A Spectral Analysis of the Decoupling Algorithm for Semiconductor Simulation

The location in the complex plane of the eigenvalues of the derivatives of the partial nonlinear maps, which constitute the fixed point mapping T defining Gummel’s method for decoupling the steady-state semiconductor equations is examined for the general, N-dimensional case. In particular, it is proven that the size of these eigenvalues is unconditionally smaller than 1. It is shown that for up to three-dimensional models all the derivatives are suitably compact. This implies that the spectral radii of these mappings are equal to the size of the largest eigenvalue. For the current continuity equations this implies an essential improvement over earlier contraction mapping results.Furthermore, the precise location of the eigenvalues of the derivatives of the maps defined through the current continuity equations for the general one-dimensional case is found. For a special one-dimensional case it is proven that the size of the eigenvalues of the total mapping T defining Gummel’s method is unconditionally smal...