# CNT-PUFs: Highly Robust and Heat-Tolerant Carbon-Nanotube-Based Physical Unclonable Functions +
## 5 Evaluation of the Fabricated CNT-PUFs  
### CNT-PUF Design with Highly Distinguishable Responses  
A basic requirement for a robust PUF is the generation of highly stable responses. Therefore, an important design goal of CNT-PUFs is to be able to reliably classify CNT-PUF cells into conductive and non-conductive cells among repeated measurements. To determine the best CNT-PUF design that yields the optimal distribution of responses, we studied cells with three different channel widths \(W_{ch}\), namely 500 nm, 1000 nm, and 2000 nm. The channel's position and width are shown in Figure 2e. The channel width, due to its geometry, has a decisive influence during the fabrication of the PUF on whether the corresponding cell becomes conductive or non-conductive, as the probability for a conductive cell increases with increasing width at a constant density of deposited CNTs. The electrical behaviour of cells having different channel widths is shown in Figure 5. The number of cells conducting a certain maximum drain current \(I_{D,\text{max},\text{p}}\) is plotted on a logarithmic scale. It can be seen that, despite the different distribution of conducting and non-conducting cells for each channel width, there is a clear separation between these two classes independent of the channel width. The conducting cells are located in the nA to \(\mu\)A range (\(10^{-9}\text{A}<I_{D,\text{max},\text{p}}<10^{-5}\text{A}\)) and the non-conducting devices in the sub-nA range (\(10^{-11}\text{A}\) and lower). In between these two major groups of cells, we can also see a few cells with ambiguous drain current values. Since the maximum drain current of these cells is near the threshold voltage for quantisation, they can cause bit flips and, thus, instabilities in the PUF response. Section 6 discusses the treatment of these cells. Figure 5 also shows the decisive influence of the channel width on the ratio between conducting and non-conducting cells. The distribution of the maximum drain current for conducting devices (\(I_{D,\text{max},\text{p}}>1.0\,\text{nA}\)) shows a log-normal behaviour with an average of \((-7.0\pm 0.6)\) Log(A) for a channel width of 500 nm, \((-7.0\pm 0.7)\) Log(A) for a channel width of 1000 nm, and \((-6.6\pm 0.5)\) Log(A) for a channel width of 2000 nm. For a channel width of 2000 nm, a slight increase in the maximum observed \(I_{D,\text{max},\text{p}}\) can be seen (Figure 5). This stems from the increased prob  
Figure 4: Distribution of 28 measurements of the source–drain current of each cell when applying \(V_{GS}=-2.5\,\text{V}\) and \(V_{DS}=-1\,\text{V}\). One can retrieve the stability of each cell from this figure, as well as the overall distribution of conducting and non-conducting cells of the examined device, based on a broad window within which the threshold \(I_{ll}\) can be set. The window is indicated by light grey lines and an exemplary threshold by a dark grey line. This figure was adapted from [41]. ability of parallel conductive paths occurring when multiple CNTs are deposited so that they bridge the CNT-FET channel and confirmed that their occurrence correlated with the channel width. The described phenomenon can be observed by looking at the two CNTs in Figure 2e, which form a bridge between the source and drain electrodes. Another process parameter we investigated during the fabrication of the PUF cells was the relationship between the channel width, the CNT density, and the probability that the relevant CNT-FET is conductive. For this study, the threshold current was set at 20 pA. Figure 6 shows the probability for conductive cells depending on the CNT density and different channel widths. We see that, regardless of the channel width, as the density of the assembled CNTs increased, the frequency of conducting cells increased as well. Initially we assumed a CNT density \(\rho_{\mathrm{CNT}}=0.5\cdot W_{\mathrm{ch}}^{-1}\) to target the highest binary uniformity for a given \(W_{\mathrm{ch}}\). However, the behaviour was not as linear as expected.