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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  [69, 68], 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′,m′Y l,mZ(ϵ)dΩ, (3.26)
jl,ml′,m′(ϵ) = ∮ Y l′,m′vgY 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  [47, 67] have been used.

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