On quantum hydrodynamic and quantum energy transport models

Communications in Mathematical Sciences

Volumn 5, Issue 4, 2007, Pages 887-908

  Degond, Pierre ^a   Gallego, Samy ^a   Méhats, Florian ^b  

^a INSTITUT DE MATHÉMATIQUES DE TOULOUSE   (France)

^b UNIVERSITÉ DE RENNES 1   (France)

Author keywords

Commutators; Density operator; Gauge invariance; Quantum energy transport; Quantum entropy; Quantum hydrodynamics; Quantum Liouville equation; Quantum local equilibrium

Indexed keywords

EID: 39049114717     PISSN: 15396746     EISSN: None     Source Type: Journal    
DOI: 10.4310/CMS.2007.v5.n4.a8     Document Type: Article

References (34)

1. M.G. Ancona and G.J. Iafrate, Quantum correction of the equation of state of an electron gas in a semiconductor, Phys. Rev. B, 39, 9536-9540, 1989.
2. M.G. Ancona and H.F. Tiersten, Macroscopic physics of the silicon inversion layer, Phys. Rev. B, 39, 7959-7965, 1987.
3. A. Arnold, J.L. Lopez, P.A. Markowich and J. Soler, An analysis of quantum Fokker-Planck models: A Wigner function approach, Rev. Mat. Iberoamericana, 20, 771-814, 2004.
4. A. Arnold and F. Nier, The two-dimensional Wigner-Poisson problem for an electron gas in the charge neutral case, Math. Methods Appl. Sci., 14(9), 595-613, 1991.
5. P. Bhatnagar, E. Gross and M. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Review, 94, 511-525, 1954.
6. N. Ben Abdallah and A. Unterreiter, On the stationary quantum drift-diffusion model, Z. Angew. Math. Phys., 49(2), 251-275, 1998.
7. L.L. Chang, L. Esaki and R. Tsu, Resonant tunneling in semiconductor double barriers, Appl. Phys. Lett., 24, 593-595, 1974.
8. R.C. Chen and J.L. Liu, A quantum corrected energy-transport model for nanoscale semiconductor devices, J. Comput. Phys., 204, 131-156, 2005.
9. S. Datta, Nanoscale device modeling: the Green's function method, superlattices and microstructures, 28, 253-278, 2000.
10. P. Degond, S. Gallego and F. Méhats, An entropic quantum drift-diffusion model for electron transport in resonant tunneling diodes, J. Comput. Phys., 221, 226-249, 2007.
11. P. Degond, S. Gallego and F. Méhats, Isothermal quantum hydrodynamics: derivation, asymptotic analysis and simulation, SIAM Multiscale Model. Simul., 6(1), 246-272, 2007.
12. P. Degond, S. Gallego, F. Méhats and C. Ringhofer, Quantum hydrodynamic and diffusion models derived from the entropy principle, Springer Lecture Notes in Mathematics, to be published.
13. P. Degond, F. Méhats and C. Ringhofer, Quantum hydrodynamic models derived from the entropy principle, Contemp. Math., 371, 107-331, 2005.
14. P. Degond, F. Méhats and C. Ringhofer, Quantum energy-transport and drift-diffusion models, J. Stat. Phys., 118(3-4), 625-665, 2005.
15. P. Degond and C. Ringhofer, Quantum moment hydrodynamics and the entropy principle, J. Stat. Phys., 112(3-4), 587-628, 2003.
16. W. Dreyer, M. Junk and M. Kunik, On the approximation of the Fokker-Planck equation by moment systems, Nonlinearity, 14, 881-906, 2001.
17. F. Dubin, R. Melet, T. Barisien, R. Grousson, L. Legrand, M. Schott and V. Voliotis, Macroscopic coherence of a single exciton state in an organic quantum wire, Nature Phys., 2, 32-35, 2006.
18. S. Gallego and F. Méhats, Entropic discretization of a quantum drift-diffusion model, SIAM J. Numer. Anal., 43(5), 1828-1849, 2005.
19. C. Gardner, The quantum hydrodynamic model for semiconductor devices, SIAM J. Appl. Math., 54(2), 409-427, 1994.
20. C. Gardner, Resonant tunneling in the quantum hydrodynamic model, VLSI Design, 3, 201-210, 1995.
21. C. Gardner and C. Ringhofer, The smooth quantum potential for the hydrodynamic model, Phys. Rev. E, 53, 157-167, 1996.
22. C. Gardner and C. Ringhofer, The Chapman-Enskog expansion and the quantum hydrodynamic model for semiconductor devices, VLSI Design, 10, 415-435, 2000.
23. I. Gasser and A. Jügel, The quantum hydrodynamic model for semiconductors in thermal equilibrium, Z. Angew. Math. Phys., 48(1), 45-59, 1997.
24. I. Gasser and P.A. Markowich, Quantum hydrodynamics, Wigner transforms and the classical limit, Asympt. Analysis, 14(2), 97-116, 1997.
25. A. Jügel, Quasi-Hydrodynamic Semiconductor Equations, Progress in Nonlinear Differential Equations, Birkhäuser, 2001.
26. A. Jügel and D. Matthes, A derivation of the isothermal quantum hydrodynamic equations using entropy minimization, Z. Angew. Math. Mech., 85, 806-814, 2005.
27. A. Jügel, D. Matthes and J.P. Milisic, Derivation of new quantum hydrodynamic equations using entropy minimization, SIAM J. Appl. Math., 67, 46-68, 2006.
28. A. Jüngel and R. Pinnau, A positivity preserving numerical scheme for a fourth-order parabolic equation, SIAM J. Num. Anal., 39(2), 385-406, 2001.
29. C.D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83, 1021-1065, 1996.
30. F. Nier, A stationary Schrödinger-Poisson system arising from the modeling of electronic devices, Forum Math., 2(5), 489-510, 1990.
31. A. Unterreiter and R. Pinnau, The stationary current-voltage characteristics of the quantum drift-diffusion model, SIAM J. Num. Anal., 37(1), 211-245, 1999.
32. E. Polizzi, Modélisation et simulations numériques du transport quantique balistique dans les nanostructures semi-conductrices, Ph.D. thesis, INSA, Toulouse, 2001.
33. E. Polizzi and N. Ben Abdallah, Self-consistent three dimensional models for quantum ballistic transport in open systems, Phys. Rev. B, 66, 245-301, 2002.
34. D. Robert, Autour de l'Approximation Semi-Classique, Birkhäuser, Boston, 1987.