Numerical model for degenerate and heterostructure semiconductor devices

Abstract

A one-dimensional finite-difference numerical model appropriate for devices which require degenerate statistics to describe the carrier concentrations is reported. The Joyce–Dixon approximations [Appl. Phys. Lett. 31, 354 (1977)] to Fermi–Dirac statistics were used and found to be efficient and accurate methods to determine carrier concentrations, to calculate derivatives required in the iterative solution of the equations in the numerical model, and to compare to nondegenerate (i.e., Boltzmann statistics) models. For degenerate carrier statistics and heterostructure devices the Scharfetter–Gummell equations [IEEE Trans. Electron Devices ED-16, 64 (1969)] for the current densities must be modified to include the degenerate Einstein relations and variation of the density of states with position. These modifications, as well the possibilities of using the thermodynamic formulation of current density, are described.

Silicon-Integrated III–V Light Emitters and Absorbers Using Bipolar Diffusion

“…1 is studied with the drift-diffusion (DD) model, which relates the electrostatic potential φ and the quasi-Fermi potentials φ n and φ p to the electron and hole densities n and p (calculated from Fermi-Dirac statistics) and the total net recombination and generation-rate densities R and G as (for background information and physical justification, see, e.g., Ref. [40])…”

On thermodynamic consistency of a Scharfetter–Gummel scheme based on a modified thermal voltage for drift-diffusion equations with diffusion enhancement

“…This kind of extension of the Scharfetter-Gummel scheme to Fermi-Dirac statistics has been suggested using arithmetic (Purbo et al 1989;Jüngel 1995) or geometric (Stodtmann et al 2012) averaging, leading in general to schemes which are not consistent with the thermodynamic equilibrium ( j = 0 for ϕ n = const). Bessemoulin-Chatard (2012) introduced for drift-diffusion equations with nonlinear diffusion a proper local average guaranteeing consistency with thermodynamic equilibrium.…”

“…Strictly monotonically increasing non-Boltzmann distribution functions lead to diffusion enhancement. Various extensions of the Scharfetter-Gummel scheme are suggested to account for this effect, see Purbo et al (1989), Jüngel (1995), Stodtmann et al (2012). Unfortunately, they are not thermodynamically consistent.…”

Exponential decay of a finite volume scheme to the thermal equilibrium for drift–diffusion systems

“…It exploits the exponential dependence on the chemical potential and allows to recover the correct large time behavior. Various extensions of the Scharfetter-Gummel scheme have been suggested to account the diffusion enhancement induced by non Boltzmann statistics [25,37,39]. Unfortunately, they are not thermodynamically consistent.…”