A new Control Volume Finite Element Method for the stable and accurate solution of the drift–diffusion equations on general unstructured grids

Abstract

We present a new Control Volume Finite Element Method with multi-dimensional Scharfetter–Gummel upwinding (CVFEM-SG) for the drift–diffusion equations. The method combines a conservative formulation of the carrier density continuity equations with an edge element lifting of the one-dimensional Scharfetter–Gummel edge currents into curl-conforming elemental currents. These elemental currents combine the upwind effect from all element edges and enable accurate computation of the flux on arbitrary surfaces inside the elements. In so doing, we obtain a formulation that is stable and accurate on general unstructured finite element grids. This approach sets our formulation apart from other methods, which require the control volumes to be topologically dual to the primal grid. Numerical studies of the CVFEM-SG for a suite of scalar advection–diffusion test problems confirm the accuracy and the robustness of the new formulation. Simulations of a PN diode and an n-channel MOSFET device demonstrate the performance of the method for the fully coupled drift–diffusion system.