4.4.3 Analytical Profiles

Gaussian functions are used to simplify the rather complex results from the two-dimensional approach as discussed earlier in Section 3.1. The parameters describing these analytical doping models are used as optimization parameters for further optimizations.

The doping peak in the channel region resulting from the two-dimensional optimization is substituted by a Gaussian peak. To provide means of preventing punchthrough two methods are utilized: In the first method the background substrate doping is one additional optimization parameter (Method 1). The second method uses a Gaussian peak located under the source well (Method 2).

In Listing 4.7 and Listing 4.8 the specific parts of the device generation sub-model mkdev.mod used within the SIESTA framework are shown for the two different methods. The used initial, minimum, and maximum values for the doping parameters are listed in Table 4.2. It is to note that the doping values in this table are given in units (cm$^{-3}$) whereas the logarithmic format is used within the optimization process and, therefore, also in the listed sub-models. This linearizes the dependence of the drain current on the doping parameters and, therefore, improves the convergence behavior.

Listing 4.7: The optimization parameter setup using one Gaussian function, Method 1.

          .
          .
          .
          (Nsub           bound-float      ...   ...   ...)
          (N              bound-float      ...   ...   ...)
          (x0             bound-float      ...   ...   ...) ; um
          (y0             bound-float      ...   ...   ...) ; um
          (deltax         bound-float      ...   ...   ...) ; um
          (sigx           bound-float      ...   ...   ...) ; um
          (sigy           bound-float      ...   ...   ...) ; um
          .
          .
          .

Listing 4.8: The optimization parameter setup using two Gaussian functions, Method 2.

          .
          .
          .
          (Nsub           float            15.)
          (N              bound-float      ...   ...   ...)
          (x0             bound-float      ...   ...   ...) ; um
          (y0             bound-float      ...   ...   ...) ; um
          (deltax         bound-float      ...   ...   ...) ; um
          (sigx           bound-float      ...   ...   ...) ; um
          (sigy           bound-float      ...   ...   ...) ; um
          (N_2            bound-float      ...   ...   ...)
          (x0_2           float            0.0) ; um
          (y0_2           bound-float      ...   ...   ...) ; um
          (deltax_2       bound-float      ...   ...   ...) ; um
          (sigx_2         bound-float      ...   ...   ...) ; um
          (sigy_2         bound-float      ...   ...   ...) ; um
          .
          .
          .

Table 4.2: Parameter setup for optimizations with Gaussian functions.

		Device Generation A			Device Generation B
param.	unit	init	min	max	init	min	max
$N_{\mathrm{sub}}$	cm$^{-3}$	1$\cdot$10$^{16}$	1$\cdot$10$^{15}$	1$\cdot$10$^{17}$	3.16$\cdot$10$^{16}$	3.16$\cdot$10$^{15}$	3.16$\cdot$10$^{17}$
Peak 1
$N$	cm$^{-3}$	5$\cdot$10$^{18}$	3.16$\cdot$10$^{17}$	1$\cdot$10$^{19}$	1$\cdot$10$^{19}$	1$\cdot$10$^{18}$	3.16$\cdot$10$^{19}$
$x_0$	$\mu$m	0.25	0.2	0.29	0.1	0.08	0.11
$y_0$	$\mu$m	0.012	0	0.05	0.01	0	0.03
$\Delta x$	$\mu$m	0	0	0.05	0.005	0	0.02
$\sigma_x$	$\mu$m	0.017	0.01	0.05	0.01	0.004	0.02
$\sigma_y$	$\mu$m	0.01	0.01	0.05	0.005	0.004	0.02
Peak 2
$N$	cm$^{-3}$	1$\cdot$10$^{18}$	1$\cdot$10$^{17}$	1$\cdot$10$^{19}$	3.16$\cdot$10$^{18}$	3.16$\cdot$10$^{17}$	3.16$\cdot$10$^{19}$
$y_0$	$\mu$m	0.055	0.15	0.2	0.065	0.05	0.08
$\Delta x$	$\mu$m	0.16	0.04	0.1	0.025	0.02	0.04
$\sigma_x$	$\mu$m	0.04	0.01	0.05	0.015	0.004	0.02
$\sigma_y$	$\mu$m	0.015	0.01	0.05	0.006	0.004	0.02

As already mentioned in Section 3.2 the initial parameters are obtained from a manual fit to the two-dimensional optimization results. Reasonable values for the minimum and maximum ranges are chosen to avoid unrealistic doping structures. In case of Method 2 the substrate doping $N_{\mathrm{sub}}$ is not an optimization parameter and is kept at 10$^{15}$ cm$^{-3}$.

The Makedevice input deck contains an additional Peak3 section given in Listing 4.9 accounting for the doping peak in the channel. For Method 2 the input deck is completed by another doping peak definition Peak4 shown in Listing 4.10 accounting for the doping peak under the source well.

Listing 4.9: The Peak3 section in the Makedevice input deck for optimizations with Gaussian functions, Method 1 and Method 2.

Peak 
{
  .
  .
  .
  Peak3 {                                  // channel Peak
     on      = yes;                        // switch on or off
     mode    = "gauss";                    // gauss, cosine, or pearson mode
     dopType = if (~type == "NMOS", "acceptor", "donor");
                                           // acceptor or donor doping
     N         = pow10(<(N)>) *1"cm^-3";   // peak doping value
     x         = <(x0)> um;                // x-position of the peak
     xLength   = <(deltax)> um;            // x-length of the peak, must be >= 0
     y         = <(y0)> um;                // y-position of the peak
     yLength   = 0 um;                     // y-length of the peak, must be >= 0
     xSigLeft  = <(sigx)> um;              // left x-sigma of the peak
     xSigRight = xSigLeft;                 // right x-sigma of the peak
     ySigUpper = <(sigy)> um;              // upper y-sigma of the peak
     ySigLower = ySigUpper;                // lower y-sigma of the peak
  }
  .
  .
  .
}

Listing 4.10: The additional Peak4 section in the Makedevice input deck for optimizations with Gaussian functions, Method 2.

Peak 
{
  .
  .
  .
  Peak4 {                                  // peak underneath the source well
     on      = yes;                        // switch on or off
     mode    = "gauss";                    // gauss, cosine, or pearson mode
     dopType = if (~type == "NMOS", "acceptor", "donor");
                                           // acceptor or donor doping
     N         = pow10(<(N_2)>) *1"cm^-3"; // peak doping value
     x         = <(x0_2)> um;              // x-position of the peak
     xLength   = <(deltax_2)> um;          // x-length of the peak, must be >= 0
     y         = <(y0_2)> um;              // y-position of the peak
     yLength   = 0 um;                     // y-length of the peak, must be >= 0
     xSigLeft  = <(sigx_2)> um;            // left x-sigma of the peak
     xSigRight = xSigLeft;                 // right x-sigma of the peak
     ySigUpper = <(sigy_2)> um;            // upper y-sigma of the peak
     ySigLower = ySigUpper;                // lower y-sigma of the peak
  }
}

Fig. 4.8 and Fig. 4.9 show the acceptor doping profiles as the results of the optimization approach using only one Gaussian function (Method 1) for the Device Generation A and Device Generation B, respectively. Fig. 4.10 and Fig. 4.11 show the acceptor doping profiles as the results of the optimization approach using two Gaussian functions (Method 2). The resulting parameters of the Gaussian functions are summarized in Table 4.3.

Figure 4.8: The result of the optimization using Gaussian functions for Device Generation A, Method 1.

Figure 4.9: The result of the optimization using Gaussian functions for Device Generation B, Method 1.

Figure 4.10: The result of the optimization using Gaussian functions for Device Generation A, Method 2.

Figure 4.11: The result of the optimization using Gaussian functions for Device Generation B, Method 2.

Table 4.3: Optimized parameters after drive current optimizations with Gaussian functions.

		Device Generation A		Device Generation B
param.	unit	Method 1	Method 2	Method 1	Method 2
$N_{\mathrm{sub}}$	cm$^{-3}$	3.04$\cdot$10$^{16}$		1.21$\cdot$10$^{17}$
Peak 1
$N$	cm$^{-3}$	2.18$\cdot$10$^{18}$	2.20$\cdot$10$^{18}$	5.73$\cdot$10$^{18}$	7.81$\cdot$10$^{18}$
$x_0$	$\mu$m	0.23977	0.24980	0.10068	0.10283
$y_0$	$\mu$m	0.01907	0.01613	0.00919	0.00912
$\Delta x$	$\mu$m	0.00390	0.00000	0.01058	0.00665
$\sigma_x$	$\mu$m	0.01141	0.01364	0.01161	0.00924
$\sigma_y$	$\mu$m	0.01193	0.01029	0.00462	0.00400
Peak 2
$N$	cm$^{-3}$		1.14$\cdot$10$^{18}$		3.63$\cdot$10$^{18}$
$y_0$	$\mu$m		0.05670		0.02151
$\Delta x$	$\mu$m		0.15976		0.06301
$\sigma_x$	$\mu$m		0.03972		0.01548
$\sigma_y$	$\mu$m		0.01566		0.00825