An efficient solution of the Boltzmann transport equation which includes the Pauli exclusion principle

Abstract

Nonequilibrium distribution functions for both degenerate and nondegenerate electron ensembles are obtained from efficient solutions to a Boltzmann transport equation for Si. The distribution function is taken to be expressible as a Legendre polynomial expansion, which is substituted into a Boltzmann equation that incorporates the effects of nonparabolic band structure, inelastic phonon scattering and the Pauli exclusion principle. The resulting Boltzmann equation takes on a nonlinear form which is quickly solved by means of the numerical Newton-Raphson method with appropriate boundary conditions. The resulting distribution function approaches the Fermi-Dirac function in the limit of high electron concentration and low electric field. In low probability ranges, which are found at high energies, the distribution function takes on the character of the Boltzmann distribution. Using average energy as a guideline, the values of electric field and carrier concentration which require consideration of the Pauli exclusion principle in electron transport investigations for Si are provided. In addition to providing a technique for investigating the effects of degeneracy, the efficiency of the new method facilitates its use in CAD tools for semiconductor devices.