TEMPERATURE EFFECTS AND TRANSPORT PHENOMENA INTERAHERTZ QUANTUM CASCADE LASERS

BY

PHILIP C. SLINGERLANDB.A.(2004)

SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTSFOR THE DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF PHYSICSUNIVERSITY OF MASSACHUSETTS LOWELL

Signature of Author

Signature of Dissertation Chair: Dr. Christopher S. Baird

Signatures of Dissertation Committee Members:

Dr. Robert H. Giles

Dr. Viktor A. Podolskiy

TEMPERATURE EFFECTS AND TRANSPORT PHENOMENA INTERAHERTZ QUANTUM CASCADE LASERS

BY

PHILIP C. SLINGERLAND

ABSTRACT OF A DISSERTATION SUBMITTED TO THE FACULTY OFDEPARTMENT OF PHYSICS

IN PARTIAL FULFILLMENT OF THE REQUIREMENTSFOR THE DEGREE OF

DOCTOR OF PHILOSOPHYPHYSICS

UNIVERSITY OF MASSACHUSETTS LOWELL2011

Dissertation Supervisor: Dr. Christopher S. Baird

Abstract

Quantum cascade lasers (QCL’s) employ the mid- and far-infrared intersubband ra-

diative transitions available in semiconductor heterostructures. Through the precise

design and construction of these heterostructures the laser characteristics and out-

put frequencies can be controlled. When fabricated, QCL’s offer a lightweight and

portable alternative to traditional laser systems which emit in this frequency range.

The successful operation of these devices strongly depends on the effects of electron

transport. Studies have been conducted on the mechanisms involved in electron trans-

port and a computational model has been completed for QCL performance prediction

and design optimization. The implemented approach utilized a three period model

of the laser active region with periodic boundary conditions enforced. All of the

wavefunctions within these periods were included in a self-consistent rate equation

model. This model employed all relevant types of scattering mechanisms within three

periods. Additionally, an energy balance equation was studied to determine the set

of individual subband electron temperatures. This equation included the influence

of both electron-LO phonon and electron-electron scattering. The effect of differ-

ent modeling parameters within QCL electron temperature predictions are presented

along with a description of the complete QCL computational model and comparisons

with experimental results.

ii

Acknowledgements

I would like to thank Christopher Baird for giving me the opportunity to study with

him. He is an outstanding advisor and, more importantly, a great teacher. All of the

progress that I made would not have been possible without his ability to explain the

often frustrating physics of quantum cascade lasers.

I would also like to thank the faculty and staff at both The Submillimeter-wave

Technology Laboratory and The Photonics Center at the University of Massachusetts

Lowell. My work owes its birth to the ongoing research of source technologies at

these facilities. In particular, I am grateful for the generous support of both Robert

Giles and William Goodhue who provided funding and resources for the duration

of my PhD. Additionally, I am grateful to Xifeng Qian and Shiva Vangala for their

many marathon growth campaigns, Neelima Chandrayan for her skilled processing of

samples, and Andriy Danylov for his pain-staking characterization of devices. And

of course, I am grateful to all the other colleagues who answered my questions both

during the writing of this thesis and throughout my research.

I must also thank all my friends and family members who were so patient and

understanding during my graduate studies. Lastly, I would like to thank my wife,

Elizabeth. Her constant support and encouragement has inspired me to do more than

I ever could have without her.

iii

for Elizabeth

iv

TABLE OF CONTENTS

LIST OF TABLES viii

LIST OF FIGURES ix

I INTRODUCTION 1

II METHODOLOGY 62.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.a Schrodinger Equation . . . . . . . . . . . . . . . . . . . . . 72.1.b Poisson Equation . . . . . . . . . . . . . . . . . . . . . . . 92.1.c Total Potential . . . . . . . . . . . . . . . . . . . . . . . . 102.1.d Space Charge Density . . . . . . . . . . . . . . . . . . . . 122.1.e Individual Fermi Levels . . . . . . . . . . . . . . . . . . . . 202.1.f Waveguide Effects . . . . . . . . . . . . . . . . . . . . . . . 232.1.g Photon Scattering . . . . . . . . . . . . . . . . . . . . . . . 292.1.h Phonon Scattering . . . . . . . . . . . . . . . . . . . . . . 352.1.i Electron-Electron Scattering . . . . . . . . . . . . . . . . . 382.1.j Electron-Electron Screening . . . . . . . . . . . . . . . . . 472.1.k Rate Equations . . . . . . . . . . . . . . . . . . . . . . . . 532.1.l Electron Temperature . . . . . . . . . . . . . . . . . . . . 602.1.m Output Power . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.2 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . 692.2.a Non-uniform Location Grid . . . . . . . . . . . . . . . . . 722.2.b Initial Fermi Levels . . . . . . . . . . . . . . . . . . . . . . 752.2.c Poisson Equation . . . . . . . . . . . . . . . . . . . . . . . 752.2.d Space Charge Density . . . . . . . . . . . . . . . . . . . . 762.2.e Schrodinger Equation . . . . . . . . . . . . . . . . . . . . . 782.2.f Copy Wavefunctions . . . . . . . . . . . . . . . . . . . . . 812.2.g Waveguide Numerical Analysis . . . . . . . . . . . . . . . . 832.2.h Photon Scattering Implementation . . . . . . . . . . . . . 892.2.i Phonon Scattering Numerical Implementation . . . . . . . 902.2.j Electron-Electron Computational Implementation . . . . . 902.2.k Electron-electron screening implementation . . . . . . . . . 942.2.l Rate Equation Implementation . . . . . . . . . . . . . . . 952.2.m Multi-subband SCEB Algorithm . . . . . . . . . . . . . . . 972.2.n Average Electron Temperature Implementation . . . . . . 100

2.3 Experimental setups . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

v

2.3.a QCL production methods . . . . . . . . . . . . . . . . . . 1022.3.b Device characterization methods . . . . . . . . . . . . . . . 1052.3.c Details of grown devices . . . . . . . . . . . . . . . . . . . 106

III RESULTS 1103.1 Average Electron Temperature . . . . . . . . . . . . . . . . . . . . . 1103.2 The effect of approximations on e-e scattering rates . . . . . . . . . . 114

3.2.a Convergence and integration types . . . . . . . . . . . . . . 1183.2.b State-blocking and screening . . . . . . . . . . . . . . . . . 1193.2.c Symmetric and asymmetric transitions . . . . . . . . . . . . 123

3.3 Scaling Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1263.4 Screening Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1283.5 Multi-subband electron temperature model . . . . . . . . . . . . . . 129

3.5.a 3.4 THz, three-well design . . . . . . . . . . . . . . . . . . . 1293.5.b 2.8 THz, four-well design . . . . . . . . . . . . . . . . . . . 131

3.6 Temperature optimization . . . . . . . . . . . . . . . . . . . . . . . . 133

IV DISCUSSIONS 1364.1 Average Electron Temperature . . . . . . . . . . . . . . . . . . . . . 1364.2 The effect of approximations on e-e scattering rates . . . . . . . . . . 138

4.2.a Convergence and integration types . . . . . . . . . . . . . . 1384.2.b State-blocking and screening . . . . . . . . . . . . . . . . . 1394.2.c Symmetric and asymmetric transitions . . . . . . . . . . . . 140

4.3 Scaling Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1414.4 Screening Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1424.5 Multi-subband electron temperature model . . . . . . . . . . . . . . 143

4.5.a 3.4 THz, three-well design . . . . . . . . . . . . . . . . . . . 1434.5.b 2.8 THz, four-well design . . . . . . . . . . . . . . . . . . . 144

4.6 Temperature optimization . . . . . . . . . . . . . . . . . . . . . . . . 145

V CONCLUSIONS 1465.1 Average Electron Temperature Calculations . . . . . . . . . . . . . . 1465.2 Electron-electron scattering rate approximations . . . . . . . . . . . . 1465.3 Device scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1475.4 Electron-electron screening . . . . . . . . . . . . . . . . . . . . . . . . 1475.5 Multi-subband energy balance model . . . . . . . . . . . . . . . . . . 1485.6 Temperature optimization . . . . . . . . . . . . . . . . . . . . . . . . 148

vi

VI RECOMMENDATIONS 150

VII REFERENCES 151

VIII BIOGRAPHY 160

vii

LIST OF TABLES

1 Structures based on a 2.9 THz Barbieri QCL design which lased. Thepredicted and measured frequencies are both shown for comparison. . 107

2 Structures based on a 1.9 THz Freeman QCL design which lased. Thepredicted and measured frequencies are both shown for comparison. . 108

3 Structures based on a 2.83 THz Vitiello QCL design which lased. Thepredicted and measured frequencies are both shown for comparison. . 109

4 Average electron temperatures calculated for the mid-IR Sirtori QCLdesign using different sets of electron states. All input parameters intothe QCL computational model were the same as those used by a pub-lished study of the same structure. The left column contains the 15electron states, ψ’s, while the remaining columns contain the calculatedTe from using the corresponding collection of ψ’s. The lattice tempera-ture TL and calculated Te from the published study are included aboveeach column. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5 The e-e scattering rates in the bound-to-continuum structure for allpossible transitions which significantly contribute to W11,10 at a tem-perature of 40 K. The e-e rates were calculated using the relativewavevector form with 106 integration points and with state-blockingand screening included. . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6 The e-e scattering rates in the resonant phonon structure for all possi-ble transitions which significantly contribute to W4,3 at a temperatureof 80 K and a bias of 14.2 kV/cm. The e-e rates were calculated us-ing the relative wavevector form with 106 integration points and withstate-blocking and screening included. . . . . . . . . . . . . . . . . . . 125

viii

LIST OF FIGURES

1 Flow chart of the complete QCL computational model. Conditionalsteps are indicated by a question mark. After the Schrodinger-Poissonbox, electron wavefunctions are checked for convergence. After theelectron populations box, electron populations are checked for conver-gence. After the electron temperature box, the electron temperature ischecked for convergence. Finally, the electron wavefunctions are againchecked for convergence before the process is repeated again. . . . . . 70

2 The energy transition rates for each subband as a function of electrontemperature for each subband. The five subbands shown are thosefrom the 1.8 THz Kumar structure. All subband temperatures start atthe lattice temperature of 150 K (as indicated by the vertical dashedline) and end when the energy transition rate reaches zero (as indicatedby the horizontal dashed line). . . . . . . . . . . . . . . . . . . . . . 100

3 Plot of total energy change ∆ as a function of electron temperaturefor a typical configuration, showing the correct electron temperatureas the one at the zero crossing. In this case, Te = 83 K. Also shown arethe contributions from phonon emission (ph em), phonon absorption(ph abs) and electron-electron scattering (e-e) to the energy balanceequation. These data are calculated for the Sirtori mid-IR structure ata bias voltage of 48 kV/cm and at a lattice temperature of 77 K. . . . 102

4 Wavefunctions found in 3 periods of the active region design of theSirtori mid-IR QCL using an applied bias voltage of 48 kV/cm and ata lattice temperature of 77 K. . . . . . . . . . . . . . . . . . . . . . . 111

5 Plot of standard deviation of the calculated electron temperatures as afunction of the number of electron states included. Results are shownfor three lattice temperatures applied to the Sirtori mid-IR structure ata bias voltage of 48 kV/cm. The standard deviation reaches a minimumwhen 20 out of the 27 total states are included. . . . . . . . . . . . . 113

6 Plot of standard deviation of the calculated electron temperatures as afunction of the number of electron states included. Results are shownfor six lattice temperatures applied to the Page mid-IR structure ata bias voltage of 53 kV/cm. There is a minimum standard deviationwhen 18 out of the 24 total states are included. . . . . . . . . . . . . 113

7 Plot of Te vs. TL over a range of bias voltages applied to the Pagemid-IR QCL. These voltages are all above threshold. . . . . . . . . . 115

ix

8 Wavefunctions for the bound-to-continuum structure at a bias voltageof 2.246 kV/cm and lattice temperature of 10 K. The subbands men-tioned in this thesis are labeled. Only the middle period is fully shownin order to reduce confusion. . . . . . . . . . . . . . . . . . . . . . . 116

9 Wavefunctions for the resonant phonon structure at a bias voltage of13.25 kV/cm and lattice temperature of 10 K. The subbands mentionedin this thesis are labeled. All three periods of the active region used inthe computation are shown. . . . . . . . . . . . . . . . . . . . . . . . 117

10 Convergence of the scattering rate as more integration points are usedfor the infinite quantum well at a temperature of 300 K and a subbandpopulation density of 1015 (1/m2). The form using the non-relativewavevector definition and the form using the relative wavevector defi-nition are compared. All integrals were performed with state-blockingand screening included. . . . . . . . . . . . . . . . . . . . . . . . . . 119

11 Convergence of scattering rates within the bound-to-continuum struc-ture as more points are used in the integrals at different temperatures.The percent error is averaged over all symmetric transitions at eachtemperature and all integrals are performed with state-blocking andscreening included. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

12 Convergence of scattering rates within the resonant phonon structureas more points are used in the integrals at different temperatures. Thepercent error is averaged over all symmetric transitions at each temper-ature and all integrals are performed with state-blocking and screeningincluded. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

13 The W1,1→0,0 scattering rates in the infinite quantum well as a functionof temperature and population density. The legend indicates whichapproximations were used in each plot: state-blocking included (SBon), state-blocking not included (SB off), screening included (SC on)and screening not included (SC off). For all points, 106 integrationpoints were used in the relative wavevector form of e-e scattering. . . 122

14 The W16,16→15,15 scattering rates in the bound-to-continuum structureas a function of electron temperature and the initial and final 2D sub-band population densities. For all points, 106 integration points wereused in the relative wavevector form of e-e scattering. . . . . . . . . 123

15 The W6,6→5,5 scattering rates in the resonant phonon structure as afunction of electron temperature and the initial and final 2D subbandpopulation densities. For all points, 106 integration points were usedin the relative wavevector form of e-e scattering. . . . . . . . . . . . 124

x

16 On the left is a section of the transmittance spectrum for atmosphericwater vapor at 5% relative humidity obtained in-house using a BrukerFTIR spectrometer. On the right is the output frequency of a 1.9 THzQCL as a function of a layer width scaling factor applied to the originaldevice design. Both graphs together indicate the desired emission fre-quency for this device, 1.97 THz, since that frequency is in a region ofhigh transmittance. The blue dots indicate the scaling factor and emis-sion frequencies of devices which were grown by the UML PhotonicsCenter and the red line indicates the model-predicted trends. . . . . 127

17 On the left is a section of the transmittance spectrum for atmosphericwater vapor at 5% relative humidity obtained in-house using a BrukerFTIR spectrometer. On the right is the output frequency of a 2.83THz QCL as a function of a layer width scaling factor applied to theoriginal device design. Both graphs together indicate the desired emis-sion frequency for this device, 2.53 THz, since that frequency is in aregion of high transmittance. The blue dots indicate the scaling fac-tor and emission frequencies of devices which were grown by the UMLPhotonics Center and the red line indicates the model-predicted trends. 128

18 The model-predicted output power as a function of bias voltage ofa 2.83 THz QCL for several scaling factors. On the top graph, thepower was calculated without screening included in the e-e scatteringrates. The bottom graphs shows the calculated power when screeningis included in the computational model. . . . . . . . . . . . . . . . . 130

19 Calculated potential profile and squared wavefunctions of the 3.4 THzstructure at a bias of 64 mV/module for three periods of the activeregion. The middle period states are labeled such that state 4 and 3are the lasing states, while 2 and 1 are responsible for depopulationand injection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

20 Model-predicted device characteristics as a function of lattice tempera-ture at a bias of 70 mV/module for the 3.4 THz structure. (a) Electrontemperature of the upper lasing state 4. (b) Electron-electron scatter-ing rates of the injection (1′ → 4) transition as well as the lifetime ofthe upper lasing state. (c) Population inversion between lasing levels4 and 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

21 Calculated potential profile and squared wavefunctions of the 2.8 THzstructure at a bias of 65 mV/module for three periods of the activeregion. The middle period states are labeled such that state 5 and 4are the lasing states, while 3, 2 and 1 are responsible for depopulationand injection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

xi

22 Model-predicted device characteristics as a function of bias at a latticetemperature of 100 K for the 2.8 THz structure. (a) Electron temper-atures of the upper lasing state (5) and the two injector states (1 and2). (b) Population inversion between the lasing levels 5 and 4. . . . . 134

23 On the left is a portion of the 1.8 THz Kumar QCL structure. Theupper lasing state is indicated as blue (u) and the lower state is indi-cated as red (l). The barrier that was modified lies between the arrowsindicating the shift directions. The figure on the upper right showsthe upper lasing state electron temperature as a function of the barrieredge shift and the lower right figure shows the optical power of thelasing transition. The power at a barrier shift of -2 A was found to be0.48 W but was omitted from the graph as it was an outlier. . . . . 135

xii

1

I. INTRODUCTION

Terahertz (THz) radiation is a portion of the electromagnetic spectrum which

lies between the infrared and microwave frequencies and is typically defined to be

between the wavelengths of 600 to 30 µm (i.e. 0.5 - 10 THz). This region is of par-

ticular interest due to the large number of potential applications which have recently

been suggested.1,2 These applications are typically related to two characteristics of

THz radiation: its absorption by water and its transparency to dielectric materials.

This leads to uses in medical imaging, where differences in water content between

tissue types can be used to differentiate healthy from cancerous tissues;3–7 security

screening, where the transparency of clothing to terahertz radiation can allow the

imaging of concealed objects;8,9 and quality assurance, where the transparency of

paper and cardboard allows for the imaging of packaged contents.10,11 However, it

should also be noted that many other molecules aside from water have strong spec-

tral signatures at terahertz frequencies. This has led to further interest in remote

detection applications,12,13 provided that atmospheric absorption can be minimized,

and satellite based communication and spectroscopy,14,15 where atmospheric effects

are not a problem.

All of these applications would benefit greatly from a compact, coherent and

continuous-wave (cw) source of THz radiation that provides an adequate signal-to-

noise ratio for real-time imaging and allows integration into large-array detectors for

remote-sensing.16 However, development in this area has been slow. In terms of solid-

2

state devices, such as Gunn oscillators and Schottky diode multipliers, the output

power has not been able to exceed the milliwatt level above 1 THz.1,17–20 Photonic

approaches to terahertz generation are limited by the small band gaps required at

these frequencies. For example, lead salt laser diodes cannot emit at frequencies

lower than 15 THz. These frequencies can also be generated through multiplication

up from millimeter-wave sources such as optically pumped molecular gas lasers or

free-electron lasers. However, these systems are not attractive for the mentioned

applications due to their size, cost and complexity. Further discussions of the current

state and limitations of terahertz technologies can be found in the literature.21,22

Terahertz quantum cascade lasers (QCL’s) have helped to fill this gap of viable

THz sources. A QCL is a type of semiconductor laser whose emission frequency can be

chosen by proper design of the epitaxial layers. They are made by growing alternating

layers of material with varying thicknesses onto a substrate. Each layer is only a few

nanometers thick but still maintains a band-gap between the conduction and valence

bands. At the junction of the layers the difference of the conduction band energies

forms a potential barrier. These heterostructures establish a series of finite quantum

wells that confine electrons. Additionally, a bias voltage is applied which provides

electronic pumping. In each period of the active region, electrons tunnel through the

energy barriers, de-excite, emit a photon, and continue tunneling through to the next

period to repeat the process.

The first suggestion of such a device came in 1971 by Kazarinov23 after which

followed a long period of experimental investigation. However, QCL’s experienced

3

rapid development once the first successful mid-IR device was created by Faist24

and the first THz device was made by Kohler.25 Currently, QCL’s can emit at any

frequency in the THz range, but this flexibility is not matched with high power or

temperature performance. The maximum power from a THz QCL so far is 250 mW

in pulsed mode and 130 mW in cw mode, as demonstrated by Williams.26 But this

success has only been seen above 4 THz and lower frequencies see a reduction in the

optical power. Similarly, the highest operating temperature is 186 K for a 3.9 THz

laser designed by Kumar,27 but at 1.2 THz the highest operating temperature is only

69 K, as demonstrated by Walther.28 There is a need to improve their performance

if the potential applications are to be successful.

Further improvement of these devices requires a complete understanding of the

many mechanisms which affect their performance. In particular, electron-electron

(e-e) scattering is a crucial element to the operation and design of QCL’s since it con-

tributes heavily to the transport of electrons in the laser active region.29 Specifically,

the injection and depopulation of lasing states is often designed to be dependent upon

e-e scattering. This is especially true in the THz regime where the energy spacing

between states is small and, therefore, very sensitive to e-e scattering.

Also, electron heating is another important effect since all of the electron scatter-

ing rates are strongly temperature dependent. Therefore, electron or carrier transport

will by strongly affected by any increase in temperature that the electrons may ex-

perience. Due to the small energy spacing between electron energy levels, thermal

effects may be particularly harmful to the carefully designed electron populations. In

4

order for devices to operate at higher temperatures, either the electrons have to be

kept as cool as possible or the laser must be designed with electron heating in mind.

Both methods require device modeling to be performed.

Device modeling is an important element in meeting this need to improve QCL

properties and several methods exist for producing QCL predictions. One such model,

the self-consistent energy balance (SCEB) model was proposed by Harrison,30 and has

been shown to predict lasing characteristics without the computational demands of

Monte Carlo31–33 or non-equilibrium Green’s function34–38 methods. Another model,

the density-matrix model,39,40 which has been suggested as an alternative to the self-

consistent approach, has the appeal of being relatively easy to implement and with low

computational demand. However, since this model considers scattering events within

a smaller portion of the active region as compared to the self-consistent model, many

vital effects can be lost as discussed by Beji.41

An improvement to the SCEB model has been suggested by Jovanovic,42 which

determines the temperature of every subband within the QCL conduction band. Such

a model would provide a means to determine thermal effects, which have a significant

impact on the transport mechanisms within a QCL (through thermal backfilling,

parasitic transitions, etc.).

In the sections that follow, a full description of the computational model em-

ployed (based on the SCEB approach) will be given in addition to a theoretical de-

scription of the various components. Particular emphasis will be placed on electron-

electron interactions and electron heating due to the reasons mentioned above. In

5

addition, the results of these models will be presented along with experimental find-

ings, demonstrating the role thermal effects play in QCL physics.

6

II. METHODOLOGY

2.1 Theory

The QCL computational model is comprised of many components which are all based

on physical first principles wherever possible. In order to accomplish this, many

theoretical aspects had to be derived and understood so that they could be properly

implemented numerically. In the following subsections, a detailed explanation of

these various theoretical aspects can be found. In Section 2.2, a description of their

numerical implementations is also given.

First, the appropriate form of the Schrodinger and Poisson equations are derived

in the context of a conduction band heterostructure to find the electron states. Sup-

plemental information on the total potential, space charge density and Fermi levels

is also given in this context. Then, the waveguide effects are derived to help deter-

mine the fields that can exist in QCL structures. These are followed by derivations

for the three primary forms of carrier scattering discussed in this thesis: electron-

photon, electron-phonon, and electron-electron scattering. Additionally, the effect of

screening on electron-electron scattering is derived to further enhance the accuracy

of this vital particle interaction. Electron-impurity and electron-interface scattering

effects were not included. Although there has been evidence both for and against

the significance of these scattering channels,43–47 it was decided for this particular

research effort, where low-doped, THz QCL’s are involved, that the effects of impu-

7

rity and interface scattering are negligible. The rate equations, which determine the

electron populations densities, are then derived. These are followed by the energy

balance equations, which determine the electron temperatures. Finally, the necessary

equations for calculating the output properties of a QCL are given.

2.1.a Schrodinger Equation

Quantum cascade lasers are fabricated by stacking up alternating layers of semicon-

ductor material with nanoscale thicknesses. This heterostructure of layers forms a

series of conduction-band quantum wells in the z direction which quantize the elec-

trons into eigenenergies or subband states.

The eigenstate of an electron in the unperturbed Hamiltonian of a QCL is the

product of the Bloch envelope function B(x, y, z), the free electron wavefunction in

the x and y direction, and the bound quantum well eigenfunctions ψn(z) in the z

direction:

〈r|ψ〉 = 〈x|ψ〉〈y|ψ〉〈z|ψ〉 = B(x, y, z)1√Lxeikxx

1√Lyeikyyψn(z). (1)

The Bloch function factor contains the effects on the electron state due to the non-

uniform nature of the crystal potential on the atomic scale. The semiconductor

layer widths are assumed to be large compared to the atoms. Therefore, the Bloch

function factor is assumed to be negligible. Each electron is pseudo-free in the x

and y dimensions because the material is uniform in those dimensions. Even though

each electron is bound to the crystal in these dimensions, the electrons are essentially

free if the effective mass of the electron is used. The bound-state z-component wave

8

functions ψn(z) are found by numerically solving the one-dimensional Schrodinger

equation when the potential profile is known (see Section 2.2.e). As developed in

Section 2.1.c, the potential profile is a combination of the conduction-band edge

quantum well profile of the material layers, the bias voltage, and the built-in potential

which accounts for the effects of space charge. The built-in potential is found by

solving the Poisson equation (Section 2.1.b).

The one-dimensional, time-independent Schrodinger equation for a single electron

in the potential V (z) is the Hamiltonian eigenvalue equation, where E is the total

z-dimensional energy of the electron in the eigenstate:

Hψ = Eψ. (2)

The Hamiltonian operator H is just the total energy operator, so it is expanded into

a sum of the kinetic energy operator T and the potential energy operator V . The

kinetic energy operator T , which contains the effective mass m∗ is also expanded.

The effective mass is a function of z because the material is non-uniform along this

dimension. As a result, Schrodinger’s equation is written as:(1

2m∗(z)zz

)ψ(z) = (E − V (z))ψ(z). (3)

Expanding the velocity operators in terms of momentum operators, we find:

d

dz

(1

m∗(z)

dψ(z)

dz

)= − 2

~2(E − V (z))ψ(z). (4)

Schrodinger’s equation in the form of Eq. (4) is the fundamental physical equation to

be solved. However, the potential V (z) is too complicated to allow the Schrodinger

9

equation to be solved analytically (see Section 2.1.c for how the total potential is

determined). Therefore numerical methods must be employed (see Section 2.2.e).

2.1.b Poisson Equation

The potential function in the one-dimensional, one-electron Schrodinger equation in

Eq. (4), which determines the wavefunctions in a QCL, includes the built-in poten-

tial. Physically, the electrons in the doped layers of a QCL are readily ionized into

the conduction band and then migrate through the QCL. The mobile electrons and

the positive ions they leave behind in the doped layers constitute the space charge

ρ(z) that creates the built-in potential (see Section 2.1.d for how the space charge is

determined). In order to avoid the intractable problem of a many-body Schrodinger

equation involving innumerable electrons, the built-in potential, Φ, is calculated clas-

sically using the Poisson equation.

The charges are uniform in the x and y dimensions, so Φ is independent of these

dimensions. In the z dimension, the sequence of quantum wells in a QCL confines the

electrons into quantized states with differing transition rates, so that the overall charge

density becomes non-uniform and gives rise to the built-in potential. Therefore, only

the one-dimensional Poisson equation is needed.

Gauss’s Law in differential form states that a space charge density ρ(z) gives rise

to a diverging electric field D:

∇ ·D = ρ(z). (5)

Assuming linear dielectric materials throughout the QCL, the D field is expressed in

10

terms of the electrical permittivity as εE. Since the material composition changes

across the QCL in the z direction, the permittivity is z-dependent and cannot be

taken out of the divergence operator. With these positional dependencies, Gauss’s

Law becomes:

∇ · (ε(z)E) = ρ(z). (6)

The built-in scalar potential Φ is defined according to E = −∇Φ. The potential only

depends on z, which allows the use of the one-dimensional Poisson equation:

− d

dz

(ε(z)

dΦ(z)

dz

)= ρ(z). (7)

Since the space charge profile ρ(z) is typically complicated, this equation can only be

solved numerically (see Section 2.2.c).

2.1.c Total Potential

Before the one-dimensional Schrodinger equation in Eq. (4) is solved to find the bound

electron states, the potential energy profile of the QCL must be known. The potential

energy V (z) is the sum of the conduction-band edge energy Ec, the externally applied

bias voltage eVapp, and the built-in potential eΦ due to space charge. The conduction-

band edge is a series of quantum wells and the built-in potential is found by solving

the Poisson equation in Eq. (7) when the space charge is known. The bias voltage is

contained in the built-in potential, because it is a constant, linear function which can

be accounted for by applying the proper boundary conditions to the Poisson equation.

11

With this in mind, the total potential energy becomes:

V (z) = Ec(z)− eΦ(z). (8)

Here, e is the electron charge needed to convert the electrostatic potential to a po-

tential energy.

Because the Poisson equation takes care of space charge effects arising from the

heterojunction of thin semiconductor layers, the conduction band-edge is assumed to

be the same as that for a large bulk piece of material. The location of the conduction

band edge is calculated as the sum of the valence-band edge Ev,abs on an absolute

scale plus the energy band gap Eg:

Ec = Ev,abs + Eg. (9)

The valence-band edge energies on an absolute scale are taken from the Van de Walle48

values, and are calculated as a function of alloy concentration according to a linear

model:

Ev,abs = Ev,0 + Ev,1x. (10)

Here x is the alloy concentration, Ev,0 is the Van de Walle absolute-scale valence

band edge of the pure material (e.g. GaAs), and (Ev,0 + Ev,1) is the Van de Walle

absolute-scale valence band edge of the fully alloy material (e.g. AlAs).

The energy gap is temperature and alloy dependent and so the Varshni empirical

form49 is used to calculate it:

Eg = (E0 + E1x+ E2x2)− (α0 + α1x)T 2

T + (β0 + β1x). (11)

12

Here, T is the lattice temperature, x is the alloy concentration, E0, E1, and E2 are

the zero-temperature band gap parameters, and α and β are the Varshni form pa-

rameters. All of these parameters are different for each material and can be found

in the literature. For ease of use, the model interface presents the most common

QCL semiconductor materials to the user as layer material options, and their corre-

sponding material parameters are automatically loaded by the model from a material

parameters file.

2.1.d Space Charge Density

The Poisson equation in Eq. (7) depends on the space charge present. The space

charge, including ionized donor atoms in the valence band as well as electrons in

the conduction band, gives rises to the built-in electrostatic potential described in

the Poisson equation. The built-in potential is an effect which is additional to the

conduction band-edge quantum-well profile. The sum of both of these potentials is

the potential that an electron experiences and is what determines its wavefunction via

the Schrodinger equation. The space charge density must be found before the Poisson

and Schrodinger equations can be solved. But the space charge density ultimately

depends on the Schrodinger equation because it dictates the wavefunctions’ shapes,

which for a large ensemble of electrons becomes the electron density. These equations

must therefore be solved self-consistently; each equation is applied iteratively until

the solution converges to the physical solution. However, some initial guess must be

used for the space charge density before it can be iterated to the correct solution. One

13

possible guess is to place the free electrons in the wells before the injection barrier.

Because the device material is uniform in the x and y dimensions, the space

charge is also trivially uniform in these dimensions. The space charge density is

therefore only dependent on the z dimension. The space charge density is represented

by the three-dimensional charge density ρ(z), by the two-dimensional charge density

ρ2D(z), or by the three- and two-dimensional number densities n(z) and n2D(z). It is

often useful to use many of these forms at once, but the Poisson equation depends

on the three-dimensional charge density ρ(z). The different forms are related in the

following way:

ρ(z) = qn(z)

ρ2D(z) = qn2D(z)

ρ2D(z) = ρ(z)L

n2D(z) = n(z)L (12)

where q is the charge on one charge carrier (−e for electrons and +e for ionized donor

atoms) and L is the length of one period of the QCL structure.

Hole Densities

By construction, QCL’s are structures where the dynamics are completely determined

by electrons moving between states in the conduction band. In order to get a sub-

stantial number of electrons into the conduction band to sustain a current, some of

the layers are doped with donor atoms. QCL’s are rarely doped with acceptor atoms.

As a result, the number of holes in the valence band is negligible and is assumed to

14

make no contribution to space charge effects. While there may be some electrons in

the conduction band because they left a hole behind, the great majority of conduction

electrons come from the donor atoms and leave behind fixed positively-charged donor

ions. In the rare instances when holes are created in the valence band by regular

thermal action, the abundance of excess electrons in the conduction band means that

these holes are very quickly filled and destroyed. In the end, the dynamics of holes

can be completely ignored in QCL’s.

Total Space Charge Density

The total charge density ρ(z), as needed in the Poisson equation, is the sum of the

positively charged ionized donor atoms ρdonor(z), and the negatively-charged electrons

ρelec(z) that have left the donor atoms and been freed into the conduction band:

ρ(z) = ρdonor(z) + ρelec

ρ(z) = endonor(z)− enelec(z). (13)

Only the semiconductor layers that are doped experience significant ionization levels.

The doped atoms have an extra electron that is very loosely bound and very easily

excited into the conduction band. An electron in the conduction band becomes de-

localized and is pseudo-free within the effective-mass approximation because it is

bound to the crystal as a whole rather than any local atom. The electron leaves

behind a positively-charged atom that is fixed in the crystal lattice. However, the

electron is described by an associated wavefunction within the quantum well structure

and can move from state to state through scattering. In this model, the donor density

15

need only be found once at the beginning of the model because the donors are fixed,

but the electron density must be self-consistently determined as part of the iterative

process.

Note that the total charge in the QCL is zero because it is part of a grounded

electric circuit. Integrating the charge density over the entire device must yield zero:∫ρ(z)dz = 0. (14)

This condition leads to the result:∫nelec(z)dz =

∫ndonor(z)dz. (15)

Because the donor density is fixed and found at the beginning, this equation is used

repeatedly to normalize the electron density to its proper magnitude.

Space Charge Electron Density

The space charge electron density is defined as the number of conduction electrons

per unit volume at a point z in space. All conduction electrons are assumed to be in

the wavefunctions of the quantum well structure. It is also assumed that there are

enough electrons in each wavefunction state (remembering the electrons have different

wavevectors in the x and y dimensions, so that the Pauli exclusion principle does not

have a significant impact) that the wave’s probability density becomes the average

charge density of that state. The electron space charge density is then just the sum

over all wavefunctions:

nelec(z) =

1 period∑i

ni2D,elec|ψi(z)|2. (16)

16

Here, ni2D,elec is the overall electron number sheet density in the ith quantum level

and ψi is the wavefunction of that level. The two-dimensional density must be used

because the wavefunction squared is the density in the third dimension. In practice,

the charge density is found in the central period where the wavefunctions are the most

accurate and then copied to the outer periods to ensure periodicity. The electron

densities in each level are referred to as populations and are found by solving the

rate equations (see Section 2.1.k). The rate equations depend on the scattering rate

calculations, which depend on the wavefunctions as well, so there are several iterative

loops that must be carried out to ensure self-consistency.

Space Charge Ion Density

The positive ions left behind when the electron leaves a donor atom for the conduction

band are fixed in space and do not move. As a result, the ionized donor space charge

density ρdonor(z) needs to be calculated only once at the beginning of the model

and then stored for future use. The material layers are assumed to be thick enough

that they essentially act as infinite bulk pieces of material when it comes to donor

ionization. The z-dependent donor density is therefore calculated by considering one

z grid point at a time in the model’s data structure, looking up the material and

doping at that grid point as specified by the user, and calculating the ionization

using a bulk model.

The ionization process actually involves a complex interaction of holes being

created and destroyed, and donors being ionized and de-ionized according to the

17

lattice temperature. Therefore the bulk density of states and bulk Fermi energy level

must be found first in order to determine the equilibrium point. All of the following

derivations assume an infinite bulk uniform material, so there is no concept of subband

levels or QCL periods. In the end, the bulk model ionization is applied point by point

to the QCL structure.

Derivation of ionized donor density

Define the Fermi energy level EF as the bulk-material (non-junctioned) Fermi level

that includes doping effects. Using the free-electron/quasi-particle model for three-

dimensional bulk material, the density of allowed states in k-space is given by:

g(k) = 2

(2π

L

)−3= 2

V

(2π)3. (17)

The number of states with wave number less than k, using the quasi-free particle

relation k =√

2m∗E/~2 is the density times the volume in k-space, N(k) = Vkg(k):

N(E) =

(2m∗E

~2

)3/2(V

3π2

). (18)

The density of states as a function of energy is the derivative of the number of states

with respect to energy:

g(E) =dN(E)

dE=

(2m∗

~2

)3/2√E

(V

2π2

). (19)

When applying this to the negatively-charged electrons in the conduction band the

energy is replaced with E → E − Ec(z), whereas for positively-charged holes in the

18

valence band the energy becomes E → Ev(z)− E:

gn(E) =

(2m∗n~2

)3/2√E − Ec

(V

2π2

)gp(E) =

(2m∗p~2

)3/2√Ev − E

(V

2π2

). (20)

The density of conduction-band states occupied by electrons using Fermi-Dirac statis-

tics becomes:

gn,occ(E) = gn(E)fD(E) (21)

where fD(E) and gn,occ(E) are defined according to:

fD(E) =1

1 + e(E−EF )/kBT

gn,occ(E) =

(2m∗n~2

)3/2(V

2π2

)√E − Ec

1

1 + e(E−EF )/kBT. (22)

The density of valence-band states occupied by holes using Fermi-Dirac statistics

becomes:

gp,occ(E) = gp(E)(1− fD(E))

gp,occ(E) =

(2m∗p~2

)3/2(V

2π2

)√Ev − E

1

1 + e(EF−E)/kBT. (23)

Using the approximation:

1

1 + e(E−EF )/kBT≈ e(EF−E)/kBT , (24)

the electron number density in the conduction band is calculated:

n =N

V

n =1

V

∫ ∞Ec

gn,occ(E)dE

n = 2

(m∗n(z)kBT

2π~2

)3/2

e(EF−Ec)/kBT . (25)

19

Using the same approximation again, the density of holes in the valence band is

calculated:

p =P

V

p =1

V

∫ Ev

−∞gp,occ(E)dE

p = 2

(m∗pkBT

2π~2

)3/2

e(Ev−EF )/kBT . (26)

The density of positive donor ions at the donor level ED, using the same approxima-

tion, is now found to be:

N+D = ND(z)

(1− gDe(EF−ED)/kBT

). (27)

Using the charge neutrality of the crystal, n = p+N+D , the Fermi energy is solved for

by substituting and applying the quadratic equation:

EF = kBT ln

G+

√G2 + 4

[(m∗n)3/2 e−Ec/kBT +GgDe−ED/kBT

] (m∗p)3/2

eEv/kBT

2[(m∗n)3/2 e−Ec/kBT +GgDe−ED/kBT

]

(28)

where the term G is defined as:

G =1

2

(2π~2

kBT

)3/2

ND. (29)

Because this is the bulk model, the conduction band edge energy Ec is just the

valence band energy plus the temperature-dependent band gap, as calculated early

in the model, and does not include the bias potential or the built-in potential. The

donor doping density at each grid point ND as well as the lattice temperature T is

20

provided directly by the user as a design parameter at runtime. The electron effective

mass m∗n, the hole effective mass m∗p, the donor valence band edge ED, and the donor

valley degeneracy gD are all material parameters that can be found in the literature.

Because the material and doping varies from layer to layer in the QCL core structure,

all of these parameters are position dependent. At each z grid point, the material

and doping must be received from the user, and then all these material-dependent

parameters must be loaded from a material parameters file. The Fermi energy is

therefore calculated at each grid point.

Once these Fermi levels are known, the ionized donor space charge density is

calculated at each grid point:

ρdonor(z) =eND

1 + gDe(EF−ED)/kBT. (30)

2.1.e Individual Fermi Levels

Every quantized level in a QCL has a different electron population and temperature.

It is assumed that each subband thermalizes much quicker than electrons transition

out of the subband, so that all subbands are always thermalized. This means that

all the electrons in a subband are in a Fermi distribution, which depends on the sub-

band’s Fermi level (and therefore the population density) and electron temperature.

The populations are determined by the rate equations (Section 2.1.k) and the temper-

atures are determined by the energy balance equations (Section 2.1.l). The individual

Fermi levels are found from the populations and temperatures, and the Fermi levels

are used in the scattering calculations. Because the rate equations depend on scat-

21

tering calculations, all these properties are circularly dependent and must be found

iteratively.

When dealing with individual levels, the approximation that the material acts

as bulk can no longer be made. Instead a set of initial level populations are used

and from there the corresponding Fermi levels are found (see Section 2.2.b). The

assumption that subbands can be represented by Fermi-Dirac distributions has been

shown to be valid for intersubband devices by Kinsler50 and Lee.51

Using the two-dimensional quasi-free electron model, the density of allowed states

in k-space is:

g(k) = 2

(2π

L

)−2= 2

A

(2π)2. (31)

The number of states with wave number less than k, using the quasi-free particle

relation k =√

2m∗(z)E/~2 is the area in k-space of the circle containing the points

where the wave number is less than k, times the density N(k) = Akg(k):

N(E) =Aπ4m∗E

(2π)2~2. (32)

The density of states as a function of energy g(E) is the change in the number of

states with respect to energy:

g(E) =dN(E)

dE=Am∗

π~2. (33)

The density of subband states occupied by electrons gn,occ(E) is found by multiplying

the density of states by a Fermi distribution:

fD(E) =1

1 + e(E−EF,i)/kBT(34)

22

The density of subband states then becomes:

gn,occ(E) = gn(E)fD(E)

gn,occ(E) =Am∗

π~21

1 + e(E−EF,i)/kBT. (35)

The spatial sheet density of electrons n2D in the subband is defined as the total

number of electrons N in the subband divided by the spatial area of interest A:

n2D =N

A. (36)

The total number of electrons N in the subband is the integral over all occupied

states:

n2D =1

A

∫ ∞Ei

gn,occ(E)dE

n2D =m∗

π~2[EF,i − Ei + kBT ln(1 + e((Ei−EF,i)/kBT ))

]. (37)

Now inverting this to solve for the Fermi energy yields the expression:

EF,i = Ei + kBT ln[eπ~2n2D/kBTm

∗ − 1]. (38)

The sheet density is related to the regular density by n2D = Ln where L is the length

of one period of the QCL (which represents the average width of the wavefunctions).

When the level’s population and electron temperature T are known, this equation

is applied directly to calculate the individual Fermi energy. Note that the subband

energy minimum Ei and the Fermi energy EF,i are measured on the same absolute

energy scale. For high temperatures and low populations, the Fermi level necessarily

becomes smaller than the subband minimum. These individual Fermi levels are then

used in calculating scattering rates.

23

2.1.f Waveguide Effects

The internal waveguide of a quantum cascade laser will only support certain modes

of the laser field. The internal waveguide typically consists of reflective or semi-

reflective material layers above and below the active region layers, and may include

the substrate. The reflective layers are typically metal or heavily-doped semicon-

ductor material. The waveguide algorithm is tasked with receiving any series of

layers from the user (material, width, and doping), and calculating the fundamental

mode for that structure at a certain frequency. The waveguide loss, wavenumber,

and confinement factor that the laser radiation will experience is then found from

the waveguide mode. Because the waveguide calculations depend only on the fixed

waveguide structure and a frequency, they are carried out at the beginning of the

model before entering the iterative procedures. However, the lasing frequency is not

known in advance. The solution is to run the waveguide calculations for several pos-

sible frequencies and generate a lookup table. Then, later in the iterative loops when

the waveguide parameters are needed and the frequency is known, the parameters are

simply interpolated from the lookup table.

The computational model currently employs a simple one-dimensional slab wave-

guide model, similar to that described by Williams.52 The width and depth of a QCL

is typically so much larger than its height that the QCL is approximated to be uniform

and infinite in these dimensions, which reduces the problem down to one dimension.

Note that individual layers within the active region are so thin compared to the

waveguide layers, that their effects are assumed to be negligible. Instead, the entire

24

active region is modeled as one waveguide layer with a doping equal to the average of

the actual layer dopings. It should be noted that in order to conform to the accepted

waveguide equations, the Cartesian coordinate system used in this section is defined

so that z is the direction of propagation, rather than the growth direction.

Complex Permittivity

The Drude model is used for the complex permittivity εc of each layer:

εc(ω) = ε+ne2τ 2

m∗(1 + (ωτ)2)

(i

ωτ− 1

)(39)

where n is the free carrier density, ε is the material permittivity, and τ is the electron

momentum relaxation time. For gold, nGold = 5.6 × 1028 and τGold = 5 × 10−14.

For GaAs, the free carrier density is just the ionized doping density and the electron

relaxation time is found from experiment to be:

τGaAs = 10−13 +0.71× 1010

n+ 2.2× 1021(40)

where the free carrier density is in units of m−3 and the relaxation time is in seconds.

Similar expressions for other materials can be found in the literature.

Mirror Losses

Any loss causes the intensity of the electromagnetic wave to attenuate in space ac-

cording to:

I(z) = I0e−αz. (41)

After radiation makes one pass through the device, the intensity lost out the front mir-

ror (M2) diminishes the wave, so that the resultant intensity is the original intensity

25

times the reflectivity R2 of the second mirror:

I(2L) = R2I0 (42)

Comparing this to the first equation and solving for the loss yields:

R2I0 = I0e−αM2

2L

αM2 = − ln(R2)

2L. (43)

For a GaAs/air interface, R = 0.32. Here L is the length of the QCL cavity in the

direction that the radiation is emitted. The mirror loss due to the back mirror has

the exact same form.

General Waveguide Equations

It is assumed that the free currents and free charges are negligible and that in any

one region the material is uniform, linear and isotropic so that D = εE and B =

µH. Additionally, it is assumed that the waveguide has a uniform shape along the

z direction. All of the fields therefore have a harmonic free-wave solution in this

dimension with wave number kz. If all of the fields are also oscillating harmonically

in time at the same frequency ω they have the form:

E = E(x, y)eikzz−iωt

H = H(x, y)eikzz−iωt. (44)

26

Using these forms, Maxwell’s equations are solved to link the tangential and parallel

components of the fields:

Ht =i

κ2(kz∇tHz + ωεz×∇tEz)

Et =i

κ2(kz∇tEz − ωµz×∇tHz) . (45)

where the variable:

κ2 = εµω2 − k2z (46)

is determined by the boundary conditions in terms of the waveguide’s material and

geometry. The transverse fields can now be calculated if the parallel fields are known.

Therefore, either the transverse or the parallel fields can now be solved for.

Now taking the curl of Faraday’s law and the Maxwell-Ampere Law, and assum-

ing harmonic time and z-dependence gives:

(∇2t + κ2

)E = 0(

∇2t + κ2

)H = 0. (47)

These apply separately to each component of the vector fields. The general solution

to any component i in rectangular coordinates in any one particular region of uniform

linear material is:

Ei =∑kx,ky

(Aeikxx +Be−ikxx)(Ceikyy +De−ikyy)

Hi =∑kx,ky

(Eeikxx + Fe−ikxx)(Geikyy +He−ikyy). (48)

Boundary conditions must be applied at each interface between regions of uniform

linear material in order to determine the coefficients and wavenumbers. The boundary

27

conditions will cause the span of possible wave numbers kx and ky to form a discrete

set of modes. The lowest-order modes are typically the ones excited first and are the

ones of most interest.

QCL Slab Waveguide Equations

A QCL’s active region consists of a sequence of planar epitaxial layers grown on top

of each other in the x direction. A quantum selection rule dictates that all coherent

radiation generated by any QCL is polarized such that the electric field points in

the x direction, normal to the epitaxial layers. The active region structure thus

automatically dictates that Ey = 0 and Hx = 0.

Often the width of the QCL is much larger than the height of the QCL. As an

approximation, the waveguide is assumed to be infinite and uniform in the y-direction,

which removes any y-dependence. This automatically forbids TE modes. The fields

in the TM modes are already independent of y.

For the TM modes, there is a choice of three approaches to solving the problem:

either solve for Ex, Ez or Hy. Since the magnetic field only has one component, the

simplest approach would be to solve for Hy. All of the relevant equations are then

found according to the following steps:

• Apply the boundary conditions and solve the expression:

Hy =∑kx

(Aeikxx +Be−ikxx) (49)

28

• Use the expression for Hy in the perpendicular components of the E-field:

Ex(x) =kzωεHy(x)

Ez(x) =i

ωε

∂

∂xHy(x) (50)

• Use the fact that Ey = Hx = Hz = 0 and then put the field equations into

explicit form to find:

Ex(x) =∑kx

kzωε

(Aeikxx +Be−ikxx)

Ez(x) =∑kx

kxωε

(−Aeikxx +Be−ikxx) (51)

• The final solutions are then:

H = yHy(x)eikzz−iωt

E = (xEx(x) + zEz(x))eikzz−iωt (52)

where the wavevector is of the form:

k2x = εµω2 − k2z . (53)

In the one-dimensional QCL waveguide approach, all of the material boundaries

are planes parallel to the y − z plane. The applicable boundary conditions then

are that the tangential components of the magnetic field H must be continuous and

the tangential components of the electric field E field must be continuous across

the boundary. Stated more formally, each region of linear uniform material has its

solutions with its own H and E as a function of its own kx, A, B, ε, and µ. The

29

index i will be used to denote the ith region so that properties in the ith region are

denoted Hi and Ei, kx,i, Ai, Bi,εi, and µi. Note that the boundary conditions require

that all the regions have the same z-directional wave number kz and frequency ω.

Let the i = 0th region be the semi-infinite substrate at the bottom of the stack, the

i = 1st region be the one directly above the substrate, and so on. Also denote the

known location of the boundaries between regions as xi where x0 = 0 is the origin

of the x-coordinate and is also the location of the zeroth boundary, the one between

the substrate and the next layer. With these definitions, the boundary conditions

become:

x×Hi(xi) = x×Hi+1(xi)

Hy,i(xi) = Hy,i+1(xi)

Aieikx,ixi +Bie

−ikx,ixi = Ai+1eikx,i+1xi +Bi+1e

−ikx,i+1xi (54)

and the fields and wavevector become:

x× Ei(xi) = x× Ei+1(xi)

Ez,i(xi) = Ez,i+1(xi)

kx,iεi

(Aieikx,ixi −Bie

−ikx,ixi) =kx,i+1

εi+1

(Ai+1eikx,i+1xi −Bi+1e

−ikx,i+1xi). (55)

where i = 0, 1 . . . N − 1 and N is the number of layers, not including the substrate.

2.1.g Photon Scattering

Electrons in a QCL can transition between quantum states through photon scattering.

Photon scattering rates are used in the rate equations (Section 2.1.k) to determine

30

electron and photon populations and, in the end, the laser power (Section 2.1.m).

Photon scattering events include spontaneous photon emission, stimulated photon

emission, and stimulated photon absorption.

The computational model is most complete if there are no assumptions made in

advance as to which transition is the laser transition. This means that the model

calculates the photon scattering rate for all possible transitions. The laser transition

is identified as the one with the most laser power emitted, which is essentially the

one with the highest photon population. The photon population of each transition is

a function of electron population inversion and the photon scattering rate.

General Derivation

Fermi’s “golden rule” describes the transition rate Wi→f from an initial quantum

state i in the z dimension, with wave vector ki in the x − y dimension, and photon

state nq,σ with polarization index σ, to a final quantum state f in the z dimension,

with wavevector kf in the x− y dimension, and photon state mq,σ:

W ems,absi→f (q,ki,kf ) =

2π

~|〈f,kf ;mq,σ|H ′|i,ki;nq,σ〉|2δ(Ef (kf )− Ei(ki)± Eq). (56)

The delta function is a statement of the conservation of energy. If a photon is emitted,

then Ei = Ef + Eq, and if a photon is absorbed, then Ei + Eq = Ef .

The interaction Hamiltonian in SI units between an electron and a photon is

given by:

H ′ = − e

m∗A · p (57)

31

where A is the electromagnetic vector potential operator describing the photons in-

teracting with the electron, and p is the momentum operator of the electron. The

effective mass is taken from the well material since this is where the electron has the

highest location probability.

In SI units, The Lorentz-Gauge vector potential for a harmonic interaction with

a quantized EM field at a single wave vector q is a sum of creation and annihilation

operators:

A =2∑

σ=1

√~

2εV qc/n

(aq,σεq,σe

iq·r + a†q,σεq,σe−iq·r) (58)

where V is the cavity volume, ε is the dielectric constant, ε is the polarization vector

and q is the wave vector of the photon. Now substitute this expanded Hamiltonian

in Fermi’s “golden rule”:

W ems,absi→f (q,ki,kf ) =

2π

~e2

(m∗)2~

2εV qc/n

(m∗)2(Ef − Ei)2

~2|〈f(z)|z|i(z)〉|2

×

∣∣∣∣∣2∑

σ=1

εσ,z

√nq,σ + 1/2± 1/2

∣∣∣∣∣2

δ(Ef − Ei ± Eq)δkf ,ki. (59)

Summing over all possible final electron momenta and using the Kronecker delta to

ensure conservation of momentum, we find:

W ems,absi→f (q) =

2π

~e2

(m∗)2~

2εV qc/n

(m∗)2(Ef − Ei)2

~2|〈f(z)|z|i(z)〉|2

×

∣∣∣∣∣2∑

σ=1

εσ,z

√nq,σ + 1/2± 1/2

∣∣∣∣∣2

δ(Ef − Ei ± Eq). (60)

By defining some parameters and simplifying the constants, the scattering rate can

also be written as:

W ems,absi→f (q) =

e2hωif4m∗√εrε0V qc

fi→f

∣∣∣∣∣2∑

σ=1

εσ,z

√nq,σ + 1/2± 1/2

∣∣∣∣∣2

δ(Ef − Ei ± Eq) (61)

32

where the oscillator strength is defined as:

fi→f =2m∗ωif

~

∣∣∣∣∫ ψ∗f (z)zψi(z)dz

∣∣∣∣2. (62)

and the resonant frequency is:

ωif =Ei − Ef

~. (63)

The integral matches that found in Smet53 and Harrison54 and is done numerically

using the non-uniform-grid trapezoidal method.

There are two relevant cases that are handled differently: spontaneous photon

emission into all modes, and stimulated photon emission and absorption into a narrow

mode distribution.

Spontaneous Photon Emission into all modes

In this case, Eq. (61) becomes:

W spi→f (q) =

e2hωif4m∗√εrε0V qc

fi→f

∣∣∣∣∣2∑

σ=1

εσ,z

∣∣∣∣∣2

δ(Ef − Ei + Eq). (64)

Summing over all modes in the cavity (assuming an effectively infinite cavity in all

three dimensions) leads to:

W spi→f =

1

(2π/L)3

∫dqW sp

i→f (q)

=1

(2π/L)3

∫dqq2

∫dθ sin θ

∫dφ

e2hωif4m∗√εrε0V qc

fi→f

×

∣∣∣∣∣2∑

σ=1

εσ,z

∣∣∣∣∣2

δ(Ef − Ei + Eq). (65)

33

If the axis is chosen such that ε1 lies in the plane defined by k and q, then ε2,z = 0

and ε1,z = sin θ, which leads to:

W spi→f =

e2nω2if

6πm∗ε0c3fi→f . (66)

This expression is used in the rate equations simply as another scattering mechanism

that effects electron populations. The radiation due to spontaneous emission into

all modes does not contribute to the laser radiation. Note that for typical QCL’s,

the spontaneous photon emission rate into all modes is so small compared to other

scattering mechanisms that it is essentially negligible. However, this calculation is

still included in the computational model because it is easy to implement, runs quickly

and establishes a more complete model.

Stimulated Photon Emission/Absorption into a narrow mode distribution at onepolarization

Starting with the one-mode expression before integrating over all energies, Eq. (61)

becomes:

W sti→f (q) =

e2hωif4m∗√εrε0V qc

fi→fMifδ(Ef − Ei ± Eq). (67)

To mimic reality where the energy levels have finite widths due to their finite lifetimes,

the Dirac delta is replaced with a normalized line shape function in the form of a

Lorentzian (see the appendix of Williams’ thesis52):

W st,bandi→f (q) =

e2ωif4m∗√εrε0V qc

fi→fMifγif (ν) (68)

34

where

γif (ν) =∆νif/2π

(ν − νif )2 + (∆νif/2)2

νif =ωif2π

=Ei − Ef

h(69)

and the full-width half maximum line width of the transition is:

∆νif =1

π

(1

2τi+

1

2τf+

1

T ∗2

). (70)

The parameters τi and τf are the initial and final state total lifetimes and T ∗2 is

the pure dephasing time. The pure dephasing time is a characteristic lifetime that

describes the process of phase randomization that occurs to a collection of oscillators,

as described by Williams.52

The stimulated emission rate must be used in the rate equations in order to

determine the photon populations. The rate equations require a single number for

the transition rate, but the expression above is an entire function. This is solved by

taking the scattering rate at the peak frequency. Setting ν = νif , Eq. (68) becomes:

W st,bandi→f = Mif

e2

2πm∗εV∆νiffi→f (71)

where

fi→f =2m∗ωif

~

∣∣∣∣∫ ψf*(z)zψi(z)dz

∣∣∣∣2∆νif =

1

π

(1

2τi+

1

2τf+

1

T ∗2

). (72)

35

2.1.h Phonon Scattering

Only longitudinal optical (LO) phonon scattering is assumed to be significant in

QCL’s. The phonon spectrum available to a QCL electron for scattering is approx-

imated to be the phonon spectrum of a bulk sample of the material used in the

quantum wells. The assumption is also made that the bulk crystal is dispersionless

such that every LO photon that can be created or destroyed in a transition is at the

frequency ωLO(q) = ωLO(0) regardless of the wavevector q. This is equivalent to a

constant phonon energy of ELO = ~ωLO, which is different for each material, but can

be determined experimentally and found in the literature.

Inside a QCL, the electron is quantized in the z dimension, described by a wave-

function state, and pseudo-free in the x and y dimensions according to the effective

mass model. Scattering between an electron and a phonon is described by the Frohlich

interaction.

General derivation

Fermi’s “golden rule” is written for the transition rate Wi→f from an initial quantized

intersubband state i in the z dimension with wavevector ki in the x − y dimension

and crystal phonon state ni,q to a final intersubband state f in the z dimension with

wavevector kf in the x− y dimension and crystal phonon state nf,q:

W ems,absi→f (ki,kf ) =

2π

~|〈f,kf ;nf,q|H ′|i,ki;ni,q〉|2δ(Ef (kf )− Ei(ki)± ELO). (73)

The delta factor is a statement of the conservation of energy. If a phonon is emitted,

then Ei = Ef + ELO and if a phonon is absorbed, then Ei + ELO = Ef .

36

The interaction Hamiltonian between an electron and a phonon is given by:

H ′ =∑q

[α(q)(eiq·rbq + e−iq·rb†q)

](74)

where b†q and bq are the phonon creation and annihilation operators, respectively.

The interaction strength α(q) is given by the Frohlich representation:

|α(q)|2 =ELO

2

e2

V q2

(1

ε∞− 1

εs

)(75)

where ε∞ and εs are the high and low frequency permittivities, respectively. The

interaction Hamiltonian is a sum of all the possible phonon wavevectors (modes) q.

Although the modes are discreet, the crystal is large enough that the modes are

assumed to be infinitesimally close and the sum is approximated by an integral:

H ′ =L3

(2π)3

∫dqx

∫dqy

∫dqzα(qxi+ qy j + qzk)

×(ei(qxx+qyy+qzz)b(qx i+qy j+qz k) + e−i(qxx+qyy+qzz)b†

(qx i+qy j+qz k)

). (76)

Plugging the Hamiltonian into Fermi’s “golden rule” and moving the integrals

outside of the bra and ket vectors we find:

W ems,absi→f (ki,kf ) =

Le2

~V(nLO + 1/2± 1/2)

ELO

2

(1

ε∞− 1

εs

)×δ(Ef (kf )− Ei(ki)± ELO)

×∫dqz

1

(q2 + q2z)

∣∣〈f(z)|e∓iqzz|i(z)〉∣∣2 . (77)

The transition rate to a quantum state f in the z dimension but any state in the

x− y dimensions is the integral over all possible final states in the x− y dimension:

W ems,absi→f (ki) =

1

(2π/L)2

∫W ems,absi→f (ki,kf )dkf . (78)

37

Applying this sum, and expanding the two-dimensional final wavevector integral into

polar coordinates, the transition rate becomes:

W ems,absi→f (ki) =

1

2

(2m∗

~2

)1

(2π/L)22π

L

(2π)4

L4

L6

(2π)62π

~e2

V(nLO + 1/2± 1/2)

×ELO

2

(1

ε∞− 1

εs

)π

∫ 2π

0

A(q)

qdθ. (79)

The form factor A(q) is the same one that appears in electron-electron scattering

calculations (see Section 2.1.i). To avoid recalculating the same parameters, the form

factors are calculated in advance, before the electron-electron scattering or phonon

scattering calculations. The form factor depends only on the wavefunctions and the

transverse interaction wavenumber q. The model’s runtime is improved significantly

by pre-calculating the form factor outside of the other integrals to form a look-up

table in q. When the phonon scattering integrals are calculated and specific form

factor values are needed for certain q values, they are interpolated from the look-up

table.

After simplifying the constants, the final expression is:

W ems,absi→f (ki) =

m∗e2ELO

8π~3

(1

ε∞− 1

εs

)(nLO + 1/2± 1/2)

∫ 2π

0

A(q)

qdθ (80)

where

A(q) =

∫dz

∫dz′ψi(z)ψf (z)ψf (z

′)ψi(z′)e−q|z−z

′|

q2 = k2i + k2f − 2kikf cos(θ)

k2f = k2i +2m∗

~2(Ei(0)− Ef (0)∓ ELO) . (81)

38

Averaging over all initial wavevectors in Eq. (80) leads to:

Wi,j→f,g =

∫Lx

2πdki,x

∫ Ly

2πdki,yWi,j→f,g(ki)fi(ki)∫

Lx

2πdki,x

∫ Ly

2πdki,yfi(ki)

. (82)

Final Expression

Apply the average of Eq. (82) to get the final expression:

W ems,absi,j→f,g =

m∗e2ELO

8π~3

(1

ε∞− 1

εs

)(nLO + 1/2± 1/2)

∫dkikifi(ki)

∫ 2π

0dθA(q)

q∫dkikifi(ki)

(83)

where

A(q) =

∫dz

∫dz′ψi(z)ψf (z)ψf (z

′)ψi(z′)e−q|z−z

′|

q2 = k2i + k2f − 2kikf cos(θ)

k2f = k2i +2m∗

~2(Ei(0)− Ef (0)∓ ELO) . (84)

The LO phonon energy ELO is taken to be a constant of the material. The phonon

occupation number nLO is taken to be its bulk material value, which is given by

Bose-Einstein statistics to be:

nLO =1

eELO/kT − 1. (85)

These equations match those found in Smet53 and Harrison.54 The integrands are

evaluated computationally as described in Section 2.2.i.

2.1.i Electron-Electron Scattering

Scattering will first be derived without screening. The details of the screening deriva-

tion can be found in Section 2.1.j. Also, the electron-electron exchange interaction will

39

not be discussed here as there are many explanations available in the literature.55–57

The exchange effect was not included in the QCL computational model for this work

but will be included in future versions.

The eigenstate of an electron in the unperturbed Hamiltonian of a QCL (neglect-

ing the Bloch envelope function) is projected into coordinate space:

〈r|ψ〉 = 〈x|ψx〉〈y|ψy〉〈z|ψz〉 =1√Lxeikxx

1√Lyeikyyψn(z) (86)

where Lx and Ly are the lengths of the crystal in the dimension perpendicular to

the growth dimension z and are needed to normalize the effectively free-electron

components of the wave function. The bound-state z-component wavefunctions ψn are

found by numerically solving the one-dimensional Schrodinger equation (Section 2.2.e)

and the Poisson equation (Section 2.2.c) and are understood to be already normalized.

In the most general form, electron-electron scattering in a QCL involves the inter-

action of an electron in an initial quantum state i in the z dimension and wavevector

ki in the x and y dimensions with an electron in state j with wavevector kj so that

they end up, respectively, in states f and g and with wavevectors kf and kg. In the

x and y dimensions, the electron is considered pseudo-free within the effective mass

approximation for bulk semiconductor materials. In these dimensions, the material

is considered to be a bulk volume of the well material.

Variable Definitions

It is worthwhile to anticipate some aspects of the formal derivation and make some

variable definitions in advance. This will simplify the mathematics later on as well

40

as bring the notation in line early on with the literature.

Initially there are four independent, two-dimensional wavevectors: ki, kj, kf

and kg. Because the electrons are pseudo-free in the x-y plane, which is the same

plane in which all these wavevectors lie, the law of conservation of momentum holds

and removes both components of kg as independent variables. In addition, the law of

conservation of energy applies to the whole three-dimensional interaction and removes

kf as an independent variable. The remaining components (ki, θki , kj, θkj and θkf )

are independent and any intermediate variables should be functions of these.

The transition wavevector is defined as q = ki − kf , which leads to:

q =√k2i + k2f − 2kikf cos(θki − θkf ). (87)

The dot product of ki and kj is written as:

k2u = kikj cos(θki − θkj). (88)

This quantity is negative and therefore ku is complex valued. The transition energy

wavenumber g0, which can also be complex, is defined as:

g20 =4m∗

~2[Ei(0) + Ej(0)− Ef (0)− Eg(0)]. (89)

The sum of the initial wavevectors dotted by the unit vector kf is defined as:

ks = ki cos(θki − θkf ) + kj cos(θkj − θkf ). (90)

The initial relative wavevector is defined as g = kj − ki (it is also referred to as kij).

41

Written in terms of magnitude and phase, this wavevector is expressed as:

g =√k2j + k2i − 2k2u

tan θg =kj sin θkj − ki sin θkikj cos θkj − ki cos θki

. (91)

Similarly, the final relative wavevector is defined as g′ = kg − kf (it is also referred

to as kfg). Written in terms of magnitude and phase, the wavevector becomes:

g′ =√k2g + k2f − 2kgkf cos(θkg − θkf )

tan θg′ =kg sin θkg − kf sin θkfkg cos θkg − kf cos θkf

. (92)

Finally, the relative angle θ between the initial and final relative wavevectors is defined

as θ = θg − θg′ . All of the above definitions lead to the energy difference relation:

g2 − g′2 = k2i + k2j − k2f − k2g − 2k2u + 2kfkg cos(θkf − θkg). (93)

Conservation of energy (Ei + Ej = Ef + Eg) is now applied and used to fur-

ther simplify the variables defined above. Expanding the energy into x, y, and z

components yields:

Ei(0) +~2k2i2m∗

+ Ej(0) +~2k2j2m∗

= Ef (0) +~2k2f2m∗

+ Eg(0) +~2k2g2m∗

(94)

and using the energy wavenumber from Eq. (89), the energy conservation condition

reduces to:

1

2g20 + k2i + k2j = k2f + k2g . (95)

The dependence on kg is removed using the conservation of momentum to obtain:

g20 − 4k2f − 4k2u + 4kfks = 0. (96)

42

There are two ways to solve Eq. (96): using kf directly or using g′, which depends on

kf . Both methods are shown and implemented below as a check that no errors have

been made.

Solving Eq. (96) for kf leads to:

kf =1

2ks ±

1

2

√k2s + g20 − 4k2u. (97)

Note that there are two possible solutions in this equation and sometimes both are

valid, so each possibility must be calculated and summed together.

Alternatively, a solution to Eq. (96) can be found for g′. This is done by first

transforming all variables into g and g′. The last three terms in Eq. (96) are equal to

(g2 − g′2):

g20 + g2 − g′2 = 0. (98)

This leads to the intuitive statement that the transition energy equals the energy

difference. Solving for g′ leads to the result:

g′ =√g20 + g2. (99)

General Derivation of the Scattering Rate for one ki

Now that the preliminary definitions have been made a formal derivation can now be

done. The conservation laws will appear automatically and will be applied as they

appear. All of the variable definitions and relations found in the previous section will

be used directly here without much need of further comment.

43

Within the approximation of first-order perturbation theory, the electron-electron

scattering rate is found using Fermi’s “golden rule”:

Wi,j→f,g(ki,kj,kf ,kg) =2π

~|M |2δ(Ef (kf ) + Eg(kg)− Ei(ki)− Ej(kj)) (100)

where the matrix element is:

M = 〈f,kf ; g,kg|H ′|i,ki; j,kj〉. (101)

The perturbed Hamiltonian is just Coulomb’s Law:

H ′ =e2

4πε |rj − ri|, (102)

where the position variables become operators. Here SI units are being used, which

therefore lead to the appearance of 4π in the denominator. For now, the permittivity

ε is the unscreened permittivity for a bulk volume of material consisting of the well

material, but will become the screened version in Section 2.1.j.

Following an approach similar to that found in the work of Goodnick58 and

Smet,53 a 2D Fourier transform of the Coulombic potential is taken, which leads to:

Wi,j→f,g(ki,kj,kf ,kg) =1

(LxLy)3e4

(4πε)2(2π)4

q22π

~|A(q)|2

×δ(Ef (kf ) + Eg(kg)− Ei(ki)− Ej(kj))

×δ(ki − kf + kj − kg). (103)

where the form factor, A(q), is defined as:

A(q) =

∫ Lz

0

dz

∫ Lz

0

dz′ψ∗f (z)ψ∗g(z′)ψi(z)ψj(z

′)e−|z−z′|q. (104)

The Dirac deltas ensure conservation of energy and momentum. For them to be

applied, an integration must be performed.

44

Integrate over all Possible States

All possible events that can happen to one electron state must be considered, so a

summation over all possible interactions weighted by the carrier distributions must

be done to account for occupied states. To get the total number of states in some

interval dk = dkxdky, the probability of a state being occupied is multiplied by the

density of states. The density of states here is the two-dimensional density of states

for an infinite rectangular crystal, which is 2LxLy/(2π)2. The factor of two is present

because of the two possible spin states of an electron at each level, thus doubling the

number of possible states in the interval. However, because of exchange effects, only

scattering of electrons with anti-parallel spins are considered and therefore the two

must be dropped. Therefore, a scattering rate over all possible transition would be:

Wi,j→f,g(ki) =

∫LxLy(2π)2

dkj

∫LxLy(2π)2

dkf

∫LxLy(2π)2

dkg

×Wi,j→f,g(ki,kj,kf ,kg)fj(kj)(1− fg(kg))(1− ff (kf )).

(105)

The final form of the scattering rate for a given ki using the kf approach is:

Wi,j→f,g(ki) =1

(2π)2m∗e4

(4πε)22π

~3∑kf

∫dkj

∫dθkf

kf|ks − 2kf |

1

q2|A(q)|2

×fj(kj)(1− fg(kg))(1− ff (kf )) (106)

45

where

A(q) =

∫ Lz

0

dz

∫ Lz

0

dz′ψ∗f (z)ψ∗g(z′)ψi(z)ψj(z

′)e−|z−z′|q (107)

q =√k2i + k2f − 2kfki cos(θki − θkf ) (108)

kg =√k2i + k2j + k2f − 2kfks + 2k2u (109)

kf =1

2ks ±

1

2

√k2s + g20 − 4k2u (110)

ks = ki cos(θki − θkf ) + kj cos(θkj − θkf ) (111)

k2u = kikj cos(θki − θkj) (112)

g20 =4m∗

~2[Ei(0) + Ej(0)− Ef (0)− Eg(0)]. (113)

Apply Conservation of Energy Using the Approach that Solves for g′

In this case, the variables inside the Dirac delta of Eq. (103) are switched to the g and

g′ notation. Switching the variables inside the Dirac delta to the g and g′ notation

yields:

Wi,j→f,g(ki) =1

(2π)24m∗e4

(4πε)22π

~3

∫dkj

∫dkf

1

q2|A(q)|2

×δ(g20 + g2 − g′2)fj(kj)(1− fg(kg))(1− ff (kf )). (114)

Also, the factor dkf = dkfxdkfy becomes:

dkf =1

4d(kix + kjx − g′x)d(kiy + kjy − g′y)

=1

4d(g′x)d(g′y)

=1

4dg′. (115)

46

This result was arrived at by taking advantage of the relation:

kf =ki + kj − g′

2. (116)

Making this transformation to Eq. (114) leads to:

Wi,j→f,g(ki) =1

(2π)2m∗e4

(4πε)22π

~3

∫dkj

∫dθg′

∫dg′

1

q2|A(q)|2g′

×δ(g′2 − (g20 + g2))fj(kj)(1− fg(kg))(1− ff (kf )). (117)

The conservation of energy delta function is applied and the transformation θ =

θg − θg′ , dθ = −dθg′ (the negative sign goes away when the limits are flipped on the

integral) is made:

Wi,j→f,g(ki) =1

(2π)2m∗e4

(4πε)22π

~31

2

∫dkj

∫dθ

1

q2|A(q)|2fj(kj)(1− fg(kg))(1− ff (kf ))

(118)

where

A(q) =

∫ Lz

0

dz

∫ Lz

0

dz′ψ∗f (z)ψ∗g(z′)ψi(z)ψj(z

′)e−|z−z′|q (119)

kf =

√k2i + k2j − k2g +

1

2g20 (120)

kg =

√q2 + k2u +

√1 + (g0/g)2[(k2j − k2u) cos θ − kikj sin(θki − θkj) sin θ] (121)

q =1

2

√g20 + 2g2 − 2g

√g20 + g2 cos θ (122)

g2 = k2j + k2i − 2k2u (123)

k2u = kikj cos(θki − θkj) (124)

g20 =4m∗

~2[Ei(0) + Ej(0)− Ef (0)− Eg(0)]. (125)

47

This form agrees with that of Goodnick58 and Smet53 when errors in these works are

corrected according to Moskova’s suggestions.55

Scattering Rate Averaged Over ki

The actual wavevector ki of the specific electron of interest is not known, so it must

be averaged over a typical distribution of ki to get an average scattering rate. This is

done by summing up all the scattering rates for all the different states and dividing

by the number of states. As before, the density of occupied states in an interval dki

is the density of states at this value times the probability of occupation, which is the

Fermi function:

Wi,j→f,g =

∫ LxLy

(2π)2dkiWi,j→f,g(ki)fi(ki)∫ LxLy

(2π)2dkifi(ki)

. (126)

Canceling terms and expanding the integral into polar coordinates leads to:

Wi,j→f,g =

∫dk i∫kidθkiWi,j→f,g(ki)fi(ki)∫dk i∫kidθkifi(ki)

. (127)

2.1.j Electron-Electron Screening

The scattering of one electron with one other electron has already been derived in

Section 2.1.i. While screening involves a multi-electron interaction, the multi-electron

Hamiltonian is too complicated to be solved and is therefore approximated. If it is

assumed that two electrons scatter in the absence of any other charged particles, then

the previous treatment still applies. If it is assumed that all the other electrons are

not negligible, but are far enough way that they behave semi-classically as a sea of

electrons that exert a net influence, then screening must be taken into account.

48

The presence of other electrons tends to weaken the interaction between two elec-

trons and therefore the effect is described as screening. Classically, this macroscopic

screening effect in matter is handled by replacing the permittivity of free space with

a permittivity value that is representative of the material. Some level of screening

was already incorporated in Section 2.1.i since the material’s permittivity was used.

However, that permittivity is for a bulk volume of material. In QCL’s, nano-scale

layers of differing materials lead to quantum wells and quantized states for the elec-

trons which do the screening. In order to incorporate a more exact form of screening,

the bulk permittivity should be replaced with a new permittivity that depends on the

quantized states. Once the screened permittivity has been found, it is incorporated

into the innermost integrand of the e-e scattering equation from Eq. (118).

A general expression for the permittivity of a solid is derived using a self-

consistent field (SCF) approach, which has been shown to be equivalent to a random-

phase approximation (RPA) approach by Haug.59

General Screening Expression

By definition, the relative permittivity connects the polarization field P with the total

electric field E inside a material as described by the relation:

P = ε0(εr − 1)E. (128)

Causality dictates that induced polarization fields at different times and places

add together to a total field at a certain time and place. This means that the solution

would be an integral over many time points. Instead, a transformation to frequency

49

space is made, which reduces the polarization dependency to just one wave number

component q:

P(q, t) = εs(εr(ω,q)− 1)E(q, t). (129)

Here, εs is the static permittivity of the material and εr is a relative permittivity

screening constant that represents how the total permittivity deviates from the static

value. Solving for the screening constant leads to:

εr(ω,q) = 1 +P(q, t)

εsE(q, t). (130)

The electric field is expressed in terms of the negative gradient of a total potential V .

Doing this and performing a Fourier transform yields:

E(x, t) = −∇V (x, t)

E(q, t) = −iqV (q, t). (131)

The polarization field is created by the induced charge density ρpol. Using this and

performing a Fourier transform leads to:

∇ ·P(x, t) = −ρpol(x, t)

iq ·P(q, t) = −ρpol(q, t). (132)

Plugging both the electric and polarization field into the permittivity relation of

Eq. (130) yields:

εr(ω,q) = 1− P(q, t)

εsiqV (q, t)

εr(ω,q) = 1 +iq ·P(q, t)

εsq2V (q, t)

εr(ω,q) = 1− 1

εsq2ρpol(q, t)

V (q, t). (133)

50

The classical induced charge density is the sum over the different energy states of the

quantum density matrix:

ρpol(q, t) = − e

L3

∑k

〈k|ρ1|k + q〉. (134)

In order to get the ratio of the density matrix to the potential in terms of quantum

statistical distributions, the Louisville equation and first-order perturbation theory is

applied, which yields:

ρpol(q, t)

V (q, t)=〈k|ρ1|k + q〉V (q, t)

= ef(Ek+q)− f(Ek)

Ek+q − Ek − ~ω. (135)

This ratio is plugged back into Eq. (133):

εr(ω,q) = 1 +e2

εsq2L3

∑k

f(Ek+q)− f(Ek)

Ek+q − Ek − ~ω. (136)

This is the general expression for the relative permittivity in terms of quantum distri-

butions, in SI units. For a two-dimensional system with a quantized third dimension,

and in the static limit, this becomes:

εr(q) = 1 +e2

εsqL2Aiiii(q)

∑k

fi(Ei,k+q)− fi(Ei,k)

Ei,k+q − Ei,k(137)

where Aiiii(q) is the intrasubband form factor for the initial electron state. A polar-

izability function is defined as:

Πii(q) =2

L2

∑k

fi(Ei,k+q)− fi(Ei,k)

Ei,k+q − Ei,k(138)

and the permittivity is rewritten as:

εr(q) = 1 +e2

2εsqΠii(q)Aiiii(q). (139)

51

In reality, an electron is screened by all the electrons in all subbands, not just its own.

Therefore, a sum over all subbands is performed:

εr(q) = 1 +e2

2εsq

∑i

Πii(q)Aiiii(q). (140)

Expanding out the Fermi distributions and energies of Eq. (138), the polarizabil-

ity becomes:

Πii(q) =2

L2

∑k

[1 + exp

(~2|ki+q|2

2m∗ −EF,i

kBT ie

)]−1−[1 + exp

(~2k2i2m∗ −EF,i

kBT ie

)]−1~2|ki+q|2

2m∗− ~2k2i

2m∗

. (141)

Polarizability at Zero Temperature

For the special case of zero temperature, T = 0, this reduces to:

Πii(q, kF,i, T = 0) =m∗

~2π

[1− θ(q − 2kF,i)

√1− (2kF,i/q)2

]. (142)

This polarization function contains the Fermi wavevector, which is defined as:

kF,i =√

2m∗EF,i/~. (143)

This also assumed that the crystal is large enough that the set of quantum states for

the electron becomes a continuous band of states.

Polarizability at Any Temperature

To solve for any temperature, the Fermi functions must be expressed differently.

Starting with the definition of the hyperbolic tangent:

tanhx =ex − e−x

ex + e−x(144)

52

and expanding tanh into its integral form leads to the relationship:

1

1 + ex=

1

2

∫ ∞x/2

dx′

cosh2 x′. (145)

The integral can span all positive numbers if the lower contribution is forced to be

zero using the Heaviside step function (θ(x) = 1 if x > 0, θ(x) = 0 if x < 0):

1

1 + ex=

1

2

∫ ∞0

θ(x′ − x/2)dx′

cosh2 x′. (146)

Using this identity, the general polarizability expression becomes:

Πii(q) =4

(2π)2

∫ ∞0

dk′k′

4kBT ie cosh2(

~2(k′2−k2F,i)

4m∗kBT ie

) [∫ ∞0

dk

∫ 2π

0

kdθ

(1

q2 + 2kq cos(θ)

)× (θ

(k′2 − k2 − q2 − 2kq cos θ

)− θ

(k′2 − k2

))]. (147)

Comparing the factor in brackets to the T = 0 expression, it is seen that they match

except that this factor is a function of the integration variable k′:

Πii(q) =~2

m∗

∫ ∞0

dk′k′Πii(q, k

′, T = 0)

4kBT ie cosh2(

~2(k′2−k2F,i)

4m∗kBT ie

) . (148)

Final Form of Screening

In summary, there is one screening constant used for all subbands and it is found

using the expression:

εr(q) = 1 +∑i

e2

2εsqΠii(q)Aiiii(q) (149)

53

where

Πii(q) =

∫ ∞0

dEΠii(q, E, T = 0)

4kBT ie cosh2(E−EF,i

2kBT ie

)Πii(q, k

′, T = 0) =m∗

~2π

[1− θ(q − 2k′)

√1− 4k′2/q2

]E =

~2k′2

2m∗. (150)

This polarization function has the added benefit of accommodating individual sub-

band temperatures, which allows the results of the electron temperature model (see

Section 2.1.l) to be used. The integral is computed numerically, as described in Sec-

tion 2.2.k.

Comparison with Literature

This derivation is consistent with the general one put forth by Maldague60 and Ando.61

Harrison also derives a form of e-e screening,54 but makes the assumption that

E = EF in the zero-temperature polarizability. He therefore takes the polarizability

out of the integral, so that the integral is evaluated analytically. Harrison finds the

polarizability to be:

Πii(q) =m∗

2~2π(1 + tanh(EF/2kBT ))

[1− θ(q − 2kf )

√1− 4k2f/q

2]

(151)

This form was found to lead to significant errors by our team.

2.1.k Rate Equations

Once all of the scattering rates are known, the rate equations are applied to determine

the level populations as described by Donovan.62 Use of the rate equations assumes

54

that the system is in a state of equilibrium. This means that the total rate at which

electrons transition into a given level is equal to the total rate at which electrons

transition out. The total rate is the the transition rate of one electron times the

number of electrons in the initial state. The rate equations therefore form a series of

coupled equations. In this model, the photon populations are solved self-consistently

with the electron populations. There is a photon population associated with each

possible transition, and a rate equation for each photon population. In order to

attain completeness and accuracy, all possible electron and photon populations in the

three period structure are included in the rate equations.

Definitions and Notation

There are N quantum states in the three repetitions of the QCL core period, spanning

from i = 0 to i = N − 1. The computational model finds the electron population

ni of each level as well as the photon population mij of each possible transition.

At this point in the calculations, the model has already found all transition rates

Wij from level i to level j: electron-photon scattering, electron-electron scattering,

and electron-phonon scattering (see Sections 2.1.g, 2.1.h and 2.1.i). Also known is

the total electron density ntot which is the sum of all the level populations in one

period and remains constant. The model has already found the waveguide loss αw,

the mirror loss αm, the confinement factor Γ, and the photon cavity loss rate W pij at

many frequencies (see Section 2.1.f).

The current cannot be known since it depends on the populations. Therefore,

55

electrons flowing into and out of one period of the QCL must be accounted for by

assuming the structure is periodic and at equilibrium so that all the electrons flowing

out of the period must flow back into the period. The middle of the three numerically

calculated QCL periods is taken to be the most accurate and is the one used. This

raises the dilemma of how to handle wavefunctions and transitions that span several

periods.

One way to accomplish this is to find the center of mass of each wavefunction

and treat the wavefunction as located in the period containing its center of mass. For

the purpose of enforcing periodicity, transitions are considered to happen between

the center-of-mass points of each state. Based on the center-of-mass approach, this

reduces the periodicity problem to four cases:

1. A transition starting in the central period and ending in the central period.

2. A transition starting in the central period and ending in the outer periods and

vice versa. This must be handled by matching up corresponding wavefunctions

in different periods and tying them to the same population variable.

3. A transition starting in one outer period and ending in the other outer pe-

riod. These are two-period transitions and are again handled by matching up

equivalent wavefunctions.

4. A transition starting and ending in the same outer period. These are ignored

as they are redundant to the first case.

56

Derivations

At equilibrium, the sum of all possible transition rates into the level and out of the

level are equal. The sum is done over every type of transition (electron-electron,

electron-phonon, electron-photon) and over every possible combination of initial and

final level in all three periods:

0 =N−1∑j 6=i

Wjinj − niN−1∑j 6=i

Wij +N−1∑j 6=i

W st,1modeji nj − ni

N−1∑j 6=i

W st,1modeij . (152)

The first term in Eq. (152) is the sum over all non-stimulated photon emission

transitions coming into level i. The second term is the sum over all non-stimulated

photon emission transitions leaving level i. The third term is all of the stimulated pho-

ton emission transitions coming into level i. The fourth term is all of the stimulated

photon emission transitions leaving level i.

The stimulated absorption and emission rates are equal, W st,1modeji = W st,1mode

ij ,

leading to:

0 =N−1∑j 6=i

Wjinj − niN−1∑j 6=i

Wij −N−1∑j=0

W st,1modeij (ni − nj) . (153)

The total stimulated emission rate is just the one-mode spontaneous rate times the

total number of photons Mij available to stimulate:

W st,1modeij = MijW

sp,1modeij . (154)

There is a photon numberMij corresponding to every possible transition and each

photon population also has a rate equation to be solved. Because both the stimulated

absorption rate and stimulated emission rate are equal, they are both dependent

57

on the one-mode spontaneous emission rate. This is handled in the computational

model by making the one-mode spontaneous emission matrix symmetric so that Wij =

Wji. Even though spontaneous emission cannot happen for transitions which increase

energy, this is still valid because the one-mode spontaneous emission rates are only

used to determine the stimulated rates, which can both increase and decrease in

energy.

The number of photons in a certain mode in the cavity that are available to

stimulate emission are set equal to the one-mode photon population density mij

times the volume V of the active region:

Mij = mijV. (155)

Inserting this into the Eq. (153) yields:

0 =N−1∑j 6=i

Wjinj − niN−1∑j 6=i

Wij −N−1∑j=0

mijVWsp,1modeij (ni − nj) . (156)

This expression represents a set of N coupled equations. This is solved numerically

by using fixed point iteration (i.e. solve for ni in terms of nj in order to establish iter-

ation equations). Start with an estimated set of populations and apply the iteration

equation repeatedly, treating ni as the next iterated value of the population densities

and nj as the current value:

ni =

∑N−1j 6=i nj

[Wji +mijVW

sp,1modeij

]∑N−1

j 6=i

[Wij +mijVW

sp,1modeij

] . (157)

This equation is repeated iteratively, setting nj = ni after each step, until the

population densities converge to a value. After each iteration, the populations must be

58

normalized so that their sum equals the total average electron density. One of the rate

equations is redundant and the populations are under-specified by the rate equations.

Applying the normalization is required to find a unique solution. The coupled rate

equations could also be solved using matrix inversion techniques. However, fixed-

point iteration was found to be easier to implement, and not significantly slower or

less accurate then matrix techniques.

Periodic boundary conditions, which take into account the total current entering

and leaving a period, are implemented by treating each level of the three period

numerical array of data as independent. Therefore, the rate equations are solved as

if they are independent, and, at the end of each iteration, the population densities

are copied from the levels in the central period to the equivalent levels in the outer

periods. This requires knowing which wavefunctions are equivalent to each other.

Each photon population present also has a rate equation at equilibrium:

0 = VW st,1modeij ni − VW st,1mode

ji nj + VW sp,1modeij ni − VpmijW

pij. (158)

The first term is the total rate at which photons are added to the cavity in mode

ij due to stimulated emission. The second term is the total rate at which photons

are removed from the cavity in mode ij due to stimulated absorption. The third

term is the total rate at which photons are added to the cavity in mode ij due to

spontaneous emission. The fourth term is the rate at which photons are removed due

to the waveguide losses and mirror losses, where W pij is the total photon cavity loss

rate at the frequency corresponding to the ij transition.

59

The photon loss rate is a function of the waveguide loss, mirror loss and group

velocity according to W pij = (αw + αm)vg. These values are pre-calculated in the

waveguide model as a frequency-dependent look-up table (see Section 2.2.g). In these

rate equation calculations, the frequency of the transition is used to look up the pho-

ton cavity loss rate for that photon population. The electrons able to be stimulated

are only in the active region, so that the active region volume V must be multiplied

by the electron population densities to get the total number. However, all photons

can be lost (those in the active region and in the outer waveguide layers) so that the

cavity volume Vp must be used to find the total number of photons lost.

Dividing every term by Vp and recognizing the known confinement factor Γ =

V/Vp:

0 = ΓW st,1modeij ni − ΓW st,1mode

ji nj + ΓW sp,1modeij ni −mijW

pij. (159)

This equation exists for every possible pair of initial electron level i and final electron

level j. The density of photons mij with a frequency and polarization corresponding

to the ij transition is assumed to be in equilibrium.

A photon population corresponds to a pair of levels involved in the transition and

includes both i to j and j to i transitions. In order to avoid redundant calculations,

this equation is only implemented when the initial level is higher in energy than the

final level. The opposite calculation (lower to higher) should not be done because it

gives redundant information (because mij = mji) and requires a different equation.

The computational model simply calculates only down transitions when finding the

photon populations, then copies the populations to the up transition slots for use in

60

future calculations.

Again using W st,1modeij = mijVW

sp,1modeij and W st,1mode

ji = W st,1modeij where V is

the active region volume leads to:

0 = ΓmijVWsp,1modeij (ni − nj) + ΓW sp,1mode

ij ni −mijWpij. (160)

Solving for the photon populations yields:

mij =ni

W pij/(ΓW

sp,1modeij )− V (ni − nj)

. (161)

2.1.l Electron Temperature

All electrons on average can acquire a temperature (Te) different from the lattice

temperature (TL) due to the applied bias and exchanged energy between the elec-

tron subbands and the lattice as described by Troccoli.63 Additionally, each electron

subband can have a different temperature. These differences are vital to the inter-

nal physics of QCL’s since the subband energy distributions are dependent on the

subband temperature. Therefore, the scattering rates, population densities and all

subsequent calculations are also affected by electron temperature. The harmful ef-

fects of temperature on device performance have been studied extensively by Indjin,

Jirauschek and Lever.64–66

The electron temperatures for each subband can be found through a MC proce-

dure, but not from a simple self-consistent model. When self-consistent models are

modified to account for electron heating, typically only an average Te for the entire

conduction band is found as seen in several papers.67–70 However, Monte Carlo (MC)

61

simulations in the literature39,44,71–76 and experimental measurements by Vitiello77,78

have shown that there is considerable variation among subband temperatures, espe-

cially in resonant phonon QCL designs. In order to determine a separate Te for every

subband using the self-consistent model, a multi-subband energy balance condition

must be implemented as proposed by Ikonic and Jovanovic.42,79

A self-consistent energy balance (SCEB) condition which determines the tem-

perature for electron conduction subbands is described below. This additional com-

ponent to the computational model permits the determination of individual electron

subband temperatures. The effects of electron subband heating on carrier transport

within QCL’s can then be assessed. The specific implementation in the model as well

as the associated algorithm is described in Section 2.2.m.

Subband electron temperature

If a lasing QCL is assumed to reach a steady-state condition, as was the case when

considering the electron population densities, then the rate of change of energy for

a subband is assumed to be constant. Therefore, the subband electron temperatures

(T ie) can be determined by a set of energy balance equations as described by Jo-

vanovic.42 This is similar to the average electron temperature equation put forth by

Harrison.67 In the energy balance equations, the potential energy is defined as the

subband energy minimum (or the quantized eigenenergy). The kinetic energy, which

is related to the electron temperature, is defined as any additional energy above that

minimum. Although the average electron temperature equation can be solved using

62

just the electron eigenenergies, a multi-subband equation also requires the electrons’

kinetic energy. An average subband kinetic energy is found by taking advantage of

the fact that in a QCL, electrons are quasi-free in two-dimensions. This permits the

use of the classical result that the average kinetic energy per degree of freedom is

12kBT . Therefore, the average electron kinetic energy in a subband i is:

βi = kBTie (162)

where kB is Boltzmann’s constant.

The multi-subband energy balance condition is met when the rate of energy

lost from transitions out of each subband is equal to the rate of energy gained from

transitions into each subband. For a single particle transition between two states,

i→ f , the rate of change of energy out of subband i is the average kinetic energy of

an electron in subband i (measured from the minimum of subband i) multiplied by

Wif . This energy rate is also multiplied by the population density of the initial state ni

within the balance equations to properly weight each subband energy rate. However

the same single particle transition also affects subband f . From the perspective of

subband f , the rate of change of energy of subband f is the average kinetic energy of

an electron coming into subband f multiplied by Wif . The average kinetic energy of

this particle must be determined using conservation of energy by taking into account

the energy Ei as well as any additional scattering particles (such as phonons), which

also contribute to the energy transition. In general, the energy balance condition for

63

a subband f is:

dEfdt

=∑i

ni〈Ekf 〉Wif − nf

∑i

βfWfi (163)

where 〈Ekf 〉 is the average kinetic energy in subband f as determined using conser-

vation of energy. For transitions involving one electron, the average kinetic energies

are related according to:

〈Ekf 〉 = Ei + βi − Ef + δE (164)

where δE represents the energy of the scattering particle. The single particle scatter-

ing mechanisms included in the computational model were:

δE =

ELO for phonon absorption

−ELO for phonon emission.

(165)

The balance condition for e-e transitions is more complex since not only does it

involve four particle states, but it also has to be determined from a four-dimensional

matrix of scattering events. In this case, the energy transition rate is:

dEe-ef

dt=∑ijg

ni〈Ekf 〉W e-e

ijfg − nf∑gij

βfWe-efgij (166)

where W e-eijfg is the rate for two electrons in the initial states i and j that scatter to

the states f and g, respectively, and:

〈Ekf 〉 = Ei + βi + Ej + βj − Ef − Eg − βg. (167)

Finally, all forms of scattering which contribute to the rate of change of energy

into and out of a subband f must meet the energy balance condition:

dEemf

dt+dEabs

f

dt+dEe-e

f

dt= 0. (168)

64

Since the scattering rates, population densities and kinetic energies in Eq. (168) are

temperature dependent, the temperature of each subband has to be varied until the

balance condition is met for all subbands. This is accomplished using an iterative

procedure with the entire multi-subband SCEB model, as described in Section 2.2.m).

Average electron temperature

Due to it’s simplicity and successful application in mid-infrared QCL’s (see works by

Harrison67 and Spagnolo80), an average electron temperature is often calculated for

the entire device. This has the advantage of considerably reducing the computation

time for an iteration of the computational model; however, potentially valuable in-

formation is lost since there may be considerable variation among subband electron

temperatures. Additionally, Terazzi has suggested that using a single temperature

to describe electron heating can lead to overly pessimistic predictions of laser perfor-

mance.81

The average electron temperature energy balance condition is found by simply

summing the subband energy balance condition of Eq. (168) over all subbands. It is

often rewritten as:

∆ =∑

em,abs,e−e

∑f

∑i

niτif

(Ei − Ef − δE) = 0 (169)

where δE is equal to −ELO for phonon emission (em), +ELO for phonon absorption

(abs), and zero for electron-electron (e-e) scattering.

It should be noted that electron-photon scattering does not have to be considered

in either Eqs. (168) or (169) since the absorption or emission of a photon by an electron

65

negligibly changes its kinetic energy due to conservation of momentum. However, the

indirect effects of photon scattering on energy balance through electron population

alterations are automatically included.

2.1.m Output Power

Finding the laser output power at a certain frequency is the end purpose of the QCL

computational model. The photon population at a certain frequency is found self-

consistently using the rate equations (see Section 2.1.k). With the photon population

known, it is straight-forward to find the output power from the population and the

device characteristics. In theory, the output power only needs to be calculated from

the population at the end of the calculations, after all self-consistent loops have

converged. In practice, it is helpful to the user to have an order-of-magnitude estimate

of the optical power before the loops converge. For this reason, the output power

calculations are placed inside the loops and calculated every time a new photon

population is found. Considering the fact that the output power calculations are

very simple and quick to perform, placing them inside the loops does not incur any

substantial run-time degradation.

The QCL computational model does not ask the user to specify in advance which

transition is the laser transition. Rather, it is the task of the model to determine

the laser transition. In order to do this, the rate equations calculate the photon

populations for all possible transitions. Further, the output powers are calculated for

all possible transitions. The laser transition is then identified as the one with the

66

highest output power.

Large photon populations are likely for transitions with frequencies below 1 THz

in addition to the frequency of interest. However, such laser frequencies do not

occur experimentally. The reason for this is that traditional QCL waveguides do not

support modes at frequencies below 1 THz, as the loss is too high and the confinement

too low. It is therefore imperative for the QCL computational model to properly

calculate the waveguide effects at low frequencies and apply them as part of the photon

population and output power calculations in order to realistically damp down the high

power response of low frequency transitions. Slight inaccuracies in the waveguide

calculations can lead to false predictions of high-power, low-frequency lasing.

Derivation

The laser radiated output power Pout at a certain frequency is the energy emitted per

unit time out of the front facet. The radiated power at a frequency equals the total

number of photons M at that frequency emitted per unit time, times the energy per

photon E:

Pout(ω) = ME = M~ω. (170)

The number of photons of a certain frequency being emitted per unit time out

the front surface of a laser must equal the total number of photons present mtot at

that frequency inside the laser cavity times the rate Wm at which a single electron is

emitted out of the front surface mirror, M = mtotWm, so that:

Pout(ω) = mtotWm~ω. (171)

67

Also, the total number of photons at a frequency equals the volume of the photon

region Vp times the photon population density m, mtot = Vpm, so that:

Pout(ω) = VpmWm~ω. (172)

The volume of the photon cavity is defined as the volume of the active region V

divided by the confinement factor Γ, Vp = V/Γ, so that:

Pout(ω) =V mWm~ω

Γ. (173)

Mirror Effects

One photon can be thought of as bouncing back and forth between the two end

mirrors of the cavity. Every time it hits the front surface mirror, it has a certain

chance of being emitted based on the mirror’s reflectivity. The effects of the mirror

reflectivity are averaged over one round trip of the photon. The rate W at which

a single photon is emitted out the front surface equals the total probability γ2 of a

photon being emitted out of the second mirror during one round trip divided by the

time it takes the photon to traverse one round trip:

Wm =γ2∆t. (174)

The time ∆t it takes to make a round trip is just the distance traveled in one round

trip (twice the length of the cavity l) divided by the velocity v of the light in the

material:

∆t =2l

v=

2ln

c(175)

68

so that:

Wm =γ2c

2ln. (176)

Defining the loss per unit length αM2 due to emission out mirror two as the total loss

γ2 in one round trip divided by the length of one round trip 2l,

αM2 =γ22l

(177)

so that:

Wm = αM2c

n. (178)

The mirror loss αM2 and the effective index of refraction n were found in the waveguide

calculations (in Section 2.1.f).

Final Equation

Plugging in the mirror loss rate into the output power equation yields:

Pout(ω) =V mαM2c~ω

nΓ. (179)

This defines a power spectrum as a function of frequency. Typically there is only

one frequency with non-negligible output power. However, the power is calculated

using this equation for all possible transitions to ensure completeness. Note that the

photon population m, the effective index of refraction n, and the confinement factor

Γ are all frequency dependent and must be calculated separately for each possible

transition. The above equations agrees with similar results from Williams.52

69

Finally, if the total power radiated into all frequencies is desired, a sum is per-

formed over the individual frequencies:

P totout =

∑ω

Pout(ω). (180)

2.2 Numerical Implementation

Computational modeling was established using a set of sequential calculations and

convergence conditions which ensured all electron wavefunctions, eigenenergies, pop-

ulations, lifetimes and temperatures were consistent with one another. This sequence

(called the self-consistent energy balance or SCEB model), which was repeated until

all convergence conditions were met, provided an accurate representation of the lasing

device (see Fig. 1).

The wavefunctions and eigenenergies for three full QCL periods were found using

a solution of the one electron Schrodinger and Poisson equations. Doing so took into

account space charge effects, which have been shown by Jirauschek to significantly

affect model predictions.74 The wavefunctions from the central period were then

copied to the surrounding two periods since they most accurately represented an

infinite cascade structure, as argued by Ikonic.79 The subbands were evenly populated

for the first SCEB iteration; however, all later iterations used the population densities

resulting from the solution of population rate equations.

A waveguide mode solver was used to determine all field coefficients and the

axial wavenumber of the fundamental mode (see Section 2.2.g). A one-dimensional

70

FIG. 1. Flow chart of the complete QCL computational model. Conditional steps areindicated by a question mark. After the Schrodinger-Poisson box, electron wavefunctionsare checked for convergence. After the electron populations box, electron populations arechecked for convergence. After the electron temperature box, the electron temperature ischecked for convergence. Finally, the electron wavefunctions are again checked for conver-gence before the process is repeated again.

71

slab waveguide model was used, which coupled the field coefficients with the axial

wavenumber via a transfer-matrix approach, which is described in detail by Baird.82

The total waveguide loss, confinement factor and gain threshold were then found.

Transition rates were calculated using Fermi’s “golden rule” for electron-photon

(e-p), electron-longitudinal-optical phonon (e-LO), and electron-electron (e-e) scat-

tering (see Sections 2.2.h, 2.2.i and 2.2.j, respectively). These rates were averaged

over the in-plane wavevector assuming Fermi-Dirac statistics. All possible transi-

tions within three periods of the active region were found, which included a four-

dimensional matrix of rates to describe all possible e-e scattering events. The effect

of screening on e-e scattering rates was also taken into account (see Section 2.2.k).

A set of iterative rate equations were then solved (see Section 2.2.l) for each

subband in order to modify the subband population densities using the calculated

transition rates as described by Donovan.62 The rate equations took into account

every transition within three periods of the active region which could possibly affect

the subband populations. This was a refinement over rate equation models which

simply used the current density to account for scattering into and out of the lasing

transition as done by Kohler.83 Once the electron and photon population densities

reached convergence, the energy balance condition was applied to determine subband

electron temperatures (see Section 2.2.m). This required calculating scattering rates

and population densities at a number of different electron temperatures, until the

temperature convergence condition was met.

Finally, after the first iteration of the complete multi-subband SCEB model, the

72

electron population densities were checked for convergence by comparing them to the

results of the previous iteration. If the densities differed by more then a predefined

threshold value, the model was repeated, starting with the Schrodinger-Poisson solver,

using the results of the previous iteration to improve the input parameters. If the

populations differed by less than the threshold value on the next iteration of the

model, then the output parameters were determined, such as lasing frequency, power

and current.

2.2.a Non-uniform Location Grid

A quantum cascade laser consists of a stack of nanoscale epitaxial semiconductor

planar layers. The material is uniform and near-infinite in the directions transverse

to crystal growth (the x and y dimensions). As a result, all of the properties in a QCL

depend only on the growth direction (the z dimension). Because the material changes

in the growth direction, all material properties, such as the dielectric constant and

effective mass, are functions of z and cannot be treated as constant. The conduction

band edge, the built-in potential, the electron wavefunctions, and the space charge are

all functions of z. In order to solve the physical equations numerically, every function

of z must be represented as a set of values over a discrete grid of z location points.

Practicality, efficiency, and accuracy dictate that all properties that are functions of

z should be defined across the same grid of points. The model therefore establishes

a location grid data structure before solving any physical equations. The location of

all z points is held fixed through-out the entire QCL computation and all variables

73

are calculated at these z points.

Create Location Grid

Three repeated periods of the core QCL structure are used in the model. While

a typical QCL has over 70 periods, three periods are sufficient for computations if

periodic boundary conditions are enforced. (The potential is not strictly periodic

because there is a net potential drop across one period due to the applied bias. More

accurately, all of the variables are periodic after the bias has been divided out.)

Multiple periods are needed to represent the structure with any reasonable amount

of accuracy since many electron wavefunctions are longer than one period. In order

to enforce periodicity and avoid edge effects, all variables are solved across all three

periods, and then the more accurate values in the central period are copied to the

outer periods.

The location grid starts at z = 0, marking the point where electrons begin

cascading down the structure, and extends to z = 3L, where L is the length of one

period. The positive z direction is the direction of the electrons’ net motion as they

cascade down through the structure.

One option for choosing location grid points is to simply distribute them uni-

formly between the endpoints according to some user-defined density. This option

would allow standard integration techniques such as Simpson’s rule to be used, as

well as simplify the numerical representation of differential equations. However, this

option leads to poor accuracy. The inaccuracy occurs because material layer widths in

74

a QCL are highly non-uniform, and the wavefunctions depend strongly on the layer

widths. Therefore, using a uniform grid of z points would require rounding layer

interface locations to the nearest grid point, significantly altering their widths, and

therefore degrading all subsequent calculations.

An alternative option is used by the present computational model: grid points

are set exactly at the location of every material layer interface. Grid points are then

spread uniformly within each layer according to a density specified by the user. By

using this approach, the grid is very close to uniform, but still preserves the exact

layer widths. Because the grid is non-uniform, the non-uniform trapezoidal method

must be used to perform numerical integrations over z instead of Simpson’s rule.

Also, when expanding differential equations, the step size h of any derivative depends

on the location and is not a constant.

Populate Grid with Material Properties

All of the position-dependent properties that stay constant are calculated at the start

of the computational model. These properties are essentially material properties

that are assumed to behave as in bulk. The band-gap, band alignment, and the

donor ionization density are discussed elsewhere. All other material properties are

calculated at each grid point using a linear model such as:

m = m0 +m1x (181)

Here x is the alloy concentration at the grid point, m0 is the property value when

there is no alloy present, and m1 is the property value change that occurs at a full

75

alloy concentration.

2.2.b Initial Fermi Levels

The first time the individual Fermi levels are calculated (see Section 2.1.e for the

derivation), the populations have not yet been determined. As an initial set of values

from which the model can iteratively converge, all of the populations are set to be

equal and then normalized so that the total of all the populations equals the overall

crystal electron density. This is based on the assumption that the electrons are

uniformly distributed across all of the states as an initial setting.

2.2.c Poisson Equation

The fourth-order Runge-Kutta (RK4) numerical method was used to solve the Poisson

equation (see Section 2.1.b for derivation). This method requires transforming one

second-order differential equation into two coupled first-order differential equations.

Applying this to Poisson’s equation we define:

dΦ(z)

dz=

1

ε(z)Φ2(z) where

dΦ2(z)

dz= −ρ(z). (182)

Boundary Conditions

This method requires knowledge of the initial conditions Φ(z0) and Φ2(z0). Instead,

the boundary conditions Φ(z0) = 0 and Φ(zN) = ∆Vbias are known. The variable

∆Vbias is the external bias voltage applied across the grid of location points (three

periods in the current model). In order to get around this problem, the shooting

76

method is used. This involves guessing some value for Φ2(z0), computing the entire

function corresponding to this value using the methods above, then computing the

error as the difference between the resulting boundary value and the desired boundary

condition. The value of Φ2(z0) is then refined using a binary search method until the

error is minimized.

2.2.d Space Charge Density

The Poisson equation (see Section 2.2.c) depends on the space charge present. How-

ever, charge density is a dynamic quantity in a QCL and special care must be taken

throughout the computational model to ensure that it is properly calculated and im-

plemented. A derivation of the various quantities described below can be found in

Section 2.1.d.

Initial Space Charge Electron Density

The populations are found using the rate equations, but when the model begins, the

rate equations have not yet been applied. An initial space charge density must be

used so that the model has a starting point before it converges to the true solution. In

theory, the choice of initial electron density should be irrelevant to the final solution.

In practice, using an initial electron density that is close to the final solution will

dramatically reduce run-time and will lessen the effects of compounding numerical

error. The model used for the initial space charge electron density is one that assumes

the electrons tend to bunch up behind the largest barrier, but extend backwards

somewhat from this barrier because of thermal smearing. The model therefore carries

77

out the following algorithm:

• Find the space charge donor density ndonor(z).

• Find the location of the thickest barrier in the QCL core structure.

• Set up the initial space charge electron density profile as a decaying exponential

curve nelec(z) = f(z) with a peak at the barrier, extending uphill away from the

barrier (in the opposite direction from the cascading electron motion).

• Scale the electron density profile until the area under its curve matches the area

under the donor density profile curve to ensure the overall total charge of the

QCL is zero, according to:

∫nelec(z)dz =

∫ndonor(z)dz. (183)

• Find the total charge density as the sum of the donor and electron densities,

according to:

ρ(z) = endonor(z)− enelec(z). (184)

• Using this initial charge density, find the built-in potential using the Poisson

equation, add the built-in potential to the conduction band-edge potential to

get the total potential, insert this into Shrodinger’s equations and find the

wavefunctions.

• Refine the initial charge density profile by using the wavefunctions multiplied

78

by the exponential model:

nelec(z) = f(z)

1 period∑i

|ψi(z)|2, (185)

and again scale to proper magnitude.

• The refined initial charge density is then the starting point when the main

iterative loop of the model is begun. For all further iterations, the populations

are found using the rate equations and applied as described in Section 2.1.d.

2.2.e Schrodinger Equation

The fourth-order Runge-Kutta (RK4) method was used to numerically solve the

Schrodinger equation (see Section 2.1.a for a derivation and Section 2.1.c for an expla-

nation of V (z)). First, the single second-order differential equation must be broken

into two first-order differential equations before applying the RK4 method:

dψ(z)

dz= m∗(z)ψ2(z) (186)

dψ2(z)

dz= − 2

~2(E − V (z))ψ(z). (187)

Boundary Conditions

This method requires the initial conditions ψ(z0) and ψ2(z0)to be known. However,

the boundary conditions that are known are ψ(z0) = 0 and ψ(zlast) = 0. Therefore

the initial conditions ψ(z0) = 0 and ψ2(z0) = 1 are used, understanding that the

entire wavefunction will be off by an overall scale factor. This scale factor error goes

away however when the wavefunction is normalized at the end to ensure that there

79

is 100% probability of finding the electron anywhere in its wavefunction. The ac-

tual boundary conditions that should be implemented are pseudo-periodic boundary

conditions because a typical QCL has over 100 repetitions of the same core period

structure. However, periodic boundary conditions are difficult to implement directly

without requiring substantial computational cost. Instead, the Dirichlet boundary

conditions are applied, and solved for the wavefunctions across three periods of the

QCL core structure. After finding all valid wavefunctions, all wavefunctions with a

center of mass that lies in the outer periods are thrown away, and all wavefunctions

with a center of mass in the central period are copied to the outer periods for future

use. This process ensures periodic boundary conditions.

Eigenvalues

These equations depend on the energy E of the electron in its state, but E is not

known beforehand. There are actually several possible solutions to a certain poten-

tial: the eigenstate wavefunctions ψn of the system, each with its own energy En. The

correct eigenenergies En are the ones that yield valid bound wavefunctions; wavefunc-

tions that do not blow up towards infinity outside the wells. Initially, a En must be

guessed, and the corresponding wavefunction must be calculated using the algorithm

listed above. The guess is then iteratively refined for En by minimizing the error in

the wavefunction. Because the wavefunction is supposed to match the boundary con-

dition ψ(zlast) = 0, the actual amplitude squared at the boundary |ψ(zlast)|2 is taken

as the error. By calculating a sweep of possible En values and their corresponding

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wavefunction errors, an error landscape is generated. The correct energies are the

minima in such an energy landscape. The minima are identified, and then refined

using a binary search method.

Normalization

Electron wave functions should always be normalized before being used in any sub-

sequent calculations. The wave functions should not be magnitude squared unless

explicitly required, as the complex phase of the wavefunction is needed is some subse-

quent calculations. The squared wave functions constitute a probability density, thus

the proper way to normalize each is by integrating.

Defining ψnorm(z) = Aψunnorm(z) and applying the concept that the probability

of being found anywhere is 1 leads to:

1 =

∫ ∞−∞|ψnorm(z)|2dz

1 =

∫ ∞−∞

A2|ψunnorm(z)|2dz

A =1√∫∞

−∞ |ψunnorm(z)|2dz

ψnorm(z) =ψunnorm(z)√∫∞

−∞ |ψunnorm(z)|2dz(188)

The integral is done numerically using the non-uniform-grid trapezoidal method. The

integral cannot be done over infinity, but is instead done over the three periods of

the QCL core structure used in this model. Note that the location points z and

the wavefunctions are defined on a non-uniform grid in order to preserve material

layers widths exactly. Using a non-uniform grid means that traditional numerical

81

integration methods such as Simpson’s rule and the Boole rule cannot be used.

2.2.f Copy Wavefunctions

In order to strictly ensure periodic boundary conditions and periodic wavefunctions,

but still model three full periods of the core structure, the wavefunctions in the central

period are copied to the outer periods and replace the original wavefunctions there.

Even though the wavefunctions typically extend across multiple periods, most of the

wavefunction is contained in one period, and is therefore assigned to that period by

calculating the location of the center of mass of the wavefunction. All wavefunctions

with a center of mass lying outside the central period are discarded, and then all

remaining wavefunctions are copied and shifted left one period, and copied again and

shifted right one period. The wavefunctions are shifted in space by the length of one

period and shifted in energy by the potential energy drop across one period due to

the bias.

Note that when some wavefunctions are copied to the outer periods, part of the

wavefunction’s profile extends into regions not modeled by the three-period grid of

location points and therefore must be discarded. The natural result is that such

wavefunctions are no longer accurately normalized and are also unable to give accu-

rate form factors. Such edge effects are inevitable but have minimal overall effect.

These edge effects are minimized by always keeping only calculations performed in

the central period and copying them to the outer periods for every type of calcula-

tions. Using outer periods keeps the edge effects away from the central period where

82

the meaningful calculations are performed. Additionally, wavefunctions that reside

in the central period still extend into the outer periods.

Calculating Center of Masses

A given wavefunction ψi(z) corresponding to the ith energy level has some spatial

extent. If instead of treating its square as the quantum extent of one electron, it is

treated as the population density of a large ensemble of electrons, then it becomes

the spatially varying electron density of that level. Because the electrons all have the

same mass (neglecting the fact that the effective mass varies from material layer to

layer), the center of mass of the ith state is:

zi,com =

∫|ψi(z)|2zdz∫|ψi(z)|2dz

. (189)

The wavefunctions are properly normalized as soon as they are found, so that the

integral in the denominator is equal to one:

zi,com =

∫|ψi(z)|2zdz. (190)

The integral is done numerically using the non-uniform-grid trapezoidal method. The

integral cannot be done over infinity, but is instead done over the three periods of

the QCL core structure used in this model. Note that the location points z and

the wavefunctions are defined on a non-uniform grid in order to preserve material

layers widths exactly. Using a non-uniform grid means that traditional numerical

integration methods such as Simpson’s rule cannot be used.

Every time new wavefunctions are found using the Schrodinger equation, the

model immediately calculates the center of mass of each wavefunction, assigns it to

83

one of the three periods based on this center of mass, copies the central wavefunctions

to the outer periods, and provides this information for use by later calculations.

Equivalent Levels

After copying wavefunctions to the outer periods, all periods have been forced to

be numerically equivalent. Every wavefunction in an outer period matches exactly

with one in the central period. This information is used later on to optimize the rate

equations as well as the scattering rate calculations. For this reason, all wavefunctions

in the three periods are indexed in order of energy, the lowest energy being state zero.

A list of equivalent wavefunctions according to index number is kept for later use.

In this sense, the wavefunctions are automatically matched because we have forced

them to be matched.

2.2.g Waveguide Numerical Analysis

The equations which were derived in Section 2.1.f require numerical methods in order

to solve them. The problem is to solve for Ai and Bi in each region in terms of the

known permittivities εi, boundary locations xi, frequency ω and the guessed wave

number kz. The wave number kz is guessed and refined until the outermost boundary

conditions are met, that of fields approaching zero at positive and negative infinity

for x.

Add the two boundary condition equations together to eliminate Bi+1 and solve

84

for Ai+1:

Ai+1 =1

2

(1 +

kx,iεi+1

kx,i+1εi

)Aie

i(kx,i−kx,i+1)xi+1

2

(1− kx,iεi+1

kx,i+1εi

)Bie

−i(kx,i+kx,i+1)xi . (191)

Subtract the two equations to eliminate Ai+1 and solve for Bi+1:

Bi+1 =1

2

(1− kx,iεi+1

kx,i+1εi

)Aie

i(kx,i+kx,i+1)xi+1

2

(1 +

kx,iεi+1

kx,i+1εi

)Bie

−i(kx,i−kx,i+1)xi . (192)

This now gives a set of iteration equations. Using a trial kz and because ω, εi, and

µi are known, kx,i is also known using:

kx,i =√εiµiω2 − k2z . (193)

Thus if A0 and B0 are known, all other Ai and Bi are found using these iteration

equations.

Since the bound modes are the modes of interest, the wave must die down to

zero at negative and positive infinity. This leads to the conditions:

• if =(kx,0) > 0, then A0 = 0

• if =(kx,0) < 0, then B0 = 0

• if =(kx,N) < 0, then AN = 0

• if =(kx,N) > 0, then BN = 0

There is an overall normalization constant for the mode that needs to be found

and applied after the problem is solved. This is done by setting the non-zero coefficient

of the zeroth layer equal to one and normalizing it at the end to its proper value.

85

• if =(κ0) < 0, then A0 = 1

• if =(κ0) > 0, then B0 = 1

The system is solved iteratively for the propagation constant kz. First a guess for

kz is made (or a grid of guesses), then the iteration equations are applied to find the

fields everywhere corresponding to this kz. The amount that the coefficients AN and

BN in the last region differ from what they should be as dictated by the boundary

conditions is calculated as the error corresponding to that kz. The right kz is found

by refining its value until the error is minimized. The lowest-order mode is the one of

most importance and it is the one that is found. Note that kz is complex-valued so

that finding the physical modes amounts to finding the minima in a two-dimensional

error landscape.

Final Numerical Recipe

1. Calculate the complex permittivity of each layer using the Drude model (see

Section 2.1.f) and assume non-magnetic material µi = µ0.

2. Sweep across a grid of initial possible values for the real and imaginary part of

the wave vector kz:

(a) For each complex kz value, calculate the complex x-directional wave num-

ber of each waveguide layer using:

kx,i =√εiµiω2 − k2z (194)

86

(b) Set A0 and B0 according to the following rule:

• if =(kx,0) > 0, then A0 = 0 and B0 = 1

• else A0 = 1 and B0 = 0

(c) Calculate all remaining Ai and Bi using:

Ai+1 =1

2

(1 +

kx,iεi+1

kx,i+1εi

)Aie

i(kx,i−kx,i+1)xi

+1

2

(1− kx,iεi+1

kx,i+1εi

)Bie

−i(kx,i+kx,i+1)xi

Bi+1 =1

2

(1− kx,iεi+1

kx,i+1εi

)Aie

i(kx,i+kx,i+1)xi

+1

2

(1 +

kx,iεi+1

kx,i+1εi

)Bie

−i(kx,i−kx,i+1)xi (195)

(d) Find the error as the difference between the calculated AN or BN and the

required AN or BN according to:

• if =(kx,N) < 0, then AN = 0

• else BN = 0

3. Find all mimima in the 2D error landscape as a function of kz. A minima is a

grid point where its error value is lower than the error value of its eight nearest

neighbors.

4. Refine every minimum kz by repeating step 2 for additional guesses and keeping

the ones that give the least error. Use the method of gradient descent. Step

iteratively in the opposite direction of the gradient of the error landscape ac-

cording to: [kz]n+1 = [kz]n− γn∇f([kz]n) where f(kz) is the 2D error landscape

87

as a function of kz. Due to the complexity of the error landscape, the gradient

cannot be solved analytically but the finite difference is used. The step size γn

is optimized by finding the first value that minimizes f([kz]n+1) locally.

5. Choose the kz which corresponds to the lowest-order mode. It is the one with

the lowest value for (kx/kz) in the largest layer.

6. Once the lowest order kz has been identified and refined, use step 2 again to get

the final values for Ai and Bi

7. Using the final values of Ai and Bi calculate all the fields across a fine grid of

x locations using:

Hy,i(x) = Aieikx,ix +Bie

−ikx,ix

Ex,i(x) =kzωεi

(Aieikx,ix +Bie

−ikx,ix)

Ez,i(x) =kx,iωεi

(−Aieikx,ix +Bie−ikx,ix)

|Hi(x)|2 = |Hy,i(x)|2

|Ei(x)|2 = |Ex,i(x)|2 + |Ez,i(x)|2 (196)

8. Calculate the confinement factor for this waveguide structure at this frequency

using the trapezoidal or cubic splines method:

Γ =

∫act. reg.

|Ex,i(x)|2dx∫∞−∞ |Ei(x)|2dx

(197)

9. Calculate the associated parameters:

(a) Waveguide Loss: αw = 2=(kz)

88

(b) Total Cavity Loss: α = αw + αM1 + αM2

(c) Threshold Gain: gth = α/Γ

(d) Velocity: v = ω<(kz)

(e) Effective Index of Refraction: n = cv

(f) Total Cavity Photon loss rate: Wp = αv

(g) Total Cavity Photon lifetime: τp = 1/Wp

10. Repeat Steps 1 through 9 for many possible frequencies in order to generate

look-up tables.

Cutoff Frequency

Upon generating the look-up tables, which essentially establish loss vs. frequency

and confinement factor vs. frequency trend lines, a problem arises. Below a certain

cutoff frequency, electromagnetic modes cannot propagate. In terms of the practical

requirements, the waveguide model should return a confinement factor of zero and

a loss of infinity for these frequencies. However, the waveguide model is designed to

find modes as minima in an error landscape, and cannot find the lack of modes. As

implemented above, the waveguide model will output erroneous values for frequencies

below cutoff, and the rest of the model will use these values as if valid, leading to

widespread error.

The solution is to have the model determine the cutoff frequency and then hard-

set all confinement factors to zero and all losses to infinity below this frequency. Note

89

that if the confinement factor is set to zero, then divide-by-zero errors will occur later

on in the rate equations. In practice, each confinement factor must be set below cutoff

to a very small non-zero number; so small that it is essentially zero. Similarly, the

losses below cutoff cannot be set to infinity in a numerical calculation. Instead each

loss is set equal to a very high number that essentially behaves as infinity.

The cutoff frequency is found by starting at a high frequency on the confinement

factor vs. frequency curve, and asymptotically finding at what frequency the curve

approaches zero confinement.

2.2.h Photon Scattering Implementation

The photon scattering rate into a narrow mode distribution from Eq. (71) is calculated

without the photon number factor. The photon number is then found using the rate

equations. Note that there is great uncertainty in the literature as to what the

pure dephasing time T ∗2 should be,27,41,71 which is used to determine the line width.

The value used and published for mid-infrared QCL’s was found to give poor results

for Terahertz QCL’s. The value of T ∗2 = 15 picoseconds was found to give the best

model predictions for Terahertz QCL’s when compared to experiment and is the value

currently used by the model.

The integral in the oscillator strength of Eq. (72) is done numerically using the

non-uniform-grid trapezoidal method. The integral cannot be done over infinity, but

is instead done over the three periods of the QCL core structure used in this model.

90

2.2.i Phonon Scattering Numerical Implementation

The form factor integrands of Eq. (84) (see Section 2.1.h) are defined over a non-

uniform grid, so their integrals are done using the non-uniform trapezoidal method.

The rest of the integrals in Eq. (83) are defined on a uniform grid, so they are

calculated using Simpson’s rule or the Bode (Boole) rule. The form factor integrals are

performed over the full three periods of the QCL structure. The wavenumber integral

runs from zero to the maximum wavenumber possible. The maximum wavenumber is

the one corresponding to the well height in energy space, because wavenumbers greater

than this are not confined in the quantum well structures and do not contribute to

the process.

2.2.j Electron-Electron Computational Implementation

The two approaches (as discussed at the end of Section 2.1.i) were found to give

similar results with regards to numerical accuracy. The relative wavevector approach

was chosen because it was simpler to implement and is the only one considered in the

rest of this thesis. Also, state-blocking was found to be negligible in terahertz QCL’s

(see Section 4.2). For this reason, the final states are assumed to be empty so that

ff = 0 and fg = 0. Upon dropping state blocking, it was found that nothing else

depends on kf or kg. For this reason, they do not need to be calculated.

Since electron-electron scattering is a four-body interaction, the number of scat-

tering events which have to be calculated scales as N4, where N is the number of

states in a structure. These events include both symmetric types such as 2, 2→ 1, 1

91

(where 2 and 1 represent subbands) and asymmetric types such as 3, 2→ 2, 1. Previ-

ous studies have indicated the importance of these asymmetric scattering events (see

works by Donovan, Kinlser and Lee84–86) and are discussed further in Section 4.2.

The steps which are employed when calculating electron-electron scattering rates

are as follows:

• Calculate the form factor A(q) look-up table for a large set of q points using

the non-uniform trapezoidal method:

A(q) =

∫ Lz

0

dz

∫ Lz

0

dz′ψ∗f (z)ψ∗g(z′)ψi(z)ψj(z

′)e−|z−z′|q. (198)

• Calculate the front constant:

C =1∫

kifi(ki)dk i

m∗e4

64π3ε2~3(199)

• Calculate the energy difference constant:

g20 =4m∗

~2[Ei(0) + Ej(0)− Ef (0)− Eg(0)] (200)

• Calculate the scattering rate by performing the integrals and calling a sub-

routine to calculate the innermost integral. The integrals are calculated over a

uniform grid of points. The user chooses between 10 to 1000 points per integral,

depending on the accuracy desired. Experience has shown that using more than

50 integration points per integral has no discernible effect on the final results

of the entire computational model. The wavenumber integrals are calculated

from a lower limit of zero to an upper limit of kmax, which corresponds to the

92

height of the energy barriers. Any electrons with energies beyond the barrier

height will not be contained in the quantum wells. The angular integrals are

performed over the whole unit circle. Because the integrals are defined over

uniform grids, standard methods such Simpson’s Rule or the Boole (Bode) rule

are used. The scattering rate is in the form:

Wi,j→f,g = C

∫dkikifi(ki)

∫dkjkjfj(kj)

∫dθijU(ki, kj, θij ) (201)

• Calculate the inner integral U by performing the following steps:

1. Calculate the quantity: k2u = kikj cos θij

2. Calculate the quantity: g2 = k2j + k2i − 2k2u (If g = 0 then W = 0 for this

set of integration points)

3. Calculate the quantity: q = 12

√g20 + 2g2 − 2g

√g20 + g2 cos θ (If g20 +g2 ≤ 0

then W = 0 for this set of integration points)

4. Use these quantities to find: U(ki, kj, θij ) =∫dθ∣∣∣A(q)q ∣∣∣2

Additional Computational Optimizations

When including all antisymmetric transitions, calculating the form factor look-up

tables becomes unreasonably time intensive. For instance, a structure with 30 levels

across the three periods will require 304 = 810, 000 look-up tables to cover all possible

interactions of the states i, j to f , g. However, the number of calculations to perform

is reduced if redundant calculations are skipped.

93

Because of the periodic nature of the QCL structure, transitions completely

happening in the first period are identical to those happening solely in the second

period. If every wavefunction is located to either period 1, 2, or 3 based on its

center of mass (period 3 being the lowest in energy), then all redundant calculations

are skipped by applying the following rules. Later on, the whole data structure is

populated by copying the original data to its appropriate redundant locations. The

notation used below is ij → fg so that for instance 11 → 23 means an electron in

period 1 scatters with another electron in period 1 and they end up in periods 2 and

3 respectively.

• Include all 22→ 22 transitions.

• Skip transitions if: i 6= 1 and j 6= 1 and f 6= 1 and g 6= 1.

• Skip all 11→ 11 transitions.

• Skip double period transitions, when |i− f | = 2 or |j − g| = 2.

Scrutiny of the form factor equation reproduced below reveals that there are

several symmetries:

A(q) =

∫ Lz

0

dz

∫ Lz

0

dz′ψ∗f (z)ψ∗g(z′)ψi(z)ψj(z

′)e−|z−z′|q (202)

Aijfg = Afjig = Aigfj = Ajigf = Ajfgi = Agijf = Agfji = Afgij (203)

The way to skip these redundancies computationally is through the following rules.

Skip every transition except when (indexes now refer to subbands):

94

• j ≤ g

• i ≤ j and i ≤ f and i ≤ g

• f ≤ g if i = j.

Note that the data structure must be all filled out in order for later use, so these

rules must be used to copy the unique data to its corresponding redundant locations.

Also note that these symmetries only apply to the form factor and not the entire e-e

scattering rate.

2.2.k Electron-electron screening implementation

In order to take into account screening of the electron-electron interaction, the di-

electric constant ε in Eq. (118) is replaced by a modified form, which is a function of

the transition wavevector q, as described in Section 2.1.j. Since this quantity is no

longer a constant, it must be brought inside the integral which averages over ki before

arriving at Eq. (118). This is accomplished by making a look-up table for values of

q, similar to the one made for the form factor A(q) (see Section 2.2.j). In fact, the

permittivity look-up table uses the form factor look-up table since it is comprised of

not only the polarization function and a 1/q term, but also the intrasubband form

factor term Aiiii(q). Therefore, the inclusion of scattering requires a modified version

of the computational model to calculate the ki average integral.

The actual form of the dielectric function implemented was a modified version

of the single subband screening model. Single subband models only consider the

95

polarization of the electrons in the initial state of the e-e interaction, such as the one

employed by Smet.53 However, the modified form takes into account all subbands

within one period of the QCL active region. Additionally, the modified form of the

model has been shown by Lu and Jirauschek to provide better agreement with the full

screening tensor equation (see Eq. (138) from Section 2.1.j) than the single subband

model.73,74 This modified model is represented as Eq. (149), which was derived at the

end of Section 2.2.k. As done in that section, the temperature-dependent polarization

function of Maldague60 was found in the static limit as in Ando’s paper.61 The

polarization integral of Eq. (150) is computed numerically over a uniform grid using

standard methods such as Simpson’s rule or the Bode (Boole) rule.

2.2.l Rate Equation Implementation

In order to determine the population densities for the electron subband states, a

rate equation approach was implemented. This method determines the populations

through an iterative solution to a set of coupled equations. Each equation has the

form of Eq.(157). However, since these equations are all coupled to one another, the

system of population densities is under-specified so the additional equation:

∑i

ni = N, (204)

must be used, where i represents an electron subband and N represents the total

charge density.

In order to implement iteration equations some initial values for the electron

population densities and photon population densities must be chosen. The closer

96

that the initial values are to the final values, the more quickly the solutions will

converge. For the initial electron densities, all are set equal so that total electron

density is evenly divided among the levels. For the photon populations, an arbitrary

number is chosen that has the right order of magnitude as experimental values, such

as 1018 photons/m3.

The iterative solution of these electron rate equations is as follows:

• Initially set all populations equal to one another based on the total charge

density of the system

• Calculate the populations for each subband using Eq. (157)

• Normalize the populations so that they satisfy Eq. (204)

• Recalculate and normalize the populations using Eqs. (157) and (204)

• Repeat until the populations do not change more than a pre-defined threshold

value between multiple iterations

Together, the photon and electron population equations form another layer of

coupled equations. However, simply applying Eqs. (157) and (161) does not lead to

a converging solution as the parameter space shape is not conducive to convergence.

The solution, however, is forced to converge by taking the previous photon population

values and moving them slowly towards the photon population calculated by the above

equation, instead of setting them outright equal to the photon population calculated

by the above equation. What happens in practice is that through the course of

97

the numerical iterative process, the electron population inversion temporarily goes

higher than it ever would in the real laser. If the equation above is applied blindly,

the photon density will go negative for high population inversions and unphysical

trends will result. In reality, the population inversion asymptotically approaches a

“pinned” value. To handle this numerically, the above equation is applied and if the

photon population goes negative, the previous photon population was too low, so it

is increased in the next iteration. In real QCL’s, the stimulated emission of photons

depletes the upper level enough that it never goes passed the pinned value.

2.2.m Multi-subband SCEB Algorithm

As described in Section 2.1.l, the electron temperatures must be determined to pro-

vide a more accurate description of the dynamics within a QCL. As described in that

section, these temperatures can deviate significantly from the lattice temperature

of the device and can vary between subbands as well. However, previous publica-

tions expressed difficulties associated with generating a set of electron temperatures

reliably using the self-consistent energy balance approach such as in the paper by

Ikonic.79 Therefore, an original algorithm and an associated convergence condition

was developed to determine the electron temperature for each subband, as described

below.

• Initially, the electron temperatures were all assumed to be equal to the lattice

temperature, TL. All calculations were performed (such as the scattering rates

and population densities) based on this assumption. Then, Eq. (168) was cal-

98

culated for each subband. This resulted in an energy transition rate for each

subband, which was typically non-zero.

• Since the magnitude of the energy transition rate was assumed to be an in-

dication of how far the current subband electron temperature was from the

correct temperature, the value of the energy transition was used to generate a

new guess for the subband temperatures. For the first several iterations of this

algorithm, the new temperatures were generated by multiplying each subband

energy transition rate by a constant factor.

• Once a new set of subband temperatures was generated, these temperatures

were used to recalculate the transition rates and subband population densities.

Then, Eq. (168) was calculated for each subband again, which resulted in a new

set of subband energy transition rates. The same factor used in the previous

iteration of the algorithm was used again to generate a new set of temperatures

from the current values of the energy transition rates.

• After several iterations of this algorithm, the data from the previous iterations

were used to generate new temperature estimates more intelligently. Specifically,

the temperatures and energy transition rates for each subband were used to

create a “slope” rather than an arbitrary constant factor:

∆(dEf/dt)

∆Tf. (205)

This slope was multiplied with the current value of the energy transition rate

99

to determine a new value for each subband temperature via the formula:

(T fe )i = (T fe )i−1 +(dEf/dt)i−1

slope(206)

where (Te)i−1 represents the value of the electron temperature for subband f

from the previous iteration. This process was repeated until the value of the

energy transition rate was essentially zero, as determined by a convergence con-

dition. This is a modification to the secant method which only uses information

from the previous two iterations to determine a slope.

• The following convergence condition, which was also determined from the slope,

was applied to each subband:∣∣∣∣(dEfdt)i

∣∣∣∣ ≤ ∣∣α (T fe )i × slope∣∣ (207)

where i was the iteration number and α was a threshold factor to be chosen

by the user. For temperature convergence to within 0.1%, α was set to 0.001.

Only when this condition was met for all subbands within the same iteration of

the algorithm was it determined that the electron subband temperatures were

found.

• These final subband temperatures (and the associated transition rates and pop-

ulation densities) were then used in the calculation of the output properties of

the QCL (see Section 2.1.m).

This process is represented graphically, by displaying the energy transition rates

for each subband as a function of the electron temperature for each subband. This is

100

-4

-2

0

2

4

6

40 80 120 160 200 240

ener

gytr

ansi

tion

rate

(×10

12

J/s

)

electron temp. (K)

12345

FIG. 2. The energy transition rates for each subband as a function of electron temperaturefor each subband. The five subbands shown are those from the 1.8 THz Kumar structure.All subband temperatures start at the lattice temperature of 150 K (as indicated by thevertical dashed line) and end when the energy transition rate reaches zero (as indicated bythe horizontal dashed line).

shown in Fig. 2, where the subband energy transition rates are those from a study of

a 1.8 THz QCL designed by Kumar.16

2.2.n Average Electron Temperature Implementation

The previous section discussing the multi-subband electron temperature algorithm

deals with the temperatures for each subband individually. This method required

finding these temperatures after several convergence conditions were met simultane-

ously. This condition is simplified if there is not a need for every subband temperature

to be determined. An average electron temperature is determined, which describes

101

the effect of electron heating with a single parameter.

Similar to the case for subband electron temperatures, the scattering times from

Eq. (169) in Section 2.1.l, τif , are functions of ni and the electron temperature. There-

fore, ni and τif must be calculated over a range of average electron temperatures and

then used in Eq. (169) to solve the energy balance equation. However, only one tem-

perature has to reach convergence rather than all subband temperatures. Whichever

temperature solves the energy balance equation, by leading to the equilibrium state

where ∆ = 0, is identified as the average electron temperature Te of the device. This

process is represented graphically, using data from the Sirtori mid-IR QCL,87 in Fig. 3.

Although the average electron temperature approach is simpler, it is used only for

comparison purposes as the individual subband electron temperature approach yields

more physical information.

2.3 Experimental setups

In this section, the experimental methods associated with QCL’s are presented in or-

der to give a complete picture of the collaborative projects between the Submillimeter-

wave Technology Laboratory and the Photonics Center. First, the methods used to

create these devices are described in Section 2.3.a. Then, the characterization tech-

niques are given in Section 2.3.b. Finally, the structures which were made and used

in various applications are described in Section 2.3.c.

102

FIG. 3. Plot of total energy change ∆ as a function of electron temperature for a typicalconfiguration, showing the correct electron temperature as the one at the zero crossing. Inthis case, Te = 83 K. Also shown are the contributions from phonon emission (ph em),phonon absorption (ph abs) and electron-electron scattering (e-e) to the energy balanceequation. These data are calculated for the Sirtori mid-IR structure at a bias voltage of 48kV/cm and at a lattice temperature of 77 K.

2.3.a QCL production methods

As a typical example of how devices are made, run number 238 will be described

from start to finish. This particular wafer was grown using the active region design

of Vitiello.88 More details about this and other designs can be found in Section 2.3.c.

The QCL active regions were grown using molecular beam epitaxy (MBE), a

process by which layers of material are deposited mono-layers at a time until a desired

thickness is reached. To start, a GaAs buffer of 500 nm was grown on top of a Si

substrate, which took approximately 2000 seconds. On top of this, a GaAs bottom

contact layer, which was n+ doped with Si at a concentration of 2×1018 cm−3 was

grown to a thickness of 600 nm. Then a 16 nm thick GaAs layer, n− doped with Si

103

at a concentration of 5×1017 cm−3, was grown. Both of these layers together took

approximately 2400 seconds to grow.

On top of these layers the active region design was grown. This consisted of 90

periods of the design, with each period consisting of 18 layers of varying thickness.

The exact dimension of each layer can be found in Vitiello’s paper.88 The layers

alternated between GaAs and AlGaAs, where the Al alloy concentration was 15%.

Also, two of the GaAs layers were doped using Si to a concentration of 1.6×1016 cm−3.

However, it should be noted that when this laser was completed, it was discovered

that the active region layers were grown 3.5% thicker than the original design.

On top of the active region a GaAs top contact layer, which was n+ doped with

Si at a concentration of 5×1018 cm−3 was grown to a thickness of 80 nm. Finally,

a GaAs cap was grown with n++ doping to a thickness of 3.5 nm. Both of the top

layers together took approximately 320 seconds to grow. The thickness from the top

of the wafer surface to the bottom contact layer was measured to be 11.3 µm.

Fabrication techniques varied slightly between devices and especially between

the two different types of waveguides: semi-insulating surface plasmon (SISP) and

metal-metal (MM). However, all structures discussed in this thesis were made using

SISP waveguides and, as a typical example of this fabrication process, run number

238 will be described.

After cleaning, the first photolithography step began with baking the wafer at

110◦C for five minutes and allowed to cool for five minutes. Photoresist (AZ1512)

was spun at 2000 rpm for 30 seconds and then baked on a hot plate at 95◦C for two

104

minutes and then allowed to cool for five minutes. To form a ridge, the sample was

then developed (AZ327) for 1 minute 55 seconds, and then rinsed in de-ionized (DI)

water and hard baked for five minutes.

Next, the wet etching process consisted of applying H3PO4:H202:H2O (1:2:20) at

170 rpm’s for 26 minutes, which produced an etch depth of 12 µm. For the second

photolithography step, the pre-baking was done as in the first step but the photoresist

(P4620) was spun at 5000 rpm for 30 seconds. After baking at 95◦C for 2 minutes

and cooling for five minutes, the sample was developed for 4 minutes 55 seconds using

AZ400K. The sample was dipped in Chlorobenzene for 15 minutes and blown dry with

N2. It was then developed for 8 minutes using AZ400K, then rinsed in DI and blown

dry with N2 again.

Then, the metal deposition process was begun by dipping into 50% HCl for 15

seconds and then rinsing in DI water and blow drying with N2. E-beam deposition

was then performed with Ni (50 A), Ge (170 A), Au (330 A), Ni (150 A), and then

Au (3000 A). An ultrasonic bath with acetone was then used for lift off for 5 seconds.

After blow drying with N2, the sample was annealed at 425◦C for 150 seconds.

The final step in completing this sample was the packaging process. Thinning

was preformed by first grinding using SiC and then polishing with Alumina. Back

side metal deposition was then done with Ti (150 A) and Au (1000 A). Lastly, the

sample was cleaved to give a ridge length of 3.5 mm using a wafer scriber.

105

2.3.b Device characterization methods

Once the wafer growth and fabrication processes were completed, characterization of

the QCL’s could take place. This was typically done through a series of standard

measurements, (as described in Danylov’s thesis89) such as: the device current versus

voltage (IV ), the output optical power versus current (LI), and the emission spectra.

Lasers were characterized in both pulsed and cw mode and each sample was mounted

on the cold finger of a liquid helium (LHe) dewar. The LHe held the laser temperature

to about 5-20 K when the sample was not driven. The QCL’s were centered on a

1-inch-diameter high-density polyethylene (HDPE) external window of the dewar,

which was approximately 25 mm from the facet of the QCL.

The output power from the devices was measured by a LHe cooled silicon (Si)

bolometer. The windows of the QCL dewar and the bolometer were aligned and posi-

tioned next to each other to minimize any power loss. The bolometer was calibrated

against a Keating photo-acoustic power meter to measure the bolometer responsivity,

which was then used to calculate the output power.

In pulsed mode, lasers were driven with a high power pulse generator (Agilent

8114A) using square wave pulses with a frequency of 30 Hz, and a pulse width of

1 ms. The QCL current and voltage signals were digitized with a data acquisition

board (DAQ) NI 6212. The output voltage of the QCL was recorded by the bolometer

and read by the lock-in amplifier (LIA) SR830. The LIA was triggered with a signal

from the QCL driver. Finally, the bolometer signal was then converted to output

power using the measured bolometer responsivity. The QCL current, voltage, and

106

bolometer signal were recorded and stored on a computer. The IV , LI, and dV/dI

versus I relationships were then determined. Characterization of QCL’s in cw mode

was very similar to that for pulsed mode except for the use of a chopper and the LIA,

which was triggered with pulses from the chopper controller.

Spectra were obtained using a FTIR spectrometer Bruker 66vs with an evacu-

ated spectrometer box. This reduced the effects of atmospheric absorption due to

water vapor. To measure the QCL emission spectra in cw mode, the rapid-scan mod-

ulation regime of the spectrometer was used. In this regime, the translating mirror

was moved continuously over some distance, depending on the resolution, to gener-

ate an interferogram. A fast Fourier transform (FFT) was then performed on the

interferogram to obtain a source spectrum.

To measure the QCL emission spectra in pulsed mode, step-scan modulation of

the spectrometer was used. In this regime, the translating mirror moved in steps,

pausing after each step to record the intensity through a LIA. After a certain number

of steps, the interferogram was collected and then, after performing a FFT, a spectrum

was produced. However, since one high resolution scan took approximately 30 minutes

to complete and there was little gained from recording spectral content in pulsed

mode, spectra were measured mainly in cw mode.

2.3.c Details of grown devices

In the following tables are data which describe some of the growth and fabrication

runs of lasing QCL’s. All of the QCL’s here are of the bound-to-continuum type

107

TABLE 1. Structures based on a 2.9 THz Barbieri QCL design which lased. The predictedand measured frequencies are both shown for comparison.

run no. freq. pred. (THz) freq. meas. (THz) power (mW) scaling %

169 2.89 2.690 · · · 1.9

363 2.65 2.575 1.78 5.5

109 2.49 2.610 · · · 9.0

346 2.40 2.410 0.69 11.3

64 2.38 2.640 · · · 12.0

323 2.38 2.370 0.23 12.0

and were fabricated with SISP waveguides. All were robust designs and two of them

were near transmission windows for atmospheric water vapor. The 2.9 THz structure

was designed by Barbieri90 and devices grown from this design were found to be

robust and emit power in tens of milliwatts (see Table 1). The 1.9 THz structure was

designed by Freeman91 and although its power output was lower, it was near the 2

THz transmission window (see Table 2). Finally, the 2.83 THz structure was designed

by Vitiello and some devices had the benefit of being both high in power (in the tens

of milliwatts) and near the 2.5 THz transmission window (see Table 3).

Each table is organized by the structure on which the design was based. All

output powers shown are maximum cw power. Also, comparisons are made between

measured and predicted quantities, where the predicted frequencies took into account

the measured thicknesses of the devices. Missing values in the tables indicate that

the QCL did not lase in cw, or the measurements were not considered reliable enough

to report. The scaling % refers to a factor which was applied to all layer thicknesses.

For example, for a scale factor of 9%, each layer was 1.09 times as thick as the original

published structure.

108

TABLE 2. Structures based on a 1.9 THz Freeman QCL design which lased. The predictedand measured frequencies are both shown for comparison.

run no. freq. pred. (THz) freq. meas. (THz) power (mW) scaling %

389 2.18 2.235 4.04 -5.2

405 2.16 2.245 · · · -4.6

404 2.14 2.130 · · · -4.4

404.1 2.14 2.130 · · · -4.4

404.2 2.14 2.130 · · · -4.4

388 2.14 2.130 2.29 -3.7

388.1 2.14 2.130 9 -3.7

388.2 2.14 2.130 · · · -3.7

390 2.14 2.170 2.5 -3.7

402 2.03 2.070 · · · -2.1

365 2.10 2.086 0.89 -1.4

367 2.08 2.097 1.5 -1.2

366 2.07 2.080 0.57 -1.0

109

TABLE 3. Structures based on a 2.83 THz Vitiello QCL design which lased. The predictedand measured frequencies are both shown for comparison.

run no. freq. pred. (THz) freq. meas. (THz) power (mW) scaling %

396 2.92 2.700 · · · -3.3

343 2.75 2.700 1.87 2.0

343.136 2.75 2.700 0.83 2.0

343.250 2.75 2.700 1.87 2.0

343.500 2.75 2.700 0.03 2.0

238 2.66 2.630 19.5 3.5

238.100 2.66 2.630 1.45 3.5

238.250 2.66 2.630 0.94 3.5

348 2.66 2.640 5 3.5

403 2.63 2.500 · · · 4.0

380 2.56 2.530 1.05 5.0

382 2.41 2.370 1.5 9.0

387 2.41 2.440 · · · 9.0

383 2.40 2.410 3.35 9.1

385 2.40 2.393 5.5 9.1

400 2.28 2.310 · · · 10.2

400.1 2.28 2.325 · · · 10.2

400.2 2.28 2.310 4.6 10.2

400.3 2.28 2.314 · · · 10.2

400.4 2.28 2.318 · · · 10.2

398 2.22 2.247 · · · 11.2

335 2.77 2.780 · · · · · ·204 · · · 2.790 · · · · · ·207 · · · 2.960 · · · · · ·

110

III. RESULTS

The following sections present the results from various studies which either mod-

ified or implemented the computational model. Only background information and

data are given here, as detailed explanations of the implications of these studies are

explained in the Discussion section.

3.1 Average Electron Temperature

As originally published by Harrison,67 the method of finding an average electron

temperature was based on the use of 15 states in 3/2 periods of the QCL active region.

However, this is an odd choice for multiple reasons. First, since this temperature is

an average, by using 3/2 periods, more weight is given to the states which fall into the

extra 1/2 period when the average is calculated. Second, if more of the active region

is included in the QCL computations it is unclear what fraction of the active region

should be used in average Te calculations. Additionally, if only 3/2 periods are used,

it is unclear which combination of electron states should be used to make up those

3/2 periods. In order to resolve these issues, a study was conducted to observe the

effects of using different combinations of states to calculate the electron temperature.

As explained in the following paragraphs, it was found that these calculations were

sensitive to such combinations.

First, an attempt was made to recreate the theoretical results presented by Har-

rison.67 This work studied a mid-IR QCL structure by Sirtori87 referred to in this

111

FIG. 4. Wavefunctions found in 3 periods of the active region design of the Sirtori mid-IRQCL using an applied bias voltage of 48 kV/cm and at a lattice temperature of 77 K.

thesis as the Sirtori QCL. When the QCL computational model described in the

present thesis was applied to the mid-IR structure, all three periods were used. The

model found 27 states in three periods, as illustrated in Fig. 4. To reconcile the un-

certainty of which electron states from 3/2 periods to use, all possible combinations

of 15 consecutive states were used to calculate Te. The results are shown in Table 4.

A discussion of these data can be found in Section 4.1.

The Te data from the study of the Sirtori mid-IR QCL87 were analyzed further by

looking at their standard deviation. First, additional Te calculations were performed

by including more electron states than just 15. Collections of 16, 17, 18, . . . , 25, 26,

27 states and all of their possible combinations of consecutive states were used to find

112

TABLE 4. Average electron temperatures calculated for the mid-IR Sirtori QCL designusing different sets of electron states. All input parameters into the QCL computationalmodel were the same as those used by a published study of the same structure. Theleft column contains the 15 electron states, ψ’s, while the remaining columns contain thecalculated Te from using the corresponding collection of ψ’s. The lattice temperature TLand calculated Te from the published study are included above each column.

TL = 77 K TL = 200 K TL = 300 K

ψ’s Te = 128 K Te = 244 K Te = 363 K

1-15 79 204 315

2-16 85 211 321

3-17 85 213 326

4-18 82 214 329

5-19 84 206 321

6-20 82 214 335

7-21 81 210 327

8-22 79 208 323

9-23 79 206 318

10-24 84 204 315

11-25 85 211 321

12-26 85 213 326

13-27 89 214 329

Te as was done using just 15 states. The standard deviation from their average was

then determined from each collection of calculated Te’s. The results from this study

are shown in Fig. 5 and examined further in Section 4.1.

Additional structures (referred to as the Barbieri90 and Page mid-IR92 QCL’s)

were examined in a similar way. The standard deviation as a function of the number

of wavefunctions used for the Page structure is shown in Fig. 6 and both structures

are discussed in more detail in Section 4.1.

113

FIG. 5. Plot of standard deviation of the calculated electron temperatures as a functionof the number of electron states included. Results are shown for three lattice temperaturesapplied to the Sirtori mid-IR structure at a bias voltage of 48 kV/cm. The standarddeviation reaches a minimum when 20 out of the 27 total states are included.

FIG. 6. Plot of standard deviation of the calculated electron temperatures as a functionof the number of electron states included. Results are shown for six lattice temperaturesapplied to the Page mid-IR structure at a bias voltage of 53 kV/cm. There is a minimumstandard deviation when 18 out of the 24 total states are included.

114

In order to make comparisons with experimental measurements, the Page QCL,92

was studied in more detail. The lattice and electron temperatures of the Page QCL

have been measured experimentally and reported by Spagnolo80 where measurements

were made over a range of electronic power. However, the voltages used were well

below the alignment voltage. Two different heat sink temperatures (TH), 140 K and

243 K, were also used. Comparisons were made between the Te’s calculated using

the 3 period computational model and the reported measurements. Specifically, the

design was simulated using the highest values of electric power quoted in the paper

at both heat sink temperatures. At TH = 140 K, the highest power used was 7 W.

At this point, the measured TL was 190 K and the measured Te was 295 K. When

simulated with the 3 period QCL computational model, the calculated Te was found

to be 192 K. At TH = 243 K, the highest power used was 3 W. At this point the

measured TL was 265 K and the measured Te was 330 K. When simulated, the Te was

found to be 270 K.

The Page structure was also modeled using bias voltages above threshold. A

range of lattice temperatures were used for each bias voltage and the electron tem-

peratures were calculated under these conditions (see Fig. 7). This study is discussed

in Section 4.1.

3.2 The effect of approximations on e-e scattering rates

Electron-electron scattering is an essential transport mechanism for electrons in a

QCL. Therefore, the electron-electron scattering rates must be calculated between all

115

FIG. 7. Plot of Te vs. TL over a range of bias voltages applied to the Page mid-IR QCL.These voltages are all above threshold.

relevant states in the device structure. However, due to the fact that it is a four-body

interaction, the computational demands of finding these rates is extremely high. In

order to prioritize the most essential aspects of the electron-electron scattering rate

calculations, a study was performed to determine the effect of various approximations

on the resulting rates.

Electron-electron scattering was studied using three different structures. The

first, an infinite quantum well, was included to examine various effects in a simple

system with controlled conditions. In this case, the full machinery of the computa-

tional model was not necessary since only the eigenstates and eigenenergies, which

could be found analytically, were needed to calculate e-e rates. The well was 300

A in width and the two lowest states in the well were used to find scattering rates at

various temperatures and population densities. The effects of these parameters along

116

FIG. 8. Wavefunctions for the bound-to-continuum structure at a bias voltage of 2.246kV/cm and lattice temperature of 10 K. The subbands mentioned in this thesis are labeled.Only the middle period is fully shown in order to reduce confusion.

with approximations are discussed in the next section.

A 2.9 THz bound-to-continuum (BTC) QCL designed by Barbieri90 was mod-

eled using the full self-consistent computational model and the resulting states and

populations (shown in Fig. 8) were used to analyze the effects of e-e scattering rate

approximations. This structure was designed to have a large number of energetically

close states due to its dependence on e-e scattering as the means for depopulation.

This therefore made it a good candidate to study the effects of e-e scattering rate

approximations.

A 3.9 THz resonant-phonon (RP) QCL designed by Kumar27 with exceptional

high-temperature performance was modeled in the same way as the BTC structure

117

FIG. 9. Wavefunctions for the resonant phonon structure at a bias voltage of 13.25 kV/cmand lattice temperature of 10 K. The subbands mentioned in this thesis are labeled. Allthree periods of the active region used in the computation are shown.

and is shown in Fig. 9. Despite the depopulation scheme being based upon a phonon

resonance, the role of electron-electron scattering is still important between some

transitions in the RP active region. Therefore, this structure was also used as a

means to study the effects of approximations on the e-e scattering rate.

In the following sections, each approximation will be described and then its effect

within the infinite quantum well, BTC and RP structures will be presented. First, the

relative and non-relative wavevector definitions of the scattering rate from Eq. (118)

will each be calculated as a function of the number of integration points. Next, the

impact of including state-blocking and screening will be evaluated using several vital

transitions within the structures. Finally, the result of calculating transition rates

118

other than the symmetric type (i, i→ f, f) will be presented.

3.2.a Convergence and integration types

In order to numerically implement the integrals in Eq. (118), the step size for each

integration has to be chosen. The mean scattering rate equations consist of four nested

integrals (not including the sum over states in the denominator and not including the

two integrals in the form factor, which are pre-calculated). Therefore, if the step

size along each integration axis is defined so that there are 100 points, then the total

number of integration points used to find the scattering rate is 1004 = 108. Deciding

whether to use more or less points is a compromise between speed and accuracy and

requires some consideration.

As more points are used, the e-e rate eventually converges to some value. How-

ever, if this convergence can be reached more quickly, then fewer integration points

are necessary to achieve a certain level of accuracy. The behavior of the integrals in

Eq. (118) was studied as a function of the number of integration points to see whether

the non-relative wavevector or the relative wavevector form reached convergence with

fewer points. This was done for the infinite quantum well, the BTC structure and

the RP structure.

The scattering rate as a function of the number of integration points is shown in

Fig. 10 for the infinite quantum well at one particular temperature and population

density. The data are discussed in Section 4.2.

In order to make a general statement about the relative performance of the inte-

119

FIG. 10. Convergence of the scattering rate as more integration points are used for theinfinite quantum well at a temperature of 300 K and a subband population density of1015 (1/m2). The form using the non-relative wavevector definition and the form usingthe relative wavevector definition are compared. All integrals were performed with state-blocking and screening included.

gration types, the percent error was averaged from each scattering rate at a particular

temperature. This error was determined as the deviation from the scattering rate cal-

culated using 2.56× 1010 integration points. The average percent error created from

modeling results for the BTC design is shown in Figs. 11(a) and 11(b) and the results

for the RP design is shown in Figs. 12(a) and 12(b) as a function of the number of

integration points. Both graphs are discussed in Section 4.2.

3.2.b State-blocking and screening

When two electrons scatter with one another, they can only go into final states which

are not already occupied due to Pauli’s exclusion principle. This effect is commonly

referred to as state-blocking and is accounted for in Eq. (118) by the inclusion of the

120

FIG. 11. Convergence of scattering rates within the bound-to-continuum structure as morepoints are used in the integrals at different temperatures. The percent error is averagedover all symmetric transitions at each temperature and all integrals are performed withstate-blocking and screening included.

Fermi-Dirac distribution functions ff (kf ) and fg(kg). However, at low enough carrier

densities, the effect of state-blocking can be neglected. Considering the densities

encountered in a typical QCL (1013 to 1015 m−2), this approximation is questionable.

Additionally, the scattering rate without state-blocking is:

Wi,j→f,g(ki) =m∗e4

64π3~3

∫dkj

∫dθij

∫dθ|Aijfg(q)|2

q2ε(q)2fj(kj) (208)

which is an attractive form since it does not contain kf or kg and therefore does not

require any conversion from relative wavevectors.

Another effect dependent on the electron population density is screening. It is

treated in many texts,59,93–96 but no consensus exists yet on an optimal implementa-

tion in QCL computational model.

121

FIG. 12. Convergence of scattering rates within the resonant phonon structure as morepoints are used in the integrals at different temperatures. The percent error is averagedover all symmetric transitions at each temperature and all integrals are performed withstate-blocking and screening included.

In order to reduce complexity, a single subband model is often used with two

variations. Either the ground state (i.e. the subband with the highest population

density) is used to determine the effect of screening for all subbands, or only the

initial subband in each scattering event is used to calculate this effect as in papers

by Harrison and Smet.53,54 Both of these approaches have been shown to be flawed

by Lee, Bonno and Lu.72,73,86 However, a modified single subband model has been

proposed by Lu73 and has been used in this work to determine the effect of screening

on e-e scattering. This function is summed over all subbands in one period of the

active region along with the intrasubband form factor Aiiii(q) and then replaces the

122

FIG. 13. The W1,1→0,0 scattering rates in the infinite quantum well as a function of tem-perature and population density. The legend indicates which approximations were used ineach plot: state-blocking included (SB on), state-blocking not included (SB off), screeningincluded (SC on) and screening not included (SC off). For all points, 106 integration pointswere used in the relative wavevector form of e-e scattering.

dielectric function ε(q) in Eq. (118) so that:

εsc(q) = 1 +e2

2εq

∑i

Πii(q, T )Aiiii(q). (209)

The effects of both screening and state-blocking were studied using the three

structures mentioned so far. For the infinite quantum well, the lowest two states were

once again used to find the scattering rate W1,1→0,0. As shown in Fig. 13, this rate

was studied as a function of 2D population density Ni as well as temperature T . This

figure is discussed in more detail in Section 4.2.

The 2.9 THz BTC device designed by Barbieri90 shown in Fig. 8, was studied

in a similar manner. Two transitions were singled out to demonstrate the reduction

in scattering rates due to state-blocking and screening. First, the primary transition

123

FIG. 14. The W16,16→15,15 scattering rates in the bound-to-continuum structure as a func-tion of electron temperature and the initial and final 2D subband population densities. Forall points, 106 integration points were used in the relative wavevector form of e-e scattering.

responsible for depopulation, from the lower laser level (subband 16) to the next

lowest state (subband 15), was plotted as a function of the initial and final population

densities (Ni and Nf , respectively) as well as temperature (Fig. 14). The effects of

state-blocking and screening on these transitions are discussed in Section 4.2.

Finally, the 3.9 THz RP structure designed by Kumar27 shown in Fig. 9, was

studied. In this case, since the depopulation scheme is based on a phonon resonance,

there are fewer transitions which have an obvious dependence on the e-e scattering

rate. However, a vital injection transition was modeled (from subband 6 to subband

5) and the results are shown in Fig. 15. The approximation effects are discussed in

Section 4.2.

3.2.c Symmetric and asymmetric transitions

In order to find a transition rate Wij from subband i to subband j all scattering

events that can contribute to that transition must be considered, as mentioned by

124

FIG. 15. The W6,6→5,5 scattering rates in the resonant phonon structure as a function ofelectron temperature and the initial and final 2D subband population densities. For allpoints, 106 integration points were used in the relative wavevector form of e-e scattering.

Kinsler.85 Using a simple two-level system as an example, a 2 → 1 transition rate

is the result of the symmetric scattering event 2, 2 → 1, 1 as well as the asymmetric

scattering events 2, 1 → 1, 1 and 1, 2 → 1, 1. The situation is more complicated in a

QCL structure where many more states exist within the active region.

For the infinite quantum well, asymmetric scattering cannot occur due to the

quantum selection rules for a transition in a symmetric quantum well. However,

due to the asymmetry of the potential structure in any type of QCL, asymmetric

transitions can occur. This opens up a myriad of possible scattering events which

can contribute to a total transition rate Wij. In order to simplify the presentation of

the data somewhat, only scattering events resulting from four adjacent subbands are

considered for each transition in the present study.

As a representative example for the BTC design, the scattering events which

contributed to the upper-state injection transition rate W11,10 are listed in Table 5.

These data are discussed in more detail in Section 4.2. Results obtained for the RP

125

structure are shown in Table 6 and are discussed in more detail in Section 4.2.

TABLE 5. The e-e scattering rates in the bound-to-continuum structure for all possibletransitions which significantly contribute to W11,10 at a temperature of 40 K. The e-e rateswere calculated using the relative wavevector form with 106 integration points and withstate-blocking and screening included.

scattering event (i, j → f, g) rate (×109 1/s)

11, 12→ 10, 12 = 12, 11→ 12, 10 4.898878

11, 9→ 10, 10 = 9, 11→ 10, 10 3.499128

11, 11→ 10, 11 = 11, 11→ 11, 10 3.440731

11, 10→ 10, 10 = 10, 11→ 10, 10 3.361894

11, 11→ 10, 10 2.793062

11, 10→ 10, 9 = 10, 11→ 9, 10 2.264781

11, 9→ 10, 9 = 9, 11→ 9, 10 0.841501

11, 11→ 10, 12 = 11, 11→ 12, 10 0.655019

11, 11→ 10, 9 = 11, 11→ 9, 10 0.608250

11, 12→ 10, 10 = 12, 11→ 10, 10 0.117851

11, 10→ 10, 12 = 10, 11→ 12, 10 0.087562

TABLE 6. The e-e scattering rates in the resonant phonon structure for all possible tran-sitions which significantly contribute to W4,3 at a temperature of 80 K and a bias of 14.2kV/cm. The e-e rates were calculated using the relative wavevector form with 106 integra-tion points and with state-blocking and screening included.

scattering event (i, j → f, g) rate (×109 1/s)

4, 3→ 3, 3 = 3, 4→ 3, 3 8.911816

4, 4→ 3, 4 = 4, 4→ 4, 3 8.367449

4, 2→ 3, 2 = 2, 4→ 2, 3 7.551600

4, 5→ 3, 5 = 5, 4→ 5, 3 6.800549

4, 4→ 3, 3 3.017463

4, 3→ 3, 2 = 3, 4→ 2, 3 0.132317

126

3.3 Scaling Studies

In collaboration with the University of Massachusetts Lowell Photonics Center, a

series of structures was studied and successfully grown. These devices needed to op-

erate at specific frequencies in order to minimize the absorption of terahertz radiation

by atmospheric water vapor. These frequencies were attained by scaling pre-existing

structure designs found in the literature by a certain percentage. Structures were

chosen based on their past growth success at the Photonics Center.

One such device was a 1.9 THz QCL designed by Freeman.91 This device was

modeled over a number of scaling percentages in order to determine the proper scaling

factor to apply to the device design in order for it to emit at 1.97 THz. A smooth curve

was generated from this data by applying a spline fit to the modeling data. It should

be noted that these predictions were obtained without energy balance convergence

or population convergence effects included as they had not yet been implemented

at the time this data was generated. Also of note is that the lasing frequency at a

given scale factor depends on the applied bias. The predicted frequencies reported in

these results are for the bias originally published as the one that gave peak power,

scaled appropriately. The structure was then grown at several of these scaling factors.

The resulting devices and their associated measured scaling factors and emission

frequencies are shown in Fig. 16 along with the modeling data. Each experimental

data point in this figure represents a separately grown device. The associated run

number is labeled alongside the corresponding data point. Further information about

127

1.9

1.95

2

2.05

2.1

2.15

2.2

0 0.2 0.4 0.6 0.8 1

freq

.(T

Hz)

transmission

-3 -2 -1 0 1 2 3 4

scale %

1.97 THz

G404

G390

G388

G367G365

G366

G402

FIG. 16. On the left is a section of the transmittance spectrum for atmospheric water vaporat 5% relative humidity obtained in-house using a Bruker FTIR spectrometer. On the rightis the output frequency of a 1.9 THz QCL as a function of a layer width scaling factorapplied to the original device design. Both graphs together indicate the desired emissionfrequency for this device, 1.97 THz, since that frequency is in a region of high transmittance.The blue dots indicate the scaling factor and emission frequencies of devices which weregrown by the UML Photonics Center and the red line indicates the model-predicted trends.

each growth run can be found in the Section 2.3.c. This structure’s scaling factors

were measured experimentally using X-ray diffraction techniques.

Another device which was studied was a 2.83 THz QCL designed by Vitiello.88

This structure was modeled in order to find the appropriate scaling factor to apply

to the device design in order for it to emit at 2.53 THz. The associated data, as

shown in Fig. 17, was acquired in the same way as the 1.9 THz laser described above.

128

2

2.2

2.4

2.6

2.8

3

0 0.2 0.4 0.6 0.8 1

freq

.(T

Hz)

transmission

-2 0 2 4 6 8 10 12

scale %

2.53 THz

G343

G348G238

G403G380

G387

G382

G383G385

G400

G398

FIG. 17. On the left is a section of the transmittance spectrum for atmospheric water vaporat 5% relative humidity obtained in-house using a Bruker FTIR spectrometer. On the rightis the output frequency of a 2.83 THz QCL as a function of a layer width scaling factorapplied to the original device design. Both graphs together indicate the desired emissionfrequency for this device, 2.53 THz, since that frequency is in a region of high transmittance.The blue dots indicate the scaling factor and emission frequencies of devices which weregrown by the UML Photonics Center and the red line indicates the model-predicted trends.

Further discussion of these studies can be found in Section 4.3.

3.4 Screening Studies

In order to quantify the effect of screening on the output properties of the compu-

tational model, an additional study was conducted on the Vitiello 2.83 THz QCL

design.88 This study employed the electron-electron screening model of Eq. (149)

129

described in Section 2.1.j. In order to quantify the effect, the output power as a func-

tion of bias and device scaling (see Section 3.3) was modeled both with and without

screening. The results are shown in Fig. 18. In order to reduce the time necessary to

acquire data, this study did not employ the full SCEB model. Specifically, the energy

balance condition was excluded and no anti-symmetric transitions were included in

finding the electron-electron scattering rates.

3.5 Multi-subband electron temperature model

The electron subband temperatures were determined for two resonant phonon struc-

tures using the self-consistent energy balance (SCEB) model as described in Sec-

tion 2.1.l. Both structures exhibited a considerable variation among subband tem-

peratures in either Monte Carlo (MC) simulation or experimental measurement. The

dependence of the electron temperatures on either lattice temperature or bias voltage

is shown as well as additional quantities such as population densities and transition

rates. These quantities were chosen over other parameters (such as current density

and power) since they would provide the most direct comparison to other models as

well as minimize confounding effects.

3.5.a 3.4 THz, three-well design

The first structure studied, shown in Fig. 19, was the 3.4 THz, three-well, diagonal

design of Kumar27 which has been shown to lase up to a lattice temperature of 186 K.

Due to its exceptional temperature performance, this structure was the subject of an

130

-5-2.50

2.55

7.510

12.5

12

3

0

5

10

15

output power (mW)

scale %

bias (×105 V/m)

-5-2.50

2.55

7.510

12.5

12

3

0

5

10

15

output power (mW)

scale %

bias (×105 V/m)

FIG. 18. The model-predicted output power as a function of bias voltage of a 2.83 THz QCLfor several scaling factors. On the top graph, the power was calculated without screeningincluded in the e-e scattering rates. The bottom graphs shows the calculated power whenscreening is included in the computational model.

exhaustive MC analysis by Han,75 which allowed for a direct comparison of modeling

results from the SCEB model.

131

FIG. 19. Calculated potential profile and squared wavefunctions of the 3.4 THz structureat a bias of 64 mV/module for three periods of the active region. The middle period statesare labeled such that state 4 and 3 are the lasing states, while 2 and 1 are responsible fordepopulation and injection.

The SCEB model predicted a slightly higher optimum voltage, at 70 mV/module,

than that predicted by the Han MC study.75 At this bias, the electron temperature of

the upper lasing state as a function of lattice temperature was found and is shown in

Fig. 20(a). Relevant transition rates for the upper lasing state are shown in Fig. 20(b).

The population ratio between the two lasing states is shown in Fig. 20(c). Further

discussion of these graphs can be found in Section 4.5.

3.5.b 2.8 THz, four-well design

Experimental measurements of subband electron temperatures done by Vitiello have

provided further indications that there is significant temperature variation between

subbands in resonant phonon QCL designs.77,78 One of these measured structures was

132

FIG. 20. Model-predicted device characteristics as a function of lattice temperature at abias of 70 mV/module for the 3.4 THz structure. (a) Electron temperature of the upperlasing state 4. (b) Electron-electron scattering rates of the injection (1′ → 4) transition aswell as the lifetime of the upper lasing state. (c) Population inversion between lasing levels4 and 3.

a four well, 2.8 THz design designed by Vitiello.77 This structure was also simulated

using the SCEB model for comparison to experimental results and is shown in Fig. 21.

The electron temperatures of the upper laser level (state 5) and the two injection

states (1 and 2) are shown in Fig. 22(a). The population inversion ratios determined

from the SCEB model are shown in Fig. 22(b). These graphs are discussed in more

detail in Section 4.5.

133

FIG. 21. Calculated potential profile and squared wavefunctions of the 2.8 THz structureat a bias of 65 mV/module for three periods of the active region. The middle period statesare labeled such that state 5 and 4 are the lasing states, while 3, 2 and 1 are responsiblefor depopulation and injection.

3.6 Temperature optimization

Having the ability to determine the temperature of each subband individually opens

up the opportunity to use this additional information as a means to optimize the

performance of a device. For example, the electron temperature of the upper lasing

state of a structure could be minimized as a means to improve the performance of

the laser as a whole.

This concept was explored further through modifications to a previously pub-

lished 1.8 THz QCL designed by Kumar.16 In order to change the electron temper-

ature of the upper laser state of this structure, the barrier of the well preceding the

center-of-mass location for that state was modified. It was believed that by increasing

134

FIG. 22. Model-predicted device characteristics as a function of bias at a lattice temper-ature of 100 K for the 2.8 THz structure. (a) Electron temperatures of the upper lasingstate (5) and the two injector states (1 and 2). (b) Population inversion between the lasinglevels 5 and 4.

the barrier thickness, this would result in a decrease in the magnitude of the scatter-

ing form factors between the injection state and the upper lasing state. Therefore,

with a reduced scattering rate, there would be less scattering and less energy entering

the upper lasing state. This could potentially reduce the electron temperature of the

upper lasing state. Similarly, if the barrier thickness is decreased, then the electron

temperature could increase.

Applying this concept to the structure, the upper lasing state electron temper-

ature and the optical power of the lasing transition were recorded as a function of

the barrier thickness. It should be noted that the well thickness preceding the barrier

was also modified in order to maintain a constant thickness for a QCL period. The

135

+ −

u

l

180

190

200

210

-3 -2 -1 0 1 2 3

elec

tron

tem

p.

(K)

0.1

0.2

0.3

-3 -2 -1 0 1 2 3

pow

er(W

)

barrier shift (A)

T ue

FIG. 23. On the left is a portion of the 1.8 THz Kumar QCL structure. The upper lasingstate is indicated as blue (u) and the lower state is indicated as red (l). The barrier that wasmodified lies between the arrows indicating the shift directions. The figure on the upperright shows the upper lasing state electron temperature as a function of the barrier edgeshift and the lower right figure shows the optical power of the lasing transition. The powerat a barrier shift of -2 A was found to be 0.48 W but was omitted from the graph as it wasan outlier.

resulting model predictions are shown in Fig. 23 and discussed further in Section 4.6.

136

IV. DISCUSSIONS

4.1 Average Electron Temperature

A study of the average electron temperature as a function of the electron states was

described in Section 3.1. The results of that study are shown in Table 4, which shows

a considerable amount of variation among the calculated electron temperatures. This

was expected since, as mentioned in Section 3.1, the average electron temperature

would be weighted differently depending on which states comprise the additional 1/2

period. It should also be noted that all of the Te’s are lower than those calculated

by Harrison.67 This is most likely due to the fact that only symmetric electron-

electron scattering rates were used in this study. However, from these results, it

was concluded that any arbitrary choice of states in the computation will not lead

to the same temperature. A further study of this issue was performed in order to

determine if there is a set of states which produced a minimum deviation among

electron temperature calculations (see Section 3.1).

This optimal choice of states was found by studying the deviation among com-

binations of electron states of various sizes (i.e. consecutive groupings of 15, 16, 17,

. . . electron states). It was anticipated that the deviation would consistently decrease

as more states were included. Surprisingly, there is a clear minimum in standard

deviation when 20 states are used to calculate Te as seen in Fig. 5. This equates

to approximately 3/4 of the total number of states in three periods of the Sirtori

137

structure.87

Other QCL structures were modeled in order to examine the effect of different

electron state combinations on the calculated Te and the amount of deviation ob-

served. Structures designed by Barbieri90 and Page92 were used. Above threshold,

the Page structure exhibited very similar behavior to that of Sirtori’s structure. A

minimum in standard deviation was observed when 18 of the 24 electron states in

three periods were used to calculate Te (see Fig. 6). This corresponds to exactly 3/4

of the total number of states.

The Barbieri structure exhibited similar behavior with regards to which fraction

of electron states provides the lowest amount of deviation among the possible combi-

nations of consecutive states. However, in some cases the deviation was lowest at a

fraction higher than 3/4 of the total number of electron states. It should also be noted

that the mean Te’s that were calculated for the Barbieri structure were lower than

the TL used in the computational model, and so may have had some inaccuracies.

Comparisons between modeling results of the Page structure92 and experimen-

tal measurements by Spagnolo80 of the same device were made. First, devices were

modeled using below-threshold voltages, since these voltages were used in the ex-

perimental measurements. The calculated electron temperatures were smaller than

then the measured values. However, it should be noted that in Spagnolo’s paper,80

an offset value was applied to the measured Te’s to take into account heating by a

probe laser. Taking this offset value into account might have improved the agreement

between the measured Te’s and those simulated using the computational model. Also

138

since the lasers were operating below threshold, the steady-state assumptions of the

SCEB model may not have been applicable.

Finally, the same structure was modeled over a range of bias voltages, some

of which were at or above the threshold condition. These can be found in Fig. 7,

which shows the electron temperature results over a range of bias voltages and lattice

temperatures. In this figure, Te increases linearly with increasing TL. Also, the

calculated Te’s reached higher values compared to the Te’s calculated using below-

threshold conditions.

4.2 The effect of approximations on e-e scattering rates

Three structures (an infinite quantum well, the 2.9 THz Barbieri QCL,90 and the 3.9

THz Kumar QCL27) were studied to determine the effect of various e-e scattering rate

approximations on the calculated rates. First, the relative and non-relative wavevec-

tor definitions of the scattering rate in Eq. (118) were each calculated as a function

of the number of integration points. Then, the impact of including state-blocking

and screening was evaluated using several vital transitions within the structures. Fi-

nally, the importance of calculating transition rates other than the symmetric type

(i, i→ f, f) was examined.

4.2.a Convergence and integration types

The behavior of the integrals in Eq. (118) was studied as a function of the num-

ber of integration points to see whether the non-relative wavevector or the relative

139

wavevector form reached convergence with fewer points.

The scattering rate as a function of the number of integration points is shown in

Fig. 10 for the infinite quantum well at one particular temperature and population

density. The non-relative form seems to have converged before the relative form;

however, this was not always the case at different temperatures and populations.

The situation was equally complex in the modeled QCL structures making it difficult

to reach a conclusion as to the superiority of one form over another in terms of

convergence speed.

In order to make a general statement about the relative performance of the inte-

gration types, the percent error was averaged from each scattering rate at a particular

temperature. The average percent error created from computational modeling of the

Barbieri design is shown in Figs. 11(a) and 11(b) as a function of the number of

integration points. Despite the averaging, neither form consistently converged faster

than the other. This is also the case in the Kumar design as shown in Figs. 12(a) and

12(b). However, both Figs. 11 and 12 have lower percent error at higher numbers of

integration points and indicate that in order to guarantee a percent error of less than

1% at all temperatures, 108 points (or 100 points per integration axis) must be used.

4.2.b State-blocking and screening

The effects of both screening and state-blocking were studied using the three struc-

tures mentioned so far. For the infinite quantum well, the lowest two states were

once again used to find the scattering rate W1,1→0,0. As shown in Fig. 13, this rate

140

was studied as a function of 2D population density Ni as well as temperature T .

As more carriers were present in the well, a reduction of the scattering rates due to

screening and state-blocking became more noticeable, where screening had the most

pronounced impact with an increase in Ni.

The 2.9 THz Barbieri QCL90 was studied in a similar manner. Two transitions

were singled out to demonstrate the reduction in scattering rates due to these effects.

First, the primary transition responsible for depopulation was plotted as a function

of the initial and final population densities (Ni and Nf , respectively) as well as tem-

perature (Fig. 14). Due to the large number of states within each period of the active

region, screening had a noticeable impact and reduced the rates by 10 to 15 percent.

State-blocking reduced the rates by only 1 to 2 percent.

The 3.9 THz Kumar QCL27 was also studied. A vital injection transition was

examined in Fig. 15. Screening was once again significant, reducing the rates by 5 to

10 percent. State-blocking was only noticeable at higher carrier densities and only

reduced the rates by 1 to 2 percent.

4.2.c Symmetric and asymmetric transitions

Results were obtained to study the relative magnitude of symmetric and asymmetric

scattering rates. The scattering events which contributed to the upper-state injection

transition rate W11,10 in the Barbieri structure are listed in Table 5. The symmet-

ric scattering event 11, 11 → 10, 10 is seen to have only the fifth highest rate of

the scattering events included. Scattering events of the type 11,m → 10,m and

141

m, 11→ m, 10, where m is one of the four adjacent subbands, are seen to contribute

significantly to the transition rate W11,10. When all possible scattering events among

the four adjacent subbands are summed, a total transition rate of 4.51× 1010 (1/s) is

found. This is eight times greater than the transition rate found from just symmetric

scattering events. Therefore, it is essential that scattering events of this type along

with the symmetric one be considered when determining transition rates which will

eventually be used in the rate equations.

Similar results were observed with the RP structure as shown in Table 6, where

the symmetric transition rate (4, 4→ 3, 3) is only the fifth highest rate among those

listed. This transition is also responsible for injection into the upper lasing state

(subband 3). When all possible scattering events are summed, a total transition rate

of 6.97×1010 (1/s) is found. This is eleven times greater than the symmetric transition

rate and is verified by comparison to Han’s Monte Carlo simulations.75 Once again,

the influence of these additional scattering events must be considered to improve the

accuracy of transition rates.

4.3 Scaling Studies

The scaling studies shown in Figs. 16 and 17 from Section 3.3 indicate the ability of

the QCL computational model to be used as an accurate predictive tool. Not only

was the computational model able to predict a lasing frequency over a broad range of

scaling factors, but the agreement was very good between the predicted and measured

frequencies as well. The RMS value between the modeled and measured data was

142

0.0229 for the Freeman 1.9 THz QCL and 0.0858 for the Vitiello 2.83 THz QCL. This

indicates the potential use of the model to optimize device performance based on other

parameter studies. These results also indicate that, in general, uniformly increasing

the width of a QCL structure a small fraction will lower its lasing frequency without

necessarily destroying the lasing action. Note that the relationship between lasing

frequency and scale factor is non-linear, justifying the complex modeling approach

used in this work.

4.4 Screening Studies

During the course of the device scaling study explained in Section 3.3, the effect of

screening was also analyzed. In Fig. 18 from Section 3.4, the output power as a

function of both scaling factor and bias voltage is shown. The top portion of Fig. 18

shows the power with screening included in the e-e scattering rate calculations and

the bottom portion shows the power when screening is not included.

As discussed in Section 4.2, screening tends to reduce the magnitude of the

electron-electron scattering rates. This appears to lead to a reduction in output

power as seen by the difference in maximum voltages in both figures. Interestingly,

screening does not seem to significantly change the overall power vs. frequency vs. bias

behavior of the QCL beyond uniformly lowering the power.

Additionally, the effect of scaling on the relationship between bias voltage and

power can also be seen. With both screening on and off, a scaling factor which reduces

the size of the device leads to the need for a higher bias voltage to reach optimum

143

power output.

4.5 Multi-subband electron temperature model

The results of applying the multi-subband SCEB model to two QCL structures were

presented in Section 3.5 to illustrate the success of the model in matching other

studies of the same structures.

4.5.a 3.4 THz, three-well design

For this structure, the SCEB model predicted a slightly higher optimum voltage, at 70

mV/module, than that predicted by the MC study of Han.75 At this bias, the electron

temperature of the upper lasing state as a function of lattice temperature was found

and is shown in Fig. 20(a). At lower lattice temperatures, the electron temperature of

the upper lasing state (T 4e ) was significantly higher than TL. But as TL was increased

to 150 K and above, T 4e and TL were roughly equal. This behavior was also seen in the

MC study of this structure by Han,75 but T 4e was several degrees higher than in the

present study. This reduction in temperature was most likely due to a smaller overall

injection rate into the upper lasing state via electron-electron scattering as seen in

Fig. 20(b). Additionally, the predicted lifetime of the upper laser state was much lower

at higher TL due to increased electron-phonon emission. These discrepancies between

the SCEB and MC calculations indicated that the observed differences in electron

temperatures for this structure were most likely due to differences in the calculation

of scattering rates and screening. The reduced population inversion between the two

144

lasing states predicted by the SCEB model and shown in Fig. 20(c) can be explained

using the same arguments.

4.5.b 2.8 THz, four-well design

This structure was also modeled using the SCEB for comparison to experimental

results and is shown in Fig.21. The electron temperatures of the upper laser level

(state 5) and the two injection states (1 and 2) are shown in Fig. 22(a). At 60

mV/module, the SCEB model predicted a temperature difference between the upper

laser state and the two injector states of 25-38%. This is not as large as the almost

100% difference seen experimentally; however, MC simulations have never predicted

temperature variations of that magnitude in resonant phonon QCL designs. The

SCEB temperature variations are very close to that of MC simulations performed by

Callebaut of a similar device,39 which predicted subband temperature variations of

up to 38%. The subband temperatures reported in the MC study were 96, 111, and

102 K for the upper lasing, upper injector, and lower injector state, respectively. The

subband temperatures from the SCEB model were 99, 118, and 99 K for the same

states, which is in good agreement.

The population inversion ratios determined from the SCEB model are shown in

Fig. 22(b) and are nearly identical to the values measured by Vitiello.77 However, the

MC study of Callebaut of the similar structure39 yielded a population inversion ratio of

8 at the alignment voltage, which is considerably higher than that from measurement

or the present SCEB modeling. This higher population density of the upper laser

145

state is most likely due to the higher lifetime of 3.5 ps reported in that study as

compared to the 1.2 ps predicted by the SCEB model. However, the lifetime of the

lower lasing states predicted by the MC study and the present one were identical.

4.6 Temperature optimization

As described in Section 3.6, the barrier thickness of a THz QCL design was modified in

order to observe its effect on the upper lasing state electron temperature and optical

power. As shown in Fig. 23, as the barrier thickness was increased the electron

temperature decreased and the optical power increased. Conversely, a decrease in

barrier thickness resulted in an increase in electron temperature and a decrease in

power.

The observed relationship between optical power and upper state electron tem-

perature could be explained through two effects. First, if the temperature of the upper

lasing state is kept low, that should inhibit the parasitic transition where the upper

lasing state emits a phonon rather than a photon. Additionally, a lower temperature

would prevent electrons in the upper state from escaping quantum confinement and

entering the energy continuum above the conduction band minimum. If both of these

effects are reduced, then the population inversion should increase, which improves

the optical power for the lasing transition.

146

V. CONCLUSIONS

5.1 Average Electron Temperature Calculations

When more than 3/2 periods of a QCL active region are used in the computational

model, different combinations of 15 electron states produced different calculated val-

ues for the average electron temperature (Te). For the case of resonant phonon, mid-IR

designs in the above-threshold condition, the number of electron states which pro-

duced the least amount of variation among calculations of the electron temperature

was found to be 3/4 of the total number of states in a three-period model.

5.2 Electron-electron scattering rate approximations

The convergence of both the non-relative and relative wavevector forms of the scat-

tering rates from Eq. (118) were shown to improve with an increase in the number

of integration points. From this, it was concluded that if an error of less than 1

percent is needed, then 108 integration points (or 100 points per integration axis) are

necessary to calculate a scattering rate. Both forms were shown to lead to near iden-

tical results, so that the choice of which form to use is irrelevant. Within the QCL

structures, the scattering rate was shown to be significantly affected by screening,

with scattering rates reduced by up thirty percent in the Barbieri structure and ten

percent in the Kumar structure. In both devices, state-blocking was shown to only

reduce scattering rates by one to two percent and can therefore typically be omitted

147

from calculations. Finally, the importance of including both symmetric and asymmet-

ric scattering events when finding a transition rate was demonstrated. Asymmetric

scattering events were shown to increase the transition rate by a factor of eight in

the 11, 11→ 10, 10 transition of the Barbieri structure and by factor of eleven in the

4, 4→ 3, 3 transition of the Kumar structure.

5.3 Device scaling

The success of the QCL computational model in predicting the lasing frequency of

devices was discussed in Section 4.3. There was excellent agreement between predicted

and measured frequencies over a range of device scaling factors and frequencies. This

indicates the high accuracy of the model when determining QCL frequencies.

5.4 Electron-electron screening

As discussed in Section 4.4, a screening model was successfully implemented to im-

prove the accuracy of electron-electron scattering rate calculations. This implemen-

tation also affected the predicted power output of modeled devices. Results from

studies of the effects of screening were shown as both a function of bias voltage and

and scaling factor. These indicated that screening tended to decrease the predicted

output power.

148

5.5 Multi-subband energy balance model

A self-consistent model containing a multi-subband energy balance condition was

presented. This model was used to generate subband electron temperatures as well

other output parameters for two resonant phonon QCL designs. Detailed compar-

isons between MC simulations, experimental measurements and the SCEB model

were performed. The SCEB model predicted similar electron temperatures to those

of MC studies. The discrepancies observed in population densities were most likely

due to differences in scattering rate calculations. When applied to a structure mea-

sured experimentally, the SCEB model predicted the upper lasing state to have the

highest subband temperature within the device, which was verified experimentally.

Although the degree of subband temperature variation was not as high as that seen

in experiment, the population inversion did match. These results demonstrate that

the multi-subband energy balance model is an accurate and powerful tool for un-

derstanding electron subband dynamics, as well as for numerically optimizing QCL

designs.

5.6 Temperature optimization

A preliminary study was discussed in Section 4.6 to determine if the electron temper-

ature of the upper lasing state could be controlled through the manipulation of other

device parameters. In this study, the well thickness of a device design was modified

to both increase and decrease the upper lasing state temperature. Additionally, this

149

modification also affected the predicted output power of the device. It was observed

that as the upper lasing state temperature decreased, the output power increased.

150

VI. RECOMMENDATIONS

Future versions of the QCL computational model will include the effects of

electron-impurity and interface roughness scattering, which have been shown to be

significant in some THz QCL studies.43–47 Also, additional approximations that may

contribute to electron-electron scattering rate calculations should be examined fur-

ther, such as the impact of the exchange interaction.55–57

With regards to the energy balance equations, some improvements have been

suggested to the author which may have an impact on the the calculated electron

temperatures. Specifically, there are multiple ways in which the energy transition

rate can be determined, each of which may influence temperature calculations. The

effects of these various forms on the electron temperatures as well as the predicted

QCL output properties should be studied further.

Once the energy balance equations have been fully integrated into the QCL com-

putational code, the electron temperatures could be used as an optimization tool, as

briefly discussed in Section 4.6. As far as the author is aware, this optimization

parameter has never been pursued in the literature, and may provide insight into

increasing high temperature performance of THz QCL’s. Additionally, electron tem-

perature could be used as a figure of merit in an automated optimization process.

For example, a genetic algorithm could be applied to a QCL design using the upper

lasing state temperature as part of a fitness function. Such a procedure could yield

new and practical designs.

151

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VIII. BIOGRAPHY

Philip Slingerland graduated from Ithaca College in May 2004 with a Bachelor of

Arts degree in physics and a minor in mathematics. He then went to work for a civil

engineering consulting and testing company in Acton, Massachusetts for two years.

He then pursued his graduate education in physics at the University of Massachusetts

Lowell. His research focused on the optimization of terahertz quantum cascade lasers.

After acquiring his PhD, he will work as an analyst for Metron Scientific Solutions in

Reston, Virginia.