Radio Frequency Reflectometry of Single-Electron Box Arrays for Nanoscale Voltage Sensing Applications  
[PERSON]  
1 Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA; [EMAIL_ADDRESS] (M.J.F. ); ;  
\(10^{-6}\) e/Hz\({}^{\frac{1}{2}}\)[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]. The subjects of study in this work are metal-metal oxide Al/AlO\({}_{\text{x}}\) 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 \(T<<E_{\text{C}}/k_{B}\), 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, \(E_{\text{C}}=e^{2}/2C_{\text{z}}\) (\(C_{\text{z}}=2C_{\text{f}}+C_{\text{g}}\); \(C_{\text{f}}\) and \(C_{\text{g}}\) are junction and gate capacitances, respectively and \(e\) is an electron charge) and junction resistance \(R_{\text{f}}\), which must be large enough to enable charge localization on the island, \(R_{\text{f}}>\)_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. Despite their simplicity, SEBs are not used in practical applications due to the difficulty presented by the lack of a DC path through the device. Indeed, the example Figure 1c shows that the state of an SEB cannot be probed directly and thus requires either use of an auxiliary sensing device (SET) or an alternative measurement technique. Over the past two decades, the development of radio-frequency  
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 \(C_{\text{C}}\). (**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. (RF) reflectometry for single-electron applications [18] opened a way to directly probe the charge state of the SEB and expanding the bandwidths from less than 10 kHz, using the conventional approach [17] up to hundreds of megahertz [18]. Functional diagram of the reflectometry setup explained in detail in Sections 2 and 3 is shown in Figure 1d.