Finite element approximation to initial-boundary value problems of the semiconductor device equations with magnetic influence

Author:
Jiang Zhu

Journal:
Math. Comp. **59** (1992), 39-62

MSC:
Primary 65N30; Secondary 65N12

MathSciNet review:
1134742

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We shall consider Zlámal's approach to the nonstationary equations of the semiconductor device theory under magnetic fields, with mixed boundary conditions. Owing to the reduced smoothness of the electric potential and carrier densities *n* and *p* caused by considering the mixed boundary conditions, we must use a nonstandard analysis for this procedure. Existence as well as uniqueness of the approximate solution is proved. The convergence rates obtained in this paper are slower than those previously obtained for pure Dirichlet or Neumann boundary conditions.

**[1]**W. Allegretto, Y. S. Mun, A. Nathan, and H. P. Baltes,*Optimization of semiconductor magnetic field sensors using finite element analysis*, Proc. NASECODE IV Conf., Boole Press, Dublin, 1985, pp. 129-133.**[2]**J. Banasiak and G. F. Roach,*On mixed boundary value problems of Dirichlet oblique-derivative type in plane domains with piecewise differentiable boundary*, J. Differential Equations**79**(1989), no. 1, 111–131. MR**997612**, 10.1016/0022-0396(89)90116-2**[3]**R. E. Bank, W. M. Fichtner, Jr., D. J. Rose, and R. K. Smith,*Transient simulation of silicon devices and circuits*, IEEE Trans. Computer-Aided Design**4**(1985), 436-451.**[4]**Randolph E. Bank, Joseph W. Jerome, and Donald J. Rose,*Analytical and numerical aspects of semiconductor device modeling*, Computing methods in applied sciences and engineering, V (Versailles, 1981), North-Holland, Amsterdam, 1982, pp. 593–597. MR**784655****[5]**Philippe G. Ciarlet,*The finite element method for elliptic problems*, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR**0520174****[6]**Jim Douglas Jr., Richard E. Ewing, and Mary Fanett Wheeler,*The approximation of the pressure by a mixed method in the simulation of miscible displacement*, RAIRO Anal. Numér.**17**(1983), no. 1, 17–33 (English, with French summary). MR**695450****[7]**Jim Douglas Jr., Irene Martínez-Gamba, and M. Cristina J. Squeff,*Simulation of the transient behavior of a one-dimensional semiconductor device*, Mat. Apl. Comput.**5**(1986), no. 2, 103–122 (English, with Portuguese summary). MR**884996****[8]**Jim Douglas Jr. and Yuan Yirang,*Finite difference methods for the transient behavior of a semiconductor device*, Mat. Apl. Comput.**6**(1987), no. 1, 25–37 (English, with Portuguese summary). MR**903000****[9]**J. Douglas, Jr., Yirang Yuan, and Gang Li,*A modified method of characteristic procedure for the transient behavior of a semiconductor device*(preprint).**[10]**-,*A mixed method for the transient behavior of a semiconductor device*(preprint).**[11]**Richard E. Ewing,*Time-stepping Galerkin methods for nonlinear Sobolev partial differential equations*, SIAM J. Numer. Anal.**15**(1978), no. 6, 1125–1150. MR**512687**, 10.1137/0715075**[12]**R. E. Ewing and M. F. Wheeler,*Galerkin methods for miscible displacement problems with point sources and sinks-unit mobility ratio case*, Lectures on the Numerical Solution of Partial Differential Equations, University of Maryland, 1981, pp. 151-174.**[13]**H. Gajewski,*On existence, uniqueness and asymptotic behavior of solutions of the basic equations for carrier transport in semiconductors*, Z. Angew. Math. Mech.**65**(1985), no. 2, 101–108 (English, with German and Russian summaries). MR**841263**, 10.1002/zamm.19850650210**[14]**H. Gajewski and K. Gröger,*On the basic equations for carrier transport in semiconductors*, J. Math. Anal. Appl.**113**(1986), no. 1, 12–35. MR**826656**, 10.1016/0022-247X(86)90330-6**[15]**Irene Martínez-Gamba and Maria Cristina J. Squeff,*Simulation of the transient behavior of a one-dimensional semiconductor device. II*, SIAM J. Numer. Anal.**26**(1989), no. 3, 539–552. MR**997655**, 10.1137/0726032**[16]**Vivette Girault and Pierre-Arnaud Raviart,*Finite element methods for Navier-Stokes equations*, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR**851383****[17]**Joseph W. Jerome,*Consistency of semiconductor modeling: an existence/stability analysis for the stationary Van Roosbroeck system*, SIAM J. Appl. Math.**45**(1985), no. 4, 565–590. MR**796097**, 10.1137/0145034**[18]**-,*Evolution systems in semiconductor modeling*:*A cyclic uncoupled analysis for the Gummel map*(to appear).**[19]**Peter A. Markowich,*A singular perturbation analysis of the fundamental semiconductor device equations*, SIAM J. Appl. Math.**44**(1984), no. 5, 896–928. MR**759704**, 10.1137/0144064**[20]**-,*The stationary semiconductor device equations*, Springer-Verlag, Wien-New York, 1985.**[21]**Peter A. Markowich and C. A. Ringhofer,*A singularly perturbed boundary value problem modelling a semiconductor device*, SIAM J. Appl. Math.**44**(1984), no. 2, 231–256. MR**739302**, 10.1137/0144018**[22]**Peter A. Markowich and Miloš A. Zlámal,*Inverse-average-type finite element discretizations of selfadjoint second-order elliptic problems*, Math. Comp.**51**(1988), no. 184, 431–449. MR**930223**, 10.1090/S0025-5718-1988-0930223-7**[23]**M. S. Mock,*On equations describing steady-state carrier distributions in a semiconductor device*, Comm. Pure Appl. Math.**25**(1972), 781–792. MR**0323233****[24]**M. S. Mock,*An initial value problem from semiconductor device theory*, SIAM J. Math. Anal.**5**(1974), 597–612. MR**0417573****[25]**Michael S. Mock,*Analysis of mathematical models of semiconductor devices*, Advances in Numerical Computation Series, vol. 3, Boole Press, Dún Laoghaire, 1983. MR**697094****[26]**C. Ringhofer and C. Schmeiser,*An approximate Newton method for the solution of the basic semiconductor device equations*, SIAM J. Numer. Anal.**26**(1989), no. 3, 507–516. MR**997653**, 10.1137/0726030**[27]**S. Selberherr and C. Ringhofer,*Discretization methods for the semiconductor equations*, Proc. NASECODE III Conf., Boole Press, Dublin, 1983, pp. 31-45.**[28]**E. Stephan and J. R. Whiteman,*Singularities of the Laplacian at corners and edges of three-dimensional domains and their treatment with finite element methods*, Math. Methods Appl. Sci.**10**(1988), no. 3, 339–350. MR**949661**, 10.1002/mma.1670100309**[29]**Vidar Thomée,*Galerkin finite element methods for parabolic problems*, Lecture Notes in Mathematics, vol. 1054, Springer-Verlag, Berlin, 1984. MR**744045****[30]**Yuanming Wang,*Mathematical model and its analysis for the carrier transport in semiconductor devices*, Appl. Math. J. Chinese Univ.**2**(1987), 228-240.**[31]**Jiang Zhu,*Finite element methods for nonlinear Sobolev equations*, Northeast. Math. J.**5**(1989), no. 2, 179–196 (Chinese, with English summary). MR**1034130****[32]**Jiang Zhu,*Finite difference methods for the semiconductor device equations with magnetic influence*(to appear).**[33]**Miloš Zlámal,*Finite element solution of the fundamental equations of semiconductor devices. I*, Math. Comp.**46**(1986), no. 173, 27–43. MR**815829**, 10.1090/S0025-5718-1986-0815829-6**[34]**-,*Finite element solution of the fundamental equations of semiconductor devices*. II (to appear).

Retrieve articles in *Mathematics of Computation*
with MSC:
65N30,
65N12

Retrieve articles in all journals with MSC: 65N30, 65N12

Additional Information

DOI:
http://dx.doi.org/10.1090/S0025-5718-1992-1134742-3

Keywords:
Finite element approximation,
semiconductor device equation,
magnetic field

Article copyright:
© Copyright 1992
American Mathematical Society