Open Access
Volume 57, Number 4, July-August 2023
Page(s) 2557 - 2593
Published online 03 August 2023
  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. [CrossRef] [MathSciNet] [Google Scholar]
  2. L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L.D. Marini and A. Russo, Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23 (2013) 199–214. [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. [CrossRef] [MathSciNet] [Google Scholar]
  4. M. Bessemoulin-Chatard and C. Chainais-Hillairet, Exponential decay of a finite volume scheme to the thermal equilibrium for drift–diffusion systems. J. Numer. Math. 25 (2017) 147–168. [CrossRef] [MathSciNet] [Google Scholar]
  5. M. Bessemoulin-Chatard and C. Chainais-Hillairet, Uniform-in-time bounds for approximate solutions of the drift–diffusion system. Numer. Math. 141 (2019) 881–916. [CrossRef] [MathSciNet] [Google Scholar]
  6. 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. [CrossRef] [MathSciNet] [Google Scholar]
  7. J. Blakemore, Approximations for Fermi-Dirac integrals, especially the function # used to describe electron density in a semiconductor. Solid-State Electron. 25 (1982) 1067–1076. [CrossRef] [Google Scholar]
  8. X. Blanc and E. Labourasse, A positive scheme for diffusion problems on deformed meshes. J. Appl. Math. Mech./Z. Angew. Math. Mech. 96 (2014) 660–680. [Google Scholar]
  9. F. Brezzi, L.D. Marini and P. Pietra, Numerical simulation of semiconductor devices Comput. Methods Appl. Mech. Eng. 75 (1989) 493–514. [CrossRef] [Google Scholar]
  10. F. Brezzi, L.D. Marini and P. Pietra, Two-dimensional exponential fitting and applications to drift–diffusion models. SIAM J. Numer. Anal. 26 (1989) 1342–1355. [CrossRef] [MathSciNet] [Google Scholar]
  11. F. Brezzi, L.D. Marini, S. Micheletti, P. Pietra, R. Sacco and S. Wang, Discretization of semiconductor device problems. I, in Handbook of Numerical Analysis. Vol. XIII. Special volume: Numerical Methods in Electromagnetics. Amsterdam, Elsevier/North Holland (2005) 317–441. [Google Scholar]
  12. J.-S. Camier and F. Hermeline, A monotone nonlinear finite volume method for approximating diffusion operators on general meshes. Int. J. Numer. Methods Eng. 107 (2016) 496–519. [CrossRef] [Google Scholar]
  13. C. Cancès, Energy stable numerical methods for porous media flow type problems. Oil Gas Sci. Technol. – Rev. IFP / Énergies nouvelles 73 (2018) 78. [CrossRef] [Google Scholar]
  14. C. Cancès and C. Guichard, Convergence of a nonlinear entropy diminishing Control Volume Finite Element scheme for solving anisotropic degenerate parabolic equations. Math. Comp. 85 (2016) 549–580. [Google Scholar]
  15. C. Cancès and C. Guichard, Numerical analysis of a robust free energy diminishing finite volume scheme for parabolic equations with gradient structure. Found. Comput. Math. 17 (2017) 1525–1584. [Google Scholar]
  16. C. Cancès, C. Chainais-Hillairet and S. Krell, Numerical analysis of a nonlinear free-energy diminishing Discrete Duality Finite Volume scheme for convection diffusion equations. Comput. Methods Appl. Math. 18 (2018) 407–432. [CrossRef] [MathSciNet] [Google Scholar]
  17. C. Cancès, C. Chainais-Hillairet, M. Herda and S. Krell, Large time behavior of nonlinear finite volume schemes for convection diffusion equations. SIAM J. Numer. Anal. 58 (2020) 2544–2571. [CrossRef] [MathSciNet] [Google Scholar]
  18. X. Cao and H. Huang, An adaptive conservative finite volume method for Poisson–Nernst–Planck equations on a moving mesh. Commun. Comput. Phys. 26 (2019) 389–412. [CrossRef] [MathSciNet] [Google Scholar]
  19. C. Chainais-Hillairet, Discrete duality finite volume schemes for two-dimensional drift–diffusion and energy-transport models. Int. J. Numer. Methods Fluids 59 (2009) 239–257. [CrossRef] [Google Scholar]
  20. C. Chainais-Hillairet and F. Filbet, Asymptotic behaviour of a finite-volume scheme for the transient drift–diffusion model. IMA J. Numer. Anal. 27 (2007) 689–716. [CrossRef] [MathSciNet] [Google Scholar]
  21. C. Chainais-Hillairet, M. Herda, S. Lemaire and J. Moatti, Long-time behaviour of hybrid finite volume schemes for advection– diffusion equations: linear and nonlinear approaches. Numer. Math. 151 (2022) 963–1016. [CrossRef] [MathSciNet] [Google Scholar]
  22. K. Domelevo and P. Omnes, A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids. ESAIM Math. Model. Numer. Anal. 39 (2005) 1203–1249. [CrossRef] [EDP Sciences] [Google Scholar]
  23. J. Droniou, Finite volume schemes for diffusion equations: introduction to and review of modern methods. Math. Models Methods Appl. Sci. 24 (2014) 1575–1619. [Google Scholar]
  24. J. Droniou, R. Eymard, T. Gallouët, C. Guichard and R. Herbin, The gradient discretisation method, in Mathématiques & Applications, Vol. 82, Springer International Publishing, Cham, Switzerland (2018). [CrossRef] [Google Scholar]
  25. L.C. Evans, Partial differential equations: second edition, in Graduate Studies in Mathematics. Vol. 19. American Mathematical Society, Providence, R.I. (2010). [CrossRef] [Google Scholar]
  26. R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in Techniques of Scientific Computing (Part 3), Handb. Numer. Anal., VII, North-Holland, Amsterdam (2000) 713–1020. [Google Scholar]
  27. R. Eymard, T. Gallouët and R. Herbin, Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes. SUSHI: a scheme using stabilization and hybrid interfaces. IMA J. Numer. Anal. 30 (2010) 1009–1043. [CrossRef] [MathSciNet] [Google Scholar]
  28. P. Farrell, T. Koprucki and J. Fuhrmann, Computational and analytical comparison of flux discretizations for the semiconductor device equations beyond Boltzmann statistics. J. Comput. Phys. 346 (2017) 497–513. [CrossRef] [MathSciNet] [Google Scholar]
  29. P. Farrell, M. Patriarca, J. Fuhrmann and T. Koprucki, Comparison of thermodynamically consistent charge carrier flux discretizations for Fermi-Dirac and Gauss-Fermi statistics. Opt. Quantum Electron. 50 (2018) 1–10. [CrossRef] [Google Scholar]
  30. P. Farrell, N. Rotundo, D.H. Doan, M. Kantner, J. Fuhrmann and T. Koprucki, Drift-diffusion models, in Handbook of Optoelectronic Device Modeling and Simulation, CRC Press, 2017, 733–772. [CrossRef] [Google Scholar]
  31. F. Filbet and M. Herda, A finite volume scheme for boundary-driven convection–diffusion equations with relative entropy structure. Numer. Math. 137 (2017) 535–577. [CrossRef] [MathSciNet] [Google Scholar]
  32. 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. [CrossRef] [MathSciNet] [Google Scholar]
  33. H. Gajewski and K. Gärtner, On the discretization of van Roosbroeck’s equations with magnetic field. Z. Angew. Math. Mech. 76 (1996) 247–264. [CrossRef] [MathSciNet] [Google Scholar]
  34. H. Gajewski and K. Gröger, Semiconductor equations for variable mobilities based on Boltzmann statistics or Fermi-Dirac statistics. Math. Nachr. 140 (1989) 7–36. [CrossRef] [MathSciNet] [Google Scholar]
  35. A. Glitzky, M. Liero and G. Nika, An existence result for a class of electrothermal drift–diffusion models with Gauss-Fermi statistics for organic semiconductors. Anal. Appl. 19 (2021) 275–304. [CrossRef] [MathSciNet] [Google Scholar]
  36. R. Herbin and F. Hubert, Benchmark on discretization schemes for anisotropic diffusion problems on general grids, in Finite Volumes for Complex Applications V – Problems & Perspectives, Edited R. Eymard and J.-M. Hérard. ISTE, London (2008) 659–692. [Google Scholar]
  37. F. Hermeline, A finite volume method for the approximation of diffusion operators on distorted meshes. J. Comput. Phys. 160 (2000) 481–499. [Google Scholar]
  38. A. Jüngel, Entropy Methods for Diffusive Partial Differential Equations, in SpringerBriefs in Mathematics, Springer International Publishing, Cham, Switzerland (2016). [CrossRef] [Google Scholar]
  39. A. Jüngel and P. Pietra, A discretization scheme for a quasi-hydrodynamic semiconductor model. Math. Models Methods Appl. Sci. 7 (1997) 935–955. [CrossRef] [Google Scholar]
  40. P.A. Markowich, The stationary semiconductor device equations, in Computational Microelectronics, Springer-Verlag, Vienna (1986). [CrossRef] [Google Scholar]
  41. P.A. Markowich and A. Unterreiter, Vacuum solutions of a stationary drift–diffusion model. Ann. Sc. norm. super. Pisa Cl. Sci., Ser. 4 20 (1993) 371–386. [Google Scholar]
  42. P.A. Markowich, C.A. Ringhofer and C. Schmeiser, Semiconductor Equations. Springer-Verlag, Wien (1990). [CrossRef] [Google Scholar]
  43. M.S. Mock, An initial value problem from semiconductor device theory. SIAM J. Math. Anal. 5 (1974) 597–612. [CrossRef] [MathSciNet] [Google Scholar]
  44. D.L. Scharfetter and H.K. Gummel, Large-signal analysis of a silicon Read diode oscillator. IEEE Trans. Electron Devices 16 (1969) 64–77. [CrossRef] [Google Scholar]
  45. S. Su and H. Tang, A positivity-preserving and free energy dissipative hybrid scheme for the Poisson–Nernst–Planck equations on polygonal and polyhedral meshes. Comput. Math. Appl. 108 (2022) 33–48. [CrossRef] [MathSciNet] [Google Scholar]
  46. 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. B 78 (2008) 085207. [CrossRef] [Google Scholar]
  47. W. Van Roosbroeck, Theory of the flow of electrons and holes in germanium and other semiconductors. Bell Syst. Tech. J. 29 (1950) 560–607. [CrossRef] [Google Scholar]

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