### Abstract

The Boltzmann equation for transport in semiconductors is projected onto spherical harmonics in such a way that the resultant balance equations for the coefficients of the distribution function times the generalized density of states can be discretized over energy and real spaces by box integration. This ensures exact current continuity for the discrete equations. Spurious oscillations of the distribution function are suppressed by stabilization based on a maximum entropy dissipation principle avoiding the H transformation. The derived formulation can be used on arbitrary grids as long as box integration is possible. The approach works not only with analytical bands but also with full band structures in the case of holes. Results are presented for holes in bulk silicon based on a full band structure and electrons in a Si NPN bipolar junction transistor. The convergence of the spherical harmonics expansion is shown for a device, and it is found that the quasiballistic transport in nanoscale devices requires an expansion of considerably higher order than the usual first one. The stability of the discretization is demonstrated for a range of grid spacings in the real space and bias points which produce huge gradients in the electron density and electric field. It is shown that the resultant large linear system of equations can be solved in a memory efficient way by the numerically robust package ILUPACK.

Original language | English (US) |
---|---|

Article number | 024502 |

Journal | Journal of Applied Physics |

Volume | 100 |

Issue number | 2 |

DOIs | |

State | Published - 2006 |

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### ASJC Scopus subject areas

- Physics and Astronomy (miscellaneous)
- Physics and Astronomy(all)

### Cite this

*Journal of Applied Physics*,

*100*(2), [024502]. https://doi.org/10.1063/1.2212207

**Stable discretization of the Boltzmann equation based on spherical harmonics, box integration, and a maximum entropy dissipation principle.** / Jungemann, C.; Pham, J. A T; Meinerzhagen, B.; Ringhofer, Christian; Bollhöfer, M.

Research output: Contribution to journal › Article

*Journal of Applied Physics*, vol. 100, no. 2, 024502. https://doi.org/10.1063/1.2212207

}

TY - JOUR

T1 - Stable discretization of the Boltzmann equation based on spherical harmonics, box integration, and a maximum entropy dissipation principle

AU - Jungemann, C.

AU - Pham, J. A T

AU - Meinerzhagen, B.

AU - Ringhofer, Christian

AU - Bollhöfer, M.

PY - 2006

Y1 - 2006

N2 - The Boltzmann equation for transport in semiconductors is projected onto spherical harmonics in such a way that the resultant balance equations for the coefficients of the distribution function times the generalized density of states can be discretized over energy and real spaces by box integration. This ensures exact current continuity for the discrete equations. Spurious oscillations of the distribution function are suppressed by stabilization based on a maximum entropy dissipation principle avoiding the H transformation. The derived formulation can be used on arbitrary grids as long as box integration is possible. The approach works not only with analytical bands but also with full band structures in the case of holes. Results are presented for holes in bulk silicon based on a full band structure and electrons in a Si NPN bipolar junction transistor. The convergence of the spherical harmonics expansion is shown for a device, and it is found that the quasiballistic transport in nanoscale devices requires an expansion of considerably higher order than the usual first one. The stability of the discretization is demonstrated for a range of grid spacings in the real space and bias points which produce huge gradients in the electron density and electric field. It is shown that the resultant large linear system of equations can be solved in a memory efficient way by the numerically robust package ILUPACK.

AB - The Boltzmann equation for transport in semiconductors is projected onto spherical harmonics in such a way that the resultant balance equations for the coefficients of the distribution function times the generalized density of states can be discretized over energy and real spaces by box integration. This ensures exact current continuity for the discrete equations. Spurious oscillations of the distribution function are suppressed by stabilization based on a maximum entropy dissipation principle avoiding the H transformation. The derived formulation can be used on arbitrary grids as long as box integration is possible. The approach works not only with analytical bands but also with full band structures in the case of holes. Results are presented for holes in bulk silicon based on a full band structure and electrons in a Si NPN bipolar junction transistor. The convergence of the spherical harmonics expansion is shown for a device, and it is found that the quasiballistic transport in nanoscale devices requires an expansion of considerably higher order than the usual first one. The stability of the discretization is demonstrated for a range of grid spacings in the real space and bias points which produce huge gradients in the electron density and electric field. It is shown that the resultant large linear system of equations can be solved in a memory efficient way by the numerically robust package ILUPACK.

UR - http://www.scopus.com/inward/record.url?scp=33746818225&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33746818225&partnerID=8YFLogxK

U2 - 10.1063/1.2212207

DO - 10.1063/1.2212207

M3 - Article

AN - SCOPUS:33746818225

VL - 100

JO - Journal of Applied Physics

JF - Journal of Applied Physics

SN - 0021-8979

IS - 2

M1 - 024502

ER -