2  Adaptivity of the Finite Difference Method

The solution of the atomic mixing model for a material system of n species involves the solution of n partial differential equations. In practical situations, the equations that arise can only be solved numerically and hence the developing solution will always contain numerical error. The finite difference or finite element discretisation can be applied in the spacial coordinate, it is natural to apply a dose-stepping method along the other axis. In general, the finer the mesh (in both space and dose) the greater the accuracy of the method, but the slower the processing speed.
In reference [4], the implicit FDM that solves the partial differential equations in the mixing region is described in detail. Unlike the explicit method that is employed in the original IMPETUS code, the explicit method has no stability restriction on the dose step. One further difficulty with the explicit method is that halving the size of the spacial interval automatically required the dose step to be reduced to a quarter of its size. Hence, in cases where a finer spacial mesh is required, the dose step needs to be severely reduced and thus the processing time greatly increased.
In IMPETUS II, the size of the dose step is restricted only by accuracy considerations. There is no direct relationship between the size of the dose step and the spacial mesh size. Thus the implementation of the implicit method also gives us much greater freedom to control or adapt the method to the problem in hand. IMPETUS II contains methods for adjusting the length of both the spacial interval and the dose step.

2.1  Adjusting the Spacial Interval

In our experience with the method on a variety of material systems, it is generally found that an inter-nodal distance of 2 or 3 angstroms is generally sufficient to give a satisfactory solution. However, in the case of a thin layer (for example 20 angstroms or less), it is natural to refine the mesh so that the layer is well-represented. Atomic diameters are generally of the order of a few angstroms and hence there is a physical lower limit on the realistic layer thickness. If we assume that around ten nodes are sufficient to represent a thin layer and the layers are at least say 5 angstroms thick then the minimum inter-nodal distance expected is around 0.5 angstroms.
In the IMPETUS II algorithm, sharp interfaces are only allowed into the domain of the FDM when they have become sufficiently smoothed by the application of the linear diffusion methods. Once they enter the domain of the FDM, the mesh may need refining to satisfactorily represent the interface.
In the execution of IMPETUS II, the domain of the FDM progresses through the extent of the material (see figure 1 of reference [4]). In IMPETUS II the spacial mesh is always uniform. The adaptivity is achieved simply by continually halving the inter-nodal distance should a thin layer or sharp interface exists in the window at any particular time until the mesh is sufficiently fine to represent the material distribution. The inter-nodal distance is increased when the thin layers or interfaces become more diffused. The control of the mesh size is not directly based on a mathematical analysis of the error, rather it is guided by the desire to obtain a good node-to-node representation of the material distribution within the range of the FDM.

2.2  Adjusting the Dose Step

In IMPETUS II a measure of the accuracy of the numerical method from dose step to dose step is obtained through simply summing the concentrations in the material distribution at the new dose and observing the drift from unity. (Note that in IMPETUS II the concentrations are adjusted after each dose step so that they do sum to unity, and the method for carrying this out is described in section 5.) The drift from unity is used to control the dose step; if the measure of accuracy is larger than expected then the dose step is decreased and if it is smaller the dose step is increased.

2.3  Solution to Test Problems

The test structure of figure 1 is used to demonstrate the facility of IMPETUS II to deal effectively with thin layers. The structure consists of four layers of pure Germanium of thicknesses 40, 20, 10 and 5 angstroms with sharp interfaces in pure Silicon. The bombardment energy is 5keV and the angle of incidence is 45 degrees to the normal. The resulting yield-dose curves are given in figure 2.


Figure 1. Material consisting of thin layers.


Figure 2. Yield-depth curve for Germanium arising from the structure of figure 1.