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

C. Jungemann, J. A T Pham, B. Meinerzhagen, Christian Ringhofer, M. Bollhöfer

Research output: Contribution to journalArticle

59 Citations (Scopus)

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 languageEnglish (US)
Article number024502
JournalJournal of Applied Physics
Volume100
Issue number2
DOIs
StatePublished - 2006

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spherical harmonics
boxes
dissipation
entropy
distribution functions
grids
junction transistors
expansion
linear systems
bipolar transistors
continuity
stabilization
spacing
formulations
gradients
oscillations
electric fields
silicon
coefficients
electrons

ASJC Scopus subject areas

  • Physics and Astronomy (miscellaneous)
  • Physics and Astronomy(all)

Cite this

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.

In: Journal of Applied Physics, Vol. 100, No. 2, 024502, 2006.

Research output: Contribution to journalArticle

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