About the dependence of the convergence of Gummel’s algorithm and its linear variants on the device geometry

Abstract

The convergence of Gummel’algorithm is known for sufficiently small currents or sufficiently small MOSFET-devices. We take a closer look at the dependence of the convergence of Gummel’s algorithm and two linear variants on the device geometry. Using spectral analysis we give a sufficient condition for local convergence, where the geometry enters as the smallest eigenvalue of a certain general eigenvalue problem. Estimating the eigenvalue with a maximum principle yields an upper limit for the size of the device which guarantees local convergence of all of the three algorithms. Finally we compare à priori estimates with the computed eigenvalues.