Journal of Modern Physics, 2013, 4, 121-126
http://dx.doi.org/10.4236/jmp.2013.44A012 Published Online April 2013 (http://www.scirp.org/journal/jmp)
Comparison of High Field Electron Transport in GaAs,
InAs and In0.3Ga0.7As
B. Bouazza, A. Guen-Bouazza, C. Sayah, N. E. Chabane-Sari
Unite de Recherches Matériaux et Energies Renouvelables, Faculté des Sciences de l’Ingénieur,
Université Abou-Bekr-Belkaïd de Tlemcen, Tlemcen, Algérie
Received January 8, 2013; revised February 10, 2013; accepted February 25, 2013
Copyright © 2013 B. Bouazza et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
An ensemble Monte Carlo simulation is used to compare high field electron transport in bulk GaAs, InAs and
In0.3Ga0.7As. In particular, velocity overshoot and electron transit times are examined. We find the steady state velocity
of the electrons is the most important factor determining transit time over distances longer then 0.2 μm. Over shorter
distances velocity overshoot effects in InAs and In0.3Ga0.7 As at high fields are comparable to those in GaAs. We esti-
mate the minimum transit time across a 1 μm InAs sample to be about 4.2 ps. Similar calculations for In0.3Ga0.7As yield
6 ps (for GaAs yield 10 ps). Calculations are made using a nonparabolic effective mass energy band model, Monte
Carlo simulation that includes all of the major scattering mechanisms. The band parameters used in the simulation are
extracted from optimized pseudopotential band calculations to ensure excellent agreement with experimental informa-
tion and ab initio band models.
Keywords: Monte Carlo Method; Semiconductor Devices; Velocity Overshoot; Electron Transport
1. Introduction
The Ensemble Monte Carlo technique has been used now
for over 30 years as a numerical method to simulate
nonequilibrium transport in semiconductor materials and
devices, and has been the subject of numerous books and
reviews. In application to transport problems, a random
walk is generated to simulate the stochastic motion of
particles subject to collision processes in some medium.
This process of random walk generation may be used to
evaluate integral equations and is connected to the gen-
eral random sampling technique used in the valuation of
multi-dimensional integrals. The basic technique is to
simulate the free particle motion (referred to as the free
flight) terminated by instantaneous random scattering
events. The Monte Carlo algorithm consists of generating
random free flight times for each particle, choosing the
type of scattering occurring at the end of the free flight,
changing the final energy and momentum of the particle
after scattering, and then repeating the procedure for the
next free flight. Sampling the particle motion at various
times throughout the simulation allows for the statistical
estimation of physically interesting quantities such as the
single particle distribution function, the average drift
velocity in the presence of an applied electric field, the
average energy of the particles, etc. By simulating an
ensemble of particles, representative of the physical sys-
tem of interest, the non-stationary time-dependent evolu-
tion of the electron and hole distributions under the in-
fluence of a time-dependent driving force may be simu-
lated. The particle-based picture, in which the particle
motion is decomposed into free flights terminated by
instantaneous collisions, is basically the same picture
underlying the derivation of the semi-classical BTE. In
fact, it may be shown that the one-particle distribution
function obtained from the random walk Monte Carlo
technique satisfies the BTE for a homogeneous system in
the long-time limit [1].
The purpose of this work is to compare, using Mont
Carlo simulation, the potentialities of n-type GaAs, InAs
and In0.3Ga0.7As. We first analyse, in Section 1, we ex-
plain how to apply our band structure model to Monte
Carlo simulation. In Section 2 we present to describe in
the detail our calculation of high field transport proper-
ties of n-type GaAs, InAs and In0.3Ga0.7As by Monte
Carlo method. We shall also compare our results with
other theoretical and experimental data insofar as it is
possible highlight the accuracy of the simulation results.
C
opyright © 2013 SciRes. JMP

B. BOUAZZA ET AL. 122
1.1. Free Flight Generation
In the Monte Carlo method, the dynamics of particle mo-
tion is assumed to consist of free flights terminated by
instantaneous scattering events, which change the mo-
mentum and energy of the particle. To simulate this
process, the probability density


pt

pt t
tt
t

is required, in
which is the joint probability that a particle will
arrive at time t without scattering after the previous colli-
sion at t = 0, and then suffer a collision in a time interval
around time . The probability of scattering in the
time interval around t may be written as



kt t 




,
where


kt 


is the scattering rate of an electron or
hole of wave vector . The scattering rate, k


kt 



,
represents the sum of the contributions from each indi-
vidual scattering mechanism, which are usually calcu-
lated using perturbation theory, as described later. The
implicit dependence of


kt

k




on time reflects the
change in due to acceleration by internal and external
fields [2-4]. For electrons subject to time independent
electric and magnetic fields, the time evolution of
between collisions as
k
 
0kt keEv B t



Ev
 (1)
where the electric field, is the electron velocity,
and

is the magnetic flux density. In terms of the
scattering rate,


kt



tkt t

 , the probability that a particle
has not suffered a collision after a time t is given by
0
exp 

 




tt

t
t
k tt 

.
Thus, the probability of scattering in the time interval
after a free flight of time may be written as the
joint probability
 
0
exp
f
Ptt kt 


  






 (2)
Random flight times may be generated according to
the probability density

pt above using, for example,
the pseudo-random number generator implicit on most
modern computers, which generate uniformly distributed
random numbers in the range [0,1]. Using a direct
method, random flight times sampled from


pt

1
11
00
f
tr
tprr



may
be generated according to

f
rpt


 (3)
where r is a uniformly distributed random number and
f
t

is the desired free flight time. Integrating (3) with

pt

f
t
ktt


 




1
0
ln
f
t
rktt 

given by (2) above yields
1
0
1expr   
 (4)
Since 1 − r is statistically the same as r, (4) may be
simplified to
 


k

(5)

Equation (5) is the fundamental equation used to gen-
erate the random free flight time after each scattering
event, resulting in a random walk process related to the
underlying particle distribution function. If there is no
external driving field leading to a change of between
scattering events, the time dependence vanishes, and the
integral is trivially evaluated. In the general case where
this simplification is not possible, it is expedient to in-
troduce the so called self-scattering method [1-3,5,6], in
which we introduce a fictitious scattering mechanism
whose scattering rate always adjusts itself in such a way
that the total (self-scattering plus real scattering) rate is a
constant in time


0Self
kt


  
 (6)
where self

is the self-scattering rate. The self-scatter-
ing mechanism itself is defined such that the final state
before and after scattering is identical. Hence, it has no
effect on the free flight trajectory of a particle when se-
lected as the terminating scattering mechanism, yet re-
sults in the simplification of (5) such that the free flight is
given by

1
0
1ln
f
tr (7)
The constant total rate (including self-scattering) 0

is chosen a priori so that it is larger than the maximum
scattering encountered during the simulation interval. In
the simplest case, a single value is chosen at the begin-
ning of the entire simulation (constant gamma method),
checking to ensure that the real rate never exceeds this
value during the simulation. Other schemes may be cho-
sen that are more computationally efficient, and which
modify the choice of 0

at fixed time increments.
1.2. Selection of Scattering Rate
When the electrons are accelerated and the scattering
time is chosen, scattering must then occur at the end of
the scattering time period. The method used for this is the
rejection technique. This technique chooses the scattering
using the relative probabilities of the individual events.
To start we construct a scattering table and normalize all
scattering probabilities to the maximum scattering value,
which was found in the above section for the self scat-
tering [4]. Once the entire table is constructed, it can be
used throughout the entire simulation without need for
recompilation. The selection of the scattering now be-
comes a two-part step. As the table has already been

B. BOUAZZA ET AL. 123
normalized to one, we may use a uniform random num-
ber to select the scattering rate. The reason why this
works is simple. The choice of scattering is random, but
is also governed by the relative strength of certain scat-
tering rates in connection with others that exist in the
system. The random numbers take care of choosing the
scattering rate, and the relative strength of each scattering
rate to the total in the table controls the frequency of cer-
tain events over others.
1.3. Scattering Angle and Final State
For elastic scattering, the scattering is isotropic. There-
fore, all final states in the energy-conserving sphere have
the same probability of occupation after scattering. The
final angle is independent of the initial state , and the
angles of are proportional to
k
ksin

  
2π
cos
. Realiz-
ing that the azimuthal angle varies between 0 and , a
direct technique can be employed to obtain [2,4,7,8]:
3
1r

 and 4 2πr

 (8)
1.4. Mean Velocity and Energy Calculation
When the electric field is applied in the x direction, the
average drift velocity and the average electron energy are
given for each valley, respectively, by [9],


1
1
1
1
N
di
i
N
i
i
vvt
N
t
N















(10)
where

12
xi
i
k
m
i
vt




 (11)

114
2
it



 
  (12)
and


222
2
2
x
iyizi
kkkk
m

 (13)
where


i
vt and


i represent the electron drift
velocity and the electron energy at the end of each time
step, while
t 
,
x
i
k
yi
and
k
z
i are the wave vector com-
ponents in
k
,
x
y and direction for each electron, re-
spectively.
z
2. Simulation and Results
In each simulation, twenty thousand electrons are ini-
tially distributed in the sample according to an equilib-
rium Maxwellian distribution at 300 K. A variety of field
strengths are simulated to determine the effect on the
transient behavior of the electron ensemble. The simula-
tion steps the electric field from zero to full intensity at
the beginning of the run


0t

, after which the velocity
of the electrons is averaged at 10fs intervals. The average
traveled as a function of time is found by integrating the
drift velocity. Our Monte Carlo program is based on a
three isotropic and non parabolic valley model. The pa-
rameters for valleys are estimated from recent band
structure calculations. The scattering mechanisms in-
cluded in the simulations are polar optical phonon,
acoustic phonons, piezoelectric, intervalley scattering,
ionized impurities and alloy scattering. Values for the
various coupling constants which determine many of the
scattering rates are the same as those used in Reference
[7]. The donor concentration is set to 1.107/cm3.
Figure 1 shows electron velocity versus distance for
GaAs, InAs and In0.3Ga0.7As. Previous Monte Carlo
studies of velocity overshoot in GaAs have performed
and are in agreement with the present results. We find the
fields which produce the highest steady state velocities (2
kv/cm in InAs and 5kv/cm in In0.3Ga0.7As) are similar to
the results using the full band Monte Carlo simulation.
Furthermore, in all three materials overshoot only occurs
at field strengths larger than the peak steady state veloc-
ity field and one sees that the higher amplitude of veloc-
ity overshoot, the lower its distance (or duration). These
observations are tentatively explained in the following
manner. First of all, we have checked that, as long as all
electrons remain in the

valley, the velocity increases.
Thus, the maximum velocity is reached when the “most
rapid” electrons have gained enough energy to transfer to
L valley. These electrons are “lucky electrons”, which
have suffered no, or very few, or very inefficient scatter-
ing events. Therefore, the time needed to reach the
maximum velocity is mainly determined quasiballistic
motion and is sensitive to the scattering rates. On the
contrary, the final static velocity is obtained when the
whole electron distribution has reached its new equilib-
rium situation. This process is completed when even
“unlucky” widely scattered, electrons have gained
enough energy to transfer.
Figure 2 shows the electron transit time as a function
of distance traveled. The field strengths transit time oc-
curs when the steady state velocity is the highest. Using
the relation


12πfT

 where


is the transit
time at 1 μm, we estimate the corresponding cutoff fre-
quencies for GaAs to 29 GHz. Values as high as 20 GHz
chosen minimize the electron transit time at 1 μm. In
GaAs, InAs and In0.3Ga0.7As the minimum have been
measured in modern GaAs modulation doped field
effect transistors, not far from the upper limit predicted
from the transit time alone.
1 μm
In Figure 3 we show the transit time as a function of
distance in the overshoot regime. In this figure the ap-
plied fields were chosen to minimize the transit time

B. BOUAZZA ET AL. 124
0.0 1.0x10-7 2.0x10-7 3.0x10-7 4.0x10-7
0
1x105
2x105
3x105
4x105
5x105
5.0x10-7
0.0 2.0x10-7 4.0x 10-7 6.0x10-7 8.0x10-7 1.0x
0.0
2.0x105
4.0x105
6.0x105
8.0x105
1.0x106
10-6 1.2x10-6
0.0 2.0x10-7 4.0x10-7
0.0
2.0x105
4.0x105
6.0x105
8.0x105
6.0x10-7
Figure 1. Electron velocity as function of distance in each of the materials simulated.

B. BOUAZZA ET AL. 125
1.0 ×10-
11
8.0 ×10-12
6.0 ×10-12
4.0 ×10-12
2.0 ×10-12
0.0
Transit Time (sec)
0.0 2.0 ×10-7 4.0 ×10
-7 6.0 ×10 -7 8.0 ×10 -7 1.0 ×10 -6
Distance (m)
GaAs
In0.3Ga0.7As
InAs
Figure 2. Electron transit time as a function of distance. The field strengths chosen minimize the transit time across 1 μm.
The applied fields are 5 kV/cm for GaAs, 2 kV/cm for InAs, and 5 kV/cm for In0.3Ga0.7As.
0.0 5.0x10-8 1.0x10-7 1.5x10-7
0.0
2.0x10-13
4.0x10-13
6.0x10-13
8.0x10-13
1.0x10-12
2.0x10-7
Figure 3. Electron transit time as a function of distance 0.2 μm. The field strengths chosen minimize the transit time across
0.1 μm. The applied fields are 20 kV/cm for GaAs, InAs, and In0.3Ga0.7As.
across a 0.1 μm region. In this regime, one normally ex-
pects electrons with a lower effective mass to have greter
acceleration and therefore have greater velocity and a
smaller transit time. We predict however, that although
the effective mass is larger InAs
(In0.3Ga0.7As) its ability to operate at higher voltages
allows the transit time to be reduced below that of GaAs.
We therefore conclude that for device lengths less than
0.2 μm where velocity overshoot is important, the elec-
tronic transport properties of GaAs demonstrate no ad-
vantage over those of InAs (In0.3Ga0.7As).
3. Conclusion
The authors experience has shown that the effective mass
of the gamma valley and the relative energy gap between
the valleys has the greatest effect on velocity overshoot
and mobility. These parameters have been measured ex-
perimentally or have been obtained through band struc-
ture calculations. Several of scattering rates depend upon
coupling constants that are currently not well known.
Therefore, this constant for the acoustic deformation po-
tential was varied by ±20% and ±40%. The Monte Carlo
technique has been used to compare transit times and
velocity overshoot effects in GaAs, InAs and In0.3Ga0.7As.
We find that over distances longer than 0.2 μm the transit
times in InAs (In0.3Ga0.7As) are less than those in GaAs
due to InAs’s (In0.3Ga0.7As’s) greater peak velocity. Over
shorter distances velocity overshoot effects dominate and

B. BOUAZZA ET AL. 126
the transit time in InAs (In0.3Ga0.7As) is comparable or
even less than that of GaAs. We conclude that InAs
(In0.3Ga0.7As) devices should be capable of equal or
higher frequency performance than GaAs when transit
time is an important factor.
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