Abstract
In this chapter, various numerical techniques for stable device simulations are discussed in detail. The H-transformation and the maximum entropy dissipation scheme, which are two key ingredients for a stable higher-order SHE simulation, are expounded. Moreover, in the special case of the lowest order expansion, it is explicitly shown that the Jacobian matrix of the resultant set of equations is a non-singular M-matrix. Therefore, the non-negativeness of the solution (the isotropic component of the particle distribution function) is guaranteed. Dimensional splitting and box integration are applied. Discretization of the boundary conditions is also discussed. At the end of this chapter, stabilization of the linearized equation system for the small-signal analysis is presented.
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Notes
- 1.
This definition is used throughout the book besides Sect. 3.6.
- 2.
The free-streaming operator due to the magnetic force does not have this property of even-odd coupling as shown in [8].
- 3.
For example, this property holds for isotropic band models such as the Modena model [11], which has been extensively discussed in [8]. Moreover, the dimensional splitting used in the odd free-streaming operator effectively avoids the coupling terms between different directions.
- 4.
When the absolute value of the diagonal component is larger than the absolute sum of the off-diagonal components of this matrix, the diagonal dominance strictly holds.
- 5.
In the three dimensional real space with a Cartesian grid, there are six adjoint nodes surrounding the given direct node.
- 6.
The Pauli principle is neglected in this analysis, because it makes the BTE nonlinear, therefore, the direct relation between the linear response and the physical solution is lost. Also velocity-randomizing scattering considered in ( 2.134 ) and ( 2.135 ) is assumed.
- 7.
In our implementation, impact ionization scattering does not satisfy this relation. However, impact ionization events are usually very rare and the degradation of stability is thus negligible.
- 8.
A pair of states with energies ε − ℏωη and ε yields the opposite result. By the scaling shown in (3.35), this pair also gives exactly balanced contributions to the Jacobian matrix.
- 9.
For a Cartesian tensor grid, where n is aligned to e x or e y , this procedure results in a trivial mapping.
- 10.
The scaling shown in (3.35) is to be used.
- 11.
Compared with (3.11), l and l ′ are interchanged in order to be consistent with our parity convention.
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Hong, SM., Pham, AT., Jungemann, C. (2011). Device Simulation. In: Deterministic Solvers for the Boltzmann Transport Equation. Computational Microelectronics. Springer, Vienna. https://doi.org/10.1007/978-3-7091-0778-2_3
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