SOLUTION OF THE BOLTZMANN TRANSPORT-EQUATION IN AN ARBITRARY ONE-DIMENSIONAL-POTENTIAL PROFILE

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.