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Classical solutions of drift-diffusion equations for semiconductor devices: The two-dimensional case. (English) Zbl 1167.35390
Summary: We regard drift-diffusion equations for semiconductor devices in Lebesgue spaces. To that end we reformulate the (generalized) van Roosbroeck system as an evolution equation for the potentials to the driving forces of the currents of electrons and holes. This evolution equation falls into a class of quasi-linear parabolic systems which allow unique, local in time solution in certain Lebesgue spaces. In particular, it turns out that the divergence of the electron and hole currents is an integrable function. Hence, Gauss’ theorem applies, and gives the foundation for space discretization of the equations by means of finite volume schemes. Moreover, the strong differentiability of the electron and hole density in time is constitutive for the implicit time discretization scheme. Finite volume discretization of space, and implicit time discretization are accepted custom in engineering and scientific computing. This investigation puts special emphasis on non-smooth spatial domains, mixed boundary conditions, and heterogeneous material compositions, as required in electronic device simulation.

35K50 Systems of parabolic equations, boundary value problems (MSC2000)
35R05 PDEs with low regular coefficients and/or low regular data
35K55 Nonlinear parabolic equations
35K57 Reaction-diffusion equations
78A35 Motion of charged particles
Full Text: DOI
[1] van Roosbroeck, W., Theory of the flow of electrons and holes in germanium and other semiconductors, Bell syst. tech. J., 29, 560-607, (1950) · Zbl 1372.35295
[2] Selberherr, S., Analysis and simulation of semiconductor devices, (1984), Springer Wien
[3] Markowich, Peter A., The stationary semiconductor device equations, () · Zbl 0556.65070
[4] Markowich, P.A.; Ringhofer, C.A.; Schmeiser, C., Semiconductor equations, (1990), Springer-Verlag Vienna · Zbl 0765.35001
[5] Gajewski, H., Analysis und numerik von ladungstransport in halbleitern (analysis and numerics of carrier transport in semiconductors), Mitt. ges. angew. math. mech., 16, 1, 35-57, (1993), (in German)
[6] Degond, P.; Schmeiser, C., Macroscopic models for semiconductor heterostructures, J. math. phys., 39, 9, 4634-4663, (1998) · Zbl 0938.82049
[7] Degond, Pierre, Mathematical modelling of microelectronics semiconductor devices, (), 77-110 · Zbl 0956.35121
[8] Gummel, H.K., A self-consistent iterative scheme for one-dimensional steady state calculations, IEEE trans. electron devices, 11, 455, (1964)
[9] Mock, M.S., On equations describing steady – state carrier distributions in a semiconductor device, Comm. pure appl. math., 25, 781-792, (1972)
[10] Mock, M.S., An initial value problem from semiconductor device theory, SIAM J. math. anal., 5, 597-612, (1974) · Zbl 0254.35020
[11] Wu, Hao; Markowich, Peter A.; Zheng, Songmu, Global existence and asymptotic behavior for a semiconductor drift – diffusion-Poisson model, Math. models methods appl. sci., 18, 3, 443-487, (2008) · Zbl 1157.35013
[12] Gajewski, H.; Gröger, K., On the basic equations for carrier transport in semiconductors, J. math. anal. appl., 113, 12-35, (1986) · Zbl 0642.35038
[13] Gajewski, H.; Gröger, K., Semiconductor equations for variable mobilities based on Boltzmann statistics or fermi – dirac statistics, Math. nachr., 140, 7-36, (1989) · Zbl 0681.35081
[14] Gajewski, H.; Gröger, K., Initial boundary value problems modelling heterogeneous semiconductor devices, (), 4-53
[15] Gajewski, H.; Skrypnik, I.V., On the uniqueness of solutions for nonlinear elliptic-parabolic problems, J. evol. equ., 3, 247-281, (2003) · Zbl 1033.35016
[16] Gajewski, Herbert; Skrypnik, Igor V., Existence and uniqueness results for reaction-diffusion processes of electrically charged species, (), 151-188 · Zbl 1090.35007
[17] Gajewski, H.; Kaiser, H.-Chr.; Langmach, H.; Nürnberg, R.; Richter, R.H., Mathematical modeling and numerical simulation of semiconductor detectors, (), 355-364 · Zbl 1036.82029
[18] Kaiser, H.-Chr.; Neidhardt, H.; Rehberg, J., Classical solutions of quasilinear parabolic systems on two-dimensional domains, Nodea nonlinear differential equations appl., 13, 3, 287-310, (2006) · Zbl 1105.35047
[19] Bandelow, U.; Gajewski, H.; Kaiser, H.-Chr., Modeling combined effects of carrier injection, photon dynamics and heating in strained multi-quantum well lasers, (), 301-310
[20] Bandelow, U.; Kaiser, H.-Chr.; Koprucki, T.; Rehberg, J., Modeling and simulation of strained quantum wells in semiconductor lasers, (), 377-390 · Zbl 1161.78326
[21] Bandelow, U.; Hünlich, R.; Koprucki, T., Simulation of static and dynamic properties of edge-emitting multiple-quantum-well-lasers, IEEE J. sel. top. quantum electron., 9, 798-806, (2003)
[22] Kaiser, H.-Chr.; Rehberg, J., About a one-dimensional stationary schrödinger – poisson system with kohn – sham potential, Z. angew. math. phys. (ZAMP), 50, 3, 423-458, (1999) · Zbl 0928.35030
[23] Kaiser, H.-Chr.; Rehberg, J., About a stationary schrödinger – poisson system with kohn – sham potential in a bounded two- or three-dimensional domain, Nonlinear anal. TMA, 41, 1-2, 33-72, (2000) · Zbl 0960.35088
[24] Kaiser, H.-Chr.; Neidhardt, H.; Rehberg, J., Macroscopic current induced boundary conditions for schrödinger – type operators, Integral equations operator theory, 45, 39-63, (2003) · Zbl 1033.35079
[25] Quastel, J., Diffusion of color in the simple exclusion process, Comm. pure appl. math., XLV, 623-679, (1992) · Zbl 0769.60097
[26] Quastel, J.; Rezakhanlou, F.; Varadhan, S.R.S., Large deviations for the symmetric exclusion process in dimension \(d \geq 3\), Probab. theory related fields, 113, 1-84, (1999) · Zbl 0928.60087
[27] Giacomin, G.; Lebowitz, J.L., Phase segregation in particle systems with long-range interactions. I. macroscopic limits, J. statist. phys., 87, 37-61, (1997) · Zbl 0937.82037
[28] Giacomin, G.; Lebowitz, J.L., Phase segregation in particle systems with long-range interactions. II. interface motion, SIAM J. appl. math., 58, 1707-1729, (1998) · Zbl 1015.82027
[29] Griepentrog, Jens A., On the unique solvability of a nonlocal phase separation problem for multicomponent systems, Banach center publ., 66, 153-164, (2004) · Zbl 1235.35149
[30] Koprucki, Thomas; Kaiser, Hans-Christoph; Fuhrmann, Jürgen, Electronic states in semiconductor nanostructures and upscaling to semi-classical models, (), 367-396 · Zbl 1366.82065
[31] Wünsche, H.J.; Bandelow, U.; Wenzel, H., Calculation of combined lateral and longitudinal spatial hole burning in \(\lambda / 4\) shifted DFB lasers, IEEE J. quantum electron., 29, 6, 1751-1761, (1993)
[32] Landsberg, P.T., Recombination in semiconductors, (1991), Cambridge University Press Cambridge · Zbl 0077.23802
[33] H.-J. Wünsche, Modellierung optoelektronischer Bauelemente, NUMSIM ’91 in: H. Gajewski, P. Deuflhard, P.A. Markowich (Eds.), Konrad-Zuse-Zentrum für Informationstechnik, Technical Report TR 91-8, Berlin, 1991, pp. 18-23
[34] Grisvard, P., ()
[35] Triebel, H., Interpolation theory, function spaces, differential operators, (1978), Dt. Verl. d. Wiss. Berlin, North Holland, Amsterdam, 1978; Mir, Moscow 1980 · Zbl 0387.46032
[36] Gröger, K., A \(W^{1, p}\)-estimate for solutions to mixed boundary value problems for second order elliptic differential equations, Math. ann., 283, 679-687, (1989) · Zbl 0646.35024
[37] Gröger, K.; Rehberg, J., Resolvent estimates in \(W^{1, p}\) for second order elliptic differential operators in case of mixed boundary conditions, Math. ann., 285, 105-113, (1989) · Zbl 0659.35032
[38] Ciarlet, P.G., The finite element method for elliptic problems, () · Zbl 0285.65072
[39] Gajewski, H.; Gröger, K.; Zacharias, K., Nichtlineare operatorgleichungen und operatordifferentialgleichungen (nonlinear operator equations and operator differential equations), (1974), Akademie-Verlag Berlin, German
[40] Trudinger, N.S., On imbeddings into Orlicz spaces and some applications, J. math. mech., 17, 473-483, (1967) · Zbl 0163.36402
[41] Appell, Jürgen; Zabrejko, Petr P., Nonlinear superposition operators, (1990), Cambridge University Press · Zbl 0701.47041
[42] Recke, Lutz; Gröger, Konrad, Applications of differential calculus to quasilinear elliptic boundary value problems with non-smooth data, Nodea nonlinear differential equations appl., 13, 3, 263-285, (2006) · Zbl 1387.35185
[43] Kato, T., ()
[44] Temam, R., Navier – stokes equations — theory and numerical analysis, (1979), North Holland Publishing Company Amsterdam, New York, Oxford · Zbl 0426.35003
[45] Evans, L.C.; Gariepy, R.F., Measure theory and fine properties of functions, (1992), CRC Press Boca Raton, Ann Arbor, London · Zbl 0626.49007
[46] Gajewski, H.; Gärtner, K., On the discretization of Van roosbroeck’s equations with magnetic field, Z. angew. math. mech., 76, 247-265, (1996) · Zbl 0873.35004
[47] Fuhrmann, Jürgen; Langmach, Hartmut, Stability and existence of solutions of time-implicit finite volume schemes for viscous nonlinear conservation laws, Appl. numer. math., 37, 1-2, 201-230, (2001) · Zbl 0978.65081
[48] Eymard, Robert; Fuhrmann, Jürgen; Gärtner, Klaus, A finite volume scheme for nonlinear parabolic equations derived from one-dimensional local Dirichlet problems, Numer. math., 102, 3, 463-495, (2006) · Zbl 1116.65101
[49] Scharfetter, D.L.; Gummel, H.K., Large-signal analysis of a silicon Read diode oscillator, IEEE trans. on electron devices, 16, 64-77, (1969)
[50] Gärtner, K.; Richter, R.H., DEPFET sensor design using an experimental 3d device simulator, Nucl. instrum. methods phys. res. A, 568, 12-17, (2006)
[51] Gärtner, K., DEPFET sensors, a test case to study 3d effects, J. comput. electronics, 6, 275-278, (2007)
[52] Meyers, N., An \(L^p\)-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. scuola norm. sup. Pisa, sci. fis. mat., III. ser., 17, 189-206, (1963) · Zbl 0127.31904
[53] Elschner, Johannes; Kaiser, Hans-Christoph; Rehberg, Joachim; Schmidt, Gunther, \(W^{1, q}\) regularity results for elliptic transmission problems on heterogeneous polyhedra, Math. models methods appl. sci., 17, 4, 593-615, (2007) · Zbl 1206.35059
[54] Elschner, Johannes; Rehberg, Joachim; Schmidt, Gunther, Optimal regularity for elliptic transmission problems including \(C^1\) interfaces, Interfaces free bound., 9, 2, 233-252, (2007) · Zbl 1147.47034
[55] Dauge, Monique, Neumann and mixed problems on curvilinear polyhedra, Integral equations operator theory, 15, 2, 227-261, (1992) · Zbl 0767.46026
[56] Haller-Dintelmann, Robert; Kaiser, Hans-Christoph; Rehberg, Joachim, Elliptic model problems including mixed boundary conditions and material heterogeneities, J. math. pures appl., 89, 1, 25-48, (2008) · Zbl 1132.35022
[57] Rehberg, Joachim, Quasilinear parabolic equations in \(L^p\), (), 413-419 · Zbl 1092.35047
[58] Hieber, Matthias; Rehberg, Joachim, Quasilinear parabolic systems with mixed boundary conditions on nonsmooth domains, SIAM J. math. anal., 40, 1, 292-305, (2008) · Zbl 1221.35194
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