Mm. Dignam et Aa. Grinberg, SOLUTION OF THE BOLTZMANN TRANSPORT-EQUATION IN AN ARBITRARY ONE-DIMENSIONAL-POTENTIAL PROFILE, Physical review. B, Condensed matter, 50(7), 1994, pp. 4345-4354
We present a numerical approach to the solution of the Boltzmann trans
port equation in an arbitrary one-dimensional potential. Instead of em
ploying the usual piecewise linear approximation to the potential in t
he discretization of the equation, pre use a stepwise approximation. I
t is shown that due to absence of the electric field between mesh poin
ts, the Boltzmann transport equation can be reduced to a relatively si
mple equation involving only the spherical part of the distribution fu
nction. The method naturally describes situations where energy-band di
scontinuities take place-situations which are important in modern semi
conductor heterostructures. We demonstrate the method for a system of
finite length in a constant electric field. We observe a spatial perio
dicity in the electron drift velocity and average energy produced by i
nteraction of electrons with optical phonons. These oscillations demon
strate one of the advantages of our method over Monte Carlo calculatio
ns, which do not predict such oscillations due to the averaging of the
statistical data over distances larger than the oscillation period. W
e compare our results for the drift velocity in moderate electric fiel
ds (5 to 50 kV/cm) to a phenomenological model, finding the drift velo
city to depend roughly on the square root of the field in this field r
ange.