Solution Of The Boltzmann Transport Equation Using Spherical Harmonics Expansions

Recently，because of the fast development of semiconductor technology,theinformation technology industry has been largely promoted. Complex and highlyrepetitive design process of semiconductor devices can be accomplished bysemiconductor numerical simulation technology, which can highly improve theefficiency of semiconductor devices development. The methods of spherical harmonicexpansion for directly solving Boltzmann transport equation have been widely used insemiconductor device simulation. the semi-classical Boltzmann equation can accuratelydescribe the microscopic behavior of electrons, and by using the solved distributionfunction, the physical characteristics of the semiconductor devices including the carrierconcentration, current density and electric field can be calculated. That’s why thismethod is feasible and loved by the majority of scholars. There are three commonlyused spherical harmonic expansion method, which are the term-matching method,Galerkin method and projection method.This paper mainly focuses on the spherical expansion methods for directly solvingthe Boltzmann equation. In particular, using the Galerkin method and combined withthe one-dimensional Poisson equation, the numerical simulation ofN+NN+structureis completed. In the process, considering phonon scattering and parabolic single siliconband structure, the distribution function is expanded to the first order sphericalharmonic function, and equations are dispersed on two-dimensional uniform rectangulargrids. Finally, linear equations are solved by the uncoupled method, and the electrondensity inN+NN+structure can be obtained by the integral of zero-order distributionfunction expansion coefficients in the entire energy space. In addition, by using theGeneral-purpose Semiconductor Simulator, the electric potential distribution can beobtained, which will be an input parameter in the C++program of Galerkin method forsolving the Boltzmann equation.The main work and innovation in this paper are as follows:1. Expound the physical basis, numerical method and numerical stabilitytechnology which involved in the simulation process of spherical harmonic expansion methods solving for semiconductor devices.2. Derive the solving process of term-matching method, Galerkin method andprojection method. Moreover, sum up the similarities and differences of the threemethods, and their advantages, disadvantages as well as the scope of application.3. Provide the flow of numerically solving one-dimensional semi-classicalBoltzmann equation using the Galerkin method, including the establishment of physicalequations, meshing, numerical discretization, imposed boundary conditions andnon-coupling method to solve higher-order equations whose unknown quantity is thedistribution function.4. Combined with one-dimensional Poisson equation, the Galerkin method canhelp to solve the potential distribution, electron concentration and distribution function.Realize the above solving process by using the C++programming language in VS2010environment. Compare the results with simulation results of GSS software and solvingresults published in other literature.