## 5 Cell Imbalance Factor  
Unfortunately, the distribution of start-up values can only be obtained experimentally, that is, once the memory has been manufactured. However, having information about the percentage of reliable cells at the initial design stages is a must. Therefore, it will be very useful to have an alternative parameter that can be obtained from electrical simulation, and whose resulting distribution can be related to the start-up values distribution. This parameter is introduced in this section. As mentioned in Section 2, each SRAM cell has two stable states: (\(\mathrm{V_{DD}}\), 0) and (0, \(\mathrm{V_{DD}}\)). We can represent them in a state space diagram [34] as depicted in Figure 4a, in which each point represents the voltages of the internal nodes (\(\mathrm{V_{Q}}\), \(\mathrm{V_{QB}}\)). The state S0 corresponds to (\(\mathrm{V_{DD}}\), 0), and represents the state in which the cell stores a logic '0'; and S1 corresponds to (0, \(\mathrm{V_{DD}}\)), and represents the state in which the cell stores a logic '1'. M is a metastable point. If the cell is perfectly symmetric, this metastable point belongs to the line that goes from (0, 0) to (\(\mathrm{V_{DD}}\), \(\mathrm{V_{DD}}\)), and is located at (\(\mathrm{V_{M}}\), \(\mathrm{V_{M}}\)), where \(\mathrm{V_{M}}\) depends on the relative strength between the pull-up and pull-down devices. This diagonal line is called the separatrix and divides the diagram into two regions, each one related to one of the stable states. Each of these regions represents the area of attraction of one of the two stable points. This means that if the cell is started at a specific point inside the area, the cell will end up at its corresponding area stable point. This is represented by the grey trajectories depicted in Figure 4a. Memories are usually started up from the (0, 0) point. In the case of a perfectly balanced cell, this point is on the separatrix, which is also the boundary between the areas of attraction. In this case, the cell will theoretically go to the metastable point and will remain there. Nevertheless, any external or internal perturbation will affect it and drive the cell to one of the two stable states. However, any real cell will present a mismatch to a certain extent, and thus will be biased towards one of the stable states, as represented in Figure 4b,c. In these cases, the separatrix is not defined by the points (0, 0) and (V\({}_{\text{DD}}\), V\({}_{\text{DD}}\)), and thus it divides the phase state into two unequal regions. If the cell is started from (0, 0), it will end up in the stable state of the shadowed area of Figure 4b,c. The greater the mismatch of a cell, the greater the distance from the point (0, 0) to the separatrix and, consequently, the greater the probability of the cell starting always at the same logic value [35]. So, we can define a cell imbalance factor named SD (Separatrix Distance), which is related to the cell mismatch and can be regarded as a measure of the tendency that a cell has to start up consistently to a given value. Figure 4b,c show this distance labeled \"SD\". Note that the distance is defined along the axis that the separatrix intersects. A large distance means that the cell will be able to sustain large perturbations without changing its start-up value, and thus the cell will be reliable for PUF applications. On the other hand, a small distance means that a small perturbation will be capable of flipping the preferred start-up state, and thus that the cell will be less reliable from a PUF point of view. For convenience, we define the SD parameter as positive if the cell is biased towards S1 (logic '1' start-up, like in Figure 4c), and negative if it is biased towards S0 (logic '0' start-up, as shown in Figure 4b). The computation of the SD parameter was performed using transient Monte Carlo electrical simulations with the Spectre simulator on a commercial 65-nm CMOS technology.