## 2 Analysis and Simulations of SEB Arrays for Probing Using RF Reflectometry
(**d**) Normalized signal strength of the two derivatives: magnitude (black) and phase (red) of admittance for SEBA of varying size. Each data point is the average of 100 simulations of the same array conditions with randomization to account for process variation. (**e**) Simulations for magnitude sensitivity for SEBA with different average gate capacitances: \(C_{g}=22.9\) aF (green), and \(C_{g}=2.9\) aF (black). The solid (dashed) black and solid (dashed) green lines correspond to a \(V_{g}\) search range of 10 V (100 mV). The blue dashed line corresponds to the response from a single SET with indicated capacitances and \(R_{S}=R_{D}=R_{J}=50\) k\(\Omega\) biased at the maximum sensitivity point. the maximum derivatives of \(Y\) (both in magnitude, \(|Y|\), and phase, \(\varPhi\)) in the simulated \(V_{g}\) range is extracted. Figure 3d shows the results of SEBA sensitivity scaling with \(N\). The maxima of magnitude \(d|Y|dV_{g}\) (black) and phase \(d\varPhi/dV_{g}\) (red) are plotted for SEBAs with the following parameters for \(i\)-th SEB: \(C_{ji}=40\) aF (STD = 15%), \(C_{gi}=2.9\) aF (STD = 10%) and \(R_{ji}=50\) k\(\Omega\times 40\) aF/\(C_{ji}\) at \(T=2.4\) K, and the maxima are found within 10 V of \(V_{g}\) span. Curiously, after an initial boost in both components of the signal for \(1<N<10\), the magnitude and phase sensitivity curves diverge. The cause for this behavior is simple. At moderately low temperatures when admittance of a SEB can be approximated by a sinusoid with vertical offset, the admittance of a SEBA composed of \(N\) boxes, \(Y_{N}\), follows [21]:  
\[Y_{N}=N\times\Delta Y_{M}+\sqrt{(}N)\times\Delta Y_{M}\times\sin(2\pi V_{g}/ \Delta V_{g}) \tag{3}\]  
where \(\Delta Y_{M}\) is the averaged (for an array) SEB admittance swing in (2). Both real and imaginary parts of \(Y_{N}\) have positive average values that scale \(\sim\)\(N\). Note that the \(V_{g}\) dependent part in (3) is the only useful part for sensing applications. Magnitude sensitivity, \(d|Y|dV_{g}\), therefore, scales proportional to \(N^{1/2}\). Phase sensitivity, \(d\varPhi/dV_{g}\), however, exhibits a very different trend. While for small arrays (\(N<10\)) a peak value of \(d\varPhi/dV_{g}\) increases, for large \(N>>10\) a continuously larger fraction of each of the real and imaginary components of admittance will be composed of the constant average value corresponding to a fixed phase angle at a given frequency. As this happens, the oscillations in phase of \(Y\) will decrease. While the peak derivative of the magnitude increases proportional to \(N^{1/2}\), the derivative of the phase does the opposite, decreasing proportionally to \(N^{-1/2}\). Clearly, the ultimate sensitivity of SEB sensor is set by a level-arm factor \(\alpha\) in (2), and it increases for a larger \(C_{g}\). To better understand limits achievable by the SEBA approach, two sets of arrays are simulated using different \(C_{g}\) values shown in Figure 3e. For the first, \(C_{g}~{}=~{}2.9\) aF, which corresponds to SEBs with the same structure as the SET with two junction capacitances \(C_{j}=20\) aF, but without the drain side tunnel junction (see Figure 1a,b for the reference). The second, \(C_{g}~{}=~{}22.9\) aF, corresponds to SEBs with the same \"capacitance budget\" (42.9 aF) as the SET, with the \"spare\" 20 aF capacitance allocated to the gate, i.e., having the same averaged charging energy as the SET. In the simulations, the gate capacitances are randomized with a standard deviation of 10% of the mean (2.9 aF and 22.9 aF, respectively) to model lithographic variations between individual SEBs. Similar to Figure 3d, each array is simulated 100 times, with new randomized values each time, and the mean value extracted. Green solid (dashed) line and black solid (dashed) lines in Figure 3e corresponded to the SEBA with average gate capacitances \(C_{g}=22.9\) aF, green, and \(C_{g}=2.9\) aF, black, simulated over \(V_{g}\) span of 10 V (100 mV).