A Godunov-type stabilization scheme for the Boltzmann transport equation of III-V devices with a 3D k-space

Abstract

This paper presents a deterministic approach for solving the Boltzmann transport equation (BTE) together with the Poisson equation (PE) for III-V semiconductor devices with a three-dimensional k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf {k}}$$\\end{document}-space. The BTE is stabilized using Godunov’s scheme, whose linearity in the distribution function simplifies the application of the Newton–Raphson method to the coupled discrete BTE and PE. The formulation of the discrete equations ensures the nonnegativity of the distribution function regardless of the scattering rate, which can include the Pauli exclusion principle, and exhibits excellent numerical stability under steady state as well as transient conditions. In the latter case, both implicit and explicit time integration methods can be used and even slow processes (e.g., recombination) can be handled using this approach. In addition, the direct solution of the BTE can be easily extended to the small-signal case for arbitrary frequencies. Exemplary BTE results are shown for a GaAs N+NN+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extrm{N}}^{+}{\ extrm{NN}}^{+}$$\\end{document}-structure, revealing, inter alia, that the approximations of the drift-diffusion model can fail for large built-in fields in III-V devices.