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

ksin

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

iyizi

kkkk

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

,

i

k

yi

and

k

i are the wave vector com-

ponents in

k

,

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

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