steady-state and transient electron transport...

J Mater Sci: Mater Electron (2006) 17: 87–126

DOI 10.1007/s10854-006-5624-2

R E V I E W

Steady-State and Transient Electron Transport Within the III–VNitride Semiconductors, GaN, AlN, and InN: A ReviewStephen K. O’Leary∗ · Brian E. Foutz† ·Michael S. Shur · Lester F. Eastman

Received: 15 July 2005 / Accepted: 26 July 2005C© Springer Science + Business Media, Inc. 2006

Abstract The III–V nitride semiconductors, gallium nitride,

aluminum nitride, and indium nitride, have, for some time

now, been recognized as promising materials for novel elec-

tronic and optoelectronic device applications. As informed

device design requires a firm grasp of the material properties

of the underlying electronic materials, the electron transport

that occurs within these III–V nitride semiconductors has

been the focus of considerable study over the years. In an

effort to provide some perspective on this rapidly evolving

field, in this paper we review analyses of the electron trans-

port within the III–V nitride semiconductors, gallium nitride,

aluminum nitride, and indium nitride. In particular, we dis-

cuss the evolution of the field, compare and contrast results

determined by different researchers, and survey the current

literature. In order to narrow the scope of this review, we will

primarily focus on the electron transport within bulk wurtzite

gallium nitride, aluminum nitride, and indium nitride, for this

analysis. Most of our discussion will focus on results ob-

tained from our ensemble semi-classical three-valley Monte

∗Author to whom correspondence should be addressed.

†Present address: Cadence Design Systems, 6210 Old Dobbin Lane,Columbia, Maryland 21045, USA.

S. K. O’LearyFaculty of Engineering, University of Regina, Regina,Saskatchewan, Canada, S4S [email protected]

B. E. Foutz · Lester F. EastmanSchool of Electrical Engineering, Cornell University, Ithaca, NewYork 14853, USA

M. S. ShurDepartment of Electrical, Computer, and Systems Engineering,Rensselaer Polytechnic Institute, Troy, New York 12180-3590,USA

Carlo simulations of the electron transport within these ma-

terials, our results conforming with state-of-the-art III–V ni-

tride semiconductor orthodoxy. A brief tutorial on the Monte

Carlo approach will also be featured. Steady-state and tran-

sient electron transport results are presented. We conclude

our discussion by presenting some recent developments on

the electron transport within these materials.

1. Introduction

The III–V nitride semiconductors, gallium nitride (GaN),

aluminum nitride (AlN), and indium nitride (InN), have,

for some time now, been recognized as promising materi-

als for novel electronic and optoelectronic device applica-

tions [1–9]. In terms of electronics, their wide energy gaps,

large breakdown fields, high thermal conductivities, and fa-

vorable electron transport characteristics, make GaN, AlN,

and InN, and alloys of these materials, ideally suited for novel

high-power and high-frequency electron device applications.

On the optoelectronics front, the direct nature of the energy

gaps associated with GaN, AlN, and InN, make this family

of materials, and its alloys, well suited for novel optoelec-

tronic device applications in the visible and ultraviolet fre-

quency range. While initial efforts to study these materials

were hindered by growth difficulties, recent improvements

in the material quality have made possible the realization of

a number of III–V nitride semiconductor based electronic

[10–16] and optoelectronic [17–25] devices. These develop-

ments have fueled considerable interest in the III–V nitride

semiconductors, GaN, AlN, and InN.

In order to analyze and improve the design of III–V ni-

tride semiconductor based devices, an understanding of the

electron transport which occurs within these materials is

necessary. Electron transport within bulk GaN, AlN, and

InN has been extensively examined over the years [26–45].

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88 J Mater Sci: Mater Electron (2006) 17: 87–126

Unfortunately, uncertainty in the material parameters

associated with GaN, AlN, and InN remains a key source

of ambiguity in the analysis of the electron transport within

these materials [45]. In addition, some recent experimen-

tal [46] and theoretical [47] developments have cast doubt

upon the validity of widely accepted notions upon which our

understanding of the electron transport mechanisms within

the III–V nitride semiconductors, GaN, AlN, and InN, has

evolved. Further confounding matters is the sheer volume of

research activity being performed on the electron transport

within these materials, this presenting the researcher with a

dizzying array of seemingly disparate approaches and results.

Clearly, at this critical juncture at least, our understanding of

the electron transport within the III–V nitride semiconduc-

tors, GaN, AlN, and InN, remains in a state of flux.

In order to provide some perspective on this rapidly evolv-

ing field, we aim to review analyses of the electron transport

within the III–V nitride semiconductors, GaN, AlN, and

InN, within this paper. We start with a brief tutorial on the

electron transport mechanisms within semiconductors, and

on how the Monte Carlo approach may be used in order

to probe such mechanisms. Then, focusing on the III–V

nitride semiconductors under investigation in this analysis,

i.e., GaN, AlN, and InN, we present results obtained

from ensemble semiclassical three-valley steady-state and

transient Monte Carlo simulations of the electron transport

within these materials, these results conforming with

state-of-the-art III–V nitride semiconductor orthodoxy. We

conclude this review with a discussion on the evolution of the

field and a survey of the current literature. In order to narrow

the scope of this review, we will primarily focus on the

electron transport within bulk wurtzite GaN, AlN, and InN

for the purposes of this analysis. We hope that researchers

in the field will find this review useful and informative.

For our brief tutorial on the electron transport mechanisms

within semiconductors, we begin with an introduction to the

Boltzmann transport equation, this equation underlying most

analyses of the electron transport within semiconductors.

Then, the general principles underlying the ensemble semi-

classical three-valley Monte Carlo simulation approach, that

we employ in order to solve the Boltzmann transport equa-

tion, are presented. We conclude the tutorial by presenting

the material parameters corresponding to bulk wurtzite GaN,

AlN, and InN. We then use these material parameter selec-

tions and our ensemble semi-classical three-valley Monte

Carlo simulation approach to determine the nature of the

steady-state and transient electron transport within the III–V

nitride semiconductors. Finally, we present some recent de-

velopments on the electron transport within these materials.

This paper is organized in the following manner. In

Section 2, we present our tutorial on the electron transport

mechanisms within semiconductors. In particular, the Boltz-

mann transport equation and our ensemble semi-classical

three-valley Monte Carlo simulation approach, that we

employ in order to solve this equation for the III–V nitride

semiconductors, GaN, AlN, and InN, are presented. The

material parameters, corresponding to bulk wurtzite GaN,

AlN, and InN, are also presented in the tutorial featured

in Section 2. Then, in Section 3, using results obtained

from our ensemble semi-classical three-valley Monte Carlo

simulations of the electron transport within the III–V nitride

semiconductors, we study the nature of the steady-state elec-

tron transport that occurs within these materials. Transient

electron transport within the III–V nitride semiconductors

is also discussed in Section 3. A review of the III–V nitride

semiconductor electron transport literature, in which the

evolution of the field is discussed and a survey of the current

literature is presented, is then featured in Section 4. Finally,

conclusions are provided in Section 5.

2. Electron Transport Within Semiconductors andThe Monte Carlo Simulation Approach: A Tutorial

2.1. Introduction

The electrons within a semiconductor are in a perpetual state

of motion. In the absence of an applied electric field, this mo-

tion arises as a result of the thermal energy which is present,

and is referred to as thermal motion. From the perspective

of an individual electron, thermal motion may be viewed

as a series of trajectories interrupted by a series of random

scattering events. Scattering may arise as a result of interac-

tions with the lattice atoms, impurities, other electrons, and

defects. As these interactions lead to electron trajectories in

all possible directions, i.e., there is no preferred direction,

while individual electrons will move from one location to

another, taken as an ensemble, assuming that the electrons

are in thermal equilibrium, the overall electron distribution

will remain static. Accordingly, no net current flow occurs.

With the application of an applied electric field, �E , each

electron in the ensemble will experience a force, −q �E , qdenoting the electron charge. While this force may have a

negligible impact upon the motion of any given individual

electron, taken as an ensemble, the application of such a force

will lead to a net aggregate motion of the electron distribution.

Accordingly, a net current flow will occur, and the overall

electron ensemble will no longer be in thermal equilibrium.

This movement of the electron ensemble in response to an

applied electric field, in essence, represents the fundamental

issue at stake when we study the electron transport within a

semiconductor.

In this chapter, we provide a brief tutorial on the is-

sues at stake in our analysis of the electron transport within

the III–V nitride semiconductors, GaN, AlN, and InN. We

begin our analysis with an introduction to the Boltzmann

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J Mater Sci: Mater Electron (2006) 17: 87–126 89

transport equation, this equation describing how the electron

distribution function evolves under the action of an applied

electric field, this equation underlying the electron transport

within bulk semiconductors. We then introduce the Monte

Carlo simulation approach to solving the Boltzmann trans-

port equation, focusing on the ensemble semi-classical three-

valley Monte Carlo simulation approach used in our own

simulations of the electron transport within the III–V nitride

semiconductors. Finally, we present the material parameters

corresponding to bulk wurtzite GaN, AlN, and InN.

This chapter is organized in the following manner. In

Section 2.2, the Boltzmann transport equation is introduced.

Then, in Section 2.3, a brief discussion on the ensemble

semiclassical three-valley Monte Carlo simulation approach

to solving this Boltzmann transport equation is presented. Fi-

nally, in Section 2.4, our material parameter selections, corre-

sponding to bulk wurtzite GaN, AlN, and InN, are presented.

2.2. The Boltzmann transport equation

An electron ensemble may be characterized by its distribution

function, f (�r , �p, t), where �r denotes the position, �p repre-

sents the momentum, and t indicates time. The response of

this distribution function to an applied electric field, �E , is

the issue at stake when one investigates the electron trans-

port within a semiconductor. When the dimensions of the

semiconductor are large, and quantum effects are negligible,

the ensemble of electrons may be treated as a continuum,

i.e., the corpuscular nature of the individual electrons within

the ensemble, and the attendant complications which arise,

may be neglected. In such a circumstance, the evolution of

the distribution function, f (�r , �p, t), may be determined us-

ing the Boltzmann transport equation. In contrast, when the

dimensions of the semiconductor are small, and quantum ef-

fects are significant, then the Boltzmann transport equation,

and its continuum description of the electron ensemble, is no

longer valid. In such a case, it is necessary to adopt quan-

tum transport methods in order to study the electron transport

within the semiconductor [48].

For the purposes of this analysis, we will focus on the elec-

tron transport within bulk semiconductors, i.e., semiconduc-

tors of sufficient dimensions so that the Boltzmann transport

equation is valid. Ashcroft and Mermin [49] demonstrated

that this equation may be expressed as

∂ f

∂t= − �p · ∇p f − �r · ∇r f + ∂ f

∂t

∣∣∣∣scat

. (1)

The first term on the right-hand side of Equation (1) repre-

sents the change in the distribution function due to external

forces applied on the system. The second term on the right-

hand side of Equation (1) accounts for the electron diffu-

sion which occurs. The final term on the right-hand side of

Equation (1) describes the effects of scattering.

Owing to its fundamental importance in the analysis of

the electron transport within semiconductors, a number of

techniques have been developed over the years in order to

solve the Boltzmann transport equation. Approximate solu-

tions to the Boltzmann transport equation, such as the dis-

placed Maxwellian distribution function approach of Ferry

[27] and Das and Ferry [28] and the non-stationary charge

transport analysis of Sandborn et al. [50], have proven useful.

Low-field approximate solutions have also proven elemen-

tary and insightful [30, 33, 51]. A number of these techniques

have been applied to the analysis of the electron transport

within the III–V nitride semiconductors, GaN, AlN, and InN

[27, 28, 30, 33, 51, 52]. Alternatively, more sophisticated

techniques have been developed, these solving the Boltz-

mann transport equation directly. These techniques, while

allowing for a rigorous solution of the Boltzmann transport

equation, are rather involved, and require intense numerical

analysis. They are further discussed by Nag [53].

For studies of the electron transport within the III–V

nitride semiconductors, GaN, AlN, and InN, the most

common approach to solving the Boltzmann transport

equation, by far, has been the ensemble semi-classical

Monte Carlo simulation approach. In terms of the III–V

nitride semiconductors, using the Monte Carlo simulation

approach, the electron transport within GaN has been studied

the most extensively [26, 29, 31, 32, 34, 35, 40, 42, 45],

AlN [37, 38, 42] and InN [36, 41, 42, 44] less so. The Monte

Carlo simulation approach has also been used to study the

electron transport within the two-dimensional electron gas

of the AlGaN/GaN interface which occurs in high electron

mobility AlGaN/GaN field-effect transistors [54, 55].

At this point, it should be noted that the complete so-

lution of the Boltzmann transport equation requires a reso-

lution of both steady-state and transient responses. Steady-

state electron transport refers to the electron transport that

occurs long after the application of an applied electric field,

i.e., once the electron ensemble has settled to a new equi-

librium state [56]. As the distribution function is difficult to

quantitatively visualize, in the analysis of steady-state elec-

tron transport, researchers typically study the dependence of

the electron drift velocity [57] on the applied electric field

strength, i.e., they determine the velocity-field characteristic.

Transient electron transport, by way of contrast, refers to the

transport that occurs while the electron ensemble is evolving

into its new equilibrium state. Typically, it is characterized

by studying the dependence of the electron drift velocity on

the time elapsed, or the distance displaced, since the electric

field was initially applied. Both steady-state and transient

electron transport within the III–V nitride semiconductors,

GaN, AlN, and InN, are reviewed within this paper.

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2.3. The ensemble semi-classical Monte Carlo

simulation approach

In the study of the electron transport within a semiconduc-

tor, the Monte Carlo approach is often used in order to solve

the Boltzmann transport equation. In this approach, the mo-

tion of electrons within a semiconductor, under the action

of an applied electric field, is simulated. The acceleration

of each electron in the applied electric field, and the pres-

ence of scattering, are both taken into account in these sim-

ulations. The scattering events that an individual electron

experiences are selected randomly, the probability of each

such event being selected in proportion to the scattering rate

corresponding to that particular event. Through such an anal-

ysis, one hopes to be able to estimate the resultant distribution

function, f (�r , �p, t).In simulating the electron transport within a semicon-

ductor, there are a variety of different Monte Carlo ap-

proaches that researchers have adopted over the years. Most

of these approaches may be classified as being either single-

particle Monte Carlo simulation approaches or ensemble

Monte Carlo simulation approaches. In a single-particle

Monte Carlo approach, one simulates the motion of a single

electron, tracking its wave-vector for a sufficiently long time

so that, in steady-state conditions, this wave-vector sweeps

through all of phase space, the amount of time spent in any

particular place in phase space being a proportionate predic-

tor for the distribution function there. Ergodicity is implicitly

assumed, i.e., it is assumed that time-averages are equal to

ensemble-averages [58].

In an ensemble Monte Carlo simulation of the electron

transport within a semiconductor, the motion of a large num-

ber of electrons, under the action of an applied electric field,

is studied. The evolution over time of this distribution of elec-

trons is interpreted as being indicative of the corresponding

distribution function, the density of electrons at any point

in phase space being a proportionate predictor for the distri-

bution function there. Assuming that there are enough elec-

trons used in the simulation, the law of large numbers dic-

tates that the results will indeed correspond to those deter-

mined through an exact evaluation of the distribution func-

tion, f (�r , �p, t). This approach allows for the ready analysis

of both steady-state and transient electron transport. We have

adopted an ensemble Monte Carlo simulation approach for

the purposes of our analysis of the electron transport within

the III–V nitride semiconductors, GaN, AlN, and InN.

Before describing the algorithm used for our Monte Carlo

simulations, we first provide a brief overview of key model-

ing considerations. In particular, we present the three-valley

model that is used to represent the conduction band electron

band structure. Then, we discuss our semi-classical descrip-

tion for the motion of the electrons within this electron band

structure. The interactions of the electrons with the semicon-

ductor lattice, through the various scattering mechanisms, are

then elaborated upon. Finally, after the basic physics of the

electron transport within the III–V nitride semiconductors

has been introduced, a flow chart, describing the mechanics

of our own particular Monte Carlo simulation approach, is

presented.

2.3.1. The three-valley electron band structure model

We restrict our attention to the analysis of the electron trans-

port within the conduction band. In the absence of an applied

electric field, electrons tend to occupy the lowest energy lev-

els of the conduction band. When an electric field is applied,

the average electron energy increases. Typically, however,

only the lowest parts of the conduction band contain a sig-

nificant fraction of the electron population. This allows for

a considerable simplification in the analysis to be made. In-

stead of including the entire electron band structure for the

conduction band, only the lowest valleys need be represented.

The Monte Carlo simulation approach, used for our simula-

tions of the electron transport within the III–V nitride semi-

conductors, GaN, AlN, and InN, uses a three-valley model

for the conduction band electron band structure, representing

the three lowest energy minima of the conduction band.

Within the framework of this three-valley model, the non-

parabolicity of each valley is treated through the application

of the Kane model, the energy band corresponding to each

valley being assumed to be spherical and of the form

�2k2

2m∗ = E(1 + αE), (2)

where �k denotes the magnitude of the crystal momentum, Erepresents the electron energy, E = 0 corresponding to the

band minimum, m∗ is the effective mass of the electrons in

the valley, and the non-parabolicity coefficient, α, is given by

α = 1

Eg

(1 − m∗

me

)2

, (3)

where me and Eg denote the free electron mass and the energy

gap, respectively [59]. A schematic illustration of the three-

valley model representing the conduction band electron band

structure associated with bulk wurtzite GaN, used for the

purposes of our ensemble semi-classical three-valley Monte

Carlo simulations of the electron transport within this mate-

rial, is depicted in Fig. 1. Values for the valley parameters

corresponding to bulk wurtzite GaN, AlN, and InN, used for

the purposes of our simulations, are tabulated in Section 2.4.

2.3.2. The semi-classical motion of particles

Electrons in a periodic potential possess wave-functions that

can be distributed over volumes which are substantially larger

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m*=0.2 me

α=0.189 eV–1

Valley 1

m*=me

α=0 eV –1

Valley 3

m*=me

α=0 eV –1

Valley 2

2.1 eV 1.9 eV

GaN

Fig. 1 The three-valley model used to represent the conduction bandelectron band structure associated with bulk wurtzite GaN for our MonteCarlo simulations of the electron transport within this material. Thevalley parameters, corresponding to bulk wurtzite GaN, AlN, and InN,are tabulated in Section 2.4

than the single unit cell. The electron is thus capable of inter-

acting with many different components of the crystal simul-

taneously. It can interact with different phonons and different

crystal impurities all at once. This picture, however, is too

complex to handle directly, and several approximations are

usually made in order to render the analysis tractable.

One approximation that is commonly made is that the

electrons behave as if they were point particles, whose mo-

tion, in response to an applied electric field, is well behaved

and deterministic. The velocity of each electron may thus be

expressed as

vg = 1

�∇kε(�k), (4)

where ε(�k) denotes the electron band structure, i.e., the en-

ergy of the electron as a function of the electron wave-vector,�k [60]. In addition, the rate-of-change of the electron’s wave-

vector with time is proportional to the force that the electron

experiences from the applied electric field, �E , i.e.,

�d�kdt

= −q �E, (5)

where q denotes the electron charge.

Equations (4) and (5) collectively determine the motion of

an electron, assuming that the periodic potential associated

with the underlying crystal is static. In reality, the thermal

motion of the lattice, imperfections, and interactions with

the other electrons in the ensemble, result in the electron de-

viating from the path literally prescribed by Equations (4)

and (5). Although an individual electron’s interaction with

the lattice is very complex, the description is simplified con-

siderably through the use of the quantum mechanical notion

of “scattering events.” During a scattering event, the elec-

tron’s wave-function abruptly changes. Quantum mechanics

determines the probability of each type of scattering event,

and dictates how to probabilistically determine the change in

the wave-vector after each such event. With this information,

the behavior of an ensemble of electrons may be simulated,

this behavior being expected to closely approximate the elec-

tron transport within a real semiconductor. The probability

of scattering is introduced into the Monte Carlo simulation

approach through a determination of the scattering rates cor-

responding to the different scattering processes.

2.3.3. Scattering processes

The scattering rate corresponding to a particular interaction

refers to the expected number of scattering events of that

particular interaction taking place per unit time. Quantum

mechanics determines the scattering rates for the different

processes based on the physics of the interaction. In general,

scattering processes within semiconductors can be classi-

fied into three basic types; (1) phonon scattering, (2) car-

rier scattering, and (3) defect scattering [53]. For the III–V

nitride semiconductors, GaN, AlN, and InN, phonon scat-

tering is the most important scattering mechanism, and it is

featured prominently in our simulations of the electron trans-

port within these materials. Carrier scattering, or in our case,

electron-electron scattering, has also been taken into account

in our simulations. It should be noted, however, that as this

scattering mechanism leads to very little change in the results

with a substantial increase in the running time, in an effort to

determine our results as expeditiously as possible, electron-

electron scattering was not included in our simulations. The

final category of scattering mechanism, defect scattering,

refers to the scattering of electrons due to the imperfections

within the crystal. Throughout this work, it is assumed that

donor impurities are the only defects present. These defects,

when ionized, scatter electrons through their positive charge.

This mechanism is an important factor in determining the


and the effect of the doping concentration on the electron

transport within these materials is considered in our analysis.

Owing to their importance in determining the nature of the

electron transport within the III–V nitride semiconductors, it

is instructive to discuss the different types of phonon scatter-

ing mechanisms. Phonons naturally divide themselves into

two distinctive types, optical phonons and acoustic phonons.

Optical phonons are the phonons which cause the atoms of

the unit cell to vibrate in opposite directions. For acoustic

phonons, however, the atoms vibrate together, but the wave-

length of the vibration occurs over many unit cells. Typically,

the energy of the optical phonons is greater than that of the

acoustic phonons. For each type of phonon, two types of in-

teraction occur with the electrons. First, the deformations in

the lattice, which arise from the interaction of the lattice with

the phonons, changes the energy levels of the electrons, caus-

ing transitions to occur. This type of interaction is referred

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to as non-polar optical phonon scattering for the case of op-

tical phonons and acoustic deformation potential scattering

for the case of acoustic phonons.

In polar semiconductors, such as the III–V nitride semi-

conductors, the deformations which arise also induce local-

ized electric fields. These electric fields also interact with

the electrons, causing them to scatter. For the case of optical

phonons, the interaction of the electrons with these localized

electric fields is referred to as polar optical phonon scattering.

For acoustic phonons, however, this mechanism is referred

to as piezoelectric scattering. Owing to the extremely polar

nature of the nitride bonds within the III–V nitride semicon-

ductors, GaN, AlN, and InN, it turns out that polar optical

phonon scattering is very important for these materials. It

will be shown that this mechanism alone determines many

of the key properties of the electron transport within the III–V

nitride semiconductors.

When the energy of an electron within a valley increases

beyond the energy minima of the other valleys, it is also

possible for the electrons to scatter from one valley to an-

other. This type of scattering is referred to as inter-valley

scattering. It is an important scattering mechanism for the

III–V compound semiconductors in general, and for the III–

V nitride semiconductors, GaN, AlN, and InN, in particular.

Inter-valley scattering is believed to be responsible for the

negative differential mobility observed in the velocity-field

characteristics associated with these materials

A derivation of all of these scattering rates, as a function

of the semiconductor parameters, can be found in the lit-

erature; see, for example, [53, 61, 62]. A formalism, which

closely matches the form used in our ensemble semi-classical

three-valley Monte Carlo simulations of electron transport,

is found in Nag [53]. Many of the scattering rates that are

employed for the purposes of our Monte Carlo simulations

of the electron transport within the III–V nitride semicon-

ductors, GaN, AlN, and InN, are also explicitly tabulated in

Appendix 22 of Shur [63].

2.3.4. Our Monte Carlo simulation approach

For the purposes of our analysis of the electron transport

within the III–V nitride semiconductors, GaN, AlN, and

InN, we employ ensemble semi-classical three-valley Monte

Carlo simulations. The scattering mechanisms considered are

(1) ionized impurity, (2) polar optical phonon, (3) piezo-

electric, and (4) acoustic deformation potential. Intervalley

scattering is also considered. We assume that all donors are

ionized and that the free electron concentration is equal to the

dopant concentration. For our steady-state electron transport

simulations, the motion of three thousand electrons is exam-

ined, while for our transient electron transport simulations,

the motion of ten thousand electrons is considered. For our

simulations, the crystal temperature is set to 300 K and the

Fig. 2 The scattering rates for the lowest (�) valley as a function ofthe wave-vector for bulk wurtzite GaN. The scattering mechanisms are:(1) ionized impurity, (2) polar optical phonon emission, (3) inter-valley(1 → 3) emission, (4) inter-valley (1 → 2) emission, (5) acoustic de-formation potential, (6) piezoelectric, (7) polar optical phonon absorp-tion, (8) inter-valley (1 → 3) absorption, and (9) inter-valley (1 → 2)absorption. The most important scattering mechanisms are shown inFig. 2(a), Fig. 2(b) depicting the other scattering mechanisms

doping concentration is set to 1017 cm−3 for all cases, un-

less otherwise specified. Electron degeneracy effects are ac-

counted for by means of the rejection technique of Lugli and

Ferry [64]. Electron screening is also accounted for follow-

ing the Brooks-Herring method [65]. Further details of our

approach are discussed in the literature [29, 34–37, 42, 45].

Figs. 2 through 4 plot the scattering rates corresponding to

the various scattering mechanisms as a function of the elec-

tron wave-vector, �k, for the III–V nitride semiconductors con-

sidered in this analysis, i.e., GaN, AlN, and InN. These are the

rates corresponding to the lowest energy valley in the conduc-

tion band, i.e., the � valley for the III–V nitride semiconduc-

tors under investigation in this review. The upper valleys have

similar scattering rates. Each of the scattering mechanisms

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included in our simulations of the electron transport within

the III–V nitride semiconductors is described, in detail, by

Nag [53]. For the ionized impurity, polar optical phonon,

and piezoelectric scattering mechanisms, screening effects

are taken into account. These screening effects tend to lower

the scattering rate when the electron concentration is high.

2.3.5. The Monte Carlo algorithm

Now that the fundamentals of electron transport within a

semiconductor have been presented, a brief description of

our ensemble semi-classical three-valley Monte Carlo al-

gorithm will be provided. This description will be qualita-

tive in nature. Further quantitative details are presented in

Appendix A.

For the purposes of our analysis, we employ an en-

semble Monte Carlo approach. This approach simulates

the transport of many electrons simultaneously. Often, in

such an approach, the scattering rates are calculated once

at the beginning of the program and remain fixed. How-

ever, more sophisticated techniques have been developed

which depend upon the properties of the current electron

distribution. These scattering rate formulas can be imple-

mented using a self-consistent ensemble technique. This

technique recalculates the scattering rate table at regular

intervals throughout the simulation as the electron distri-

bution evolves. This self-consistent ensemble Monte Carlo

technique is the method employed for the purposes of our

analysis.

The essence of our Monte Carlo simulation algorithm,

used to simulate the electron transport within the III–V ni-

tride semiconductors, GaN, AlN, and InN, is as depicted

in the flow chart shown in Fig. 5. During the initialization

phase of our simulations, the initial scattering rate tables are

computed. The initial electron distribution is set by assign-

ing each electron a distinct wave-vector. The distribution of

wave-vectors is chosen using Fermi-Dirac occupation statis-

tics. As was mentioned earlier, the motion of three thousand

electrons is studied for each steady-state electron transport

simulation, while the motion of ten thousand electrons is

considered for each transient electron transport simulation,

these selections allowing us to achieve sufficient statistics.

Next, the main body of the algorithm begins. In this phase,

each electron moves through a series of time-steps, each time-

step being of duration �t . This is accomplished by mov-

ing the electron through a free-flight. During this free-flight,

the electron experiences no scattering events, and its motion

through the conduction band is determined semi-classically,

i.e., as suggested by Equations (4) and (5). The time for each

free-flight must be chosen carefully, and depends critically

on the scattering rate at the beginning of the electron’s flight,

as well as the scattering rate throughout its free-flight. Since

the scattering rate changes over the flight, the selection of

Fig. 3 The scattering rates for the lowest (�) valley as a function ofthe wave-vector for bulk wurtzite AlN. The scattering mechanisms are:(1) ionized impurity, (2) polar optical phonon emission, (3) inter-valley(1 → 2) emission, (4) inter-valley (1 → 3) emission, (5) acoustic de-formation potential, (6) piezoelectric, (7) polar optical phonon absorp-tion, (8) inter-valley (1 → 2) absorption, and (9) inter-valley (1 → 3)absorption. The most important scattering mechanisms are shown inFig. 3(a), Fig. 3(b) depicting the other scattering mechanisms. Notethat piezoelectric scattering is more pronounced in bulk wurtzite AlNthan in either bulk wurtzite GaN (see Fig. 2) or bulk wurtzite InN (seeFig. 4)

the free-flight time is complex. Methods used for generating

the free-flight time have been extensively studied, and the

algorithm employed for our simulations is further detailed

in Appendix A. At the end of each free-flight, the electron

experiences a scattering event. The scattering event is chosen

randomly, in proportion to the scattering rate for each mech-

anism. Finally, a new wave-vector for the electron is chosen,

based on conservation of momentum and conservation of en-

ergy considerations, as well as the angular distribution func-

tion corresponding to that particular scattering mechanism.

After the electron has moved, a new free-flight time is chosen

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Fig. 4 The scattering rates for the lowest (�) valley as a func-tion of the wave-vector for bulk wurtzite InN. The scattering mech-anisms are: (1) ionized impurity, (2) polar optical phonon emis-sion, (3) inter-valley (1 → 2) emission, (4) inter-valley (1 → 3) emis-sion, (5) acoustic deformation potential, (6) polar optical phonon ab-sorption, (7) piezoelectric, (8) inter-valley (1 → 2) absorption, and(9) inter-valley (1 → 3) absorption. The most important scatteringmechanisms are shown in Fig. 4(a), Fig. 4(b) depicting the other scat-tering mechanisms

and the process repeats itself until that electron reaches the

end of the current time-step.

After all of the electrons have been moved through the

time-step, macroscopic quantities are extracted from the

resultant electron distribution. The relevant macroscopic

quantities include the electron drift velocity, the average elec-

tron energy, and the number of electrons in each valley. The

entire process repeats itself, time-step after time-step, until

the end of the simulation is reached. When statistics are to be

calculated as a function of the applied electric field strength,

the applied electric field strength is also periodically updated

throughout the simulation; it should be noted, however, that

steady-state equilibrium must be achieved before the next

update to the applied electric field strength occurs. At the

Fig. 5 A flowchart corresponding to our Monte Carlo algorithm. Amore detailed flowchart is shown in Appendix A

end of the simulation, the accumulated statistics are sent to a

file for the purposes of archiving, processing, and subsequent

retrieval.

2.4. Parameter selections for bulk wurtzite GaN, AlN,

and InN

The material parameter selections, used for our simulations

of the electron transport within the III–V nitride semicon-

ductors, GaN, AlN, and InN, are tabulated in Table 1 [3,

30, 33, 37, 66–78]. Most of these parameters are from Chin

et al. [30], although we did select some values from other

references [3, 33, 37, 42, 68, 69, 73–75]; these parameter se-

lections are the same as those employed by Foutz et al. [42].

While the band structures corresponding to bulk wurtzite

GaN, AlN, and InN, are still not agreed upon, for the pur-

poses of this analysis, the band structures of Lambrecht and

Segall [79] are adopted. For the case of bulk wurtzite GaN, the

analysis of Lambrecht and Segall [79] suggests that the low-

est point in the conduction band is located at the center of the

Brillouin zone, at the � point, the first upper conduction band

valley minimum also occurring at the � point, 1.9 eV above

the lowest point in the conduction band, the second upper

conduction band valley minima occurring along the symme-

try lines between the L and M points, 2.1 eV above the lowest

point in the conduction band; see Table 2. For the case of bulk

wurtzite AlN, the analysis of Lambrecht and Segall [79] sug-

gests that the lowest point in the conduction band is located

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Table 1 The material parameterselections corresponding to bulkwurtzite GaN, AlN, and InN.Most of these parameterselections are from Chin et al.[30]; the source of the otherparameter selections isexplicitly indicated in the table.This selection of parameters isthe same as that employed byFoutz et al. [42]

Parameter GaN AlN InN

Mass density (g/cm3) 6.15 3.23 6.81

Longitudinal sound velocity (cm/s) [66] 6.56 × 105 9.06 × 105 6.24 × 105

Transverse sound velocity (cm/s) [66] 2.68 × 105 3.70 × 105 2.55 × 105

Acoustic deformation potential (eV) 8.3 9.5 7.1

Static dielectric constant 8.9 [33] 8.5 15.3

High-frequency dielectric constant. 5.35 [33] 4.77 8.4

Effective mass (�1 valley) [67] 0.20 me [33] 0.48 me 0.11 me [3, 68, 69]

Piezoelectric constant, e14 (C/cm2) [70, 71, 72] 3.75 × 10−5 9.2 × 10−5 [37] 3.75 × 10−5

Direct energy gap (eV) 3.39 [73] 6.2 [74] 1.89 [75]

Optical phonon energy (meV) 91.2 99.2 89.0

Intervalley deformation potentials (eV/cm) [76] 109 109 109

Intervalley phonon energies (meV) [77] 91.2 99.2 89.0

Table 2 The valley parameterselections corresponding to bulkwurtzite GaN, AlN, and InN.These parameter selections arefrom the band structuralcalculations of Lambrecht andSegall [79]. This selection ofparameters is the same as thatemployed by Foutz et al. [42]

Valley number 1 2 3

GaN Valley location �1 �2 L-M

Valley degeneracy 1 1 6

Effective mass 0.2 me me me

Intervalley energy separation (eV) - 1.9 2.1

Energy gap (eV) 3.39 5.29 5.49

Non-parabolicity (eV−1) 0.189 0.0 0.0

AlN Valley location �1 L-M K




Energy gap (eV) 6.2 6.9 7.2


InN Valley location �1 A �2




Energy gap (eV) 1.89 4.09 4.49


at the center of the Brillouin zone, at the � point, the first

upper conduction band valley minima occurring along the

symmetry lines between the L and M points, 0.7 eV above

the lowest point in the conduction band, the second upper

conduction band valley minima occurring at the K points,

1 eV above the lowest point in the conduction band; see

Table 2. For the case of bulk wurtzite InN, the analysis of

Lambrecht and Segall [79] suggests that the lowest point in

the conduction band is located at the center of the Brillouin

zone, at the � point, the first upper conduction band valley

minimum occurring at the A point, 2.2 eV above the lowest

point in the conduction band, the second upper conduction

band valley minimum occurring at the � point, 2.6 eV above

the lowest point in the conduction band; see Table 2. We

ascribe an effective mass equal to the free electron mass,

me, to all of the upper conduction band valleys. Thus, from

Equation (3), it follows that the non-parabolicity coefficient,

α, corresponding to each upper conduction band valley is

zero, i.e., the upper conduction band valleys are completely

parabolic. For our simulations of the electron transport within

gallium arsenide (GaAs), the material parameters employed

are from Littlejohn et al. [78] and Blakemore [80].

It should be noted that the energy gap associated with InN

has been the subject of some controversy since 2002. The

pioneering experimental results of Tansley and Foley [75],

reported in 1986, suggested that InN has an energy gap of

1.89 eV. This value, or values similar to it [81], have been

used extensively in Monte Carlo simulations of the electron

transport within this material since that time [36, 41, 42,

44]; typically, the influence of the energy gap on the electron

transport occurs through its impact on the non-parabolicity

coefficient, α, i.e., through Equation (3), and on the effective

mass associated with the lowest energy valley, m∗; see, for

example, Fig. 1 of Chin et al. [30]. In 2002, Davydov et al.[82], Wu et al. [83], and Matsuoka et al. [84] presented ex-

perimental evidence which instead suggests a considerably

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smaller energy gap for InN, around 0.7 eV. Most recently,

within the context of an experimental study on the electron

transport within InN, Tsen et al. [85] suggested an energy

gap of 0.75 eV and an effective mass of 0.045 me for this

material. As this new value for the energy gap associated

with InN is still under active investigation [86], for the pur-

poses of our present Monte Carlo simulations of the elec-

tron transport within this material, we adopt the traditional

Tansley and Foley [75] energy gap value. The sensitivity of

the velocity-field characteristic associated with bulk wurtzite

GaN to variations in the non-parabolicity coefficient, α, and

the effective mass associated with the lowest energy valley,

m∗, will be explored, in detail, in Section 3, InN being ex-

pected to exhibit a similar behavior.

The band structure associated with bulk wurtzite GaN has

also been the focus of some controversy. In particular, Brazel

et al. [87] employed ballistic electron emission microscopy

measurements in order to demonstrate that the first upper con-

duction band valley occurs only 340 meV above the lowest

point in the conduction band. This contrasts rather dramat-

ically with more traditional results, such as the calculation

of Lambrecht and Segall [79], which instead suggest that

the first upper conduction band valley minimum within this

material occurs about 2 eV above the lowest point in the

conduction band. Clearly, this will have a significant impact

upon the results. While the results of Brazel et al. [87] were

reported in 1997, most bulk wurtzite GaN electron trans-

port simulations have adopted the more traditional interval-

ley energy separation of about 2 eV. Accordingly, we have

adopted the more traditional intervalley energy separation

for the purposes of our present analysis. The sensitivity of


GaN to variations in the intervalley energy separation will be

explored, in detail, in Section 3.

3. Steady-State and Transient Electron TransportWithin Bulk Wurtzite GaN, AlN, and InN

3.1. Introduction

The current interest in the III–V nitride semiconductors,

GaN, AlN, and InN, is primarily being fueled by the tremen-

dous potential of these materials for novel electronic and op-

toelectronic device applications. With the recognition that in-

formed electronic and optoelectronic device design requires

a firm understanding of the nature of the electron transport

within these materials, electron transport within the III–V

nitride semiconductors has been the focus of intensive inves-

tigation over the years. The literature abounds with studies

on the steady-state and transient electron transport within

these materials [26–47, 51, 52, 54, 55]. As a result of this

intense flurry of research activity, novel III–V nitride semi-

conductor based devices are starting to be deployed in com-

mercial products today. Future developments in the III–V

nitride semiconductor field will undoubtably require an even

deeper understanding of the electron transport mechanisms

within these materials.

In the previous section, we presented details of our semi-

classical three-valley Monte Carlo simulation approach, that

we employ for the analysis of the electron transport within

the III–V nitride semiconductors, GaN, AlN, and InN. In this

section, a collection of steady-state and transient electron

transport results, obtained from these Monte Carlo simula-

tions, is presented. Initially, an overview of our steady-state

electron transport results, corresponding to the three III–V

nitride semiconductors under consideration in this analysis,

i.e., GaN, AlN, and InN, will be provided, and a comparison

with the more conventional III–V compound semiconductor,

GaAs, will be presented. A comparison between the temper-

ature dependence of the velocity-field characteristics asso-

ciated with GaN and GaAs will then be presented, and our

Monte Carlo results will be used in order to account for the

differences in behavior. A similar analysis will be presented

for the doping dependence. Next, detailed simulation results,

in which the sensitivity of the velocity-field characteristics

associated with AlN and InN to variations in the crystal tem-

perature and the doping concentration is explored, will be

presented. The sensitivity of the velocity-field characteris-

tic associated with bulk wurtzite GaN to variations in the

band structure will then be examined, this analysis providing

us with some insight into the range of outcomes expected

for these materials, this being a useful exercise, particularly

for those III–V nitride semiconductors which have, as yet,

unresolved band structures, i.e., GaN and InN. Finally, the

transient electron transport which occurs within the III–V ni-

tride semiconductors under investigation in this analysis, i.e.,

GaN, AlN, and InN, is determined and compared with that

corresponding to GaAs. Our Monte Carlo results conform

with state-of-the-art III–V nitride semiconductor orthodoxy,

although there have been some recent developments which

have led to mild corrections to these results. These will be

discussed in Section 4.

This section is organized in the following manner. In

Sections 3.2, 3.3, and 3.4, the velocity-field characteristics

associated with GaN, AlN, and InN are presented and an-

alyzed. For the purposes of comparison, in Section 3.5, an

analogous analysis is performed for the case of GaAs, the

velocity-field characteristics associated with the III–V ni-

tride semiconductors under consideration in this analysis,

i.e., GaN, AlN, and InN, being compared and contrasted with

that corresponding to GaAs in Section 3.6. The sensitivity of

the velocity-field characteristic associated with GaN to varia-

tions in the crystal temperature will then be examined in Sec-

tion 3.7, and a comparison with that corresponding to GaAs

presented. In Section 3.8, the sensitivity of the velocity-field

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Fig. 6 The velocity-field characteristic associated with bulk wurtziteGaN. Like many other compound semiconductors, the electron driftvelocity reaches a peak, and at higher applied electric field strengths itdecreases until it saturates

characteristic associated with GaN to variations in the dop-

ing concentration level will be explored, and a comparison

with that corresponding to GaAs presented. The sensitivity

of the velocity-field characteristics associated with AlN and

InN to variations in the crystal temperature and the doping

concentration will then be examined in Sections 3.9 and 3.10,

respectively. The sensitivity of the velocity-field character-

istic associated with bulk wurtzite GaN to variations in the

band structure will then be examined in Section 3.11. Our

transient electron transport results are then presented in Sec-

tion 3.12. Finally, the conclusions of this electron transport

analysis are summarized in Section 3.13.

3.2. Steady-state electron transport within bulk wurtzite

GaN

Our examination of results begins with bulk wurtzite GaN,

the most commonly studied III–V nitride semiconductor. The

velocity-field characteristic associated with this material is

depicted in Fig. 6. This result was obtained through a steady-

state Monte Carlo simulation of the electron transport within

this material for the GaN parameter selections specified in

Tables 1 and 2, the crystal temperature being set to 300 K

and the doping concentration being set to 1017 cm−3. We

note that initially the electron drift velocity monotonically

increases with the applied electric field strength, reaching a

maximum of about 2.9 × 107 cm/s when the applied electric

field strength is around 140 kV/cm. For applied electric fields

strengths in excess of 140 kV/cm, the electron drift veloc-

ity decreases in response to further increases in the applied

electric field strength, i.e., a region of negative differential

mobility is observed, the electron drift velocity eventually

Fig. 7 The average electron energy as a function of the applied electricfield strength for bulk wurtzite GaN. Initially, the average electron en-ergy remains low, only slightly higher than the thermal energy, 3

2kbT ,

where kb denotes Boltzmann’s constant. At 100 kV/cm, however, theaverage electron energy increases dramatically. This increase is dueto the fact that the polar optical phonon scattering mechanism can nolonger absorb all of the energy gained from the applied electric field.The energy minima corresponding to the upper valleys are depictedwith the dashed lines

saturating at about 1.4 × 107 cm/s for sufficiently high ap-

plied electric field strengths. By examining further the results

of our Monte Carlo simulation, an understanding of this re-

sult becomes clear.

First, we consider the results at low applied electric

field strengths, i.e., applied electric field strengths less than

30 kV/cm. This is referred to as the linear regime of elec-

tron transport, as in this regime, the electron drift velocity

is well characterized by the low-field electron drift mobil-

ity, μ, i.e., a linear low-field electron drift velocity depen-

dence on the applied electric field strength, vd = μE , applies

in this regime. Examining the distribution function for this

regime, we find that it is very similar to the zero-field distri-

bution function with a slight shift in the direction opposite

of the applied electric field. In this regime, the average elec-

tron energy remains relatively low, with most of the energy

gained from the applied electric field being transferred into

the lattice through polar optical phonon scattering. We find

that the low-field electron drift mobility, μ, corresponding to

the velocity field characteristic depicted in Fig. 6, is around

850 cm2/Vs.

If we examine the average electron energy as a function of

the applied electric field strength, shown in Fig. 7, we see that

there is a sudden increase at around 100 kV/cm; this result

was obtained from the same steady-state GaN Monte Carlo

simulation of electron transport as that used to determine

Fig. 6. In order to understand why this increase occurs, we

note that the dominant energy loss mechanism for many of

the III–V compound semiconductors, including bulk wurtzite

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Fig. 8 The valley occupancy as a function of the applied electric fieldstrength for the case of bulk wurtzite GaN. Soon after the average elec-tron energy increases, i.e., at about 100 kV/cm, electrons begin to trans-fer to the upper valleys of the conduction band. There were three thou-sand electrons employed for this simulation. The valleys are labeled 1,2, and 3, in accordance with their energy minima, i.e., the lowest energyvalley is valley 1, the next higher energy valley being valley 2, and thehighest energy valley being valley 3

GaN, is polar optical phonon scattering. When the applied

electric field strength is less than 100 kV/cm, all of the energy

that the electrons gain from the applied electric field is lost

through polar optical phonon scattering. The other scattering

mechanisms, i.e., ionized impurity scattering, piezoelectric

scattering, and acoustic deformation potential scattering, do

not remove energy from the electron ensemble, i.e., they are

elastic scattering mechanisms. Beyond a certain critical ap-

plied electric field strength, however, the polar optical phonon

scattering mechanism can no longer remove all of the energy

gained from the applied electric field. Other scattering mech-

anisms must start to play a role if the electron ensemble is to

remain in equilibrium. The average electron energy increases

until inter-valley scattering begins and an energy balance is

re-established.

As the applied electric field strength is increased beyond

100 kV/cm, the average electron energy increases until a

substantial fraction of the electrons have acquired enough

energy in order to transfer into the upper valleys. In Fig. 8,

we plot the occupancy of the valleys as a function of the

applied electric field strength for the case of bulk wurtzite

GaN, this result being obtained from the same steady-state

GaN Monte Carlo simulation of electron transport as that

used to determine Figs. 6 and 7, the motion of three thousand

electrons being considered for this analysis. As the effective

mass of the electrons in the upper valleys is greater than that

in the lowest valley, the electrons in the upper valleys will

be slower. As more electrons transfer to the upper valleys,

the electron drift velocity decreases. This accounts for the

negative differential mobility observed in the velocity-field

characteristic depicted in Fig. 6.

Finally, at sufficiently high applied electric fields, the num-

ber of electrons in each valley saturates. It can be shown that

in the high-field limit, the number of electrons in each valley

is proportional to the product of the density of states of that

particular valley and the corresponding valley degeneracy.

At this point, the electron drift velocity stops decreasing and

achieves saturation.


AlN

We continue our analysis with an examination of the steady-

state electron transport within bulk wurtzite AlN, a material

often used as the insulator within III–V nitride semiconduc-

tor based electron devices. AlN has the highest effective mass

of the three III–V nitride semiconductors considered in this

analysis, and therefore, it is not surprising that it exhibits

the smallest electron drift velocity and the lowest low-field

electron drift mobility of the III–V nitride semiconductors

considered in this analysis. The velocity-field characteristic

associated with this material is depicted in Fig. 9. This result

was obtained through a steady-state Monte Carlo simulation

of the electron transport within this material for the AlN pa-

rameter selections specified in Tables 1 and 2, the crystal

temperature being set to 300 K and the doping concentration

being set to 1017 cm−3. We note that initially the electron drift

velocity monotonically increases with the applied electric

field strength, reaching a maximum of about 1.7 × 107 cm/s

when the applied electric field strength is around 450 kV/cm.

As with the case of GaN, a linear regime of electron transport

Fig. 9 The velocity-field characteristic associated with bulk wurtziteAlN. Like many other compound semiconductors, the electron driftvelocity reaches a peak, and at higher applied electric field strengths itdecreases until it saturates

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Fig. 10 The average electron energy as a function of the applied elec-tric field strength for bulk wurtzite AlN. Initially, the average electronenergy remains low, only slightly higher than the thermal energy, 3

2kbT .

At 300 kV/cm, however, the average electron energy increases dramat-ically. This increase is due to the fact that the polar optical phononscattering mechanism can no longer absorb all of the energy gainedfrom the applied electric field. The energy minimum corresponding tothe first upper valley is depicted with the dashed line

is observed for the case of AlN for applied electric field

strengths less than 100 kV/cm, the low-field electron drift

mobility, μ, corresponding to the velocity-field characteris-

tic depicted in Fig. 9, being about 130 cm2/Vs. For applied

electric fields strengths in excess of 450 kV/cm, the electron

drift velocity decreases in response to further increases in

the applied electric field strength, i.e., a region of negative

differential mobility is observed, the electron drift velocity

eventually saturating at about 1.4 × 107 cm/s for sufficiently

high applied electric field strengths.

If we examine the average electron energy as a function

of the applied electric field strength, shown in Fig. 10, we

see that there is a sudden increase at around 300 kV/cm; this

result was obtained from the same steady-state AlN Monte

Carlo simulation of electron transport as that used to deter-

mine Fig. 9. As with the case of GaN, beyond a certain critical

applied electric field strength, polar optical phonon scatter-

ing can no longer remove all of the energy gained from the

applied electric field. The average electron energy increases

until inter-valley scattering begins and an energy balance is

re-established. In Fig. 11, we plot the occupancy of the val-

leys as a function of the applied electric field strength for

the case of AlN, this result being obtained from the same

steady-state AlN Monte Carlo simulation of electron trans-

port as that used to determine Figs. 9 and 10, the motion of

three thousand electrons being considered for this steady-

state electron transport analysis. This result is similar to that

found for the case of GaN.

Fig. 11 The valley occupancy as a function of the applied electricfield strength for the case of bulk wurtzite AlN. Soon after the averageelectron energy increases, i.e., at about 300 kV/cm, electrons beginto transfer to the upper valleys of the conduction band. There werethree thousand electrons employed for this simulation. The valleys arelabeled 1, 2, and 3, in accordance with their energy minima, i.e., thelowest energy valley is valley 1, the next higher energy valley beingvalley 2, and the highest energy valley being valley 3


InN

The steady-state electron transport that occurs within bulk

wurtzite InN is the next focus of our analysis. InN has the

lowest effective mass of the three III–V nitride semicon-

ductors considered in this analysis, and therefore, it is not

surprising that it exhibits the largest electron drift velocity

and the highest low-field electron drift mobility of the III–

V nitride semiconductors considered in this analysis. The

velocity-field characteristic associated with this material is

depicted in Fig. 12. This result was obtained through a steady-

state Monte Carlo simulation of the electron transport within

this material for the InN parameter selections specified in

Tables 1 and 2, the crystal temperature being set to 300 K

and the doping concentration being set to 1017 cm−3. We

note that initially the electron drift velocity monotonically

increases with the applied electric field strength, reaching a

maximum of about 4.1 × 107 cm/s when the applied electric

field strength is around 65 kV/cm [88]. As with the cases of

GaN and AlN, a linear regime of electron transport is ob-

served for the case of InN for applied electric field strengths

less than 20 kV/cm, the low-field electron drift mobility, μ,

corresponding to the velocity-field characteristic depicted in

Fig. 12, being about 3400 cm2/Vs. For applied electric fields

strengths in excess of 65 kV/cm, the electron drift velocity de-

creases in response to further increases in the applied electric

field strength, i.e., a region of negative differential mobility

is observed, the electron drift velocity eventually saturating

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Fig. 12 The velocity-field characteristic associated with bulk wurtziteInN. Like many other compound semiconductors, the electron driftvelocity reaches a peak, and at higher applied electric field strengths itdecreases until it saturates

Fig. 13 The average electron energy as a function of the applied electricfield strength for bulk wurtzite InN. Initially, the average electron energyremains low, only slightly higher than the thermal energy, 3

2kbT . At

50 kV/cm, however, the average electron energy increases dramatically.This increase is due to the fact that the polar optical phonon scatteringmechanism can no longer absorb all of the energy gained from theapplied electric field. The energy minima corresponding to the uppervalleys are depicted with the dashed lines

at about 1.8 × 107 cm/s for sufficiently high applied electric

field strengths.




result was obtained from the same steady-state InN Monte

Carlo simulation of electron transport as that used to deter-

mine Fig. 12. As with the cases of GaN and AlN, beyond

a certain critical applied electric field strength, polar optical

Fig. 14 The valley occupancy as a function of the applied electric fieldstrength for the case of bulk wurtzite InN. Soon after the average electronenergy increases, i.e., at about 50 kV/cm, electrons begin to transfer tothe upper valleys of the conduction band. There were three thousandelectrons employed for this simulation. The valleys are labeled 1, 2,and 3, in accordance with their energy minima, i.e., the lowest energyvalley is valley 1, the next higher energy valley being valley 2, and thehighest energy valley being valley 3

phonon scattering can no longer remove all of the energy

gained from the applied electric field. The average electron

energy increases until inter-valley scattering begins and an

energy balance is re-established. In Fig. 14, we plot the occu-

pancy of the valleys as a function of the applied electric field

strength for the case of InN, this result being obtained from

the same steady-state InN Monte Carlo simulation of elec-

tron transport as that used to determine Figs. 12 and 13, the

motion of three thousand electrons being considered for this

steady-state electron transport analysis. This result is similar

to that found for the cases of GaN and AlN.

3.5. Steady-state electron transport within bulk GaAs

For the purposes of comparison, a study of the steady-state

electron transport that occurs within bulk GaAs is also pre-

sented. The velocity-field characteristic associated with this

material is depicted in Fig. 15. This result was obtained

through a steady-state Monte Carlo simulation of the elec-

tron transport within this material for the GaAs parameter

selections specified by Littlejohn et al. [78] and Blakemore

[80], the crystal temperature being set to 300 K and the dop-

ing concentration being set to 1017 cm−3. We note that ini-

tially the electron drift velocity monotonically increases with

the applied electric field strength, reaching a maximum of

about 1.6 × 107 cm/s when the applied electric field strength

is around 4 kV/cm. As with the cases of GaN, AlN, and

InN, a linear regime of electron transport is observed for

the case of GaAs for applied electric field strengths less than

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Fig. 15 The velocity-field characteristic associated with bulk GaAs.Like many other compound semiconductors, the electron drift velocityreaches a peak, and at higher applied electric field strengths it decreasesuntil it saturates

Fig. 16 The average electron energy as a function of the applied elec-tric field strength for bulk GaAs. Initially, the average electron energyremains low, only slightly higher than the thermal energy, 3

2kbT . At

2 kV/cm, however, the average electron energy increases dramatically.This increase is due to the fact that the polar optical phonon scatter-ing mechanism can no longer absorb all of the energy gained from theapplied electric field. The energy minimum corresponding to the firstupper valley is depicted with the dashed line

2 kV/cm, the low-field electron drift mobility, μ, correspond-

ing to the velocity-field characteristic depicted in Fig. 15, be-

ing about 5600 cm2/Vs. For applied electric fields strengths

in excess of 4 kV/cm, the electron drift velocity decreases

in response to further increases in the applied electric field

strength, i.e., a region of negative differential mobility is ob-

served, the electron drift velocity eventually saturating at

about 1.0 × 107 cm/s for sufficiently high applied electric

field strengths.

Fig. 17 The valley occupancy as a function of the applied electric fieldstrength for the case of bulk GaAs. Soon after the average electronenergy increases, i.e., at about 2 kV/cm, electrons begin to transfer tothe upper valleys of the conduction band. There were three thousandelectrons employed for this simulation. The valleys are labeled 1, 2,and 3, in accordance with their energy minima, i.e., the lowest energyvalley is valley 1, the next higher energy valley being valley 2, and thehighest energy valley being valley 3




result was obtained from the same steady-state GaAs Monte

Carlo simulation of electron transport as that used to de-

termine Fig. 15. As with the cases of GaN, AlN, and InN,

beyond a certain critical applied electric field strength, polar

optical phonon scattering can no longer remove all of the

energy gained from the applied electric field. The average

electron energy increases until inter-valley scattering begins

and an energy balance is re-established. In Fig. 17, we plot the

occupancy of the valleys as a function of the applied electric

field strength for the case of GaAs, this result being obtained

from the same steady-state GaAs Monte Carlo simulation

of electron transport as that used to determine Figs. 15 and

16, the motion of three thousand electrons being considered

for this steady-state electron transport analysis. This result is

similar to that found for the cases of GaN, AlN, and InN.

3.6. Steady-state electron transport: A comparison of

the III–V nitride semiconductors with GaAs

In Fig. 18, we contrast the velocity-field characteristics

associated with the III–V nitride semiconductors under con-

sideration in this analysis, i.e., GaN, AlN, and InN, with that

associated with GaAs. In all cases, we have set the crys-

tal temperature to 300 K and the doping concentration to

1017 cm−3, and the material parameters are as specified in

Tables 1 and 2, i.e., these results are the same as those pre-

sented in Figs. 6, 9, 12, and 15, for the cases of GaN, AlN,

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Fig. 18 A comparison of the velocity-field characteristics associatedwith the III–V nitride semiconductors, GaN, AlN, and InN, with thatassociated with GaAs. This plot is depicted on a logarithmic scale.Adapted with permission from the American Institute of Physics; thisfigure was adapted from Figure 1 of Foutz et al. [42]

InN, and GaAs, respectively. We see that each of these III–V

compound semiconductors achieves a peak in its velocity-

field characteristic. InN achieves the highest steady-state

peak electron drift velocity, about 4.1 × 107 cm/s at an ap-

plied electric field strength of around 65 kV/cm. This con-

trasts with the case of GaN, 2.9 × 107 cm/s at 140 kV/cm,

and that of AlN, 1.7 × 107 cm/s at 450 kV/cm. For GaAs,

the peak electron drift velocity, 1.6 × 107 cm/s, occurs at a

much lower applied electric field strength than that for the

III–V nitride semiconductors considered in this analysis, only

4 kV/cm.

Fig. 19 A comparison of the crystal temperature dependence of thevelocity-field characteristics associated with bulk wurtzite GaN andbulk GaAs. GaN maintains a higher electron drift velocity with in-creased crystal temperature than does GaAs (Continued)

Fig. 19 (Continued)

3.7. The sensitivity of the velocity-field characteristic

associated with bulk wurtzite GaN to variations in the

crystal temperature

The temperature dependence of the velocity-field character-

istic associated with bulk wurtzite GaN is now examined. Fig.

19(a) shows how the velocity-field characteristic associated

with GaN varies as the crystal temperature is increased from

100 to 700 K, in increments of 200 K. The upper limit, 700 K,

is chosen as it is the highest operating temperature which may

be expected for AlGaN/GaN power devices. To highlight the

differences between the III–V nitride semiconductors with

Fig. 20 A comparison of the crystal temperature dependence ofthe peak electron drift velocity (open squares), the saturation elec-tron drift velocity (open diamonds), and the low-field electrondrift mobility for bulk wurtzite GaN and bulk GaAs. The low-field electron drift mobility of GaN drops quickly with increas-ing temperature, but its peak and saturation electron drift velocitiesare less sensitive to increases in temperature than those of GaAs

(Continued on next page)

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Fig. 20 (Continued)

more conventional III–V compound semiconductors, such

as GaAs, Monte Carlo simulations of the electron transport

within GaAs have also been performed under the same con-

ditions as GaN. Fig. 19(b) shows the results of these simu-

lations. Note that the electron drift velocity for the case of

GaN is much less sensitive to changes in temperature than

that associated with GaAs.

To quantify this dependence further, the peak electron drift

velocity, the saturation electron drift velocity, and the low-

field electron drift mobility are plotted as functions of the

crystal temperature in Fig. 20, these results being determined

from our steady-state Monte Carlo simulations of the electron

transport within these materials. For both GaN and GaAs, it is

found that all of these electron transport metrics diminish as

the crystal temperature is increased. As may be seen through

an inspection of Fig. 19, the peak and saturation electron drift

velocities do not drop as much in GaN as they do in GaAs in

response to increases in the crystal temperature. The low-field

electron drift mobility in GaN, however, is seen to fall quite

rapidly with crystal temperature, this drop being particularly

severe for temperatures at and below room temperature. This

property will likely have an impact on high-power device

performance.

Delving deeper into our Monte Carlo results yields clues

into this difference in the temperature dependence. First, we

examine the polar optical phonon scattering rate as a function

of the applied electric field strength. Fig. 21 shows that the

scattering rate only increases slightly with temperature for the

case of GaN, from 6.7 × 1013 s−1 at 100 K to 8.6 × 1013 s−1

at 700 K, for high applied electric field strengths. Contrast

this with the case of GaAs, where the rate increases from

4.1 × 1012 s−1 at 100 K to more than twice that amount at

700 K, 9.2 × 1012 s−1, at high applied electric field strengths.

This large increase in the polar optical phonon scattering rate

for the case of GaAs is one reason for the large drop in the

electron drift velocity with increases in temperature for the

case of GaAs.

A second reason for the difference in the temperature de-

pendence of the two materials is the occupancy of the upper

valleys, shown in Fig. 22. In the case of GaN, the upper val-

leys begin to become occupied at roughly the same applied

electric field strength, 100 kV/cm, independent of tempera-

ture. For the case of GaAs, however, the upper valleys are

at a much lower energy than those in GaN. In particular,

while in GaN the first upper conduction band valley mini-

mum is at 1.9 eV above the lowest point in the conduction

band [79], in GaAs, the first upper conduction band valley is

only 290 meV above the bottom of the conduction band; see

Fig. 45 of Blakemore [80]. As the upper conduction band val-

leys are so close to the bottom of the conduction band for the

case of GaAs, the thermal energy (at 700 K, kbT � 60 meV)

is enough in order to allow for a small fraction of the elec-

trons to transfer into the upper valleys even before an electric

field is applied. When electrons occupy the upper valleys,

inter-valley scattering, as well as the upper valleys’ larger

effective masses, reduce the overall electron drift velocity.

This is another reason why the velocity-field characteristic

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Fig. 21 A comparison of the polar optical phonon scattering rates asa function of the applied electric field strength for various crystal tem-peratures for bulk wurtzite GaN and bulk GaAs. Polar optical phononscattering is seen to increase much more quickly with temperature inGaAs

associated with GaAs is more sensitive to variations in the

crystal temperature than that associated with GaN.



doping concentration

The doping concentration is a parameter which can be read-

ily controlled in the fabrication of a semiconductor device.

Understanding the effect of the doping concentration on

the resultant electron transport is important. In Fig. 23, the

velocity-field characteristic associated with bulk wurtzite

GaN is presented for a number of different doping concen-

tration levels. Once again, three important electron transport

metrics are influenced by the doping concentration level, i.e.,

the peak electron drift velocity, the saturation electron drift

Fig. 22 A comparison of the number of particles in the lowest energyvalley of the conduction band, i.e., the � valley, as a function of theapplied electric field strength, for various crystal temperatures, for thecases of bulk wurtzite GaN and bulk GaAs. In GaAs, the electrons beginto occupy the upper valleys much more quickly, causing the electrondrift velocity to drop as the crystal temperature is increased. Three thou-sand electrons were employed for these steady-state electron transportsimulations

velocity, and the low-field electron drift mobility; see Fig. 24.

Our simulation results suggest that for doping concentrations

less than 1017 cm−3, there is very little effect on the velocity-

field characteristic for the case of GaN. However, for doping

concentrations above 1017 cm−3, the peak electron drift ve-

locity diminishes considerably, from about 2.9 × 107 cm/s

for the case of 1017 cm−3 doping to around 2.0 × 107 cm/s

for the case of 1019 cm−3 doping. The saturation electron drift

velocity within GaN is found to only decrease slightly in re-

sponse to increases in the doping concentration. The effect of

doping on the low-field electron drift mobility is also shown.

It is seen that this mobility drops significantly in response

to increases in the doping concentration level, from about

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Fig. 23 A comparison of the dependence of the velocity-field char-acteristics associated with bulk wurtzite GaN and bulk GaAs on thedoping concentration. GaN maintains a higher electron drift velocitywith increased doping levels than does GaAs

1200 cm2/Vs at 1016 cm−3 doping to around 400 cm2/Vs at

1019 cm−3 doping.

As with the case of temperature, we compare the sensitiv-

ity of the velocity-field characteristic associated with GaN

to doping with that associated with GaAs. Fig. 23 shows this

comparison. For the case of GaAs, it is seen that the electron

drift velocities are much more greatly reduced with increased

doping when compared with those associated with GaN. In

particular, for the case of GaAs, the peak electron drift veloc-

ity decreases from about 1.9 × 107 cm/s at 1016 cm−3 doping

to around 0.6 × 107 cm/s at 1019 cm−3 doping. For GaAs, at

the higher doping levels, the peak in the velocity-field charac-

teristic disappears altogether for sufficiently high doping con-

centrations. The saturation electron drift velocity decreases,

from about 1.0 × 107 cm/s at 1016 cm−3 doping to around

0.6 × 107 cm/s at 1019 cm−3 doping. The low-field electron

drift mobility also diminishes dramatically with increased

doping, dropping from about 7600 cm2/Vs at 1016 cm−3

doping to around 2400 cm2/Vs at 1019 cm−3 doping. The

peak electron drift velocity, the saturation electron drift ve-

locity, and the low-field electron drift mobility associated

with GaAs are plotted as functions of the doping concentra-

tion in Fig. 24.

Once again, it is interesting to determine why the doping

dependence in GaAs is so much more pronounced than it is

in GaN. Again, we examine the polar optical phonon scat-

tering rate and the occupancy of the upper valleys. Fig. 25

shows the polar optical phonon scattering rates as a func-

tion of the applied electric field strength, for both GaN and

GaAs. In this case, however, due to screening effects, the

rate drops when the doping concentration is increased. The

decrease, however, is much more pronounced for the case

of GaAs rather than GaN. It is believed that this drop in the

polar optical phonon scattering rate allows for upper valley

Fig. 24 A comparison of the doping concentration dependence ofthe peak electron drift velocity (open squares), the saturation elec-tron drift velocity (open diamonds), and the low-field electron driftmobility, for bulk wurtzite GaN and bulk GaAs. These parametersare more insensitive to increases in doping in GaN rather than GaAs


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Fig. 24 (Continued)

occupancy to occur more quickly in GaAs rather than in GaN;

see Fig. 26. For GaN, electrons begin to occupy the upper

valleys at roughly the same applied electric field strength,

independent of the doping level. However, for the case of

GaAs, the upper valleys are occupied more quickly with

greater doping. When the upper valleys are occupied, the

electron drift velocity decreases due to inter-valley scattering

and the larger effective mass of the electrons within the upper

valleys.


associated with bulk wurtzite AlN to variations in the

crystal temperature and the doping concentration

The sensitivity of the velocity-field characteristic associated

with bulk wurtzite AlN to variations in the crystal tempera-

ture may be examined by considering Fig. 27. As with the

case of GaN, the velocity-field characteristic associated with

AlN is extremely robust to variations in the crystal temper-

Fig. 25 A comparison of the polar optical phonon scattering rates asa function of the applied electric field strength, for both bulk wurtziteGaN and bulk GaAs, for various doping concentrations

ature. In particular, the peak electron drift velocity, which

is about 1.9 × 107 cm/s at 100 K, only decreases to around

1.2 × 107 cm/s at 700 K. Similarly, the saturation electron

drift velocity, which is about 1.5 × 107 cm/s at 100 K, only

decreases to around 1.0 × 107 cm/s at 700 K. The low-field

electron drift mobility associated with AlN also diminishes in

response to increases in the crystal temperature, from about

375 cm2/Vs at 100 K to around 35 cm2/Vs at 700 K. The peak

electron drift velocity, the saturation electron drift velocity,

and the low-field electron drift mobility associated with AlN

are plotted as functions of the crystal temperature in Fig. 28.

The sensitivity of the velocity-field characteristic asso-

ciated with AlN to variations in the doping concentration

may be examined by considering Fig. 29. It is noted that

the variations in the velocity-field characteristic associated

with AlN in response to variations in the doping concentra-

tion are not as pronounced as those which occur in response

to variations in the crystal temperature. Quantitatively, the

peak electron drift velocity drops from about 1.7 × 107 cm/s

at 1017 cm−3 doping to around 1.4 × 107 cm/s at 1019 cm−3

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Fig. 26 A comparison of the number of particles in the lowest valley ofthe conduction band, i.e., the � valley, as a function of the applied elec-tric field strength, for both bulk wurtzite GaN and bulk GaAs, for variousdoping concentration levels. Three thousand electrons were employedfor these steady-state electron transport simulations

Fig. 27 The velocity-field characteristic associated with bulk wurtziteAlN for various crystal temperatures. AlN exhibits its peak electron driftvelocity at very high applied electric fields. AlN has the smallest peakelectron drift velocity and the lowest low-field electron drift mobilityof the III–V nitride semiconductors considered in this analysis

Fig. 28 The variations in the bulk wurtzite AlN peak electron driftvelocity (open squares), the saturation electron drift velocity (open di-amonds), and the low-field electron drift mobility, as a function of thecrystal temperature, are shown

Fig. 29 The velocity-field characteristic associated with bulk wurtziteAlN for various doping concentrations

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Fig. 30 The variations in the bulk wurtzite AlN peak electron driftvelocity (open squares), the saturation electron drift velocity (open di-amonds), and the low-field electron drift mobility, as a function of thedoping concentration, are shown

doping. Similarly, the saturation electron drift velocity drops


1.2 × 107 cm/s at 1019 cm−3 doping. The influence of dop-

ing on the low-field electron drift mobility associated with

AlN is also observed to be not as pronounced as for the case

of crystal temperature, the low-field electron drift mobility

decreasing from about 140 cm2/Vs at 1016 cm−3 doping to

around 100 cm2/Vs at 1019 cm−3 doping. The peak elec-

tron drift velocity, the saturation electron drift velocity, and

the low-field electron drift mobility associated with AlN are

plotted as functions of the doping concentration in Fig. 30.


associated with bulk wurtzite InN to variations in the

crystal temperature and the doping concentration

The sensitivity of the velocity-field characteristic associated

with bulk wurtzite InN to variations in the crystal temperature

Fig. 31 The velocity-field characteristic associated with bulk wurtziteInN for various crystal temperatures. InN has the largest peak electrondrift velocity and the highest low-field electron drift mobility of theIII–V nitride semiconductors considered in this analysis

may be examined by considering Fig. 31. As with the cases

of GaN and AlN, the velocity-field characteristic associated

with InN is extremely robust to increases in the crystal tem-

perature. In particular, the peak electron drift velocity, which

is about 4.4 × 107 cm/s at 100 K, only decreases to around

3.2 × 107 cm/s at 700 K. Similarly, the saturation electron

drift velocity, which is about 1.9 × 107 cm/s at 100 K, only

decreases to around 1.5 × 107 cm/s at 700 K. The low-field

electron drift mobility associated with InN also diminishes in

response to increases in the crystal temperature, from about

9000 cm2/Vs at 100 K to around 800 cm2/Vs at 700 K. The

peak electron drift velocity, the saturation electron drift ve-

locity, and the low-field electron drift mobility associated

with InN are plotted as functions of the crystal temperature

in Fig. 32.

The sensitivity of the velocity-field characteristic associ-

ated with InN to variations in the doping concentration may

be examined by considering Fig. 33. These results suggest

a similar robustness to the doping concentration for the case

of InN. In particular, it is noted that for doping concentra-

tions below 1017 cm−3, the velocity-field characteristic asso-

ciated with InN exhibits very little dependence on the doping

concentration. When the doping concentration is increased

above 1017 cm−3, however, the peak electron drift velocity

diminishes. Quantitatively, the peak electron drift velocity

decreases from about 4.1 × 107 cm/s at 1017 cm−3 doping

to around 3.1 × 107 cm/s at 1019 cm−3 doping. The satu-

ration electron drift velocity only drops slightly, however,


1.6 × 107 cm/s at 1019 cm−3 doping. The low-field elec-

tron drift mobility, however, drops significantly with dop-

ing, from about 4700 cm2/Vs at 1016 cm−3 doping to around

1400 cm2/Vs at 1019 cm−3 doping. The peak electron drift

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Fig. 32 The variations in the bulk wurtzite InN peak electron driftvelocity (open squares), the saturation electron drift velocity (open di-amonds), and the low-field electron drift mobility, as a function of thecrystal temperature, are shown

Fig. 33 The velocity-field characteristic associated with bulk wurtziteInN for various doping concentrations

Fig. 34 The variations in the bulk wurtzite InN peak electron driftvelocity (open squares), the saturation electron drift velocity (open di-amonds), and the low-field electron drift mobility, as a function of thedoping concentration, are shown

velocity, the saturation electron drift velocity, and the low-

field electron drift mobility associated with InN are plotted

as functions of the doping concentration in Fig. 34.



band structure

While the electron transport that occurs within bulk wurtzite

GaN, AlN, and InN has been extensively examined over the

years [26–47, 51, 52, 54, 55], it is only recently that the sen-

sitivity of these results to variations in the band structure

parameters has been examined [45]. Unfortunately, as many

of the band structure parameters associated with GaN, AlN,

and InN remain the subject of debate, for the purposes of sim-

ulation, one is often forced to make specific selections from

wide ranges of potentially legitimate values. Therefore, the

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study of how the electron transport within the III–V nitride

semiconductors varies in response to changes in the band

structure parameters, particularly those which have yet to be

determined, represents a worthwhile endeavor.

A sensitivity analysis into how the electron transport ex-

hibited by the III–V nitride semiconductors varies in response

to changes in the band structure has the added benefit of pro-

viding insight into the nature of the electron transport that

occurs in alloys of the III–V nitride semiconductors, in which

the band structure may be engineered to specification. Many

nitride based electronic and optoelectronic devices are com-

prised of alloys. For example, alloys of AlN and GaN are used

in field-effect transistors [10–16] and alloys of AlN and GaN

and alloys of InN and GaN are used in lasers [17–25]. Despite

their widespread use, electron transport within the alloys of

the III–V nitride semiconductors has yet to be studied to any

extent, the preliminary results of Krishnamurthy et al. [47],

Albrecht et al. [89], and Foutz et al. [90] being amongst the

first to be reported. Unfortunately, the results obtained pro-

vide little insight into the independent role that each material

parameter plays in determining the electron transport within

the III–V nitride semiconductors, GaN, AlN, and InN, an

important consideration if one wishes to optimize the alloy

composition. Clearly, an electron transport sensitivity anal-

ysis, in which the individual impact of each band structure

parameter on the electron transport is assessed, would ame-

liorate these limitations.

Accordingly, in this section, we examine the sensitivity of

the steady-state electron transport within bulk wurtzite GaN

to variations in some of the important band structure param-

eters. GaN is chosen as the reference material in this study

as it is the most common III–V nitride semiconductor. For

the purposes of our analysis, we focus on the velocity-field

characteristic, this being the usual means whereby steady-

state electron transport is characterized. To determine this

characteristic, we employ Monte Carlo simulations of the

electron transport. Within the framework of our three-valley

model for the conduction band of GaN, the band structure

parameters which we focus upon in this analysis are (1) the

lowest energy conduction band valley effective mass, m∗, (2)

the upper conduction band valley effective masses, m∗u, (3)

the non-parabolicity of the lowest energy conduction band

valley, α, (4) the conduction band intervalley energy sepa-

ration, �, and (5) the degeneracy of the upper conduction

band valleys, g. Our goal is to establish the bounds within

which we expect the velocity-field characteristics associated

with GaN, AlN, and InN to lie. The band structure parameter

variations are taken over the broad range of values expected

for the III–V nitride semiconductor alloys.

We first consider how the velocity-field characteristic as-

sociated with bulk wurtzite GaN varies in response to changes

in the lowest energy conduction band valley effective mass,

m∗. In Fig. 35(a), we plot this characteristic corresponding to

a number of selections of m∗, all other parameters being held

at their nominal values, i.e., as prescribed by Tables 1 and

2, these selections spanning over the range expected for the

III–V nitride semiconductor alloys; see, for example, Chin

et al. [30] and Tsen et al. [85]. We note that this characteristic

varies considerably in response to changes in m∗. In particu-

lar, it is seen that the peak in the velocity-field characteristic

lowers, broadens, and shifts to higher electric fields as m∗ is

increased. Quantitatively, the peak electron drift velocity de-

creases from about 3.9 × 107 cm/s when m∗ is set to 0.1 me

to around 2.1 × 107 cm/s when m∗ is set to 0.4 me. Concomi-

tantly, the saturation electron drift velocity is seen to increase

from about 1.2 × 107 cm/s when m∗ is set to 0.1 me to around

Fig. 35 (a) The velocity-field characteristic associated with bulkwurtzite GaN for various selections of the lowest conduction band valleyeffective mass, m∗, all other parameters being held at their nominal val-ues, i.e., as prescribed by Tables 1 and 2. Adapted with permission fromThe Minerals, Metals, and Materials Society; this figure was adaptedfrom Fig. l(a) of O’Leary et al. [45]. (b) The dependence of the bulkwurtzite GaN peak and saturation electron drift velocities on m∗, allother parameters being held at their nominal values, i.e., as prescribedby Tables 1 and 2. The m∗ associated with bulk wurtzite GaN, AlN, andInN are depicted with the dashed lines; a range of m∗ for the case of InN,corresponding to the values suggested by Tsen et al. [85] (0.045 me)and Mohammad and Morkoc [3, 67–69] (0.11 me), is shown. The peakelectron drift velocity values are shown with the open squares whilethe saturation electron drift velocity values are shown with the opendiamonds. Adapted with permission from The Minerals, Metals, andMaterials Society; this figure was adapted from Fig. l(b) of O’Learyet al. [45]. (c) The dependence of the bulk wurtzite GaN peak field onm∗, all other parameters being held at their nominal values, i.e., as pre-scribed by Tables 1 and 2. The m∗ associated with bulk wurtzite GaN,AlN, and InN are depicted with the dashed lines. Adapted with per-mission from The Minerals, Metals, and Materials Society; this figurewas adapted from Fig. l(c) of O’Leary et al. [45]. (d) The dependenceof the bulk wurtzite GaN low-field electron drift mobility on m∗, allother parameters being held at their nominal values, i.e., as prescribedby Tables 1 and 2. The open squares represent the results obtained fromour Monte Carlo simulations. The solid line depicts our theoretical mo-bility results, obtained using the analytical expressions of Shur et al.[33]. Adapted with permission from The Minerals, Metals, and Mate-rials Society; this figure was adapted from Fig. 1(d) of O’Leary et al.[45] (Continued on next page)

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Fig. 35 (Continued)

1.5 × 107 cm/s when m∗ is set to 0.4 me. The electric field

at which the peak in the velocity-field characteristic occurs,

hereafter referred to as the peak field, is seen to increase with

m∗, varying from about 60 kV/cm when m∗ is set to 0.1 me

to around 320 kV/cm when m∗ is set to 0.4 me. The depen-

dence of the peak and saturation electron drift velocities on

m∗, and the dependence of the peak field on m∗, are shown

in Figs. 35(b) and 35(c), respectively.

One would expect that the low-field electron drift mobility

is a strong function of m∗. In Fig. 35(d), we plot the depen-

dence of this mobility on m∗, these results being obtained

from our Monte Carlo simulations of the electron transport;

in Fig. 35(d), the values for m∗ range from 0.05 me to 0.5 me,

this being the expected range for the III–V nitride semicon-

ductors considered in this analysis [30, 85]. We note that the

low-field electron drift mobility decreases dramatically as m∗

is increased. In particular, the low-field electron drift mobil-

ity is found to decrease from about 2200 cm2/Vs when m∗ is

set to 0.1 me to around 320 cm2/Vs when m∗ is set to 0.4 me;

the low-field electron drift mobility for the nominal selection

of m∗ being set to 0.2 me is found to be about 850 cm2/Vs.

Theoretical mobility results, obtained using the analytical

expressions of Shur et al. [33], are plotted alongside these

Monte Carlo results, all material parameters being held at the

same values as those used in the simulations. It is seen that

Fig. 36 (a) The velocity-field characteristic associated with bulkwurtzite GaN for various upper conduction band valley effective massselections; the upper conduction band valley effective mass, m∗

u, isas defined in the text. All other parameters are held at their nominalvalues, i.e., as prescribed by Tables 1 and 2. Adapted with permis-sion from The Minerals, Metals, and Materials Society; this figure wasadapted from Fig. 2a of O’Leary et al. [45]. (b) The dependence ofthe bulk wurtzite GaN peak and saturation electron drift velocities onm∗

u, all other parameters being held at their nominal values, i.e., asprescribed by Tables 1 and 2. The peak electron drift velocity valuesare shown with the open squares while the saturation electron driftvelocity values are shown with the open diamonds. Adapted with per-mission from The Minerals, Metals, and Materials Society; this figurewas adapted from Fig. 2b of O’Leary et al. [45]. (c) The dependenceof the bulk wurtzite GaN peak field on m∗

u, all other parameters be-ing held at their nominal values, i.e., as prescribed by Tables 1 and2. Adapted with permission from The Minerals, Metals, and MaterialsSociety; this figure was adapted from Figure 2c of O’Leary et al. [45]


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Fig. 36 (Continued)

the analysis of Shur et al. [33] forms a relatively tight upper

bound to our Monte Carlo results, exhibiting essentially the

same functional dependency.

We now consider how changes in the upper conduction

band valley effective masses, m∗u, influence the velocity-field

characteristic. In Fig. 36(a), we plot the velocity-field char-

acteristic associated with bulk wurtzite GaN for a number of

upper conduction band valley effective mass selections. In

order to simplify matters, we set the effective mass of the

first upper conduction band valley to that of all of the second

upper conduction band valleys, i.e., there is a common upper

conduction band valley effective mass, m∗u, endowed to all

upper conduction band valleys, our selections for m∗u captur-

ing the range found in the literature; see, for example [36–40].

All other parameters are held at their nominal values, i.e., as

prescribed by Tables 1 and 2. We note that the upper con-

duction band valley effective mass selection plays no role in

the low-field region of the velocity-field characteristic, i.e.,

the low-field electron drift mobility is around 850 cm2/Vs in

all cases. We also note, however, that the saturation electron

drift velocity decreases dramatically with increased m∗u. In

particular, we find that the saturation electron drift velocity

decreases from about 2.2 × 107 cm/s when m∗u is set to 0.5 me

to around 8.5 × 106 cm/s when m∗u is set to 2 me. In contrast,

the peak electron drift velocity and the peak field are found to

be relatively insensitive to variations in m∗u. The dependence

of the peak and saturation electron drift velocities on m∗u, and

the dependence of the peak field on m∗u, are depicted in Figs.

36(b) and (c), respectively.

We have treated the lowest energy conduction band val-

ley within the framework of the Kane model [59]. Ignoring

Kane’s prescription for the non-parabolicity coefficient, α, in

Fig. 37(a) we plot the band structure associated with the low-

est energy conduction band valley for various selections of

α, the effective mass being held at the nominal value we have

ascribed to bulk wurtzite GaN, i.e., 0.2 me; these selections

of α span over the range found in the literature [36–38, 40].

Fig. 37 (a) The band structure ascribed to the lowest energy conduc-tion band valley of bulk wurtzite GaN, for various selections of thenon-parabolicity coefficient, α. The effective mass, m∗, is held at thenominal value of 0.2 me. Adapted with permission from The Minerals,Metals, and Materials Society; this figure was adapted from Figure 3a ofO’Leary et al. [45]. (b) The velocity-field characteristic associated withbulk wurtzite GaN for various selections of the non-parabolicity coef-ficient, α. All other parameters are held at their nominal values, i.e., asprescribed by Tables 1 and 2. Adapted with permission from The Miner-als, Metals, and Materials Society; this figure was adapted from Figure3b of O’Leary et al. [45]. (c) The dependence of the bulk wurtzite GaNpeak and saturation electron drift velocities on α, all other parametersbeing held at their nominal values, i.e., as prescribed by Tables 1 and 2.The peak electron drift velocity values are shown with the open squareswhile the saturation electron drift velocity values are shown with theopen diamonds. The α associated with bulk wurtzite GaN, AlN, and InNare depicted with the dashed lines. Adapted with permission from TheMinerals, Metals, and Materials Society; this figure was adapted fromFigure 3c of O’Leary et al. [45]. (d) The dependence of the bulk wurtziteGaN peak field on α, all other parameters being held at their nominalvalues, i.e., as prescribed by Tables 1 and 2. The α associated withbulk wurtzite GaN, AlN, and InN are depicted with the dashed lines.Adapted with permission from The Minerals, Metals, and MaterialsSociety; this figure was adapted from Figure 3d of O’Leary et al. [45]


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Fig. 37 (Continued)

Increased non-parabolicity leads to lesser band structural cur-

vature. The impact of α on the velocity-field characteristic is

shown in Fig. 37(b), in which we plot the characteristic as-

sociated with bulk wurtzite GaN for various selections of α,

all other parameters being held at their nominal values, i.e.,

as prescribed by Tables 1 and 2. While the low-field electron

drift mobility remains around 850 cm2/Vs in all cases, we

find that the peak broadens and shifts to higher electric fields

as α is increased. In particular, while the peak and satura-

tion electron drift velocities are only weakly dependent on

Fig. 38 (a) The velocity-field characteristic associated with bulkwurtzite GaN for various conduction band intervalley energy separa-tions; the conduction band intervalley energy separation, �, is as definedin the text. All other parameters are held at their nominal values, i.e., asprescribed by Tables 1 and 2. Adapted with permission from The Min-erals, Metals, and Materials Society; this figure was adapted from Fig.4(a) of O’Leary et al. [45]. (b) The dependence of the bulk wurtzite GaNpeak and saturation electron drift velocities on �, all other parametersbeing held at their nominal values, i.e., as prescribed by Tables 1 and 2.The peak electron drift velocity values are shown with the open squareswhile the saturation electron drift velocity values are shown with theopen diamonds. The � associated with bulk wurtzite GaN, AlN, andInN are depicted with the dashed lines. Adapted with permission fromThe Minerals, Metals, and Materials Society; this figure was adaptedfrom Fig. 4(b) of O’Leary et al. [45]. (c) The dependence of the bulkwurtzite GaN peak field on �, all other parameters being held at theirnominal values, i.e., as prescribed by Tables 1 and 2. The � associatedwith bulk wurtzite GaN, AlN, and InN are depicted with the dashedlines. Adapted with permission from The Minerals, Metals, and Mate-rials Society; this figure was adapted from Fig. 4(c) of O’Leary et al. [45]


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Fig. 38 (Continued)

α, the peak field is a strong function of α; the peak field in-

creases from about 115 kV/cm when α = 0.0 eV−1 to around

225 kV/cm when α = 1.0 eV−1. The dependence of the peak

and saturation electron drift velocities on α, and the depen-

dence of the peak field on α, are depicted in Figs. 37(c) and

(d), respectively.

In Fig. 38(a), we study how the velocity-field character-

istic associated with bulk wurtzite GaN varies in response

to changes in the conduction band intervalley energy separa-

tion, all other parameters being held at their nominal selec-

tions, i.e., as prescribed by Tables 1 and 2. In this analysis,

the conduction band intervalley energy separation, �, is de-

fined as the difference in energy between the conduction band

minimum, located at the � point, and the first upper conduc-

tion band valley minimum, also located at the � point, the

energy difference between the first and second upper con-

duction band valley minima being held at our nominal selec-

tion of 0.2 eV; the intervalley energy separation ascribed to

bulk wurtzite InN is 2.2 eV [36] while that ascribed to bulk

wurtzite AlN is 0.7 eV [37], so the selections of � made

here are representative of the III–V nitride semiconductor al-

loys. We note that the velocity-field characteristic associated

with GaN varies moderately when � is increased. In particu-

lar, while the conduction band intervalley energy separation

plays no role in the low-field region of the velocity-field char-

acteristic, the low-field electron drift mobility being about

850 cm2/Vs in all cases, we see that the peak electron drift

velocity increases from about 2.3 × 107 cm/s when � is set

to 0.5 eV to around 3.0 × 107 cm/s when � is set to 2.5 eV.

The peak field is also seen to increase monotonically with �,

increasing from about 100 kV/cm when � is set to 0.5 eV to

around 150 kV/cm when � is set to 2.5 eV. The dependence

of the saturation electron drift velocity on � is considerably

milder. The dependence of the peak and saturation electron

drift velocities on �, and the dependence of the peak field

on �, are depicted in Figs. 38(b) and (c), respectively.

In Fig. 39(a), we plot the velocity-field characteristic asso-

ciated with bulk wurtzite GaN for various upper conduction

band minima degeneracy selections, all other parameters be-

ing held at their nominal values, i.e., as prescribed by Tables

1 and 2. In this analysis, we set the degeneracy of the first

upper conduction band valley minima to that of the second

Fig. 39 (a) The velocity-field characteristic associated with bulkwurtzite GaN for various upper conduction band valley degeneracyselections; the upper conduction band valley degeneracy, g, is as de-fined in the text. All other parameters are held at their nominal val-ues, i.e., as prescribed by Tables 1 and 2. Adapted with permissionfrom The Minerals, Metals, and Materials Society; this figure wasadapted from Fig. 5(a) of O’Leary et al. [45]. (b) The dependenceof the bulk wurtzite GaN peak and saturation electron drift velocitieson g, all other parameters being held at their nominal values, i.e., asprescribed by Tables 1 and 2. The peak electron drift velocity valuesare shown with the open squares while the saturation electron driftvelocity values are shown with the open diamonds. Adapted with per-mission from The Minerals, Metals, and Materials Society; this figurewas adapted from Fig. 5(b) of O’Leary et al. [45]. (c) The depen-dence of the bulk wurtzite GaN peak field on g, all other parametersbeing held at their nominal values, i.e., as prescribed by Tables 1 and2. Adapted with permission from The Minerals, Metals, and MaterialsSociety; this figure was adapted from Fig. 5(c) of O’Leary et al. [45]


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Fig. 39 (Continued)

upper conduction band valleys, i.e., there is a common degen-

eracy, g, given to all upper valleys, our selections for g being

representative of the degeneracies of the various conduction

band valley minima found in bulk wurtzite GaN, AlN, and

InN; see, for example, [36–38, 40]. We see that while the

low-field region of the velocity-field characteristic is essen-

tially independent of g, the low-field electron drift mobility

being about 850 cm2/Vs in all cases, the saturation electron

drift velocity is found to be moderately sensitive to variations

in the degeneracy. Quantitatively, we find that the saturation

electron drift velocity decreases from about 1.8 × 107 cm/s

when the degeneracy is set to one to around 1.3 × 107 cm/s

when the degeneracy is set to six. The peak electron drift ve-

locity and the peak field are seen to be relatively insensitive

to the degeneracy. The dependence of the peak and saturation

electron drift velocities on g, and the dependence of the peak

field on g, are depicted in Figs. 39(b) and 39(c), respectively.

3.12. Transient electron transport

Steady-state electron transport is the dominant electron trans-

port mechanism in devices with larger dimensions. For de-

vices with smaller dimensions, however, transient electron

transport must also be considered when evaluating device

performance. Ruch [91] demonstrated, for both silicon and

GaAs, that the transient electron drift velocity may exceed

the corresponding steady-state electron drift velocity by a

considerable margin for appropriate selections of the ap-

plied electric field strength. Shur and Eastman [92] explored

the device implications of transient electron transport, and

demonstrated that substantial improvements in the device

performance may be achieved as a consequence. Heiblum

et al. [93] made the first direct experimental observation of

transient electron transport within GaAs. Since then, there

have been a number of experimental investigations into the

transient electron transport within the III–V compound semi-

conductors; see, for example, [94–96].

Thus far, very little research has been invested into the

study of the transient electron transport within the III–V ni-

tride semiconductors, GaN, AlN, and InN. Foutz et al. [34]

examined transient electron transport within both the wurtzite

and zincblende phases of GaN. In particular, they examined

how electrons, initially in thermal equilibrium, respond to

the sudden application of a constant electric field. In devices

with dimensions greater than 0.2 μm, they found that steady-

state electron transport is expected to dominate device perfor-

mance. For devices with smaller dimensions, however, with

the application of a sufficiently high electric field strength,

they found that the transient electron drift velocity can con-

siderably overshoot the corresponding steady-state electron

Fig. 40 The electron drift velocity as a function of the distance dis-placed since the application of the electric field, for various appliedelectric field strength selections, for the cases of (a) bulk wurtziteGaN, (b) bulk wurtzite AlN, (c) bulk wurtzite InN, and (d) bulk GaAs.For all cases, we have assumed an initial zero field electron distri-bution, a crystal temperature of 300 K, and a doping concentrationof 1017 cm−3. Adapted with permission from the American Instituteof Physics; this figure was adapted from Fig. 2 of Foutz et al. [42]


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Fig. 40 (Continued)

drift velocity. This velocity overshoot was found to be com-

parable with that which occurs within GaAs.

Foutz et al. [42] performed a subsequent analysis in which

the transient electron transport within all of the III–V ni-

tride semiconductors under consideration in this analysis,

i.e., GaN, AlN, and InN, were compared with that which oc-

curs within GaAs. In particular, following the approach of

Foutz et al. [34], they examined how electrons, initially in

thermal equilibrium, respond to the sudden application of a

constant electric field. A key result of this study, presented in

Fig. 40, plots the transient electron drift velocity as a function

of the distance displaced since the electric field was initially

applied, for a number of applied electric field strength selec-

tions, for each of the materials considered in this analysis.

Focusing initially on the case of bulk wurtzite GaN (see

Fig. 40(a)), we note that for the applied electric field strength

selections 70 kV/cm and 140 kV/cm, that the electron drift

velocity reaches steady-state very quickly, with little or no ve-

locity overshoot. In contrast, for applied electric field strength

selections above 140 kV/cm, significant velocity overshoot

Fig. 41 A comparison of the velocity overshoot amongst the III–V ni-tride semiconductors considered in this analysis and GaAs. The appliedelectric field strengths chosen correspond to twice the critical appliedelectric field strength at which the peak in the steady-state velocity-fieldcharacteristic occurs (see Fig. 18), i.e., 280 kV/cm for the case of GaN,900 kV/cm for the case of AlN, 130 kV/cm for the case of InN, and8 kV/cm for the case of GaAs. The InN and GaAs results do not cross.Adapted with permission from the American Institute of Physics; thisfigure was adapted from Figure 4 of Foutz et al. [42]

occurs. This result suggests that in GaN, 140 kV/cm is a crit-

ical field for the onset of velocity overshoot effects. As was

mentioned in Sections 3.2 and 3.6, 140 kV/cm also corre-

sponds to the peak in the velocity-field characteristic asso-

ciated with GaN; recall Figs. 6 and 18. Steady-state Monte

Carlo simulations suggest that this is the point at which sig-

nificant upper valley occupation begins to occur; recall Fig. 7.

This suggests that velocity overshoot is related to the transfer

of electrons to the upper valleys. Similar results are found for

the other III–V nitride semiconductors, i.e., AlN and InN, and

GaAs, the critical fields being 450 kV/cm for AlN, 65 kV/cm

for InN, and 4 kV/cm for GaAs; see Figs. 40(b)–(d).

We now compare the transient electron transport charac-

teristics for the various materials. From Fig. 40, it is clear

that certain materials exhibit higher peak overshoot veloci-

ties and longer overshoot relaxation times. It is not possible

to fairly compare these different semiconductors by applying

the same applied electric field strength to all of the materi-

als, as the transient effects occur over such a disparate range

of applied electric field strengths for each material. In order

to facilitate such a comparison, we choose a field strength

equal to twice the critical applied electric field strength for

each material, i.e., 280 kV/cm for GaN, 900 kV/cm for AlN,

130 kV/cm for InN, and 8 kV/cm for GaAs. Fig. 41 shows

such a comparison of the velocity overshoot effects amongst

the four materials considered in this analysis, i.e., GaN, AlN,

InN, and GaAs. It is clear that among the three III–V nitride

semiconductors considered, InN exhibits superior transient

electron transport characteristics. In particular, InN has the

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largest overshoot velocity and the distance over which this

overshoot occurs, 0.3 μm, is longer than in either GaN and

AlN. GaAs exhibits a longer overshoot relaxation distance,

approximately 0.7 μm, but the electron drift velocity exhib-

ited by InN is greater than that of GaAs for all distances.

3.13. Electron transport conclusions

In this section, steady-state and transient electron transport

results, corresponding to the III–V nitride semiconductors,

GaN, AlN, and InN, were presented, these results being ob-

tained from our Monte Carlo simulations of the electron

transport within these materials. Steady-state electron trans-

port was the dominant theme of our analysis. In order to

aid in the understanding of these electron transport charac-

teristics, a comparison was made between GaN and GaAs.

Our simulations showed that GaN is more robust to varia-

tions in crystal temperature and doping concentration than

GaAs, and an analysis of our Monte Carlo simulation re-

sults showed that polar optical phonon scattering plays the

dominant role in accounting for these differences in behav-

ior. This analysis was also performed for the other III–V

nitride semiconductors considered in this analysis, i.e., AlN

and InN, and similar results were obtained. The sensitivity

of the steady-state electron transport that occurs within bulk

wurtzite GaN to variations in the band structure parameters

was then examined. Finally, we presented some key tran-

sient electron transport results, these results indicating that

the transient electron transport that occurs within InN is the

most pronounced of all of the materials under consideration

in this review, i.e., GaN, AlN, InN, and GaAs.

4. Electron Transport Within the III–V NitrideSemiconductors: A Review

4.1. Introduction

Pioneering investigations into the material properties of the

III–V nitride semiconductors, GaN, AlN, and InN, were per-

formed during the earlier half of the 20th Century [97–99].

The III–V nitride semiconductor materials available at the

time, small crystals and powders, were of poor quality, and

completely unsuitable for device applications. Thus, it was

not until the late 1960s, when Maruska and Tietjen [73] em-

ployed chemical vapor deposition to fabricate GaN, that in-

terest in the III–V nitride semiconductors experienced a re-

naissance. Since that time, interest in the III–V nitride semi-

conductors has been growing, the material properties of these

semiconductors being considerably improved over the years.

As a result of this research effort, as of the present moment,

there are a number of commercial devices available which

employ the III–V nitride semiconductors. More III–V nitride

semiconductor based device applications are currently under

development, and these should become available in the near

future.

In this section, we present a brief overview of the III–V

nitride semiconductor electron transport field. We start with a

survey, describing the evolution of the field. In particular, the

sequence of critical developments that have occurred, con-

tributing to our current understanding of the electron trans-

port mechanisms within the III–V nitride semiconductors,

GaN, AlN, and InN, is chronicled. Then, some current liter-

ature is presented, particular emphasis being placed on the

most recent developments in the field and how such develop-

ments are modifying our understanding of the electron trans-

port mechanisms within the III–V nitride semiconductors,

GaN, AlN, and InN. Finally, frontiers for further research

and investigation are presented.

This section is organized in the following manner. In

Section 4.2, we present a brief survey, describing the evo-

lution of the field. Then, in Section 4.3, some current liter-

ature is discussed. Finally, frontiers for further research and

investigation are presented in Section 4.4.

4.2. The evolution of the field

The favorable electron transport characteristics of the III–V

nitride semiconductors, GaN, AlN, and InN, have been rec-

ognized for a long time now. As early as the 1970s, Littlejohn

et al. [26] pointed out that the large polar optical phonon en-

ergy characteristic of GaN, in conjunction with its large inter-

valley energy separation, suggests a high saturation electron

drift velocity for this material. As the high-frequency elec-

tron device performance is, in large measure, determined by

this saturation electron drift velocity [27], the recognition of

this fact ignited enhanced interest in this material, and its

III–V nitride semiconductor compatriots, AlN and InN. This

enhanced interest, and the developments which have tran-

spired as a result of it, are responsible for the III–V nitride

semiconductor industry of today.

In 1975, Littlejohn et al. [26] were the first to report results

obtained from semi-classical Monte Carlo simulations of the

steady-state electron transport within bulk wurtzite GaN. A

one-valley model for the conduction band was adopted for

the purposes of their analysis. Steady-state electron transport,

for both parabolic and non-parabolic band structures, was

considered in their analysis, non-parabolicity being treated

through the application of the Kane model [59]. The pri-

mary focus of their investigation was the determination of the

velocity-field characteristic associated with GaN. All donors

were assumed to be ionized and the free electron concentra-

tion was taken to be equal to the dopant concentration. The

scattering mechanisms considered were (1) ionized impurity,

(2) polar optical phonon, (3) piezoelectric, and (4) acoustic

deformation potential. For the case of the parabolic band, in

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the absence of ionized impurities, they found that the elec-

tron drift velocity monotonically increases with the applied

electric field strength, saturating at a value of around 2.5 ×107 cm/s for the case of high applied electric field strengths.

In contrast, for the case of the non-parabolic band, in the ab-

sence of ionized impurities, a region of negative differential

mobility was found, the electron drift velocity achieving a

maximum of about 2 × 107 cm/s at an applied electric field

strength of around 100 kV/cm, further increases in the applied

electric field strength resulting in a slight decrease in the cor-

responding electron drift velocity. The role of ionized impu-

rity scattering was also investigated by Littlejohn et al. [26].

In 1993, Gelmont et al. [29] reported on ensemble semi-

classical two-valley Monte Carlo simulations of the electron

transport within bulk wurtzite GaN, this analysis improving

upon the analysis of Littlejohn et al. [26] by incorporating

intervalley scattering into the simulations. They found that

the negative differential mobility exhibited by bulk wurtzite

GaN is much more pronounced than that found by Littlejohn

et al. [26], and that intervalley transitions are responsible for

this. For a doping concentration of 1017 cm−3, Gelmont et al.[29] demonstrated that the electron drift velocity achieves a

peak value of about 2.8 × 107 cm/s at an applied electric

field strength of around 140 kV/cm. The impact of inter-

valley transitions on the electron distribution function was

also determined and shown to be significant. The impact of

doping and compensation on the velocity-field characteristic

associated with bulk wurtzite GaN was also examined.

Since these pioneering investigations, ensemble Monte

Carlo simulations of the electron transport within GaN have

been performed numerous times. In particular, in 1995 Man-

sour et al. [31] reported the use of such an approach in

order to determine how the crystal temperature influences


GaN. Later that year, Kolnık et al. [32] reported on employing

full-band Monte Carlo simulations of the electron transport

within bulk wurtzite GaN and bulk zincblende GaN, finding

that bulk zincblende GaN exhibits a much higher low-field

electron drift mobility than bulk wurtzite GaN. The peak elec-

tron drift velocity corresponding to bulk zincblende GaN was

found to be only marginally greater than that exhibited by

bulk wurtzite GaN. In 1997, Bhapkar and Shur [35] reported

on employing ensemble semi-classical three-valley Monte

Carlo simulations of the electron transport within bulk and

confined wurtzite GaN. Their simulations demonstrated that

the two-dimensional electron gas within a confined wurtzite

GaN structure will exhibit a higher low-field electron drift

mobility than bulk wurtzite GaN, by almost an order of mag-

nitude, this being in agreement with experiment. In 1998,

Albrecht et al. [40] reported on employing ensemble semi-

classical five-valley Monte Carlo simulations of the electron

transport within bulk wurtzite GaN, with the aim of deter-

mining elementary analytical expressions for a number of

the electron transport metrics corresponding to bulk wurtzite

GaN, for the purposes of device modeling.

Electron transport within the other III–V nitride semicon-

ductors, AlN and InN, has also been studied using ensem-

ble semi-classical Monte Carlo simulations of the electron

transport. In particular, employing ensemble semi-classical

three-valley Monte Carlo simulations, the velocity-field char-

acteristic associated with bulk wurtzite AlN was reported by

O’Leary et al. [37] in 1998. They found that AlN exhibits

the lowest peak and saturation electron drift velocities of

the III–V nitride semiconductors considered in this analy-

sis. Similar simulations of the electron transport within bulk

wurtzite AlN were also reported by Albrecht et al. [38] in

1998. The results of O’Leary et al. [37] and Albrecht et al.[38] were found to be quite similar. The first known simula-

tion of the electron transport within bulk wurtzite InN was

the semi-classical three-valley Monte Carlo simulations of

O’Leary et al. [36], reported in 1998. InN was demonstrated

to have the highest peak and saturation electron drift veloc-

ities of the III–V nitride semiconductors. The subsequent

ensemble full-band InN Monte Carlo simulations of Bellotti

et al. [41], reported in 1999, produced results similar to those

of O’Leary et al. [36].

The first known study of transient electron transport within

the III–V nitride semiconductors was that performed by

Foutz et al. [34], reported in 1997. In this study, ensem-

ble semi-classical three-valley Monte Carlo simulations were

employed in order to determine how the electrons within bulk

wurtzite and bulk zincblende GaN, initially in thermal equi-

librium, respond to the sudden application of a constant elec-

tric field. The velocity overshoot which occurs within these

materials was examined. It was found that the electron drift

velocities that occur within the zincblende phase of GaN are

slightly greater than those exhibited by the wurtzite phase

owing to the slightly higher steady-state electron drift veloc-

ity exhibited by the zincblende phase of GaN. A compari-

son with the transient electron transport which occurs within

GaAs was made. Using the results of this analysis, a deter-

mination of the minimum transit time, as a function of the

distance displaced since the application of the applied elec-

tric field, was performed for all three materials considered

in this study, i.e., wurtzite GaN, zincblende GaN, and GaAs.

For distances in excess of 0.1 μm, both phases of GaN were

shown to exhibit superior performance, i.e., reduced transit

time, when contrasted with that associated with GaAs.

A more general analysis, in which transient electron trans-

port within GaN, AlN, and InN, was studied, was performed

by Foutz et al. [42], and reported in 1999. As with their

previous study, Foutz et al. [42] determined how electrons,

initially in thermal equilibrium, respond to the sudden appli-

cation of a constant electric field. For GaN, AlN, InN, and

GaAs, it was found that the electron drift velocity overshoot

only occurs when the applied electric field exceeds a certain

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critical applied electric field strength unique to each mate-

rial. The critical applied electric field strengths, 140 kV/cm

for the case of bulk wurtzite GaN, 450 kV/cm for the case of

AlN, 65 kV/cm for the case of InN, and 4 kV/cm for the case

of GaAs, were shown to correspond to the peak electron drift

velocity in the velocity-field characteristic associated with

each of these materials, i.e., the peak field; recall Fig. 18. It

was found that InN exhibits the highest peak overshoot ve-

locity, and that this overshoot lasts over prolonged distances,

compared with AlN, InN, and GaAs. A comparison with the

results of experiment was performed.

In addition to Monte Carlo simulations of the electron

transport within these materials, a number of other types

of electron transport studies have been performed. In 1975,

for example, Ferry [27] reported on the determination of


GaN using a displaced Maxwellian distribution function ap-

proach. For high applied electric fields, Ferry [27] found that

the electron drift velocity associated with GaN monotoni-

cally increases with the applied electric field strength, i.e., it

does not saturate, reaching a value of about 2.5 × 107 cm/s

at an applied electric field strength of 300 kV/cm. The de-

vice implications of this result were further explored by Das

and Ferry [28]. In 1994, Chin et al. [30] reported on a de-

tailed study of the dependence of the low-field electron drift

mobilities associated with the III–V nitride semiconductors,

GaN, AlN, and InN, on the crystal temperature and the dop-

ing concentration. An analytical expression for the low-field

electron drift mobility,μ, determined using a variational prin-

ciple, was employed for the purposes of this analysis. The

results obtained were contrasted with those of experiment.

Subsequent mobility studies were reported in 1996 by Shur

et al. [33] and in 1997 by Look et al. [51]. Then, in 1998,

Weimann et al. [39] reported on a model for the determina-

tion of how the scattering of electrons by the threading dis-

locations within bulk wurtzite GaN influences the low-field

electron drift mobility. They demonstrated why the experi-

mentally measured low-field electron drift mobility associ-

ated with this material is much lower than that predicted from

Monte Carlo analyses, threading dislocations not being taken

into account in the Monte Carlo simulations of the electron

transport within the III–V nitride semiconductors, GaN, AlN,

and InN.

While the negative differential mobility exhibited by the


tride semiconductors, GaN, AlN, and InN, is widely at-

tributed to intervalley transitions, and while direct experi-

mental evidence confirming this has been presented [100],

Krishnamurthy et al. [47] suggest that instead the inflection

points in the bands, located in the vicinity of the � valley, are

primarily responsible for the negative differential mobility

exhibited by bulk wurtzite GaN. The relative importance of

these two mechanisms, i.e., intervalley transitions and inflec-

tion point considerations, were evaluated by Krishnamurthy

et al. [47], both for the case of bulk wurtzite GaN and an

AlGaN alloy.

4.3. Recent developments

There have been a number of interesting recent develop-

ments in the study of the electron transport within the III–V

nitride semiconductors which have influenced the direction

of thought in this field. On the experimental front, in 2000

Wraback et al. [46] reported on the use of a femtosecond opti-

cally detected time-of-flight experimental technique in order

to experimentally determine the velocity-field characteristic

associated with bulk wurtzite GaN. They found that the peak

electron drift velocity, about 1.9 × 107 cm/s, is achieved

at an applied electric field strength of around 225 kV/cm.

No discernible negative differential mobility was observed.

Wraback et al. [46] suggested that the large defect density,

characteristic of the GaN samples they employed, these

not being taken into account in Monte Carlo simulations of

the electron transport within this material, accounts for this

difference between this experimental result and that obtained

using simulation. They also suggested that decreasing the

intervalley energy separation, from about 2 eV to 340 meV,

as suggested by the experimental results of Brazel et al. [87],

may also account for these observations; recall Fig. 38.

The determination of the electron drift velocity from ex-

perimental measurements of the unity gain cutoff frequency,

ft , has been pursued by a number of researchers. The key

challenge in these analyses is the de-embedding of the par-

asitics from the experimental measurements so that the true

intrinsic saturation electron drift velocity may be obtained.

Eastman et al. [101] present experimental evidence which

suggests that the electron drift velocity within bulk wurtzite

GaN is about 1.2–1.3 × 107 cm/s. A more recent report, by

Oxley and Uren [102], suggests a value of 1.1 × 107 cm/s.

The role of self-heating was probed by Oxley and Uren [102]

and shown to be relatively insignificant. A completely sat-

isfactory explanation for the discrepancy between these ex-

perimental results and those of the Monte Carlo simulations

has yet to be provided.

Wraback et al. [103] performed a subsequent study on

the transient electron transport within bulk wurtzite GaN. In

particular, using their femtosecond optically detected time-

of-flight experimental technique in order to experimentally

determine the velocity overshoot that occurs within bulk

wurtzite GaN, they observed substantial velocity overshoot

within this material. In particular, a peak transient electron

drift velocity, of 7.25 × 107 cm/s, was observed within the

first 200 fs after photoexcitation, for an applied electric field

strength of 320 kV/cm. These experimental results were

shown to be reasonably consistent with the theoretical pre-

dictions of Foutz et al. [42].

Springer


On the theoretical front, there have been a number of re-

cent developments. In 2001, O’Leary et al. [43] presented an

elementary one-dimensional analytical model for the elec-

tron transport within the III–V compound semiconductors,

and applied it to the cases of bulk wurtzite GaN and GaAs.

The predictions of this analytical model were compared with

those of Monte Carlo simulations and were found to be in

satisfactory agreement. Hot-electron energy relaxation times

within the III–V nitride semiconductors were recently stud-

ied by Matulionis et al. [104], and reported in 2002. Bulutay

et al. [105] studied the electron momentum and energy re-

laxation times within the III–V nitride semiconductors, and

reported the results of their study in 2003. It is particularly

interesting to note that their arguments add considerable

credence to the earlier inflection point argument of Krish-

namurthy et al. [47]. In 2004, Brazis and Raguotis [106]

reported on the results of a Monte Carlo study involving

additional phonon modes and a smaller intervalley energy

separation for the case of bulk wurtzite GaN. Their results

were found to be much closer to the experimental results of

Wraback et al. [46] than those found previously.

The influence of hot-phonons on the electron transport

mechanisms within the III–V nitride semiconductors, GaN,

AlN, and InN, an effect not considered in our simulations

of the electron transport within these materials, i.e., we as-

sumed steady-state phonon populations, has been the focus

of considerable recent investigation. In particular, in 2004 it-

self, Silva and Nascimento [107], Gokden [108], and Ridley

et al. [109], to name just three, presented results related to

this research focus. These results suggest that hot-phonon

effects play a role in influencing the nature of the electron

transport within the III–V nitride semiconductors, GaN, AlN,

and InN. In particular, Ridley et al. [109] point out that the

saturation electron drift velocity and the peak field are both

influenced by hot-phonon effects; it should be noted, how-

ever, that Ridley et al. [109] neglect intervalley transitions

in their analysis, their analysis challenging the conventional

belief that the negative differential mobility exhibited by the


tride semiconductors, GaN, AlN, and InN, is attributable to

transitions into the upper valleys. Research into the role that

hot-phonons play in influencing the electron transport mech-

anisms within the III–V nitride semiconductors, GaN, AlN,

and InN, seems likely to continue into the foreseeable future.

Research into how the electron transport within the III–

V nitride semiconductors, GaN, AlN, and InN, influences

the performance of nitride semiconductor based devices is

ongoing. In 2004, Matulionis and Liberis [110] reported on

the role that hot-phonons play in determining the microwave

noise within AlGaN/GaN channels. More recently, in 2005,

Ramonas et al. [111] further developed this analysis, focus-

ing on how hot-phonon effects influence power dissipation

within AlGaN/GaN channels. The high-field electron trans-

port within AlGaN/GaN heterostructures was examined and

reported in 2005 by Barker et al. [112] and Ardaravicius

et al. [113]. A numerical simulation of the current-voltage

characteristics of AlGaN/GaN high electron mobility transis-

tors at high temperatures was performed by Chang et al. [114]

and reported in 2005. Other device modeling work involving

Monte Carlo simulations of the electron transport within the

III–V nitride semiconductors, GaN, AlN, and InN, was re-

ported in 2005 by Yamakawa et al. [115] and Reklaitis and

Reggiani [116]. It is evident that research into how the elec-

tron transport which occurs within the III–V nitride semicon-

ductors, GaN, AlN, and InN, influences the performance of

nitride semiconductor based devices will continue for many

years to come.

4.4. Future prospectives

It is clear that our understanding of the electron transport

within the III–V nitride semiconductors, GaN, AlN, and InN,

is, at present at least, in a state of flux. A complete under-

standing of the electron transport mechanisms within these

materials has yet to be achieved, and is the subject of intense

current research. Most troubling is the discrepancy between

the results of experiment and those of simulation. There are

a two principle sources of uncertainty in our analysis of the

electron transport mechanisms within these materials; (1) un-

certainty in the material properties, and (2) uncertainty in the

underlying physics. We discuss each of these subsequently.

Uncertainty in the material parameters associated with the

III–V nitride semiconductors, GaN, AlN, and InN, remains a

key source of ambiguity in the analysis of the electron trans-

port within these materials [45]. Even for bulk wurtzite GaN,

the most studied of the III–V nitride semiconductors consid-

ered in this analysis, uncertainty in the band structure remains

an issue [87]. The energy gap associated with InN, and the

effective mass associated with this material, continues to fuel

debate; see, for example, Davydov et al. [82], Wu et al. [83],

Matsuoka et al. [84], and Tsen et al. [85]. Variations in the

experimentally determined energy gap associated with InN,

observed from sample to sample, further confound matters.

Most recently, Shubina et al. [117] suggested that nonstoi-

chiometry within InN may be responsible for these variations

in the energy gap. Further research will have to be performed

in order to confirm this. Given this uncertainty in the band

structures associated with the III–V nitride semiconductors,

GaN and InN, it is clear that new simulations of the electron

transport within these materials will have to be performed

once researchers have settled on appropriate band structures.

We thus view the results presented in Section 3 as a baseline,

our sensitivity analysis, presented in Section 3.11, providing

some insights into how variations in the band structure will

impact upon the results. Work on finalizing a set of mate-

rial parameters suitable for the III–V nitride semiconductors

Springer


under consideration in this analysis, i.e., GaN, AlN, and InN,

and on performing the corresponding electron transport sim-

ulations, is ongoing. New experimental results, such as those

of Tsen et al. [85], will aid in this endeavor.

Uncertainty in the underlying physics is considerable.

The source of the negative differential mobility remains a

matter to be resolved. The presence of hot-phonons within

these materials, and how such phonons impact upon the

electron transport mechanisms within these materials, re-

mains another point of contention. It is clear that a deeper

understanding of these electron transport mechanisms will

have to be achieved in order for the next generation of

III–V nitride semiconductor based devices to be properly

designed.

5. Conclusions

In this paper, we reviewed analyses of the electron trans-

port within the III–V nitride semiconductors, GaN, AlN, and

InN. In particular, we have discussed the evolution of the

field, surveyed the current literature, and presented frontiers

for further investigation and analysis. In order to narrow the

scope of this review, we focused on the electron transport

within bulk wurtzite GaN, AlN, and InN for the purposes of

this paper. Most of our discussion focused upon results ob-

tained from our ensemble semi-classical three-valley Monte

Carlo simulations of the electron transport within these ma-

terials, our results conforming with state-of-the-art III–V ni-

tride semiconductor orthodoxy.

We began our review with the Boltzmann transport equa-

tion, this equation underlying most analyses of the electron

transport within semiconductors. A brief description of our

ensemble semi-classical three-valley Monte Carlo simula-

tion approach to solving the Boltzmann transport equation

was then provided. The material parameters, corresponding

to bulk wurtzite GaN, AlN, and InN, were then presented.

We then used these material parameter selections and our

ensemble semi-classical three-valley Monte Carlo simula-

tion approach to determine the nature of the steady-state and

transient electron transport within the III–V nitride semicon-

ductors. Finally, we presented some recent developments on

the electron transport within these materials, and pointed to

fertile frontiers for further research and investigation.

Appendix A: Further Details of Our Monte CarloAlgorithm

We now provide further details of the semi-classical Monte

Carlo algorithm employed for the purposes of our simulations

of the electron transport within the III–V nitride semiconduc-

tors, GaN, AlN, and InN. Initially, we overview a more de-

tailed flow chart corresponding to our approach. Then, details

of some of the trickier parts of our Monte Carlo algorithm

will be discussed. The generation of the free flight time will

then be covered and then the selection of the scattering event

(after a free flight) will be described.

A.1. Flow chart for our Monte Carlo algorithm

A more detailed flow chart for our Monte Carlo algorithm

is shown in Fig. A1. This flow chart provides a detailed de-

scription of how the dynamics of the electrons are handled,

as well as how the statistics are kept during the simulation.

When the simulation initializes, it reads the input file and

sets the simulation parameters. Next, the initial electron dis-

tribution is determined. During this stage, each electron in

the simulation is given an initial wave-vector in accordance

with a Maxwell-Boltzmann distribution. At the same time, a

rejection technique is used in order to ensure that the number

of electrons in any given region of �k-space never exceeds the

Fermi-Dirac limit. This technique provides a close approxi-

mation to an initial Fermi-Dirac distribution.

Next, the electric field is set and the scattering rate tables

are initialized. The time step is set to zero and then a loop is

entered which moves each particle through free flights and

scattering events until the end of the time step is reached.

After all of the particles are moved, macroscopic quantities,

such as the electron drift velocity, are calculated over the

distribution and stored in temporary arrays. At the end of the

simulation, the accumulated statistics are output to a file. In

the next sections, details of some of these steps are provided.

A.2. Generation of the free-flight times

An electron’s energy and wave-vector, �k, determine the prob-

ability that this electron will scatter by means of any of the

aforementioned scattering processes. In between each scat-

tering event, the electron’s motion is determined through

semi-classical physics, i.e., Equations (4) and (5). The

amount of time between each scattering event is determined

statistically, based on the total scattering rate,

λ(�k) =∑

i

λi (�k), (6)

which is just the sum of the individual scattering rates cor-

responding to each scattering mechanism. The statistically

determined time between scattering events is known as the

free flight time, t f .

Generating a proper distribution of free flight times is es-

sential in order to obtain correct simulation results. A number

of methods, used for the purposes of generating these free

flight times, have already been studied in detail [118]. A

derivation of the algorithm used in our simulations of the

Springer


Fig. A1 A more completeflowchart for our Monte Carloalgorithm used for simulatingelectron transport within theIII–V nitride semiconductors,GaN, AlN, and InN


GaN, AlN, and InN, will be provided here.

We first note that the probably distribution, P(t), for the

free flight time, of length t, is just the probability that an

electron survives without a collision to time t multipled by the

probability of a collision within a small interval, dt, around

t. The probability of a collision within dt of t is simply the

product of the scattering rate at time t and dt. The first part

of the distribution, the probability that the electron survives

to time t without a collision, can be found by assuming that

the scattering processes are Poisson in nature. For a Poisson

process, the probability of no scattering event for any interval,

δt , is exp(−λδt). If the scattering rate were constant, this

would be the distribution we require. However, the scattering

rate changes with time as the electron drifts under the action

of the applied electric field. To take into account the fact that

the scattering rates change with time, we divide the interval,

[0, t], into N small intervals. The probability, pi , that no

scattering event occurs, in interval i, is

pi = exp(−λiδt), (7)

Springer


1 + 2 + 3 + 4 + 5

1 + 2 + 3 + 4

1 + 2 + 3

1+ 2

1

t inc 2t inc 3t inc 4t inc 5t inc

t

Sca

tter

ing r

ate

0

Fig. A2 The scattering mechanism selection process

where λi is the scattering rate during interval i and δt is

duration of interval i. The probability that no scattering event

occurs in any of the i intervals, 1 through N, is the product of

the probabilities for each interval, i.e.,

p(t) =N∏

i=1

exp[−λiδt],

= exp

(−

N∑i=1

λiδt

). (8)

Letting the intervals become very small, i.e., δt → dt , the

sum of Equation (8) reduces to an integral, i.e.,

p(t) = exp

[−

∫ t

0

λ(�k(t ′)) dt ′]. (9)

The free flight time distribution then becomes the scattering

rate multiplied by p(t), i.e.,

P(t) = λ(�k(t)) exp

[−

∫ t

0

λ(�k(t ′)) dt ′]. (10)

In order to generate random free flight times, with a given

P(t), we apply a direct method [61]. In particular, we select

a random number, r, with a uniform distribution between [0,

1], and set it equal to the integrated probability distribution

function, i.e.,

r =∫ t

0

P(t ′) dt ′. (11)

Substituting Equation (10) into Equation (11), and solving

the integral, yields

r = 1 − exp

[−

∫ t

0

λ(�k(t ′))dt ′]

. (12)

Thus, we conclude that

− ln(1 − r ) =∫ t

0

λ(�k(t ′))dt ′. (13)

A time, t, must be found which satisfies the above equation

for the random number, r.

One difficulty in evaluating the integral over λ is that it is

a complicated function of t. This problem can be overcome

by introducing an artificial scattering mechanism, known as

the self-scattering mechanism, λ0 (�k). This new mechanism

makes the total scattering rate constant over some interval of

time, i.e.,

� = λ0(�k) + λ(�k). (14)

Yorston [118] discusses several algorithms for generating

the free flight times using this self-scattering concept. One

of the most efficient algorithms, and the one employed in our

Monte Carlo simulations of the electron transport within the

III–V nitride semiconductors, GaN, AlN, and InN, is the con-

stant time method. In this method, a fixed time, tinc, is chosen,

and the integral in Equation (13) is carried out over intervals

of length tinc. In each interval, a self-scattering mechanism,

λ0(�k), is added in order to make the total scattering rate con-

stant and greater than λ(�k) during the tinc interval. Fig. A2

illustrates this algorithm. The free flight time is chosen when

the total integral satisfies Equation (13). At that time, λ0(�k)

and each λi (�k) are used to determine the choice of scattering

event.

In the case that a self-scattering mechanism is chosen,

special treatment is necessary. The integral for the next free

flight time must continue where the previous one left off. In

the example shown in Fig. A2, the integral from t to 4tinc is

first used, then that from 4tinc to 5tinc is used, and so on.

A.3. Choice of scattering event

Once the electron finishes its free flight, it scatters. The choice

of the scattering event is also made with a random number.

This time, the probability that a particular scattering event is

selected is directly proportional to the scattering rate corre-

sponding to that particular mechanism. A random number,

r, uniformly distributed between [0, 1], is chosen, and the

scattering mechanism, i, which satisfies

Si < r < Si+1, (15)

where

Si =∑i

j=0 λ j (�k)

�, (16)

Springer


is selected, where

� =∑

i

λi (�k). (17)

Once the scattering mechanism is selected, the final wave-

vector of the electron must be chosen. This selection must,

of course, obey conservation of energy. With this require-

ment, there exists a sphere in �k-space into which the electron

is allowed to scatter. Therefore, by determining the angle

(azimuthal and polar) from the electron’s original direction,

we may uniquely select the final wave-vector for the elec-

tron, and at the same time select the phonon with which the

electron is scattering, in order to obey conservation of mo-

mentum considerations. For all the scattering mechanisms

selected in our Monte Carlo approach, the selection of the

azimuthal angle is done with a uniform distribution, i.e., there

is no preference in terms of the azimuthal angle. However,

many of the scattering mechanisms have a preference with

the polar angle. For each of the scattering mechanisms in the

Monte Carlo approach, the dependence of the scattering rate

with the polar angle is known, i.e.,

λi (�k) =∫ 2π

0

Pi (θ, �k) dθ. (18)

There are three different techniques available for convert-

ing random numbers with a uniform distribution into one with

an arbitrary distribution. These are the direct, rejection, and

combined techniques, which are all described by Jacoboni

and Lugli [61]. For most of the scattering mechanisms used

in our Monte Carlo approach, the rejection technique is used

to determine the polar angles. However, some of the most im-

portant mechanisms are handled differently. For polar optical

phonon and piezoelectric scattering, a combined technique is

used. For ionized impurity scattering at low energies, when

non-parabolicity can be ignored, the direct technique is used.

In other cases, the rejection technique is used, except when

the distribution is highly peaked, in which case a combined

technique is used.

The simulation continues, moving the electron through

each time step until a special time step is reached, known as

the collection time. After this special time step, the macro-

scopic averages, which are stored in temporary arrays, are

averaged and stored in permanent arrays. Each average is

simply the average over all of the electrons in the simulation.

For example, for the electron drift velocity,

v(t) =∑

vi (t)

N, (19)

where N denotes the total number of electrons. After each col-

lection time, the scattering rates tables are also recalculated.

This occurs because some of the scattering rates, i.e., po-

lar optical phonon, ionized impurity, and piezoelectric, are a

function of the electron temperature, which changes through-

out the simulation. If the simulation requires that the applied

electric field strength be updated, then it is updated after

every fourth collection time (this number can be adjusted).

The average from that fourth collection time is assumed to

be in steady-state and is associated with the electric field

during that interval. At the end of the simulation, the quan-

tities stored in the permanent arrays are written to an output

file.

A.4. Monte Carlo codes available on the internet

A variety of Monte Carlo codes, for the purposes of simu-

lating the steady-state and transient electron transport within

bulk semiconductors, are available on the internet. For ex-

ample, SDemon is available at:

https://www.nanohub.org/simulation tools/sdemon tool information

For a description of the theoretical basis of SDemon and

its implementation, please consult, M. A. Stettler, “Monte

Carlo Studies of Electron Transport in Silicon Bipolar Tran-

sistors,” MSEE Thesis, Purdue University, West Lafayette,

Indianna, December 1990 and the SDemon User’s Manual.

The program is written in Fortran 77. The program author is

M. A. Stettler of Purdue University.

Acknowledgements The administrative assistance of P. Kale is ac-knowledged. The copyright permissions granted from the AmericanInstitute of Physics, Pergamon, and The Minerals, Metals, and Materi-als Society are also acknowledged. Financial support from the Officeof Naval Research and the Natural Sciences and Engineering ResearchCouncil of Canada is gratefully acknowledged. The use of equipmentgranted from the Canada Foundation for Innovation, and equipmentloaned from the Canadian Microelectronics Corporation, is also ac-knowledged.

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