproximity to the HH2 state.

This effect is shown schematically in Fig. 3.12. As a result, the computed band structure is

usually fairly complex, and yield results such as the ones shown in Fig. 3.11


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3.6. FULL MODEL: THE VALENCE BAND 27

Figure 3.12: Schematic description of the origin of the valence band dispersion in the quan-tum well showing schematically the effects of confinement and interactions.


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Chapter 4

Intersubband absorption

4.1 Interaction Hamiltonian

Typically two different Hamiltonian can be used to compute the absorption. The first oneis the dipole given by

Hint= qR Esin(t) (4.1.1)The other one is derived from the potential vector A by

H=( P qA)2

2m (4.1.2)

Use Coulomb gauge (A= 0). For low intensity (neglect the term in A2

), we obtain as aninteraction Hamiltonian:

Hint= qm0

A P . (4.1.3)Because of the large difference between the light wavelength and the atomic dimension, thespatial dependence of A(r) is neglected inside the matrix elements. This is called the dipoleapproximation. The form commonly used is then:

i|Hint|j =qA(r)

m i(r)| P|j(r) (4.1.4)

For a plane wave, A is paralell to

E, for a wave polarized along z and propagating in the y

direction:A(r, t) =A0eze

i(kyt) + A0ezei(kyt). (4.1.5)

It is convient to use A0 being pure imaginary such both electric Eand magneticB fields arereal: that yieldE= 2iA0 and B= 2ikA0.The matrix elements of the position and momentum operators are related to each other.As the commutator between p and z is [z, p] = i, applying it to the kinetic term of theHamiltonian one can derive the relation:

n|p|m =im0nmn|z|m (4.1.6)This relation between the momentum and position matrix elements enables the use of either

one interaction Hamiltonian equivalently.

29


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30 CHAPTER 4. INTERSUBBAND ABSORPTION

4.2 Intersubband and interband transition

The intersubband and interband transitions have a very different character. The distinctioncan be easily understood when looking at the action of the momentum operator on twowavefunction taken each on a different band:

i,j(r) =fi.j(r)u,(r) (4.2.7)

where fi,j is the envelope and u, the Bloch part of the wavefunction. The matrix elementcan be written then as

i|p|j

=fiu|

p|fj

u

(4.2.8)

= fi|p|fju|u + fi|fju|p|u (4.2.9)= fi|p|fj+ fi|fju|p|u (4.2.10)

where we have taken advantage of the slow variation of the envelope function compared tothe one of the Bloch part.The first part of the last expression represents the intersubbandtransition and will be non-zero when two envelope states are taken from the same band, thesecond corresponds on the contrary to an interband transition matrix element. In this one-band model, one should be careful to use the momentum matrix element with the effectivemass m instead of the bare electron mass m0. One easy way to see this is to look at

the commutator [z, H] used to derive the relation between dipole and momentum matrixelements; this one will involve the Hamiltonian with the effective mass and not the bareelectron mass.

4.3 Selection rules and absorption geometries

It can be shown that only the component of the electric field in the growth direction willinduce a non-zero dipole matrix element. In fact, taking for the wavefunction the canonicalform:

fi= 1Seikri(z) (4.3.11)

then the matrix element corresponding to the x-component of the momentum px yield:

(Expx)ij = i|jkxExk,k = kxExk,ki,j (4.3.12)

which, because of the termi,jis zero for transitions between different states. As a result, onlythe z-component of the electric field couples to the intersubband transition. This has profoundexperimental implications as it for example rules out absorption for light at normal incidence.As a result, a number of geometries have been developed to measure the absorption, as shown

in Fig. 4.2.


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4.3. SELECTION RULES AND ABSORPTION GEOMETRIES 31

Figure 4.1: Experimental geometries allowing the measurements of intersubband transitions

The validity of the selection rule was checked in GaAs/AlGaAs samples by H.C. Liu andcoworker in a photocurrent experiment. They found that for such samples, the absorptionby the TE polarization was less than 0.2% than the TM one.

Figure 4.2: Photocurrent in a quantum well photoconductor as a function of the polarization.The residual responsivity for the TE polarization is less than 0.2% than for the electric fieldnormal to the layers. (From H.C.Liu et al, APL 1998)

The absorption in TE is non-zero when considering the wavefunctions in the multiband case,

and is discussed shortly further in the text.


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32 CHAPTER 4. INTERSUBBAND ABSORPTION

4.4 Absorption strength

4.4.1 Absorption for a 2D system

When looking at the transmission through of an essentialy two-dimentional system, oneshould realize that the correct number to represent the absorption is unitless and representsthe fraction of the light absorption by the system. To compu

qcl lecture long

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