## 3 Results and Discussion
### Delay  
#### 3.2.1 Verification  
To validate the accuracy of the proposed delay model, as well as the prediction method, delay (\(T_{d}\)) and output slew (\(S_{out}\)) of an inverter under distribution process parameters {XL, XW, TOX1, TOX2, TSI, PHIG1} variations are simulated through MC SPICE using a 22-nm industrial design kit and fitting parameters are extracted for the delay model. We define the baseline characterization as a 1000-point MC simulation sampled randomly within the process parameter space \(\mathbf{P}=\) {XL, XW, TOX1, TOX2, TSI, PHIG1}. The nominal values of process parameters with zero variation for the n-type MOSEFET in the inverter are: {\(L_{n}=26\) nm, \(W_{n}=80\) nm, \(\text{TOX1}_{n}=1.33\) nm, \(\text{TOX2}_{n}=20\) nm, \(\text{TSI}_{n}=6\) nm, \(\text{PHIG1}_{n}=4.45\) eV} and those for the p-type MOSEFET in the inverter are {\(L_{p}=24\) nm, \(W_{p}=80\) nm, \(\text{TOX1}_{p}=1.48\) nm, \(\text{TOX2}_{p}=20\) nm, \(\text{TSI}_{p}=6\) nm, \(\text{PHIG1}_{p}=4.71\) eV}. \(T_{d}\) and \(S_{out}\) versus effective current \(I_{eff}\) and their probability density using the proposed model and 1000 MC SPICE simulations under different \(V_{dd}\) values are shown in Figure 7. The mean absolute percentage error (MAPE) and the root mean square error (RMSE) versus \(V_{dd}\) between the proposed delay model and SPICE simulation are plotted in Figure 8 Non-Gaussian distributions of delay caused by nonlinear relation with delay and distribution process parameters can be found in near-threshold (0.5 V) and subthreshold regions (0.3 V) and the proposed model accurately captures non-Gaussian features. The delay prediction model was set at 0.3 V to verify the proposed method's compensation effect on the nonlinear relation between delays and distribution process parameters in the subthreshold region. A single-point transient SPICE simulation with a nominal value for the process parameter was performed to obtain \(T_{d}\Big{(}P_{i}^{(0)}\Big{)}\). One thousand seeds of process variations were generated by random sampling for both MC simulation and the prediction model to obtain statistical distributions of \(T_{d}\) and \(S_{out}\). Figure 9 shows Delay L-H's distribution using the prediction model with n = 1, 3, 5 under PHIG1 variations. Figure 6: (a) Computational time comparison between the proposed surface potential approximation and the iterative method. (b) Computational time comparison for the current calculation. Figure 8: (**a**) MAPE (**b**) RMSE between the proposed delay model and SPICE simulations. Figure 7: The comparison of inverter delay versus \(I_{eff}\) and the probability density between 1000 MC SPICE simulations and the delay model with a supply voltage of (**a**) 0.8 V (**b**) 0.5 V (**c**) 0.3 V.  
Variations in the front gate work function (PHIG1) make a significant contribution to the inverter delay variation. The coefficient of variation (CV), defined by \(\sigma/\mu\) of delay H-L with PHIG1 variation, is 74.9%. The relationship between delay and PHIG1 shows a strong nonlinearity in the subthreshold region, which can be explained by the effective voltage of the front gate:  
\[V_{fgeff}=V_{fg}-\Delta\phi_{MS1}-\Delta V_{TH} \tag{22}\]  
where \(\Delta V_{TH}\) represents the variation in the threshold voltage. The front gate work function is the basic component of the threshold voltage, which means that PHIG1 variation and \(V_{TH}\) variation have equivalent effects on \(V_{fgeff}\). When operating in the subthreshold region, the channel current is dominated by the diffusion current, and the subthreshold current expression based on threshold voltage can be expressed as  
\[I_{sub}=\mu_{0}C_{OX}\frac{W}{L}V_{tm}{}^{2}(n-1)exp\left(\frac{V_{GS}-V_{TH}} {nV_{tm}}\right)\left[1-exp\left(\frac{-V_{DS}}{V_{tm}}\right)\right] \tag{23}\]  
The subthreshold current expression shows an exponential relation with \(V_{TH}\), resulting in the nonlinear change in delay caused by PHIG1.