Semiclassical limit in a simplified quantum energy-transport model for semiconductors

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December 2011, 4(4): 1049-1062. doi: 10.3934/krm.2011.4.1049

Semiclassical limit in a simplified quantum energy-transport model for semiconductors

Li Chen ^1, , Xiu-Qing Chen ^2, and Ansgar Jüngel ^3,

1.	Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China
2.	School of Sciences,, Beijing University of Posts & Telecommunications, Beijing, 100876, China
3.	Institute for Analysis and Scientiﬁc Computing, Vienna University of Technology, Wiedner Hauptstr. 8-10, 1040 Wien

Received June 2011 Revised July 2011 Published November 2011

Abstract
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The semiclassical limit in a quantum energy-transport model for semiconductors is proved. The system consists of a nonlinear parabolic fourth-order equation for the electron density, including temperature gradients; a degenerate elliptic heat equation for the electron temperature; and the Poisson equation for the electric potential. The equations are solved in a bounded domain with periodic boundary conditions. The asymptotic limit is based on a priori estimates independent of the scaled Planck constant, obtained from entropy functionals, on the use of Gagliardo-Nirenberg inequalities, and weak compactness methods.

Keywords: semiconductors., Quantum energy transport, degenerate elliptic equation, semiclassical limit, fourth-order equation.

Mathematics Subject Classification: Primary: 35B25, 35J40, 35Q40; Secondary: 82D3.

Citation: Li Chen, Xiu-Qing Chen, Ansgar Jüngel. Semiclassical limit in a simplified quantum energy-transport model for semiconductors. Kinetic & Related Models, 2011, 4 (4) : 1049-1062. doi: 10.3934/krm.2011.4.1049

References:

[1]	M. Ancona, Diffusion-drift modeling of strong inversion layers,, COMPEL, 6 (1987), 11.
[2]	A. Asenov, G. Slavcheva, A. Brown, J. Davies and S. Saini, Increase in the random dopant induced threshold fluctuations and lowering in sub-100 nm MOSFETs due to quantum effects: A 3-D density-gradient simulation study,, IEEE Trans. Electron Dev., 48 (2001), 722. doi: 10.1109/16.915703.
[3]	N. Ben Abdallah and P. Degond, On a hierarchy of macroscopic models for semiconductors,, J. Math. Phys., 37 (1996), 3306. doi: 10.1063/1.531567.
[4]	L. Chen and M. Dreher, Quantum semiconductor models,, in, 211 (2010), 1.
[5]	L. Chen and Q.-C. Ju, Existence of weak solution and semiclassical limit for quantum drift-diffusion model,, Z. Angew. Math. Phys., 58 (2007), 1. doi: 10.1007/s00033-005-0051-4.
[6]	L. Chen and Q.-C. Ju, The semiclassical limit in the quantum drift-diffusion equations with isentropic pressure,, Chin. Ann. Math. Ser. B, 29 (2008), 369. doi: 10.1007/s11401-007-0314-9.
[7]	R.-C. Chen and J.-L. Liu, A quantum corrected energy-transport model for nanoscale semiconductor devices,, J. Comput. Phys., 204 (2005), 131. doi: 10.1016/j.jcp.2004.10.006.
[8]	X.-Q. Chen and L. Chen, Initial time layer problem for quantum drift-diffusion model,, J. Math. Anal. Appl., 343 (2008), 64. doi: 10.1016/j.jmaa.2008.01.015.
[9]	P. Degond, S. Gallego and F. Méhats, On quantum hydrodynamic and quantum energy transport models,, Commun. Math. Sci., 5 (2007), 887.
[10]	P. Degond, S. Génieys and A. Jüngel, A system of parabolic equations in nonequilibrium thermodynamics including thermal and electrical effects,, J. Math. Pures Appl. (9), 76 (1997), 991. doi: 10.1016/S0021-7824(97)89980-1.
[11]	P. Degond, F. Méhats and C. Ringhofer, Quantum energy-transport and drift-diffusion models,, J. Stat. Phys., 118 (2005), 625. doi: 10.1007/s10955-004-8823-3.
[12]	H. Doebner and G. Goldin, On a general nonlinear Schrödinger equation admitting diffusion currents,, Phys. Lett. A, 162 (1992), 397. doi: 10.1016/0375-9601(92)90061-P.
[13]	M. Dreher and A. Jüngel, Compact families of piecewise constant functions in $L^p(0,T;B)$,, preprint, (2011).
[14]	U. Gianazza, G. Savaré and G. Toscani, The Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation,, Arch. Rat. Mech. Anal., 194 (2009), 133. doi: 10.1007/s00205-008-0186-5.
[15]	H. Grubin and J. Kreskovsky, Quantum moment balance equations and resonant tunnelling structures,, Solid-State Electr., 32 (1989). doi: 10.1016/0038-1101(89)90192-5.
[16]	M. P. Gualdani, A. Jüngel and G. Toscani, A nonlinear fourth-order parabolic equation with nonhomogeneous boundary conditions,, SIAM J. Math. Anal., 37 (2006), 1761. doi: 10.1137/S0036141004444615.
[17]	A. Jüngel, "Transport Equations for Semiconductors,", Lect. Notes Phys., 773 (2009).
[18]	A. Jüngel, Dissipative quantum fluid models,, to appear in Revista Mat. Univ. Parma, (2011).
[19]	A. Jüngel and D. Matthes, The Derrida-Lebowitz-Speer-Spohn equation: Existence, nonuniqueness, and decay rates of the solutions,, SIAM J. Math. Anal., 39 (2008), 1996. doi: 10.1137/060676878.
[20]	A. Jüngel and J.-P. Milišić, A simplified quantum energy-transport model for semiconductors,, Nonlin. Anal.: Real World Appl., 12 (2011), 1033. doi: 10.1016/j.nonrwa.2010.08.026.
[21]	A. Jüngel and R. Pinnau, Global non-negative solutions of a nonlinear fourth-order parabolic equation for quantum systems,, SIAM J. Math. Anal., 32 (2000), 760. doi: 10.1137/S0036141099360269.
[22]	M. Kostin, On the Schrödinger-Langevin equation,, J. Chem. Phys., 57 (1972), 3589. doi: 10.1063/1.1678812.
[23]	P. Markowich, C. Ringhofer and C. Schmeiser, "Semiconductor Equations,", Springer-Verlag, (1990). doi: 10.1007/978-3-7091-6961-2.
[24]	J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura Appl. (4), 146 (1987), 65.

show all references

References:

[1]	M. Ancona, Diffusion-drift modeling of strong inversion layers,, COMPEL, 6 (1987), 11.
[2]	A. Asenov, G. Slavcheva, A. Brown, J. Davies and S. Saini, Increase in the random dopant induced threshold fluctuations and lowering in sub-100 nm MOSFETs due to quantum effects: A 3-D density-gradient simulation study,, IEEE Trans. Electron Dev., 48 (2001), 722. doi: 10.1109/16.915703.
[3]	N. Ben Abdallah and P. Degond, On a hierarchy of macroscopic models for semiconductors,, J. Math. Phys., 37 (1996), 3306. doi: 10.1063/1.531567.
[4]	L. Chen and M. Dreher, Quantum semiconductor models,, in, 211 (2010), 1.
[5]	L. Chen and Q.-C. Ju, Existence of weak solution and semiclassical limit for quantum drift-diffusion model,, Z. Angew. Math. Phys., 58 (2007), 1. doi: 10.1007/s00033-005-0051-4.
[6]	L. Chen and Q.-C. Ju, The semiclassical limit in the quantum drift-diffusion equations with isentropic pressure,, Chin. Ann. Math. Ser. B, 29 (2008), 369. doi: 10.1007/s11401-007-0314-9.
[7]	R.-C. Chen and J.-L. Liu, A quantum corrected energy-transport model for nanoscale semiconductor devices,, J. Comput. Phys., 204 (2005), 131. doi: 10.1016/j.jcp.2004.10.006.
[8]	X.-Q. Chen and L. Chen, Initial time layer problem for quantum drift-diffusion model,, J. Math. Anal. Appl., 343 (2008), 64. doi: 10.1016/j.jmaa.2008.01.015.
[9]	P. Degond, S. Gallego and F. Méhats, On quantum hydrodynamic and quantum energy transport models,, Commun. Math. Sci., 5 (2007), 887.
[10]	P. Degond, S. Génieys and A. Jüngel, A system of parabolic equations in nonequilibrium thermodynamics including thermal and electrical effects,, J. Math. Pures Appl. (9), 76 (1997), 991. doi: 10.1016/S0021-7824(97)89980-1.
[11]	P. Degond, F. Méhats and C. Ringhofer, Quantum energy-transport and drift-diffusion models,, J. Stat. Phys., 118 (2005), 625. doi: 10.1007/s10955-004-8823-3.
[12]	H. Doebner and G. Goldin, On a general nonlinear Schrödinger equation admitting diffusion currents,, Phys. Lett. A, 162 (1992), 397. doi: 10.1016/0375-9601(92)90061-P.
[13]	M. Dreher and A. Jüngel, Compact families of piecewise constant functions in $L^p(0,T;B)$,, preprint, (2011).
[14]	U. Gianazza, G. Savaré and G. Toscani, The Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation,, Arch. Rat. Mech. Anal., 194 (2009), 133. doi: 10.1007/s00205-008-0186-5.
[15]	H. Grubin and J. Kreskovsky, Quantum moment balance equations and resonant tunnelling structures,, Solid-State Electr., 32 (1989). doi: 10.1016/0038-1101(89)90192-5.
[16]	M. P. Gualdani, A. Jüngel and G. Toscani, A nonlinear fourth-order parabolic equation with nonhomogeneous boundary conditions,, SIAM J. Math. Anal., 37 (2006), 1761. doi: 10.1137/S0036141004444615.
[17]	A. Jüngel, "Transport Equations for Semiconductors,", Lect. Notes Phys., 773 (2009).
[18]	A. Jüngel, Dissipative quantum fluid models,, to appear in Revista Mat. Univ. Parma, (2011).
[19]	A. Jüngel and D. Matthes, The Derrida-Lebowitz-Speer-Spohn equation: Existence, nonuniqueness, and decay rates of the solutions,, SIAM J. Math. Anal., 39 (2008), 1996. doi: 10.1137/060676878.
[20]	A. Jüngel and J.-P. Milišić, A simplified quantum energy-transport model for semiconductors,, Nonlin. Anal.: Real World Appl., 12 (2011), 1033. doi: 10.1016/j.nonrwa.2010.08.026.
[21]	A. Jüngel and R. Pinnau, Global non-negative solutions of a nonlinear fourth-order parabolic equation for quantum systems,, SIAM J. Math. Anal., 32 (2000), 760. doi: 10.1137/S0036141099360269.
[22]	M. Kostin, On the Schrödinger-Langevin equation,, J. Chem. Phys., 57 (1972), 3589. doi: 10.1063/1.1678812.
[23]	P. Markowich, C. Ringhofer and C. Schmeiser, "Semiconductor Equations,", Springer-Verlag, (1990). doi: 10.1007/978-3-7091-6961-2.
[24]	J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura Appl. (4), 146 (1987), 65.

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