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

Journal of Applied Physics

Volumn 100, Issue 2, 2006, Pages

  Jungemann, C ^a   Pham, J A T ^b   Meinerzhagen, B ^b   Ringhofer, C ^c   Bollhofer M ^d  

^a BUNDESWEHR UNIVERSITY MUNICH   (Germany)

^b TECHNICAL UNIVERSITY OF BRAUNSCHWEIG   (Germany)

^c ARIZONA STATE UNIVERSITY   (United States)

^d TECHNISCHE UNIVERSITÄT BERLIN   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

BOLTZMANN EQUATION; BOX INTEGRATION; SPHERICAL HARMONICS;

CARRIER CONCENTRATION; ELECTRIC CURRENT CONTROL; ELECTRIC FIELD EFFECTS; ENTROPY; FINITE DIFFERENCE METHOD; HARMONIC ANALYSIS; INTEGRATION; NANOSTRUCTURED MATERIALS; OSCILLATIONS; TIME MEASUREMENT; TRANSPORT PROPERTIES;

SEMICONDUCTOR MATERIALS;

EID: 33746818225     PISSN: 00218979     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.2212207     Document Type: Article

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53. This definition does not include the integration over the solid angle. The generalized DOS is therefore for the case that it does not depend on the angles by a factor of 4π smaller than the conventional expression.
54. Equation (46) can be also derived for/ instead of g by neglecting the delta function in Eq. (17) and integrating only over the solid angle, but the resultant system of equations does not conserve the particle charge exactly. Due to box integration a certain density defined with f will be conserved exactly, but because the generalized DOS is missing, this density does not correspond to the particle density.
55. This stabilization scheme should not be confused with the so-called Scharfetter-Gummel scheme presented in Ref. 18, which was derived by enforcing continuity in the real space for the homogeneous part of the balance equation for l=m=0 and which can be applied only to a first order expansion. In contrast to Ref. 18 in our approach current continuity is ensured by box integration and the Scharfetter-Gummel scheme is used to stabilize the equations similar to the case of the DD model Ref. 31.