A generalized Legendre polynomial/sparse matrix approach for determining the distribution function in non-polar semiconductors

Abstract

Until recently, the Legendre polynomial (LP) expansion method for solving the Boltzmann transport equation was limited to the use of two or three Legendre polynomials. In this work we generalize the method to include an arbitrarily high order LP expansion. The expansion method consists of representing the angular dependence of the distribution function about the field direction in terms of an infinite series of Legendre polynomials with unknown coefficients. The expansion is then substituted into the Boltzmann transport equation. With the use of orthogonality and the LP recurrence relations, an infinite system of equations is then generated from the original Boltzmann equation. This system is then solved numerically, using sparse matrix algebra, for the unknown coefficients of the LP expansion. Once the coefficient are determined, the complete distribution function is readily constructed. In an example calculation the Boltzmann equation is solved to 40th order of the LP expansion. Finally, resulting values for the energy distribution, as well as average energy and average velocity, are shown to agree with Monte Carlo simulation results.