The stationary current-voltage characteristics of the quantum drift-diffusion model

SIAM Journal on Numerical Analysis

Volumn 37, Issue 1, 2000, Pages 211-245

  Pinnau, René ^a   Unterreiter, Andreas ^b  

^a DARMSTADT UNIVERSITY OF TECHNOLOGY   (Germany)

^b Rheinland Pfälzische Technische Universität Kaiserslautern Landau   (Germany)

Author keywords

Bipolar quantum drift diffusion model; Convergence; Generalized gummel iteration; Internal approximation; Projected quasi gradient method; Resonant tunneling diode; Uniqueness

Indexed keywords

EID: 0003033838     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/s0036142998341039     Document Type: Article

References (39)

1. N. B. ABDALLAH AND A. UNTERREITER, On the stationary quantum drift diffusion model, Z. Angew. Math. Phys., 49 (1998), pp. 251-275.
2. R. A. ADAMS, Sobolev Spaces, Academic Press, New York, 1975.
3. M. G. ANCONA, Diffusion-drift modelling of strong inversion layers, COMPEL, 6 (1987), pp. 11-18.
4. 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 (1989), pp. 9536-9540.
5. J.-P. AUBIN, Approximation of Elliptic Boundary-Value Problems, John Wiley & Sons, New York, 1972.
6. D. P. BERTSEKAS, On the Goldstein-Levitin-Polyak gradient projection method, IEEE Trans. Automat. Control, AC-21 (1976), pp. 174-184.
7. F. BREZZI AND G. GILARDI, Fundamentals of P.D.E for Numerical Analysis, Report 446, CNR, Via Ferrata 1, 1-27100 Pavia, Italy, 1984.
8. Z. CHEN, B. COCKBURN, C. L. GARDNER, AND J. W. JEROME, Quantum hydrodynamic simulation of hysteresis in the resonant tunneling diode, J. Comput. Phys., 117 (1995), pp. 274-280
9. J. C. DUNN, Global and asymptotic convergence rate estimates for a class of projected gradient processes, SIAM J. Control Optim., 19 (1981), pp. 368-400.
10. C. L. GARDNER, The quantum hydrodynamic model for semiconductor devices, SIAM J. Appl. Math., 54 (1994), pp. 409-427.
11. C. L. GARDNER AND C. RINGHOFER, Approximation of thermal equilibrium for quantum gases with discontinuous potentials and application to semiconductor devices, SIAM J. Appl. Math., 58 (1998), pp. 780-805.
12. I. GASSER AND P. A. MARKOWICH, Quantum hydrodynamics, Wigner transform and the classical limit, Asymptot. Anal., 14 (1997), pp. 97-116.
13. S. K. GHOSH AND B. M. DEB, Density, Density-Functionals and Electron Fluid, Phys. Rep., 92 (1982), pp. 1-44.
14. D. GILBARG AND N. S. TRUDINGER, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983.
15. P. GRISVARD, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985.
16. H. L. GRUBIN AND J. P. KRESKOVSKY, Quantum moment balance equations and resonant tunnelling structures, Solid-State Electronics, 32 (1989), pp. 1071-1075.
17. H. K. GUMMEL, A self-consistent iterative scheme for one-dimensional steady state transistor calculations, IEEE Trans. Elec. Dev., ED-11 (1964), pp. 455-465.
18. A. JÜNGEL, A note on current-voltage characteristics from the quantum hydrodynamic equations for semiconductors, Appl. Math. Lett., 10 (1997), pp. 29-34.
19. T. KERKHOVEN, A proof of convergence of Gummel's algorithm for realistic device geometries, SIAM J. Numer. Anal., 23 (1986), pp. 1121-1137.
20. N. C. KLUKSDAHL, A. M. KRIMAN, D. K. FERRY, AND C. RINGHOFER, Self-consistent study of the resonant-tunneling diode, Phys. Rev. B, 39 (1989), pp. 7720-7735.
21. L. D. LANDAU AND E. M. LIFSCHITZ, Lehrbuch der Theoretischen Physik III -Quantenmechanik. Akademie-Verlag, Berlin, 1985.
22. D. G. LUENBERGER, Linear and Nonlinear Programming, Addison-Wesley, 2nd ed., Reading, MA, 1989.
23. I. GASSER, P. A. MARKOWICH, AND C. RINGHOFER, Closure conditions for classical and quantum moment hierarchies in the small-temperature limit, Transport Theory Statist, Phys., 25 (1996), pp. 409-423.
24. P. A. MARKOWICH, The Stationary Semiconductor Device Equations, Springer-Verlag, Vienna, 1986.
25. P. A. MARKOWICH, C. A. RINGHOFER, AND C. SCHMEISER, Semiconductor Equations, Springer-Verlag, Vienna, 1990.
26. S. MUTO, T. INATA, H. OHNISHI, N. YOKOYAMA, AND S. HIYAMIZU, Effect of silicon doping profile on I-V characteristics of an AlGaAs/GaAs resonant tunneling barrier structure grown by MBE, Japanese J. Appl. Phys., 25 (1986), pp. L557-L579.
27. J. NAUMANN AND M. WOLFF, A uniqueness theorem for weak solutions of the stationary semiconductor equations, Appl. Math. Optim., 24 (1991), pp. 223-232.
28. H. OHNISHI, T. INATA, S. MUTO, N. YOKOYAMA, AND A. SHIBATOMI, Self-consistent analysis of resonant tunneling current, Appl. Phys. Lett., 49 (1986), pp. 1248-1250.
29. F. PACARD AND A. UNTERREITER, A variational analysis of the thermal equilibrium state of charged quantum fluids, Comm. Partial Differential Equations, 20 (1995), pp. 885-900.
30. R. PINNAU, A note on boundary conditions for quantum hydrodynamic models, Appl. Math. Lett., 12 (1999), pp. 77-82.
31. M. REED AND B. SIMON, Methods of Modern Mathematical Physics, Vol. I, Academic Press, San Diego, 1987.
32. C. RINGHOFER, A spectral method for the numerical simulation of quantum tunneling phenomena, SIAM J. Numer. Anal., 27 (1990), pp. 32-50.
33. C. RINGHOFER, A spectral collocation technique for the solution of the Wigner Poisson problem, SIAM J. Numer. Anal., 29 (1992), pp. 679-700.
34. D. SCHOTT, A general iterative scheme with applications to convex optimization and related fields, Optimization, 22 (1991), pp. 885-902.
35. M. STRUWE, Variational Methods, Springer-Verlag, Berlin, 1990.
36. G. M. TROIANIELLO, Elliptic Differential Equations and Obstacle Problems, Plenum Press, New York, 1987.
37. A. UNTERREITER, The thermal equilibrium solution of a generic bipolar quantum hydrodynamic model, Comm. Math. Phys., 188 (1997), pp. 69-88.
38. O. VANBÉSIEN, R. BOUREGBA, P. MOUNAIX, AND D. LIPPENS, Temperature dependence of peak to valley current ratio in resonant tunneling double barriers, in Resonant Tunneling in Semiconductors, L.L. Chang et al, eds., Plenum Press, New York, 1991, pp. 107-116.
39. E. ZEIDLER, Nonlinear Functional Analysis and Its Applications, Vol. II/A and II/B, Springer-Verlag, Berlin, 1990.