Electron transport in semiconductor superlattices
Authors:
N. Ben Abdallah, P. Degond, A. Mellet and F. Poupaud
Journal:
Quart. Appl. Math. 61 (2003), 161-192
MSC:
Primary 82D37; Secondary 35B40, 47N20, 76P05, 76X05
DOI:
https://doi.org/10.1090/qam/1955228
MathSciNet review:
MR1955228
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Abstract: In this paper, we rigorously derive a diffusion model for semiconductor superlattices, starting from a kinetic description of electron transport at the microscopic scale. Electron transport in the superlattice is modelled by a collisionless Boltzmann equation subject to a periodic array of localized scatters modeling the periodic heterogeneities of the material. The limit of a large number of periodicity cells combined with a large-time asymptotics leads to a homogenized diffusion equation which belongs to the class of so-called “SHE” models (for Spherical Harmonics Expansion). The rigorous convergence proof relies on fine estimates on the operator modeling the localized scatters.
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C. Weisbuch and B. Vinter, Quantum Semiconductor Structures, fundamentals and applications, Academic Press, Boston, 1991
R. Alexandre and K. Hamdache, Homogenization of kinetic equations in nonhomogeneous medium, preprint
G. Allaire, Homogenization and two scale convergence, SIAM Math. Anal. 23, 1482–1518 (1992)
G. Allaire and G. Bal, Homogenization of the criticality spectral equation in neutron transport, Math. Model Numer. Anal. 33, 721–746 (1999)
C. Bardos, Problémes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; Théorèmes d’approximation; application à l’équation de transport, Ann. Scient. École Norm. Sup. 4, 185–233 (1970)
H. Babovsky, C. Bardos, and T. Platkowski, Diffusion approximation for a Knudsen gas in a thin domain with accommodation on the boundary, Asymptotic Analysis 3, 265–289, (1991)
C. Bardos, F. Golse, and B. Perthame, The Rosseland approximation for the radiative transfer equations, Comm. Pure Appl. Math. 40, 691–721 (1987) and 42, 891–894 (1989)
C. Bardos, R. Santos, and R. Sentis, Diffusion approximation and computation of the critical size, Trans. Amer. Math. Soc. 284, 617–649 (1984)
N. Ben Abdallah and P. Degond, On a hierarchy of macroscopic models for semiconductors, J. Math. Phys. 37, 3306–3333 (1996)
A. Bensoussan, J. L. Lions, and G. Papanicolaou, Boundary layers and homogenization of transport processes, Publ. RIMS Kyoto Univ. 15, 53–157 (1979)
H. Brézis, Analyse Fonctionnelle, théorie et applications, Masson, Paris, 1983
C. Cercignani, The Boltzmann equation and its applications, Springer-Verlag, New York, 1998
P. Degond, Mathematical modelling of microelectronics semiconductor devices. Some current topics on nonlinear conservation laws, AMS/IP Studies in Advanced Mathematics, vol. 15, Amer. Math. Soc. and International Press, 2000, pp. 77–110
P. Degond, A model of near-wall conductivity and its application to plasma thrusters, SIAM J. Appl. Math. 58, 1138–1162 (1998)
P. Degond and S. Mancini, Diffusion driven by collisions with the boundary, Asymptotic Anal. 27, 47–73 (2001)
P. Degond and K. Zhang, Diffusion approximation of a scattering matrix model of semiconductor superlattices, preprint, 1999
P. Dmitruk, A. Saul, and L. Reyna, High electric field approximation in semiconductor devices, Appl. Math. Letters 5, 99–102, (1992)
W. E, Homogenization of linear and nonlinear transport equations, Comm. Pure Appl. Math. 45, 301–326 (1992)
L. Esaki and R. Tsu, Superlattice and negative differential conductivity in semiconductors, IBM J. Res. Develop. 14, 61 (1970)
H. Federer, Geometric Measure Theory, Springer-Verlag, Berlin, 1969
E. Frenod and K. Hamdache, Homogenisation of transport kinetic equations with oscillating potentials, Proc. Roy. Soc. Edinburgh Sect. A 126, 1247–1275 (1996)
V. Girault and P. A. Raviart, Finite element methods for the Navier-Stokes equations, Springer-Verlag, Berlin, 1986
N. Goldsman, L. Henrickson, and J. Frey, A physics based analytical numerical solution to the Boltzmann transport equation for use in device simulation, Solid State Electron. 34, 389–396 (1991)
F. Golse and F. Poupaud, Limite fluide des équations de Boltzmann des semiconducteurs pour une statistique de Fermi-Dirac, Asymptotic Analysis 6, 135–160 (1992)
T. Goudon and F. Poupaud, Approximation by homogenization and diffusion of kinetic equations, Comm. Partial Differential Equations 26, 537–569 (2001)
H. T. Grahn (ed.), Semiconductor Superlattices, growth and electronic properties, World Scientific, Singapore, 1995
E. W. Larsen, Neutron transport and diffusion in inhomogeneous media I, J. Math. Phys. 16, 1421–1427 (1975)
E. W. Larsen, Neutron transport and diffusion in inhomogeneous media II, Nuclear Sci. Engrg. 60, 357–368 (1976)
E. W. Larsen and J. B. Keller, Asymptotic solution of neutron transport problems for small mean free paths, J. Math. Phys. 15, 75–81 (1974)
G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM Math. Anal. 20, 608–623 (1989)
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983
F. Poupaud, Diffusion approximation of the linear semiconductor equation: Analysis of boundary layers, Asymptotic Analysis 4, 293–317 (1991)
F. Poupaud, Étude de l’opérateur de transport $Au = a\nabla u$, manuscript, unpublished
D. Ventura, A. Gnudi, G. Baccarani, and F. Odeh, Multidimensional spherical harmonics expansion of Boltzmann equation for transport in semiconductors, Appl. Math. Letters 5, 85–90 (1992)
C. Weisbuch and B. Vinter, Quantum Semiconductor Structures, fundamentals and applications, Academic Press, Boston, 1991
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