Driven by applications like organic semiconductors there is an increased interest in numerical simulations based on drift-diffusion models with arbitrary statistical distribution functions. This requires numerical schemes that preserve qualitative properties of the solutions, such as positivity of densities, dissipativity and consistency with thermodynamic equilibrium. An extension of the Scharfetter–Gummel scheme guaranteeing consistency with thermodynamic equilibrium is studied. It is derived by replacing the thermal voltage with an averaged diffusion enhancement for which we provide a new explicit formula. This approach avoids solving the costly local nonlinear equations defining the current for generalized Scharfetter–Gummel schemes.
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Bank, R.E., Rose, D.J.: Some error estimates for the box method. SIAM J. Numer. Anal. 24(4), 777–787 (1987)
Bessemoulin-Chatard, M.: A finite volume scheme for convection-diffusion equations with nonlinear diffusion derived from the Scharfetter–Gummel scheme. Numer. Math. 121, 637–670 (2012)
Blakemore, J.: The parameters of partially degenerate semiconductors. Proc. Phys. Soc. Lond. A 65, 460–461 (1952)
Eymard, R., Fuhrmann, J., Gärtner, K.: A finite volume scheme for nonlinear parabolic equations derived from one-dimensional local dirichlet problems. Numer. Math. 102, 463–495 (2006)
Jüngel, A.: Numerical approximation of a drift-diffusion model for semiconductors with nonlinear diffusion. ZAMM 75(10), 783–799 (1995)
Koprucki, T., Gärtner, K.: Discretization scheme for drift-diffusion equations with strong diffusion enhancement. Opt. Quantum Electron. 45(7), 791–796 (2013a)
Koprucki T, Gärtner K: Generalization of the Scharfetter–Gummel scheme. In: 2013 13th International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD), pp. 85–86 (2013b)
Macneal, R.H.: An asymmetrical finite difference network. Q. Math. Appl. 11, 295–310 (1953)
Purbo, O.W., Cassidy, D.T., Chisholm, S.H.: Numerical model for degenerate and heterostructure semiconductor devices. J. Appl. Phys. 66(10), 5078–5082 (1989)
Scharfetter, D.L., Gummel, H.K.: Large signal analysis of a silicon Read diode. IEEE Trans. Electron. Dev. 16, 64–77 (1969)
Si, H., Gärtner, K., Fuhrmann, J.: Boundary conforming Delaunay mesh generation. Comput. Math. Math. Phys. 50, 38–53 (2010)
Stodtmann, S., Lee, R.M., Weiler, C.K.F., Badinski, A.: Numerical simulation of organic semiconductor devices with high carrier densities. J. Appl. Phys. 112(11), 114–909 (2012)
van Mensfoort, S.L.M., Coehoorn, R.: Effect of Gaussian disorder on the voltage dependence of the current density in sandwich-type devices based on organic semiconductors. Phys. Rev. B 78(8), 085–207 (2008)
The work has been supported by ERC-2010-AdG no. 267802 Analysis of Multiscale Systems Driven by Functionals (N.R.) and by Deutsche Forschungsgemeinschaft (DFG) within the collaborative research center 787 Semiconductor Nanophotonics (T.K.).
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Koprucki, T., Rotundo, N., Farrell, P. et al. On thermodynamic consistency of a Scharfetter–Gummel scheme based on a modified thermal voltage for drift-diffusion equations with diffusion enhancement. Opt Quant Electron 47, 1327–1332 (2015). https://doi.org/10.1007/s11082-014-0050-9
- Scharfetter–Gummel scheme
- Thermodynamic consistency