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3.4  Full-band Effects

In Monte Carlo simulators full-band effects can be integrated without further approximations in a straight-forward fashion  [25]. For a SHE of the BTE this is not possible, since the generalized density of states is only well-defined if there is a bijective mapping between k and energy per band (cf. Equation (3.6)). For silicon there is no straight-forward bijective mapping between energy and wave vector (cf. Figure 2.1) and thus approximations need to be applied. The simplest approximation for silicon is the parabolic band approximation or the Modena model together with the Herring-Vogt transform, both of which are bijective. Aside from these two approximations, other ways to include full-band effects have been investigated and are summarized in  [47]. In this work the anisotropic band model and the extended Vecchi model will be explained briefly. Both band models deliver the same amount of accuracy, whilst the extended Vecchi model requires less computational resources. A third technique to cover full-band effects is described in  [47]. In this model the first conduction band an anisotropic model is used, whilst for higher bands an isotropic model is employed. During the derivation of said model, the assumption of isotropic vallies for higher bands, generally throughout the derivation of the extended Vecchi model, is delayed to the last moment of the derivation. However, this third technique is more complex then the extended Vecchi model and will not be further discussed in this work in favour of the simpler extended Vecchi model.

3.4.1  The Anisotropic Band Model

The anisotropic band model, developed by  [72], was an attempt to increase the accuracy of the Modena model by taking the anisotropic nature of the bandstructure into account. The main idea of the anisotropic band model is to expand the inverse dispersion relation k(ϵ,θ,φ) for all bands per valley n into spherical harmonics,

kfitn(ϵ,θ,φ) = ∑ l=0L ∑ m=-llk l,mnY l,m(θ,φ), (3.30)
where the expansion coefficients kl,mn are obtained by Least-Squares fitting  [73, 74] to the real bandstructure. This expansion can then be inserted into Equation (3.6) to obtain the anisotropic density of states for the SHE equations. Although being numerically quite challenging, the model only delivers moderate accuracy as shown in  [47] and was thus not used in the course of this thesis.

3.4.2  The extended Vecchi Model

Building upon the Herring-Vogt transformed dispersion relation, Vecchi et al.  [58] found that for first-order SHE the full density of states and group velocity (cf. Figure 2.2) can directly be used in the equation assembly. This model has later been extended to arbitrary order SHE by Jin et al.  [48]. They found that under the assumption of spherically symmetric dispersion relations one can write

2Z --- ℏk = ∂vZ ---- ∂ ϵ (3.31)
and eliminate the dependence on k in Equation (3.28) and directly use the discretized density of states and group velocity. In their approach the resulting quantities, e.g. drift velocity, match better with the results from a full-band Monte Carlo simulaton than the original Vecchi model  [58]. In more detail, the derivation of the extended Vecchi model starts from the Herring-Vogt transformed density of states and group velocity
gν(E) and vg′(E) = ∘ ---- m-*c m *dvν g, (3.32)
where mc* is the conductivity effective mass and md* is the DOS effective mass  [48]. Now the generalized density of states Z is approximated such that ∂||k||∕∂ϵ disappears
Z = 2||k||2 ----3- (2π)∂||k|| ----- ∂ϵ ≈∂(gν(E )v′(E)) --------g----- ∂ ϵ. (3.33)
This assumption is only justified by numerical results and shows the same quality as the anisotropic band model, but requires significantly less computational time. Due to its minimal runtime penalty, this full-band model for SHE has been employed in the course of this thesis.

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