# Radio Frequency Reflectometry of Single-Electron Box Arrays for Nanoscale Voltage Sensing Applications

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## Abstract

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^{2}), a large number of devices (>1000) can be assembled into an array occupying just a few square microns. We show that it is possible to design SEB arrays that may compete with an SET in terms of sensitivity. In this, we tested SETs using RF reflectometry in a configuration with no DC through path (“DC-decoupled SET” or DCD SET) along with SEBs connected to the same matching network. The experiment shows that the lack of a path for a DC current makes SEBs and DCD SETs highly electrostatic discharge (ESD) tolerant, a very desirable feature for applications. We perform a detailed analysis of experimental data on SEB arrays of various sizes and compare it with simulations to devise several ways for practical applications of SEB arrays and DCD SETs.

## 1. Introduction

^{−6}e/Hz

^{½}[2]. Single-electron devices represent a natural choice for measurement applications that require sensing of the charge or/and potential at the nanoscale, such as scanning probes [3,4,5,6] and readouts of qubits [7,8,9,10,11,12,13,14].

_{x}SEBs and SETs fabricated using the so called Niemeyer–Dolan bridge technique [15,16]. They are composed of a nanoscale “island” coupled to the outside world by two (SET), Figure 1a, or one (SEB), Figure 1b, tunnel junctions (TJs) and a non-leaky capacitive gate. Electron transport through the SET island from source to drain at temperatures T << E

_{C}/k

_{B}, where k

_{B}is Boltzmann’s constant, is controlled by a gate voltage periodically enabling/disabling carrier transfer and resulting in the Coulomb blockade oscillations (CBOs) of conductance. Here the key parameters are the charging energy, E

_{C}= e

^{2}/2C

_{Σ}(C

_{Σ}= 2C

_{J}+ C

_{g}; C

_{J}and C

_{g}are junction and gate capacitances, respectively and e is an electron charge) and junction resistance R

_{J}, which must be large enough to enable charge localization on the island, R

_{J}> h/e

^{2}[1]. For either device as the gate voltage is swept, the energy cost to add or remove an electron periodically reaches zero, at which point the total electron population of the island changes by one. Figure 1c shows a schematic diagram of experiment from [17] where an SET was used as a sensor to probe single-electron charging in an SEB. As electrons added one by one to the SEB by applying positive gate voltage V

_{g}to the SEB gate capacitor, the resulting sawtooth-like oscillations of the island potential were detected by a capacitively coupled SET biased with a source–drain voltage V

_{SD}at a sensitive point of the SET response, in turn resulting in oscillations of source–drain current I

_{SD}, measured by an ammeter.

_{Σ}and thus increases the charging energy, which increases the maximal operating temperature and sensitivity of the device at k

_{B}T<<E

_{C}. When designing single electron devices to operate at a desired temperature, the primary constraint is the total island capacitance, C

_{Σ}. At the same time, to increase voltage gain in the SET, G

_{V}= −C

_{g}/C

_{J}, its gate capacitance needs to be maximized [19]. Therefore, depending on the application, SEBs can be designed to either allocate more of its “capacitance budget” to C

_{g}while keeping the same total capacitance C

_{Σ}as an SET with two junctions, or to achieve the highest operating temperature by minimizing C

_{Σ}. Second, since no DC current flows across the junctions, there is no contribution from shot noise, the currently recognized limiting noise of SETs for fast measurements at RF frequencies [20]. Third, the DC path through the SET makes them far more susceptible to electrostatic discharge (ESD) damage. While SETs are notoriously prone to ESD and must be handled with extreme caution, the SEBs we experimentally studied survived tens of thermal cycles and electrical connections and disconnections to various experimental setups. Moreover, components were soldered and de-soldered to the boards with devices already connected, without any noticeable degradation of performance. We will discuss the physical reasons for this below in Section 6.

_{gi}for an i-th box in the array results in a change of a period of CBO oscillations ΔV

_{gi}= e/C

_{gi}, leading to a beating pattern in the transfer function. By optimally biasing the array in a region with constructive interference, giving higher values of slope of the transfer function, $d\left|Y\right|/d{V}_{g}$, it was estimated that the sensitivity can be increased by a factor of ≈ 2 above ${N}^{1/2}$ [21]. This, however, is not always practically possible if only one gate is used both for sensing and tuning. In this work we demonstrate how this optimization can be achieved with an extra “tuning “gate using arrays of parallel SEBs sharing the common “sensing” gate for applications such as a scanning probe, where the surface potential is of interest and a small gate voltage is required to avoid perturbing the sensed object.

## 2. Analysis and Simulations of SEB Arrays for Probing Using RF Reflectometry

_{J}and C

_{g}in series, while when the Coulomb blockade is lifted the SEB admittance is maximized. Near charge degeneracy points, two mechanisms associated with the underlying physics of single electron charge transfer are responsible for enhancement of admittance of the SEB. One is the so-called Sisyphus resistance [22]—excess dissipation at RF frequencies ω approaching or exceeding tunneling rate γ through the junction with resistance R

_{J}, ω ≥ γ. For a given temperature T the tunneling rate at the degeneracy point is given by

_{B}T/e

^{2}R

_{J}.

_{g}. These two components of total admittance change as the device goes in and out of the blockade, resulting in the phase and magnitude oscillations of the SEB admittance yielding and equivalent circuit of a parallel combination G

_{Sis}(V

_{g}) and C

_{Dyn}(V

_{g}) [26]:

_{g}is the gate voltage relative to an SEB population degeneracy point.

_{ds}≈ 0 oscillates between the constant value determined by the resistances of two SET junctions, R

_{J}

_{1}and R

_{J}

_{2}: G

_{max}= 1/(2(R

_{J}

_{1}+ R

_{J}

_{2})) [27] and at minima experience similar exponential suppression of conductance as in the SEB ~ cosh

^{−2}(−eαΔV

_{g}/2k

_{B}T). By contrast, admittance in the SEB contains both real and imaginary parts whose relationship is a strong function of temperature and probing frequency.

_{Sis}and C

_{dyn}by ~3 and ~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 ~90 degrees of admittance (red curve in Figure 2b) and oscillations of admittance, while reduced from the low temperature case, are primarily caused by oscillations in C

_{dyn}. Note that even at low temperature the maximal swing of admittance oscillations (~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 kΩ; C

_{g}= 2.88 aF and test frequency f

_{RF}= 198.9 MHz.

_{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 ~ ${N}^{1/2}$ 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.

_{g}(black) and phase dΦ/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 kΩ × 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]:

_{N}= N × ΔY

_{M}+ √(N) × ΔY

_{M}× sin(2πV

_{g}/ΔV

_{g})

_{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 ~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, dΦ/dV

_{g}, however, exhibits a very different trend. While for small arrays (N < 10) a peak value of dΦ/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 ${N}^{-1/2}$.

_{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.

_{g}= 22.9 aF, green, and C

_{g}= 2.9 aF, black, simulated over V

_{g}span of 10 V (100 mV). It is apparent that a larger gate capacitance increases the sensitivity because (a) the peak value of both resistive and reactive components of admittance Y in (2) are proportional to ${\alpha}^{2}$, and (b) the larger ${C}_{g}$ value results in stronger coupling to the gate and thus the values of the derivative dY/dV

_{g}increase. From these, it is clear that for both cases (with large and small gate capacitances) the peak sensitivity scales proportional to ${N}^{1/2}$. However, searching across a wider ${V}_{g}$ range gives more of an opportunity to find a high sensitivity region, resulting in a higher sensitivity for all array sizes. For the smallest arrays, those less than around 5 SEBs, the scaling is proportional to $N$ when measuring a wide ${V}_{g}$ range. This indicates that for small arrays, it is likely to find a region where all $N$ boxes converge in the ${V}_{g}$ ranges investigated.

_{g}for such SET biased at a maximum sensitivity point. From this we conclude that optimized SEBAs could compete with SETs for $N>8$. However, if average admittance of the array needs to approach 1/Z

_{0}(similar to [21]), respectively larger number of SEBs in the array (>1000) will be required.

## 3. Experimental Setup: Reflectometer and Matching Network Considerations

#### 3.1. Hardware Configuration

^{−}, to the incident RF voltage wave V

^{+}, on a load, which is a function of the load impedance: Γ = V

^{−}/V

^{+}= (Z

_{load}− Z

_{o})/(Z

_{load}+ Z

_{o}). In this case, the load is an SEB coupled to a MN (red box in Figure 4a, see below for details). Therefore, the reflection coefficient, Γ, will track the impedance of the SEB as a function of the gate voltage. We used a single port RF homodyne reflectometer where the probing RF signal is attenuated on the way to the sample (approximately 60 dB) by warm and cold attenuators and a directional coupler (ZFDC-20-50-S+ by Minicircuits). The attenuated signal is sent down the transmission line to the sample through the MN. A matching Π network, composed of input capacitor C

_{in}, surface mount inductor L with associated parasitic components, and pad capacitor C

_{pad}(Figure 4a) is tuned to achieve a sensitive response to small variations in admittance schematically represented as Y(V

_{g1}, V

_{g2}). The photography of a typical printed circuit board used in experiments with a fabricated sample and surface mounted components is shown in Figure S1 (Supplementary Materials). Note that the input of the reflectometer is a DC ground because of the internal design of the directional coupler (R

_{DC}< 10 Ω)

_{C}/e). Second, the heating caused by the RF signal must be less than the environmental heating (k

_{B}T/e). If these considerations are made, then the RF signal will not “smear” the response of the SEBs. Practically, this point is determined by ramping the RF voltage until smearing is observed. The measurement voltage is then chosen to be below this point while still retaining a sufficient signal-to-noise ratio. This usually results in RF amplitudes at the device on the order of tens to hundreds of microvolts.

#### 3.2. Matching Network Design

^{1/2}, yielding a minimal expected impedance ≥0.7 MΩ for N ≤ 200 studied here.

_{0}= 50 Ω. Without an impedance transformation, the changes in the reflected signal induced by changing device impedances are extremely small. Therefore, a carefully designed MN becomes crucially important for the signal extraction. The main purpose of the MN that performs this impedance transformation is therefore to convert the load impedance to the value close to a characteristic impedance of the transmission line, Z

_{0}, so that changes in the load impedance Z

_{load}will result in noticeable changes of the reflected signal. Note that strictly speaking all of the following considerations are only applicable if the measurement system is calibrated against known standards. This means, for example, that at a calibrated reference plane, both open and short standards are expected to produce |Γ| = 1 and a matched load should produce |Γ| = 0. Experimental setups never produce these results without the use of error correction because of a wide range of non-idealities in the signal paths (standing wave resonances in the transmission lines, deviations from exact ${Z}_{0}$, matched values in components used, frequency dependent transfer characteristics and phase shifts in the amplifiers and couplers, etc.). If calibration cannot be performed, the experimental results can only qualitatively be compared with the theoretical predictions. The optimization of MN that uses a critically coupled resonator is discussed in [29]. The secondary role of MN is to act as a band-pass filter to enable propagation of the carrier and sidebands while attenuating the out-of-band signals and thus improving the signal to noise ratio. This bandwidth consideration needs to be taken into account for proper design of the MN.

_{pad}-L-C

_{IN}) shown in red box Figure 4a. In our experiment a typical value of C

_{pad}≈ 0.5 pF is determined by the size of the bond pads and stray capacitance of the bond wire to ground. We use off the shelf surface mount size 0805 ceramic core inductors (220–820 nH) by Coilcraft and ceramic surface mount capacitors by Johanson Technology. The lower frequency limit of the reflectometry setup (≈100 MHz) is chosen to increase the SEBA admittance in accordance with equation (2); the upper frequency limit is determined by the homodyne detector used for carrier demodulation (600 MHz in our experiment, set by a frequency limit of ZI UHF).

_{IN}, the MN can be adjusted to convert Z

_{SEBA}to Z

_{load}≈ Z

_{0}= 50 Ω. To determine C

_{IN}that ensures the operation near a match point, the response of the MN is simulated using a realistic 5-element model of the inductor provided by the manufacturer (Coilcraft). However, to accurately simulate the MN these parameters need to be adjusted for low ambient temperature; failing to do so leads to gross errors and non-functioning MN. Note that due to large values of device impedance (>10 MΩ), the RF power is dissipated almost entirely in the parasitic resistances of the inductor coil rather than the SEBA components. This factor ultimately limits the sensitivity of the MN [15].

_{SE}in MN circuit, inside the red box in Figure 4a) as the pre-factor scales down proportional to the resistivity. Experiment shows that by contrast, the capacitance of 0805 size Johansson Technology chip capacitors used in experiments changes only by about 1% from 300 K down to low temperature (Figure 5, black curve).

_{IN}are considered. The blue curve in Figure 6b corresponded to a slightly overcoupled case, for which Z(f

_{res}) < 50 Ω; the red curve corresponded to a slightly undercoupled case, Z(f

_{res}) > 50 Ω and the green curve corresponded to the critically coupled case, Z(f

_{res}) ≈ 50 Ω. Next, the oscillations of admittance of a single SEB connected to the MN in Figure 6a in response to sweeping V

_{g}are simulated (Figure 6(c1),(c2)). Note that both magnitude and phase of admittance are simulated for each resonant frequency corresponding to the respective minima in Figure 6b (462.309 MHz for C

_{IN}= 20 pF, 461.288 MHz for C

_{IN}= 28.2 pF and 460.439 MHz for C

_{IN}= 40 pF) but they are indistinguishable in Figure 6(c1),(c2). By contrast, the oscillations in the reflected signal resulting from SEB admittance oscillations differed drastically for closely matched (C

_{IN}= 28.2 pF) vs. the other two (C

_{IN}= 20 pF and C

_{IN}= 40 pF) cases. In Figure 6(c3),(c4) we plotted respective derivatives d|Γ|/dV

_{g}and dΘ/dV

_{g}of the reflected signals as they represent sensitivity as is explained in Section 2. It is straightforward to see that the MN with C

_{IN}= 28.2 pF yields a much stronger signal and therefore it offered a significant improvement in both the magnitude and phase response. For the chosen parameters of the devices and temperature the resulting enhancement of oscillations is far more prominent in the dΘ/dV

_{g}signal than in d|Γ|/dV

_{g}for the reasons discussed below in the experimental section.

_{0}[30]. In the literature it is common to estimate the bandwidth available for gate modulation using −3 dB reduction in the magnitude of reflection coefficient (e.g., [31]) near the resonant frequency, however this definition is oversimplified because a 3 dB change in |Γ| does not always correspond to a true resonance (cancellation of positive and negative reactances). Moreover, a resonant match that is not critically coupled may not even have a 3 dB change in |Γ|(e.g., about −1 dB is was reported [26]). Therefore, to accurately evaluate the bandwidth of the experimental setup the response of the SEB to a modulating small signal (equivalent to 0.01e in magnitude) applied to the gate of SEB is simulated using Keysight ADS (the detailed description of the simulations will be published elsewhere) and it gives the result shown in the table within Figure 6b thus confirming the bandwidth requirement (>1 MHz) for fast SEBA sensor.

## 4. Fabrication

_{x}tunnel junctions (more details on fabrication can be found in [32]). The EBL double stacks used are composed of 200 nm PMGI (polymethyl-glutarimide) and 100 nm PMMA (polymethyl methacrylate) as the first and second layers, respectively. The use of such double layer resist ensures a reliable undercut needed to form a suspended bridge since they use two different developers that selectively remove the respective resists (917-MIF for PMGI and a solution of IPA:MIBK:MEK (3:1:1.5% volume) for PMMA). In addition to a resist stack, a conductive polymer “E-spacer” produced by the Showa Denko Group is used to reduce charging effects during the EBL process performed on non-conductive substrates.

_{2}throttled through a leak valve at a pressure in the range 20–40 µTorr for about 10 min. After deposition, the unwanted metal and resists are removed in a liftoff process using mr-REM 700 at 90 °C.

^{3}nm

^{2}(Figure 1b). Source wires from individual SEBs are bunched together and connected to 10 micron-wide Pt “fingers” providing interconnect between optically defined Ti/Au wires leading to 150 × 150 μm

^{2}bonding pads, which connect the array to the surface mount inductor of the MN. Gate electrodes are placed at a distance 0.5–10 µm away from the islands. More images of the fabricated devices will be presented below.

## 5. Experimental Results and Discussion

#### 5.1. Characterization of Individual SEBs

_{IN}= 39 pF, a 560 nH inductor and a pad capacitance, C

_{pad}. The response of the MN measured at a low temperature (about 50 mK) is shown in Figure 7b (solid line) along with the simulated response curve based on the model in Figure 7a. The values of the inductor’s DC resistance (R

_{DC}) and skin-effect resistance (R

_{SE}) are adjusted for experimentally obtained temperature dependence of the coil resistance. The value of the parallel capacitor of the inductor, C

_{L}, is not changed, and the resistor that accounted for loss in C

_{L}is adjusted from its room temperature value (R

_{C}= 61 Ω) to obtain best fit to the data. The value of C

_{pad}= 901 fF is calculated from the resonant frequency using the parasitics listed above. The results of reflectometry measurements of a single SEB are presented in Figure 8 where the magnitude (Figure 8a) and phase (Figure 8b) of the reflected signal are plotted as a function of gate bias V

_{g}. After significant curve averaging resulting in the SNR improvement (by a factor ≈55), delta-function like features emerged in both the magnitude and phase of the reflected signal.

_{g}= 2.88 aF is extracted from the period of oscillations, ΔV

_{g}in Figure 8: C

_{g}= e/ΔV

_{g}. To evaluate directly immeasurable SEB parameters (R

_{J}, C

_{J}) we measured the resistance of an SET fabricated along with SEB with the same junction design R

_{T}~45 kΩ. The value of the junction capacitance is obtained by comparing the experiment with the simulations, with the value of C

_{J}= 65 aF providing the best fit for the experiment. The simulations of SEB admittance oscillations (Figure 8c,d) using (1) show that despite significant oscillations of “intrinsic” (i.e., occurring within the device) SEB admittance in both magnitude and phase, the measured signals exhibited much smaller deviations from constant values corresponding to blockaded states. The observed relative magnitude variation of |Γ/Γ

_{0}| < 0.7% along with ΔΘ < 0.3° correlated well with the simulated response. The reason for such small variations from average values is due to the fact that change in the dynamic capacitance, C

_{Dyn}< 100 aF (see Figure 2) the maximal values of Sisyphus conductance (<100 nS) even at a low temperature of 50 mK are occurring in parallel with much larger pad capacitance (C

_{pad}= 901 fF) with an absolute value of admittance of ~1 mS at the frequency of the experiment. Note that the choice of C

_{IN}= 39 pF brought the Π-network close to the matching point. The use of the L-C

_{pad}section alone, without C

_{IN}, would have resulted in the magnitude of oscillations |Γ/Γ

_{0}| < 0.01% and Δφ < 0.025 degrees, respectively. Despite close matching, the signal in Figure 8a,b could only be obtained after massive curve averaging (in fact, single scan yielded a noise exceeding signal by a factor of ~14). This example illustrates the major difficulty facing “gate reflectometry” where changes in the admittance of the device are measured through the gate port: it yielded very small SNR, which stemmed from extreme impedance mismatch. The reduction of C

_{pad}would have greatly improved the sensitivity, yet in the experimental setup it is very difficult to reduce it significantly below a fraction of a picofarad using bonding pads to wire the devices. The use of a single bond pad for connection of all the sources for an entire array with N >> 1 devices significantly alleviates this problem, as will be shown below.

#### 5.2. Characterization of SEB Arrays

_{g}is applied to the “tuning gate” to choose an appropriate operating point and V

_{p}is the voltage applied to the “sensing gate”. For a sensing application this electrode will be connected to a point at which the voltage needs to be measured. The micrograph in Figure 9b shows an SEB array designed to feature thousands of SEBs situated atop of source lines and Figure 9c illustrates a 9 SEB self-aligned array with an extremely small footprint yet strongly coupled to the gate.

#### 5.2.1. Characterization of a Small (N = 3) SEBA

_{IN}= 28.2 pF, and Figure 10c presents the results of MN characterization vs. carrier frequency at T = 11.5 K, which shows a nearly perfect match.

_{g}and phase dΘ/dV

_{g}, derivatives (solid lines in two subplots). The beating pattern, expected for the addition of asynchronous periodic signals with similar periods is clearly visible. The FFT spectrum of the beating pattern of oscillations acquired over the 16 V V

_{g}span (Figure 11b) unveiled the presence of three primary “frequencies” in 1/V units (A1, B1 and C1) corresponding to the three distinct periods of Coulomb blockade oscillations (respective gate capacitances): A1: 1/53.3 mV (3.0 aF), B1: 1/57.2 mV (2.8 aF) and C1: 1/56.7 mV (2.7 aF). Indeed, post-experimental inspection of this array under SEM revealed a fabrication defect for one SEB (delineated by a dashed white line in Figure 10a) reducing the number of functional devices to three. The FFT spectrum also indicates that at T = 11.5 K signals generated by SEBs are almost sinusoidal, i.e., with clear dominance of the 1st harmonic. The magnitudes of the 2nd harmonics are hardly visible and signals from higher harmonics fell below the noise floor. Superimposed with experimental data are oscillations of d|Γ|/dV

_{g}and phase dΘ/dV

_{g}(red in Figure 11a) simulated for 3 SEB using Formula (2), which are then passed through the MN depicted in Figure 10b. For simulations, the values of the three gate capacitances are extracted from the FFT plot acquired over the 16 V V

_{g}span. The values of junction capacitances 32, 34 and 28 aF and respective junction resistances, 80, 70 and 100 kΩ for the three respective SEBs are obtained by generating oscillations of SEBA admittance using (2), passing it through the MN shown in Figure 10b and then fitting the magnitude of oscillations in the reflected signal to the experimental data. The initial value of junction resistance is evaluated based upon junction resistance of SETs fabricated in a close proximity to SEBA with the same geometry of island and junction (see Figure 1a,b). The close match of relative amplitudes in FFT (Figure 11b) confirmed a reasonable choice of junction parameters. It is worth noting that at this relatively high temperature, the dominant contribution in formula (2) came from the second term (“C

_{box}”) related to dynamic capacitance since tunneling rates (f

_{T}= γ/2π ≈ 20 GHz at 11.5 K for a 100 kΩ tunneling resistor (2)) are much higher than the RF excitation frequency (461 MHz). To investigate temperature limits for the use of SEB arrays we studied how oscillations of reflected signal wash away with increasing temperature, Figure 11c. It can be seen from this plot that oscillations became hardly distinguishable from the background noise at highest T ≈ 25 K. The device with the smallest total capacitance is expected to exhibit oscillations until temperature fluctuations overcame the charging energy E

_{C}~k

_{B}T. The estimation of charging energy based upon the comparison experiment and simulations in Figure 11a yields C

_{J}+ C

_{g}= 31 aF neglecting all other capacitors. This gives the value for charging energy E

_{C}≈ e

^{2}/2(C

_{g}+ C

_{J}) = k

_{B}30 K or 2.58 meV, in good correlation with the experiment.

^{(i)}

_{S}= C

_{S}, it might be difficult to choose the operating point at a specific gate bias V

_{g_sens}due to random charge offset. For instance, it is difficult to obtain high sensitivity if sensing needs to occur near V

_{g}= 1.05 V in Figure 11a, because of the destructive interference. To enable flexibility of choice for the operating point, a second, tuning gate, with capacitance C

^{(i)}

_{t}that couples the i-th SEB in the array can be added (see Figure 9a). These capacitors (i.e., C

^{(i)}

_{t}) can be designed to be intentionally dissimilar (e.g., linearly decreasing) to ensure the placement of the operating point to the steepest slopes of constructive interference peaks within the reachable span of the tuning gate voltage, V

_{g tune}. To illustrate this technique, an example of the reflectometry signal obtained from the same array, now coupled to two gates is presented in Figure 12. In Figure 12a, in the top and bottom subplots sharp lines represent respective derivatives that define sensitivity in magnitude and phase with respect to changing gate bias, d|Γ|/dV

_{g}(upper left plot) and dΘ/dV

_{g}(bottom left plot) of the reflected signal. Each SEB generates a set of lines with a distinct slope $\partial {V}_{{g}_{tune}}/\partial {V}_{{g}_{sens}}=-{C}_{{g}_{sens}}/{C}_{{g}_{tune}}$. Constructive interference peaks appear at the points of line crossings. Note that the spacing between lines in each SEB along the ${V}_{{g}_{sens}}$ axis is almost the same, while along the ${V}_{{g}_{tune}}$ axis it is distinctly different, indicating dissimilar capacitance ${C}_{{g}_{tune}}$ for each SEB. To achieve weak and dissimilar coupling tuning gate can be simply positioned on the side of the array [33]. This combination ensures the appearance of line crossings within an easily accessible span of ${V}_{{g}_{tune}}$. Near these line crossings, one can find the points of maximum sensitivity where derivatives reach their extrema clearly visible at “knots” on the plot (there are eight such “knots” in Figure 12a). Clearly, the use of a second gate makes it easier to reach a high-sensitivity crossing point where the signals add up (and extrema of respective derivatives can be chosen as “sweet spots” for the sensing application) within a given V

_{g}span, since to find such a point with a single gate would require a much broader span of ${V}_{{g}_{sens}}$. This example illustrates that the presence of the secondary gate greatly reduced the necessary ${V}_{{g}_{sens}}$ span in the search for a constructive interference peak as a point of highest sensitivity.

_{C}/k

_{B}T ≥ 100 signals generated by each SEB start looking like periodic “delta-functions like” (Figure 8) with a very narrow voltage span for each peak leading to a collapse of overlapped regions and a disappearance of constructive interference peaks.

_{g}signal (lower panels in Figure 12) one of the lines, with the steepest slope, had a significantly stronger appearance compared to the other two while in d|Γ|/dV

_{g}plots the strength of the signal from each SEB are almost equal. We performed simulations of the response for this array using parameters of SEBs obtained from higher temperature measurements in Figure 11. However, to obtain a good fit to the experimental data in the simulated dΘ/dV

_{g}response, the resistance of the two junctions, corresponding to the two lines with poor contrast in this plot need to be increased by almost an order of magnitude. The reason for that likely stems from incomplete suppression of superconductivity in the two SEBs in the array at T < 0.5 K. Opening of the gap in the density of states at the metal-superconductor transition drastically reduces the tunneling rate at zero bias due to a lack of available quasiparticles. This is confirmed by the disappearance of the reflected signal from the SEB if no magnetic field is applied for temperatures T < T

_{C}≈ 1.05 K, where T

_{C}is critical temperature for Al. Indeed, in our experimental setup with small permanent magnets the superconducting gap is not completely suppressed in about 50% of the samples. Due to tunnel rate reduction in the two SEB with developing the superconducting gap, the admittance there started to be dominated by Sisyphus resistance while significant reduction in dynamic capacitance term in (2), in turn, resulted in the greatly reduced phase swings at the charge degeneracy points.

#### 5.2.2. Comparative Characterization of Larger (“N = 40” vs. “N = 200”) SEBAs

_{21}and C

_{12}need to be much smaller than sensing gate capacitances C

_{11}and C

_{22}in Figure 4c. In the experiment, this is achieved by picking two arrays separated by about 5 mm on a chip. A block-diagram of the experiment where two important parameters characterizing sensitivity, d|Γ|/dV

_{g}and dΘ/dV

_{g}are acquired simultaneously for two arrays with a nominal size N = 40 and N = 200 SEB is presented in Figure 13. Note that since the SEB is a two-terminal device, its admittance can be probed either way (see Section 5.2.3 for further discussion); in this experiment both arrays had their sensing gates tied together. In addition to the ramp voltage, V

_{g}, small modulation signals with a magnitude of 0.5 mV and frequency of f

_{1}= 1.45 kHz and f

_{2}= 2.48 kHz are applied to the common sources of 40 SEB and 200 SEB arrays, respectively. The reflectometry setup utilized the MN tuned at 305.7 MHz and homodyne reflectometer (labeled “R” in Figure 12a, see Figure 4a for details) followed by two synchronous demodulators operating at frequencies f

_{1}and f

_{2}. For this purpose, the bandwidth of ZI UHF demodulator is set to pass the signals at frequencies f

_{1}and f

_{2}without attenuation. The signals d|Γ|/dV

_{g}and dΘ/dV

_{g}are obtained for both 40 SEB and 200 SEB arrays by performing a second demodulation using two low-frequency lock-ins operating at f

_{1}and f

_{2}. The results of the experiment are presented in Figure 13b where d|Γ|/dV

_{g}and dΘ/dV

_{g}are shown for both the 40 SEB array (blue) and 200 SEB array (red). Curve averaging (58 ramps, about 25 s each) is used to boost SNR by a factor ≈7.6; the equivalent BW of the signals presented in Figure 13b is 26 Hz. The results, however, indicate that the ratios of magnitudes of peaks in both derivative signals deviate from the expected value of (200/40)

^{1/2}≈ 2.23 as the 200SEB array signal appeared to be >4 times stronger than that from the 40 SEB array. The likely cause for this discrepancy is the presence of fabrication defects, which might result in a reduction of the number of functional devices. An SEM micrograph Figure 13c shows several examples of such defects (observed in a different device). It is clear that only half of the inspected devices on that sample, namely the SEBs above the gate line, are functional. Simulations indicate that the observed ratio of 4 in signal strength suggests that the actual ratio of functional devices in the 200 SEB vs. 40 SEB array is close to 16, rather than 5.

#### 5.2.3. Experimental Characterization of SEB Arrays with Different Matching Networks

_{IN}. To verify this, we perform an experiment to compare the response from the same array of 200 SEBs for different values of input capacitance C

_{IN}

_{,}Figure 14. In this example, the MN is composed of L = 270 nH inductor, pad capacitor C

_{pad}~0.55 pF and two selected values of C

_{IN}. The results of measurements for the two derivatives, dR/dV

_{g}and dΘ/dV

_{g}at two different values of this capacitor are presented in Figure 14a,b. The results show more than 10 times enhancement of the (phase-gate voltage) sensitivity factor, dΘ/dV

_{g}. Note that the enhancement of the magnitude response is less pronounced, leading to about a 5-fold boost in the signal in good agreement with simulations in Figure 6. It is also worth noting that according to the simulations the accessible bandwidth is narrower for the network settings close to a match point C

_{IN}= 27 pF (≈2.2 MHz) compared to the overcoupled case C

_{IN}= 18 pF (≈3.3 MHz).

_{g}versus two gate voltages, V

_{g sens}and V

_{g tune}, with an apparent moire pattern resulting from multiple line crossings. In comparison with Figure 12, discrimination between individual lines representing individual SEBs is no longer possible. Moreover, multiple line crossings in Figure 15a produced “bunching”—an effect that appears to look like a sharp line over a smaller 2D region but visibly fades away on a larger scale—an effect typical for moire patterns. Indeed, crossing lines create that appears as oscillations with a period much shorter than that of a single SEB despite the same gate coupling for each individual box.

#### 5.2.4. Characterization of Cross-Coupled SEB Arrays

_{g}= C

_{1}V

_{1}+ C

_{2}V

_{2}. The “counter-parallel” connection of devices shown schematically in Figure 16a ensures that charging processes in array SEBA 1 and SEBA 2 (for simplicity represented as one junction) go in the opposite directions. For instance, by increasing positive voltage V

_{g}

_{1}electrons are added to SEBA 1 and at the same time removed from SEBA 2. This configuration is therefore expected to yield lines with both positive and negative slopes thus greatly increasing the number of interfering points. A way to design such a device using Dolan bridge technique with a small footprint and a large density of SEBs is to use the interdigitated design with alternating placement of SEB islands, as schematically shown in Figure 16b. One of the electrodes labeled “source” is connected to a MN while the other labeled “gate” is used as a sensing gate. The tuning gate would be located farther away from the structure and it needs to have much weaker coupling to the SEBA. In the experiment we use two separate SEBA with nominal N = 200 in each and simply connect them as shown in Figure 16a. The expected line crossing pattern is observed in the experiment, Figure 16c, where multiple outstanding extrema are occurring all over the map.

#### 5.3. Comparison of Performance between SEB Array and DC-Decoupled SET

_{SET}> 50 kΩ, so it has little effect on the measured impedance at these frequencies. To compare performance of DCD SET with a SEBA, we set up an experiment a circuit diagram for which is shown in Figure 17b and a physical layout is represented by a micrograph Figure 17c. The performance of the array composed of four SEBs is directly compared with that of two SETs AC drain-coupled to the same resonator. In an SEB the capacitance between the gate and the SEB island is a part of the total “island to outside world” capacitance, and thus while larger gate capacitance has the benefits of enhancing the total admittance of the device Y(V

_{g}), it also leads to a reduction of charging energy. By contrast, in the DCD SET the maximal admittance is set primarily by the resistance of the tunnel junctions and is independent of the gate capacitance. Note that for the structure shown in Figure 17c the gate capacitance for the SETs is approximately 2 times smaller than that of SEB due to different gate proximity.

^{−1}and 10.5 V

^{−1}, red curve in Figure 18b) and several higher frequency components of much smaller magnitude bunching around 18 V

^{−1}and 36 V

^{−1}. These higher frequency components are also present in the FFT of phase (blue curve in Figure 18b), while one of the lower frequency components (10.5 V

^{−1}) became practically undetectable.

_{g}and dΘ/dV

_{g}in coordinates V

_{gS}and V

_{gT}are obtained (Figure 18c). The pattern on the left panel shows the appearance of two distinct sets of lines: the one with a positive slope and another one with a negative slope, corresponding to a 10.5 V

^{−1}and 8.5 V

^{−1}spectral component in the FFT in Figure 18b. This set of lines is exactly what is expected from the counter-parallel connection of SETs. The signal stemming from a “fully-floating” DCD device yields a different phase compared to the SET DC-drain coupled to V

_{gT}in Figure 17c. On closer inspection a series of faint lines are also visible in the phase plot of Figure 18c. One can also speculate that these lines correspond to signatures of SEBs visible in FFT of Figure 18b with inverse periods of 16–18 V

^{−1}(first harmonic) and 32–26 V

^{−1}(second harmonic), since the physical layout for SEBs in this experiment is exactly the same as in for the device with three SEBs discussed above (Figure 10, Figure 11 and Figure 12). However, the closeness of the second harmonic of the signal 2 × 8.5 = 17 V

^{−1}generated by the DCD SET makes it impossible to accurately identify the SEB signature in FFT.

_{g}is shown in Figure 19c where only lines of negative slopes are observed as expected for SEBA with DC grounded sources, and a typical oscillation “frequency” of about 20 V

^{−1}. It is clear that the four SEBs device yields much weaker (by more than an order of magnitude) signal in magnitude of reflection but exhibited noticeable signal strength in phase of the reflected signal. Taking into account sensitivity scaling ~ N

^{1/2}a nine times larger SEBA (N ≥ 4 × 9) is needed to reach the phase sensitivity, and N ≥ 4 × 400 SEBA is needed to reach the magnitude sensitivity of a single DCD SET. (Here we also took into consideration the doubling of the sensitivity in the maxima for the two SETs, counterbalanced by approximately two times smaller C

_{g}coupling to the SETs). This confirms the trends discussed in Section 2 (Figure 3e).

## 6. Conclusions

_{g sense}~ 3 aF) this number is fairly large, N > 1000. Our simulations show, however, that within the same “capacitive budget” as in the studied SETs, a 100 times improvement in performance for SEBs could be obtained simply by enhancing the sensing gate capacitance. The practicality of an array of SEBs is greatly improved by including a second tuning gate to find a maximum sensitivity point over a smaller voltage range. A considerable improvement in the sensitivity of an optimally designed MN is also demonstrated. The advancements we developed in the area of SEB applications created greater potential to use these sensors in a wider array of applications. The experimentally observed relative ESD immunity compared to traditional SETs stems from two factors: a) lack of DC path through the device, which prevents device destruction upon application of large biases (up to 40 V tested) and b) capacitive voltage division in series C

_{g}C

_{J}chain, which for the case of C

_{J}>> C

_{g}leads to significant (>10) attenuation of any spurious voltage across C

_{J}and thus prevents its destruction. A conceptually similar device, a “DC-decoupled SET” appears to provide a reasonable compromise between SEB and traditional SET and has to be further explored for use as more rugged sensors in applications where the devices cannot be protected from exposure to large potentials. Increasing the charging energy through further reduction in the total capacitance may also yield devices that can provide the highly sensitive electrometers at elevated temperatures, potentially up to the range accessible by liquid nitrogen (77 K).

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic diagrams and scanning electron micrographs of a single electron transistor (

**a**) and single electron box (

**b**) fabricated by the Dolan bridge technique. The scale bar is the same for both cases. (

**c**) Simplified circuit diagram of the experiment in [17] where the single-electron transistor (SET), delineated by a red dashed line, is employed to probe single-electron charging in single-electron box (SEB), delineated by a blue dashed line. The SEB is coupled to the SET using a coupling capacitor C

_{C}. (

**d**) Functional diagram of the SEB measurement by an RF reflectometry setup. R

_{Sis}and C

_{dyn}represent contribution to the change in total impedance of SEB from Sisyphus resistance and dynamic capacitance. These two effects associated with single electron tunneling in and out of SEB cause variations in the matching network and thus affect the reflected signal.

**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).

**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|Y|/dV

_{g}and dΦ/dV

_{g}as a function of the number of SEB in the arrays at 2.4 K with junction parameters listed in the text. (

**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 kΩ biased at the maximum sensitivity point.

**Figure 4.**(

**a**) Simplified circuit diagram of the single-port reflectometry setup. Circuit delineated by a dashed line is the sample attached to the PCB board with the MN and cooled to a temperature defined by the setup; the MN circuitry is outlined in a red box with port 1 facing the sample and port 2 facing the directional coupler. Blue rectangle delineates the cryogenic part of the measurement apparatus. Amplifier at 40 K, ZX60-P33ULN+ is thermally anchored to the 1st stage of the cryocooler, T ≈ 40 K; two stage amplifier at room temperature uses ZX60-P103LN+; both types of amplifiers are by Mini-Circuits. Directional coupler ZFDC-20-50-S+ and cold attenuator are located at the second stage of cryocooler @2.7 K (Janis He3, Bluefors) or 3.5 K (ARS). The PCB with the sample is bolted to the He3 pot (Janis He3); mixing chamber sample exchange platform (Bluefors) or 2nd stage of cryocooler (ARS). (

**b**) Block diagram of the experiment with a SEB array coupled to two gates with capacitances C

_{11}…C

_{1n}to V

_{g}

_{1}(“sensing gate”) and C

_{21}…C

_{2n}to V

_{g}

_{2}(“tuning gate”). (

**c**) Block diagram of the experiment for comparative measurements of two arrays. The junction sides of the SEBs for both arrays are connected to the same MN. For simplicity each array is shown as a single SEB. The devices are spatially separated on the chip to minimize capacitive crosstalk.

**Figure 5.**Temperature dependence of the DC resistance of the 390 nH Coilcraft 0805 inductor (red) and 43 pF Johansson technology capacitor (black).

**Figure 6.**(

**a**) Circuit diagram of a Π-MN used in measurements. C

_{IN}is a tuning element; C

_{PAD}is a bond pad capacitor; horizontally places elements between C

_{In}and C

_{PAD}represent the realistic inductor at low temperature; a single SEB with parameters similar to the experiment is attached to a bond pad. (

**b**) Magnitude and phase response of the MN for three different values of C

_{IN}indicated in the plot: blue, 20 pF; green, 28.2 pF; and red, 40 pF. Inset table represents −3 dB small signal gate modulation bandwidth calculated in ADS software. (

**c**) Simulated response of the network with an SEB with parameters shown in subplot (

**a**) at T = 2.4 K: Oscillations in magnitude (

**c1**) and phase (

**c2**) of SEB admittance vs. gate bias V

_{g}calculated for carrier frequencies corresponding to the minima in the magnitude of the reflected signal in subplot (

**b**). Derivatives d|Γ|/dV

_{g}(

**c3**) and dΘ/dV

_{g}(

**c4**) of oscillations in the reflected signal.

**Figure 7.**(

**a**) MN used for probing single SEB. The values for R

_{DC}= 0.02 Ω and R

_{SE}= 0.35 Ω at 199 MHz are adjusted for low temperature. The series resistance of parasitic parallel capacitor (57.5 Ω) is tuned to obtain the best fit. The value of parasitic parallel capacitor (260 fF) used in simulations is equal to its room temperature value, while R

_{C}(57.5 Ω) is an adjustable parameter. (

**b**) Comparison between the simulated and measured response of the network shown in Figure 7a; upper panel—magnitude and lower panel—phase of reflection. Dashed line—simulations; solid line—experiment.

**Figure 8.**Experimentally obtained reflectometry data (blue lines) along with simulations (black lines) for magnitude (

**a**) and phase (

**b**) of the RF signal reflected from SEB. Base temperature of the fridge is 25 mK. Both curves are obtained by averaging of 3049 scans; effective bandwidth is 2.7 kHz. Caclulated oscillations of admittance magnitude (

**c**) and phase (

**d**) using Formula (2) from Section 2. Parameters used in simulations: C

_{g}= 2.88 aF, C

_{J}= 65 aF, R

_{J}= 45 kΩ, T = 50 mK. The calculated phase response is offset by a constant ( −40 degrees) for easy comparison.

**Figure 9.**(

**a**) High resolution SEM image of an array composed of 8 SEBs. Probing gate is located in close proximity to the array and has nominally equal coupling to each SEB. Tuning gate is located farther away and is unequally coupled to each SEB element. (

**b**) example of an array composed of a large number of SEBs (> 1000). Small dots below the lines are secondary islands resulting from the Dolan bridge process, and the two horizontal lines leading to the left are the gate. (

**c**) An example of a self-aligned 9 SEB array with a footprint less than 1 × 1 μm

^{2}.

**Figure 10.**(

**a**) High resolution SEM image of the 3 SEBA (only 3 out of 4 SEBs are operational). The three lines leading to the right are drains that are bonded in parallel to the MN. The vertical line situated about 0.5 μm away from the islands is the probing gate. The tuning gate is situated much farther (about 10 μm) below the entire array (not shown). A fabrication defect for one SEB is delineated by a dashed white line. The data for the 3 SEBA presented below are obtained from this very device. (

**b**) Model of the MN used in simulations. The values of C

_{L}= 86 fF and L = 215 nH are from the Coilcraft datasheet. The values of R

_{DC}and R

_{SE}are both adjusted based on measured resistivity reduction at low temperature; C

_{PAD}is calculated by fitting the resonant frequency to the experimental data, and the value of R

_{C}is adjusted to obtain the observable depth in the magnitude response. (

**c**) Experimentally obtained (solid lines) and simulated (dashed lines) frequency response of the reflectometry setup using the MN in (

**b**), T = 11.5 K.

**Figure 11.**(

**a**) Oscillations of derivatives d|Γ|/dV

_{g}and phase dΘ/dV

_{g}vs. voltage of sensing gate V

_{g}of the 3 SEB array at 11.5 K superimposed with simulations. Black—experiment, red—simulations. (

**b**) FFT of dΘ/dV

_{g}data over 16 V V

_{g}span (black) along with the spectrum of simulated data (red in Figure 11a). (

**c**) Oscillations of the derivative dRe(Γ)/dV

_{g}vs. V

_{g}acquired at different temperatures (from 3.9 to 23.5 K).

**Figure 12.**(

**a**) Experimentally obtained two dimensional maps of derivatives d|Γ|/dV

_{g}(top) and dΘ/dV

_{g}(bottom) for the 3 SEB array from Figure 9a. The maps are acquired without curve averaging using an equivalent noise bandwidth of 53 Hz. T = 0.305 K and RF carrier excitation ≈ 50 μV measured at the input of the MN. (

**b**) Simulations of respective derivatives using MN parameters obtained from fitting to the experimentally obtained resonant curve. Junction capacitances for simulations parameters are obtained from Figure 10a: C

_{Ji}= [32 34 28] aF C

^{(i)}

_{S}= [2.59, 2.7, 3.16] aF; C

^{(i)}

_{t}= [0.057 0.129, 0.248] and R

_{Ji}= [70, 400, 500] kΩ, where i = 1,2,3. The calculation is performed for T = 0.5 K. MN parameters are obtained by a procedure similar to that for Figure 10c, adjusted for lower temperature.

**Figure 13.**Comparison of the performance between two SEBA with a nominal number of SEB N = 40 (“40 SEB”, here and below marked blue) and N = 200 (“200 SEB”, here and below marked red) at T = 2.8 K. (

**a**) Simplified block diagram of the experiment with 40 SEB and 200 SEB connected to the same resonator and sharing the same ramp voltage V

_{g}while modulated at two different frequencies, f

_{1}= 1.45 kHz for 40 SEB and f

_{2}= 2.48 kHz for 200 SEB. MN—matching network operating at 305.7 MHz; circles with “+” sign represent adders; R—reflectometer that includes ultra-high frequency lock-in amplifier by Zurich Instruments (ZI UHF) and SR830 lock-ins to extract derivatives by further demodulating the downconverted signals of the ZI lock-in. (

**b**) Comparison of oscillation patterns for derivatives d|Γ|/dV

_{g}and dΘ/dV

_{g}representing magnitude and phase sensitivity for 40 SEB (blue) and 200 SEB (red) (

**c**) Typical fabrication defects found in the devices inspected under SEM. The number of devices in the micrograph by design is 6, but the entire set of bottom source lines to individual SEB is missing.

**Figure 14.**Comparison between the response of the same array of 200 SEB at T = 4.5 K with two different settings of the Π-network (Figure 4a) is composed of L = 270 nH Coilcrtaft 0805CS inductor, C

_{pad}= 0.55 pF and C

_{IN}: red—C

_{IN}= 27 pF, blue—C

_{IN}= 18 pF. (

**a**) Comparison of derivatives d|Γ|/dV

_{g}; (

**b**) Comparison of derivatives dΘ/dV

_{g}. Comparison of the magnitude (

**c**) and phase (

**d**) characteristics of the two MNs.

**Figure 15.**(

**a**) Experimentally acquired two–dimensional map of derivative dRe(Γ)/dV

_{g}from a source-coupled 200 SEB array at 0.3 K; (

**b**) simulated response of an array of 200 SEB composed of two 100 SEB subarrays with different coupling to tuning gate. C

_{g sense}= 3 aF with 1% STD (horizontal axis, V

_{g sens}) and C

_{g tune 1}= 0.2 aF STD 10%, C

_{g tune 2}= 0.4 aF STD 10% (vertical axis, V

_{g tune}). The contribution of each SEB to the signal is assumed to be equal. The development of the moire pattern is clearly visible. Red (blue) regions correspond to the maxima (minima) of the derivative, where the constructive interference occurs.

**Figure 16.**(

**a**) Simplified circuit diagram of the counter-parallel connection of SEB arrays. (

**b**) A sketch of the design featuring the interdigitated design with alternating placement of islands overlapping either source or gate regions. The different shades of blue represent the two evaporation steps of the Dolan bridge technique. The tuning gate is not shown and it can be placed above or below the structure to ensure dissimilar coupling. (

**c**) Experimental 2D map obtained from connecting two nominally identical arrays connected as shown in the circuit diagram. Both arrays nominally contain 200 SEB each.

**Figure 17.**(

**a**) Simplified circuit diagram of the DC decoupled-SET (DCD SET). Either one or both junctions are isolated with one or two non-leaky capacitors. (

**b**) Circuit diagram of the experiment: two SETs and four SEBs are connected together to a single MN, with the SETs in counter-parallel configuration, and SEBs in parallel configuration. The lower SET in the diagram is a ”DC-decoupled.” (

**c**) The electron micrograph of the structure used in the experiment. Four SEBs are clearly visible near V

_{gs}electrode The SET on the left is fully floating; the SET on the right is source-coupled to the V

_{gT}source (=AC ground).

**Figure 18.**Experimental results from 4 SEBs and 2 DCD SETs obtained at T = 1.45 K (

**a**) Oscillations of R and Θ as a function of the sensing gate, V

_{gS}. (

**b**) Fast Fourier spectra of the magnitude and phase of the reflected signal over ±4 V V

_{gS}range. Two lower frequency components in magnitude stem from the SETs, while smaller peaks marked by green arrows point to the 1st and 2nd harmonics of signals stemming from 4 SEBs charging. (

**c**) 2D maps of dR/dV

_{g}and dΘ/dV

_{g}oscillations. Phase derivative plot reveals the presence of a much weaker signal (faint steep lines in dΘ/dV

_{g}map) that corresponds to charging in the SEBs. There is a visible random telegraph noise near V

_{gS}= −0.62 V, likely due to random trap charging near the “fully floating “DCD SET.

**Figure 19.**(

**a**) Comparison of performance between combined structure (2 DCD SETs +4 SEB) devices (red) and 4 SEB devices only (blue); (

**b**) FFT for the respective cases: 2 DCD SETs +4 SEB) devices (red) and 4 SEB devices only (blue) and (

**c**) 2D map of dΘ/dV

_{g}oscillations reveals distinct pattern composed of four crossing lines, characteristic of SEB. T = 0.305 K.

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**MDPI and ACS Style**

Zirkle, T.A.; Filmer, M.J.; Chisum, J.; Orlov, A.O.; Dupont-Ferrier, E.; Rivard, J.; Huebner, M.; Sanquer, M.; Jehl, X.; Snider, G.L.
Radio Frequency Reflectometry of Single-Electron Box Arrays for Nanoscale Voltage Sensing Applications. *Appl. Sci.* **2020**, *10*, 8797.
https://doi.org/10.3390/app10248797

**AMA Style**

Zirkle TA, Filmer MJ, Chisum J, Orlov AO, Dupont-Ferrier E, Rivard J, Huebner M, Sanquer M, Jehl X, Snider GL.
Radio Frequency Reflectometry of Single-Electron Box Arrays for Nanoscale Voltage Sensing Applications. *Applied Sciences*. 2020; 10(24):8797.
https://doi.org/10.3390/app10248797

**Chicago/Turabian Style**

Zirkle, Thomas A., Matthew J. Filmer, Jonathan Chisum, Alexei O. Orlov, Eva Dupont-Ferrier, Joffrey Rivard, Matthew Huebner, Marc Sanquer, Xavier Jehl, and Gregory L. Snider.
2020. "Radio Frequency Reflectometry of Single-Electron Box Arrays for Nanoscale Voltage Sensing Applications" *Applied Sciences* 10, no. 24: 8797.
https://doi.org/10.3390/app10248797