QUANTUM TRANSPORT IN SEMICONDUCTOR HETEROSTRUCTURES USING

DENSITY-MATRIX AND WIGNER-FUNCTION FORMALISMS

by

Olafur Jonasson

A dissertation submitted in partial fulfillment of

the requirements for the degree of

Doctor of Philosophy

(Physics)

at the

UNIVERSITY OF WISCONSIN–MADISON

2016

Date of final oral examination: 12/13/2016

The dissertation is approved by the following members of the Final Oral Committee:

Irena Knezevic, Professor, Electrical & Computer Engineering

Dan Botez, Professor, Electrical & Computer Engineering

Luke Mawst, Professor, Electrical & Computer Engineering

Mark A. Eriksson, Professor, Physics

Maxim G. Vavilov, Professor, Physics

c© Copyright by Olafur Jonasson 2016

All Rights Reserved

i

ACKNOWLEDGMENTS

I would like to express my special appreciation and thanks to my advisor Irena Knezevic. I am

grateful for the effort and time you invest into mentoring your graduate students and I thank you

for the great deal of freedom you have given me over the course of my PhD studies, allowing me

to grow as a research scientist.

I want to thank my groupmates for their company and interesting discussions during group

meetings. The work we do in Prof. Knezevic’s group is broad and interacting with groupmates

with other specialties has broadened my perspective of physics and engineering. Out of my fellow

group members, I worked most closely with Farhad Karimi and I am indebted to him for his help

with the derivation of the dissipator within the density-matrix formalism. I also want to thank Song

Mei for fruitful discussions about the k.p formalism and its use in describing heterostructures. In

addition I want to thank all my other groupmates during the course of my PhD: Alex Gabourie,

Amanda ZuVerink, Amirhossein Davoody, Bozidar Novakovic, Jason Hsu, Kyle Gag, Leon Mau-

rer, Michelle King, Nishant Shule, Sina Soleimanikahnoj, Yanbing Shi, and Zlatan Aksamija.

I also want to thank Professor Vidar Gudmundsson, who was my advisor during my master’s

studies at the University of Iceland. I am grateful for the opportunity you gave me by taking me in

as a summer research assistant during my undergraduate, and providing me with experience in the

field of condensed matter physics and computational physics.

I ended up getting a lot more than just a degree from my time here in Madison. I made a lot of

friends of different nationalities, many of which are from my physics class of 2012 and the small

(but closely knit) Icelandic community in Madison. Last, but not least, I also met my wife Brittany

Kaiser early in my PhD studies. I want to thank Brittany for her support during the course of my

PhD, and for making my stay here in Madison even more enjoyable.

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TABLE OF CONTENTS

Page

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

1 Overview of Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Simulation of Quantum Cascade Lasers within the Density-Matrix Formalism . . . 3

2.1 Introduction and Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.1 Quantum Cascade Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.2 Simulation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.3 Organization of Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.1 Electronic States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.2 Dissipator for Electron-Phonon Interaction . . . . . . . . . . . . . . . . . 182.2.3 Dissipator for Elastic Scattering Mechanism . . . . . . . . . . . . . . . . . 252.2.4 Steady-State Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.2.5 Optical Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.2.6 Comparison With Semiclassical Models . . . . . . . . . . . . . . . . . . . 35

2.3 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.3.1 Steady-State Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 372.3.2 Linear-Response Density Matrix . . . . . . . . . . . . . . . . . . . . . . . 392.3.3 Low-Energy Thermalization . . . . . . . . . . . . . . . . . . . . . . . . . 402.3.4 Time-Dependent Calculations . . . . . . . . . . . . . . . . . . . . . . . . 41

2.4 Steady-State Results for a Mid-IR QCL . . . . . . . . . . . . . . . . . . . . . . . 442.4.1 Bandstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.4.2 Current Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.4.3 Optical Gain and Threshold Current Density . . . . . . . . . . . . . . . . 512.4.4 In-Plane Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.5 Results for a Terahertz QCL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

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2.5.1 Steady-State Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592.5.2 Time-Resolved Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.6 Conclusion to Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3 Coulomb-Driven THz-Frequency Intrinsic Current Oscillations in a Double-BarrierTunneling Structure within the Wigner Function Formalism . . . . . . . . . . . . . 73

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.2 System and Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.3.1 DBTS With a Uniform Doping Profile . . . . . . . . . . . . . . . . . . . . 783.3.2 Traditionally Doped RTD . . . . . . . . . . . . . . . . . . . . . . . . . . 813.3.3 Effects of Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.3.4 The Quasi-Bound-State Picture. The Coulomb Mechanism Behind the

Current-Density Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . 833.3.5 Frequency of the Current-Density Oscillations. Effect of Varying Doping

Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883.3.6 Experimental Considerations Relevant for Observing Persistent Oscillations 91

3.4 Conclusion to Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4 Dissipative Transport in Superlattices within the Wigner Function Formalism . . . 97

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.2 The Wigner Transport Equation for Superlattices . . . . . . . . . . . . . . . . . . 98

4.2.1 The Wigner Transport Equation . . . . . . . . . . . . . . . . . . . . . . . 1004.2.2 The Quantum Evolution Term for a Periodic Potential . . . . . . . . . . . 1014.2.3 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.3 Modeling Dissipation and Decoherence in Superlattices: The Collision Operator . . 1044.3.1 The Relaxation Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.3.2 The Momentum-Relaxation Term . . . . . . . . . . . . . . . . . . . . . . 1064.3.3 The Spatial Decohrence (Localization) Term . . . . . . . . . . . . . . . . 108

4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.4.1 Comparison With Experiment . . . . . . . . . . . . . . . . . . . . . . . . 1094.4.2 Effects of the Different Terms in the Collision Integral (4.21) on the J−E

Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104.5 Conclusion to Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

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AppendixPage

LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

APPENDICES

Appendix A: Detailed Calculation of Scattering Rates . . . . . . . . . . . . . . . . . 129Appendix B: Bias Treatment within the Wigner-function formalism . . . . . . . . . . 137

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LIST OF TABLES

Table Page

2.1 Parameters relating to bandstructure (see Section 2.2.1) for both In0.47Ga0.53As andIn0.52Al0.48As. All values are obtained using the procedure described in Ref. [1], andare given for a lattice temperature of 300 K. Parameters for which no units are givenare unitless. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.2 Various parameters used for the In0.47Ga0.53As/In0.52Al0.48As material system. . . . . 45

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LIST OF FIGURES

Figure Page

2.1 Schematic diagram showing the conduction-band profile for the QCL design fromRef. [2] under an applied bias. The dashed rectangle contains a single stage; adjacentstages are separated by thick injection barriers. The upper lasing state (u) is denotedby the red curve, the lower lasing state by the blue curve (`), the extractor state (e) bythe purple curve, and the injector state (i) by the orange curve. The green-shaded areadenotes the miniband, which is a region with many closely spaced states. . . . . . . . 6

2.2 Kane energy as a function of position for a single period of the InGaAs/InAlAs-basedmid-IR QCL described in section 2.4. The dashed curve shows the piecewise constantversion and the dashed curve is the smoothed version given by Eq. (2.12), with LML =0.29 nm. The region of higher Kane energy corresponds to InGaAs. . . . . . . . . . . 15

2.3 (a) Conduction band Ec(z) and valance band Ev(z) used for the calculation of eigen-functions. Dashed rectangle shows a single period of the conduction band. Potentialbarriers are imposed to the far left and far right for both conduction and valence bands.Also show, are two of the Hermite functions (n = 0 and n = 10) used as basis func-tions. (b) After discarding the high and low-energy states, about 80 states are left,most of which have a energy high above the conduction band or are greatly affectedby the boundary conditions on the right. (c) After choosing 8 states with the lowestEn and discarding copies from adjacent stages (see main text), the states in the cen-tral stage (bold curves) are left. States belonging to adjacent periods (thin curves) arecalculated by translating states from the central stage in position and energy. . . . . . 17

2.4 Conduction-band edge (thin curve bounding gray area) and probability densities forthe 8 eigenstates used in calculations (bold curves). States belonging to neighboringperiods are denoted by thin gray curves. The states are numbered in increasing or-der of energy, starting with the ground state in the injector. Using this convention,the lasing transition is from state 8 to state 7. The length of one period is 44.9 nm,with a layer structure (in nanometers), starting with the injector barrier (centered atthe origin) 4.0/1.8/0.8/5.3/1.0/4.8/1.1/4.3/1.4/3.6/1.7/3.3/2.4/3.1/3.4/2.9, with barriersdenoted bold. Underlined layers are doped to 1.2×1017 cm−3, resulting in an averagecharge density of n3D = 1.74× 1016 cm−3 over the stage. . . . . . . . . . . . . . . . 47

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Figure Page

2.5 Calculated energy differences ∆8,7 (lowest red curve), ∆8,6 (middle green curve) and∆8,5 (top blue curve) as a function of field. The symbols show results based on exper-imentally determined electroluminescence spectra from Ref. [2]. . . . . . . . . . . . 48

2.6 Calculated current density vs field based on this work and comparison to results usingNEGF [3]. Also shown are experimental results from Ref. [2] (E1) and a regrown de-vice from Ref. [3] (E2). Experimentally determined threshold fields of 48 kV/cm (E1)and 52 kV/cm (E2) are denoted by dashed vertical lines. Both theoretical results aregiven for a lattice temperature of 300 K. Experimental results are provided for pulsedoperation, minimizing the effect of lattice heating above the heatsink temperature of300 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.7 Comparison between the current density obtained using the full density matrix (solidblue curve) and only including diagonal elements (short-dash red curve). The relativedifference between the two results is also shown (long-dash black curve). (a) Resultsfor the same device considered in Fig. 2.6. (b) Results for a device with a thickerinjection barrier (5 nm). (c) Results for a device with a thinner injection barrier (3 nm). 51

2.8 (a) Peak gain vs electric field. (b) Energy position corresponding to peak gain as afunction of electric field. Results are given for calculations both with off-diagonaldensity-matrix elements (green circles) and without them (violet squares), as well as acomparison with NEGF (dashed blue line) and DM 2nd (solid red line). Experimentalresults from Ref. [2] are denoted by a thick solid black line. Both the NEGF and DM2nd results are from Ref. [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.9 In-plane energy distribution for the ULS (solid red) and LLS (dashed blue). For com-parison, a Maxwellian thermal distribution with a temperature of 300 K is also shown(thin black). Results are shown for electric fields of 30 kV/cm (top), 55 kV/cm (mid-dle), and 70 kV/cm (bottom). For each value of the electric field, the electron tem-peratures of the ULS and LLS calculated using Eq.(2.77) are shown, as well as theweighted average electron temperature, Tavg, obtained using Eq. (2.76). . . . . . . . . 55

2.10 (a) Average excess electron temperature Te − T as a function of field, where Te iscalculated using Eq. (2.77). Results are shown for T = 200 K (blue stars), T = 300 K(orange squares) and T = 400 K (red circles). The two dashed vertical lines representthreshold (left line) and onset of NDR (right line) for the room-temperature results.(b) Average excess electron temperature Te−T as a function of electrical input powerper electron, defined by Eq. (2.117). The black line shows a best fit with the energy-balance equation (2.118). Vertical lines have the same meaning as in panel (a). . . . . 58

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2.11 Conduction-band edge (solid black line) and probability densities for the upper lasingstate (u), lower lasing state (`), injector state (i), and extractor state (e). Also shownis the extractor state (eL) for the previous stage to the left, the injector (iR) state forthe next stage to the right, and a high-energy state (h). The high-energy state wasincluded in numerical calculations, however, it had a small occupation and a negligibleeffect on physical observables. The dashed rectangle represents a single stage with thelayer structure (from the left) 44/62.5/10.9/66.5/22.8/84.8/9.1/61 A, with barriersin bold font. The thickest barrier (injector barrier) is doped with Si, with a dopingdensity of 7.39× 1016 cm−3 such that the average electron density in a single stage is8.98× 1015 cm−3. Owing to low doping, the potential drop is approximately uniform. 60

2.12 Current density vs electric field for density-matrix results (blue circles) and NEGF(green triangles) for a lattice temperature TL = 50 K. Also shown are experimentalresults (red squares) for a heat-sink temperature TH = 10 K. Experimental and NEGFresults are both from Ref. [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.13 Current density vs electric field for different values of the strength parameter α. Thebest agreement with experimental data is for α = 0.1 (green squares). The results athigh fields are not sensitive to the strength parameter, while low-bias results are. Thehigher peak at 9 kV/cm for the α = 0.1 data is a result of a finer electric-field meshfor that data set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2.14 log10(|ρNM |), where ρNM are density matrix elements after integration over parallelenergy. Normalization is chosen so that the highest occupation is one (upper las-ing level). Results are for an electric field of 21 kV/cm and a lattice temperature of50 K. Coherences and occupations are given for all combinations of states shown inFig. 2.11, except for the high energy h state, which had very small coherences andoccupation. The highest occupations are the upper lasing level (1.0), extractor state(0.60), lower lasing level (0.39), and injector state (0.33). The largest coherences arebetween the extractor state e and injector state iR (0.21), and between the upper andlower lasing levels (0.05). Other coherences were smaller than 0.01. . . . . . . . . . 64

2.15 The magnitude of the matrix element |ρEkNM | as a function of in-plane energy, for Ncorresponding to the upper lasing level (top panel) and extractor state (bottom panel).These two states were chosen because they have the greatest occupations. Note thatuR corresponds to the upper lasing level in the next period to the right (not shown inFig. 2.11), which is equal to the coherence between eL and u, owing to periodicity. . . 65

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2.16 Current density vs time for two values of electric fields. The upper panel shows thefirst two picoseconds and the lower the next 10 picoseconds. Note the different rangeson the vertical axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

2.17 Occupation vs time for the lower lasing, upper lasing, injector, and extractor states.Normalization is chosen such that all occupations add up to one. Note the longer timescale compared with the current density in Fig. 2.16. . . . . . . . . . . . . . . . . . . 68

2.18 In-plane energy distribution for all eigenstates shown in Fig. 2.11, except for the high-energy state (h). Results are shown for four values of time, starting in thermal equi-librium (t = 0 ps) and ending in the steady state (t = 100 ps). Also shown are thecorresponding electron temperatures, calculated from Te = 〈Ek〉 /kB. . . . . . . . . . 70

3.1 A schematic of the double-barrier GaAs/AlGaAs tunneling structure. The barrier andquantum-well widths are denoted by d and w, respectively, and L is the system length.We assume that the device dimensions in the x and y directions are much greater thanits length L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.2 (Color online) (a) Time-averaged potential profile (solid) and charge density (dashed)as a function of position for an applied bias of 96 mV (bias for which intrinsic currentoscillations are observed). The potential profile is obtained from a self-consistentsolution of the coupled WBTE and Poisson’s equation. (b) Time-averaged currentdensity vs applied bias for RTDs with 6-nm-wide (solid red) and 7-nm-wide (dashedblue) quantum wells. The time average is obtained over 1 ps intervals. Arrows inpanel (b) are guides for Fig. 3.3. Arrow 2 also corresponds to the bias value of 96 mV,which is used in (a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.3 Current density vs time (left column) and the corresponding Fourier transform ampli-tudes (right column) for the uniformly doped DBTS from Sec. 3.3.1 at an applied biasof 1: 50 mV (top row), 2: 96 mV (middle row) and 3: 130 mV (bottom row). Panelnumbers 1–3 refer to the bias values marked in Fig. 3.2. . . . . . . . . . . . . . . . . 80

3.4 (a) Time-averaged potential profile (solid line) and charge density (short dashed line)as a function of position for the traditionally doped RTD at an applied bias of 180 mV.Long dashed line shows the doping profile of the device. The potential profile isobtained from a self-consistent coupled solution of the WBTE and Poisson’s equation.(b) Time-averaged current density vs applied bias. The time average is obtained over1 ps intervals. The arrow marks the bias value 180 mV, which is used in (a). . . . . . 81

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3.5 (a) Current density as a function of time for a bias of 180 mV (peak current) forthe traditionally doped RTD (lower blue curve) and a comparison with the uniformlydoped DBTS from Sec. 3.3.1 for a bias of 0.096 mV (upper red curve). (b) Fourierspectrum of the current density oscillations for the traditionally doped (blue curve)and uniformly doped DBTS (red curve). Fourier transform amplitudes are normalizedso the maximum equals one. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.6 Fourier transform amplitude of the time-dependent current density for the uniformlydoped DBTS from Sec. 3.3.1 with the full scattering rates (blue squares), 50% of thefull scattering rates (green triangles) and 25% of the full scattering rates (red circles).The inset shows the corresponding time-averaged potential profiles for a part of thedevice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.7 (a) Snapshot of the potential profile (thick black curve) for the uniformly doped DBTSfrom Sec. 3.3.1 at an applied bias of 96 mV (bias for which intrinsic oscillations areobserved). The solid red and the dashed blue curves correspond to the ground stateprobability densities in the well and emitter regions, respectively. (b) Snapshot of thepotential profile (thick black curve) of the traditionally doped RTD from Sec. 3.3.2 ina steady state for an applied bias of 180 mV (bias for which intrinsic oscillations areobserved). The solid red curve corresponds to the ground state probability density ofthe main quantum well bound state. The emitter quantum well is very shallow andcontains no bound states. In both (a) and (b), the two arrows define the emitter andwell regions (the computational domain for the wave functions). The two domainsoverlap in the region of the left barrier. . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.8 Fourier transforms of the potential on the left side of the first barrier at zb = 6 nm (redopen circles), the middle of the well at zw = 0 (green closed circles), and the potentialdifference ∆(t) = V (zb, t) − V (zw, t) (blue squares) for the uniformly doped DBTSfrom Sec. 3.3.1. In the legend, the tilde denotes a Fourier transform from the time tofrequency (ν) domain. The inset shows a snapshot of the potential profile where thepositions zb (red downward-pointing triangle) and zw (green upward-pointing triangle)are marked. The figure shows that the largest Fourier components of the emitter andwell potential oscillations are for frequencies above the intrinsic current oscillations.However, these high-frequency plasmonic oscillations [Eq. (3.9)] are in phase, as arethe low frequency oscillations around 0.4 THz. In contrast, the potential differencebetween the barrier and well region oscillates with the same frequency as the currentdensity (4.5 THz). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

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3.9 (a) Time evolution of ∆, the energy difference between the MQW ground state and theEQW ground state (red circles) and current density (blue triangles) for the uniformlydoped DBTS from Sec. 3.3.1. The energy difference between the ground states in theEQW and the MQW is oscillating with the same frequency as the current density andis close to zero when ∆ is at a minimum. The current density leads ∆ in phase by 49.(b) Probability density for the emitter ground state at a few points in time (the corre-sponding values of ∆ are denoted in the inset). We see that the EQW wavefunctionvaries weakly over time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.10 (a) Frequency of current oscillation fmax vs doping density. (b) Amplitude of currentoscillations JA vs doping density. The amplitude is calculated using JA = 2

√2σ,

where σ is the standard deviation in current density. (c) Relative amplitude of currentoscillations JA/Jav where Jav is the time-averaged current density. (d) Operating biasVop (see main text for definition) as a function of doping density. (e) Frequency of cur-rent oscillations vs the ratio of time averaged charge in the MQW and the EQW (seemain text for details). All panels are for the uniformly doped DBTS from Sec. 3.3.1. . 92

3.11 Time-averaged potential profile in the uniformly doped DBTS from Sec. 3.3.1 at adoping density of 1.0 × 1018 cm−3 (solid red) and 2.0 × 1018 cm−3 (dashed blue)for applied bias of 96 mV and 77 mV, respectively. For the higher doping density,the potential well on the collector side is deep enough to form a bound state, whichdisrupts the current density oscillations that are governed by charge transfer betweenthe emitter and main quantum wells. . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.12 Fourier transform of the time-dependent potential value at the center of the MQW re-gion, corresponding to V (zw, t) in Fig. 3.8, for doping densities of 1.0 × 1018 cm−3

(blue squares), 1.5×1018 cm−3 (filled circles) and 1.8×1018 cm−3 (red circles). Verti-cal lines mark the corresponding electron–plasma frequencies, 9.65 THz, 11.81 THz,and 12.94 THz, respectively. Plasma frequences are calculated using Eq. (3.9), as-suming an electron density equal to doping density. We see that, as the electron den-sity is increased, plasma oscillations increase both in frequency and amplitude. Thelower-frequency Fourier component, corresponding to the frequency of intrinsic cur-rent oscillations, varies more slowly with doping and remains between 4 and 5 THz. . 94

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4.1 Steady-state conduction band profile obtained from the WTE (solid thick black line).The dashed rectangle marks the extent of a single period, where the first barrier is re-ferred to as the injection barrier. Also shown are probability densities correspondingto wavefunction of the upper (u) and lower (`) lasing levels (thick blue and thick redrespectively). Also shown is one of the injector (i) states (thick green) as well as othersubbands (thin gray) that are localized near the considered period. Starting at the injec-tion barrier, the layer thicknesses (in nanometers) are 4.0/8.5/2.5/9.5/3.5/7.5/4.0/15.5,with the widest well n-doped to 2.05 × 1016 cm−3 resulting in an average doping of5.7 × 1015 cm−3 per period. (The layer structure has been rounded to the nearesthalf-nm to match the simulation mesh spacing of ∆z = 0.5 nm.) Owing to the lowelectron concentration, the potential (not including the barriers) is almost linear. . . . . 99

4.2 Wigner functions for an injector state (i), the upper lasing level (u), and the lowerlasing level (`). See figure 4.1 for the spatial profile of the corresponding wave functions.107

4.3 Current density vs electric field from theory (solid blue curve) and experimental re-sults from Ref. [5] (dashed red curve). τ−1

R = 1012 s−1, τ−1M = 2 × 1013 s−1, and

Λ = 2× 109 nm−2s−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.4 Steady-state WF for three values of the electric field. The lattice temperature is 10 K.Barriers are shown in black. All parameters are the same as in Fig 4.3. . . . . . . . . . 111

4.5 Current density vs electric field for different values of the relaxation rate τ−1R with

Λ = 0 and τ−1M = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.6 Current density vs electric field for different values of the relaxation rate τ−1R . τ−1

M =2× 1013 s−1 and Λ = 2× 109 nm−2s−1, as in Fig. 4.3. . . . . . . . . . . . . . . . . . 112

4.7 Current density vs electric field for different values of the momentum relaxation rateτ−1M . τ−1

R = 1012 s−1 and Λ = 2× 109 nm−2s−1, as in Fig. 4.3. . . . . . . . . . . . . . 113

4.8 Current density vs electric field for different values of the localization rate Λ, withτ−1R =

1012 s−1 and τ−1M = 2 × 1013 s−1. High localization rates wash out some of the fine

features of the J − E curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

B.1 (Color online) Solid black curve shows the potential profile for the uniformly-dopedRTD described in Sec. 3.3.1. Long-dashed red curve shows the bias potential V∆(z),with parameters β = 0.035 nm−1, z0 = 0, and V0 = 0.096 eV. Short-dashed bluecurve shows Vδ(z), which is the potential remaining when the bias and barriers havebeen subtracted from the total potential V (z). . . . . . . . . . . . . . . . . . . . . . 140

xiii

ABSTRACT

Modern epitaxy methods have made it possible to grow semiconductor heterostructures with

monolayer precision. Owing to the abrupt change in the conduction and valance bands at material

interfaces, quantum effects such as tunneling and confinement can arise. Some devices rely on

quantum effects for their operation. An example is the double-barrier tunneling structure (DBTS),

in which electrons tunnel into a quasibound state, localized between two potential barriers. By

tuning the bias, the quasibound state can be brought into or out of alignment with electron coming

from the injector, resulting in a bias regime of negative differential resistance. Another practical

application is the quantum cascade laser (QCL), where population inversion between quasibound

quantum states is achieved by applying a bias to a periodic arrangement of potential wells and

barriers. In this dissertation, we employ the density-matrix and Wigner-function formalisms to

tackle the challenge of modeling both classes of devices (QCLs and DBTS) in a way that captures

all the necessary physics, but it still tractable to numerical calculations.

In order to describe periodic structures such as QCLs, we derive a Markovian master equation

for the single-electron density matrix. The master equation conserves the positivity of the density

matrix, includes off-diagonal elements (coherences) as well as in-plane dynamics, and accounts

for electron scattering with all the relevant scattering mechanisms. We use the model to simulate

QCLs in both the terahertz regime and the mid-infrared range. We validate the model by compar-

ing the results with experiment, as well with theoretical work based on nonequilibrium Green’s

functions (NEGF) and simplified density-matrix models. For the considered devices, we show

that the magnitude of coherences (off-diagonal elements of the density matrix) can be a significant

xiv

fraction of the diagonal matrix elements, which demonstrates their importance when describing

QCLs. We show that the in-plane energy distribution can deviate far from a thermal distribution in

both terahertz and mid-IR QCLs, which suggests that the assumption of thermalized subbands in

simplified density-matrix models is inadequate.

Using the Wigner-function formalism, we investigate time-dependent, room-temperature quan-

tum electronic transport in GaAs/AlGaAs DBTSs. The open-boundary Wigner-Boltzmann trans-

port equation is solved by the stochastic ensemble Monte Carlo technique, coupled with Poisson’s

equation and including electron scattering with phonons and ionized dopants. We observe well-

resolved and persistent THz-frequency current-density oscillations in uniformly doped, dc-biased

DBTSs at room temperature. We show that the origin of these intrinsic current oscillations is not

consistent with previously proposed models, which predicted an oscillation frequency given by the

average energy difference between the quasibound states localized in the emitter and main quantum

wells. Instead, the current oscillations are driven by the long-range Coulomb interactions, with the

oscillation frequency determined by the ratio of the charges stored in the emitter and main quantum

wells. We discuss the tunability of the frequency by varying the doping density and profile.

We use the Wigner-function formalism to simulate time-dependent, partially coherent, dis-

sipative electron transport in biased semiconductor superlattices. We introduce a model collision

integral with terms that describe energy dissipation, momentum relaxation, and the decay of spatial

coherences (localization). Based on a particle-based solution to the Wigner transport equation with

a model collision integral, we simulate quantum electronic transport at 10 K in a GaAs/AlGaAs

superlattice and accurately reproduce its current density vs. field characteristics obtained in exper-

iment.

1

Chapter 1

Overview of Dissertation

The goal of this dissertation is to describe both time-dependent and steady-state quantum trans-

port in semiconductor heterostructures such as quantum cascade lasers, tunneling structures, and

superlattices. The dissertation is divided into three standalone chapters (excluding this overview),

where each chapter covers a different class of devices. The techniques employed are the density-

matrix and Wigner-function formalisms.

In Chapter 2, we derive a Markovian master equation for the single-electron density matrix

that is applicable to simulation of quantum cascade lasers. The master equation conserves the

positivity of the density matrix, includes off-diagonal elements (coherences) as well as in-plane

dynamics, and accounts for electron scattering with phonons, interface roughness, ionized impuri-

ties, and random alloy distribution. We use the model to simulate QCLs in both the terahertz and

the midinfrared range. We validate the model by comparing the results with both experiment and

simulations via nonequilibrium Green’s functions (NEGF) and simplified density-matrix models.

The material presented in this chapter is based on the work published in Refs. [6, 7, 8].

In Chapter 3, we investigate time-dependent, room-temperature quantum electronic transport in

GaAs/AlGaAs double-barrier tunneling structures (DBTSs). The open-boundary Wigner-Boltzmann

transport equation is solved by the stochastic ensemble Monte Carlo technique, coupled with Pois-

son’s equation and including electron scattering with phonons and ionized dopants. We observe

well-resolved and persistent terahertz-frequency current-density oscillations in uniformly doped,

dc-biased DBTSs at room temperature. We show that the origin of these intrinsic current oscilla-

tions is not consistent with previously proposed models, which predicted an oscillation frequency

given by the average energy difference between the quasibound states localized in the emitter and

2

main quantum wells. Instead, we show that the current oscillations are driven by the long-range

Coulomb interactions. The material presented in this chapter is based on the work published in

Ref. [9]

In Chapter 4, we use the Wigner-function formalism to simulate partially coherent, dissipative

electron transport in biased semiconductor superlattices. We introduce a model collision integral

with terms that describe energy dissipation, momentum relaxation, and the decay of spatial co-

herences (localization). Based on a particle-based solution to the Wigner transport equation with

a model collision integral, we simulate quantum electronic transport at 10 K in a GaAs/AlGaAs

superlattice and accurately reproduce its current density vs field characteristics obtained in experi-

ment. The material presented in this chapter is based on the work published in Ref. [10].

3

Chapter 2

Simulation of Quantum Cascade Lasers within the Density-MatrixFormalism

2.1 Introduction and Background

2.1.1 Quantum Cascade Lasers

Quantum cascade lasers (QCLs) are high-power light-emitting sources in the midinfrared (mid-

IR) and THz parts of the electromagnetic spectrum. They are electrically driven, unipolar semicon-

ductor devices that achieve population inversion based on quantum confinement and tunneling [11].

The QCL core (the gain medium) is a vertically grown semiconductor heterostructure made up of

a periodic arrangement of stages, where each stage consist of alternating layers of quantum wells

and barriers. Typically, the layers are several nanometers thick. This periodic arrangement is

sometimes called a superlattice [12], because it can be thought of an artificial crystal with a lat-

tice constant equal to the length of a single stage. The use of different III-V semiconductors and

their alloys enables great flexibility through precise engineering of material composition and layer

widths. Common material systems include GaAs/AlGaAs grown on the GaAs substrate [4], In-

GaAs/InAlAs grown on the InP substrate [2], and AlSb/InAs grown on the InAs substrate [13].

The first QCL was developed at Bell Laboratories in 1994 [14]. The device lased at a wave-

length of 4.6 µm, but only produced about 8 mW of radiated power (in pulsed operation) at a low

temperature of 10 K. The wall-plug efficiency was less than 0.15%, which means about 99.85% of

the input electrical power was converted into heat and only 0.15% into radiation. Since then, a great

deal of progress has been made through optimization of the QCL core, waveguide improvements,

better growth techniques, and improved thermal management [15]. Important milestones include

4

the development of the first continuous wave midinfrared QCL at room temperature with about

∼17 mW emitted power in 2002 [16] and the first terahertz QCL in the same year [17]. Quan-

tum cascade lasers are no longer just an academic laboratory experiment and are now available

commercially through various vendors, such as Alpes Lasers, Pranalytica, and Daylight Solutions.

Quantum cascade lasers span wavelengths in the range 2.6 µm to 250 µm (1.2-115 THz) [15]

except in the reststrahlen band ∼34 µm (9 THz), where photon energies are similar to the polar-

optical-phonon energy in GaAs and InP [15]. The lasing wavelength is limited from below by the

conduction-band discontinuity between the well and barrier materials, where larger discontinuity

is required for low-wavelength devices. The commonly used In0.53Ga0.47As/In0.52Al0.48As mate-

rial system (lattice-matched to InP) has a conduction-band discontinuity of about ∼520 meV and

higher offset can be obtained using strain-balanced designs based on different alloy concentration.

For example, the In0.669Ga0.331As/In0.362Al0.638As material system has a conduction-band offset

of ∼800 meV, which is useful for low-wavelength devices such as the 4.6 µm (270 meV) QCL

proposed in Ref. [18].

High performance at room temperature has been reached for midinfrared lasers based on the

InGaAs/InAlAs material system, where multiple Watts of output power have been demonstrated

in continuous-wave operation with wall-plug efficiencies of 21% at room temperature and 50% at

77 K [19, 20, 21]. Achieving high-temperature operation of THz QCLs has been less successful

owing to thermally activated relaxation methods that make it difficult to maintain a population

inversion at high temperatures [22]. To date, all THz QCLs are limited to cryogenic tempera-

tures [23]. The best-performing THz QCLs are based on the GaAs/AlGaAs material system, with

a high-temperature record (without magnetic field assistance) of 200 K in pulsed operation [24]

and 129 K in continuous-wave operation [23].

Owing to the great deal of flexibility in choice of materials and layer widths, there exist many

different QCL-core designs. Comparison of the merits of different designs is not the focus of this

dissertation. However, in order to illustrate the basic working principle of QCLs, Fig. 2.1 shows a

schematic conduction-band diagram (under bias) for a 8.5-µm QCL design proposed in [2]. The

dashed rectangle shows the extent of a single stage, where stages are separated by thick injection

5

barriers. The most important states are the upper lasing state (ULS, denoted by u), the lower lasing

state (LLS, denoted by `), the extractor state (denoted by e), and the injector state (denoted by i).

The green-shaded rectangle represents a region with many closely spaced states (miniband). Above

the miniband, between the lower and upper lasing states, is a region with no states (minigap). The

basic idea behind this design is to bias the structure in such a way that the bottom on the miniband

(the injector state) is aligned with the ULS to facilitate efficient injection from the bottom of the

miniband (mostly from the injector state) to the ULS. After an electron is injected into the ULS,

it can contribute to lasing via stimulated emission from the ULS to the LLS. Once an electron

is in the LLS, it is quickly scattered into the bottom of the miniband, usually by optical-phonon

emission; from there, it is injected again into the upper lasing state of the next stage, where it

can emit another photon. As an electron traverses the active core, it can in principle emit the

same number of photons as the number of stages in the device, although in practice many of the

transitions from an ULS to a LLS will be nonradiative (e.g., due to phonon emission).

In order to maintain a population inversion between the upper and lower lasing states (necessary

for lasing), the lifetime of an electron in the ULS must be greater than the lifetime in the LLS. This

condition is satisfied by designing a layer structure in such a way that the energy spacing ∆`,e

between the LLS and the extractor state is close to the optical-phonon energy. Another factor in

the lower lifetime of the LLS is the large number of states in the miniband with large overlap with

the LLS (many states to scatter to). This specific design is called a bound-to-continuum design,

named after the lasing transition from the ULS (the bound state) to the miniband composed of

many closely spaced states (quasicontinuum).

2.1.2 Simulation Methods

Numerical simulations can provide detailed insights into the microscopic physics of QCL op-

eration and are essential in the design process [25, 26]. Electron transport in GaAs-based quantum

cascade lasers in both mid-IR and THz regimes has been successfully simulated via semiclassical

(rate equations [27, 28, 29] and Monte Carlo [30, 31, 32, 33]) and quantum techniques (density

matrix [34, 35, 36, 25, 37] and nonequilibrium Green’s functions (NEGF) [38, 39, 40]). InP-based

6

Figure 2.1: Schematic diagram showing the conduction-band profile for the QCL design from

Ref. [2] under an applied bias. The dashed rectangle contains a single stage; adjacent stages are

separated by thick injection barriers. The upper lasing state (u) is denoted by the red curve, the

lower lasing state by the blue curve (`), the extractor state (e) by the purple curve, and the injector

state (i) by the orange curve. The green-shaded area denotes the miniband, which is a region with

many closely spaced states.

mid-IR QCLs have been addressed via semiclassical [41] and quantum-transport approaches (8.5-

µm [3] and 4.6-µm [39, 42] devices). Semiclassical approaches are appealing owing to their low

computational requirements. They go beyond the effective-mass approximation [32] and can ex-

plore phenomena such as nonequilibrium phonons [33]. However, semiclassical models can pro-

vide an inadequate descriptions of QCLs when off-diagonal density-matrix (DM) elements (coher-

ences) are a significant fraction of the diagonal ones. Including coherences is especially important

for QCLs working in the THz range, where the role of coherence cannot be ignored [43, 35]. While

NEGF simulations accurately and comprehensively capture quantum transport in these devices,

7

they are computationally demanding. Density-matrix approaches offer a good compromise: they

have considerably lower computational overhead than NEGF, but are still capable of describing

coherent-transport features.

In order to maximize the performance of QCLs, optimization methods such as the genetic al-

gorithm have been employed, where the simulation converges on a layer structure that maximizes

the gain of the device [4]. These simulations require repeated calculations of device performance

for a large number of parameters, so computational efficiency plays an important role. This fact

makes density-matrix-based approaches advantageous over the relatively high computational bur-

den of NEGF [3]. However, previously proposed density-matrix-based approaches have significant

drawbacks. One drawback is the assumption of thermalized subbands, where the electron temper-

ature is either an input parameter [35, 3] or determined using an energy-balance method [44]. This

approximation may not be warranted, because QCLs operate far from equilibrium, so the in-plane

energy distribution can (as will be shown later in this work) deviate far from a heated thermal

distribution (Maxwellian or Fermi-Dirac), which makes electron temperature an ill-defined quan-

tity. Density matrix models that include in-plane dynamics have been proposed [45]. However,

coupling between the transport and in-plane direction is ignored (the density matrix is assumed

to be separatable into the cross-plain and in-plane directions), which may not be warranted. An-

other drawback is the phenomenological treatment of dephasing across thick injection barriers:

transport is treated semiclassically within a single stage of a QCL (or a part of a single stage)

while interstage dynamics are treated quantum-mechanically, using phenomenological dephasing

times [35, 3, 37, 43].

2.1.3 Organization of Chapter

In Sec. 2.2, we propose a computationally efficient density-matrix model based on a rigorously

derived Markovian master equation, where the aforementioned simplifications (separation of the

density matrix or phenomenological dephasing times) are not made. The Markovian master equa-

tion conserves the positivity of the density matrix, includes off-diagonal matrix elements as well as

full in-plane dynamics and time dependence, and accounts for the relevant scattering mechanisms

8

with phonons, impurities, interface roughness, and random alloy scattering. In Sec. 2.3, we discuss

details regarding the numerical solution of the master equation and derive an iterative scheme to

efficiently solve the steady-state density matrix. In Sec. 2.4, we validate the model by simulating a

midinfrared QCL proposed in [2] and, in Sec. 2.5, we do the same for the terahertz QCL proposed

in Ref. [4]. Concluding remarks and possible extensions to the current work are given in Sec. 2.6.

2.2 Theory

Electrons interacting with an environment can be described using a Hamiltonian

H = H0 +Hi +HE, (2.1)

where H0 is the unperturbed electron Hamiltonian (potential and kinetic energy terms), HE is

the Hamiltonian of the environment (e.g., free-phonon Hamiltonian), and Hi contains interaction

terms (e.g., electron-phonon interaction or scattering due to interface roughness). In this work, we

will use the eigenstates of H0 as a basis. We treat the eigenstates under an applied bias as bound

states, even though, strictly speaking, the states are better described as resonances with some en-

ergy spread [46]. The bound-state approximation is good if the energy spread is much smaller than

other characteristic energies, and if the dynamics are mostly limited to the subspace of resonance

states. For more discussion on the validity of this approximation, see Ref. [26]. Nonparabolicity

in the bandstructure is accounted for by calculating eigenstates using a 3-band k · p Hamiltonian,

which is described in Sec. 2.2.1. We will assume that the device area in the direction perpendicular

to transport is macroscopic compared to the the length of a single period. The eigenstates can then

be labelled as |n,k〉 = |n〉 ⊗ |k〉, where the discrete index n labels the eigenstates in the trans-

port direction (z-direction) and the continuous parameter k labels the wavevector associated with

free motion in the x-y plane (in-plane motion). Nonparabolicity has previously been included in

density-matrix models using an energy-dependent effective mass [2, 47, 3], however, that method

has the undesirable feature of nonorthogonal eigenstates. With our treatment of nonparabolicity

using k · p, the eigenstates are orthogonal, with 〈n′,k′|n,k〉 = δn′nδ(k′ − k).

9

Throughout this work, we assume translational invariance in the in-plane direction (we ignore

edge effects). Using this approximation, the reduced density matrix of the electrons is diagonal

in k and only depends on the magnitude k = |k|. The central quantity of interest is then ρEkN,M ,

the coherence between eigenstates N and M , at the in-plane energy Ek. When working with

periodic systems, it is more convenient to work with the density matrix in relative coordinates

fEkN,M ≡ ρEkN,N+M , with M = 0 terms giving occupations and terms with |M | > 0 represent

coherences between states |N〉 and |N +M〉. Relative coordinates are convenient because fEkN,M

is periodic in N , with a period of Ns, where Ns is the number of eigenstates in a single period.

We also have fEkN,M → 0 as |M | → ∞, providing an obvious truncation scheme |M | ≤ Nc,

where Nc is a number we refer to as a coherence cutoff. Sections 2.2.2 and 2.2.3 are devoted to

the derivation of a Markovian master-equation (MME) for fEkN,M that preserves the positivity of

the density matrix. Section 2.2.2 covers the electron-phonon interaction with acoustic and optical

phonons and Sec. 2.2.3 covers other elastic scattering mechanisms (interface roughness, ionized

impurities, and random-alloy scattering). We will derive a MME for the matrix elements fEkN,M of

the form

∂fEkN,M∂t

= −i∆N,N+M

~fEkN,M +DEkN,M , (2.2)

where ∆N,M is the energy spacing between states N and M and we have defined the dissipator

DEkN,M = −∑

n,m,s,±

Γout,g,±NMnmEk

fEkN,n +∑

n,m,s,±

Γin,g,±NMnmEk

fEk∓E0,g+∆N+M,N+M+m

N+n,M+m−n + h.c.. (2.3)

Here, Γin,g,±NMnmEk

and Γout,g,±NMnmEk

are the quantum-mechanical generalizations of semiclassical scat-

tering rates [32, 26]. In this context, h.c. should be understood as the matrix elements of the

Hermitian conjugate (it is not simply the complex conjugate because indices need to be swapped

as well). In this work we will refer to them as scattering rates (or simply rates). The g index

represents the different scattering mechanisms, +(−) refers to absorption (emission), and E0,g is

the energy associated with scattering mechanism g (i.e., phonon energy). For elastic scattering

mechanisms such as ionized-impurity scattering, we have E0,g = 0.

10

Equation (2.2) improves upon previous theoretical work on QCLs [31, 37, 3], where transport

was considered as semiclassical within a single period (or a subregion of a single period), while

coupling between different stages (separated by a thick injector barrier) was treated quantum me-

chanically, using phenomenological dephasing times. In addition, we include full in-plane dynam-

ics, while in Refs. [31, 37, 3], a thermal distribution was assumed. Here, we derive a completely

positive Markovian evolution for the density matrix, which supplants the need for phenomenologi-

cal dephasing times and captures all relevant coherent transport features, not just tunneling between

stages. We note that Eq. (2.2) can be derived as the low-density limit of the more general but still

completely positive Markovian equation that includes electron-electron interaction [48]; in QCLs

considered in this work, carrier density is low, so the low-density limit is appropriate. We also note

that our main equation (2.2) is equivalent to the low-density, Markovian, single-particle density-

matrix equations Eqs. (112)-(116) from Iotti et al. [49]; however, Iotti et al. derived these from a

more general Markovian set of equations that are not completely positive [49]. We also emphasize

that the formulation presented here is tailored toward efficient numerical implementation.

In this work, we are interested in the steady-state solution of Eq. (2.2), although we will pro-

vide some time-dependent results for a terahertz QCL in Sec. 2.5.2. Once the steady-state density

matrix is known, observables can easily be calculated. Detailed information regarding the differ-

ent observables, such as the current density and charge density, are given in Section. 2.2.4. In

Section 2.2.5, we derive an expression for the optical gain by considering the linear response of

the steady-state density matrix to an optical field, treated as a small perturbation. This section

concludes with Sec. 2.2.6, where we provide a comparison with semiclassical models (the Pauli

master equation [50]), where only the diagonal values of the density matrix are included.

2.2.1 Electronic States

Within the envelope function approximation, the electron eigenfunctions near a band extremum

k0 (Γ valley in this work) can be written as a combination of envelope functions from different

11

bands [51, 52]

Ψnk(r, z) =∑b

φ(b)n,k(r, z)u

(b)k=k0

(r, z), (2.4)

where the b sum is over the considered bands, u(b)k0

(k, z) is a rapidly varying Bloch function that has

the period of the lattice, and φ(b)n,k(r, z) are slowly varying envelope functions belonging to band

b. We make the approximation that the envelope functions factor into cross-plane and in-plane

direction according to

φ(b)n,k(r, z) = Nnormψ

(b)n (z)eik·r, (2.5)

where ψ(b)n (z) = φ

(b)n,k=0(r, z) is the envelope function for k = 0 and Nnorm is a normalization con-

stant. This separation of in- and cross-plane motion is a standard approximation when describing

QCLs and other superlattices [26]. Without it, a separate eigenvalue problem for each k would

need to be calculated, which is very computationally expensive.

In order to model QCLs based on III-V semiconductors, accurate calculation the Γ-valley

eigenstates is crucial. The simplest approach is the Ben Daniel–Duke model, in which only the

conduction band is considered [51]. The Schrodinger equation for the conduction-band envelope

function ψ(c)n (z) in Eq. (2.5) reduces to the 1D eigenvalue problem [26][−~2

2

d

dz

1

m∗(z)

d

dz+ V (c)(z)

]ψ(c)n (z) = Enψ

(c)n (z), En,k = En + Ek, (2.6)

where V (c)(z) is the conduction-band profile, and m∗(z) is the spatially dependent cross-plane ef-

fective mass of the conduction band. Typically, a parabolic dispersion relation Ek = ~2k2/(2m∗‖)

is assumed. A single-band approximation such as Eq. (2.6) can be justified when electrons have

energy close to the conduction-band edge (compared with the bandgap Egap) and quantum well

widths are larger than ∼10 nm [53, 51]. This is the case for THz QCLs, where single-band ap-

proximations can give satisfactory results [4, 6]. However, in mid-IR QCLs, the well thicknesses

can be below 2 nm [2] and the energy of the lasing transition can exceed 150 meV, which is a sig-

nificant fraction of typical bandgaps of the semiconductor materials, especially for low-bandgap

materials containing InAs. In this case, including nonparabolicity in the electron Hamiltonian is

important.

12

Nonparabolicity has previously been included in semiclassical [41, 26], density-matrix [47, 3]

and NEGF-based [54, 3, 39] approaches using single-band models, where only the conduction band

is modeled, but the influence of the valence bands is included via an energy-dependent effective

mass. One of the most important effects of nonparabolicity is the energetic lowering of the high-

energy states, such as the upper lasing level. This effect results from higher effective mass at high

energies, reducing the kinetic energy term in the Schrodinger equation [55]. However, an energy-

dependent effective mass has an undesirable trait in density-matrix and semiclassical models, the

resulting eigenfunctions are not orthogonal [26]. In this work, we will calculate the bandstructure

using a 3-band k · p model, which includes the conduction (c), light hole (lh) and spin-orbit split-

off (so) bands [56]. The 3-band k · p model has previously been used in semiclassical work on

QCLs [32, 33], however present work is the first quantum mechanical simulation of QCLs using

such a model.

The 3-band Hamiltonian can be derived from an 8-band Hamiltonian for bulk zinc-blende crys-

tals [57], which, at in-plane wavevector k = 0, reduces to 3 bands due to spin degeneracy and the

heavy-hole band factoring out. The 3× 3 Hamiltonian for an unstrained crystal can be written as

HBulk =

Ec + (1 + 2F )~

2k2z

2m0

√~2EP3m0

kz

√~2EP6m0

kz√~2EP3m0

kz Ev − (γ1 + 2γ2)~2k2z

2m0−√

2γ2~2k2

z

m0√~2EP6m0

kz −√

2γ2~2k2

z

m0Ev −∆so − γ1

~2k2z

2m0

, (2.7)

whereEc (Ev) is the unstrained conduction (valence) band edge, ∆so the spin-orbit split-off energy,

EP the Kane-energy [58], m0 the free electron mass, F is a correction to the conduction band ef-

fective mass due to higher energy bands [58], γ1 and γ2 are the modified Luttinger parameters [59],

given by [57]

γ1 = γL1 −EP

3Egap + ∆so

γ2 = γL2 −1

2

EP3Egap + ∆so

,

(2.8)

where γL1 and γL2 are the standard Luttinger parameters [60], and Egap is the material bandgap.

For a brief overview as well as numerical values for the different material parameters, we refer

13

the reader to Ref. [1] (note that the Luttinger parameters given within this reference correspond to

the standard Luttinger parameters γL1 /γL2 ). Bandstructure parameters for both In0.47Ga0.53As and

In0.52Al0.48As are given in Section 2.4.

Equation (2.7) is valid for a bulk III-V semiconductor, where kz refers to crystal momentum

in the [001] direction. However, for a heterostructure under bias, kz is no longer a good quantum

number and must be replaced by a differential operator kz → −i ∂∂z . In addition, all the material

properties are functions of position. Care must be taken when applying Eq. (2.7) to heterostructures

because the proper operator symmetrization must be performed to preserve the Hermicity of the

Hamiltonian. In this work, we follow the standard operator symmetrization procedure in k · p

theory for heterostructures [61]

f(z)kz →1

2(f(z)kz + kzf(z)) ,

g(z)k2z → kzg(z)kz,

(2.9)

where f(z) and g(z) are arbitrary functions of position (e.g., Kane energy, or Luttinger parameter).

The electron Hamiltonian for a heterostructure can then be written as

Hhet =

Hc Hc,lh Hc,so

H†c,lh Hlh Hlh,so

H†c,so H†lh,so Hso

, (2.10)

with

14

Hc = Ec(z)− ~2

2m0

∂

∂z(1 + 2F (z))

∂

∂z,

Hlh = Ev(z) +~2

2m0

∂

∂z(γ1(z) + 2γ2(z))

∂

∂z,

Hso = Ev(z)−∆so(z) +~2

2m0

∂

∂zγ1(z)

∂

∂z,

Hc,lh = − i2

√~2EP (z)

3m0

∂

∂z− i

2

∂

∂z

√~2EP (z)

3m0

,

Hc,so = − i2

√~2EP (z)

6m0

∂

∂z− i

2

∂

∂z

√~2EP (z)

6m0

,

Hlh,so =√

2~2

m0

∂

∂zγ2(z)

∂

∂z.

(2.11)

From Eq. (2.11) we see that the electron Hamiltonian now depends on spatial derivatives of mate-

rial functions such as the Kane energy EP (z) and modified Luttinger parameters γ1(z) and results

will be sensitive to how an interface between two different materials is treated. One choice is to

treat the material functions as piecewise-constant. However, with that choice, spatial derivatives

become delta functions (and derivatives of delta functions) that are difficult to treat numerically. In

this work we will assume that material parameters are smooth functions that have slow variations

on length scales shorter than a single monolayer LML (in devices grown on InP, LML = 0.29 nm

at room temperature). This assumption can be justified on the grounds that any spatial variation in

envelope functions faster than a single monolayer is unphysical (in the envelope function approx-

imation it is assumed that envelope functions are approximately constant over a single unit cell).

The material parameters that we use are obtained using

f(z) =1√πLML

∫fpc(z

′)e−(z−z′)2/L2MLdz′ (2.12)

where LML is the monolayer width of the considered material and fpc(z) is a piecewise constant

material function that is discontinuous at material interfaces. As an example, Fig. 2.2 shows the

Kane energy as well as the piecewise-constant variant for a single period of the QCL studied in

Sec. 2.4, with LML = 0.29 nm.

15

Figure 2.2: Kane energy as a function of position for a single period of the InGaAs/InAlAs-based

mid-IR QCL described in section 2.4. The dashed curve shows the piecewise constant version and

the dashed curve is the smoothed version given by Eq. (2.12), with LML = 0.29 nm. The region of

higher Kane energy corresponds to InGaAs.

The eigenfunctions of the Hamiltonian Hhet given in Eqs. (2.10)-(2.11) satisfy

Hhetψn(z) = Enψn(z), (2.13)

where ψn are three-dimensional vectors of envelope functions ψn(z) =(ψcn(z), ψlh

n (z), ψson (z)

)T .

In later sections, we will use the Dirac notation |n〉, with 〈z|n〉 = ψn(z) and |n,k〉with 〈r, z|n,k〉 =

Aψn(z)eik·r and normalization 〈n,k|n′,k′〉 = δn,n′δ(k − k′). The electron probability density is

obtained by summing over the different bands

|ψn(z)|2 = | 〈z|n〉 |2 =∑b

|ψbn(z)|2 (2.14)

16

and matrix elements of an operator B such as 〈n|B|n′〉 are understood as

〈n|B|n′〉 =∑b

∫dz[ψ(b)n (z)

]∗B(b)ψ

(b)n′ (z) , (2.15)

where the sum is over the considered bands, with

B =

B(c) 0 0

0 B(lh) 0

0 0 B(so)

. (2.16)

Note that B(b) is an operation working on component b of the envelope function ψn(z). In this

work all operators are assumed to be identical for all three bands (B(c) = B(lh) = B(so)) except

for the interaction potential due to interface roughness. The interaction potential due to interface

roughness is not the same for all bands because the band discontinuities at material interfaces are

not the same for different bands [see Sec. 2.2.3.1].

The eigenfunctions |n〉 are calculated in a similar manner as in previous semiclassical work [32,

33, 26]. A typical computational domain is shown in Fig. 2.3(a), where the conduction band of

a single period of the QCL from Ref. [2] is marked by a dashed rectangle. The computational

domain contains 3–4 periods, padded with tall potential barriers, placed far away from the central

region in order for the boundary conditions not to effect the eigenfunctions near the center of

the computational domain. We solve the eigenvalue problem in (2.13) using a basis of Hermite

functions (eigenfunctions of the harmonic oscillator), with two example basis functions shown

in Fig. 2.3(a). We used Hermite functions because they are easy to compute and required less

computational resources than finite-difference methods. We typically use a basis of∼ 400 Hermite

functions for each band, resulting in a matrix with dimension 1200 × 1200. The diagonalization

procedure results in 1200 eigenstates, most of which are far above the conduction band or far below

the valance bands. For transport calculations, we only include the states close to the conduction-

band edge. This bound-state approach is known to produce spurious solutions, which are states

with high amplitude far above the conduction band edge. In order to systematically single out the

relevant states, we first “throw away” the states that have energies far (∼ 1 eV) below or far above

the conduction-band edge. This step typically reduces the number of states to∼ 100. Figure 2.3(b)

17

Figure 2.3: (a) Conduction band Ec(z) and valance band Ev(z) used for the calculation of eigen-

functions. Dashed rectangle shows a single period of the conduction band. Potential barriers are

imposed to the far left and far right for both conduction and valence bands. Also show, are two

of the Hermite functions (n = 0 and n = 10) used as basis functions. (b) After discarding the

high and low-energy states, about 80 states are left, most of which have a energy high above the

conduction band or are greatly affected by the boundary conditions on the right. (c) After choosing

8 states with the lowest En and discarding copies from adjacent stages (see main text), the states

in the central stage (bold curves) are left. States belonging to adjacent periods (thin curves) are

calculated by translating states from the central stage in position and energy.

18

shows the conduction-band edge along with the remaining eigenstates. Next we remove the states

that are obtained by translation of a state from the previous/next period. We do accomplish this by

calculating each eigenstate’s ”center of mass”,

zn =

∫z|ψn(z)|2dz, (2.17)

and only keep states within a range z ∈ [zc, zc + LP ], where LP is the period length. We will refer

to this range as the center period. The choice of zc is arbitrary and we typically use zc = −LP/4.

Lastly, we follow the procedure in Ref. [26] and, for the remaining states, calculate the energy with

respect to the conduction-band edge using

En = En −∫Ec(z)|ψn(z)|2dz. (2.18)

States with En < 0 are valence-band states and are discarded. We keep Ns states, with the lowest

positive values of En. The value of Ns is chosen such that all eigenstates below the conduction

band top are included. After this elimination step, Ns is the number of states used in transport

calculations. The remaining states for Ns = 8 are shown in bold in Fig. 2.3(c). States belonging to

other periods are finally obtained by translation in space and energy of the Ns states belonging to

the center period. These states are identified by thin curves in Fig. 2.3(c).

2.2.2 Dissipator for Electron-Phonon Interaction

The total Hamiltonian, describing electrons, phonons and their interaction can be written as

H = H0 +Hph +Hi, (2.19)

where H0 is the electron Hamiltonian described in Sec. 2.2.1. Hph is the free-phonon Hamiltonian

and Hi describes the interaction between electrons and phonons. In the interaction picture (with

~ = 1), the equation of motion for the density matrix can be written as

∂ρ

∂t= −i

[Hi(t), ρ(t)

], (2.20a)

ρ(t) = ρ(0)− i∫ t

0

[Hi(s), ρ(s)

]ds, (2.20b)

19

where A(t) = ei(H0+Hph)tAe−i(H0+Hph)t denotes operator A in the interaction picture. We will

assume that the electron-phonon interaction is weak enough to warrant the assumption that the

phonon distribution is negligibly affected by the electrons and write the density matrix as a product

ρ(t) = ρe(t) ⊗ ρph, where ρe(t) is the electron density matrix and ρph is the thermal equilibrium

density matrix for the phonons. This approximation is called the Born approximation. Inserting

Eq. (2.20a) into Eq. (2.20b) and tracing over the phonon degrees of freedom gives

∂ρe∂t

= −iTrph

[Hi(t), ρe(0)⊗ ρph

]−∫ t

0

dsTrph

[Hi(t),

[Hi(s), ρe(s)⊗ ρph

]]. (2.21)

The first term on the RHS is zero for interaction Hamiltonians that are linear in creation/annihilation

operators [62], which is the case for the electron–phonon interaction Hamiltonians considered in

this work. If we are only interested in the density matrix for long times t, an alternative justifica-

tion for its omission is that we can assume that after a sufficiently long time, the density matrix

has no memory of its initial state ρe(0). Next we assume that the time evolution of the density

matrix is memoryless (Markov approximation), and replace ρe(s) in the integrand of Eq. (2.21) by

the density matrix at the current time ρe(t), giving

∂ρe∂t

= −∫ t

0

dsTrph

[Hi(t),

[Hi(s), ρe(t)⊗ ρph

]]. (2.22)

Equation (2.22) is the Redfield equation and is not completely positive. Making the change of

variables s → t − s and extending the range of integration to infinity (in doing that we assume

environmental correlations decay fast with increasing |t− s|) gives

∂ρe∂t

= −∫ ∞

0

dsTrph

[Hi(t),

[Hi(t− s), ρe(t)⊗ ρph

]]. (2.23)

Equation (2.23) is a Markovian master equation but still contains time dependence in the interaction

Hamiltonians. We can remove the time dependence by switching back to the Schrodinger picture

∂ρe∂t

= −i [H0, ρe(t)]−∫ ∞

0

dsTrph

[Hi,[e−i(H0+Hph)sHie

i(H0+Hph)s, ρe(t)⊗ ρph

]], (2.24)

which can be written as a sum consisting of unitary time evolution as well as a dissipative term

that includes interaction with phonons,

∂ρe∂t

= −i [H0, ρe(t)] +D[ρe(t)], (2.25)

20

where the superoperator D will be referred to as the dissipator. Expanding the commutators in

Eq. (2.24) gives 4 terms, which can be split into two Hermitian conjugate pairs

D[ρe(t)] =

∫ ∞0

dsTrph+Hiρe(t)⊗ ρphe−i(H0+Hph)sHie

i(H0+Hph)s

−Hie−i(H0+Hph)sHie

i(H0+Hph)sρe(t)⊗ ρph+ h.c.. (2.26)

In order to proceed, we assume a Frolich-type Hamiltonian for the description of an electron inter-

acting with a phonon bath in the single electron approximation [63]

Hi =1√V

∑Q

M(Q)(bQeiQ·R − b†Qe

−iQ·R). (2.27)

Here, b†Q (bQ) is the creation (annihilation) operator for a phonon with wavevector Q and the

associated matrix elementM(Q) and V is the quantization volume. Note that the matrix elements

are anti-HermitianM(Q)∗ = −M(Q), making the interaction Hamiltonian Hermitian.

Inserting Eq. (2.27) into Eq. (2.26) gives

D =1

V

∫ ∞0

ds∑Q,Q′

M(Q)∗M(Q′)Trph

−(bQe

iQ·R − b†Qe−iQ·R

)ρe(t)⊗ ρphe

−i(H0+Hph)s(bQ′e

iQ′·R − b†Q′e−iQ′·R

)e−i(H0+Hph)s

+(bQe

iQ·R − b†Qe−iQ·R

)e−i(H0+Hph)s

(bQ′e

iQ′·R − b†Q′e−iQ′·R

)ei(H0+Hph)sρe(t)⊗ ρph

+ h.c..

(2.28)

In order to proceed, we use the fact that the thermal equilibrium-phonon density matrix is diagonal,

so only terms with Q = Q′ survive the trace over the phonon degree of freedom. We also use that

only terms containing both creation and annihilation operators are nonzero after performing the

trace. Using the two simplifications, we get

D =1

V

∫ ∞0

ds∑Q

|M(Q)|2Trph

+ bQe

iQ·Rρe(t)⊗ ρphe−iH0se−iHphsb†Qe

iHphse−iQ·ReiH0s

+ b†Qe−iQ·Rρe(t)⊗ ρphe

−iH0se−iHphsbQeiHphseiQ·ReiH0s

− bQeiQ·Re−iH0se−iHphsb†QeiHphse−iQ·ReiH0sρe(t)⊗ ρph

− b†Qe−iQ·Re−iH0se−iHphsbQe

iHphseiQ·ReiH0sρe(t)⊗ ρph

+ h.c.. (2.29)

21

Performing the trace, and going from a sum over Q to an integral, we get

D =1

(2π)3

∑±

∫ ∞0

ds

∫d3Q|M(Q)|2

×

(1/2∓ 1/2 +NQ) e∓iEQseiQ·Rρe(t)e−iH0se−iQ·ReiH0s

− (1/2∓ 1/2 +NQ) e±iEQse−iQ·Re−iH0seiQ·ReiH0sρe(t)

+ h.c., (2.30)

whereEQ refers to the energy of a phonon with wavevector Q and we have defined the expectation

value of the phonon number operator Trphρphb†QbQ = NQ. Note that the upper (lower) sign

refers to absorption (emission) terms. In the following, we will assume phonons are dispersionless,

so EQ = E0 and NQ = N0 = (eE0/kBT − 1)−1, where T is the lattice temperature, and kB is the

Boltzmann constant. In order to keep expressions more compact, we define the scattering weight

W±(Q) =

(1

2∓ 1

2+N0

)|M(Q)|2. (2.31)

We can then write the dissipator due to a single dispersionless phonon type as (after reintroduction

of ~),

D =1

~2(2π)3

∑±

∫ ∞0

ds

∫d3Q

W±(Q)e∓iE0s/~eiQ·Rρe(t)e

−iH0s/~e−iQ·ReiH0s/~

−W±(Q)e±iE0s/~e−iQ·Re−iH0s/~eiQ·ReiH0s/~ρe(t)

+ h.c.,

(2.32)

where the total dissipator will be obtained later by summing over the different relevant phonon

branches. The dissipator can be written as D = Din + Dout, corresponding to the positive and

negative terms in (2.32) respectively. We will derive the form of the matrix elements of the in-

scattering term. The out-scattering term is less complicated and can be obtained by inspection of

the in-scattering term and only the final results will be given.

22

By using the completeness relation 4 times, we can write

Din =1

~2(2π)3

∫d3Q

∫ ∞0

ds∑

n1234,±

∫d2k1234

×⟨n1,k1|eiQ·R|n2,k2

⟩e∓iE0s/~W±(Q)

×⟨n3,k3|e−iH0s/~e−iQ·ReiH0s/~|n4,k4

⟩× 〈n2,k2|ρe|n3,k3〉 |n1,k1〉 〈n4,k4|+ h.c., (2.33)

where∑n1234 and

∫d2k1234 refer to summation over n1 through n4 and integration of k1 through

k4 respectively. Integration of k2 through k4 and shifting the integration variable Q → −Q + k1

gives

Din =1

~2(2π)3

∫d3Q

∫ ∞0

ds∑

n1234,±

∫d2k1

×W±(Q− k1) (n1|n2)Qz (n3|n4)∗Qz ρEqn2,n3

× e−is~ (∆n3,n4−Ek1

±E0+Eq) |n1,k1〉 〈n4,k1|+ h.c., (2.34)

where Ek is the in-plane dispersion relation, ∆nm = En − Em is the energy spacing of the cross-

plane eigenfunctions and

(n|m)Qz =∑b

∫ ∞−∞

[ψ(b)n (z)]∗eiQzzψ(b)

m (z)dz , (2.35)

where the sum is over the considered bands (conduction, light-hole and split-off bands in this

work).

In order to perform the s integration, we use∫ ∞0

e−i∆s/~ = π~δ(∆)− i~P 1

∆, (2.36)

where P denotes the Cauchy principal value, which causes a small shift to energies called the

Lamb shift [64]. By omitting the Lamb shift term and dropping the subscript on the k1 integration

variable, we get

Din =π

~(2π)3

∑n1234,±

∫d2k

∫d3QW±(Q− k) (n1|n2)Qz (n3|n4)∗Qz (2.37)

× ρEk+∆n4n3∓E0n2,n3 δ[Eq − (Ek + ∆n4n3 ∓ E0)] |n1,k〉 〈n4,k|+ h.c..

23

Multiplying by 〈N | from the left and |N +M〉 from the right and renaming the sum variables

n2 → n and n3 → m gives the matrix elements of the in-scattering contribution of the dissipator,

giving (∂fEkN,M∂t

)in

=π

~(2π)3

∑n,m,±

∫d3QW±(Q− k) (N |n)Qz (m|N +M)∗Qz (2.38)

× ρEk+∆N+M,m∓E0n,m δ[Eq − (Ek + ∆N+M,m ∓ E0)] + h.c..

The sum over n and m is over all integers. However, it is convenient to numerical reasons to shift

the sum variables in such a way that terms with small |n| and |m| dominate and terms with large

|n| and |m| quickly approach zero. This can be done by making the switch n → N + n and

m→ N +M +m, giving(∂fEkN,M∂t

)in

=π

~(2π)3

∑n,m,±

∫d3QW±(Q− k) (N |N + n)Qz (N +M +m|N +M)∗Qz

× ρEk+∆N+M,N+M+m∓E0

N+n,N+M+m δ[Eq − (Ek + ∆N+M,N+M+m ∓ E0)] + h.c.. (2.39)

In order to write the RHS in Eq. (2.39) in terms of fEkN,M , we use

ρEkN+n,N+M+m = ρEkN+n,N+n+(M+m−n) = fEkN+n,M+m−n (2.40)

and get(∂fEkN,M∂t

)in

=π

~(2π)3

∑n,m,±

∫d3QW±(Q− k) (N |N + n)Qz (N +M +m|N +M)∗Qz

× fEk+∆N+M,N+M+m∓E0

N+n,M+m−n δ[Eq − (Ek + ∆N+M,N+M+m ∓ E0)] + h.c.. (2.41)

or (∂fEkN,M∂t

)in

=∑n,m,±

Γ±,inNMnmEkfEk+∆N+M,N+M+m∓E0

N+n,M+m−n + h.c., (2.42)

where we have defined the in-scattering rates

Γ±,inNMnmEk=

π

~(2π)3

∫d3QW±(Q− k) (N |N + n)Qz (N +M +m|N +M)∗Qz

× δ[Eq − (Ek + ∆N+M,N+M+m ∓ E0)]. (2.43)

24

Note that the RHS in Eq. (2.43) looks like it depends on the direction of k. However, the scattering

weight only depends on the magnitude |Q−k|, so the coordinate system for the Q integration can

always been chosen relative to k, which means the RHS only depends on the magnitude |k|.

In order to proceed, we need to specify the in-plane dispersion relation Ek, as well as the

scattering mechanism, which determines the form of the scattering weight W±(Q − k). In this

work, we will assume a parabolic dispersion relation Ek = ~2k2/2m∗‖, with a single in-plane

effective mass m∗‖, defined as the average effective mass of the Ns considered eigenstates

1

m∗‖=

1

Ns

Ns∑n=1

∑b

∫|ψ(b)n (z)|2

m∗b(z)dz, (2.44)

where m∗b(z) is the position-dependent effective mass of band b. The approximation of a parabolic

in-plane dispersion relation can fail in short-wavelength QCLs, where electrons can have in-plane

energies comparable to the material bandgap [41]. It is possible to go beyond a parabolic dispersion

relation by including a term proportional to k4 (in-plane nonparabolicity). However, the inclusion

of a k4 term would significantly complicate the integration over q in Eq. (2.43) and is beyond the

scope of this work.

Since we do not have an analytical expression for the eigenfunctions, the products (n|m)Qz

have to be calculated numerically. For a more compact notation, we define

In,m,k,`(Qz) = (n|m)Qz (k|`)∗Qz . (2.45)

Switching to cylindrical coordiates d3Q → qdqdθdQz, making the change of variables Eq =

~2q2/2m∗‖ and integrating over Eq gives

Γ±,inNMnmEk=

m∗‖2π~3

u[Ek + ∆N+M,N+M+m ∓ Es]

×∫ ∞

0

dQzIN,N+n,N+M+m,N+M(Qz)

×[

1

2π

∫ 2π

0

dθW±(Q− k)

]Eq=Ek+∆N+M,N+M+m∓E0

, (2.46)

where the term in the square brackets is the scattering weight averaged over the angle between Q

and k, with the in-plane part of Q constrained by energy conservation, and u[x] is the Heaviside

25

function, which is zero when no positive value of Eq can satisfy energy conservation. Note that we

have also taken advantage of the fact that the real part of the integrand is even in Qz, allowing us

to limit the range of integration to positive Qz. The resulting expressions from the angle-averaged

scattering weight can be quite cumbersome, so we define the angle-averaged scattering weight

W±(Ek, Qz, Eq) =1

2π

∫ 2π

0

dθW±(Q− k), (2.47)

simplifying the in-scattering rate to

Γ±,inNMnmEk=

m∗‖2π~3

u[Ek + ∆N+M,N+M+m ∓ E0]

×∫ ∞

0

dQzIN,N+n,N+M+m,N+M(Qz)

× W±(Ek, Qz, Ek + ∆N+M,N+M+m ∓ E0). (2.48)

In order to proceed we need to specify the scattering weight, which depends on the kind of phonon

interaction being considered. Calculations of scattering rates for longitudinal acoustic phonons

and polar optical phonons is performed in Appendix A.

2.2.3 Dissipator for Elastic Scattering Mechanism

In addition to electron–phonon scattering, there are numerous elastic scattering mechanisms

that are relevant to QCLs. The most important ones are interface roughness, ionized impurities,

and alloy scattering [38, 40, 26]. The derivation of the corresponding scattering rates is slightly

different from the case for phonons and will be demonstrated for the case of interface roughness,

which is a very important elastic scattering mechanism for both THz and mid-IR QCLs [65, 66, 67]

and has been found to decrease the upper lasing state lifetime by a factor of 2 at room temperature in

mid-IR QCLs [67]. For other elastic scattering mechanisms, we only briefly discuss the interaction

potential and give final expressions for rates in Appendix A.

26

2.2.3.1 Interface Roughness

For an electron in band b, deviation from a perfect interface between two different semicon-

ductor material can be modeled using an interaction Hamiltonian on the form [68]

H(b)IR =

∑i

∆i(r)∂V (b)

∂z

∣∣∣∣z=zi

'∑i

∆V(b)i δ(z − zi)∆i(r) , (2.49)

where i labels an interface located at position zi, V (b)(z) the band edge corresponding to band b

and ∆V(b)i is the band discontinuity at interface i. The function ∆i(r) represents the deviation from

a perfect interface at different in-plane positions r. For the conduction band of an InGaAs/InAlAs

heterostructure, the sign of ∆V(b)i is positive when going from well material to barrier material

(from the left) and negative when going from barrier to well material. For the light-hole and split-

off bands, the signs are opposite (with respect to the conduction band).

The introduction of the interaction Hamiltonian in Eq. (2.49) breaks translational invariance in

the xy-plane, and the resulting density matrix will no longer be diagonal in the in-plane wavevector.

In order to get around this limitation, we follow the treatment in Refs. [69, 38, 26] and only consider

statistical properties of the interface roughness, which is contained in the correlation function

〈∆i(r)∆j(r′)〉 = δijC(|r− r′|) (2.50)

where 〈...〉 denotes an average over the macroscopic interface area, C(|r|) is the spatial autocorre-

lation function and the Kronecker delta function means that we neglect correlations between dif-

ferent interfaces. In this work, we employ a Gaussian correlation function C(|r|) = ∆2e−|r|2/Λ2 ,

characterized by standard deviation ∆ and correlation length Λ. However, calculations will be

taken as far as possible without specifying a correlation function. When calculating rates, we will

mainly use its Fourier transform,

C(|q|) =

∫d2C(|r|)e−ir·q. (2.51)

The derivation for the interface-roughness scattering rates proceeds the same way as for electron–

phonon scattering until Eq. (2.26). For the case of interface roughness, there is no phonon degree

27

of freedom to trace over and we get

DIR =

∫ ∞0

ds+HIRρe(t)e−iH0sHIRe

iH0s

−HIRe−iH0sHIRe

iH0sρe(t)+ h.c.. (2.52)

Using the completeness relation 4 times and performing the s integration gives the in-scattering

term in Eq. (2.52) becomes

DinIR =

π

~∑n1234

∫d2k1234δ[E(n4,k4)− E(n3,k3)] 〈n2,k2|ρe|n3,k3〉

× 〈n1,k1|HIR|n2,k2〉 〈n3,k3|HIR|n4,k4〉 |n1,k1〉 〈n4,k4|+ h.c.. (2.53)

Integrating over k3 gives

DinIR =

π

~∑n1234

∫d2k124δ[E(n4,k4)− E(n3,k2)]ρ

Ek2n2,n3

× 〈n1,k1|HIR|n2,k2〉 〈n3,k2|HIR|n4,k4〉 |n1,k1〉 〈n4,k4|+ h.c. (2.54)

Now let us focus on the part of the integrand containing the interaction Hamiltonian

〈n1,k1|HIR|n2,k2〉 〈n3,k2|HIR|n4,k4〉

=∑i,j

(n1|n2)zi,IR (n3|n4)∗zj ,IR

∫d2rd2r′ 〈r|k1〉∗ 〈r|k2〉 〈r′|k2〉∗ 〈r′|k4〉∆i(r)∆j(r

′), (2.55)

where we have defined

(n|m)zi,IR =∑b

∆V(b)i ψ∗n(zi)ψm(zi), (2.56)

where the sum is over the conduction (c), light hole (lh), and spin-orbit split-off bands (so). We use

a convention where the band discontinuity ∆V(b)i is positive for the conduction band. In order to

proceed, we replace the product ∆i(r)∆j(r′) with the Fourier decomposition of its spatial average

using Eqs. (2.50)-(2.51) and get

〈n1,k1|HIR|n2,k2〉 〈n3,k2|HIR|n4,k4〉

=∑i

(n1|n2)zi,IR (n3|n4)∗zi,IR1

(2π)2

∫d2qδ[k2 − (k1 − q)]δ[k4 − (k2 + q)]C(|q|) . (2.57)

28

Inserting Eq. (2.57) into Eq. (2.54) and integrating over k4 and k2, dropping the subscript index of

k1 and shifting the integration variable q→ −q + k gives

DinIR =

π

~(2π)2

∑n1234

∫d2k

∫d2qδ[Eq −∆n4,n3 − Ek]∑

i

(n1|n2)zi,IR (n3|n4)∗zi,IRC(|q− k|)ρEqn2,n3|n1,k〉 〈n4,k1|+ h.c.. (2.58)

Switching to polar coordinates d2q → qdqdθ and performing the change of variables Eq =

~2q2/(2m∗‖) gives

DinIR =

m∗‖2~3

∑n1234

∫d2k

∑i

(n1|n2)zi,IR (n3|n4)∗zi,IR

× 1

2π

∫ 2π

0

∫ ∞0

dEqδ[Eq −∆n4,n3 − Ek]

× C(|q− k|)ρEqn2,n3|n1,k〉 〈n4,k1|+ h.c.. (2.59)

Performing the Eq integration, renaming the dummy variables n3 → m and n2 → n and multiply-

ing from the left by 〈N | and from the right with |N +M〉 gives(∂fEkN,M∂t

)in

IR

=m∗‖2~3

∑n,m

u[∆N+M,m + Ek]×∑i

(N |n)zi,IR (m|N +M)∗zi,IR (2.60)

C(Ek,∆N+M,m + Ek)ρ∆N+M,m+Eknm + h.c.. (2.61)

where we have defined the angle-averaged correlation function

C(Ek, Eq) =1

2π

∫ 2π

0

dθC

√2m∗‖~2

√Ek + Eq − 2

√EkEq cos(θ))

. (2.62)

Shifting the sum variables n→ N + n, m→ N +M +m and using Eq. (2.40), we can write(∂fEkN,M∂t

)in

IR

=m∗‖2~3

∑n,m

u[∆N+M,N+M+m + Ek]

×∑i

(N |N + n)zi,IR (N +M +m|N +M)∗zi,IR

× C(Ek,∆N+M,N+M+m + Ek)f∆N+M,N+M+m+EkN+n,M+m−n + h.c.

= 2∑n,m

Γin,IR,±NMnmEk

f∆N+M,N+M+m+EkN+n,M+m−n + h.c., (2.63)

29

where we have defined Γin,IR,±NMnmEk

, the in-scattering rate due to interface roughness. The purpose

of the factor of two in the definition of the rate is so we can write(∂fEkN,M∂t

)in

IR

=∑n,m,±

Γin,IR,±NMnmEk

f∆N+M,N+M+m+EkN+n,M+m−n + h.c.. (2.64)

In other words, we have split the interface roughness scattering into two identical absorption and

emission terms to be consistent with the notation for phonon scattering in Section 2.2.2. In order

to calculate the rates, we need to choose a correlation function. The rates are calculated in Ap-

pendix A for a Gaussian correlation function, which is often used to model QCLs [3, 26]. The

reason we choose a Gaussian correlation function is that it results in a simple analytical expres-

sion for the angle-averaged correlation function C, given by Eq. (2.62). Another popular choice

is an exponential correlation function [38, 40, 26]. However, angle-averaged expressions for the

correlation function are more complicated, involving elliptical integrals [38, 26].

2.2.3.2 Ionized Impurities

The effects of the long-range Coulomb interaction from ionized impurities is already included

to lowest order with a mean-field treatment (Poisson’s equation). The goal of this section is to in-

clude higher order effects by including scattering of electrons with ionized impurities. We assume

a 3D doping density of the form

N(z) = N2Dδ(z − z`) , (2.65)

which is a delta-doped sheet at cross-plane position z`. The advantage of Eq. (2.65) is that an

arbitrary doping density can be approximated by a linear combination of closely spaced delta-

doped sheets. Typically, in QCLs, only one [4] or a few [70, 2] layers are doped and the doping

is kept away from the active region to minimize nonradiative transitions from the upper to lower

lasing level. Note that, in this context, a layer refers to a barrier or a well, not atomic layers. The

corresponding scattering rates due to the ionized-impurity scattering distribution in Eq. (2.65) is

calculated in Appendix A.4.

30

2.2.3.3 Random Alloy Scattering

In ternary semiconductor alloys AxB1−xC, the energy of band b can be calculated using the

virtual crystal approximation [68], where the alloy band energy is obtained by averaging over the

band profiles of the two materials AC and BC

Vb,VCA(R) =∑n

[xVb,AC(R−Rn) + (1− x)Vb,BC(R−Rn)] , (2.66)

where the sum is over all unit cells n and Vb,AC(R−Rn) and Vb,BC(R−Rn) are the corresponding

bands of the AC and BC material respectively. The difference between an alloy and a binary

compound is that, in alloys, periodicity of the crystal is broken by the random distribution of AC

and BC unit cells. This deviation from a perfect lattice can be modeled as a scattering potential,

which can be considered much weaker than Eq. (2.66). The scattering potential is given by the

difference between the real band energy and the band energy obtained using the virtual crystal

approximation. We will follow the treatment of alloy scattering in Refs. [71, 40, 26] and write the

scattering potential (which we assume is the same for all bands) as

Valloy(R) =∑n

Cxnδ(R−Rn), (2.67)

where the sum is over all unit cells and Cxn is a random variable, analogous to the interface-

roughness correlation function in Eq. (2.50), with

〈Cxn〉 = 0

〈CxnC

xm〉 = δnmxn(1− xn)(VnΩ)2,

(2.68)

where Ω = a3/4 is equal to one-fourth of the volume of a unit cell and Vn the alloy scattering

potential in unit cell n, which can be approximated as the conduction band offset of the two binary

materials, corresponding to x = 0 and x = 1. The presence of the Kronecker delta function

δnm means that the location of A and B atoms are assumed to be completely uncorrelated. The

derivation of corresponding scattering rates proceeds in the same way as for interface roughness

scattering and results are given in Appendix A. The final expressions for the rates are considerably

simpler than the ones due to interface roughness due to the local nature of the interaction potential

in Eq. (2.67).

31

2.2.4 Steady-State Observables

Once the steady-state density matrix elements have been calculated from Eq. (2.2), the expec-

tation value of any observable A can be calculated using

〈A〉 =Ns∑N=1

Nc∑M=−Nc

∫dEk 〈N |A|N +M〉 fEkN,M , (2.69)

where Ns is the number of eigenstates per period and Nc is a coherence cutoff (see further discus-

sion in Sec. 2.3.1). The observable A can depend on position z, any derivative of position ∂n

∂zn, as

well as in-plane energy. The most important steady-state observable is the drift velocity v, which is

used to calculate the expectation value of current density J = qn3D 〈v〉, where n3D is the average

3D electron density in the device

n3D =1

LP

∫ LP

0

ND(z)dz, (2.70)

with LP the period length of the QCL and ND(z) the doping density. Matrix elements of the drift

velocity operator are obtained from the time derivative of the position operator using

〈n|v|m〉 =

⟨n

∣∣∣∣dzdt∣∣∣∣m⟩ =

i

~〈n| [H, z] |m〉 =

i

~〈n| [H0, z] |m〉

=i

~(En − Em) 〈n|z|m〉 =

i

~(En − Em)zn,m,

(2.71)

where we have used 〈n| [Hi, z] |m〉 = 0 because the interaction potentials in this work only contain

the position operator z, but not momentum pz. The steady-state current density can be written as

J = qn3D

Ns∑N=1

Nc∑M=−Nc

∫dEkvN,N+Mf

EkN,M . (2.72)

In this work, electron–electron interaction is treated on a mean-field level by obtaining a self-

consistent solution with Poisson’s equation (see further discussion Sec. 2.3.1). We therefore need

to calculate the spatially resolved electron density nel(z) using

nel(z) = n3D

Ns∑N=1

Nc∑M=−Nc

∫dEk

∑b

[ψ

(b)N (z)

]∗ψ

(b)N+M(z)fEkN,M , (2.73)

32

where the sum involving b is over the considered bands. We define the occupation of state N using

fN =

∫dEkf

EkN,0 (2.74)

and the average coherence between states N and M as

ρN,M =

∫dEkρ

EkN,M =

∫dEkf

EkN,M−N . (2.75)

The magnitude of the coherences relative to the occupations give us an idea how important is it to

include off-diagonal elements of the density matrix and when semiclassical treatment is sufficient.

In order to quantify the amount of electron heating that takes place under operating conditions,

we define the expectation value of in-plane energy using

〈Ek〉 =Ns∑N=1

∫dEkEkf

EkN,0 (2.76)

and the expectation value of in-plane energy of a single subband using

〈Ek〉N =

∫dEkEkf

EkN,0∫

dEkfEkN,0

. (2.77)

For low bias, we have 〈Ek〉N ' kBT , where T is the lattice temperature. However, as will be

shown in Section 2.4.4, for electric fields around and above threshold, 〈Ek〉N can greatly exceed

kBT at lattice temperature and differ substantially between subbands. It is possible to define elec-

tronic temperature of state N using TN = 〈Ek〉N /kB. The electron temperature can be a useful

quantity because it is an input parameter for simplified DM models, where a thermal in-plane dis-

tribution is assumed [37, 3]. However, we note that the in-plane energy distribution under typical

operational electric fields can deviate strongly from a Maxwellian. The term electron temperature

should therefore only be interpreted as the temperature equivalent of the average in-plane energy.

2.2.5 Optical Gain

The z-component of an electric field propagating in the x-direction, with frequency ω and

amplitude E0, can be written as a plane wave of the form [72]

Ez(x, ω) = E0einBωx/ce−α(ω)x/2, (2.78)

33

where c is the speed of light in vacuum, nB is the background index of refraction (which we

assume is frequency independent in the frequency range of interest in this work), and α(ω) is the

absorption coefficient. We note that the electric field in typical QCL waveguides (operating in a

TM mode) also has a nonzero x-component [26]. However, only the dipole matrix element in the

z direction are nonzero, so the x-component is not amplified.

Optical gain is defined as g(ω) = −α(ω) and can be obtained from the complex susceptibility

using [72]

g(ω) = − ω

cnBIm[χ(ω)]. (2.79)

The basic idea of a laser is to create a region of positive gain (negative absorption) so that the

power P of an electromagnetic field propagating through the gain medium increases according to

P ∝ |Ez(x, ω)|2 ∝ eg(ω)x.

In this work, we follow Ref. [38] and treat the optical field as a small perturbation and consider

the linear response of the steady state. We assume the wavelength of the optical field is long

and ignore the positional dependence of the electric field across the QCL. We can then write the

perturbation potential as

Vω(z) = qE0ze−iωt . (2.80)

Using linear response theory, the density matrix can be written as

ρ(t) = ρ0 + ρωe−iωt, (2.81)

where ρ0 is the steady-state solution for E0 = 0. The equation of motion for ρ(t) can be written as

i~∂ρ

∂t= [H, ρ] = [H0 + Vω, ρ0 + ρω] +D[ρ0 + ρω]

= [H0, ρ0] +D[ρ0] + [Hω, ρω] + [H0, ρω] + [Vω, ρ0] +D[ρω]

' [H0, ρω] + [Vω, ρ0] , (2.82)

where the first two terms in the second line of Eq. (2.82) add up to zero in steady state and the third

term is proportional to e−2iωt (second harmonic). The second-harmonic term is a product of two

34

small quantities (Vω and ρω) and is ignored. Combining Eqs. (2.80)-(2.82) and dividing both sides

by e−iωt gives an Equation for ρω

~ωρω = [H0, ρω] +D[ρω] + eE0 [z, ρ0] . (2.83)

In the same way we defined the matrix elements fEkNM , we can define the matrix elements of ρω in

relative coordinates as fEkNM(ω) = ρEkN,N+M(ω). Equation (2.83) gives the following equation for

the matrix elements

ρEkN,M(ω)

[i∆N,N+M

~− iω

]= AEkN,M +DEkN,M(ω), (2.84)

where DEkN,M(ω) are matrix elements of the dissipator acting on the linear-response density matrix

and

AEkN,M = −ieE0

~∑n

zN,N+nfEkN+n,M−n + h.c., (2.85)

where zn,m is a matrix elements of the z-position operator. Using Eqs. (2.84) and (2.85), we can

calculate fEkNM(ω) for relevant values of ω. Note that the dissipator term depends on many of the

matrix elements ρEkN,M(ω), so solving for ρEkN,M(ω) is not as trivial as it looks in Eq. (2.84). Our

procedure for solving Eq. (2.84) is given in Sec. 2.3.

Once fEkNM(ω) is known, we can calculate the field-induced polarization using

p(ω) = Tr [dρω] = e

∫dEk

∑N,M

zN,N+MfEkN,M(ω) , (2.86)

where d is the dipole operator d = qz. Note that the density matrix is normalized to one, so

p(ω) gives the induced polarization per electron. To get the polarization per unit volume, we use

P(ω) = n3Dp(ω), where n3D is the average electron density. The polarization is then used to

calculate the complex susceptibility χ(ω) using [72, 38]

χ(ω) =P(ω)

ε0E0

. (2.87)

Alternatively, the complex susceptibility can be calculated using the induced current density, which

is the time derivative of the polarization J(ω) = −iωP , giving

χ(ω) =i

ω

J(ω)

ε0E0

. (2.88)

35

We have calculated the susceptibility using both Eq. (2.87) and Eq. (2.88), and both expressions

give the same result (within numerical accuracy). Once the complex susceptibility has been calcu-

lated, the gain is obtained using Eq. (2.79).

2.2.6 Comparison With Semiclassical Models

Equation (2.2) can be thought of as a generalization of the semiclassical Pauli master equation,

with rates calculated using Fermi’s golden rule [32, 26]. This fact can be seen by looking at the

diagonal terms (M = 0) and limiting the out-scattering sum to n = 0, and the in-scattering sum to

n = m, giving

∂fEkN,0∂t

= −2∑m,g,±

Γout,g,±N,0,0,m,Ek

fEkN,0 + 2∑n,g,±

Γin,g,±N,0,n,n,Ek

fEk+∆N,N+n∓EsN+n,0 . (2.89)

By defining

fEkN = fEkN,0,

Γout,g,±N,n,Ek

= 2Γout,g,±N,0,0,n,Ek

,

Γin,g,±N,n,Ek

= 2Γin,g,±N,0,n,n,Ek

,

(2.90)

we can write Eq. (2.89)

∂fEkN∂t

= −∑n,g,±

Γout,g,±N,n,Ek

fEkN +∑n,g,±

Γin,g,±N,n,Ek

fEk+∆N,N+n∓EsN+n , (2.91)

which is equivalent to the Boltzmann equation for the occupation of state n with in-plane energy

Ek (See for example Equation (129) in Ref. [26]). The factor of 2 in front of the rates comes

from the addition of the Hermitian conjugate (which is identical for the diagonal of the density

matrix). Because of the similarities between the semiclassical Boltzmann equation in (2.91) and

the quantum mechanical results in Eq. (2.2), we can easily compare results obtained using both

models and investigate when semiclassical models give satisfactory results and when off-diagonal

density matrix elements must be included.

When calculating current density using semiclassical models, we cannot directly use Eq. (2.72)

because matrix elements of the velocity operator given in Eq. (2.71) are zero for n = m (in other

36

words, all the current is given by off-diagonal matrix elements). In order to calculate drift velocity,

we first define

ΓoutNnEk

=∑g,±

Γout,g,±N,n,Ek

, (2.92)

which gives the total rate at which an electron in subband N with in-plane energy Ek is scattered

to subband N + n. We then define the semiclassical drift velocity as

vc =

∫dEk

Ns∑N=1

Nc∑n=−Nc

(zN − zN+n)fEkN ΓoutNnEk

, (2.93)

with zN = 〈ψN |z|ψN〉. The current density is then calculated using Jc = qn3Dvc. Equation (2.93)

can be interpreted as summing up the flux of electrons from position zN to zN+n. This intuitive

expression for vc can be derived from Eq. (2.72) by approximating the off-diagonal density matrix

elements using diagonal ones [73].

If the optical frequency ω is much larger than typical scattering rates (due to phonons, impuri-

ties etc.), the gain can be calculated from the diagonal density matrix elements using [74, 65, 26]

g(ω) =q2ωm∗‖

~3cε0nBLP

Ns∑N=1

Nc∑M=−Nc

∆N,N+M

|∆N,N+M ||zN,N+M |2

∫dEk(f

EkN − f

EkN+M)LN,N+M(Ek, ω),

(2.94)

with the Lorentzian broadening function

Ln,m(Ek, ω) =γn,m(Ek)

γ2n,m(Ek) + [ω −∆nm/~]2

. (2.95)

The broadening of a transition γn,m(EK) is calculated using [26]

γn,m(Ek) =1

2[γn(Ek) + γm(Ek)] , (2.96)

where γn(Ek) is the total intersubband scattering rate from subband n at energy Ek, calculated

using

γn(Ek) =∑n6=0

ΓoutNnEk

. (2.97)

37

2.3 Numerical Method

2.3.1 Steady-State Density Matrix

The central quantities of interest are the matrix elements of the steady-state density matrix

ρEkN,N+M = fEkN,M . Because of periodicity, we have fEkN,M = fEkN+Ns,M, where Ns is the number of

states in a single period. The index M strictly runs from −∞ to +∞ but is truncated according

to |M | ≤ Nc, where Nc is a coherence cutoff. The coherence cutoff needs to be large enough

such that overlaps between states |N〉 and |N +Nc〉 is small for all N . Typical values for Nc are

slightly larger than Ns, however, the specific choice of Nc is highly system dependent. We also

have to discretize the continuous in-plane energyEk intoNE evenly spaced values. This truncation

scheme results in Ns(2Nc + 1)NE matrix elements.

In order to calculate the steady-state matrix elements fEkN,M , we set the time derivative in

Eq. (2.2) to zero and get

0 = −i∆N,N+M

~fEkN,M

+

[−∑

n,m,g,±

Γout,g±NMnmEk

fEkN,n +∑

n,m,g,±

Γin,g,±NMnmEk

fEk∓E0+∆N+M,N+M,m

N+n,M+m−n + h.c.

]. (2.98)

Note that the sums in the Eq. (2.98) run over matrix elements and energies outside the center period

(e.g., N > Ns or N < 1), which are calculated using periodicity with the modulo operation

fEkN,M = fEkN ′,M

EN = EN ′ +N −N ′

Ns

E0 (2.99)

N ′ = mod(N − 1, Ns) + 1 ,

with mod(n,Ns) = n−Nsbn/Nsc and E0 the potential energy drop over a single period (intrinsic

function MOD in Matlab and MODULUS in gfortran). We also encounter terms for which |M | >

Nc, where we assume fEkN,M = 0. The exact values of Ns, Nc, and NE are very system specific,

for example, in order to simulate the QCL considered in Sec. 2.4, we use Ns = 8, Nc = 12 and

NE = 176 resulting in 35, 200 matrix elements. This high number of variables makes it impractical

38

to solve Eq. (2.98) as a matrix equation, so we will resort to iterative methods [75]. In order to

solve Eq. (2.98) iteratively, we need to write fEkN,M in terms of all other terms fE′k

N ′,M ′ for N ′ 6= N ,

M ′ 6= M , and E ′k 6= Ek. From the dissipator term in (2.98), we see that the terms containing fEkN,M

in the out-scattering part are the n = M terms. As for the in-scattering part, only the elastic terms

(E0 = 0) with n = m = 0 contain fEkN,M . Note that the Hermitian conjugate part also contains

terms proportional to fEkN,M , which are obtained using fEkN,M = [fEkN+M,−M ]∗, which is equivalent to

ρEkN,M = [ρEkM,N ]∗. It is convenient to define the reduced dissipator DEkN,M , where the terms containing

fEkN,M are excluded. Summing up all the terms multiplying fEkN,M gives

γNMEk = −∑m,g,±

[Γout,g±N,M,M,m,Ek

+ Γout,g±N+M,−M,−M,m,Ek

]+∑

E0=0,±

[Γin,g,±N,M,0,0,Ek

+ Γin,g,±N+M,−M,0,0,Ek

], (2.100)

where E0 = 0 represents summing over all elastic scattering mechanisms. Using Eq. (2.100), we

can write the reduced dissipator as

DEkN,M = DEkN,M − γNMEkfEkNM , (2.101)

which can depend on all matrix elements of the density matrix except for fEkNM . Using Eqs. (2.101)-

(2.100), we can write Eq. (2.98) as

fEkN,M =DEkNM

i~∆N,N+M − γN,M,Ek

, (2.102)

where the RHS does not depend on fEkN,M . Equation (2.102) is then used as a basis for an iterative

solution for fEkNM using a weighted Jacobi scheme [75]

fEk,(j)NM =

DEk,(j−1)NM − γNMEkf

Ek,(j−1)NM

i~∆N,N+M − γN,M,Ek

w + (1− w)fEk,(j−1)NM , (2.103)

where w ∈ [0, 1] is a weight. The quantity DEk,(j)NM should be understood as matrix elements of

the dissipator acting on the j-th iteration of the density matrix. As an initial guess, we assume a

diagonal density matrix, where all subbands are equally occupied with a thermal distribution

fEk,(1)NM = AδM,0e

−Ek/(kBT ), (2.104)

39

where A is normalization constant. In the current work, we used w = 0.2. However, we did not

find convergence speed to depend strongly on the specific value. We iterate Eq. (2.103) until the

follow convergence criteria is satisfied∑N,M

∫dEk|fEk,(j)NM − fEk,(j−1)

NM | < δiter (2.105)

where we used δiter = 10−3 in this work. Note that the density matrix is normalized to one, making

δiter unitless. Typically, 20− 100 iterations are needed to reach the convergence criteria.

In order to include corrections due to electron–electron interaction on a mean-field level, the

iterative solution method described above is supplemented with a self-consistent solution with

Poisson’s equation. The solutions to Poisson’s equation is obtained using an outer self-consistency

loop (the inner loop is the iterative solution described earlier). At the end of the inner (iterative)

loop, corrections to the bandstructure VP (z) are calculated based on a self-consistent solution with

Poisson’s equation. If the change in VP (z) is above a certain threshold, the eigenfunctions and rates

are recalculated and the inner iterative loop is repeated with the new eigenfunctions and rates. The

process is repeated until VP (z) converges. To quantify when convergence is reached, we define the

largest first-order correction to the eigenfunctions

δself = maxm

∣∣∣∣∫ dz|ψm(z)|2[V newP (z)− V old

P (z)]∣∣∣∣ , (2.106)

where |ψm(z)|2 is the probability density of eigenfunction m. In this work we use δself = 0.1 meV,

which is small compared with a typical energy spacing in QCLs. The number of iterations needed

to satisfy the convergence criterion in Eq. (2.106) depends on the electron density in the device

(higher density means more iterations). For the device considered in Section. 2.4, 2− 5 iterations

are needed for the first value of applied electric field. For other values of electric field, the number

of iteration needed can be kept to a minimum by sweeping the field and using results from previous

fields as the initial guess for the next value of the electric field.

2.3.2 Linear-Response Density Matrix

Once we have calculated the steady-state density matrix using the procedure described in

Sec. 2.3.1, we can calculate the linear-response density matrix elements fEkNM(ω) using Eqs. (2.84)

40

and (2.85). By following the same steps as in Section 2.3.1, we get

fEk,(j)NM (ω) =

DEk,(n−1)NM (ω)− γNMEkf

Ek,(j−1)NM (ω) + A

Ek,(j−1)N,M

i~(∆N,N+M − ~ω)− γNMEk

w + (1− w)fEk,(n−1)NM (ω). (2.107)

where the matrix elements AEk,(j)N,M only depend on the steady-state density matrix and are given by

Eq. (2.85). Note that the matrix elements DEk,(j)NM (ω) cannot be calculated directly using Eq. (2.3)

because Fourier components of the linear-reponse density matrix are not Hermitian and the result-

ing dissipator cannot be written as a pair of Hermitian conjugate terms. The calculations will not

be shown here but the result is

DEkNM(ω) = −∑

n,m,g,±

Γout,g±NMnmEk

fEkN,n(ω) +∑

n,m,g,±

Γin,g,±NMnmEk

fEk∓Es+∆N+M,N+M+m

N+n,M+m−n (ω)

−∑

n,m,g,±

Γout,g±N+M,−M,n,m,Ek

fEkN+M+n,−n(ω) +∑

n,m,g,±

Γin,g,±N+M,−M,n,m,Ek

fEk∓Es+∆N,N+m

N+m,M−m+n (ω), (2.108)

which would reduce to Eq. (2.3) if the linear response DM were Hermitian. Equation (2.107) is

iterated until

|g(n)(ω)− g(n−1)(ω)| < 10−3 cm−1, (2.109)

where g(n)(ω) is the gain calculated from fEk,(n)NM (ω), using Eq. (2.87). This process is repeated for

multiple values of ω in the frequency range of interest.

2.3.3 Low-Energy Thermalization

Out of the included scattering mechanisms only polar-optical-phonon (POP) and acoustic-

phonon (AP) scattering are inelastic. However, at low temperatures, AP scattering is weak and the

POP energy is typically larger than kBT and this lack of a low-energy inelastic scattering mech-

anism leads to numerical difficulties, where in-plane energy distributions can vary abruptly (this

problem is often encountered in density-matrix models; see, for example, Ref. [36]). A detailed

inclusion of electron–electron interaction would solve this issue, where arbitrarily low energy can

be exchanged between electrons. The small energy exchanges involved in electron–electron in-

teraction also play a crucial role in thermalization within a subband. However, electron-electron

41

interaction is a two-body interaction that is not straightforward to include in a single-electron pic-

ture. For this reason, in the present work we will include low-energy thermalization (LET) in a

simplified manner, by adding a scattering mechanism with an energy equal to the minimal in-plane

energy spacing ∆E in the simulation. The purpose of this extra scattering mechanism is to help

smoothen the in-plane energy distribution. We treated the LET as an additional POP-like scattering

term, with energy exchange equal to ∆E and an effective strength denoted by the dimensionless

quantity α. The matrix element is

|MLET(Q)|2 = αe2∆E

2ε0

(1

ε∞r− 1

εr

)Q2

(Q2 +Q2D)2

, (2.110)

where Q2D = ne2/(εkBT ) is the Debye wave vector. The reason for the choice a POP-like matrix

element is its preference for small-Q scattering, just like electron–electron interaction. The role of

the LET term is mainly to smoothen the in-plane energy distribution. As we will show in Sec. 2.5,

the results are not very sensitive to the value of α. This LET term is employed in Sec. 2.5, where

we model a terahertz QCL for a lattice temperature of 50 K. However it is not needed in Sec. 2.4,

where we model a mid-IR QCL at room temperature.

2.3.4 Time-Dependent Calculations

The iterative solution method described in Sec. 2.3.1 is simple and quickly gives the steady-

state density matrix. However, it does not give us any information on the time evolution which is

needed to model the initial transient when the electron field is turned on. Another approach to solve

Eq. (2.2) is to start with a thermal-equilibrium density matrix at t = 0 and time-step Equation (2.2)

until a steady state is reached. The main drawback to this approach is that many more time steps

are needed (∼ 105) to reach steady a state compared with the number of iterations (∼ 50).

In this chapter, all time-dependent results are obtained using a one-band Hamiltonian in (2.6),

which can be written as(−~2

2

d

dz

1

m(z)

d

dz+ VSL(z) + VB(z) + VH(z, t)

)ψn(z, t)

= En(t)ψn(z, t) ,

(2.111)

42

where m∗(z) is a position-dependent effective mass, VSL is the superlattice potential (wells and

barriers), VB the linear potential drop due to an applied bias, and VH the mean-field Hartree poten-

tial, which is obtained by solving Poisson’s equation. Upon the application of bias, we assume the

field and the associated linear potential drop are established instantaneously. The Hartree potential

VH depends on the electron density and is therefore time-dependent. However, its time evolution is

weak owing to low doping and is typically very slow, so we can assume that the adiabatic approx-

imation holds and the concept of eigenstates and energies is well defined during the transient. The

eigenfunctions are calculated in the same way as described in Sec. 2.2.1, but with only one band.

As the system evolves, the eigenfunctions and rates have to be recalculated, which is computa-

tionally expensive, taking about 10 to 50 times longer than a single time step. However, it does not

need to be done in every time step owing to the slow temporal and spatial variation of VH. In order

to recalculate the eigenfunctions only when needed, we calculate

δ = maxm

∣∣∣∣∫ dz|ψm(z, t`)|2(VH(z, t`)− VH(z, t`−1))

∣∣∣∣ , (2.112)

where t` is the time at the current time step `. The quantity δ is the magnitude of the maximal first-

order energy correction to the eigenstates. If δ is above a certain threshold energy, we recalculate

the eigenfunctions and the corresponding rates. If the threshold is not met, we do not update the

eigenfunctions nor the Hartree potential. The procedure of calculating δ is very cheap in terms of

computational resources and does not noticeably affect performance. Typically, the eigenfunctions

are recalculated frequently during the initial transient and much less frequently near the steady

state. In the present work, we used a threshold energy of δ = 0.1 meV. This choice of threshold

energy typically leads to ∼ 100 recalculations of eigenfunctions while the total number of time

steps is on the order of 105.

When recalculating the eigenfunctions, an issue arises when numbering the updated states and

choosing their phase. The time evolution of the eigenfunctions must be adiabatic so the same phase

must be chosen for each state when the eigenfunctions are recalculated. Since the eigenfunctions

are chosen to be real, there are only two choices of phase. A very simple assigning method is to

43

calculate

αnm =

∫dzψn(z, t`)ψm(z, t`−1) . (2.113)

States with the highest overlap |αnm| ' 1 are “matched” according to n → m, which ensures the

proper numbering of the new states and ψn(z, t`) → sign(αnm)ψn(z, t`) takes care of the choice

of phase.

We choose an initial state corresponding to thermal equilibrium. Assuming Boltzmann statis-

tics, the density matrix factors into the in-plane and cross-plane terms and we can write

fEkNM

∣∣∣eq

= CNMe−Ek/kBT . (2.114)

To calculate the expansion coefficients CNM , we first solve for the Bloch states φs,q(z) (s labels

the band and q ∈ [−π/Lp, π/Lp] labels the wave vector in the Brillouin zone associated with the

structure’s period Lp) by diagonalizing the Hamiltonian in (2.111) with VB(z) = 0, using a basis

of plane waves. We can then calculate the cross-plane equilibrium density matrix using

ρeq(z1, z2) =∑s

∫ π/Lp

−π/Lpφs,q(z1)φ∗s,q(z2)e−Es,q/kBTdq . (2.115)

Using the above result, we can calculate the expansion coefficients

CNM =

∫dz1dz2ψN(z1)ψN+M(z2)ρeq(z1, z2) . (2.116)

This choice of initial condition works well with an electric field that is turned on instantaneously

at time t = 0+; this is the limiting case of an abruptly turned-on bias. If only the steady state

is sought, all terms with M 6= 0 can be artificially set equal to zero in the initial density matrix;

this initial condition avoids high-amplitude coherent oscillations during the transient and leads to

a faster numerical convergence towards the steady state. However, we note that if only the steady-

state is sought, the iterative scheme presented in Sec. 2.3.1 is much faster.

44

2.4 Steady-State Results for a Mid-IR QCL

In order to verify the accuracy of our model, we simulated a mid-IR QCL proposed in Ref. [2].

We chose this device because it has previously been successfully modeled using both NEGF and

simplified density matrix approaches [3], allowing us to compare our results to their models. The

QCL was designed for wide voltage tuning, emitting around 8.5 µm (146 meV), with a tuning

range of almost 100 cm−1 (12 meV). The device is based on the In0.47Ga0.53As/In0.52Al0.48As

material system, which is lattice-matched to InP. With this material composition, In0.47Ga0.53As

acts as potential wells and In0.52Al0.48As as barriers with approximately a 520 meV conduction-

band discontinuity between materials. All results in this Section are obtained using the 3-band

k · p model described in Section 2.2.1. All material parameters relating to band-structure (e.g.,

Kane energy and band offsets) are calculated using the procedure described in Ref. [1] and given

in Table 2.1.

All other parameters such as LO-phonon energies and relative permittivity are given in Ta-

ble 2.2. The parameters (except those relating to interface roughness) are obtained by a weighted

average of bulk values, with the weight being the total layer thickness of a material in a pe-

riod, divided by the total period length. We used the same interface-roughness correlation length

Λ = 9 nm as in Ref. [3, 39]. We estimate the RMS value for interface roughness (∆) by using

the fact that interface-roughness scattering is the main broadening mechanism of the gain spec-

trum [11]. We follow the procedure in Ref. [39] and fix the correlation length at 9 nm and vary ∆

such that the gain spectra produces a peak with a FWHM of 25 meV, matching experimental results

for electroluminescence spectra at a field strength of 53 kV/cm [2]. We note that the resulting value

of ∆ = 0.07 nm is considerably smaller than the value of ∆ = 0.1 nm used in Ref. [3] used to

simulate the same device. We attribute the difference to the different electron Hamiltonian used in

this work (3-band k · p), resulting in different amplitude of eigenfunctions at material interfaces.

45

Table 2.1: Parameters relating to bandstructure (see Section 2.2.1) for both In0.47Ga0.53As and

In0.52Al0.48As. All values are obtained using the procedure described in Ref. [1], and are given for

a lattice temperature of 300 K. Parameters for which no units are given are unitless.

Parameter In0.47Ga0.53As In0.52Al0.48As

Egap [eV] 0.737 1.451

Ev [eV] −0.594 −0.785

EP [eV] 25.30 22.51

∆so [eV] 0.330 0.300

γL1 11.01 6.18

γL2 4.18 1.89

F −2.890 −0.630

Table 2.2: Various parameters used for the In0.47Ga0.53As/In0.52Al0.48As material system.

LO-phonon energy 36 meV

relative dielectric constant (low frequency) 13.5

relative dielectric constant (high frequency) 11.2

acoustic deformation potential 7 eV

mean density 5.24 g/cm3

sound velocity 4.41 km/s

interface roughness correlation length 9 nm

interface roughness RMS 0.07 nm

46

2.4.1 Bandstructure

Figure 2.4 shows the conduction-band diagram and the probability density of the relevant

eigenfunctions slightly above threshold (52 kV/cm). The figure shows the 8 considered states

belonging to a single period in bold and states belonging to neighboring periods are represented

as thin curves. The lasing transition is between states 8, and 7 and state 6 is the main extractor

state. State number 1 is the energetically lowest state in the injector (often denoted as the ground

state). In order to verify the accuracy of the bandstructure shown in Fig. 2.4, we can compare

various calculated energy differences ∆n,m to results obtained from experimentally determined

electroluminescence spectra [2]. Figure 2.5 shows the calculated energy differences ∆8,7, ∆8,6

and ∆8,5 as a function of electric field and a comparison to experimentally obtained values. From

Fig. 2.5 we can see that for low field strength (41.6 and 45.9 kV/cm), the deviation from experi-

ment is 2–8 meV, which is less than a 5% relative difference. For higher field strengths (54.7 and

62.7 kV/cm), all energy differences are overestimated by 5–20 meV, with a maximum relative dif-

ference of 11% for ∆7,5 at a field strength of 62.7 kV/cm. The reason for the discrepancy between

experiment and theory at high fields is not clear. A possible reason is the low bandgap of the well

material (∼ 0.74 meV) compared with the lasing transition at higher fields (∼ 0.15 eV), limiting

the accuracy of the 3-band k · p model.

2.4.2 Current Density

Figure 2.6 shows results for current density vs electric field calculated using the density-matrix

formalism in this work, along with a comparison with two different experiments and theoretical

results based on the NEGF formalism. Note that neither theoretical result takes the laser field

into account, so we can only expect results to agree with experiment below and around thresh-

old. The two experimental results are based on the same QCL-core design but with different

waveguide designs. Experiment 1 (E1) refers to the sample from Ref. [2], which is based on a

buried-heterostructure waveguide design. Experiment 2 (E2) refers to a sample based on a double-

trench waveguide design with higher estimated losses than experiment 1 [3]. From Figure 2.6, we

can see that our results based on the density-matrix formalism shows excellent agreement with the

47

Figure 2.4: Conduction-band edge (thin curve bounding gray area) and probability densities for

the 8 eigenstates used in calculations (bold curves). States belonging to neighboring periods are

denoted by thin gray curves. The states are numbered in increasing order of energy, starting with

the ground state in the injector. Using this convention, the lasing transition is from state 8 to state

7. The length of one period is 44.9 nm, with a layer structure (in nanometers), starting with the in-

jector barrier (centered at the origin) 4.0/1.8/0.8/5.3/1.0/4.8/1.1/4.3/1.4/3.6/1.7/3.3/2.4/3.1/3.4/2.9,

with barriers denoted bold. Underlined layers are doped to 1.2×1017 cm−3, resulting in an average

charge density of n3D = 1.74× 1016 cm−3 over the stage.

NEGF results for all considered fields strengths. Both NEGF and DM models are in quantitative

agreement with experiments up to about 44 kV/cm and qualitative agreement up to threshold. For

fields between 44 and 52 kV/cm, both theoretical models underestimate current density by about

0.2 kA/cm2. This field range is below threshold for E2 and therefore cannot be attributed to a lack

of inclusion of the laser field. This difference between theory and experiment can be explained by

the overestimation of the optical transition energy ∆8,7 (see Fig. 2.5), which shifts the J–E curve

48

Figure 2.5: Calculated energy differences ∆8,7 (lowest red curve), ∆8,6 (middle green curve) and

∆8,5 (top blue curve) as a function of field. The symbols show results based on experimentally

determined electroluminescence spectra from Ref. [2].

to higher fields because a higher field is needed to bring the upper lasing level into alignment with

the states localized on the left side of the injection barrier.

Figure 2.7 shows a comparison between the current density obtained using the full density

matrix and results obtained using only the diagonal matrix elements (semiclassical results). The

importance of including off-diagonal matrix elements is known to be greater for thicker injection

barriers, where transport is limited by tunneling through the injection barrier [43]. For this reason,

results are also plotted for different injection barrier widths in Figs. 2.7(b) and (c). Figure 2.7(a)

shows the results for the same device as in Fig. 2.6 (4-nm-thick injection barrier), while (b) and

(c) show the results for a thicker (5 nm) and thinner (3 nm) injection barrier, respectively (but

otherwise identical devices). In Fig. 2.7(a), we see that even for a mid-IR QCL, where transport

49

Figure 2.6: Calculated current density vs field based on this work and comparison to results using

NEGF [3]. Also shown are experimental results from Ref. [2] (E1) and a regrown device from

Ref. [3] (E2). Experimentally determined threshold fields of 48 kV/cm (E1) and 52 kV/cm (E2)

are denoted by dashed vertical lines. Both theoretical results are given for a lattice temperature

of 300 K. Experimental results are provided for pulsed operation, minimizing the effect of lattice

heating above the heatsink temperature of 300 K.

has previously been reported to be mostly incoherent [30], the difference is noticeable and the cur-

rent density is overestimated by about 13–37% by semiclassical methods. The relative difference

around threshold (∼50 kV/cm) is about 20%. This is a much bigger difference than the previously

reported value of only a few percent for a different mid-IR QCL design [30]. The difference could

be a result of the different device design, or the assumption of a factorized (in-plane and cross-

plane) density matrix employed in Ref. [30], while we do not make that assumption. Figure 2.7(b)

shows that, for a thicker injection barrier (5 nm), a semiclassical model fails completely, with a

50

relative difference ranging between 20 and 97% and about 50% difference around threshold. Fig-

ure 2.7(c) shows that, for the thinner barrier (3 nm), the difference is small (between 8 and 31%)

and about 12% around threshold. We can see that the considered device design with a 4-nm-thick

injection barrier is on the border of admitting a semiclassical description.

51

Figure 2.7: Comparison between the current density obtained using the full density matrix (solid

blue curve) and only including diagonal elements (short-dash red curve). The relative difference

between the two results is also shown (long-dash black curve). (a) Results for the same device

considered in Fig. 2.6. (b) Results for a device with a thicker injection barrier (5 nm). (c) Results

for a device with a thinner injection barrier (3 nm).

2.4.3 Optical Gain and Threshold Current Density

Figure 2.8(a) shows the calculated peak gain as a function of electric field, both with and

without inclusion of the off-diagonal density-matrix (DM) elements. Also shown is a comparison

52

with the results based on NEGF and a simplified density-matrix model, referred to as DM 2nd in

Ref. [3], where a constant electron temperature is assumed. The estimated threshold gain Gth '

10 cm−1 [2, 3] from experiment E1 is denoted by a horizontal dashed line. From Fig. 2.8(a), we see

that our DM results predict a threshold field ofEth = 49.5 kV/cm and a threshold current density of

Jth = 1.53 kA/cm2. The results are close to experiment (Eth = 48 kV/cm and Jth = 1.50 kA/cm2).

Our results are also in fairly good agreement with the NEGF results (Eth = 47.6 kV/cm and Jth =

1.20 kA/cm2), as well as with the DM 2nd results (Eth = 47.3 kV/cm and Jth = 1.30 kA/cm2). In

Fig. 2.8(a), we see that the semiclassical model gives the same qualitative behavior as the density-

matrix results. However, the gain is overestimated for all field values. The most striking feature

of Fig. 2.8(a) is our overestimation of gain at high fields (> 60 kV/cm) compared with the NEGF

results. We attribute the difference to leakage from the quasi-bound states into the continuum at

high fields, which is captured with the NEGF formalism. On the other hand, our results for the

high-field gain are considerably lower than the DM 2nd model. We attribute the difference to the

assumption of a constant electron temperature of Te = 430 K in the DM 2nd model, whereas, as

will be shown in the next section, considerably more electron heating takes place for fields higher

than 50 kV/cm and the in-plane distribution deviates far from a thermal distribution.

Figure 2.8(b) shows the energy corresponding to the maximum gain in Fig 2.8(a). We see that

all models give similar results, where the energetic position of the gain maximum is higher than the

experimentally observed photon energy for all field values and the difference is greater at higher

fields. Our results agree well with the results obtained using NEGF, but the energy of the peaks is

higher by about 5 meV than the DM 2nd results.

53

Figure 2.8: (a) Peak gain vs electric field. (b) Energy position corresponding to peak gain as a

function of electric field. Results are given for calculations both with off-diagonal density-matrix

elements (green circles) and without them (violet squares), as well as a comparison with NEGF

(dashed blue line) and DM 2nd (solid red line). Experimental results from Ref. [2] are denoted by

a thick solid black line. Both the NEGF and DM 2nd results are from Ref. [3].

54

2.4.4 In-Plane Dynamics

Figure 2.9 shows the in-plane distribution fEkN,0 for N corresponding to the upper lasing state

(ULS) and lower lasing state (LLS). Results are shown far below threshold (30 kV/cm), slightly

above threshold (55 kV/cm), and far above threshold (70 kV/cm). For comparison, a Maxwellian

thermal distribution with a temperature of 300 K is also shown. For each value of electric field, the

electron temperatures calculated using Eq. (2.77) are shown, as well as the weighted average for all

subbands using (2.76). For low fields (30 kV/cm), the in-plane distribution is close to a Maxwellian

distribution. For higher fields, significant electron heating takes place and in-plane distributions

strongly deviate from a Maxwellian. There is a striking difference in shape and temperature be-

tween the ULS and LLS distributions at high fields. Above threshold (55 and 77 kV/cm), the ULS

is slightly (37 K) “hotter” than average, with a distribution that bears some resemblance to a ther-

mal distribution, with most electrons having energy below 50 meV. However, the ULS in-plane

distribution has a much thicker tail than the thermal distribution, resulting in a high electron tem-

perature. The LLS is considerably hotter than average electron temperature, with a distribution

that is flat up to around 150 meV (approximate lasing energy) and then slowly decays with a thick

tail. These results are intuitive because the LLS has a low occupation and most of the occupation

comes from electrons that recently scattered from the ULS with a large excess in-plane energy

of ' 150 meV (corresponding to about 1740 K), making the LLS very hot. On the other hand,

electrons in the ULS come from long-lifetime states in the injector, where electrons have had time

to emit many LO-phonons and get rid of excess energy.

We note that the inclusion of the optical-field feedback (not included in this work) is known

to reduce the temperature of the LLS, because electrons can then scatter from the ULS to the LLS

via stimulated emission without acquiring any excess in-plane energy [41]. Inclusion of electron-

electron scattering (also not included in this work) would also reduce the difference between the

in-plane distributions of different subbands [76], even for the relatively low sheet doping density of

n2D = 7.8× 1010 cm−2 in the considered device. However, we do not expect electron–electron in-

teraction to significantly change the average electron temperature, because electron–electron scat-

tering conserves the total energy of the electron system.

55

Figure 2.9: In-plane energy distribution for the ULS (solid red) and LLS (dashed blue). For com-

parison, a Maxwellian thermal distribution with a temperature of 300 K is also shown (thin black).

Results are shown for electric fields of 30 kV/cm (top), 55 kV/cm (middle), and 70 kV/cm (bot-

tom). For each value of the electric field, the electron temperatures of the ULS and LLS calculated

using Eq.(2.77) are shown, as well as the weighted average electron temperature, Tavg, obtained

using Eq. (2.76).

56

The electron temperatures shown in Fig. 2.9 are considerably higher than previously reported

for the same device using NEGF [3]. For example, at a field of 55 kV/cm, we obtain TULS = 621 K

and TLLS = 1161 K, compared with TULS = 398 K and TLLS = 345 K using NEGF in Ref. [3].

Despite the disagreement, our results are similar to experimental and theoretical results for different

QCL designs with similar wavelengths. For example, at threshold, electronic temperature of 800 K

(at room temperature) has been experimentally estimated [77] for the GaAs-based QCL proposed

in Ref. [78], which had a similar lasing wavelength λ ∼ 9 µm. Even higher excess electron

temperatures have been estimated by Monte Carlo calculations for a 9.4-µm QCL [76], where the

authors obtained electron temperature of about 700 K at threshold for a lattice temperature of 77 K.

Figure 2.10(a) shows the average excess electron temperature Te − T as a function of electric

field for different values of lattice temperature T . The average electron temperature is calculated

using Eq. (2.76). In Fig. 2.10(a), we see that considerable electron heating takes place for fields

higher than ∼ 40 kV/cm. For example, at a lattice temperature of 300 K, we observe an excess

electron temperature of 162 K (Te = 462 K) around threshold (49.5 kV/cm) and a maximum value

of 682 K (Te = 982 K) for E = 74 kV/cm. We see that, for fields below ∼ 50 kV/cm, electron

heating is more pronounced at higher temperatures, while at higher fields there is no clear trend.

Every time an electron traverses a single stage of length LP , it gains energy E0, equal to the

potential drop over a single period. It is therefore instructive to consider the excess electron tem-

perature as a function of the normalized input power density

pinput =v

LPE0 =

JE

n3D

(2.117)

where v is the drift velocity, E the electric field, J the current density and n3D the average electron

density. The normalized electrical power input gives the power transferred to each electron and is

more convenient to compare between different devices than the (unnormalized) electrical power

input Pinput = JE, which depends linearly on the electron density (for low doping, as long as

scattering rates and the Hartree potential are weakly affected). Figure 2.10(b) shows a scatter

plot of the excess electron temperature as a function of the normalized input power for several

values of the lattice temperature. The reason for a scatter plot is that the electron temperature is

not a single-valued function of pinput because J is not an invertible function of in-plane energy

57

(J is not a monotonically increasing function of electric field, as there are regions of negative

differential resistance, so it cannot be inverted). Figure 2.10(b) shows that, for input power up

to about 60 meV/ps, the results for all considered temperatures can be described with a simple

energy-balance equation (EBE)

Te − T = τEpinput

kB(2.118)

with an energy dissipation time τE . Results at 300 K can be described with an EBE up to 74 meV/ps

and the 400 K results for all considered values of input power. The reason an EBE is valid up to

higher input power for higher temperature is the increased POP occupation, which is the main

source of energy dissipation (acoustic phonons also dissipate energy but are not as effective). A

least-squares fit to Eq.(2.118) for the 400 K results gives τE = 0.50 ps, which is of the same order

of magnitude as typical POP scattering times [79].

58

Figure 2.10: (a) Average excess electron temperature Te − T as a function of field, where Te is

calculated using Eq. (2.77). Results are shown for T = 200 K (blue stars), T = 300 K (orange

squares) and T = 400 K (red circles). The two dashed vertical lines represent threshold (left

line) and onset of NDR (right line) for the room-temperature results. (b) Average excess electron

temperature Te − T as a function of electrical input power per electron, defined by Eq. (2.117).

The black line shows a best fit with the energy-balance equation (2.118). Vertical lines have the

same meaning as in panel (a).

59

2.5 Results for a Terahertz QCL

In order to demonstrate the validity of our model for long-wavelength device, we simulated

a THz QCL proposed in Ref. [4]. The authors used a phonon-assisted injection and extraction

design based on a GaAs/Al0.25Ga0.75As material system and achieved lasing at 3.2 THz, up to

a heatsink temperature of 138 K. We chose this specific device because both experimental and

theoretical results are readily available for comparison [4]. The results presented in this section

were published in Ref. [6]. Figure 2.11 shows the conduction band profile and most important

eigenfunctions of the considered device at the design electric field of 21 kV/cm. We will split this

section into two parts, starting with steady-state results in Section 2.5.1 and time resolved results

in section 2.5.2.

2.5.1 Steady-State Results

Figure 2.12 shows a steady-state current density vs electric field, as well as comparison with

experiment and theoretical results based on NEGF [4]. The experimental data is for a heat-sink

temperature of TH = 10 K. The actual lattice temperature TL is expected to be higher [33]. Both

the density matrix and NEGF results are for a lattice temperature of 50 K. We included interactions

with polar optical phonons, acoustic phonons (using elastic and equipartition approximations), and

ionized impurities. In addition, we included a LET scattering mechanism discussed in section 2.3.3

with a strength parameter of α = 0.1. This choice of α gave the best agreement with experiment.

However, results around the design electric field did not depend strongly on α, as can be seen

in Fig. 2.13. From Fig. 2.12 we see a very good agreement with experiment and NEGF around

the design electric field of 21 kV/cm. For electric fields lower than 17 kV/cm, neither NEGF

nor our density-matrix results accurately reproduce experimental results. However, our density-

matrix results and the NEGF results both show a double-peak behavior. The difference between

the density-matrix results and NEGF can be attributed to collisional broadening (not captured with

density-matrix approaches) and our calculation not including interface-roughness scattering.

60

Figure 2.11: Conduction-band edge (solid black line) and probability densities for the upper

lasing state (u), lower lasing state (`), injector state (i), and extractor state (e). Also shown

is the extractor state (eL) for the previous stage to the left, the injector (iR) state for the next

stage to the right, and a high-energy state (h). The high-energy state was included in numeri-

cal calculations, however, it had a small occupation and a negligible effect on physical observ-

ables. The dashed rectangle represents a single stage with the layer structure (from the left)

44/62.5/10.9/66.5/22.8/84.8/9.1/61 A, with barriers in bold font. The thickest barrier (in-

jector barrier) is doped with Si, with a doping density of 7.39 × 1016 cm−3 such that the average

electron density in a single stage is 8.98 × 1015 cm−3. Owing to low doping, the potential drop is

approximately uniform.

61

Figure 2.12: Current density vs electric field for density-matrix results (blue circles) and NEGF

(green triangles) for a lattice temperature TL = 50 K. Also shown are experimental results (red

squares) for a heat-sink temperature TH = 10 K. Experimental and NEGF results are both from

Ref. [4].

62

Figure 2.13: Current density vs electric field for different values of the strength parameter α. The

best agreement with experimental data is for α = 0.1 (green squares). The results at high fields are

not sensitive to the strength parameter, while low-bias results are. The higher peak at 9 kV/cm for

the α = 0.1 data is a result of a finer electric-field mesh for that data set.

63

In order to visualize the occupations and coherences of all combinations of the states, it is

instructive to plot density-matrix elements after integrating out the parallel energy

ρNM =

∫ρEkNMdEk . (2.119)

Figure 2.14 shows a plot of log10(|ρNM |), with occupations and coherences of all combinations of

the states shown in Fig 2.11, except for the high-energy (h) state, which had negligible occupation

and coherences. Normalization is chosen such that the largest matrix element is 1 (the occupation

of the upper lasing level). From the figure, we can see that the magnitude of the coherences can

be quite large. For example, the largest coherence is between the eL extractor state and the i

injector state, with a magnitude of about 0.21; this is a significant fraction of the largest diagonal

element and demonstrates the importance of including coherences in calculations. The second

largest coherence is between the upper and lowing lasing states, with a magnitude of 0.05. Other

coherences are smaller than 1% of the largest diagonal element and all coherences more than 4

places off the diagonal were smaller than 10−3, justifying our coherence cutoff of Nc = 5.

The magnitude of the matrix elements ρNM gives information about the importance of includ-

ing off-diagonal matrix elements in QCL simulations. However, these matrix elements do not

give us information about the dependence on in-plane energy. In order to visualize the in-plane

dependence, Fig. 2.15 shows plots of ρEkNM as a function of the in-plane energy for multiple pairs

of N and M . In the top (bottom) panel, N = u (N = e) is fixed and M is varied; we show the

three largest coherences |ρEkNM |, as well as the diagonal term |ρEkNN |. The figure shows the energy

dependence of the two largest coherences mentioned earlier (e-iR and u-l), along with the second

and third largest coherences for each state. We see that most off-diagonal elements are more than

two orders of magnitude smaller than the diagonal terms. Both the diagonal elements and the co-

herences have an in-plane distribution that deviates strongly from a Maxwellian distribution, with

a sharp drop around 35 meV due to enhanced POP emission. This result suggests that simplified

density-matrix approaches, where a Maxwellian in-plane distribution is assumed, are not justified

for the considered system.

64

Figure 2.14: log10(|ρNM |), where ρNM are density matrix elements after integration over parallel

energy. Normalization is chosen so that the highest occupation is one (upper lasing level). Results

are for an electric field of 21 kV/cm and a lattice temperature of 50 K. Coherences and occupations

are given for all combinations of states shown in Fig. 2.11, except for the high energy h state,

which had very small coherences and occupation. The highest occupations are the upper lasing

level (1.0), extractor state (0.60), lower lasing level (0.39), and injector state (0.33). The largest

coherences are between the extractor state e and injector state iR (0.21), and between the upper and

lower lasing levels (0.05). Other coherences were smaller than 0.01.

65

Figure 2.15: The magnitude of the matrix element |ρEkNM | as a function of in-plane energy, for N

corresponding to the upper lasing level (top panel) and extractor state (bottom panel). These two

states were chosen because they have the greatest occupations. Note that uR corresponds to the

upper lasing level in the next period to the right (not shown in Fig. 2.11), which is equal to the

coherence between eL and u, owing to periodicity.

66

2.5.2 Time-Resolved Results

Figure 2.16 shows the current density vs time at the design electric field (21 kV/cm) and at

a lower electric field (5 kV/cm). All results in this section are for a lattice temperature of 50 K.

The top panel shows the initial transient (first 2 ps) and the bottom panel shows the next 10 ps,

which is long enough for the current density to reach a steady state. In the first 2 ps, we observe

high-amplitude coherent oscillations in current, with a period of 100 to 200 fs. The rapid coherent

oscillations decay on a time scale of a few picoseconds, with the high-bias oscillations decaying

more slowly. Note that the peak value of current early in the transient can be more than 10 times

higher than the steady-state value. In the bottom panel, we see a slow change in current, which is

related to the redistribution of electrons within subbands, as well as between different subbands.

Figure 2.17 shows the time evolution of occupations for the same values of bias as in Fig 2.16,

in addition to results slightly below the design electric field. Occupations are very important quan-

tities because the optical gain of the device is directly proportional to the population difference of

the upper and lower lasing level (ρuu − ρ``). The time evolution of the occupations tells us how

long it takes the device to reach its steady-state lasing capability. In Fig 2.16, we can see that the

occupations take a much longer time to reach a steady state (20-100 ps) than the current density,

and the time needed to reach a steady state is not a monotonically increasing function of the electric

field: the 20-kV/cm results take more than twice as long to reach a steady state than the 21-kV/cm

results. In Fig. 2.17, we see that a population inversion of ρuu−ρ`` = 0.26 is obtained at the design

electric field, while lower-field results show no population inversion.

Figure 2.18 shows the time evolution of the in-plane energy distribution for all the subbands

shown in Fig. 2.11. Also shown is the equivalent electron temperature of each subband calculated

using Te = 〈Ek〉 /kB. The top panel shows the initial (thermal equilibrium) state, where all sub-

bands have a Maxwellian distribution with the extractor having the highest occupation. At time

t = 2.5 ps, the in-plane distribution has heated considerably for all subbands, with the lower lasing

level being hottest at Te = 136 K. At t = 10 ps, the lower lasing level has cooled down while

the other states have heated up, and the injector state has the highest temperature of 220 K. At

t = 100 ps, the system has reached a steady state, where the lower lasing level is coolest (92 K)

67

Figure 2.16: Current density vs time for two values of electric fields. The upper panel shows the

first two picoseconds and the lower the next 10 picoseconds. Note the different ranges on the

vertical axis.

68

Figure 2.17: Occupation vs time for the lower lasing, upper lasing, injector, and extractor states.

Normalization is chosen such that all occupations add up to one. Note the longer time scale com-

pared with the current density in Fig. 2.16.

69

and the injector is hottest (203 K). A noticeable feature in Fig. 2.18 is the big difference in tem-

perature of the different subbands, with a temperature difference of 110 K between the injector

and lower lasing level. In addition to having very different temperatures, the in-plane energy dis-

tributions are very different from a heated Maxwellian. A weighted average (using occupations as

weights) of the steady-state electron temperatures is 159 K, which is 109 K higher than the lattice

temperature.

70

Figure 2.18: In-plane energy distribution for all eigenstates shown in Fig. 2.11, except for the

high-energy state (h). Results are shown for four values of time, starting in thermal equilibrium

(t = 0 ps) and ending in the steady state (t = 100 ps). Also shown are the corresponding electron

temperatures, calculated from Te = 〈Ek〉 /kB.

71

2.6 Conclusion to Chapter 2

We have derived a Markovian master equation for the single-electron density matrix that is

applicable to electron transport in both terahertz and midinfrared quantum cascade lasers. The

model is an improvement of previous work employing density matrices because it does not rely on

a phenomenological treatment of dephasing across thick injection barriers [37] or a factorization

of the density matrix into cross-plane and in-plane distribution [80]. We demonstrated the validity

of the model by simulating a midinfrared QCL at room temperature, as well as a terahertz QCL at

a cryogenic temperature [6].

For the midinfrared QCL, nonparabolicity in the band structure was accounted for by using a

3-band k · p model, which includes the conduction band, valance band, and the spin-orbit split-off

band. We calculated the current density vs electric field strength and obtained excellent agree-

ment with experiments, below threshold, as well as with theoretical results based on NEGF for all

field values. At room temperature, we obtained a threshold electric field of Eth = 49.5 kV/cm

and threshold current density of Jth = 1.53 kA/cm2, in good agreement with experimental re-

sults of Eth = 48 kV/cm and Jth = 1.50 kA/cm2 and in fair agreement with NEGF results of

Eth = 47.6 kV/cm and Jth = 1.2 kA/cm2. Comparison with a semiclassical model shows that

off-diagonal matrix elements play a significant role. The semiclassical model was found to overes-

timate current density by 20% around threshold and up to 37% for other electric field values. Our

results for maximum gain vs field were in good agreement with NEGF up to and around thresh-

old, while for high fields, where tunneling into the continuum (not captured using density-matrix

models) is important, the density-matrix results give a considerably higher gain than NEGF. We

predict significant electron heating, with in-plane distribution deviating strongly from a thermal

distribution, especially the lower lasing state. We obtained an average electron temperature of

463 K around threshold, which is 163 K above the lattice temperature of 300 K. From a sim-

ple energy-balance equation, we obtained an energy-dissipation time of 0.5 ps for the considered

device, which is similar to the scattering time due to polar optical phonons.

72

Because of the lower energies involved higher wavelength devices, a single band model was

sufficient to describe the terahertz QCL. Close to lasing (around the design electric field), our

results for current density are in good agreement with both experiment and theoretical results

based on NEGF. The differences between NEGF and density matrix at low fields are small and can

be attributed to the omission of interface-roughness scattering in our simulation and the effects of

collisional broadening. We showed that the magnitude of the off-diagonal density-matrix elements

can be a significant fraction of the largest diagonal element. With the device biased for lasing,

the greatest coherence was between the injector and extractor levels, with a magnitude of 21% of

the largest diagonal element (the upper lasing level). This result demonstrates the need to include

coherences when describing QCLs in the THz range. We have found that significant electron

heating takes place at the design electric field, with in-plane distributions deviating far from a

heated Maxwellian distribution. The electron temperature was found to vary strongly between

subbands, with an average subband temperature about 109 K hotter than the lattice temperature

of 50 K. This result demonstrates the need to treat in-plane dynamics in detail. Time-resolved

results showed that, early in the transient, the current density exhibits high-amplitude coherent

oscillations with a period of 100-200 fs, decaying to a constant value on a time scale of 3-10

picoseconds. The amplitude of current oscillations could be over 10 times larger than the steady-

state current. Occupations of subbands and in-plane energy distributions took considerably longer

(20-100 ps) than the current density to reach the steady state.

73

Chapter 3

Coulomb-Driven THz-Frequency Intrinsic Current Oscillationsin a Double-Barrier Tunneling Structure within the Wigner Func-tion Formalism

3.1 Introduction

The THz-frequency range of the electromagnetic spectrum (0.1 – 10 THz) has received con-

siderable attention is recent years [81]. THz-frequency radiation is a valuable tool in the character-

ization of doped semiconductors, whose typical electron relaxation rates and plasma frequencies

are in this range [82, 83]. Other uses of THz radiation include non-destructive imaging of samples

that are too sensitive for x-rays, opaque at optical and near-infrared frequencies, but transparent to

THz-frequency illumination [84, 85]. Unfortunately, a lack of compact and efficient sources in this

frequency range, referred to as the THz gap, is an impediment to the development and application

of THz-based technologies [86].

Semiconductor double-barrier tunneling structures (DBTSs) in which an undoped region, sev-

eral tens of nanometers wide, envelops the main quantum well and barriers, are usually referred to

as resonant-tunneling diodes (RTDs) and can act as a source of power with frequency up to sev-

eral hundred GHz when biased in the negative-differential-resistance regime [87]. Recently, RTD

oscillators working in the THz-range have been experimentally realized at room temperature by

coupling RTDs with external circuit elements such as slot antennas [88, 89, 90, 91]. However, these

experimental systems have low output power, typically in the microwatt range [88, 89, 90, 91], be-

cause the oscillations are induced by exchanging energy with external circuit elements [92].

74

Intrinsic current oscillations in dc-biased RTDs were first observed in numerical calculations

by Jensen and Buo [93], who found that the oscillations were persistent (did not decay over time).

Theoretical work by several other groups predicted the same phenomenon [93, 94, 95, 96]. It

has been suggested that the RTD intrinsic current oscillations could circumvent the low-power

problem and lead to THz-frequency power in the milliwatt range [92]. However, the effect has

not yet been observed in experiment. It is therefore critical to understand the mechanism behind

the intrinsic current oscillations, in order to predict the optimal experimental conditions for its

observation. Zhao et al. [97] proposed a mechanism that emphasized the importance of the

formation of a quantum well on the emitter side of the device (the emitter quantum well, EQW)

and connected the frequency of the oscillations to the average energy spacing between the bound

states in the EQW and the main quantum well (MQW). The work of Jensen and Buo [93] and

others [94, 95, 96, 98, 99] on intrinsic current oscillations focused on RTDs with a traditional

doping profile (no doping in the region around the well and barriers), where the EQW only forms

at low temperature (77 K or below) and in a narrow bias window. For this reason, and because of

thermal broadening effects, it has been argued that intrinsic current oscillations could only occur

at low temperatures [97].

In this chapter, we investigate room-temperature intrinsic current oscillations in GaAs/AlGaAs-

based DBTSs under dc bias. We show that the intrinsic oscillations stem from long-range Coulomb

interactions and are associated with a periodic charge redistribution between the emitter and main

quantum wells. The oscillations have the highest amplitude and are closest to single-frequency

(harmonic) when the two wells are both deep enough to support well-localized quasibound states

close in energy; otherwise, there is a continuum on the emitter side and the oscillations are low in

amplitude and have a considerable frequency spread. However, the frequency of the oscillations is

not determined by the level spacing between the quasibound states [97], but instead by the amounts

of charge stored in the two quantum wells. Considering the Coulomb origin of the oscillations, it is

not surprising that a structure different than a traditional RTD is required for observing the oscilla-

tions with a high amplitude and sharp frequency. Therefore, instead of the traditionally doped RTD

structures, we focus on DBTSs with a uniform doping profile, which ensures the formation of a deep

75

EQW for all values of applied bias and at room temperature. While the uniformly doped DBTSs

have a low peak-to-valley ratio (a commonly employed figure-of-merit of RTD performance), a

wide peak plateau, and would generally perform poorly in traditional RTD applications, they are

superior as intrinsic THz-frequency oscillators. We simulate time-dependent room-temperature

quantum transport in these systems by solving the Wigner-Boltzmann transport equation (WBTE)

by the stochastic ensemble Monte Carlo technique with particle affinities [100, 101], coupled self-

consistently with Poisson’s equation. Scattering is treated microscopically within the WBTE, with

rates obtained from time-dependent perturbation theory (Fermi’s golden rule) [79]. Our treatment

of scattering is more detailed than in earlier work, where scattering was either neglected [102] or

was treated using the relaxation-time approximation with a constant (energy-independent) relax-

ation time [93, 94, 95]. We show that the current oscillations are associated with the EQW and

MQW ground states being periodically and adiabatically tuned in and out of alignment due to a

charge redistribution. The frequency of current oscillations depends linearly on the ratio of the

charges stored in the two wells and is lower than the plasma frequency. We discuss the dependence

of the current-density-oscillation amplitude (which can be as high as 50–80% of the time-averaged

current density), frequency, and bandwidth on the doping density and profile. The work in this

Chapter was published in Ref. [9]

3.2 System and Model Description

The generic structure we consider is a GaAs-based DBTS with two AlGaAs barriers. A

schematic of the device is shown in Fig. 3.1. To simulate the considered system, we solve the

open-boundary WBTE [103, 104, 101], which governs the time evolution of the Wigner quasiprob-

ability distribution fw [105]. We employ the effective-mass approximation (we assume uniform

effective mass throughout the structure) with effective mass m∗ = 0.067m0, with m0 being the

free-electron rest mass, and assume translational invariance in the x − y plane. The WBTE for

quasi-1D transport along z, with k denoting the wave number along z, can be written as [104, 101]

∂fw∂t

+~km∗

∂fw∂z− 1

~∂Vcl

∂z

∂fw∂k

= QVqm [fw] + C[fw] , (3.1)

76

Figure 3.1: A schematic of the double-barrier GaAs/AlGaAs tunneling structure. The barrier and

quantum-well widths are denoted by d and w, respectively, and L is the system length. We assume

that the device dimensions in the x and y directions are much greater than its length L.

where QVqm [fw] is the quantum evolution term defined by

QVqm [fw](z, k, t) =

∞∫−∞

Vw(z, k − k′, t)fw(z, k′, t)dk′ , (3.2)

while Vw(z, k) is the non-local Wigner potential

Vw(z, k, t) =1

h

∞∫−∞

sin(kz′)[Vqm(z + z′/2, t)

−Vqm(z − z′/2, t)]dz′ . (3.3)

Note that in Eqs. (3.1)-(3.3), we have assumed that the total potential can be split into a spatially

slowly varying term Vcl(z, t) and a fast-varying term Vqm(z, t). It can be shown [101] that poten-

tial terms linear or quadratic in position can be included in the slowly varying term without any

approximations, but higher-order terms must be included in the fast-varying term. (In Appendix B,

we explain how we perform the separation.)

77

In Eq. (3.1), scattering is incorporated via the Boltzmann collision operator, defined by

C[fw](z, k, t) =∑i

∫[si(k

′,k)fw(z, k′, t)

− si(k,k′)fw(z, k, t)]d3k′ , (3.4)

where si(k′,k) is the transition rate from state k′ to k due to scattering mechanism i. In this work,

we consider a GaAs-based device, so the relevant scattering mechanisms are with polar optical

phonons and ionized donors [68].

Well-established methods for solving the open-boundary WBTE include the finite-difference

approaches [93, 94, 106] and Monte Carlo methods [100, 107, 108, 109, 110]. We solve Eq. (3.1)

using a particle-based ensemble Monte Carlo approach, where particles are assigned a property

called affinity [100, 101]. Quantum effects are accounted for via a time evolution of particle affini-

ties, which are induced by the quantum evolution term in Eq. (3.1). The method was originally

proposed by Shifren and Ferry [100, 111, 112] and later improved upon by Querlioz and Doll-

fus [101]. Our approach deviates from that of Querlioz and Dollfus in our treatment of contacts

and bias. Here, contacts are assumed to be Ohmic, where charge neutrality and current continuity

are enforced by injecting carriers with a velocity-weighted, drifted Maxwell-Boltzmann proba-

bility density (Querlioz and Dollfus assumed an equilibrium Maxwell-Boltzmann density [101])

defined by

pL(k) ∝ k exp

−~2(k − kL)2

2m∗kBT

, k ≥ 0, (3.5a)

pR(k) ∝ |k| exp

−~2(k − kR)2

2m∗kBT

, k ≤ 0 , (3.5b)

where L (R) refers to the left (right) contact. The drift wave numbers kL and kR are drift wave

numbers in the left and right contacts, calculated as the expectation values of k on the device side

of the contact-device interface,

kL,R =

∫∞−∞ kfw(zL,R, k, t)dk∫∞−∞ fw(zL,R, k, t)dk

, (3.6)

where zL (zR) is the position of the left (right) contact. The in-plane wave numbers kx and ky are

generated randomly from a Maxwell-Boltzmann distribution.

78

Electron–electron interaction is accounted for on a mean-field level, by a self-consistent solu-

tion of Poisson’s equation. The resulting time-dependent electrostatic potential is included in the

Vqm term in Eq. (3.3). Bias is incorporated through the boundary conditions in Poisson’s equation

(see Appendix B).

We calculate the current, I , that would be measured by an external ammeter based on the

Shockley-Ramo theorem [113, 114], given by Eq. (3.7a):

I(t) =q

L

∑i

Aivi (3.7a)

=qS

L

∫ L/2

−L/2dz 〈v〉 (z, t)n(z, t). (3.7b)

Here, q is the elementary charge, L is the device length, vi = ~ki/m∗ is the velocity of i-th

particle, and Ai is the particle’s affinity in the WMC simulation. The above expression (3.7a) has

been widely used in current calculations and is equivalent to tracking the particles that exit/enter

each contact, but with less numerical noise because the entire ensemble partakes [101, 115, 116].

In one dimension, expression (3.7a) is also proportional to the current density averaged over the

device, as captured by Eq. (3.7b), where S is the device area, 〈v〉 (z, t) is the average z-component

of the electron velocity, and n(z, t) is the electron density.

3.3 Results

3.3.1 DBTS With a Uniform Doping Profile

The device we consider in this section is a 150-nm-long GaAs DBTS with a 6-nm-wide quan-

tum well, sandwiched between two 2-nm-thick Al0.26Ga0.74As barriers with a conduction band

offset of 0.35 eV [117, p. 260]. The doping profile is uniform throughout the device, with a

density of 1018cm−3. The sample potential profile of the 6-nm-well device for an applied bias of

96 mV is shown in Fig. 3.2a. Intrinsic oscillations are observed in the bias range 94 mV to 98 mV

for the 6-nm-well device. In Fig. 3.2b, two I-V diagrams, one for the 6-nm device and another for

a 7-nm-well but otherwise identical device are shown. Arrow 2 in Fig. 3.2b points to 96 mV, the

79

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14Applied bias [V]

0

1

2

3

4

5

6

Cu

rren

t d

en

sity

[105

A/c

m2]

1 2

3

6 nm well

7 nm well

60 40 20 0 20 40 60Position [nm]

0.10

0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30P

ote

nti

al

[eV

]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

n [

1018

cm−

3]

(a)

(b)

Figure 3.2: (Color online) (a) Time-averaged potential profile (solid) and charge density (dashed)

as a function of position for an applied bias of 96 mV (bias for which intrinsic current oscillations

are observed). The potential profile is obtained from a self-consistent solution of the coupled

WBTE and Poisson’s equation. (b) Time-averaged current density vs applied bias for RTDs with

6-nm-wide (solid red) and 7-nm-wide (dashed blue) quantum wells. The time average is obtained

over 1 ps intervals. Arrows in panel (b) are guides for Fig. 3.3. Arrow 2 also corresponds to the

bias value of 96 mV, which is used in (a).

bias value from Fig. 3.2a. In contrast to the 6-nm device, no intrinsic oscillations are observed at

any bias for the 7-nm-well device.

80

0

2

4

6

8 1

0

2

4

6

8

Cu

rren

t d

en

sity

[10

5A

/cm

2]

2

8 9 10 11 12Time [ps]

0

2

4

6

8 3

Frequency [THz]0.0

0.5

1.0

Frequency [THz]0.0

0.5

1.0

0 2 4 6 8 10Frequency [THz]

0.0

0.5

1.0

Figure 3.3: Current density vs time (left column) and the corresponding Fourier transform ampli-

tudes (right column) for the uniformly doped DBTS from Sec. 3.3.1 at an applied bias of 1: 50 mV

(top row), 2: 96 mV (middle row) and 3: 130 mV (bottom row). Panel numbers 1–3 refer to the

bias values marked in Fig. 3.2.

Figure 3.3 shows the current density vs time for the three values of applied bias denoted in

Fig. 3.2b, along with the corresponding Fourier transform amplitudes of the current density. We

see that the current density fluctuates over time for all considered bias values. However, only for

bias values in the range 94 mV to 98 mV do the fluctuations have a well-resolved frequency (i.e.

a narrow peak in the frequency domain, middle row, right panel). The current oscillations do not

diminish when the number of particles in the simulation is increased, eliminating the possibility

that the current oscillations are artifacts of the stochastic nature of the Monte Carlo method.

81

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

Applied bias [V]

0.0

0.5

1.0

1.5

Cu

rren

t d

en

sity

[10

5A

/cm

2]

60 40 20 0 20 40 60

Position [nm]

0.2

0.1

0.0

0.1

0.2

0.3

Pote

nti

al

pro

file

[eV

]

0.0

0.2

0.4

0.6

0.8

1.0

Ch

arg

e d

en

sity

[10

18cm

−3](a)

(b)

Figure 3.4: (a) Time-averaged potential profile (solid line) and charge density (short dashed line)

as a function of position for the traditionally doped RTD at an applied bias of 180 mV. Long dashed

line shows the doping profile of the device. The potential profile is obtained from a self-consistent

coupled solution of the WBTE and Poisson’s equation. (b) Time-averaged current density vs ap-

plied bias. The time average is obtained over 1 ps intervals. The arrow marks the bias value

180 mV, which is used in (a).

3.3.2 Traditionally Doped RTD

For comparison, we have considered a traditionally doped RTD with a 30-nm-wide undoped

region that includes the center of the device. Figure 3.4a shows the potential profile, charge density

and doping profile for an applied bias of 180 mV (see Fig. 3.4b for IV-diagram). The main quantum

82

8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

Time [ps]

0

1

2

3

4

5

6

7

8

Cu

rren

t d

en

sity

[1

05A

/cm

2]

2 3 4 5 6 7

Frequency [THz]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Fou

rier

tran

sform

am

pli

tud

e

Traditional dopingUniform doping

(a)

(b)

Figure 3.5: (a) Current density as a function of time for a bias of 180 mV (peak current) for the

traditionally doped RTD (lower blue curve) and a comparison with the uniformly doped DBTS

from Sec. 3.3.1 for a bias of 0.096 mV (upper red curve). (b) Fourier spectrum of the current

density oscillations for the traditionally doped (blue curve) and uniformly doped DBTS (red curve).

Fourier transform amplitudes are normalized so the maximum equals one.

well is 7-nm-wide (this well width provided the most pronounced current oscillations), with other

parameters the same as the uniformly doped structure considered in Fig. 3.2. As in the case of the

uniformly doped DBTS considered in Sec. 3.3.1, we observe persistent current oscillations for all

values of bias, but only in a limited bias window (175 mV to 185 mV) do the oscillations have a

single, well-pronounced Fourier component.

In Fig. 3.5, we observe intrinsic current oscillations when the traditionally doped RTD is under

peak bias of 180 mV. In Fig. 3.5a, we can see that the current density is smaller by a factor of ∼ 3

compared with the uniformly doped device at a bias of 96 mV. The amplitude of current oscillations

(peak to bottom) is also smaller in the traditionally doped case (1.5 × 105 A/cm2 compared with

83

4.0 × 105 A/cm2 in uniformly doped device). As can be seen in Fig. 3.5b, owing to the low

current density in the traditionally doped RTD, the main Fourier component of the current density

is considerably widened by scattering, with respect to the uniformly doped case.

3.3.3 Effects of Scattering

In order to investigate the role of scattering, we ran simulations where the conventional scatter-

ing rates [79] were artificially modified via multiplication by a factor of 0.5 and 0.25. The results,

as well as a comparison with the full scattering case (unchanged rates), are presented in Fig. 3.6,

which shows the Fourier transform amplitude of the current density when the uniformly doped

DBTS from Sec. 3.3.1 is biased in such a way that intrinsic oscillations are well resolved (96 mV).

With full scattering, the main frequency component has a full width at half-maximum (FWHM)

of 0.2 THz. Multiplying the scattering rates by a factor of 0.5 results in a reduced FWHM of about

0.08 THz. However, reducing the scattering rates further does not decrease the peak FWHM.

3.3.4 The Quasi-Bound-State Picture. The Coulomb Mechanism Behind theCurrent-Density Oscillations

To get a qualitative picture of the mechanism behind the current oscillations, several authors

have previously solved the time-independent Schrodinger equation (obtained eigenstates and ener-

gies) for a subregion of the device in question [97, 98, 118]. Here, we do the same. The potential

we use for the eigenvalue problem is the potential that was obtained by solving the coupled WBTE

and Poisson’s equation. Note that, due to the time-dependent distribution of charge in the device,

the electrostatic potential is time-dependent. For a general time-dependent potential, solving the

time-independent Schrodinger equation for different times is not meaningful. However, if the wave

functions of the bound states vary slowly in time, an eigenvalue equation of the following form is

valid [97]

H(t)ψi(z, t) = Ei(t)ψi(z, t) , (3.8)

where H(t) is the system Hamiltonian, which includes the effects of the time-dependent “electro-

static” potential, ψi is the i-th eigenstate, and Ei(t) is its corresponding energy.

84

3.5 4.0 4.5 5.0 5.5 6.0Frequency [THz]

0.0

0.2

0.4

0.6

0.8

1.0F

ou

rier

tran

sform

am

pli

tud

e25% scattering

50% scattering

100% scattering

60 40 20 0 20Position [nm]

140

120

100

80

60

40

20

0

Pote

nti

al

[meV

]

Figure 3.6: Fourier transform amplitude of the time-dependent current density for the uniformly

doped DBTS from Sec. 3.3.1 with the full scattering rates (blue squares), 50% of the full scatter-

ing rates (green triangles) and 25% of the full scattering rates (red circles). The inset shows the

corresponding time-averaged potential profiles for a part of the device.

Following such a procedure, we consider the emitter quantum well (EMQ) and main quantum

well (MQW) regions of the uniformly doped DBTS from Sec. 3.3.1 seperately and solve the eigen-

value problem in each region (see Fig. 3.7a). This approach is only valid if the eigenfunctions are

well localized within each region. For example, in calculating the MQW bound states, we assume

that the two barriers extend to ±∞ and use the boundary condition ψ(±∞) = 0. A condition for

the validity of the approach is that the resulting wavefunction does not penetrate deeper than 2 nm

(the thickness of the barriers) into the barriers. From Fig. 3.7a, we can see that this is indeed the

case — the well ground state does not extend into the emitter side and the emitter ground state

does not penetrate into the well. Our motivation for the splitting of the emitter and well regions is

to elucidate the nature of the temporal evolution of the device potential, shown in Fig. 3.8. We see

that the largest Fourier components of potential oscillations in both the EQW and MQW regions

are at frequencies around 9.6 THz, considerably higher than the frequency of current oscillations

85

Position [nm]

0.1

0.0

0.1

0.2P

ote

nti

al

[eV

]

EmitterWell

40 30 20 10 0 10 20 30 40

Position [nm]

0.2

0.1

0.0

0.1

0.2

0.3

0.4

Pote

nti

al

[eV

]

EmitterWell

(a)

(b)

Figure 3.7: (a) Snapshot of the potential profile (thick black curve) for the uniformly doped DBTS

from Sec. 3.3.1 at an applied bias of 96 mV (bias for which intrinsic oscillations are observed).

The solid red and the dashed blue curves correspond to the ground state probability densities in

the well and emitter regions, respectively. (b) Snapshot of the potential profile (thick black curve)

of the traditionally doped RTD from Sec. 3.3.2 in a steady state for an applied bias of 180 mV

(bias for which intrinsic oscillations are observed). The solid red curve corresponds to the ground

state probability density of the main quantum well bound state. The emitter quantum well is very

shallow and contains no bound states. In both (a) and (b), the two arrows define the emitter and

well regions (the computational domain for the wave functions). The two domains overlap in the

region of the left barrier.

86

0 2 4 6 8 10 12

Frequency [THz]

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Fou

rier

tran

sform

am

pli

tud

e [

arb

. u

nit

s]

|∆(ν)|+2

|V (zw ,ν)|+1

|V (zb ,ν)|

20 15 10 5 0 5 10

Position [nm]

120100

806040200

Pote

nti

al

[meV

]

Figure 3.8: Fourier transforms of the potential on the left side of the first barrier at zb = 6 nm (red

open circles), the middle of the well at zw = 0 (green closed circles), and the potential difference

∆(t) = V (zb, t) − V (zw, t) (blue squares) for the uniformly doped DBTS from Sec. 3.3.1. In the

legend, the tilde denotes a Fourier transform from the time to frequency (ν) domain. The inset

shows a snapshot of the potential profile where the positions zb (red downward-pointing triangle)

and zw (green upward-pointing triangle) are marked. The figure shows that the largest Fourier

components of the emitter and well potential oscillations are for frequencies above the intrinsic

current oscillations. However, these high-frequency plasmonic oscillations [Eq. (3.9)] are in phase,

as are the low frequency oscillations around 0.4 THz. In contrast, the potential difference between

the barrier and well region oscillates with the same frequency as the current density (4.5 THz).

(4.5 THz). We note that the high-frequency oscillations correspond to the bulk electron plasma

87

frequency in GaAs, given by

fp =1

2π

√nq2

εm∗' 9.65 THz , (3.9)

with an electron density equal to the doping density n = 1018 cm−3 and ε = 12.8ε0 is the dielectric

constant of GaAs. However, the high-frequency potential oscillations of the two wells are in phase;

in contrast, the potential difference between the two regions oscillates with the same frequency as

the current, implying that the current oscillations are Coulombic in nature and that the electrostatic

potential difference between the two regions plays a central role in the mechanism. The Coulomb

nature of the mechanism was qualitatively mentioned in [119] in connection with a simplified 1D

version of the DBTS, featuring a linear potential drop across the well and barriers and a constant

density of states in the contacts. However, we found that the oscillations were impossible to ob-

serve with a linear potential drop; a self-consistent solution of coupled Poisson’s equation and a

transport kernel is necessary to capture the oscillations. In fact, current oscillations in the DBTS

appear to be closely related to the experimentally observed phenomena of charge bistability [120]

and self-sustained oscillations in doped superlattices [121, 122], which are associated with the for-

mation of electric-field domains [123]. The doped DBTS might be considered as the ultra-short

limit of a doped superlattice, which also implies a higher frequency of oscillations (THz in the

DBTS, as opposed to the MHz – GHz frequency range observed in superlattices [122]).

Figure 3.9a shows the results of the same bound state analysis as in Fig. 3.7a, performed at

different times over a time period of 4 ps. The figure shows that the energy difference between the

ground states in the EQW and MQW, defined by ∆(t) = EMQW(t)−EEQW(t), is oscillating with

the same frequency as the current density oscillations (also plotted in Fig. 3.9) with a minimum

near zero. With increased current, charge accumulates in the MQW, raising the energy of the

MQW state out of alignment with the EQW state via the self-consistent electrostatic potential.

Charge then tunnels from the MQW region into the collector, lowering the well bound state back

into alignment with the emitter state. The process is cyclic and is the cause of the periodic current

oscillations.

88

There is a phase difference of about 49 between the peak in the current density and ∆. The

current actually takes the highest value when the MQW energy is slightly (by about 10 meV ≈

kBT/2) higher than the EQW energy. The phase difference has to do with considerable inelastic

scattering and the fact that the system is nondegenerately doped, with the Fermi level below the

conduction band edge; the states are populated according to the distribution-function tail, so the

concept of level alignment holds with an energy uncertainty on the order of kBT/2.

Figure 3.9b shows that temporal variation in the EQW ground state wavefunction is small and

time variation of the MQW state (not shown) would not be visible on a similar graph, justifying

our use of Eq. (3.8). The reason the wave functions are only weakly affected is that most of the

time-dependent potential drop between the bottom of the EQW and the center of the MQW takes

place inside the left barrier, where the EQW and MQW ground states have very small amplitudes

and are therefore only weakly affected.

In the case of the traditionally doped RTD from Sec. 3.3.2, the EQW is too shallow to form a

bound state (see Fig. 3.7b). However, as already mentioned in Sec. 3.3.2, we still observe intrinsic

current oscillations when the RTD is biased in such a way that it exhibits peak current (resonant

condition). As can be seen in Fig. 3.7b, under these resonant conditions, the MQW quasibound

ground state is slightly (on the order of kBT ) higher than the emitter conduction band edge and

a large portion of incoming electrons have energy matching the MQW ground state and current

is at a maximum. With increased current, the amount of charge in the well goes up, bringing the

MQW state out of the transport window via the self-consistent electrostatic potential. As in the

case for the uniformly doped DBTS, the process is cyclic and is the cause for the intrinsic current

oscillations.

3.3.5 Frequency of the Current-Density Oscillations. Effect of Varying DopingDensity

In a model proposed by Zhao et al. [97], the frequency of intrinsic current oscillations is

given by ∆/h, where ∆ is the time average of ∆(t). From the data presented in Fig. 3.9a, we get

∆/h ' 1.7 THz, which is well below 4.5 THz, the frequency of the current density-oscillations

89

4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0

Time [ps]

0

10

20

30

40E

nerg

y [m

eV

]∆

2

0

2

4

6

8

Cu

rren

t d

en

sity

[10

5A

/cm

2]

Current density

50 40 30 20 10

Position [nm]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

|ψE

(z)|2

[a.u

.]

6.4 6.5 6.6 6.7 6.8

Time [ps]

5

0

5

10

15

20

∆ [

meV

]

(a)

(b)

Figure 3.9: (a) Time evolution of ∆, the energy difference between the MQW ground state and

the EQW ground state (red circles) and current density (blue triangles) for the uniformly doped

DBTS from Sec. 3.3.1. The energy difference between the ground states in the EQW and the

MQW is oscillating with the same frequency as the current density and is close to zero when ∆ is

at a minimum. The current density leads ∆ in phase by 49. (b) Probability density for the emitter

ground state at a few points in time (the corresponding values of ∆ are denoted in the inset). We

see that the EQW wavefunction varies weakly over time.

90

(Fig. 3.8). Another important difference between our results and the results of Zhao et al. is that

they found the potential oscillations in the EQW and MQW to be completely in-phase, and thus

concluded that the current oscillations were not driven by the charge exchange between the EQW

and MQW, i.e., that the source of current oscillations is not the long-range Coulomb interaction.

In contrast, as shown earlier in this section, we found that current oscillations are driven by the

time-dependent potential difference between the EQW and MQW. Sakurai and Tanimura worked

within a hierarchy-equation-of-motion formalism and also observed intrinsic current oscillations

in RTDs [99, 98, 99]. To explain their results, they performed a similar bound-state analysis.

However, for their eigenvalue problem, they used a time average of the device potential. When

we employed the method proposed by Sakurai and Tanimura, we were not able to predict the

frequency of current oscillations. This result is not surprising, considering that, in time-averaging

the device potential, we lose all information about the potential variation between the EQW and

MQW, which plays a pivotal role in the current oscillations in the present work. On the other hand,

Ricco and Azbel [119], who qualitatively discussed the Coulomb origin of the oscillations, argued

that the oscillation frequency would be on the order of the carrier lifetime in the well, related to

the peak width in the coherent transmission coefficient.

Here, we show that the frequency of current oscillations is in fact determined by the ratio of

charges stored in the MQW and EQW, and that it can be tuned by varying the doping density. We

observe well-resolved (near-single-frequency) intrinsic current oscillations in the uniformly doped

DBTS presented in Sec. 3.3.1 for doping density in the range 0.8 − 1.8 × 1018 cm−3, where fre-

quency can be tuned between 4.2 THz and 4.9 THz (see Fig. 3.10a). Figure 3.10 shows how several

physical quantities vary with doping density. In Fig. 3.10a, we see that the frequency of oscillation

is a nonmonotonically increasing function of doping density. The amplitude of current oscillation

JA (Fig. 3.10b) has a maximum between 1.5×1018 cm−3 and 1.7×1018 cm−3 and in Fig. 3.10c we

see that the relative current-oscillation amplitude JA/Jav (Jav is the time-averaged current density)

varies weakly once doping is above 1.0× 1018 cm−3. Below doping of 1.0× 1018 cm−3, the rela-

tive current-oscillation amplitude decreases rapidly with decreasing doping density. Modifying the

doping density affects the I–V-diagram of the device and shifts the operating bias. By operating

91

bias, we mean the bias in which the device exhibits the clearest current oscillations (lowest FWHM

of the main Fourer transform amplitude). Figure 3.10d shows how the operating bias changes with

doping density. Figure 3.10e shows that the frequency of current oscillations is roughly a linear

function of the ratio QMQW/QEQW , where QMQW is the time-averaged total charge (per unit area)

in the MQW region and QEQW is the time-averaged total charge in the EQW region. In calculating

QEQW, we intergrate the charge density over a region extending 20 nm to the left of the left barrier,

corresponding roughly to the extent of the EQW shown in Fig. 3.7a.

Outside of the doping range 0.8 − 1.8 × 1018 cm−3, current still oscillates, but with a much

greater frequency spread, similar to the top and bottom panels in Fig. 3.3. When scattering rates

are artificially lowered, we observe the intrinsic current oscillations for doping densities lower

than 0.8 × 1018 cm−3, so we conclude that for doping densities lower than 0.8 × 1018 cm−3, the

long-range Coulomb force cannot sustain periodic current oscillations owing to the dampening

effects from phonon and ion scattering. As the doping density is increased, the potential well on

the collector side of a uniformly doped DBTS gets deeper and a large amount of charge builds up

in this well (see Fig. 3.11). We suspect that the charge exchange between the collector well and the

main quantum well is what disturbs intrinsic current oscillations for higher doping densities. To

prevent the collector quantum well from forming, other doping profiles can be considered, such as

a gradual doping density towards the collector side. Another possible explanation for the lack of

well-resolved intrinsic current oscillations at higher doping densities is that, as the doping density

is increased, potential variations due to the electron plasma oscillations are stronger (have a larger

Fourier amplitude) and have a higher frequency (see Fig. 3.12), so the quasistatic picture presented

in Sec. 3.3.4 may not be warranted.

3.3.6 Experimental Considerations Relevant for Observing Persistent Oscilla-tions

Since the well-resolved current oscillations are only observed when the ground states in the

EQW and MQW have similar energies, it is expected that they will be notable only in a limited

bias window in which this alignment takes place. For the 6-nm-well uniformly doped DBTS, this

92

4.2

4.4

4.6

4.8

5.0

f max [

TH

z]

2

3

4

5

6JA

[105

A/c

m2

]

0.5

0.6

0.7

0.8

JA/J

av

0.8 1.0 1.2 1.4 1.6 1.8

Doping density [1018 cm−3 ]

0.08

0.09

0.10

Vop

[V

]

0.19 0.20 0.21 0.22 0.23 0.24 0.25QMQW/QEQW

4.2

4.4

4.6

4.8

5.0

f max [

TH

z]

(a)

(b)

(c)

(d)

(e)

Figure 3.10: (a) Frequency of current oscillation fmax vs doping density. (b) Amplitude of current

oscillations JA vs doping density. The amplitude is calculated using JA = 2√

2σ, where σ is the

standard deviation in current density. (c) Relative amplitude of current oscillations JA/Jav where

Jav is the time-averaged current density. (d) Operating bias Vop (see main text for definition) as

a function of doping density. (e) Frequency of current oscillations vs the ratio of time averaged

charge in the MQW and the EQW (see main text for details). All panels are for the uniformly

doped DBTS from Sec. 3.3.1.

93

60 40 20 0 20 40 60

Position [nm]

0.10

0.05

0.00

0.05

0.10

Pote

nti

al

[eV

]ND =1×1018 cm−3

ND =2×1018 cm−3

Figure 3.11: Time-averaged potential profile in the uniformly doped DBTS from Sec. 3.3.1 at a

doping density of 1.0×1018 cm−3 (solid red) and 2.0×1018 cm−3 (dashed blue) for applied bias of

96 mV and 77 mV, respectively. For the higher doping density, the potential well on the collector

side is deep enough to form a bound state, which disrupts the current density oscillations that are

governed by charge transfer between the emitter and main quantum wells.

range is 94–98 mV (Fig. 3.2); below 94 mV, the MQW ground state is considerably above the

EQW one, while above 98 mV the order reverses.

The necessity of this alignment is manifested in the sensitivity of the oscillation prominence on

the DBTS geometry. While the 6-nm-well uniformly doped DBTS supports really sharp-frequency,

large-amplitude oscillations, a similar 7-nm-well uniformly doped DBTS (Fig. 3.2) does not. The

reason is the strong dependence of the MQW ground state energy on the well width, where a wider

well translates into a lower ground-state energy. A natural follow-up question is why the current

oscillations are simply not observed at a lower bias, but at a low bias the EQW is too shallow

to support a well-localized bound state. Current oscillations are thus very sensitive to quantum-

well-width fluctuations. As the lattice parameter for GaAs is about 0.5 nm, monolayer precision is

fabrication in required to observe the oscillations.

Barrier thickness is another parameter that can fluctuate in fabricated devices. We performed

simulations for uniformly doped DBTSs with the same well width (6 nm) but with thicker (3 nm)

94

4 6 8 10 12 14

Frequency [THz]

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5F

ou

rier

tran

sform

am

pli

tud

e [

arb

. u

nit

s]ND =1.0×1018 cm−3

ND =1.5×1018 cm−3

ND =1.8×1018 cm−3

Figure 3.12: Fourier transform of the time-dependent potential value at the center of the MQW re-

gion, corresponding to V (zw, t) in Fig. 3.8, for doping densities of 1.0× 1018 cm−3 (blue squares),

1.5 × 1018 cm−3 (filled circles) and 1.8 × 1018 cm−3 (red circles). Vertical lines mark the corre-

sponding electron–plasma frequencies, 9.65 THz, 11.81 THz, and 12.94 THz, respectively. Plasma

frequences are calculated using Eq. (3.9), assuming an electron density equal to doping density.

We see that, as the electron density is increased, plasma oscillations increase both in frequency and

amplitude. The lower-frequency Fourier component, corresponding to the frequency of intrinsic

current oscillations, varies more slowly with doping and remains between 4 and 5 THz.

and thinner (1 nm) barriers than the 2-nm one, which had been found as ideal for the minimal fre-

quency spread and highest amplitude. We found that varying the barrier thickness had a negligible

95

effect on the frequency of current oscillations. The main effect of the increased barrier width was

a drastic decrease in the amplitude of both current oscillations and the time-averaged current, be-

cause thicker barrier impede tunneling. The effect of barrier thinning washed out the oscillations,

because with very thin wells the EQW and MQW states cease to be localized in their respective

wells (Sec. 3.3.4). Consequently, the barrier thickness also has to be achieved with monolayer

precision.

The doping density is a parameter that critically affects the characteristic frequency of intrinsic

current oscillations. As the current oscillations are driven by a cyclic redistribution of charge in the

device, the frequency will be on the same order of magnitude as the 3D plasma frequency for the

electron density equal to the doping density, Eq. (3.9). An accurate prediction of the frequency,

however, requires a detailed numerical calculation.

3.4 Conclusion to Chapter 3

By solving the WBTE self-consistently coupled with Poisson’s equation, we analyzed the per-

sistent THz current oscillations in GaAs/AlGaAs-based DBTSs. We showed that a uniform doping

profile results in a higher amplitude of current oscillations, as well as a reduced frequency spread,

than the traditional doping profiles, where the well region is undoped. By varying device param-

eters, we found that the barrier and well widths of 2 nm and 6 nm, respectively, provided the

greatest amplitude of current oscillations as well as the lowest frequency spread (FWHM of the

main Fourier component). We provided a qualitative explanation for the source of current os-

cillations by employing a quasistatic picture, where localized states in the EQW and MQW are

periodically tuned in and out of alignment because of a redistribution of charge between the EQW

and MQW. The frequency of current oscillations was found to depend linearly on the ratio of the

time average of the charge stored in the EQW and MQW regions. We found that the frequency

of current oscillations in a uniformly doped DBTS can be tuned by varying the doping density

between 0.8 × 1018 cm−3 and 1.8 × 1018 cm−3. In this doping range, frequency is a nonmono-

tonically increasing function of doping density and can be varied between 4.2 and 4.9 THz, with

a typical frequency spread of 0.2 THz. The highest amplitude of current oscillations we observed

96

was 5.5 × 105 A/cm2, for a doping density in the range 1.5 to 1.7 × 1018 cm−3. However, above

the doping density of 1.0 × 1018 cm−3, the ratio of the current-density-oscillation amplitude and

the average current density did not vary considerably and had values in the range 0.7 to 0.8.

We conclude that utilization of intrinsic current oscillations in uniformly doped DBTSs shows

potential for solid-state-based ac-current generation in the THz range at room temperature. How-

ever, as we have demonstrated, tunability of the frequency of oscillation is limited (4.2 to 4.9 THz).

The limited tuning range is a consequence of the fact that the ratio of time-averaged charge in the

EQM and MQW does not vary considerably in the doping range in which intrinsic current os-

cillations are manifested. In order to provide more tunability, it is worth considering the effect

of a more complicated doping profile and variation of other device parameters. Another limita-

tion is the ∼ 0.2 THz frequency spread of the main Fourier component of current oscillations.

We showed that artificially reducing scattering rates reduces the frequency spread and allows us

to observe well-resolved intrinsic current oscillations at lower doping densities. Experimentally,

lowering the scattering rates can be accomplished by lowering the temperature and doping den-

sity. However, tuning the temperature also changes the distribution of electrons in the contacts and

modification of the doping profile results in changes in the electrostatic potential. Optimization

of the RTD performance as a THz power source with respect to temperature and doping profile is

therefore an involved process and is beyond the scope of this publication. Another possible exten-

sion to the current work is a self-consistent solution of the WBTE with Maxwell’s equations, to

estimate the radiated power and account for losses due to attenuation. Last but not least, the effect

of nonequilibrium phonons and lattice heating on intrinsic oscillations in DBTSs is an interesting

open question.

97

Chapter 4

Dissipative Transport in Superlattices within the Wigner Func-tion Formalism

We employ the Wigner function formalism to simulate partially coherent, dissipative electron

transport in biased semiconductor superlattices. We introduce a model collision integral with terms

that describe energy dissipation, momentum relaxation, and the decay of spatial coherences (lo-

calization). Based on a particle-based solution to the Wigner transport equation with the model

collision integral, we simulate quantum electronic transport at 10 K in a GaAs/AlGaAs superlat-

tice and accurately reproduce its current density vs field characteristics obtained in experiment.

4.1 Introduction

The Wigner function (WF) enables the phase-space formulation of quantum mechanics [124,

105, 125, 126]. It shares many of the characteristics of classical phase-space distribution func-

tion, but can take negative values in phase-space regions of volume smaller than a few ~, where

the Heisenberg uncertainly relation plays an important role [127, 128, 129]. The Wigner function

formalism has found use in many fields of physics, such as quantum optics [130, 131], nuclear

physics [132, 133], particle physics [134], and the semiclassical limits of quantum statistical me-

chanics [135].

The Wigner transport equation (WTE) is the equation of motion for the Wigner function (sim-

ilar to the semiclassical Boltzmann transport equation) and it has been used to model quantum

transport in many semiconductor nanostructures [136, 137, 93, 94, 138, 139, 140, 141, 100, 108,

129, 142, 143, 144, 96, 9]. However, the exact form of the collision integral for the Wigner

98

transport equation remains an outstanding open question. In the past, the effects of scattering

with phonons and impurities in the Wigner transport equation were accounted for via a semiclas-

sical Boltzmann collision operator, which employs transition rates calculated based on Fermi’s

golden rule [142]. This approach could be justified for the devices considered (resonant tunnel-

ing diodes [136, 137, 142], double-gate MOSFETs [107], semiconductor nanowires [144], and

carbon nanotube FETs [107]), where throughout most of the device region the electrons could

be considered semiclassical, pointlike particles with a well-defined kinetic energy vs wave vector

relationship E(k) from the envelope-function and effective-mass approximation. In contrast, elec-

trons in a biased superlattice may occupy quasibound states, with no simple relationship between

the particle energy and wave vector. Considering that the rates derived based on Fermi’s golden

rule rely on a well-defined kinetic energy pre- and post-scattering, it is clear that a Boltzmann

collision integral fails to capture scattering in superlattices. The material presented in this chapter

was published in Ref. [10].

4.2 The Wigner Transport Equation for Superlattices

We are interested in simulating electronic transport in biased III-V semiconductor superlat-

tices. Superlattices are systems formed by epitaxial growth of different materials with a certain

material sequence reperated tens or hundreds of times, which results in the periodic profile of

the conduction and valence band edges in equilibrium, with a period many times greater than the

crystalline lattice constant [123]. Here, we chose an experimentally well-characterized superlat-

tice structure from Ref. [5]; this structure has been proposed as a THz quantum cascade laser at

a frequency of 1.8 THz and temperatures up to 168 K. The structure consists of alternating GaAs

(wells) and Al0.15Ga0.85As (barriers) with a conduction band offset of 0.12 eV. Figure 4.1 shows

the layer structure, as well as the conduction-band profile obtained from the WTE simulation, cou-

pled with Poisson’s equation. The figure also shows several confined states obtained by solving

the Schrodinger equation at the electric field of 15 kV/cm (this field is also the design electric field

for the laser). The effective mass is m∗ = 0.067m0, where m0 is the free-electron rest mass.

99

50 40 30 20 10 0 10 20 30

Position [nm]

0.05

0.00

0.05

0.10

0.15

0.20

Pot

entia

l [eV

]

i

u

Figure 4.1: Steady-state conduction band profile obtained from the WTE (solid thick black line).

The dashed rectangle marks the extent of a single period, where the first barrier is referred to as

the injection barrier. Also shown are probability densities corresponding to wavefunction of the

upper (u) and lower (`) lasing levels (thick blue and thick red respectively). Also shown is one

of the injector (i) states (thick green) as well as other subbands (thin gray) that are localized near

the considered period. Starting at the injection barrier, the layer thicknesses (in nanometers) are

4.0/8.5/2.5/9.5/3.5/7.5/4.0/15.5, with the widest well n-doped to 2.05 × 1016 cm−3 resulting in

an average doping of 5.7 × 1015 cm−3 per period. (The layer structure has been rounded to the

nearest half-nm to match the simulation mesh spacing of ∆z = 0.5 nm.) Owing to the low electron

concentration, the potential (not including the barriers) is almost linear.

100

4.2.1 The Wigner Transport Equation

In order to simulate time-dependent electronic tranport in a generic semiconductor structure

with one-dimensional (1D) transport, we assume translational invariance in the x − y plane per-

pendicular to the transport direction, z. The 1D Wigner function f(z, k, t) is defined as the Wigner-

Weyl transform of the single-particle density matrix ρ via [105]

f(z, k, t) =1

2π

∫ρ(z + z′/2, z − z′/2)e−ikz

′dz′ . (4.1)

The device potential is split into a spatially slowly-varying term Vc(z), treated as a classical drift

term, and a rapidly varying term Vq(z). In a structure such as a biased superlattices, the applied

bias and the Hartree potential are part of the slowly-varying Vc(z), while the barriers are the only

terms in the fast-varying Vq(z). This splitting is justified by the low electron concentration in the

structure.

The Wigner transport equation describes the time evolution of the WF in the effective mass

approximation and can be written as [104, 108]

∂f

∂t= − ~k

m∗∂f

∂z+

1

~∂Vc∂z

∂f

∂k+Q[Vq, f ] + C[f ] , (4.2)

where C[f ] is the collision integral (discussed further below) and Q[V, f ] is the quantum evolution

term defined by

Q[Vq, f ] =

∫ ∞−∞

Vw[Vq](z, k − k′)f(z, k′)dk′ . (4.3)

Here, Vw[Vq](z, k) is the Wigner potential, which can be written as [145]

Vw[Vq](z, k) =2

π~Ime2ikzVq(2k)

, (4.4)

where Vq(k) is the spatial Fourier transform of Vq with the convention

Vq(k) =

∫ ∞−∞

Vq(z)e−ikzdz . (4.5)

101

4.2.2 The Quantum Evolution Term for a Periodic Potential

As the structure we consider has a periodic fast-varying potential, it can be written as

Vq(z) =∞∑

n=−∞

Vp(z − nLp) , (4.6)

where Vp(z) is the potential for a single period and Lp is the period length. The corresponding

Wigner potential is given by

Vw[Vq](z, k) =∞∑

n=−∞

Vw[Vp](z − nLp, k) . (4.7)

It is tempting to truncate the sum in Eq. (4.7) and only consider a few neighboring periods. How-

ever, with this approach, the resulting Wigner potential becomes very “spiky,” with rapid variation

in the k-direction and requiring a prohibitively fine mesh. A better approach is to use Eqs. (4.4)

and (4.7) and get

Vw[Vq](z, k) =2

π~Im

Vq(2k)e2izk

∞∑n=−∞

e−2inLpk

. (4.8)

Using∞∑

n=−∞

e−2inLpk =π

Lp

∞∑m=−∞

δ(k −mπ/Lp) , (4.9)

we get

Vw[Vq] =2

~LpImVq(2k)e2izk

∞∑m=−∞

δ(k −mπ/Lp) . (4.10)

The quantum evolution term in Eq. (B.5) then simplifies to

2

~Lp

∞∑m=−∞

ImVq(2mπ/Lp)e

2imπz/Lpf(z, k −mπ/Lp)

=∞∑

m=−∞

Wm(z)f(z, k −mπ/Lp) , (4.11)

where Wm is a quantum weight, defined as

Wm(z) ≡ 2

~LpImV (2mπ/Lp)e

2imπz/Lp. (4.12)

102

Because the superlattice potential is real, we have Vq(−k) = V ∗q (k). In addition, we have V (0) =

0, as the bias is included in the slowly varying potential, so the result in Eq. (4.12) can be simplified

to

Q[Vq, f ](z, k, t) =∞∑m=1

Wm(z) [f(z, k −mπ/Lp, t)

−f(z, k +mπ/Lp, t)] . (4.13)

The definition of the quantum weight Wm(z) is useful because it does not depend on k and can be

computed and stored in memory for all relevant values of m and z. It only needs to be recomputed

when the self-consistent potential is updated. In practice, the infinite sum over m is limited to the

values for which f(z, k ±mπ/Lp) is nonvanishing.

Here, we will be working with smoothed square barriers, so an analytical expression for the

corresponding quantum weightWm(z) is useful. A single smoothed square barrier of width 2a and

height V0, centered at the origin, is approximated by (erf is the error function)

VB(z) =V0

2− erf[β(z − a)] + erf[β(z + a)] . (4.14)

By performing a Fourier transform we get

VB(k) =2V0

ke−k

2/(4β2) sin(ka) , (4.15)

so

Wm[VB](z) =2

~πV0

me−m

2π2

β2L2p sin

(2πma

Lp

)sin

(2πmz

Lp

). (4.16)

The above equation represents a single barrier of height V0 centered at the origin. More general

structures can be made by using

Wm[V ](z) =2

~πe−m

2π2

β2L2p

NB∑i=1

Vim

sin

(2πmaiLp

)(4.17)

× sin

(2πm(z − zi)

Lp

), (4.18)

where Vi, zi and 2ai are the height, center and width of barrier i and NB is the total number of

barriers in a single period.

103

Before we continue, we will explain our choice of smoothed barriers. The reason is that for

more smoothing (smaller β), we can truncate the sum in Eq (4.18) at smaller m, avoiding the

rapidly oscillating high-m terms. This simplification allows use to use a coarser phase-space mesh,

speeding up numerical calculations. In this work we used a smoothing length β−1 = 1.0 nm. The

current-field characteristics are not very sensitive to the value of β and we see negligible difference

for β−1 in the range 0.5 to 1.0 nm. The choice of smoothed barriers can also be justified by physical

reasoning. Indeed, the lattice constant of GaAs is about 0.5 nm, so any variation in conduction band

will happen on that length scale.

4.2.3 Numerical Implementation

We solve the WTE given by Eq. (4.2) using a particle-based method with affinities [100, 142].

The WF is written as a linear combination of delta functions on the form

f(z, k, t) =∑p

Ap(t)δ(z − zp(t))δ(k − kp(t)) , (4.19)

where Ap(t) is the affinity of particle p. The diffusive and drift terms [the first and second terms in

Eq. (4.2), respectively] represent free flow and acceleration of particles via a time evolution of the

particle position zp(t) and wave vector kp(t), respectively. The quantum evolution and collision

terms do not affect particle position or wave vector, but induce time evolution of particle affinities

given by ∑p∈M(z,k)

dApdt

= Q[f, Vq](z, k) + C[f ] , (4.20)

where the sum is over all particles p belonging to a mesh point (z, k). We note that by including

the collision term in the time evolution of affinities, the particle method is entirely deterministic,

and does not require discretization of the rapidly oscillating diffusive term, which is difficult to

deal with using finite-difference methods [146].

In order to simulate superlattices, periodic boundary conditions are implemented by simply

removing particles when they exit the simulation domain and injecting them with the same wave

vector at the other side. This approach is valid as long as the simulation domain is large enough

104

that the coherences are not artificially cut off [147]. This condition is tested by increasing the

number of simulated periods until the simulation converges, typically 100 nm (two periods).

A good choice of the initial state f(z, k, t = 0) is important for faster convergence to a steady

state, as well as to ensure the steady-state WF is a valid WF [127, 148]. Here, we will note that,

given a valid initial state, the WF will remain valid at later times as long as the model collision

integral does not violate the positivity of the density matrix. In this work, we choose the thermal

equilibrium WF as the initial state. The steady state is reached in ∼ 1 − 5 ps, after which the

simulation is continued for additional ∼ 20 ps to calculate averages of physical observables, such

as the current density.

4.3 Modeling Dissipation and Decoherence in Superlattices: The CollisionOperator

In this section, we take a closer look at the collision integral C[f ] for the Wigner transport

equation, because superlattices pose challenges that are not nearly as dire when addressing other

common structures.

Thus far, considerable work has been done on resonant-tunneling diodes (RTDs) using the

Wigner transport equation [136, 137, 142]. This structure has a relatively small region where

tunneling or the formation of bound states is important, and where the relationship between k and

the kinetic energy cannot be written. Therefore, throughout most of the simulation domain, the

concept of the kinetic energy is well defined and the use of the scattering rates based on Fermi’s

golden rule is justified.

In contrast, in superlattices, tunneling and quasibound-state formation are important in the en-

tirety of the structure, and the rates calculated based on Fermi’s golden rule cannot be realiably

used. Employing these rates for superlattices can result in unphysically high currents and negative

particle densities. For example, with the use of the common rates for all standard scattering mech-

anisms [79], we have been unable to match the experiment for the structure from Fig. 1: the closest

match obtained for current densities – even after allowing for very high electronic temperatures or

artificially increased deformation potentials – is still several times higher than experiment. Yet,

105

we know that materials parameters are not the issue, because the calculations for the RTDs on the

same GaAs/AlGaAs system are quite accurate [101].

In this chapter, we abandon the semiclassical rates based on Fermi’s golden rule and instead

present a simple model collision operator comprising three terms, each with a solid intuitive foun-

dation and no requirement for an E vs k relationship. All three are necessary in order to match

the measured current–field characteristics, which is indictive of the interactions that the electrons

undergo in superlattices at relatively low tempertures.

The model collision integral is given by

C[f ] = − f(z, k, t)− feq(z, k)

τR(4.21a)

− f(z, k, t)− f(z,−k, t)τM

(4.21b)

+ Λ∂2f

∂k2. (4.21c)

As pointed out earlier, the time evolution induced by the collision term is included in the time

evolution of particle affinities according to Eq. (4.20). The second-order k-derivative is evaluated

using a second-order centered finite-difference scheme.

4.3.1 The Relaxation Term

The first term, (4.21a), has the form of the relaxation-time approximation collision integral in

semiclassical transport. This term describes relaxation towards equilibrium with a relaxation time

τR. This term dissipates energy and also partially randomizes momentum. Even in semiclassical

transport, this term technically holds only in the linear regime [79]. It turns out that superlattices

have a particular feature that make this approximation – relaxation to equilibrium as opposed to an

a priori unknown steady state – applicable up to high current densities.

Figure 4.2 shows the WFs for a few relevant subbands (upper lasing, lower lasing, and injector

states from Fig. 4.1) at 10 K and the electric field of 15 kV/cm. We find the occupations by taking

the subband wavefunctions ψn(z) and calculate the corresponding WFs fn(z, k), using Eq. (4.1)

with ρ(z1, z2) = ψn(z1)ψ∗n(z2). The noteworthy feature here is the width of each of the subband

WFs in the k-direction. The spread over k is of order inverse well width and is considerably

106

greater than the drift wave vector (the approximate shift of the state WFs along the k-axis) up

to very high current densities. Therefore, for the purpose of forming an approximate collision

integral, the nonequilibrium steady-state Wigner function in a superlattice can be approximated by

the equilibrium form even under appreciable current.

To calculate the equilibrium Wigner function feq(z, k), we first calculate the equilibrium den-

sity matrix. After the Weyl-Wigner transformation in the parallel r‖ variables and integration over

parallel wave vector k‖, the equilibrium density matrix ρeq(z1, z2) is given by

ρeq(z1, z2) = Nnorm

∑s

∫ π/Lp

−π/Lpψs,q(z1)ψ∗s,q(z2)

× ln(1 + e−(Es,q−µ)/kT )dq ,

(4.22)

where Nnorm is a normalization factor and ψs,q(z) are the envelope Bloch wave functions obtained

from a solution of the time-independent Schrodinger equation for the considered periodic potential.

The Fermi level µ is determined from

n =

∫D(E)(1 + e(E−µ)/kT )−1dE , (4.23)

where n is the electron density averaged over a period and D(E) is the density of states for the

superlattice. The equilibrium WF is then calculated using Eq. (4.1).

4.3.2 The Momentum-Relaxation Term

In general, momentum and energy relax at different rates, and the discrepancy is accommodated

by an explicit term (4.21b) that randomizes momentum at a rate of τ−1M , but does not dissipate

energy. This term would be very imporant in the presence of nearly elastic mechanisms, especially

if they are efficient at randomizing momentum, such as in the case of acoustic phonons that should

be accounted for down to a few Kelvin [79]. Without this term, current density can be order of

magnitude higher than experimental results, regardless of the choice of the relaxation time τR. This

term also preserves the positivity of the density matrix.

107

75 50 25 0 25 50 75

1.0

0.5

0.0

0.5

1.0

k [1

/nm

]

1.0 0.5 0.0 0.5 1.0

75 50 25 0 25 50 75

1.0

0.5

0.0

0.5

1.0

k [1

/nm

]

75 50 25 0 25 50 75

Position [nm]

1.0

0.5

0.0

0.5

1.0

k [1

/nm

]

lower lasing level ( )

upper lasing level (u)

injector state (i)

Figure 4.2: Wigner functions for an injector state (i), the upper lasing level (u), and the lower lasing

level (`). See figure 4.1 for the spatial profile of the corresponding wave functions.

108

4.3.3 The Spatial Decohrence (Localization) Term

The third term in Eq. (4.21) describes the decay of the spatial coherences of the density matrix,

namely,

Λ∂2f(z, k, t)

∂k2↔ −Λ(z1 − z2)2ρ(z1, z2, t) , (4.24)

This term describes the decay of the off-diagonal terms in the density matrix, i.e., spatial deco-

herence [101], while preserving the positive-definiteness of the density matrix [149]. As a result

of the action of this term, the density matrix becomes increasingly diagonal over time and we can

speak of well-defined position. Therefore, we will use the terms spatial decoherence and local-

ization of the density matrix inerchangaby; Λ has previously been referred to as the localization

rate [150], and we adopt the term here as well, even though it is technically a misnomer (Λ has

the units of inverse length squared and time, rather than the units of inverse time, which a quantity

called rate generally has). In the Wigner function formalism, coherence over large spatial distances

corresponds with fast oscillations of the WF in the k-direction [150]. The localization term (4.21c)

depicts diffusion in phase space and tends to smooth out the WF. In simple cases, it is possible to

calculate the localization rate Λ [151, 150]. Here, we will treat it as a phenomenological parameter

that is determined based on comparison with experiment.

There is an additonal, fairly subtle reason why this form is particularly well suited for the

description of spatial decoherence in nanostructures that carry current. Namely, nanostructures that

carry current are open systems with densely spaced energy levels and generally strong coupling

with the contacts [62]. Within the framework of the open systems theory [150, 64], this limit –

smaller energy spacing (per ~) than the typical relaxation rate of the system – corresponds to the

Brownian motion limit. 1 The spatial decoherence term (4.21c) has indeed been used to study

quantum Brownian motion [150, 64] due to coupling of an open system with an Ohmic bosonic

bath by Caldeira and Legget [152, 149, 150].

1This case is to be contrasted with the optical limit, in which a system is assumed only weakly coupled with theenvironment and energy levels spaced so far apart that the secular or rotating wave approximation (RWA) can be em-plyed. While tenuous in current-carrying nanostructures, the weak coupling approximation and RWA are nonethelessoften applied to derive master equations in quantum transport [62].

109

0 5 10 15 20 25

Electric field [kV/cm]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Cur

rent

den

sity

[kA

/cm

2]

TheoryExperiment

Figure 4.3: Current density vs electric field from theory (solid blue curve) and experimental results

from Ref. [5] (dashed red curve). τ−1R = 1012 s−1, τ−1

M = 2× 1013 s−1, and Λ = 2× 109 nm−2s−1.

4.4 Results

4.4.1 Comparison With Experiment

Figure 4.3 shows the calculated current density versus electric field and comparison with exper-

iment [5] for the GaAs/AlGaAs superlattice (Fig. 4.1) at a temperature of 10 K. The experimental

data is for a nonlasing structure (without a waveguide), so electron dynamics is unaffected by the

optical field. The experimental J − E curve was obtained from the measured J − V curve upon

incorporating a reported Schottky contact resistance of 4 V [5]. The best agreement with experi-

ment was achieved with τ−1R = 1012 s−1, τ−1

M = 2 × 1013 s−1, and Λ = 2 × 109 nm−2s−1. The

simulation gives current densities very close to experimental results, with plateaus around 8 kV/cm

and 15 kV/cm.

Figure 4.4 depicts the steady-state WF for different values of the electric field at a lattice tem-

perature of 10 K, the same temperature and with the same parameters as in Fig. 4.5. In the figure

we see that, at zero field, due to the low temperature, electrons almost exclusively occupy the

ground state in the widest well. When the electric field is increased to 7.5 kV/cm, tunneling be-

tween the widest well and the well adjacent to its right is enhanced. This back-and-fourth motion

110

results in very little net current, as we can see on the corresponding J − E curve in Fig. 4.3. At

the electric field of 15 kV/cm, the WF has an even higher amplitude in the narrower wells, with

pronounced negative values around k = 0, which is a signature of high occupation of states that

are delocalized between two wells.

4.4.2 Effects of the Different Terms in the Collision Integral (4.21) on the J−ECurves

All three terms in Eq. (4.21) are necessary to achieve good agreement with experiment. In

Fig. 4.5, only the relaxation term in (4.21a) is retained and τR is swept (τM = 0 and Λ = 0).

Despite varying τR over many orders of magnitude, we were unable to reproduce experimental

results with the term (4.21a) alone for any value of τR, with the closest results shown in Fig. 4.5.

The current density is overestimated by about an order of magnitude and the J − E dependence

has features that are absent from the experimental results.

While the relaxation term (4.21a) is responsible for energy dissipation and for relaxation of the

distribution function, it also somewhat relaxes momentum, but not efficiently enough. Figure 4.6

shows the effect of varying the relaxation rate τ−1R , while Fig. 4.7 shows the dependence of the

current density on the momentum relaxation rate τ−1M . In each figure, the other two parameters are

kept at their best-fit values, as in Fig. 4.3. We see that the effect of varying τ−1R by a factor of 4

does not have a pronounced effect on the J −E curve. However, the J −E curve is quite sensitive

to the momemtun relaxation rate, with a higher rate resulting in lower current, as expected. A

less obvious effect is that a high momentum relaxation rate destroys the fine features in the J −E

diagram, making the relationship almost linear for τ−1M = 4× 1013 s−1.

Figure 4.8 shows the dependence of the current density on the localization rate Λ. As intuitively

plausible, lower values of Λ result in more prominent fine features in the J −E diagram; these get

washed out at higher localization rates and the curve is smooth at Λ = 8× 109 nm−2s−1.

111

75 50 25 0 25 50 75

1.0

0.5

0.0

0.5

1.0

k [1

/nm

]

1.0 0.5 0.0 0.5 1.0

75 50 25 0 25 50 75

1.0

0.5

0.0

0.5

1.0

k [1

/nm

]

75 50 25 0 25 50 75

Position [nm]

1.0

0.5

0.0

0.5

1.0

k [1

/nm

]

15.0 kV/cm

7.5 kV/cm

0.0 kV/cm

Figure 4.4: Steady-state WF for three values of the electric field. The lattice temperature is 10 K.

Barriers are shown in black. All parameters are the same as in Fig 4.3.

112

0 5 10 15 20 25

Electric field [kV/cm]

1

0

1

2

3

4

5

6

Cur

rent

den

sity

[kA

/cm

2]

1R =4 ×1012 s 1

1R =8 ×1012 s 1

Experiment

Figure 4.5: Current density vs electric field for different values of the relaxation rate τ−1R with

Λ = 0 and τ−1M = 0.

0 5 10 15 20 25

Electric field [kV/cm]

0.0

0.5

1.0

1.5

2.0

Cur

rent

den

sity

[kA

/cm

2]

1R =0.5 ×1012 s 1

1R =1.0 ×1012 s 1

1R =2.0 ×1012 s 1

Figure 4.6: Current density vs electric field for different values of the relaxation rate τ−1R . τ−1

M =

2× 1013 s−1 and Λ = 2× 109 nm−2s−1, as in Fig. 4.3.

113

0 5 10 15 20 25

Electric field [kV/cm]

0.0

0.5

1.0

1.5

2.0

Cur

rent

den

sity

[kA

/cm

2]

1M =1 ×1013 s 1

1M =2 ×1013 s 1

1M =4 ×1013 s 1

Figure 4.7: Current density vs electric field for different values of the momentum relaxation rate

τ−1M . τ−1

R = 1012 s−1 and Λ = 2× 109 nm−2s−1, as in Fig. 4.3.

0 5 10 15 20 25

Electric field [kV/cm]

0.0

0.5

1.0

1.5

2.0

Cur

rent

den

sity

[kA

/cm

2]

=8.0 ×109 nm 2 s 1

=4.0 ×109 nm 2 s 1

=2.0 ×109 nm 2 s 1

=1.0 ×109 nm 2 s 1

=0.5 ×109 nm 2 s 1

Figure 4.8: Current density vs electric field for different values of the localization rate Λ,

withτ−1R = 1012 s−1 and τ−1

M = 2 × 1013 s−1. High localization rates wash out some of the

fine features of the J − E curves.

114

4.5 Conclusion to Chapter 4

We showed that the Wigner transport equation can be used to model electron transport in semi-

conductor superlattices. As an example, we considered a GaAs/AlGaAs superlattice of Ref. [5], in

which transport in partially coherent. The collision integral commonly used with the Boltzmann

transport equation is not adequate for use with the WTE in superlattices. Instead, we introduced a

model collision integral that comprises three terms: one that captures the dissipation of energy and

relaxation towards equilibrium, another that only relaxes momentum, and a third that describes lo-

calization, i.e., the decay of spatial coherences due to scattering. The steady-state J–E curves were

found to be in agreement with experiment, but are fairly sensitive to the values of the localization

and momentum relaxation rates.

115

Chapter 5

Outlook

In this work, we have employed two different, but closely related formalisms: the Wigner-

function and density-matrix formalisms. We will end this dissertation by briefly discussing the

merits and outlook for each formalism in the context of dissipative quantum transport in semicon-

ductor heterostructures.

As demonstrated in Chapter 2, the density-matrix formalism is ideal for devices with a discrete

energy spectrum, such as QCLs, where transport is driven by scattering between bound quantum

states. We showed that the density-matrix formalism can be used to accurately model QCLs, while

keeping computational complexity to a manageable level. In Chapter 2, we considered relatively

simple QCL designs, with only two different materials (the well and barrier materials), grown on a

substrate without strain. However, in state-of-art devices based in InGaAs/InAlAs [153], designs

with multiple alloy concentrations (not necessarily lattice-matched to InP) are used to maximize

performance by minimizing leakage into the states above the upper-lasing state. Although not

included in the present work, it is straightforward to model strain and multiple materials with the

k · p formalism [32]. We believe that the density-matrix model presented in this dissertation will

prove to be an invaluable asset for future optimization of the active core of such state-of-the-art

QCLs.

The Wigner-function formalism can be used to describe coherent time-dependent quantum-

transport in open quantum systems (connected to contacts) with a continuous energy spectrum.

Its particle-based numerical implementations offer an excellent technique for analyzing ballistic

time-dependent quantum transport in 1D systems (with recent exciting recent contributions on

higher-dimensional systems [154, 155, 156, 157]). If transport is approximately semiclassical in

116

most of the device region, effects of dissipation and decoherence can be accounted for by using

a Boltzmann scattering operator. This was the case for the double-barrier tunneling structures

studied in Chapter 3, where the Boltzmann collision operator was employed. As discussed in

Chapter 4, the Boltzmann collision operator is not applicable for transport in superlattices and we

proposed a model collision integral that helps accurately capture electronic transport in an experi-

mentally relevant superlattice system. However, the development of systematic approximations for

the collision term in the Wigner transport equation without fitting parameters is an open problem

that presently limits the applications of this intuitive and efficient transport formalism to realistic

systems with scattering.

117

LIST OF REFERENCES

[1] I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, “Band parameters for III-V compoundsemiconductors and their alloys,” J. Appl. Phys., 89(11), 5815–5875 (2001).

[2] A. Bismuto, R. Terazzi, M. Beck, and J. Faist, “Electrically tunable, high performance quan-tum cascade laser,” Appl. Phys. Lett., 96(14), 141105 (2010).

[3] M. Lindskog, J. M. Wolf, V. Trinite, V. Liverini, J. Faist, G. Maisons, M. Carras, R. Aidam,R. Ostendorf, and A. Wacker, “Comparative analysis of quantum cascade laser model-ing based on density matrices and non-equilibrium Green’s functions,” Appl. Phys. Lett.,105(10), 103106 (2014).

[4] E. Dupont, S. Fathololoumi, Z. R. Wasilewski, G. Aers, S. R. Laframboise, M. Lindskog,S. G. Razavipour, A. Wacker, D. Ban, and H. C. Liu, “A phonon scattering assisted injec-tion and extraction based terahertz quantum cascade laser,” J. Appl. Phys., 111(7), 073111(2012).

[5] S. Kumar, C. W. I. Chan, Q. Hu, and J. L. Reno, “A 1.8-THz quantum cascade laser operatingsignificantly above the temperature of ~ω/kB,” Nature Phys., 7, 166–171 (2011).

[6] O. Jonasson, F. Karimi, and I. Knezevic, “Partially coherent electron transport in terahertzquantum cascade lasers based on a Markovian master equation for the density matrix,” J.Comput. Electron., 15(4), 1192–1205 (2016).

[7] O. Jonasson, S. Mei, F. Karimi, J. Kirch, D. Botez, L. Mawst, and I. Knezevic, “Quantumtransport simulation of High-Power 4.6-µm quantum Cascade Lasers,” Photonics, 3(2), 38(2016).

[8] O. Jonasson and I. Knezevic, “Density-matrix approach to mid-infrared quantum cascadelasers (working title),” (in preparation) (2016).

[9] O. Jonasson and I. Knezevic, “Coulomb-driven terahertz-frequency intrinsic current oscil-lations in a double-barrier tunneling structure,” Phys. Rev. B, 90, 165415 (2014).

[10] O. Jonasson and I. Knezevic, “Dissipative transport in superlattices within the Wigner func-tion formalism,” J. Comput. Electron., 14(4), 879–887 (2015).

118

[11] J. Faist, Quantum cascade lasers, Oxford University Press (2013).

[12] L. Esaki and R. Tsu, “Superlattice and negative differential conductivity in semiconductors,”IBM J. Res. Dev., 14(1), 61–65 (1970).

[13] M. Bahriz, G. Lollia, P. Laffaille, A. N. Baranov, and R. Teissier, “InAs/AlSb quantumcascade lasers operating near 20 µm,” Electron. Lett., 49(19), 1238–1240 (2013).

[14] J. Faist, F. Capasso, D. L. Sivco1, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantumcascade laser,” Science, 264, 553–556 (1994).

[15] M. S. Vitiello, G. Scalari, B. Williams, and P. D. Natale, “Quantum cascade lasers: 20 yearsof challenges,” Opt. Express, 23(4), 5167–5182 (2015).

[16] M. Beck, D. Hofstetter, T. Aellen, J. Faist, U. Oesterle, M. Ilegems, E. Gini, and H. Mel-chior, “Continuous wave operation of a mid-infrared semiconductor laser at room tempera-ture,” Science, 295(5553), 301–305 (2002).

[17] R. Kohler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A.Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature,417, 156–159 (2002).

[18] Y. Bai, S. R. Darvish, S. Slivken, W. Zhang, A. Evans, J. Nguyen, and M. Razeghi, “Roomtemperature continuous wave operation of quantum cascade lasers with watt-level opticalpower,” Appl. Phys. Lett., 92(10), 101105 (2008).

[19] Y. Bai, S. Slivken, S. Kuboya, S. R. Darvish, and M. Razeghi, “Quantum cascade lasers thatemit more light than heat,” Nat. Photon., 4, 99–102 (2010).

[20] P. Q. Liu, A. J. Hoffman, M. D. Escarra, K. J. Franz, J. B. Khurgin, Y. Dikmelik, X. Wang,J.-Y. Fan, and C. F. Gmachl, “Highly power-efficient quantum cascade lasers,” Nat. Photon.,4, 95–98 (2010).

[21] Y. Bai, N. Bandyopadhyay, S. Tsao, S. Slivken, and M. Razeghi, “Room temperature quan-tum cascade lasers with 27% wall plug efficiency,” Appl. Phys. Lett., 98(18), 181102 (2011).

[22] B. S. Williams, “Terahertz quantum-cascade lasers,” Nat. Photon., 1, 517–525 (2007).

[23] M. Wienold, B. Roben, L. Schrottke, R. Sharma, A. Tahraoui, K. Biermann, and H. T.Grahn, “High-temperature, continuous-wave operation of terahertz quantum-cascade laserswith metal-metal waveguides and third-order distributed feedback,” Opt. Express, 22(3),3334–3348 (2014).

[24] S. Fathololoumi, E. Dupont, C. Chan, Z. Wasilewski, S. Laframboise, D. Ban, A. Matyas,C. Jirauschek, Q. Hu, and H. C. Liu, “Terahertz quantum cascade lasers operating up to200 K with optimized oscillator strength and improved injection tunneling,” Opt. Express,20(4), 3866–3876 (2012).

119

[25] E. Dupont, S. Fathololoumi, and H. Liu, “Simplified density-matrix model applied to three-well terahertz quantum cascade lasers,” Phys. Rev. B, 81, 205311 (2010).

[26] C. Jirauschek and T. Kubis, “Modeling techniques for quantum cascade lasers,” Appl. Phys.Rev., 1, 011307 (2014).

[27] D. Indjin, P. Harrison, R. W. Kelsall, and Z. Ikonic, “Self-consistent scattering theory oftransport and output characteristics of quantum cascade lasers,” J. Appl. Phys., 91(11),9019–9026 (2002).

[28] D. Indjin, P. Harrison, R. W. Kelsall, and Z. Ikonic, “Influence of leakage current on tem-perature performance of GaAs/AlGaAs quantum cascade lasers,” Appl. Phys. Lett., 81(3),400–402 (2002).

[29] A. Mircetic, D. Indjin, Z. Ikonic, P. Harrison, V. Milanovic, and R. W. Kelsall, “Towardsautomated design of quantum cascade lasers,” J. Appl. Phys., 97(8), 084506 (2005).

[30] R. Iotti and F. Rossi, “Nature of charge transport in quantum-cascade lasers,” Phys. Rev.Lett., 87, 146603 (2001).

[31] H. Callebaut, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, “Importance of electron-impurity scattering for electron transport in terahertz quantum-cascade lasers,” Appl. Phys.Lett., 84(5), 645–647 (2004).

[32] X. Gao, D. Botez, and I. Knezevic, “X-valley leakage in GaAs-based midinfrared quantumcascade lasers: A Monte Carlo study,” J. Appl. Phys., 101(6), 063101 (2007).

[33] Y. B. Shi and I. Knezevic, “Nonequilibrium phonon effects in midinfrared quantum cascadelasers,” J. Appl. Phys., 116(12), 123105 (2014).

[34] H. Willenberg, G. H. Dohler, and J. Faist, “Intersubband gain in a Bloch oscillator andquantum cascade laser,” Phys. Rev. B, 67, 085315 (2003).

[35] S. Kumar and Q. Hu, “Coherence of resonant-tunneling transport in terahertz quantum-cascade lasers,” Phys. Rev. B, 80, 245316 (2009).

[36] C. Weber, A. Wacker, and A. Knorr, “Density-matrix theory of the optical dynamics andtransport in quantum cascade structures: The role of coherence,” Phys. Rev. B, 79, 165322(2009).

[37] R. Terazzi and J. Faist, “A density matrix model of transport and radiation in quantumcascade lasers,” New J. Phys., 12(3), 033045 (2010).

[38] S.-C. Lee and A. Wacker, “Nonequilibrium Green’s function theory for transport and gainproperties of quantum cascade structures,” Phys. Rev. B, 66, 245314 (2002).

120

[39] M. Bugajski, P. Gutowski, P. Karbownik, A. Kolek, G. Haldas, K. Pierscinski, D. Pierscin-ska, J. Kubacka-Traczyk, I. Sankowska, A. Trajnerowicz, K. Kosiel, A. Szerling, J. Grzonka,K. Kurzydlowski, T. Slight, and W. Meredith, “Mid-IR quantum cascade lasers: Devicetechnology and non-equilibrium Green’s function modeling of electro-optical characteris-tics,” Phys. Status Solidi B, 251 (2014).

[40] A. Wacker, M. Lindskog, and D. O. Winge, “Nonequilibrium Green’s function model forsimulation of quantum cascade laser devices under operating conditions,” IEEE J. Sel. Top.Quantum Electron., 19(5), 1–11 (2013).

[41] A. Matyas, P. Lugli, and C. Jirauschek, “Photon-induced carrier transport in high efficiencymidinfrared quantum cascade lasers,” J. Appl. Phys., 110(1), 013108 (2011).

[42] A. Kolek, G. Haldas, M. Bugajski, K. Pierscinski, and P. Gutowski, “Impact of injector dop-ing on threshold current of mid-infrared quantum cascade laser- non-equilibrium Green’sfunction analysis,” IEEE J. Sel. Topics Quantum Electron., 21(1), 124–133 (2015).

[43] H. Callebaut and Q. Hu, “Importance of coherence for electron transport in terahertz quan-tum cascade lasers,” J. Appl. Phys., 98(10), 104505 (2005).

[44] P. Harrison, D. Indjin, and R. W. Kelsall, “Electron temperature and mechanisms of hotcarrier generation in quantum cascade lasers,” J. Appl. Phys., 92(11), 6921–6923 (2002).

[45] R. C. Iotti and F. Rossi, “Microscopic theory of semiconductor-based optoelectronic de-vices,” Rep. Prog. Phys., 68(11), 2533 (2005).

[46] N. Moiseyev, “Quantum theory of resonances: calculating energies, widths and cross-sections by complex scaling,” Phys. Rep., 302(5-6), 212–293 (1998).

[47] R. L. Terazzi, Transport in quantum cascade lasers, Ph.D. thesis, ETH Zurich (2012).

[48] F. Karimi, A. H. Davoody, and I. Knezevic, “Dielectric function and plasmons in graphene:A self-consistent-field calculation within a Markovian master equation formalism,” Phys.Rev. B, 93, 205421 (2016).

[49] R. C. Iotti, E. Ciancio, and F. Rossi, “Quantum transport theory for semiconductor nanos-tructures: A density-matrix formulation,” Phys. Rev. B, 72, 125347 (2005).

[50] M. V. Fischetti, “Master-equation approach to the study of electronic transport in smallsemiconductor devices,” Phys. Rev. B, 59, 4901–4917 (1999).

[51] G. Bastard, Wave mechanics applied to semiconductor heterostructures, Monographies dephysique, Les Editions de Physique (1988).

[52] S. L. Chuang, Physics of optoelectronic devices (Wiley series in pure and applied optics)(1995).

121

[53] D. F. Nelson, R. C. Miller, and D. A. Kleinman, “Band nonparabolicity effects in semicon-ductor quantum wells,” Phys. Rev. B, 35, 7770–7773 (1987).

[54] A. Kolek, G. Haldas, and M. Bugajski, “Nonthermal carrier distributions in the subbands of2-phonon resonance mid-infrared quantum cascade laser,” Appl. Phys. Lett., 101(6), 061110(2012).

[55] U. Ekenberg, “Nonparabolicity effects in a quantum well: Sublevel shift, parallel mass, andLandau levels,” Phys. Rev. B, 40, 7714–7726 (1989).

[56] C. Sirtori, F. Capasso, J. Faist, and S. Scandolo, “Nonparabolicity and a sum rule associatedwith bound-to-bound and bound-to-continuum intersubband transitions in quantum wells,”Phys. Rev. B, 50, 8663–8674 (1994).

[57] T. B. Bahder, “Eight-band k · p model of strained zinc-blende crystals,” Phys. Rev. B, 41,11992–12001 (1990).

[58] E. O. Kane, Semiconductors and semimetals, volume 1, Academic, New York (1966).

[59] C. R. Pidgeon and R. N. Brown, “Interband magneto-absorption and Faraday rotation inInSb,” Phys. Rev., 146, 575–583 (1966).

[60] J. M. Luttinger and W. Kohn, “Motion of electrons and holes in perturbed periodic fields,”Phys. Rev., 97, 869–883 (1955).

[61] L. C. L. Y. V. Morten Willatzen, The k p Method: Electronic Properties of Semiconductors,Springer-Verlag Berlin Heidelberg, 1 edition (2009).

[62] I. Knezevic and B. Novakovic, “Time-dependent transport in open systems based on quan-tum master equations,” J. Comput. Electron., 12(3), 363–374 (2013).

[63] H. Frohlich, “Theory of electrical breakdown in ionic crystals,” Proc. R. Soc. Lond. A, 160,220–241 (1937).

[64] F. P. Heinz-Peter Breuer, The theory of open quantum systems, Oxford University Press(2002).

[65] C. Jirauschek and P. Lugli, “Monte-Carlo-based spectral gain analysis for terahertz quantumcascade lasers,” J. Appl. Phys., 105(12), 123102 (2009).

[66] T. Kubis, C. Yeh, and P. Vogl, “Quantum theory of transport and optical gain in quantumcascade lasers,” Phys. Status Solidi C, 5(1), 232–235 (2008).

[67] Y. Chiu, Y. Dikmelik, P. Q. Liu, N. L. Aung, J. B. Khurgin, and C. F. Gmachl, “Importanceof interface roughness induced intersubband scattering in mid-infrared quantum cascadelasers,” Appl. Phys. Lett., 101(17), 171117 (2012).

122

[68] D. K. Ferry, Semiconductors, 2053-2563, IOP Publishing (2013).

[69] P. Roblin, R. C. Potter, and A. Fathimulla, “Interface roughness scattering in AlAs/InGaAsresonant tunneling diodes with an InAs subwell,” J. Appl. Phys., 79(5), 2502–2508 (1996).

[70] A. Evans, S. R. Darvish, S. Slivken, J. Nguyen, Y. Bai, and M. Razeghi, “Buried heterostruc-ture quantum cascade lasers with high continuous-wave wall plug efficiency,” Appl. Phys.Lett., 91(7), 071101 (2007), layer structure given for the high wallplug device.

[71] T. Ando, “Self-Consistent Results for a GaAs/AlxGa1-xAs Heterojunciton. II. Low Tem-perature Mobility,” J. Phys. Soc. Jpn., and J. Phys. Soc., 51(12), 3900–3907 (1982).

[72] S. W. K. Hartmut Haug, Quantum Theory of Optical and Electronic Properties of Semicon-ductors (4th edition), World Scientific Publishing Company (2004).

[73] D. Calecki, J. F. Palmier, and A. Chomette, “Hopping conduction in multiquantum wellstructures,” J. Phys. C: Solid state, 17(28), 5017 (1984).

[74] B. Gelmont, V. Gorfinkel, and S. Luryi, “Theory of the spectral line shape and gain inquantum wells with intersubband transitions,” Appl. Phys. Lett., 68(16), 2171–2173 (1996).

[75] Y. Saad, Iterative Methods for Sparse Linear Systems (2 ed.), SIAM (2003).

[76] R. C. Iotti and F. Rossi, “Carrier thermalization versus phonon-assisted relaxation inquantum-cascade lasers: A Monte Carlo approach,” Appl. Phys. Lett., 78(19), 2902–2904(2001).

[77] V. Spagnolo, G. Scamarcio, H. Page, and C. Sirtori, “Simultaneous measurement of the elec-tronic and lattice temperatures in GaAs/Al0.45Ga0.55As quantum-cascade lasers: Influenceon the optical performance,” Appl. Phys. Lett., 84(18), 3690–3692 (2004).

[78] H. Page, C. Becker, A. Robertson, G. Glastre, V. Ortiz, and C. Sirtori, “300 K operation ofa GaAs-based quantum-cascade laser at 9 m,” Appl. Phys. Lett., 78(22), 3529–3531 (2001).

[79] M. Lundstrom, Fundamentals of carrier transport (2nd edition), Cambridge UniversityPress (2000).

[80] R. C. Iotti and F. Rossi, “Microscopic theory of quantum-cascade lasers,” SemiconductorScience and Technology, 19(4), S323 (2004).

[81] M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photon., 1, 97–105 (2007).

[82] D. M. Mittleman, J. Cunningham, M. Nuss, and M. Geva, “Noncontact semiconductor wafercharacterization with the terahertz Hall effect,” Appl. Phys. Lett., 71, 16–18 (1997).

[83] D. Hashimshony, I. Geltner, G. Cohen, Y. Avitzour, A. Zigler, and C. Smith, “Characteriza-tion of the electrical properties and thickness of thin epitaxial semiconductor layers by THzreflection spectroscopy,” J. Appl. Phys., 90(11), 5778–5781 (2001).

123

[84] A. Dobroiu, M. Yamashita, Y. N. Ohshima, Y. Morita, C. Otani, and K. Kawase, “TerahertzImaging System Based on a Backward-Wave Oscillator,” Appl. Opt., 43(30), 5637–5646(2004).

[85] K. Kawase, Y. Ogawa, Y. Watanabe, and H. Inoue, “Non-destructive terahertz imaging ofillicit drugs using spectral fingerprints,” Opt. Express, 11(20), 2549–2554 (2003).

[86] J. H. Booske, “Plasma physics and related challenges of millimeter-wave-to-terahertz andhigh power microwave generationa),” Phys. Plasmas, 15(5), 055502 (2008).

[87] T. C. L. G. Sollner, E. R. Brown, W. D. Goodhue, and H. Q. Le, “Observation of millimeterwave oscillations from resonant tunneling diodes and some theoretical considerations ofultimate frequency limits,” Appl. Phys. Lett., 50(6), 332–334 (1987).

[88] N. Orihashi, S. Suzuki, and M. Asada, “One THz harmonic oscillation of resonant tunnelingdiodes,” Appl. Phys. Lett., 87(23), 233501 (2005).

[89] S. Suzuki, M. Asada, A. Teranishi, H. Sugiyama, and H. Yokoyama, “Fundamental oscil-lation of resonant tunneling diodes above 1 THz at room temperature,” Appl. Phys. Lett.,97(24), 242102 (2010).

[90] M. Feiginov, C. Sydlo, O. Cojocari, and P. Meissner, “Resonant-tunnelling-diode oscillatorsoperating at frequencies above 1.1 Hz,” Appl. Phys. Lett., 99(23), 233506 (2011).

[91] H. Kanaya, S. Suzuki, and M. Asada, “Terahertz oscillation of resonant tunneling diodeswith deep and thin quantum wells,” IEICE Electron. Express, 10(18), 20130501–20130501(2013).

[92] D. L. Woolard, E. R. Brown, F. A. Buo, X. J. Lu, D. L. Rhodes, and B. S. Perlman, “In-trinsic oscillations in resonant tunneling structures,” in “Device Research Conference, 1995.Digest. 1995 53rd Annual,” (1995).

[93] K. L. Jensen and F. A. Buo, “Numerical simulation of intrinsic bistability and high-frequency current oscillations in resonant tunneling structures,” Phys. Rev. Lett., 66, 1078–1081 (1991).

[94] B. A. Biegel and J. D. Plummer, “Comparison of self-consistency iteration options forthe Wigner function method of quantum device simulation,” Phys. Rev. B, 54, 8070–8082(1996).

[95] Z. Dai and J. Ni, “Tuning bistability and current hysteresis in resonant-tunneling structures,”Phys. Rev. B, 73, 113309 (2006).

[96] P. Wojcik, J. Adamowski, M. Wołoszyn, and B. J. Spisak, “Intrinsic oscillations of spincurrent polarization in a paramagnetic resonant tunneling diode,” Phys. Rev. B, 86, 165318(2012).

124

[97] P. Zhao, D. L. Woolard, and H. L. Cui, “Multisubband theory for the origination of intrinsicoscillations within double-barrier quantum well systems,” Phys. Rev. B, 67, 085312 (2003).

[98] A. Sakurai and Y. Tanimura, “Self-excited current oscillations in a resonant tunneling diodedescribed by a model based on the CaldeiraLeggett Hamiltonian,” New J. Phys., 16, 015002(2014).

[99] A. Sakurai and Y. Tanimura, “An approach to quantum transport based on reduced hierarchyequations of motion: Application to a resonant tunneling diode,” J. Phys. Soc. Jpn., 82(3),033707 (2013).

[100] L. Shifren and D. Ferry, “Particle Monte Carlo simulation of Wigner function tunneling,”Phys. Lett. A, 285(34), 217 – 221 (2001).

[101] D. Querlioz and P. Dollfus, The Wigner Monte-Carlo method for nanoelectronic devices: Aparticle description of quantum transport and decoherence, ISTE, Wiley (2013).

[102] P. Wojcik, B. J. Spisak, M. Woloszyn, and J. Adamowski, “Tuning of terahertz intrinsicoscillations in asymmetric triple-barrier resonant tunneling diodes,” J. Appl. Phys., 111,124310 (2012).

[103] W. R. Frensley, “Boundary conditions for open quantum systems driven far from equilib-rium,” Rev. Mod. Phys., 62, 745–791 (1990).

[104] C. Jacoboni, R. Brunetti, P. Bordone, and A. Bertoni, “Quantum transport and its simulationwith the Wigner-function approach,” Int. J. Hi. Spe. Ele. Syst., 11, 387 (2001).

[105] E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev., 40,749–759 (1932).

[106] P. Wojcik, B. Spisak, M. Woloszyn, and J. Adamowski, “Application of non-classical distri-bution function to transport properties of semiconductor nanodevices,” Acta. Phys. Pol. A,114, 1431 (2008).

[107] D. Querlioz, H.-N. Nguyen, J. Saint-Martin, A. Bournel, S. Galdin-Retailleau, and P. Doll-fus, “Wigner-Boltzmann Monte Carlo approach to nanodevice simulation: from quantum tosemiclassical transport,” J. Comput. Electron., 8, 324–335 (2009).

[108] M. Nedjalkov, H. Kosina, S. Selberherr, C. Ringhofer, and D. K. Ferry, “Unified particleapproach to Wigner-Boltzmann transport in small semiconductor devices,” Phys. Rev. B, 70,115319 (2004).

[109] M. Nedjalkov, P. Schwaha, S. Selberherr, J. M. Sellier, and D. Vasileska, “Wigner quasi-particle attributes - An asymptotic perspective,” Appl. Phys. Lett., 102, 163113 (2013).

[110] M. Nedjalkov, S. Selberherr, and I. Dimov, Numerical methods and applications, Springer,Berlin, Heidelberg (2011).

125

[111] L. Shifren and D. Ferry, “Wigner function quantum Monte Carlo,” Phys. B, 314, 72 – 75(2002), proceedings of the Twelfth International Conference on Nonequilib rium CarrierDynamics in Semiconductors.

[112] L. Shifren, C. Ringhofer, and D. K. Ferry, “A Wigner function-based quantum ensembleMonte Carlo study of a resonant tunneling diode,” IEEE Trans. Electron Dev., 50, 769–773(2003).

[113] W. Shockley, “Currents to conductors induced by a moving point charge,” J. Appl. Phys.,9(10), 635–636 (1938).

[114] S. Ramo, “Currents induced by electron motion,” Proceedings of the IRE, 27(9), 584–585(1939).

[115] D. Querlioz, J. Saint-Martin, and P. Dollfus, “Implementation of the Wigner-Boltzmanntransport equation within particle Monte Carlo simulation,” J. Comput. Electron., 9, 224–231 (2010).

[116] D. Vasileska, K. Raleeva, S. M. Goodnick, C. Ringhofer, S. S. Ahmed, N. Ashraf,A. Hossain, R. Hathwar, A. Ashok, and B. Padmanabhan, Monte Carlo Device Simu-lations, chapter Nanohub Resources on Monte Carlo Device Simulations Study, 1–76,http://www.nanohub.org/resources/10579/download.

[117] H. Kressel and J. K. Butler, Semiconductor lasers and heterojunction LEDs, Academic Pr(1977).

[118] P. Wojcik, B. J. Spisak, M. Woloszyn, and J. Adamowski, “Intrinsic current oscillationsin an asymmetric triple-barrier resonant tunnelling diode,” Semicond. Sci. Technol., 25, 12(2010).

[119] B. Ricco and M. Y. Azbel, “Physics of resonant tunneling. The one-dimensional double-barrier case,” Phys. Rev. B, 29, 1970–1981 (1984).

[120] Y. Zhang, R. Klann, K. H. Ploog, and H. T. Grahn, “Observation of bistability in GaAs/AlAssuperlattices,” Appl. Phys. Lett., 70(21), 2825–2827 (1997).

[121] J. Kastrup, R. Klann, H. T. Grahn, K. Ploog, L. L. Bonilla, J. Galan, M. Kindelan,M. Moscoso, and R. Merlin, “Self-oscillations of domains in doped GaAs-AlAs superlat-tices,” Phys. Rev. B, 52, 13761–13764 (1995).

[122] J. Kastrup, R. Hey, K. H. Ploog, H. T. Grahn, L. L. Bonilla, M. Kindelan, M. Moscoso,A. Wacker, and J. Galan, “Electrically tunable GHz oscillations in doped GaAs-AlAs super-lattices,” Phys. Rev. B, 55, 2476–2488 (1997).

[123] A. Wacker, “Semiconductor superlattices: a model system for nonlinear transport,” Phys.Rep., 357(1), 1 (2002).

126

[124] H. Weyl, “Quantenmechanik und gruppentheorie,” Physik. Z., 46, 1–46 (1927).

[125] H. Groenewold, “On the principles of elementary quantum mechanics,” Physica, 12(7), 405– 460 (1946).

[126] J. E. Moyal, “Quantum mechanics as a statistical theory,” Math. Proc. Cambridge, 45, 99–124 (1949).

[127] V. I. Tatarskii, “The Wigner representation of quantum mechanics,” Sov. Phys. Uspekhi,26(4), 311 (1983).

[128] M. Hillery, R. O’Connell, M. Scully, and E. Wigner, “Distribution functions in physics:Fundamentals,” Phys. Rep., 106(3), 121 – 167 (1984).

[129] C. Jacoboni and P. Bordone, “The Wigner-function approach to non-equilibrium electrontransport,” Rep. Prog. Phys., 67(7), 1033 (2004).

[130] U. Leonhardt, Measuring the quantum state of light, Cambridge University Press, Cam-bridge (1997).

[131] W. P. Schleich, Quantum optics in phase space, Wiley-VCH, Weinheim (2002).

[132] G. A. Baker, I. E. McCarthy, and C. E. Porter, “Application of the Phase Space Quasi-Probability Distribution to the Nuclear Shell Model,” Phys. Rev., 120, 254–264 (1960).

[133] S. Shlomo and M. Prakash, “Phase space distribution of an N-dimensional harmonic oscil-lator,” Nuclear Physics A, 357(1), 157 – 170 (1981).

[134] A. Belitsky, X. Ji, and F. Yuan, “Quark imaging in the proton via quantum phase-spacedistributions,” Phys. Rev. D, 69, 074014 (2004).

[135] K. Imre, E. Ozizmir, M. Rosenbaum, and P. F. Zweifel, “Wigner method in quantum statis-tical mechanics,” J. Math. Phys., 8(5), 1097–1108 (1967).

[136] W. R. Frensley, “Transient response of a tunneling device obtained from the Wigner func-tion,” Phys. Rev. Lett., 57, 2853–2856 (1986).

[137] N. C. Kluksdahl, A. M. Kriman, D. K. Ferry, and C. Ringhofer, “Self-consistent study ofthe resonant-tunneling diode,” Phys. Rev. B, 39, 7720–7735 (1989).

[138] A. Bertoni, P. Bordone, R. Brunetti, and C. Jacoboni, “The Wigner function for electrontransport in mesoscopic systems,” J. Phys.: Condens. Matter, 11, 5999 (1999).

[139] P. Bordone, M. Pascoli, R. Brunetti, A. Bertoni, C. Jacoboni, and A. Abramo, “Quantumtransport of electrons in open nanostructures with the Wigner-function formalism,” Phys.Rev. B, 59, 3060–3069 (1999).

127

[140] F. A. Buot, P. Zhao, H. L. Cui, D. L. Woolard, K. L. Jensen, and C. M. Krowne, “Emitterquantization and double hysteresis in resonant-tunneling structures: A nonlinear model ofcharge oscillation and current bistability,” Phys. Rev. B, 61, 5644–5665 (2000).

[141] J. Garcia-Garcia and F. Martin, “Simulation of multilayered resonant tunneling diodes usingcoupled Wigner and Boltzmann distribution function approaches,” Appl. Phys. Lett., 77,3412 (2000).

[142] D. Querlioz, P. Dollfus, V.-N. Do, A. Bournel, and V. L. Nguyen, “An improved WignerMonte-Carlo technique for the self-consistent simulation of RTDs,” J. Comput. Electron., 5,443–446 (2006).

[143] Z. H. Dai, J. Ni, Y. M. Sun, and W. T. Wang, “Dynamical behavior of electron transport inAlGaAs/GaAs double-barrier structures under a high-frequency radiation field,” Eur. Phys.J. B, 60(4), 439–446 (2007).

[144] S. Barraud, “Dissipative quantum transport in silicon nanowires based on Wigner transportequation,” J. Appl. Phys., 110(9), 093710 (2011).

[145] M. Nedjalkov, H. Kosina, and P. Schwaha, “Device modeling in the Wigner picture,” J.Comput. Electron., 9, 218–223 (2010).

[146] K.-Y. Kim and B. Lee, “On the high order numerical calculation schemes for the Wignertransport equation,” Solid-State Electronics, 43(12), 2243–2245 (1999).

[147] O. Jonasson and I. Knezevic, “On the boundary conditions for the Wigner transport equation(in prep),” .

[148] N. C. Dias and J. N. Prata, “Admissible states in quantum phase space,” Annals of Physics,313(1), 110 – 146 (2004).

[149] L. Diosi, “Calderia-Leggett master equation and medium temperatures,” Phys. A, 199(3-4),517 – 526 (1993).

[150] E. Joos, H. D. Zeh, C. Kiefer, D. J. W. Giulini, J. Kupsch, and I.-O. Stamatescu, Decoher-ence and the appearance of a classical world in quantum theory, Springer (2003).

[151] H. Dekker, “Classical and quantum mechanics of the damped harmonic oscillator,” Phys.Rep., 80(1), 1 – 110 (1981).

[152] A. Caldeira and A. Leggett, “Path integral approach to quantum Brownian motion,” Phys.A, 121(3), 587–616 (1983).

[153] J. D. Kirch, C.-C. Chang, C. Boyle, L. J. Mawst, D. Lindberg, T. Earles, and D. Botez, “86%internal differential efficiency from 8 to 9 µm-emitting, step-taper active-region quantumcascade lasers,” Opt. Express, 24(21), 24483–24494 (2016).

128

[154] J. M. Sellier, M. Nedjalkov, I. Dimov, and S. Selberherr, “Decoherence and time reversibil-ity: The role of randomness at interfaces,” J. Appl. Phys., 114(17), 174902 (2013).

[155] J. Sellier, M. Nedjalkov, I. Dimov, and S. Selberherr, “Two-dimensional transient wignerparticle model,” in “Simulation of Semiconductor Processes and Devices (SISPAD), 2013International Conference on,” 404–407 (2013).

[156] J. Sellier and I. Dimov, “The Wigner-Boltzmann Monte Carlo method applied to electrontransport in the presence of a single dopant,” Comput. Phys. Commun., 185(10), 2427 –2435 (2014).

[157] P. Ellinghaus, J. Weinbub, M. Nedjalkov, S. Selberherr, and I. Dimov, “Distributed-memoryparallelization of the Wigner Monte Carlo method using spatial domain decomposition,” J.Comput. Electron., 14(1), 151–162 (2015).

[158] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products (7th edition),Elsevier/Academic Press, Amsterdam (2007).

[159] A. Arnold and F. Nier, “Numerical analysis of the deterministic particle method applied tothe Wigner equation,” Math. Comp., 58, 645–669 (1992).

129

Appendix A: Detailed Calculation of Scattering Rates

A.1 Longitudinal Acoustic Phonons

We will start with longitudinal acoustic (LA) phonons, which have the following scattering

weight for absorption [68]

W+LA(Q) = NQ

~2D2acQ

2

2ρEQ=(eEQ/kT − 1

)−1 D2acEQ

2ρv2s

, (A.1)

where we have used the LA phonon dispersion relationEQ = ~vsQ, where vs is the speed of sound

in the medium, ρ the mass density and Dac is the acoustic derformation potential. We can not

directly use the expression in Eq. (A.1) because the phonon energy is not constant, which was an

approximation we used to derive Eq. (2.48). It is possible to discretize the relevant range of Q and

treat LA phonons with different magnitude Q as independent scattering mechanism. However, we

would need to calculate and store the corresponding rates for each value of Q, which is prohibitive

in terms of cpu time as well as memory. Another option is to consider the limit EQ kT and

treat the LA phonons as an elastic scattering mechanism. However, because the LA phonons can

exchange an arbitarily small energy with the lattice, they play the important role of low energy

thermalization at energies much smaller than the polar optical phonon. A second reason to inclute

a nonzero energy for the LA phonons is that they tend to smooth the in-plane energy distribution

of the density matrix, greatly facilitating numerical convergence. For this reason we follow the

treatment in Ref. [38] and approximate acoustic phonons as having a single energy ELA kT .

Typically we will asume the LA phonon energy is equal to the energy spacing in the simulation

(0.5-2 meV). Using this approximation, we get

W±LA(Q) =

(1

2∓ 1

2+NLA

)D2

acELA

2ρv2s

, (A.2)

130

with NLA = (exp(ELA/kT )− 1)−1. The expression in Eq. (A.2) is isotropic (does not depend on

Q), so the angular integration in Eq. (2.48) is trivial and we get

Γ±,in,LANMnmEk

=

(1

2∓ 1

2+NLA

)D2

acELAm∗‖

4π~3ρv2s

u[Ek + ∆N+M,N+M+m ∓ ELA]

×∫ ∞

0

dQzIN,N+n,N+M+m,N+M(Qz). (A.3)

The corresponding out scattering rates are

Γ±,out,LANMnmEk

=

(1

2∓ 1

2+NLA

)D2

acELAm∗‖

4π~3ρv2s

u[Ek + ∆N+n,N+M+m ± ELA]

×∫ ∞

0

dQzIN+M,N+M+m,N+M+m,N+n(Qz). (A.4)

A.2 Polar Longitudinal-Optical Phonons

The scattering weight for polar longitudinal-optical (LO) phonons is [68]

W±LO(Q) =

(1

2∓ 1

2+NLO

)βLO

Q2

(Q2 +Q2D)2

, (A.5)

with

βLO =e2ELO

2ε0

(1

ε∞s− 1

ε0s

), (A.6)

where ε0 is the permitivity of vacuum, ε∞s and ε0s are the high-frequency and low-frequency rela-

tive bulk permittivities respectively, ELO is the polar-optical phonon energy and QD is the Debye

screening wavevector defined byQ2D = n3De

2/(ε∞s kBT ), with n3D the average 3D electron density

in the device. We note that QCLs are typicially low-doped, so the effect of screening is minimal.

However, the screening facilitates numerical calculations by avoiding the singularity when |Q| = 0

131

in Eq. (A.5). Integration over the angle between Q and k gives

W±LO(Ek, Qz, Eq) =

(12∓ 1

2+NLO

)βLO

2π

∫ 2π

0

|Q− k|2

(|Q− k|2 +Q2D)2

(A.7)

=

(12∓ 1

2+NLO

)βLO

2π

∫ 2π

0

dθQ2z + q2 + k2 − 2qk cos(θ)

(q2z + q2 + k2 − 2kq cos(θ) +Q2

D)2(A.8)

=

(12∓ 1

2+NLO

)βLO

Q2z + q2 + k2

[1 +

Q2D

Q2z + q2 + k2

− 4k2q2

(Q2z + q2 + k2)2

]

×

[(1 +

Q2D

Q2z + q2 + k2

)2

− 4k2q2

(Q2z + q2 + k2)2

]− 32

=~2

2m∗

(12∓ 1

2+NLO

)βLO

Ez + Ek + Eq

×[1 +

EDEz + Eq + Ek

− 4EkEq(Ez + Eq + Ek)2

]

×

[(1 +

EDEz + Eq + Ek

)2

− 4EkEq(Ez + Eq + Ek)2

]− 32

, (A.9)

with Ez = ~2Q2z/(2m

∗‖). Note that this definition of Ez is only for convenience, it has nothing to

do with energy quantization in the z-direction. The in and out scattering rates can now be written

as

Γ±,in,LONMnmEk

=m∗‖

2π~3u[Ek + ∆N+M,N+M+m ∓ ELO]

×∫ ∞

0

dQzIN,N+n,N+M+m,N+M(Qz)

× W±LO(Ek, Qz, Ek + ∆N+M,N+M+m ∓ ELO) (A.10)

Γ±,out,LONMnmEk

=m∗‖

2π~3u[Ek + ∆N+n,N+M+m ± ELO]

×∫ ∞

0

dQzIN+M,N+M+m,N+M+m,N+n(Qz)

× W±LO(Ek, Qz, Ek + ∆N+n,N+M+m ± ELO), (A.11)

with the angle-averaged scattering weights W±LO given by Eq. (A.9).

132

A.3 Interface Roughness

In order to calculate the scattering rates due to interface roughness in Eq. (2.63), we need to

calculate the angle-averaged correlation function C(Ek, Eq). In this work, we employ a Gaus-

sian correlation function C(|r|) = ∆2e−|r|2/Λ2 , with a corresponding Fourier transform C(|q|) =

π∆2Λ2e−|q|2Λ2/4. The angle-averged correlation function can be calculated using

C(Ek, Eq) =1

2π

∫ 2π

0

C(|q− k|)dθ

=1

2π

∫ 2π

0

dθC

√m∗‖2~2

√Ek + Eq − 2

√EkEq cos(θ)

= ∆2Λ2

∫ π

0

dθe−m∗‖Λ

2

2~2 (Ek+Eq−2√EkEq cos(θ))

= π∆2Λ2e−m∗‖Λ

2

2~2 (Ek+Eq)I0(√EkEqm

∗‖Λ

2/~2), (A.12)

where we have used [158, Eq. 3.339]∫ π

0

e−x cos(θ)dθ = πI0(x), (A.13)

with I0(x), the modified Bessel function of the first kind of order zero. The scattering rates are

given by

Γin,IR,±NMnmEk

=m∗‖4~3

u[∆N+M,N+M+m + Ek]

×∑i

(N |N + n)zi,IR (N +M +m|N +M)∗zi,IR

× C(Ek,∆N+M,N+M+m + Ek)

Γout,IR,±NMnmEk

=m∗‖4~3

u[∆N+n,N+M+m + Ek]

×∑i

(N +M |N +M +m)zi,IR (N +M +m|N + n)∗zi,IR

× C(Ek,∆N+n,N+M+m + Ek), (A.14)

where we have split the rates into indentical absorption (+) and emission (−) terms.

133

A.4 Ionized Impurities

The potential due to a single ionized impurity situated at Ri, can be written as

V (R) = − e2

4πε

e−β|R−Ri|

|R−Ri|, (A.15)

where β is the inverse screening length. Now suppose there are many impurities, situated on a

sheet Ri = (ri, z`). The resulting interaction Hamiltonian can be written as

HII = − e2

4πε

∑i

e−β√

(r−ri)2+(z−z`)2√(r− ri)2 + (z − z`)2

=−e2

8π2ε

∑i

∫d2qeiq·(r−ri)

e−√β2+q2|z−z`|√β2 + q2

, (A.16)

where the second line in Eq. (A.16) is the 2D Fourier decombosition of the first line. We now

proceed as we did with interface roughness scattering up until Eq. (2.54) and get the in-scattering

part of the dissipator due to ionized-impurities

DinII =

π

~∑n1234

∫d2k124δ[E(n4,k4)− E(n3,k2)]ρ

Ek2n2,n3

× 〈n1,k1|HII|n2,k2〉 〈n3,k2|HII|n4,k4〉 |n1,k1〉 〈n4,k4|+ h.c.. (A.17)

Inserting Eq. A.16 into Eq. (A.17), we can simplify the integrand using

〈n1,k1|HII|n2,k2〉 〈n3,k2|HII|n4,k4〉

=e4

64π4ε2

∑ri,rj

∫d2qd2q′

1√β2 + q2

1√β2 + q′2

× e−iq·rie−iq′·rj (n1|n2)IIq,` (n2|n3)II

q′,`

× 〈k1|k2 + q〉 〈k2|k4 − q′〉 . (A.18)

with

(n|m)IIq,` =

∑b

∫dz[ψ(b)

n (z)]∗ψ(b)m (z)

e−√β2+q2|z−z`|√β2 + q2

, (A.19)

134

where the sum is over the considered bands. Equation (A.18) depends the detailed distribution

of impurities, just as Eq. (2.55) depends on the detailed in-plane interface roughness ∆i(r). By

averaging over all posible distribution of impurities ri and rj , we can see that terms with ri = rj

and q = q′ dominate. Limiting the sum to ri = rj and going from sum over ri to integral gives∑i

e−iri·(q−q′) '

∫d2riN2De

−iri·(q−q′) = (2π)2N2Dδ[q− q′]. (A.20)

Inserting Eq. (A.20) into Eq. (A.18) gives

〈n1,k1|HII|n2,k2〉 〈n3,k2|HII|n4,k4〉

=e4

16π2ε2N2D

∫d2q

1

β2 + q2(n1|n2)II

q,` (n2|n3)IIq,`

× δ[k2 − (k1 − q)]δ[k4 − (k2 + q)]. (A.21)

Inserting Eq. (A.21) into Eq. (A.17) and integrating over k4 and k2, dropping the subscript on k1

and shifting the integration variable q→ −q + k gives

DinII = − e4N2D

16πε2~∑n1234

∫d2k

∫d2qδ(Eq −∆n4,n3 − Ek)

× (n1|n2)II|q−k|,` (n3|n4)II

|q−k|,` ρEqn2,n3

. (A.22)

Calculating the scattering rates from Eq. (A.21) proceeds the same way as for interface roughness

in Eqs. (2.58)-(2.63). The total rate due to ionized impurities is obtained by summing over all

delta-doped sheets located at positions z = z` with sheet densities N2D,`

Γ±,in,IINMnmEk=

e2m∗‖16ε2~3

u[Ek + ∆N+M,N+M+m]∑`

N2D,`

2π

×[∫ 2π

0

(N |N + n)II|q−k|,` (N +M +m|N +M)II∗

|q−k|,`

]Eq=∆N+M,N+M+m+Ek

Γ±,out,IINMnmEk

=e2m∗‖

16ε2~3u[Ek + ∆N+n,N+M+m]

∑`

N2D,`

2π

×[∫ 2π

0

(N +M |N +M +m)II|q−k|,` (N +M +m|N + n)II∗

|q−k|,`

]Eq=∆N+n,N+M+m+Ek

.

(A.23)

135

The products involving the wavefunctions (n|m)II|q−k|,` are more complicated than in the case for

both the phonon and interface roughness interaction. This is because the interaction potential

for ionized impurities in Eq. (A.16) does not factor into the cross plane and in-plane directions

and the products involving eigenfunctions ψn(z) are coupled with the in-plane motion. First we

numerically calculate and store the quantity

Fnm`(b) =1

b

∫ψ∗n(z)ψm(z)e−b|z−z`|dz (A.24)

using an evently spaced mesh [bmin, bmax] for the relevant range of b. Typical values are bmin = β

and bmax = 10 nm−1. We then perform the θ integration numerically with the stored values of

Fnm`(b) using a linear interpolation from θ to b using

b(θ) =

√2m∗‖~2

√ED + Ek + Eq − 2

√EkEq cos(θ). (A.25)

with ED = ~2β2/(2m∗‖) the Debye energy.

A.5 Alloy Scattering

Inserting the alloy interaction potential given by Eq. (2.55) into Eq. (2.54) gives the in-scattering

part of the dissipator due to random alloy scattering

Dinalloy =

π

~∑n1234

∫d2k124δ[E(n4,k4)− E(n3,k2)]ρ

Ek2n2,n3

× 〈n1,k1|Valloy|n2,k2〉 〈n3,k2|Valloy|n4,k4〉 |n1,k1〉 〈n4,k4|+ h.c.. (A.26)

The matrix elements involving the interaction potential can be simplified using

〈n1,k1|Valloy|n2,k2〉 〈n3,k2|Valloy|n4,k4〉

=∑i,j

(n1|n2)alloy (zi) (n3|n4)alloy (zj)Cxi C

xj 〈k1|δ(r− ri)|k2〉 〈k2|δ(r− rj)|k4〉

= Ω0

∑i

(n1|n2)alloy (zi) (n3|n4)alloy (zi)xi(1− xi)V 2i 〈k1|δ(r− ri)|k2〉 〈k2|δ(r− ri)|k4〉 ,

(A.27)

136

where we have replaced the product Cxi C

xj with the correlation function in Eq. (2.68) and defined

the alloy matrix element

(n|m)alloy (zi) =∑b

[ψ(b)n (zi)]

∗ψ(b)m (zi), (A.28)

where the sum is over the considered bands. Going from sum to integral using∑

i →1

Ω0

∫d3Ri

and following the same procedure as for interface roughness scattering in Eqs. (2.54)-(2.63) gives

the scattering rates

Γ±,in,alloyNMnmEk

=m∗‖Ω0

4~3u[Ek + ∆N+M,N+M+m]

×∫dz (N |N + n)alloy (z) (N +M +m|N +M)alloy (z)

× x(z)(1− x(z))V (z)2

Γ±,out,alloyNMnmEk

=m∗‖Ω0

4~3u[Ek + ∆N+n,N+M+m]

×∫dz (N +M |N +M +m)alloy (z) (N +M +m|N + n)alloy (z)

× x(z)(1− x(z))V (z)2, (A.29)

where x(z) is the alloy fraction at position z and V (z) is the alloy strength parameter at position

z, which we assume is the same for all bands.

137

Appendix B: Bias Treatment within the Wigner-function formal-ism

The method we use to solve the WBTE in Eq. (3.1) requires use to perform a Wigner transform

of the fast varying-potential term Vqm defined in Eq. (3.3). A more computationally efficient way

is to use [145]

Vw(z, k) =2

π~Ime2ikzVqm(2k)

, (B.1)

where Vqm(k, t) is the spatial Fourier transform of Vqm(z, t) with the convention

Vqm(k, t) =

∞∫−∞

Vqm(z, t)e−ikzdz . (B.2)

Equation (B.2) represents a problem when Vqm does not decay to zero as z → ±∞, such in the

case of an applied bias. As an example, let us take a prototype bias potential of the form (erf is the

error function)

V∆(z) = −V0

2erf[β(z − z0)] , (B.3)

which represents a bias drop of V0, antisymmetric around z0 and β controls how abruptly the

potential changes (see Fig. B.1). Because V∆ does not decay to zero as z → ±∞, its Fourier trans-

form is only defined in the distributional sense which makes a numerical evaluation troublesome.

However, the Wigner transform of V∆ can be done analytically using Eq. (B.2), giving

Vw,∆(z, k) =V0

~πP 1

ke−k

2/β2

cos[2k(z − z0)] , (B.4)

138

where P denotes the principal value. The quantum evolution term resulting from Vw,∆ can be

calculated using Eq. (3.2), giving

QV∆[fw](z, k) =

∞∫−∞

Vw,∆(z, k′)fw(z, k − k′)dk′

=V0

π~

∞∫−∞

P 1

k′e−k

′2/β2

cos[2k′(z − z0)]fw(z, k − k′)dk′

=V0

π~

kc∫−kc

P 1

k′e−k

′2/β2

cos[2k′(z − z0)]fw(z, k − k′)dk′

+V0

π~

∫|k′|>kc

P 1

k′e−k

′2/β2

cos[2k′(z − z0)]fw(z, k − k′)dk′

≡ QV∆,c +QV∆,∞ (B.5)

where kc > 0 is a cutoff wave number. In Eq. (B.5), we have split the k′ integral into the long

wavelength part QV∆,c and short wavelength part QV∆,∞. If we choose kc small enough such that

fw(z, k) varies weakly in the range [k − kc, k + kc], we can replace fw(z, k) with its first order

Taylor expansion in the neighborhood of k and get

QV∆,c[fw] = − V0

π~∂fw∂k

∫ kc

−kce−k

′2/β2

cos[2k′(z − z0)]dk′

= −1

~∂fw∂k

βV0√πe(z−z0)2β2

Re erf[kc/β + i(z − z0)β]

≡ −1

~∂fw∂k

F∆(z) . (B.6)

where Re refers to the real part. Comparison with Eq. (3.1) shows that the quantum evolution term

in Eq. (B.6) has reduced to a semi-classical drift term with force F∆ defined in the bottom line of

Eq. (B.6) and can absorbed into the slowly varying potential term such that F∆(z) = −∂Vcl/∂z.

The short wavelength part QV∆,∞ is still treated quantum mechanically. When choosing a length

for the considered device, care must be taken that the device is long enough such that F∆(z) is

small outside the device region.

Now consider the potential profile of the uniformly-doped RTD from section 3.3.1 under an

applied bias of 0.096 V (see Fig. B.1). Define the total potential as V (z). Now define a new

139

quantity

Vδ(z) = V (z)− V∆(z)− VB(z) , (B.7)

where VB(z) is the potential of the barriers. Figure B.1 shows a plot of Vδ(z). The main idea is

to compute the Wigner transform of VB and V∆ analytically, so that only the Wigner transform of

Vδ needs to be performed numerically. Any numerical difficulties are avoided because Vδ varies

slowly and decays rapidly to zero for z → ±∞, due to the fact that we have removed the barriers

through VB and the potential drop through V∆.

Analytic evaluation of the Wigner transform of a barrier potential can be done by considering

it as a linear combination of error functions. For example the potential of the left barrier can be

written as

VLB(z) =VL2 − erf[βL(z − zL − aL)]

+ erf[βL(z − zL + aL)] , (B.8)

where the subscript L refers to the left potential barrier, 2aL is the barrier width, VL the barrier

height, zL its center position and βL controls how much the barrier edge is smoothed. The barrier

becomes abrupt in the limit βL → ∞, however a finite value must be used because the numerical

method we use is only convergent for continuous potentials [159]. The corresponding Wigner

transform is

Vw,LB(z, k) =2VL~π

1

ke−k

2/β2L sin[2k(z − zL)] sin[2kaL] . (B.9)

We note that this approach does not only work for RTDs, but any device with an applied potential

where computation of the Wigner potential is required. The specific choice of parameters β and z0

in the bias potential is not important, however, a good guess will minimize Vδ, which contains the

part of the total potential which must be Wigner transformed numerically.

140

60 40 20 0 20 40 60

Position [nm]

0.05

0.00

0.05

0.10

0.15

Pote

nti

al

[eV

]

V(z)

V∆ (z)

Vδ (z)

Figure B.1: (Color online) Solid black curve shows the potential profile for the uniformly-doped

RTD described in Sec. 3.3.1. Long-dashed red curve shows the bias potential V∆(z), with param-

eters β = 0.035 nm−1, z0 = 0, and V0 = 0.096 eV. Short-dashed blue curve shows Vδ(z), which

is the potential remaining when the bias and barriers have been subtracted from the total potential

V (z).