3.1.5 Analytical Profiles

Once the result from the general two-dimensional optimization has been analyzed and the performance gain can be attributed to specific doping regions, analytical functions can be used to reduce the complexity of the result. The optimization process is repeated, this time with a much smaller number of doping parameters.

In this work two-dimensional Gaussian functions are used because they are simple in their structure and close to real doping profiles. In general, a two-dimensional Gaussian pulse for doping profile modeling can be described by six parameters which are listed in Table 3.2.

Table 3.2: Gaussian model parameters.

Parameter	Description
$N$	peak doping
$x_0$	lateral peak position
$y_0$	vertical peak position
$\Delta x$	lateral peak length
$\sigma_x$	lateral standard deviation
$\sigma_y$	vertical standard deviation

The one-dimensional distribution in $x$-direction (at $y=y_0$) reads:

$$\begin{displaystyle} N_{\mathrm{Gauss}}(x)=\left\{ \begin{ar... ...t) & x>x_0+\Delta x\\ \end{array} \right. \end{displaystyle}$$ (3.3)

and in $y$-direction (at $x=x_0$):

$$\begin{displaystyle} N_{\mathrm{Gauss}}(y) = 10^N\cdot\exp\l... ...\frac{(y-y_0)^2}{2\cdot\sigma_y^2}}\right) . \end{displaystyle}$$ (3.4)

Fig. 3.6 shows a Gaussian peak with the used model parameters indicated. An arbitrary number of Gaussian functions can be used to define the doping. The overall doping will result from a superposition of all the Gaussian functions plus a constant background doping.

Figure 3.6: The two-dimensional Gaussian function.