Comparison of thermodynamically consistent charge carrier flux discretizations for Fermi--Dirac and Gauss--Fermi statistics
- Farrell, Patricio
- Patriarca, Matteo
- Fuhrmann, Jürgen
- Koprucki, Thomas
2010 Mathematics Subject Classification
- 65N08 35K55
- Scharfetter--Gummel schemes, (organic) semiconductors, nonlinear diffusion, ermodynamic consistency, finite volume scheme, Gauss--Fermi integral, Fermi--Dirac integral
We compare three thermodynamically consistent Scharfetter--Gummel schemes for different distribution functions for the carrier densities, including the Fermi--Dirac integral of order 1/2 and the Gauss--Fermi integral. The most accurate (but unfortunately also most costly) generalized Scharfetter--Gummel scheme requires the solution of an integral equation. We propose a new method to solve this integral equation numerically based on Gauss quadrature and Newton's method. We discuss the quality of this approximation and plot the resulting currents for Fermi--Dirac and Gauss--Fermi statistics. Finally, by comparing two modified (diffusion-enhanced and inverse activity based) Scharfetter--Gummel schemes with the more accurate generalized scheme, we show that the diffusion-enhanced ansatz leads to considerably lower flux errors, confirming previous results (J. Comp. Phys. 346:497-513, 2017).
- Opt. Quantum Electron., 50 (2018), pp. 101/1--101/10, DOI 10.1007/s11082-018-1349-8 .