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Open AccessArticle

Maximum Entropy Closure of Balance Equations for Miniband Semiconductor Superlattices

G. Millán Institute for Fluid Dynamics, Nanoscience and Industrial Mathematics, Universidad Carlos III de Madrid, Avenida de la Universidad 30, Leganés 28911, Spain
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Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Academic Editors: Vittorio Romano and Giovanni Mascali
Entropy 2016, 18(7), 260; https://doi.org/10.3390/e18070260
Received: 27 May 2016 / Revised: 1 July 2016 / Accepted: 11 July 2016 / Published: 14 July 2016
(This article belongs to the Special Issue Maximum Entropy Principle and Semiconductors)
Charge transport in nanosized electronic systems is described by semiclassical or quantum kinetic equations that are often costly to solve numerically and difficult to reduce systematically to macroscopic balance equations for densities, currents, temperatures and other moments of macroscopic variables. The maximum entropy principle can be used to close the system of equations for the moments but its accuracy or range of validity are not always clear. In this paper, we compare numerical solutions of balance equations for nonlinear electron transport in semiconductor superlattices. The equations have been obtained from Boltzmann–Poisson kinetic equations very far from equilibrium for strong fields, either by the maximum entropy principle or by a systematic Chapman–Enskog perturbation procedure. Both approaches produce the same current-voltage characteristic curve for uniform fields. When the superlattices are DC voltage biased in a region where there are stable time periodic solutions corresponding to recycling and motion of electric field pulses, the differences between the numerical solutions produced by numerically solving both types of balance equations are smaller than the expansion parameter used in the perturbation procedure. These results and possible new research venues are discussed. View Full-Text
Keywords: maximum entropy principle; Chapman–Enskog method; Boltzmann–Poisson kinetic equation; balance equation; semiconductor superlattices maximum entropy principle; Chapman–Enskog method; Boltzmann–Poisson kinetic equation; balance equation; semiconductor superlattices
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Bonilla, L.L.; Carretero, M. Maximum Entropy Closure of Balance Equations for Miniband Semiconductor Superlattices. Entropy 2016, 18, 260.

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