3  Spherical Harmonics Expansion

Recent advances in the spherical harmonics expansion (SHE) method of the Boltzmann transport equation (BTE) allow for the accurate solution of the BTE on arbitrary three-dimensional device geometries  [57], full-band effects  [485859], rare scattering events (e.g. impact ionization)  [60], carrier-carrier scattering  [61], charge carrier recombination/generation  [62], small signal analysis  [6347], quantum mechanics  [6449], and hot carrier degradation  [46]. The SHE method has been developed to a point where it is now an attractive alternative to the common Monte Carlo (MC) method, which has a square root dependence of its accuracy on CPU time  [44]. As discussed in Section 2.4, the Monte Carlo method suffers from inherent stochastic noise in the solution and the requirement of sufficiently small time steps to achieve self-consistency with Poisson’s equation. An expansion of the BTE using spherical harmonics does not impose such restrictions, since it is a deterministic approach. In the course of this work the SHE simulator ViennaSHE  [65] has been extended and used to solve BTE (cf. Equation (2.27)).

 3.1  Theory
  3.1.1  Expansion of the Free-Streaming Operator
  3.1.2  Expansion of the Scattering-Streaming Operator
 3.2  The H-Transform
 3.3  Discretization
 3.4  Full-band Effects
  3.4.1  The Anisotropic Band Model
  3.4.2  The extended Vecchi Model
 3.5  Recombination and Generation
 3.6  Time-dependent SHE of the BTE
  3.6.1  Comparison to Drift Diffusion
  3.6.2  Stability
  3.6.3  Energy Grid Interpolation
  3.6.4  Probable Violation of Gauss’ Law
  3.6.5  The Shockley-Haynes Experiment