# American Institute of Mathematical Sciences

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

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

 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

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.
Citation: Li Chen, Xiu-Qing Chen, Ansgar Jüngel. Semiclassical limit in a simplified quantum energy-transport model for semiconductors. Kinetic and 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), 11-18. [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), 722-729. 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), 3306-3333. doi: 10.1063/1.531567. [4] L. Chen and M. Dreher, Quantum semiconductor models, in "Partial Differential Equations and Spectral Theory" (eds. M. Demuth, B.-W. Schulze and I. Witt), Operator Theory: Advances and Applications, 211 (2010), 1-72. [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), 1-15. 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), 369-384. 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), 131-156. 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), 64-80. 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), 887-908. [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), 991-1015. 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), 625-667. 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), 397-401. 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, TU Wien, 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), 133-220. 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), 1701. 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), 1761-1779. doi: 10.1137/S0036141004444615. [17] A. Jüngel, "Transport Equations for Semiconductors," Lect. Notes Phys., 773, Springer-Verlag, Berlin, 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), 1996-2015. 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), 1033-1046. 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), 760-777. doi: 10.1137/S0036141099360269. [22] M. Kostin, On the Schrödinger-Langevin equation, J. Chem. Phys., 57 (1972), 3589-3591. doi: 10.1063/1.1678812. [23] P. Markowich, C. Ringhofer and C. Schmeiser, "Semiconductor Equations," Springer-Verlag, Vienna, 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), 65-96.

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##### References:
 [1] M. Ancona, Diffusion-drift modeling of strong inversion layers, COMPEL, 6 (1987), 11-18. [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), 722-729. 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), 3306-3333. doi: 10.1063/1.531567. [4] L. Chen and M. Dreher, Quantum semiconductor models, in "Partial Differential Equations and Spectral Theory" (eds. M. Demuth, B.-W. Schulze and I. Witt), Operator Theory: Advances and Applications, 211 (2010), 1-72. [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), 1-15. 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), 369-384. 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), 131-156. 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), 64-80. 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), 887-908. [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), 991-1015. 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), 625-667. 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), 397-401. 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, TU Wien, 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), 133-220. 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), 1701. 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), 1761-1779. doi: 10.1137/S0036141004444615. [17] A. Jüngel, "Transport Equations for Semiconductors," Lect. Notes Phys., 773, Springer-Verlag, Berlin, 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), 1996-2015. 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), 1033-1046. 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), 760-777. doi: 10.1137/S0036141099360269. [22] M. Kostin, On the Schrödinger-Langevin equation, J. Chem. Phys., 57 (1972), 3589-3591. doi: 10.1063/1.1678812. [23] P. Markowich, C. Ringhofer and C. Schmeiser, "Semiconductor Equations," Springer-Verlag, Vienna, 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), 65-96.
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