Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Asymptotic stability of the stationary solution for a new mathematical model of charge transport in semiconductors

Authors: A. M. Blokhin and D. L. Tkachev
Journal: Quart. Appl. Math. 70 (2012), 357-382
MSC (2010): Primary 35G61, 35D30; Secondary 82D37
DOI: https://doi.org/10.1090/S0033-569X-2012-01251-7
Published electronically: February 29, 2012
MathSciNet review: 2953108
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Abstract | References | Similar Articles | Additional Information

Abstract: We study an initial boundary value problem for a system of quasilinear equations which are effectively used for finding by the stabilization method numerical stationary solutions of the hydrodynamical model of charge transport in the silicon MESFET (metal semiconductor field effect transistor). This initial boundary value problem has the following peculiarities: the PDE system is not a Cauchy-Kovalevskaya-type system; the boundary is a nonsmooth curve and has angular points; nonlinearity of the problem is mainly connected with squares of gradients of the unknown functions. By using a special representation for the solution of a model problem we reduce the original problem to a system of integro-differential equations. This allows one to prove the local-in-time existence and uniqueness of a weakened solution. Using a constructed energy integral and the Schauder fixed-point theorem, we prove the global-in-time solvability of the initial boundary value problem and justify the stabilization method under additional assumptions on the problem's data.

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  • 1. S. Selberherr, Analysis and Simulation of Semiconductor Devices, Springer-Verlag, New York, Vienna, 1984.
  • 2. W. Hansch, The drift diffusion equations and its applications in MESFET modeling, Springer-Verlag, Vienna, 1991.
  • 3. P. Markowich, C.A. Ringhofer, and C. Schmeiser, Semiconductor equations, Springer-Verlag, Vienna, 1990. MR 1063852 (91j:78011)
  • 4. D. Chen, E.C. Kan, U. Ravaioli, C-W. Shu, and R. Dutton, An improved energy-transport model including nonparabolicity and non-Maxwellian distribution effects, IEEE on Electron Device Letters, 13 (1992) pp.26-28.
  • 5. E. Lyumkis, B. Polsky, A. Shir, and P. Visocky, Transient semiconductor device simulation including energy balance equation, COMPEL, 11 (1992), pp. 311-325.
  • 6. N.B. Abdallah, P. Degond, On a hierarchy of macroscopic models for semiconductors, J. Math. Phys. 37 (1996), pp. 3308-3333. MR 1401227 (98b:82091)
  • 7. A.M. Anile, V. Romano, Hydrodynamical modeling of charge carrier transport in semiconductors, MECCANICA, 35 (2000), pp. 249-296.
  • 8. A.M. Anile, G. Mascali, and V. Romano, Recent developments in hydrodynamical modeling of semiconductors, Lecture Notes in Mathematics, no. 1823, Springer, 2003. MR 2073497
  • 9. A.M. Anile, V. Romano, Non-parabolic band transport in semiconductors: closure of the moment equations, Cont. Mech. Thermodyn., 11 (1999), pp. 307-325. MR 1723705 (2000j:82044)
  • 10. V. Romano, Non-parabolic band transport in semiconductors: closure of the production terms in the moment equations, Cont. Mech. Thermodyn., 12 (2000), pp. 31-51. MR 1753005 (2001g:82117)
  • 11. I. Müller, T. Ruggeri, Rational Extended Thermodynamics, Springer-Verlag, Berlin, 1998. MR 1632151 (99h:80001)
  • 12. D. Jon, J. Casas-Vazques, and G. Lebon, Extended irreversible thermodynamics, Springer-Verlag, Berlin, 1993. MR 1271780 (95a:80004)
  • 13. C.D. Levermore, Moment Closure Hierarchies for Kinetic Theories, J. Stat. Phys., 83 (1996), pp. 331-407. MR 1392419 (97e:82041)
  • 14. A.M. Anile, O. Muscato, and V. Romano, Moment Equations with maximum entropy closure for carrier transport in semiconductor devices: validation in bulk silicon, VLSI Design 10 (2000), pp. 335-354.
  • 15. O. Muscato, V. Romano, Simulation of submicron silicon diode with a non-parabolic hydrodynamical model based on the maximum entropy principle, VLSI Design 13 (2001), pp. 273-279.
  • 16. V. Romano, Nonparabolic band hydrodynamical model of silicon semiconductors and simulation of electron devices, Math. Meth. Appl. Sci., 24 (2001), pp. 439-471. MR 1829038 (2002c:82083)
  • 17. V. Romano, 2D simulation of a silicon MESFET with a non-parabolic hydrodynamical model based on the maximum entropy principle, J. Comp. Phys., 176 (2002), pp. 70-92.
  • 18. V. Romano, 2D numerical simulation of the MEP energy-transport model with a finite difference scheme, J. Comp. Phys., 221 (2007), pp. 439-468. MR 2293138 (2008b:82083)
  • 19. A.M. Blokhin, R.S. Bushmanov, A.S. Rudometova, and V. Romano, Linear asymptotic stability of the equilibrium state for the 2D MEP hydrodynamical model of charge transport in semiconductors, Nonlinear Analysis, 65 (2006), pp. 1018-1038. MR 2232491 (2006m:76153)
  • 20. A.M. Blokhin, R.S. Bushmanov, and V. Romano, Nonlinear asymptotic stability of the equilibrium state for the MEP model of charge transport in semiconductors, Nonlinear Analysis, 65 (2006), pp. 2169-2191. MR 2266431 (2007i:35225)
  • 21. A.M. Blokhin, A.S. Ibragimova, and B.V. Semisalov, Construction of a computational algorithm for a system of moment equations describing charge transfer in semiconductors, Mathematical Modelling, 21:4 (2009), pp. 15-34 (in Russian). MR 2547344 (2010e:82111)
  • 22. A.M. Blokhin, A.S. Ibragimova, Numerical method for 2D Simulation of a Silicon MESFET with a Hydrodynamical Model Based on the Maximum Entropy Principle, SIAM Journal on Scientific Computing, 31:3 (2009), pp. 2015-2046. MR 2516142 (2010i:82188)
  • 23. J. L. Lions, E. Magenes, Non-homogeneous boundary value problems and applications, Springer, New York, 1972.
  • 24. M.A. Shubin, Pseudodifferential operators and spectral theory, Nauka, Moscow, 1978 (in Russian). MR 509034 (80h:47057)
  • 25. G.V. Demidenko, S.V. Uspenskij, Embedding theorems and their applications to differential equations, Nauka, Novosibirsk, 1984 (in Russian).
  • 26. O.A. Ladyzhenskaya, N.N. Ural'tseva, Linear and quasilinear elliptic equations, Academic Press, New York, 1968. MR 0244627 (39:5941)
  • 27. D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin, 1977. MR 0473443 (57:13109)
  • 28. A.G. Sveshnikov, A.B. Alshin, M.O. Korpusov, and Yu.D. Pletner, Linear and nonlinear Sobolev-type equations, Moscow, Fizmatlit, 2007 (in Russian).
  • 29. A.M. Blokhin, D.L. Tkachev, Representation of the solution to a model problem in semiconductor physics, J. Math. Anal. Appl. 341 (2008), 1468-1475. MR 2398541 (2008m:35048)
  • 30. M.A. Lavrentjev, B.V. Shabat, Methods of Theory of Complex-Valued Functions, Gos. Izdatelstvo Tekhniko-Teoreticheskoj Literatury, Moscow-Leningrad, 1951 (in Russian).
  • 31. H. Ghaevsky, K. Grier, and K. Zakharias, Nonlinear Operator Equations and Operator Differential Equations, Mir, Moscow, 1978 (in Russian). MR 0636413 (58:30524b)
  • 32. S. Mizohata, The theory of partial differential equations, Cambridge University Press, New York, 1973. MR 0599580 (58:29033)
  • 33. I.G. Petrovsky, Lectures on Partial Differential Equations, Dover Publications, New York, 1991. MR 1160355 (93a:35001)
  • 34. L.V. Kantorovich, G.P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982. MR 664597 (83h:46002)
  • 35. K.I. Babenko, Fundamentals of numerical analysis, Moscow-Izhevsk, Regular and chaotic dynamics, 2002 (in Russian).
  • 36. A.M. Blokhin, A.S. Ibragimova, 1D Numerical Simulation of the MEP Mathematical Model in ballistic diode problem, Journal of Kinetic and Related Models, 2:1 (2009), 81-107. MR 2472150 (2010e:82110)
  • 37. O.V. Besov, V.P. Il'in, S.M. Nikol'skiĭ, Integral representations of functions and embedding theorems, Moscow, Fizmatlit, 1996 (in Russian). MR 1450401 (98b:46037)
  • 38. S.L. Sobolev, Applications of functional analysis in mathematical physics. Translations of Mathematical Monographs, vol. 7, American Mathematical Society, Providence, 1963. MR 0165337 (29:2624)
  • 39. K. Iosida, Functional Analysis, Springer-Verlag, Berlin-New York, 1980. MR 617913 (82i:46002)

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Additional Information

A. M. Blokhin
Affiliation: Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia; Novosibirsk State University, Novosibirsk, 630090, Russia
Email: blokhin@math.nsc.ru

D. L. Tkachev
Affiliation: Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia; Novosibirsk State University, Novosibirsk, 630090, Russia
Email: tkachev@math.nsc.ru

DOI: https://doi.org/10.1090/S0033-569X-2012-01251-7
Keywords: Non-Cauchy-Kovalevskaya-type system, weakened solution, local- and global-in-time solvability, asymptotic (Lyapunov’s) stability, stabilization method
Received by editor(s): October 13, 2010
Published electronically: February 29, 2012
Additional Notes: The authors are indebted to Yu. L. Trakhinin and S. A. Boyarsky for their help in the preparation of the manuscript of this paper. This work was financially supported by RFBR project, 10-01-00320-a, the interdisciplinary project of basic research SB RAS-2009 (No.91), and the project “Development of scientific potential of the Higher School” 2009-2010 (No. 2.1.1/4591).
Article copyright: © Copyright 2012 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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