## 2 Analysis and Simulations of SEB Arrays for Probing Using RF Reflectometry
Red curves in Figure 2c,d show the reduction in the maximal values of \(G_{Sis}\) and \(C_{dyn}\) by \(\sim\)3 and \(\sim\)1 orders of magnitude, respectively, as the temperature increased by roughly two orders of magnitude. As a result, the admittance became predominantly capacitive at higher temperatures. This corresponds to an almost constant phase \(\sim\)90 degrees of admittance (red curve in Figure 2b) and oscillations of admittance, while reduced from the low temperature case, are primarily caused byoscillations in \(C_{dyn}\). Note that even at low temperature the maximal swing of admittance oscillations (\(\sim\)40 nS in Figure 2a, blue curve) is much smaller than the respective magnitude of conductance oscillations in the SET with the junction parameters of \(C_{j}\) = 30 aF; \(R_{j}\) = 45 k\(\Omega\); \(C_{g}\) = 2.88 aF and test frequency \(f_{RF}\) = 198.9 MHz. Measurements of high impedance devices are challenging and greatly limit the bandwidth and speed of device operation. A simple way to alleviate this problem and to enhance the bandwidth and SNR is to use a parallel connection of SEBs to form arrays of boxes (SEBA). To understand the sensitivity trends expected in SEBAs with various sizes we studied numerically arrays of different size, \(N\). Arrays are composed of individual SEBs (Figure 3a) each represented by a parallel combination of \(1/G_{Sis}\) and \(C_{Dyn}\) (Figure 3b), in accordance with (2). An equivalent circuit of an SEBA is presented in Figure 3c. No interaction between individual SEBs within an array is considered. This assumption that proved to be experimentally valid for the arrays we investigated in this work due to the significant distance between adjacent SEBs. To account for unavoidable and uncontrollable fixed charges always present in real devices, individual SEBs within the array are assigned a random offset in the phase of the simulated oscillations uniformly distributed across one period in \(V_{g}\). This random charge offsets set the limit on the scaling, which is expected to be \(\sim N^{1/2}\) for the incoherent sum of the oscillations. Unavoidable process variations during the fabrication result in variations in gate capacitances for individual boxes. This leads to a beating oscillations pattern in the SEBA admittance. Figure 2: Complex admittance oscillations vs. \(V_{g}\) and respective components of it (**a**) magnitude; (**b**) phase; (**c**) inverse of Sisyphus resistance and (**d**) dynamic capacitance for single SEB calculated using Formula (1) at two temperatures: 50 mK (blue) and 4.2 K (red). The primary response of interest is the derivative with respect to \(V_{g}\) of the magnitude of the admittance (\(\gamma\)) of the array (\(d|\gamma|/dV_{g}\)). To maximize sensitivity, an SEBA detector would be biased at \(V_{g}\) for which \(d|\gamma|/dV_{g}\) is at maximum. Therefore, when investigating how different arrays perform, the peak of this derivative is the extracted value relating to the signal strength of that array. The responses of the arrays are simulated using Monte Carlo simulations of the equations in (2) to account for various process variations. To account for random background charge each box is assigned a random phase shift uniformly distributed across a period. Therefore, it is most useful to look at the statistics of the simulated results, rather than individual array responses. The simulation for each data point is run 100 times and the average and other statistical values, are extracted. The maximum slope of the response curve corresponds to the maximum sensitivity. Therefore, for each simulated array,  
Figure 3: (**a**) Single SEB circuit symbol. (**b**) Frake model of an SEB as parallel frequency dependent conductance and capacitance. (**c**) Circuit schematic of the SEB array. (**d**,**e**) Simulated trends of signal strength (derivatives) \(d|\gamma|/dV_{g}\) and \(d\phi dV_{g}\) as a function of the number of SEB in the arrays at 2.4 K with junction parameters listed in the text.