# CNT-PUFs: Highly Robust and Heat-Tolerant Carbon-Nanotube-Based Physical Unclonable Functions +
## 5 Evaluation of the Fabricated CNT-PUFs
### Evaluation of CNT-PUFs under Normal Conditions
#### 5.2.1 Uniformity  
We first checked whether ones and zeros were equally distributed in each PUF response. This property was evaluated by applying the quantisation algorithm introduced in Section 4.3 to the raw measurements of each individual CNT-PUF. Afterwards, the number of non-conducting cells (logical 0) and conducting cells (logical 1) was counted for each individual CNT-PUF, from which the Hamming weight was calculated. To optimally set the threshold \(I_{th}\) so that a nearly uniform distribution of PUF bits can be achieved, this metric was evaluated for different thresholds \(I_{th}\). Figure 7 illustrates the Hamming weight of three PUF responses with different channel widths, denoted as \(W_{ch}\).  
For all measurements, \(V_{\mathrm{D}}=-1\) V and \(V_{\mathrm{GS}}=-2.5\) V were applied to the circuit. As we can see, almost all CNT-PUFs, except the ones with \(W_{ch}=2000\) nm, achieved an almost uniform distribution of PUF bits when the threshold was set within the range between \(I_{\mathrm{D,max,p}}=7\) pA and \(I_{\mathrm{D,max,p}}=100\) pA (indicated by dashed vertical lines in the figure). By adjusting the threshold, more cells were classified as conductive cells when choosing a lower threshold, while higher thresholds led to more non-conductive cells and, thus, to a lower Hamming weight. As a consequence, Figure 7 exhibits a step-like shape. Moreover, we can see that a channel width of 2000 nm led to a higher amount of conducting cells and, thus, a bias towards logical 1s. The fact that the PUFs can maintain this almost ideal ratio of ones and zeros over a wide range of threshold values turned out to be a good indicator for robust PUF responses, as further indicated in Section 5.2.4.  
Figure 6: Probability of conducting cells depending on CNT density \(\rho_{\mathrm{CNT}}\) and channel width \(W_{\mathrm{ch}}\), for \(V_{\mathrm{GS}}=-2.5\) V and \(V_{\mathrm{DS}}=-1.0\) V. This figure was adapted and modified from our previous work [19], which this work extends.  
We can derive that, by using a threshold value of \(I_{th}=20\,\mathrm{pA}\), an almost uniform distribution of non-conducting (44.71%) and conducting cells (55.29%) for cells with \(W_{ch}=1000\,\mathrm{nm}\) can be achieved. These values are close to the optimal value of 50% for each class. In Figure 8, the ratio of non-conducting and conducting cells for each of the 30 measured CNT-PUFs is visualised for a threshold of \(I_{th}=20\,\mathrm{pA}\). It can be seen that no CNT-PUF exhibited an average fractional Hamming weight higher than 62.5% or lower than 43.75%.  
Figure 8: Histogram of the relative amount of conductive cells of 30 CNT-PUFs with a \(1000\,\mathrm{nm}\) channel width. The classification of conductive and non-conductive cells is based on a threshold of \(I_{th}=20\,\mathrm{pA}\). This figure was adapted and modified from our previous work [19], which this work extends.  
Figure 7: Hamming weight calculated for the PUF responses of three CNT-PUFs, one for each of the three different channel widths, depending on a threshold value of \(0.1\,\mathrm{pA}\) to \(1\,\mathrm{pA}\). The grey lines indicate the range where the amount of both \(0\,\mathrm{s}\) and \(1\,\mathrm{s}\) remains almost constant and is the same as in Figure 4, giving the best stability.