Discontinuous Galerkin Deterministic Solvers for a Boltzmann-Poisson Model of Hot Electron Transport by Averaged Empirical Pseudopotential Band Structures

TL;DR

This paper develops a discontinuous Galerkin solver for a Boltzmann-Poisson model of hot electron transport in silicon, incorporating a more accurate spherical average of empirical pseudopotential band structures to improve physical realism.

Contribution

It introduces a novel DG solver that uses a spherical average of empirical pseudopotential data for the conduction band, enhancing the accuracy of electron transport simulations.

Findings

Quantitative differences in mean velocity, average energy, and electric current compared to traditional models.

Improved physical description of electron group velocity and conduction energy band structure.

Successful comparison of the new model with classical analytic band models.

Abstract

The purpose of this work is to incorporate numerically, in a discontinuous Galerkin (DG) solver of a Boltzmann-Poisson model for hot electron transport, an electronic conduction band whose values are obtained by the spherical averaging of the full band structure given by a local empirical pseudopotential method (EPM) around a local minimum of the conduction band for silicon, as a midpoint between a radial band model and an anisotropic full band, in order to provide a more accurate physical description of the electron group velocity and conduction energy band structure in a semiconductor. This gives a better quantitative description of the transport and collision phenomena that fundamentally define the behaviour of the Boltzmann - Poisson model for electron transport used in this work. The numerical values of the derivatives of this conduction energy band, needed for the description of the electron group velocity, are obtained by means of a cubic spline interpolation. The EPM-Boltzmann-Poisson transport with this spherically averaged EPM calculated energy surface is numerically simulated and compared to the output of traditional analytic band models such as the parabolic and Kane bands, numerically implemented too, for the case of 1D $n^+-n-n^+$ silicon diodes with 400nm and 50nm channels. Quantitative differences are observed in the kinetic moments related to the conduction energy band used, such as mean velocity, average energy, and electric current (momentum).