3.1.4 Two-Dimensional Doping Characterization

This method offers a very general characterization of the doping profile in the bulk. It covers the area between and under the source and drain wells including the channel. This area is assumed to control the device performance, whereas for the rest of the bulk a constant doping is used. Fig. 3.4 shows how this region is discretized using an orthogonal optimization grid with uniform spacing. The discretization area has the shape of an inverted-T. The doping values at the grid points are the optimization parameters which will be varied.

Figure 3.4: The two-dimensional doping discretization.

To obtain a smooth two-dimensional doping profile, an interpolation method is utilized between the grid points. This method uses raised-cosine shaped fragments, one placed at each of the grid points, which are superposed in the logarithmic domain. Fig. 3.5 gives an example of a possible profile in one dimension using three fragments frag$_1$, frag$_2$, and frag$_3$, with peak concentrations $N_1$, $N_2$, and $N_3$, placed at the grid points $x_1$, $x_2$, and $x_3$, respectively. Each of the raised-cosines reaches exactly to the next closest grid points, for example, frag$_2$ reaches from $x_1$ to $x_3$, so that the doping fragment placed at a grid point influences only the area within its neighbors. Additionally, with raised-cosine shaped fragments it is possible to achieve a constant doping which is sometimes necessary, as shown by frag$_2$ and frag$_3$.

Figure 3.5: The raised-cosine interpolation method.

The total doping concentration within the inverted-T area can be calculated from

$$\log(N_\mathrm{total}(x,y)) = \sum_{i,j} \log(\tilde{N}_{i,j}(x,y))$$ (3.1)

with

$$\log(\tilde{N}_{i,j}(x,y)) = \log(N_{i,j}) \cdot \frac{1}{4}... ...t( \frac{y-y_j}{\Delta y} \cdot \frac{\pi}{2} \right) \right).$$ (3.2)

In general, it is also possible to use ortho-splines for interpolation, but then the influence area of one grid point would not be restricted as firmly as with the raised-cosine method. This would increase the cross-correlation of the optimization parameters which has a negative impact on the optimization procedure. Besides, the raised-cosine method guarantees that the interpolation function stays within a certain range defined by the minimum and maximum doping parameters. Splines usually cannot guarantee that which can be a problem in the logarithmic domain where negative doping values in the device description file would cause simulation errors.