The dopant atoms inside of semiconductor devices are usually introduced by ion implantation in several processing steps, with different masks, to restrict the implantation to certain regions, and with various implant energies and doses. Low energetic ion implantation results in very sharp profiles located only a few nanometers below the surfaces . To spread the distributions and to activate the dopants, the device structures are exposed to high temperature annealing steps. During these high temperature processes, the dopants diffuse. As the final dopant distributions determine the locations of the pn-junctions and therefore determine the device behavior, an accurate simulation of these effects is important .
When the doping concentrations are low, the diffusion process follows the linear diffusion law with constant diffusivity. This diffusivity depends on the involved materials and the processing temperatures. For higher concentrations, the behavior differs from the linear model. The diffusivity depends on the values of the concentrations. In this case, other models must be applied .
Even with the linear diffusion equation, the simulation procedure is computationally expensive. The differential equation is time variant and therefore a transient simulation must be applied. The time steps of these simulations may be very small, at least at the beginning of the simulation when high diffusion gradients are expected. As time passes, the distributions smooth out and the time steps can also be enlarged. Concurrently, at regions with high doping gradients the grid density must be high at least in the directions of the gradients. And as the domains of high gradients move, the refined domains must also move. This refinement is often done by red-green refinement or directly by hierarchical splitting of grid edges. The possibility of coarsening the refined areas is eased by the hierarchical methods. This is necessary as the high gradient locations move and temporarily produced high point densities can be removed. Under certain circumstances, another discretization method than Box Integration must be selected because the Delaunay criterion is not fulfilled. Box Integration with Delaunay grids gives the advantage that caused by the maximum principle negative concentrations can never appear . With other discretization methods this non-physical behavior must be prevented by applying special grid criteria .
However, often quick and simple predictions of layout problems have to be chosen. Especially within three-dimensional simulations, the time horizon of a transient diffusion simulation may be exceeded, difficulties in the solving procedure, or plausibility flaws may occur, such as negative concentrations by applying Finite Element methods on badly fitted grids. Simplifications concerning the models and geometries have to be assumed. As the simulation domains for dopant diffusion, usually the silicon segment, can be simplified, the use of a Green's Function approach may be a possible solution to model the diffusion process, without negligence of a three-dimensional simulation. This approach has the advantage that it requires significantly less computational time, while by simulating the time variant differential equation a lot of time is wasted for computing the dopant distributions at several time steps. But in fact, only the final distribution at the end of the high temperature process is of interest. With the use of Green's Functions, it is possible to calculate the dopant distribution at any time step of interest by only one iteration. No transient simulation is necessary .
The requirements for the applicability of this method will be examined later on after the method has been explained and the final algorithm has been derived.