2.2.1 Numerical Range Scaling

  The numerical range scaling (NRS) method [Rys83] seems to be appropriate to cover the multilayer problem. The probability density functions as well as the partial doses themselves are assembled at the material interfaces according to dose and normalization constraints. Let be the position of the material interface starting with , the surface coordinate, and be the thickness of this given material layer i, two characteristic scaling factors , have to be calculated for each material layer as given from (2.2-1) to (2.2-3).

   

gives the already consumed area of the piecewise assembled probability density functions. The final profile can then be obtained by (2.2-4).

 

  
Figure 2.2-1: Application of the NRS model and dose matching criterion to a one-dimensional multilayer structure. Material layer specific parameters , , and according to (2.2-1) - (2.2-4) are also given.

Figure 2.2-1 explains the NRS method for a one-dimensional arsenic implantation profile. The profile is discontinuous at the material interfaces due to different characteristic parameters of the adjacent material.