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

Full-text PDF

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.

**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. A. Markowich, C. A. Ringhofer, and C. Schmeiser,*Semiconductor equations*, Springer-Verlag, Vienna, 1990. MR**1063852****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. Ben Abdallah and P. Degond,*On a hierarchy of macroscopic models for semiconductors*, J. Math. Phys.**37**(1996), no. 7, 3306–3333. MR**1401227**, https://doi.org/10.1063/1.531567**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*, Mathematical problems in semiconductor physics, Lecture Notes in Math., vol. 1823, Springer, Berlin, 2003, pp. 1–56. MR**2073497**, https://doi.org/10.1007/978-3-540-45222-5_1**9.**Angelo Marcello Anile and Vittorio Romano,*Non-parabolic band transport in semiconductors: closure of the moment equations*, Contin. Mech. Thermodyn.**11**(1999), no. 5, 307–325. MR**1723705**, https://doi.org/10.1007/s001610050126**10.**Vittorio Romano,*Non parabolic band transport in semiconductors: closure of the production terms in the moment equations*, Contin. Mech. Thermodyn.**12**(2000), no. 1, 31–51. MR**1753005**, https://doi.org/10.1007/s001610050121**11.**Ingo Müller and Tommaso Ruggeri,*Rational extended thermodynamics*, 2nd ed., Springer Tracts in Natural Philosophy, vol. 37, Springer-Verlag, New York, 1998. With supplementary chapters by H. Struchtrup and Wolf Weiss. MR**1632151****12.**D. Jou, J. Casas-Vázquez, and G. Lebon,*Extended irreversible thermodynamics*, Springer-Verlag, Berlin, 1993. MR**1271780****13.**C. David Levermore,*Moment closure hierarchies for kinetic theories*, J. Statist. Phys.**83**(1996), no. 5-6, 1021–1065. MR**1392419**, https://doi.org/10.1007/BF02179552**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.**Vittorio Romano,*Non-parabolic band hydrodynamical model of silicon semiconductors and simulation of electron devices*, Math. Methods Appl. Sci.**24**(2001), no. 7, 439–471. MR**1829038**, https://doi.org/10.1002/mma.220**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. Comput. Phys.**221**(2007), no. 2, 439–468. MR**2293138**, https://doi.org/10.1016/j.jcp.2006.06.028**19.**A. M. Blokhin, R. S. Bushmanov, A. S. Rudometova, and V. Romano,*Linear asymptotic stability of the equilibrium state for the 2-D MEP hydrodynamical model of charge transport in semiconductors*, Nonlinear Anal.**65**(2006), no. 5, 1018–1038. MR**2232491**, https://doi.org/10.1016/j.na.2005.09.045**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 Anal.**65**(2006), no. 11, 2169–2191. MR**2266431**, https://doi.org/10.1016/j.na.2006.01.030**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*, Mat. Model.**21**(2009), no. 4, 15–34 (Russian, with English and Russian summaries). MR**2547344****22.**A. M. Blokhin and A. S. Ibragimova,*Numerical method for 2D simulation of a silicon MESFET with a hydrodynamical model based on the maximum entropy principle*, SIAM J. Sci. Comput.**31**(2009), no. 3, 2015–2046. MR**2516142**, https://doi.org/10.1137/070706537**23.**J. L. Lions, E. Magenes, Non-homogeneous boundary value problems and applications, Springer, New York, 1972.**24.**M. A. Šubin,*\cyr Psevdodifferentsial′nye operatory i spektral′naya teoriya*, “Nauka”, Moscow, 1978 (Russian). MR**509034****25.**G.V. Demidenko, S.V. Uspenskij, Embedding theorems and their applications to differential equations, Nauka, Novosibirsk, 1984 (in Russian).**26.**Olga A. Ladyzhenskaya and Nina N. Ural’tseva,*Linear and quasilinear elliptic equations*, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, Academic Press, New York-London, 1968. MR**0244627****27.**David Gilbarg and Neil S. Trudinger,*Elliptic partial differential equations of second order*, Springer-Verlag, Berlin-New York, 1977. Grundlehren der Mathematischen Wissenschaften, Vol. 224. MR**0473443****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 and D. L. Tkachev,*Representation of the solution to a model problem in semiconductor physics*, J. Math. Anal. Appl.**341**(2008), no. 2, 1468–1475. MR**2398541**, https://doi.org/10.1016/j.jmaa.2007.11.010**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.**Herbert Gajewski, Konrad Gröger, and Klaus Zacharias,*Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen*, Akademie-Verlag, Berlin, 1974 (German). Mathematische Lehrbücher und Monographien, II. Abteilung, Mathematische Monographien, Band 38. MR**0636412**

Kh. Gaevskiĭ, H. Gaevskiĭ, K. Greger, K. Greger, K. Zakharias, and K. Zaharias,*\cyr Nelineĭnye operatornye uravneniya i operatornye differentsial′nye uravneniya*, Izdat. “Mir”, Moscow, 1978 (Russian). Translated from the German by A. I. Perov and V. G. Zadorožniĭ; Edited by V. I. Sobolev. MR**0636413****32.**Sigeru Mizohata,*The theory of partial differential equations*, Cambridge University Press, New York, 1973. Translated from the Japanese by Katsumi Miyahara. MR**0599580****33.**I. G. Petrovsky,*Lectures on partial differential equations*, Dover Publications, Inc., New York, 1991. Translated from the Russian by A. Shenitzer; Reprint of the 1964 English translation. MR**1160355****34.**L. V. Kantorovich and G. P. Akilov,*Functional analysis*, 2nd ed., Pergamon Press, Oxford-Elmsford, N.Y., 1982. Translated from the Russian by Howard L. Silcock. MR**664597****35.**K.I. Babenko, Fundamentals of numerical analysis, Moscow-Izhevsk, Regular and chaotic dynamics, 2002 (in Russian).**36.**Alexander Blokhin and Alesya Ibragimova,*1D numerical simulation of the MEP mathematical model in ballistic diode problem*, Kinet. Relat. Models**2**(2009), no. 1, 81–107. MR**2472150**, https://doi.org/10.3934/krm.2009.2.81**37.**O. V. Besov, V. P. Il′in, and S. M. Nikol′skiĭ,*\cyr Integral′nye predstavleniya funktsiĭ i teoremy vlozheniya*, 2nd ed., Fizmatlit “Nauka”, Moscow, 1996 (Russian, with Russian summary). MR**1450401****38.**S. L. Sobolev,*Applications of functional analysis in mathematical physics*, Translated from the Russian by F. E. Browder. Translations of Mathematical Monographs, Vol. 7, American Mathematical Society, Providence, R.I., 1963. MR**0165337****39.**Kôsaku Yosida,*Functional analysis*, 6th ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 123, Springer-Verlag, Berlin-New York, 1980. MR**617913**

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC (2010):
35G61,
35D30,
82D37

Retrieve articles in all journals with MSC (2010): 35G61, 35D30, 82D37

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.