Investigating the Numerical Solution of the BoltzmannTransport Equation in Silicon in Momentum Space Using Computational Systems of Different Dimensions

Abstract

The Boltzmann transport equation is the basic equation for solving the transport of charge carrier (electrons, holes) problems in semiconductor devices. The distribution function has been obtained from the solution of this equation. The distribution function is important in calculating semiconductor properties, which can be used to calculate the average electron energy, the charge carrier concentration, and other properties. In this work the semi-classical Boltzmann transport equation in silicon was solved using analytical / numerical methods in steady state case in momentum space. The analytical solution is requires expressing the distribution function using Legendre polynomials expansion the first two terms of the expansion, by taking into account the effect of both acoustic elastic scattering and nonelastic scattering in addition to the effect of non-parabolic energy band structure. In order to obtain the numerical solution of Boltzmann transport equation the finite difference method is used. The differential equation is transformed to linear difference equation which can represented by matrices. Numerical systems with different dimensions are designed to calculate the distribution function with the least possible time to maintain the accuracy of the solution for different applied electric field which represent the low and high field regions at temperatures T= (77, 300) K. The obtained results showed good agreement with published data that used other calculation methods such as the Monte Carlo simulation method for all the system used in this work.