Domain decomposition methods in semiconductor device modeling Technical Report TR/PA/01/51

In this paper, we present some parallel implementations of domain decomposition techniques for the solution of the drift diffusion equations involved in 2D semiconductor device modeling. The model describes the stationary state of a device when biases are applied to its bounds. The mixed dual formulation is retained. Therefore, we have to deal with a system of six totally coupled nonlinear partial differential equations. This system is decoupled and discretized in time by a semi-implicit nonlinear scheme using local time stepping. At each time step, we have to solve three systems of two nonlinear partial differential equations. The first system is associated with electrostatic potential, the second with the negative charges (electrons) and the third with the positive charges (holes). Each pair of equations is naturally discretized in space by mixed finite elements defined on 2D unstructured meshes and then solved by a NewtonRaphson method [6]. At each step of the Newton-Raphson method, a linear system of equations has to be solved. Depending upon which nonlinear system is being solved, these linear systems can be either symmetric positive definite or unsymmetric. These systems are sparse with a maximum of five nonzero entries per row due to the mixed finite element triangulation. A complete simulation is decomposed into two phases: first the solution of the equilibrium problem, then the solution of the static problem. The equilibrium problem consists of applying a zero potential to the bounds of the device and its numerical solution only involves the solution of symmetric positive definite linear systems. In this paper, we consider only the solution of the equilibrium problem. Our objective is to obtain a fully parallel code in a distributed memory environment with MPI as message-passing library.