Numerical methods for kinetic equations in semiconductor superlattices

We compare three different numerical methods for solving the Boltzmann-Poisson kinetic equation describing electron transport in semiconductor superlattices. The associated initial-boundary value problem is computationally intensive, and it requires the use of efficient and accurate numerical methods and a large integration time to observe the Gunn-type self-oscillations of the current that characteristically appear among its stable solutions. The first two numerical methods solve the kinetic equation using finite differences and particles, respectively. The third method solves by finite differences a less costly drift-diffusion partial differential equation that can be derived from the Boltzmann-Poisson equation using the Chapman-Enskog perturbation method. We show the convergence of the methods by means of numerical simulations with parameter values corresponding to superlattices used in experiments. Comparing the results obtained with the three methods for a wide miniband superlattice used in experiments (for which the small dimensionless parameter in the Chapman-Enskog expansion is about 0.15), we show that the error of the Chapman-Enskog method is less than 0.8% despite a ten times shorter computation time. Thus, for this superlattice, the Chapman-Enskog perturbation method provides a very accurate solution with very low computational cost compared with directly solving the kinetic equation by either finite differences or particles methods.