Discretization scheme for drift-diffusion equations with strong diffusion enhancement

Inspired by organic semiconductor models based on hopping transport introducing Gauss-Fermi integrals a nonlinear generalization of the classical Scharfetter–Gummel scheme is derived for the distribution function $$\mathcal F_\gamma (\eta ) = 1/(\exp (-\eta )+\gamma )$$. This function provides an approximation of the Fermi-Dirac integrals of different order and restricted argument ranges. The scheme requires the solution of a nonlinear equation per edge and continuity equation to calculate the edge currents. In the current formula the density-dependent diffusion enhancement factor, resulting from the generalized Einstein relation, shows up as a weighting factor. Additionally the current modifies the argument of the Bernoulli functions.

[1]  H. Gummel,et al.  Large-signal analysis of a silicon Read diode oscillator , 1969 .

[2]  R. Coehoorn,et al.  Effect of Gaussian disorder on the voltage dependence of the current density in sandwich-type devices based on organic semiconductors , 2008 .

[3]  H. Gajewski,et al.  Thermodynamic design of energy models of semiconductor devices , 2002 .

[4]  Onno W. Purbo,et al.  Numerical model for degenerate and heterostructure semiconductor devices , 1989 .

[5]  G. Paasch,et al.  Charge carrier density of organics with Gaussian density of states: Analytical approximation for the Gauss–Fermi integral , 2010 .

[6]  R. Coehoorn,et al.  Charge-carrier concentration dependence of the hopping mobility in organic materials with Gaussian disorder , 2005 .

[7]  J. S. Blakemore,et al.  The Parameters of Partially Degenerate Semiconductors , 1952 .

[8]  Marianne Bessemoulin-Chatard,et al.  A finite volume scheme for convection–diffusion equations with nonlinear diffusion derived from the Scharfetter–Gummel scheme , 2010, Numerische Mathematik.

[9]  K. Gartner.  Existence of Bounded Discrete Steady-State Solutions of the Van Roosbroeck System on Boundary Conforming Delaunay Grids , 2007 .

[10]  Uwe Bandelow,et al.  Fabry-Perot Lasers: Thermodynamics-Based Modeling , 2005 .

[11]  Robert Eymard,et al.  A finite volume scheme for nonlinear parabolic equations derived from one-dimensional local dirichlet problems , 2006, Numerische Mathematik.

[12]  Piprek.  Optoelectronic Devices , 2005 .