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 & 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.

[1]

Benoît Pausader. The focusing energy-critical fourth-order Schrödinger equation with radial data. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1275-1292. doi: 10.3934/dcds.2009.24.1275

[2]

José A. Carrillo, Ansgar Jüngel, Shaoqiang Tang. Positive entropic schemes for a nonlinear fourth-order parabolic equation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 1-20. doi: 10.3934/dcdsb.2003.3.1

[3]

Chunhua Jin, Jingxue Yin, Zejia Wang. Positive periodic solutions to a nonlinear fourth-order differential equation. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1225-1235. doi: 10.3934/cpaa.2008.7.1225

[4]

Changchun Liu. A fourth order nonlinear degenerate parabolic equation. Communications on Pure & Applied Analysis, 2008, 7 (3) : 617-630. doi: 10.3934/cpaa.2008.7.617

[5]

Zongming Guo, Long Wei. A fourth order elliptic equation with a singular nonlinearity. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2493-2508. doi: 10.3934/cpaa.2014.13.2493

[6]

Zongming Guo, Long Wei. A perturbed fourth order elliptic equation with negative exponent. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4187-4205. doi: 10.3934/dcdsb.2018132

[7]

Wolfgang Wagner. Some properties of the kinetic equation for electron transport in semiconductors. Kinetic & Related Models, 2013, 6 (4) : 955-967. doi: 10.3934/krm.2013.6.955

[8]

Pablo Álvarez-Caudevilla, Jonathan D. Evans, Victor A. Galaktionov. Gradient blow-up for a fourth-order quasilinear Boussinesq-type equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3913-3938. doi: 10.3934/dcds.2018170

[9]

Luca Calatroni, Bertram Düring, Carola-Bibiane Schönlieb. ADI splitting schemes for a fourth-order nonlinear partial differential equation from image processing. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 931-957. doi: 10.3934/dcds.2014.34.931

[10]

Baishun Lai, Qing Luo. Regularity of the extremal solution for a fourth-order elliptic problem with singular nonlinearity. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 227-241. doi: 10.3934/dcds.2011.30.227

[11]

M. Ben Ayed, K. El Mehdi, M. Hammami. Nonexistence of bounded energy solutions for a fourth order equation on thin annuli. Communications on Pure & Applied Analysis, 2004, 3 (4) : 557-580. doi: 10.3934/cpaa.2004.3.557

[12]

Gabriele Bonanno, Beatrice Di Bella. Fourth-order hemivariational inequalities. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 729-739. doi: 10.3934/dcdss.2012.5.729

[13]

Harald Friedrich. Semiclassical and large quantum number limits of the Schrödinger equation. Conference Publications, 2003, 2003 (Special) : 288-294. doi: 10.3934/proc.2003.2003.288

[14]

A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044

[15]

Zhong Wang. Stability of Hasimoto solitons in energy space for a fourth order nonlinear Schrödinger type equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4091-4108. doi: 10.3934/dcds.2017174

[16]

Xueke Pu, Boling Guo. Global existence and semiclassical limit for quantum hydrodynamic equations with viscosity and heat conduction. Kinetic & Related Models, 2016, 9 (1) : 165-191. doi: 10.3934/krm.2016.9.165

[17]

Jaime Angulo Pava, Carlos Banquet, Márcia Scialom. Stability for the modified and fourth-order Benjamin-Bona-Mahony equations. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 851-871. doi: 10.3934/dcds.2011.30.851

[18]

Feliz Minhós, João Fialho. On the solvability of some fourth-order equations with functional boundary conditions. Conference Publications, 2009, 2009 (Special) : 564-573. doi: 10.3934/proc.2009.2009.564

[19]

Haitao Che, Haibin Chen, Yiju Wang. On the M-eigenvalue estimation of fourth-order partially symmetric tensors. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-16. doi: 10.3934/jimo.2018153

[20]

Shi Jin, Min Tang, Houde Han. A uniformly second order numerical method for the one-dimensional discrete-ordinate transport equation and its diffusion limit with interface. Networks & Heterogeneous Media, 2009, 4 (1) : 35-65. doi: 10.3934/nhm.2009.4.35

2017 Impact Factor: 1.219

Metrics

  • PDF downloads (3)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]