4.5 Initial Values and Stepping Methods

We use a linesearch which determines an optimal value for the Newton step size. If the Newton method does not converge, the ``optimal'' step can be additionally multiplied by a fixed damping factor.

An alternative, computationally more intense approach compares predicted and actual decrease of the error norm to ensure that we are closely following the Newton trajectories. If there is still no convergence another starting point has to be chosen.

In one dimension the standard initial values are derived by linear interpolation between the Dirichlet boundary values. The higher dimensional analogon is the solution of a Laplace equation with Dirichlet boundary conditions. For the Poisson equation another good initial guess is the builtin potential.

As an alternative one can start from the equilibrium values which are determined from the doping and the lattice temperature. For the examples used here there were only a few cases where this second set of initial values leads to convergence while the first one fails.

If ``cheap'' initial values which lead to convergence cannot be found, then stepping methods are applied. We use bias stepping or charge stepping. Here charge stepping refers to stepping in the value of the carrier charge. This value enters the Poisson equation and by multiplication with the electrical field in the moment equations. Charge stepping has the advantage that it can also be used for the calculation of the equilibrium solution.

We also experimented with other methods. For example we tried to use the results from a different closure or from a different discretization as an initial guess. These should be very good initial guesses. However, it turned out, that in practice they rarely converged. A similar observation is that starting with results from a coarse grid often fails to converge on a refined mesh.