Current coupling of drift-diffusion models and schrödinger - Poisson systems: Dissipative hybrid models

SIAM Journal on Mathematical Analysis

Volumn 37, Issue 3, 2006, Pages 941-981

  Baro, M ^a   Neidhardt, H ^a   Rehberg, J ^a  

^a WEIERSTRASS INSTITUTE FOR APPLIED ANALYSIS AND STOCHASTICS   (Germany)

Author keywords

Current coupling; Dissipative Schr dinger systems; Drift diffusion models; Hybrid models; Poisson equation; Quantum classical coupling; Semiconductors

Indexed keywords

BOUNDARY CONDITIONS; DIFFUSION; QUANTUM THEORY; SEMICONDUCTOR MATERIALS;

CURRENT COUPLING; DISSIPATIVE SCHRÖDINGER SYSTEMS; DRIFT-DIFFUSION MODELS; HYBRID MODELS; QUANTUM-CLASSICAL COUPLING;

POISSON EQUATION;

EID: 33646551764     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/040611690     Document Type: Article

References (54)

1. 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 (1991), pp. 595-613.
2. M. BARO, N. BEN ABDALLAH, P. DEGOND, AND A. EL AYYADI, A 1-D coupled Schrödinger drift-diffusion model including collisions, J. Comput. Phys., 203 (2005), pp. 129-153.
3. M. BARO, H.-CHR. KAISER, H. NEIDHARDT, AND J. REHBERG, Dissipative Schrödinger-Poisson systems, J. Math. Phys., 45 (2004), pp. 21-43.
4. M. BARO, H.-CHR. KAISER, H. NEIDHARDT, AND J. REHBERG, A quantum transmitting Schrödinger-Poisson system. Rev. Math. Phys., 16 (2004), pp. 281-330.
5. M. BARO AND H. NEIDHARDT, Dissipative Schrödinger-type operators as a model for generation and recombination, J. Math. Phys., 44 (2003), pp. 2373-2401.
6. N. BEN ABDALLAH, A hybrid kinetic-quantum model for stationary electron transport, J. Statist. Phys., 90 (1998), pp. 627-662.
7. N. BEN ABDALLAH, P. DEGOND, AND I. GAMBA, Coupling one-dimensional time-dependent classical and quantum transport models, J. Math. Phys., 43 (2002), pp. 1-24.
8. N. BEN ABDALLAH, P. DEGOND, AND P. A. MARKOWICH, On a one-dimensional Schrödinger-Poisson scattering model, Z. Angew. Math. Phys., 48 (1997), pp. 135-155.
9. N. BEN ABDALLAH AND O. PINAUD, A mathematical model for the transient evolution of a resonant tunneling diode, C. R. Math. Acad. Sci. Paris, 334 (2002), pp. 283-288.
10. F. BREZZI AND P. A. MARKOWICH, The three-dimensional Wigner-Poisson problem: Existence, uniqueness and approximation, Math. Methods Appl. Sci., 14 (1991), pp. 35-61.
11. V. BUSLAEV AND V. FOMIN, An inverse scattering problem for the one-dimensional Schrödinger equation on the entire axis, Vestnik Leningrad Univ., 17 (1962), pp. 56-64 (in Russian).
12. 2-solutions to the Schrödinger-Poisson system: Existence, uniqueness, time behaviour, and smoothing effects, Math. Models Methods Appl. Sci., 7 (1997), pp. 1051-1083.
13. PH. CAUSSIGNAC, J. DESCLOUX, AND A. YAMNAHAKKI, Simulation of some quantum models for semiconductors, Math. Models Methods Appl. Sci., 12 (2002), pp. 1049-1074.
14. F. CHEVOIR AND B. VINTER, Scattering-assisted tunneling in double-barrier diodes: Scattering rates and valley current, Phys. Rev. B, 47 (1993), pp. 7260-7274.
15. J. H. DAVIES, The Physics of Low Dimensional Semiconductors, Cambridge University Press, Cambridge, UK, 1997.
16. P. DEGOND AND A. EL AYYADI, A coupled Schrödinger drift-diffusion model for quantum semiconductor device simulations, J. Comput. Phys., 181 (2002), pp. 222-259.
17. P. EXNER, Open Quantum System and Feynman Integrals, Fundamental Theories of Physics, D. Reidel, Dordrecht, 1985.
18. D. K. FERRY AND S. M. GOODNICK, Transport in Nanostructures, Cambridge University Press, Cambridge, UK, 1997.
19. M. V. FISCHETTI, Theory of electron transport in small semiconductor devices using the Pauli master equation, J. Appl. Phys., 83 (1998), pp. 270-291.
20. M. V. FISCHETTI, Master-equation approach to the study of electronic transport in small semiconductor devices, Phys. Rev. B, 59 (1999), pp. 4901-4917.
21. C. FOIAS AND B. SZ.-NAGY, B, Harmonic Analysis of Operators on Hilbert Space, Akadémia Kiadó, Budapest, North-Holland, Amsterdam, London; American Elsevier, New York, 1970.
22. W. R. FRENSLEY, Boundary conditions for open quantum systems driven far from equilibrium, Rev. Modern Phys., 62 (1990), pp. 745-791.
23. H. GAJEWSKI, Analysis and numerics of carrier transport in semiconductors, Mitt. Ges. Angew. Math. Mech., 16 (1993), pp. 35-57 (in German).
24. H. GAJEWSKI, K. GRÖGER, AND K. ZACHARIAS, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Mathematische Lehrbücher und Monographien, II. Abteilung, Mathematische Monographien, Band 38, Akademie-Verlag, Berlin, 1974.
25. D. GILBARG AND N. S. TRUDINGER, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition. Classics in Math., Springer-Verlag, Berlin, 2001.
26. J. B. GARNETT, Bounded Analytic Functions, Pure and Appl. Math. 96, Academic Press, New York, 1981.
27. H. K. GUMMEL, A self-consistent iterative scheme for one-dimensional steady state transistor calculations, IEEE Trans. Education 11 (1964), pp. 455-465.
28. E. HEWITT AND K. STROMBERG, Real and Abstract Analysis. A Modern Treatment of the Theory of Functions of a Real Variable, Springer-Verlag, New York, 1965.
29. R. ILLNER, The Wigner-Poisson and Schrödinger-Poisson systems, Transport. Theory Statist. Phys., 21 (1992), pp. 753-767.
30. R. ILLNER, Existence, uniqueness and asymptotic behavior of Wigner-Poisson and Vlasov-Poisson systems: A survey, Transport. Theory Statist. Phys., 26 (1997), pp. 195-207.
31. R. ILLNER, P. F. ZWEIFEL, AND H. LANGE, Global existence, uniqueness and asymptotic behaviour of solutions of the Wigner-Poisson and Schrödinger- Poisson systems, Math. Methods Appl. Sci., 17 (1994), pp. 349-376.
32. G. JONA-LASINIO, C. PRESILLA, AND J. SJÖSTRAND, On Schrödinger equations with concentrated nonlinearities, Ann. Phys., 240 (1995), pp. 1-21.
33. H.-CHR. KAISER, H. NEIDHARDT, AND J. REHBERG, Density and current of a dissipative Schrödinger operator, J. Math. Phys., 43 (2002), pp. 5325-5350.
34. H.-CHR. KAISER, H. NEIDHARDT, AND J. REHBERG, Macroscopic current induced boundary conditions for Schrödinger-type operators, Integral Equations Operator Theory, 45 (2003), pp. 39-63.
35. H.-CHR. KAISER, H. NEIDHARDT, AND J. REHBERG, On 1-dimensional dissipative Schrödinger-type operators, their dilations and eigenfunction expansions, Math. Nachr., 252 (2003), pp. 51-69.
36. H.-CHR. KAISER AND J. REHBERG, On stationary Schrödinger-Poisson equations, Z. Angew. Math. Mech., 75 (1995), Suppl. 2, pp. 467-468.
37. H.-CHR. KAISER AND J. REHBERG, On stationary Schrödinger-Poisson equations modelling an electron gas with reduced dimension, Math. Methods Appl. Sci., 20 (1997), pp. 1283-1312.
38. H.-CHR. KAISER AND J. REHBERG, About a one-dimensional stationary Schrödinger-Poisson system, with Kohn-Sham potential, Z. Angew. Math. Phys., 50 (1999), pp. 423-458.
39. H.-CHR. KAISER AND J. REHBERG, About a stationary Schrödinger- Poisson system with Kohn-Sham potential in a bounded two- or three-dimensional domain, Nonlinear Anal. Ser. A Theory Methods, 41 (2000), pp. 33-72.
40. TH. G. KURTZ, Extensions of Trotter's operator semigroup approximation theorems, J. Funct. Anal., 3 (1969), pp. 354-375.
41. H. LANGE, B. TOOMIRE, AND P. F. ZWEIFEL, An overview of Schrödinger-Poisson problems, Rep. Math. Phys., 36 (1995), pp. 331-345.
42. P. A. MARKOWICH, Boltzmann distributed quantum steady states and their classical limit, Forum Math., 6 (1994), pp. 1-33.
43. P. A. MARKOWICH, The Stationary Semiconductor Device Equations, Springer-Verlag, Vienna, 1986.
44. M. S. MOCK, Analysis of Mathematical Models of Semiconductor Devices, Boole Press, Dun Laoghaire, Ireland, 1983.
45. P. MOUNAIX, O. VANBESIEN, AND D. LIPPENS, Effect of cathode space layer on the current-voltage characteristic of resonant tunneling diodes, Appl. Phys. Lett., 57 (1990), pp. 1517-1519.
46. F. NIER, A stationary Schrödinger-Poisson system arising from the modelling of electronic devices, Forum Math., 2 (1990), pp. 489-510.
47. F. NIER, A variational formulation of Schrödinger-Poisson systems in dimension d ≤ 3, Comm. Partial Differential Equations, 18 (1993), pp. 1125-1147.
48. F. NIER, Schrödinger-Poisson systems in dimension d ≦ 3: The whole-space case, Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), pp. 1179-1201.
49. F. NIER, The dynamics of some quantum open systems with short range nonlinearities, Nonlinearity, 11 (1998), pp. 1127-1172.
50. C. PRESILLA AND J. SJÖSTRAND, Transport properties in resonant tunneling heterostructures, J. Math. Phys., 37 (1996), pp. 4816-4844.
51. W. VAN ROOSBROECK, Theory of flow electrons and holes in germanium and other semiconductors, Bell Systems Tech. J., 29 (1950), pp. 560-607.
52. S. SELBERHERR, Analysis and Simulation of Semiconductor Devices, Springer-Verlag, Vienna, 1984.
53. H. STEINRÜCK, The one-dimensional Wigner - Poisson problem and its relation to the Schrödinger - Poisson problem, SIAM J. Math. Anal., 22 (1991), pp. 957-972.
54. C. WEISSBUCH AND B. VINTER, Quantum Semi-conductor Strucutres, Academic Press, New York, 1991.