11.1 Introduction

After a Monte Carlo simulation of ion implantation on an ortho-grid, the question arises how to translate the resulting values, i.e., concentrations, to an unstructured grid suitable for diffusion simulation. In the Monte Carlo simulation an ortho-grid is commonly used in order to achieve workable simulation times, since calculating point locations, i.e., tracing the position of ions, dominates performance. Furthermore, the resulting values have to be smoothed in order to provide suitable input for the simulation of subsequent process steps.

Thus an algorithm for smoothing Monte Carlo ion implantation results has to meet the following demands:

• It has to work with unstructured target grids.
• It must provide suitable smoothing.
• Since the number of grid points in the target grid is usually large, it must not be computationally expensive.

One simple approach is to perform a least squares fit of a multivariate polynomial of fixed degree, usually $2$, and to hope that this polynomial is a suitable approximation providing proper smoothing. This is known as the RSM (response surface methodology) [21] approach, but it does not work satisfactorily (cf. Figure 11.6), the reason is given in Section 7.2. In order to solve this problem, the generalizations of Bernstein polynomials from Chapter 7 were used. Hence a fast algorithm based on these polynomials was developed and applied to real world examples, where its advantages can be seen (cf. Figure 11.4).

In the following the algorithm is described in detail. Finally the two approaches are compared by looking at the results of a three-dimensional ion implantation example with about 80000 grid points.