## 5 Cell Imbalance Factor
The procedure used is as follows:  
Figure 4: State space diagrams for: (a) a perfectly symmetric cell; (b) a cell biased towards logic ‘0’ state; (c) a cell biased towards logic ‘1’ state. Separatrix is represented in green. Each red trajectory (test 0) represents the initial transient analysis used to determine the cell bias, and each blue line is the trajectory with less SD voltage shift that makes the cell start up to the opposite state determined with test 0. To determine the state towards the cell is biased to (red trajectories labeled as test 0 in Figure 4b,c), a transient analysis is performed with the cell starting from (0, 0). This simulation indicates which one of the axes it is necessary to explore to compute the SD parameter. For instance, if the cell is biased towards S1 (Figure 4c), we have to search for a starting point (VQB0, 0) on the horizontal axis that makes the cell flip its starting-up state to S0. This has been automated by means of a search algorithm using the Cadence Ocean scripting framework. The algorithm will try different starting points in the axis of interest, and determine the starting point with the minimum voltage that makes the cell start up to the opposite state it was biased to (flip its nominal start-up state) [35]. This is represented by blue trajectories in Figure 4b,c. The procedure is performed over all memory cells. To sum up, a perfectly symmetric cell will have an SD parameter equal to zero, a cell biased towards logic '1' will have a positive SD, and a cell biased towards logic '0' will have a negative SD parameter. Since cells are designed to be as symmetric as possible, the SD parameter is expected to have zero average value. In addition, only a small number of cells in a chip are expected to present high imbalance factors, since SRAM are circuits in which design for manufacturing techniques are applied with special care to minimize parameter variations. The SD parameter was computed for a total of 16,384 6T cell Monte Carlo iterations. This number was chosen because it is the number of cells that are available in the memory described in Section 3. From the simulation data, the histogram of Figure 5 was computed. It shows the relative frequency of cells as a function of the cell SD parameter. Figure 5 shows that the resulting SD distribution is Gaussian-shaped. As expected, most cells show a relative good matching (SD around zero), and only few cells are highly mismatched (high SD absolute value). For instance, 90.6% of the cells lie between \(-\)0.04 V and 0.04 V and, consequently, only 9.4% of cells have an SD absolute value factor greater than 0.04 V.  
As a consequence of the dynamic behavior of SRAM cells, highly mismatched cells are expected to start up consistently to the same logic value (logic '0' or logic '1') with high probability, while low mismatched cells are expected to start up to a given value with less probability. Figure 5: Histogram of the relative frequency of cells as a function of the SD parameter obtained from Monte Carlo analysis. The key question is determining the SD absolute value from which a cell can be considered as a highly reliable cell, i.e., which is the threshold value (SD\({}_{\text{th}}\)) that makes a cell start up, for instance, more than 95%, 98% or 99% of times to the same value. Figure 6 is obtained from the data of Figure 5, but considering the absolute value of SD and accumulating the values from left to right. Figure 6 depicts the frequency of cells (vertical axis) with an SD absolute value greater than SD\({}_{\text{th}}\) (horizontal axis). For instance, if obtaining a successful start-up rate greater than 99% required SD\({}_{\text{th}}\) to be 0.05 V, only 3% of the cells would meet the condition. If SD\({}_{\text{th}}\) had to be 0.04, 9.4% of cells would satisfy the condition. This percentage would increase to 21% if the condition was relaxed to SD\({}_{\text{th}}\) = 0.03 V. This question will be addressed in the next section with experimental data.