Arclength continuation methods and applications to 2D drift‐diffusion semiconductor equations

Abstract

In this paper, the homotopy deformation method to solve the nonlinear stationary semiconductor equations with Fermi-Dirac statistic is used. This method introduces an arti cial transient problem. The time discretization is based on the nonlinear implicit scheme with local time steps. In order to have an automatic adaptation of local time step parameters, we introduce arclength predictor-corrector continuation methods. The fondamental goal of these methods is to overcome the unstabilities or the failure of the classical Newton-Raphson's schemes which appear when the nonlinearity is Strong or near Limit or Bifurcation points. The approximate procedure of our system using a Galerkin method that makes use of a mixed nite element approach is used. A peculiar feature of this mixed formulation is that the electric displacement D and the current densities j n and j p for electrons and holes, are taken as unknowns, together with the potential and quasi Fermi levels n and p. This allows D, j n and j p to be determined directly and accurately. The above algorithms appear to be e cient, robust and to give satisfactory results. Numerical results are presented, in one and two dimension, for some realistic devices : an Heterojunction Diode (quasi-1D problem) and an Heterojunction Bipolar Transistor (HBT) working in ampli er mode.

Simulation of visible and ultra-violet group-III nitride light emitting diodes

“…In the present paper, the approach based on the homotopy method is adopted, following [50]. The idea of this method called also the continuation method or path following [51] is to consider the following underdetermined equation system:…”

“…The choice of the local fictious time steps s k u ðzÞ, s k u n ðzÞ, and s k u p ðzÞ for the discretized equations is discussed in detail in [50].…”

Numerical simulation of chemotactic bacteria aggregation via mixed finite elements

“…Equation (12) is similar to the Poisson equation and equation (13) to the electron continuity equation in the semiconductor framework. Following [5,6,10,11], and assuming that Dirichlet boundary condition on a part of the boundary (Γ d ) -of the computational domain Ω -and Neumann boundary condition on the complementary boundary part (Γ n ) hold for the variablesc and ϕ n , we introduce the flux variables…”

“…Refering again to the publications related to semiconductor area [5,6,10,14,15], various algorithms are described for a "stationary" version of system ( (25)- (30) The fact that we associate an "evolution" equation and the use of implicit or semi-implicit schemes is an other way to consider augmented Lagrangian techniques for the solution of saddle point problems [7] arising from the mixed formulation. Local time steps are closely connected with the penalty functions in the framework of Augmented Lagrangian techniques.…”