Early theoretical analysis has suggested that asymmetry in random dopant induced VT variation [144, 145] can be attributed to the Poisson distribution governing the number of dopants in the gate depletion region [57]. It is known that the number of dopants in the channel of a device affects the threshold
voltage, as discrete dopants cause a localised increase of the potential in the channel. However, the number of dopants is not the only source of VT variation. The position of dopants [146, 69] must also be considered, as many different configurations of dopant position will occur for a fixed number of dopants. Modelling VT variation using only total dopant number neglects variation due to dopant position and leads to an unphysical truncation of the lower tail of the distribution [74]. Thus, the first step in a more detailed analysis is to determine in which region variations in local dopant density have the greatest effect on the threshold voltage. This will help us to define the statistically significant region (SSR) of the transistor, in which random dopants dominate the statistical behaviour of the device ensemble.
In order to accomplish this we have calculated the correlation coefficient between VT and the total dopant number in a series of 1 nm deep horizontal slabs bounded by the source and the drain, starting from the oxide interface and ranging down through the device body. A similar procedure is repeated for 1 nm wide vertical slabs ranging through the channel from source to drain, as illustrated schematically in Figure 4.4. The correlation between VT and dopant x position is shown in Figure 4.5(a) for both 1 nm thick slices d nm from the interface and for slices d nm thick (i.e. the cumulative sum of the 1 nm slices up to and including that position). From this figure we see that the largest calculated correlation is for dopants between the PN junctions of the device. Figure 4.5(b) shows the correlation between threshold voltage and dopant z position, again for 1 and d nm thick slices and clearly indicates that dopants near the interface make the most significant contribution to VT fluctuations. By combining the 1D correlation in x and z from the 1 nm slices, a two-dimensional map of the correlation between the position of an individual dopant within the SSR and VT can be constructed, which is plotted in Figures 4.6(a) and (b) for both the 35 and 13 nm devices.
The SSR is bounded by the metallurgical junctions of the source and drain and extends approximately 20 nm down from the interface in the 35 nm device and approximately 10 nm from the interface in the 13 nm transistor. These values compare closely with the depletion depths for these devices, which are ∼ 25 nm and ∼ 9 nm in the 35 and 13 nm devices, respectively. This conclusion
(a)
(b)
Figure 4.5: Correlation between dopant position and threshold voltage for the 35 nm device (a) in the X axis and (b) in the Z axis. Note that z = 0 nm is at the oxide interface.
(a)
(b)
Figure 4.6: The two dimensional correlations of dopant position and VT for (a) the 35 nm device and (b) the 13 nm device. The statistically significant region can be determined visually from these plots.
is consistent with previous theoretical studies [26] and experimental observa- tions [47] that it is dopants in the channel that have the greatest influence on VT fluctuations. It is interesting to note that for the two simulated devices, the maximum correlation between dopant position and VT is not at the oxide interface, but approximately 1.5 nm below it. This is due to the density gra- dient quantum corrections used in the simulations, which force the maximum of the carrier distribution away from the surface, and determining the posi- tion at which the device is most sensitive to the presence of random dopants within the channel. The density gradient carrier distribution is consistent with that obtained from the self-consistent solution of the 1D Poisson-Schrödinger equation [104].
By choosing devices with a fixed number of dopants within the SSR (NSSR) it is possible to estimate the distribution of the threshold voltage caused solely by the random position of dopants. Figures 4.7(a) and (b) illustrate the evo- lution of the distribution of VT as a function of NSSR for the 35 and 13 nm transistors respectively. As illustrated in Figure 4.8 both the mean and stan- dard deviation of the threshold voltage distributions increase linearly with NSSR. For densely populated samples with a constant number of dopants in the SSR (around the mean value NSSR = 44 for the 35 nm and NSSR = 20 for the 13 nm transistor) the calculated skew and kurtosis are small, leading to the conclusion that the distributions of threshold voltages due to random dopant position for fixed NSSR are Gaussian. In order to verify this hypothesis, we use the Mann-Whitney test [147], which tests the null hypothesis that two samples are drawn from the same underlying population. Several of the positional dis- tributions for the 35 nm device were tested against 10,000 samples randomly generated from a Gaussian distribution with the same mean and standard de- viation as the data and the statistics of the p-values obtained can be seen in Table 4.2. Taking the standard statistical significance level of α = 0.05, we see that there are no p-values close to or below this level, therefore we accept the null hypothesis that the distributions are Gaussian.
(a) (b)
Figure 4.7: The distribution of VT as a function of number of dopants, NSSR, in the SSR for (a) the 35 nm transistor and (b) the 13 nm transistors. For a fixed NSSR, the distribution of VT is determined by dopant position. Note the increasing mean and standard deviation as a function of NSSR.
Figure 4.8: The dependence of the VT mean and standard deviation as a function of NSSR for both devices. The linear dependence allows positional effects on VT to be extrapolated out to larger values of σ.
NSSR Mean St. Dev. Min Max 40 0.856 0.108 0.330 0.999 41 0.765 0.145 0.210 0.999 42 0.771 0.145 0.262 0.999 43 0.790 0.141 0.244 0.999 44 0.798 0.138 0.246 0.999 45 0.518 0.148 0.119 0.999 46 0.836 0.120 0.351 0.999 47 0.847 0.113 0.365 0.999 48 0.733 0.143 0.210 0.999
Table 4.2: Statistics of the p-values obtained by conducting Mann-Whitney tests for the positional distributions for the 35 nm device against 10,000 ran- dom Gaussians. As there are no p-values below 0.05, we accept the null hy- pothesis that the positional distributions are Gaussian.