## 5 Experimental Results
### Temperature Dependence  
To evaluate the reliability of the random ID system over temperature, the change in \(\Delta V_{T}\) was measured for all the ID cells over 30 \({}^{\circ}\)C and 50 \({}^{\circ}\)C increases in temperature. We observed that for each ID cell, the temperature dependence of \(\Delta V_{T}\) varies, in part proportionally to \(\Delta V_{T}\), and in part  
Figure 10: Maximum number of addressable chips under the masked addressing system for a given number of random ID bits, \(N\). Figure 9: Normalized histogram showing measured and simulated distribution of \(\Delta V_{T}\). Measured sample size is 15,360 bits, and simulated sample size is 10,000 bits. randomly between cells. The change in \(\Delta V_{T}\) over the observed range is roughly linear with temperature, and so can be expressed as  
\[\frac{d\Delta V_{T}}{dT}=K \tag{21}\]  
where \(K\) is a random variable, with mean value proportional to \(\Delta V_{T}\):  
\[\mu_{K}=a\Delta V_{T} \tag{22}\]  
The temperature coefficient, \(a\), was found to be \(-\)6.1 \(\times\) 10\({}^{-3}\)\({}^{\circ}\)C\({}^{-1}\). Because \(a\) is negative, the magnitude of \(\Delta V_{T}\) tends to decrease as temperature increases. The standard deviation of \(K\) was found to be \(\sigma_{K}=0.22\) mV/\({}^{\circ}\)C. A plot of the mean and standard deviation of the temperature dependence is shown in Figure 11, over a range of \(\Delta V_{T}\) values. A possible explanation for the apparent randomness of the temperature dependence is that the devices M1 and M2 in Figure 1 are assumed to differ only in their values of \(V_{T}\), when in fact other device parameters such as the mobility \(\mu_{n}\), gate oxide capacitance \(C_{ox}\), and effective dimensions \(W\) and \(L\) will vary as well [19]. For example, in the preceding characterization scheme, if devices M1 and M2 are determined to have equal drain currents when driven with equal gate voltages, it is assumed that their values of \(V_{T}\) are equal. However, it could also be that their values of \(V_{T}\) are unequal, and other devices parameters are unequal in such a way that the drain currents remain equal. These other device parameters will then have their own, unequal temperature dependence [20], resulting in the apparent random temperature dependence. A possible improvement to this work would be a more elaborate characterization scheme in which the mismatch between devices M1 and M2 could be more completely determined, although this would increase the complexity of the characterization scheme. The practical effect of this is that a larger value of \(V_{R}\) must be selected for reliable operation, due to the combined effects of electronic noise and temperature dependence. For positive values of \(\Delta V_{T}\), and if all temperature dependencies are assumed to fall within \(6\sigma\), then the worst case temperature shift from \(\Delta V_{T}\) to \(\Delta V_{T}^{\prime}\) over an increase in temperature \(\Delta T\) is  
Figure 11: Mean and standard deviation of the change in \(\Delta V_{T}\) per 1 \({}^{\circ}\)C change in temperature, derived from measurement of 15,360 bits of a single chip. \[\Delta V_{T}^{\prime}-\Delta V_{T}=(a\Delta V_{T}-6\sigma_{K})\Delta T \tag{23}\]  
Rearranging Equation (23) and substituting \(\Delta V_{T}^{\prime}=V_{R}\) and \(\Delta V_{T}=V_{R}^{\prime}\) gives the value of \(V_{R}^{\prime}\) measured at nominal temperature that is required to ensure \(|\Delta V_{T}^{\prime}|\geq V_{R}\) after a temperature increase of \(\Delta T\). \[\Delta V_{R}^{\prime}=\frac{V_{R}+6\sigma\Delta T}{1+a\Delta T} \tag{24}\]  
That is, by only selecting ID cells with \(|\Delta V_{T}|\geq V_{R}^{\prime}\), it is ensured that after a \(\Delta T\) increase in temperature, the new threshold voltage difference will satisfy the original reliability requirement, \(|\Delta V_{T}^{\prime}|\geq V_{R}\).