Accessible Requires Authentication Published by De Gruyter July 31, 2016

Exponential decay of a finite volume scheme to the thermal equilibrium for drift–diffusion systems

Marianne Bessemoulin-Chatard and Claire Chainais-Hillairet

Abstract

In this paper, we study the large-time behavior of a numerical scheme discretizing drift–diffusion systems for semiconductors. The numerical method is finite volume in space, implicit in time, and the numerical fluxes are a generalization of the classical Scharfetter–Gummel scheme which allows to consider both linear or nonlinear pressure laws.

We study the convergence of approximate solutions towards an approximation of the thermal equilibrium state as time tends to infinity, and obtain a decay rate by controlling the discrete relative entropy with the entropy production. This result is proved under assumptions of existence and uniform in time L-estimates for numerical solutions, which are then discussed. We conclude by presenting some numerical illustrations of the stated results.

JEL Classification: 65M08; 82D37

Funding statement: The first author thanks the project ANR-12-IS01-0004 GeoNum and the project ANR-14-CE25-0001 Achylles for their partial financial contributions. The second author thanks the team Inria/Rapsodi, the ANR MOONRISE and the Labex CEMPI (ANR-11-LABX-0007-01) for their support.

References

[1] A. Arnold, J. A. Carrillo, L. Desvillettes, J. Dolbeault, A. Jüngel, C. Lederman, P. A. Markowich, G. Toscani, and C. Villani, Entropies and equilibria of many-particle systems: an essay on recent research, Monatsh. Math. 142 (2004), 35–43.10.1007/s00605-004-0239-2 Search in Google Scholar

[2] A. Arnold, P. Markowich, G. Toscani, and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker–Planck type equations, Comm. Partial Differ. Equ.26 (2001), 43–100.10.1081/PDE-100002246 Search in Google Scholar

[3] M. Bessemoulin-Chatard, A finite volume scheme for convection–diffusion equations with nonlinear diffusion derived from the Scharfetter–Gummel scheme, Numer. Math.121 (2012), 637–670.10.1007/s00211-012-0448-x Search in Google Scholar

[4] M. Bessemoulin-Chatard, C. Chainais-Hillairet, and F. Filbet, On discrete functional inequalities for some finite volume schemes, IMA J. Numer. Anal.35 (2015), 1125–1149.10.1093/imanum/dru032 Search in Google Scholar

[5] M. Bessemoulin-Chatard, C. Chainais-Hillairet, and M.-H. Vignal, Study of a finite volume scheme for the drift–diffusion system. Asymptotic behavior in the quasi-neutral limit, SIAM J. Numer. Anal. 52 (2014), 1666–1691.10.1137/130913432 Search in Google Scholar

[6] J. A. Carrillo, A. Jüngel, P. Markowich, G. Toscani, and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math. 133 (2001), 1–82.10.1007/s006050170032 Search in Google Scholar

[7] C. Chainais-Hillairet, and F. Filbet, Asymptotic behavior of a finite volume scheme for the transient drift–diffusion model, IMA J. Numer. Anal. 27 (2007), 689–716.10.1093/imanum/drl045 Search in Google Scholar

[8] M. Chatard, Asymptotic behavior of the Scharfetter–Gummel scheme for the drift–diffusion model, Finite Volumes for Complex Applications VI Problems & Perspectives (Eds. J. Fořt, J. Füřst, J. Halama, R. Herbin, and F. Hubert), Springer Proc. in Mathematics 4, Springer Berlin Heidelberg, 2011, pp. 235–243. Search in Google Scholar

[9] R. Eymard, J. Fuhrmann, and K. Gärtner, A finite volume scheme for nonlinear parabolic equations derived from one-dimensional local Dirichlet problems, Numer. Math. 102 (2006), 463–495.10.1007/s00211-005-0659-5 Search in Google Scholar

[10] R. Eymard, T. Gallouët, and R. Herbin, Finite Volume Methods, Handbook of Numerical Analysis, VII, North-Holland, Amsterdam, 2000, pp. 713–1020. Search in Google Scholar

[11] J. M. Foster, T. Snaith, H. J. Leijtens, and G. Richardson, A model for the operation of Perovskite based hybrid solar cells: formulation, analysis, and comparison to experiment, SIAM J. Appl. Math.74 (2014), 1935–1966.10.1137/130934258 Search in Google Scholar

[12] 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), 101–108.10.1002/zamm.19850650210 Search in Google Scholar

[13] H. Gajewski and K. Gärtner, On the discretization of Van Roosbroeck’s equations with magnetic field, Z. Angew. Math. Mech. 76 (1996), 247–264.10.1002/zamm.19960760502 Search in Google Scholar

[14] H. Gajewski and K. Gröger, On the basic equations for carrier transport in semiconductors, J. Math. Anal. Appl. 113 (1986), 12–35.10.1016/0022-247X(86)90330-6 Search in Google Scholar

[15] H. Gajewski and K. Gröger, Semiconductor equations for variable mobilities based on Boltzmann statistics or Fermi–Dirac statistics, Math. Nachr. 140 (1989), 7–36.10.1002/mana.19891400102 Search in Google Scholar

[16] H. Gajewski and K. Gröger, Reaction–diffusion processes of electrically charged species, Math. Nachr. 177 (1996), 109–130.10.1002/mana.19961770108 Search in Google Scholar

[17] K. Gärtner, Existence of bounded discrete steady-state solutions of the van Roosbroeck system on boundary conforming Delaunay grids, SIAM J. Sci. Comput. 31 (2009), 1347–1362.10.1137/070710950 Search in Google Scholar

[18] A. Glitzky, Exponential decay of the free energy for discretized electro–reaction–diffusion systems, Nonlinearity21 (2008). Search in Google Scholar

[19] A. Glitzky, Uniform exponential decay of the free energy for Voronoi finite volume discretized reaction–diffusion systems, Math. Nachr. 284 (2011), 2159–2174.10.1002/mana.200910215 Search in Google Scholar

[20] A. Glitzky and K. Gärtner, Energy estimates for continuous and discretized electro–reaction–diffusion systems, Nonlinear Analysis70 (2009), 788–805.10.1016/j.na.2008.01.015 Search in Google Scholar

[21] A. Glitzky, K. Gröger, and R. Hünlich, Free energy and dissipation rate for reaction–diffusion processes of electrically charged species, Applicable Analysis60 (1996), 201–217.10.1080/00036819608840428 Search in Google Scholar

[22] M. Gruber, B. Stickler, S. Possanner, K. Zojer, and F. Schurrer, Simulation of the performance of organic electronic devices based on a two-dimensional drift–diffusion approach, Comm. Appl. Ind. Math.2 (2011). Search in Google Scholar

[23] A. Il’ln, A difference scheme for a differential equation with a small parameter multiplying the highest derivative, Math. Zametki6 (1969), 237–248. Search in Google Scholar

[24] A. Jüngel, On the existence and uniqueness of transient solutions of a degenerate nonlinear drift–diffusion model for semiconductors, Math. Models Methods Appl. Sci. 4 (1994), 677–703.10.1142/S0218202594000388 Search in Google Scholar

[25] A. Jüngel, Numerical approximation of a drift–diffusion model for semiconductors with nonlinear diffusion, Z. Angew. Math. Mech. 75 (1995), 783–799.10.1002/zamm.19950751016 Search in Google Scholar

[26] A. Jüngel, Qualitative behavior of solutions of a degenerate nonlinear drift–diffusion model for semiconductors, Math. Models Methods Appl. Sci. 5 (1995), 497–518.10.1142/S0218202595000292 Search in Google Scholar

[27] A. Jüngel and P. Pietra, A discretization scheme for a quasi-hydrodynamic semiconductor model, Math. Models Methods Appl. Sci. 7 (1997), 935–955.10.1142/S0218202597000475 Search in Google Scholar

[28] T. Koprucki and K. Gärtner, Discretization scheme for drift–diffusion equations with strong diffusion enhancement, Optical and Quantum Electronics45 (2013), 791–796 (English).10.1007/s11082-013-9673-5 Search in Google Scholar

[29] T. Koprucki and K. Gärtner, Generalization of the Scharfetter–Gummel scheme, In: 13th Int. Conf. on Numerical Simulation of Optoelectronic Devices, pp. 85–86, 2013. Search in Google Scholar

[30] T. Koprucki, N. Rotundo, P. Farrell, D. Doan, and J. Fuhrmann, On thermodynamic consistency of a Scharfetter–Gummel scheme based on a modified thermal voltage for drift–diffusion equations with diffusion enhancement, Optical and Quantum Electronics47 (2015), 1327–1332 (English).10.1007/s11082-014-0050-9 Search in Google Scholar

[31] R. D. Lazarov, I. D. Mishev and P. S. Vassilevski, Finite volume methods for convection–diffusion problems, SIAM J. Numer. Anal. 33 (1996), 31–55.10.1137/0733003 Search in Google Scholar

[32] P. A. Markowich, C. A. Ringhofer, and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990. Search in Google Scholar

[33] P. A. Markowich and A. Unterreiter, Vacuum solutions of a stationary drift–diffusion model, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)20 (1993), 371–386. Search in Google Scholar

[34] P. Markowich, The Stationary Semiconductor Device Equations, Computational Microelectronics, Springer-Verlag, Vienna, 1986. Search in Google Scholar

[35] M. S. Mock, An initial value problem from semiconductor device theory, SIAM J. Math. Anal. 5 (1974), 597–612.10.1137/0505061 Search in Google Scholar

[36] A. Prohl and M. Schmuck, Convergent discretization for the Nernst–Planck–Poisson system, Numer. Math. 111 (2009), 591–630.10.1007/s00211-008-0194-2 Search in Google Scholar

[37] O. W. Purbo, D. T. Cassidy, and S. H. Chisholm, Numerical model for degenerate and heterostructure semiconductor devices, J. Appl. Physics66 (1989), 5078–5082.10.1063/1.343733 Search in Google Scholar

[38] D. Scharfetter and H. Gummel, Large signal analysis of a silicon Read diode, IEEE Trans. Elec. Dev. 16 (1969), 64–77.10.1109/T-ED.1969.16566 Search in Google Scholar

[39] S. Stodtmann, R. M. Lee, C. K. F. Weiler, and A. Badinski, Numerical simulation of organic semiconductor devices with high carrier densities, J. Appl. Physics112 (2012). Search in Google Scholar

[40] S. L. M. van Mensfoort and R. Coehoorn, Effect of Gaussian disorder on the voltage dependence of the current density in sandwich-type devices based on organic semiconductors, Phys. Rev. B78 (2008), 085207.10.1103/PhysRevB.78.085207 Search in Google Scholar

Received: 2016-1-15
Revised: 2016-7-21
Accepted: 2016-7-31
Published Online: 2016-7-31
Published in Print: 2017-9-26

© 2016 by Walter de Gruyter Berlin/Boston