## 6 Start-Up Value Probability vs. SD Relation
The plot corresponds to the function defined in Equation (5) after a fitting process (m = 0.158; k\({}_{1}\) = 101.2 V\({}^{-1}\); k\({}_{2}\) = 2348 V\({}^{-1}\)). To improve the modeling of the transition between the different regions, a function with 2 logistic components is considered:  
\[\mathit{SUP_{1}}=\frac{m_{1}}{1+e^{-k_{1}\mathit{SD}}}+\frac{m_{2}}{1+e^{-k_{2} \mathit{SD}}} \tag{5}\]  
Each component has its own exponential coefficient (\(k_{1}\) and \(k_{2}\)) and its own weight (\(m_{1}\) and \(m_{2}\)). To meet the requirements, it is necessary that \(m_{1}+m_{2}=1\). So, we impose that \(m_{1}=m\), and \(m_{2}=1-m\), and Equation (5) can be written as:  
\[\mathit{SUP_{1}}=\frac{m}{1+e^{-k_{1}\mathit{SD}}}+\frac{1-m}{1+e^{-k_{2} \mathit{SD}}} \tag{6}\]  
Observe that the new weights add to 1. As a consequence, it can be easily proved that when \(\mathit{SD}\gg 0\), \(\mathrm{SUP_{1}}\) tends to 1; when \(\mathit{SD}\ll 0\), \(\mathrm{SUP_{1}}\) tends to 0; and \(\mathit{SUP_{1}}(0)=0.5\), so it meets the requirements. A representation of this function can be seen in Figure 7b, where it can be observed that the curve meets all of the requirements and that the function shows odd symmetry with respect to the (0, 0.5) point. This is a consequence of the symmetry of both \(\mathrm{SUP_{1}}\) (Figure 3a) and SD (Figure 5) distributions. Parameters \(m\), \(k_{1}\) and \(k_{2}\) must be determined by fitting the simulated \(\mathrm{SUP_{1}}\) distribution, obtained from the Equation (6) when using the \(\mathit{SD}\) values resulting from the Monte Carlo analysis, to the experimental \(\mathrm{SUP_{1}}\) distribution. After completing the fitting, the following result has been obtained:  
\[m=0.158\quad k_{1}=101.2\;V^{-1}\quad k_{2}=2348\;V^{-1} \tag{7}\]  
The curve in Figure 7b plots Equation (6) with the fitted parameter values. Figure 8b shows the relative frequency of cells as a function of \(\mathrm{SUP_{1}}\) for both the experimental data and the simulated data converted into start-up probabilities using Equation (6) and properly fitted. As can be seen, unlike the fitting with Equation (3), now with Equation (6) the agreement between the two sets of data is good. Figure 8: Histogram of the relative frequency of cells as a function of the \(\mathit{SUP_{1}}\) (start up at logic ’1’ probability) obtained from both experimental data and from modeled simulation results using: (**a**) Equation (3) with the fitted parameter of Equation (4) (\(k=195.4\;\mathrm{V^{-1}}\)), and \(a=1\); (**b**) Equation (6) with the fitted parameters of Equation (7) (\(m=0.158\); \(k_{1}=101.2\;\mathrm{V^{-1}}\); \(k_{2}=2348\;\mathrm{V^{-1}}\)). Notice that the two exponential constants (Equation (7)) show very different values, which is a consequence of modeling the two different regions (A and B) labeled in Figure 3a. In Figure 7b, this can also be seen as two different regions: one near \(SD=0\), which has a very steep slope, and the regions far away \(SD=0\), where the slope decreases and eventually becomes zero. In fact, if the slope of the curve of Figure 7b near \(SD=0\) is calculated, it gives approximately 500 V\({}^{-1}\), which roughly means that increasing \(SD\) by 0.2 mV rises the start-up at '1' probability by 0.1. This result is a numeric quantification of the fact that small differences in cell unbalance lead to great differences in start-up probability. For this reason, there are few cells that start up at logic '1' with near 0.5 probability, because small differences of only a fraction of mV in the \(SD\) parameter are translated into great differences in start-up probability. This means that even slightly asymmetric cells are quite biased towards one specific start-up value, and thus can be good candidates for PUF. From the curve depicted in Figure 7b, it can be calculated that absolute values of SD greater than 0.03 V lead to start-up at '1' probabilities higher than 0.99 (and by symmetry the same happens for start at '0').