Analytic solution of the Boltzmann equation with applications to electron transport in inhomogeneous semiconductors

Abstract

An analytic solution has been obtained to the one-dimensional Boltzmann transport equation in the relaxation-time approximation that is exact to first order in an arbitrary position- and time-dependent potential. This is the first practical method for studying time-dependent transport in inhomogeneous systems. The solution is correct for general relaxation times and arbitrarily large external electric fields, including the ballistic limit. This solution is used to study steady-state transport in two submicron structures: a doping superlattice, where the doping density is a sinusoidal function of position, and an $N^{+}$ $N^{\mathrm{-}}$ $N^{+}$ GaAs junction. The distribution function for the $N^{+}$ $N^{\mathrm{-}}$ $N^{+}$ device exhibits a ballistic peak by a new mechanism, in a regime where the potential has no local maximum to ‘‘skim’’ electrons. At zero temperature, the ballistic peak is a true singularity, rather than just a qualitative feature. Evidence is presented that an inhomogeneous potential causes a coherent excitation of plasmons. Analytic solutions have also been obtained, within these approximations, for initially nonequilibrium carrier distributions in the presence of potentials with arbitrary time and position dependence.