Klaus Gärtner

Existence of Bounded Steady State Solutions to Spin-Polarized Drift-Diffusion Systems

We study a stationary spin-polarized drift-diffusion model for semiconductor spintronic devices. This coupled system of continuity equations and a Poisson equation with mixed boundary conditions in all equations has to be considered in heterostructures. In 3D we prove the existence and boundedness of steady states. If the Dirichlet conditions are compatible or nearly compatible with thermodynamic equilibrium, the solution is unique. The same properties are obtained for a space discretized version of the problem: Using a Scharfetter–Gummel scheme on 3D boundary conforming Delaunay grids, we show existence, boundedness, and, for small applied voltages, the uniqueness of the discrete solution.

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 F(η) = 1/(exp(−η)+γ). 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.

Generalization of the Scharfetter-Gummel scheme

For Blakemore-type distribution functions F(η) = 1/(exp(-η)+γ) describing the carrier density in semiconductors a generalization of the classical Scharfetter-Gummel scheme can be derived resulting in a nonlinear equation per edge to calculate the edge current. This approach provides a good approximation of the carrier density in degenerate semiconductors for values of the chemical potential η <; 1.3kBT. We discuss an extension of this approach based on a piecewise approximation of the distribution function by functions of that type in order improve the approximation for larger values of the chemical potential.

On modifications of the Scharfetter-Gummel scheme for drift-diffusion equations with Fermi-like statistical distribution functions

Driven by applications in fields like organic semiconductors there is an increased interest in numerical simulations based on drift-diffusion models with general statistical distribution functions. It is important to keep the well known qualitative properties of the Scharfetter-Gummel finite volume scheme, like positivity of solutions, dissipativity and consistency with thermodynamic equilibrium. A proper generalization to general statistical distribution functions is a topic of current research. The paper presents different state-of-the-art approaches to solve this problem. Their issues and advantages are discussed, and their practical performance is evaluated for real device structures.

Electro-thermal modeling of organic semiconductors describing negative differential resistance induced by self-heating

We discuss self-heating of organic semiconductor devices based on Arrhenius-like conductivity laws. The self-consistent calculation of charge and heat transport explains thermal switching, bistability, and hysteresis resulting from S-shaped current-voltage curves with regions of negative differential resistance (NDR). For large area thin film organic devices we study the appearance of a spatially localized NDR region and the spatial evolution of this NDR region in dependence on the total current. We propose that in organic light emitting diodes (OLEDs) these effects are responsible for spatially inhomogeneous current flow and inhomogeneous luminance at high power.

A Multigrid Method for the Solution of a Convection — Diffusion Equation with Rapidly Varying Coefficients

Consider the following equations

Energy estimates for continuous and discretized electro-reaction-diffusion systems

- div\,(gradn\, - \,n\,grad\,\psi)\, = \,f

Self-heating effects in organic semiconductor crossbar structures with small active area

and (for u = e −ψ n)