Functions Method

The Green's Functions Method can only be applied if the diffusivity is constant in the whole simulation domain. This condition is usually satisfied, if only one segment has to be simulated. Normally different segments consist of different materials where the dopants also have different diffusivities. A further fundamental assumption of the chosen method is that the diffusion domain must be unbounded. This is usually not satisfied and every segment boundary will violate this condition. However, when the boundaries are far away from domains of interest, their influence is only marginal and the existence of these boundaries can be neglected. Neglecting the influence is usually possible for the boundaries of the silicon wafer except the top boundary where the initial concentration was implanted. At this boundary, the distance of the implanted ions to the top of the wafer is not large enough. However, if the top of the wafer is flat and if it can be postulated that no dopant diffusion will occur through this boundary, the wafer can be mirrored above its top. With this trick, it can be achieved that no conduction of dopants will occur at the top and this boundary can be removed. Because the partial conduction coming from the original wafer and the conduction resulting from the mirrored wafer will cancel each other due to symmetry considerations. This implies, if possible simplifications make it feasible to place the dopant ions directly on the top of the wafer, that the entire dopant concentrations must be set twice than the original concentration to account for the dopants diffusing to the upper and lower half of the wafer.

Certainly, it must be guaranteed that the grid density of the initial distribution or the desired time where the final distribution will be calculated are convenient in terms of the relaxation processes between the grid points. If the final time is not far enough and the diffusion between the grid points is high, the discretization of the initial distribution must be denser. However, if the final time and the diffusion ranges become wider, the initial discretization of the distribution can be chosen cruder. Even by this diffusion of dopants, the influence of marginal areas must be accounted for in the calculation of the active areas. Which means that the initial dopant concentration of a larger domain, than where the final distribution has to be evaluated, has to be known and included in the diffusion process.

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