A GENERALIZED LEGENDRE POLYNOMIAL SPARSE-MATRIX APPROACH FOR DETERMINING THE DISTRIBUTION FUNCTION IN NONPOLAR SEMICONDUCTORS

Until recently, the Legendre polynomial (LP) expansion method for solv ing the Boltzmann transport equation was limited to the use of two or three Legendre polynomials. In this work we generalize the method to i nclude an arbitrarily high order LP expansion. The expansion method co nsists of representing the angular dependence of the distribution func tion about the field direction in terms of an infinite series of Legen dre polynomials with unknown coefficients. The expansion is then subst ituted into the Boltzmann transport equation. With the use of orthogon ality and the LP recurrence relations, an infinite system of equations is then generated from the original Boltzmann equation. This system i s then solved numerically, using sparse matrix algebra, for the unknow n coefficients of the LP expansion. Once the coefficients are determin ed, the complete distribution function is readily constructed. In an e xample calculation the Boltzmann equation is solved to 40th order of t he LP expansion. Finally, resulting values for the energy distribution , as well as average energy and average velocity, are shown to agree w ith Monte Carlo simulation results.