3.2  The H-Transform

The spherical harmonics expansion applied to the BTE yields a set of equations for the unknown expansion coefficients fl,m in real space and energy. For the assembly of this set of equations, numeric stabilization techniques, such as the H-transform  [6968], are needed  [67]. The H-transform is a linear transformation of coordinates, where the kinetic energy ϵ is translated to the total energy H by

    {
H =   ϵ-  ∥q∥φ(x,t), for electrons,
      ϵ+  ∥q∥φ(x,t), for holes.
(3.24)

The electrostatic potential φ(x,t) used in the transformation is obtained from a solution of Poisson’s equation (cf. Figure 3.1).


PIC

Figure 3.1: Sketch of the H-grid after an H-transform. The grid starts (black lines) at the band edge (thick red line) and is uniformly spaced.

In a SHE of the BTE, the H-transform is used to eliminate the derivative with respect to energy in Equation (3.12), rendering the set of equations numerically stable. Although the H-transform simplifies the free streaming operator, it unfortunately results in a potential-dependent energy grid, which needs to be recalculated during each iteration of the self-consistent solution process. Thus, with the H-transform and Equation (3.2) the full Boltzmann Transport Equation by collecting each term from Equation (3.10) to Equation (3.13) for either electrons or holes at (x,H,t) reads

∑   (  ′ ′           ′ ′               ′ ′           ′ ′   )
     Zll,,mm ∂tfl,m + qZll,,mm ∂φ-∂fl,m-+  ∇xjll,,mm fl,m - F ⋅All,,mm fl,m
l′,m′                    ∂t  ∂H
      1  ∑  (                                             ′ ′   )
  =  ----    Zl′,m ′(H )σ(H ′,H )Z0,l,m0fl,m - Z0,0(H ′)σ(H,H ′)Zll,,mm fl,m  ,
     Y0,0 l′,m ′
(3.25)

where the shorthands

Zl,ml,m(ϵ) = Y l,mY l,mZ(ϵ)dΩ, (3.26)
jl,ml,m(ϵ) = Y l,mvgY l,mZ(ϵ)dΩ, (3.27)
Al,ml,m(ϵ) = --1--
ℏ||k||(                      )
  ∂Yl′,m′e +  -∂Yl′,m′-e
    ∂θ   θ   sin(θ)∂φ  φY l,mZ(ϵ)dΩ, (3.28)
from  [4767] have been used.