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# 4.3 Advantages of the Green's Functions Method

The Green's Functions method gives a relatively simple and fast tool for calculating diffusion processes. Compared to direct simulation of the transient diffusion process, this method has the advantage that no iteration over small time steps is necessary until the final result is reached. Therefore, the frequent solution of a large equation system is not necessary. The final concentration can be directly written as given by (4.7). With this method the value of the concentration can be evaluated at any point in time and at any point in space.

Additional simplifications can be introduced, which speed up the calculation time extremely. Every partial initial Dirac like concentration sustains a broadening during time. The speed of broadening is proportional to and implies that in the first few moments the distribution broadens fastest. If the initial distribution is steep delimited within a certain area the error of replacing this distribution by a layer of Dirac functions will be marginal.

For ion implantation with low implantation energies, the concentration maxima that are located close to the surface flow away due to diffusion. If the time interval is long and the broadening of the initial distribution is large relative to the initial thickness, the initial distribution can be simplified as an areal distribution. The Dirac like initial distributions are placed along surfaces inside the wafer, at little distance from the top of the wafer or certainly direct at the top. In this simplified case, only the top of the wafer has to be meshed. Utilizing with the adequate dose the final distribution can be achieved by only a single sweep over the surface of the wafer, which is much faster than iterating within a three-dimensional grid.

If the implantation doses are known and the diffusion ranges are large compared to the transversal implantation depths and expansions widths a relatively simple mask based tool for implantation and following diffusion based on this method can be developed. The dopant concentration can be evaluated and provided to the device simulation in every desired point.

As an additional feature of this method, the grids used for the initial distribution and the final profile can be designed independently and may be adapted to their individual requirements. The final concentration distribution is not mandatorily connected to a grid, as the concentration can be calculated in any place of interest, which may be an advantage. However, most of the tools applied later on require a grid. Even visualization tools often require a grid to display the dopant profiles or calculate iso-surfaces (as shown in the following example). As the demands on the grids for electrical simulation and visualization may differ seriously, an adapted dopant profile on the desired grid can be easily obtained by reapplying the diffusion simulation on the new grid.

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J. Cervenka: Three-Dimensional Mesh Generation for Device and Process Simulation