2.5  The Spherical Harmonics Expansion Method

Inherent numerical noise in the Monte Carlo solutions of the Boltzmann Transport Equation, as well as the (N^-1∕2) dependence of the integration error inherent in Monte Carlo methods has motivated the development of new techniques to solve Boltzmann’s Transport Equation. Thus special attention has been devoted to deterministic efforts for solving Boltzmann’s Equation. A very attractive deterministic method is to expand the distribution function into spherical harmonics and to project the k-space onto a single energy axis. This way a seven dimensional system is reduced to a five dimensional system, which does not have the limitations of a Monte Carlo approach. Nevertheless, a spherical harmonics expansion (SHE) of the BTE is still challenging  [46] and it is, compared to Monte Carlo, difficult to consider as many full-band effects in the simulation  [47] as possible. However, a quite statisfying approach to consider full-band effects has been found by  [48], where current of diodes could be predicted using a fifth order expansion within an error margin of 8% in saturation compared to the results of a full-band Monte Carlo simulation. In addition to being free of stochastic noise, a spherical harmonics expansion allows to straightforward and self-consistently solve the BTE with any other partial differential equation, especially Poisson’s Equation using a Newton-Raphson solver  [49]. Thus special attention will be paid to the SHE method in Chapter 3 and for hot-carrier degradation modelling in the last chapter of this thesis.