Introduction

Recently, new types of computing technologies that use nanodot devices have been an active field of research; such as quantum1,2,3,4, reconfigurable5,6,7, and reservoir-computing8,9. A single-electron device (SED) can enable these technologies because of its high functionality and ultralow power consumption10,11,12,13. A SED exhibits a unique characteristic, known as the Coulomb blockade oscillation, with which the drain current (ID) is periodically modulated by the gate voltage (VG). This Coulomb blockade oscillation is caused by the charging effect of a nanodot (also known as the single-electron island) contained in the SED and is not observed in a conventional metal–oxide–semiconductor transistor. In previous studies, efforts were made to arrange a single dot between the SED electrodes, and the basic characteristics of the device14,15,16,17,18 were evaluated for applications associated with conventional logic circuits11,12,19,20,21,22,23,24. Using Si-based SEDs, the inverter operation20, half-sum and carry-out operation21, and multivalue memories22 have been demonstrated thus far.

Considering the single-nanodot SED known as the single-electron transistor (SET), which contains one nanodot between the source and drain electrodes, multiple SETs must be connected via wiring to form logic circuits. In this case, the one-by-one electron tunneling that occurs in SETs requires time for the wires to charge and the operation is slow. Additionally, the size variability of the nanodots induces the scattering of the SET characteristics. To overcome these difficulties, a nanodot-array-based SED that contains multiple nanodots without wiring was proposed, and its functionality was demonstrated under multigate operation5,6,25,26,27,28. For example, a flexible logic gate operation was enabled by using an Si nanodot array SED, indicating that nonuniform capacitive coupling between the dots and gate electrodes is a key factor for achieving logic operations25,26,27.

Besides nanodot fabrication methods that use lithography as reported in previous work, dispersion of chemically synthesized nanodots or self-assembled nanodot growth by thin-film deposition can be used to form nanodot array devices5,29,30,31,32,33,34,35. In these cases, the array comprises randomly distributed nanodots and numerous conduction paths with different properties must form. Although single-electron tunneling properties might be difficult to detect in such complex systems, Coulomb blockade properties are frequently detected;29,30,31,32,33,34,35,36,37,38 and the modulation that is caused by the gate voltage is evident and straightforward in some cases32,35,37,38. To determine the potential of these complex array SEDs, further investigations using multigate operation are required. However, single-electron properties controlled by multiple gate voltages, which are important fundamentals for functional SEDs, have not been studied yet. In a randomly distributed nanodot array, nonuniform capacitive coupling between the dots and gate is expected due to the variation of the dot shape and size of the surrounding dots. In accordance with numerical simulations in Ref.25, such variations of the gate capacitance are hypothesized to provide complex Coulomb blockade oscillations characteristics, which might enable a multidot SED to achieve logic operations.

In this study, an Fe nanodot single layer embedded in MgF2 and sandwiched between two gate electrodes was used for SED fabrication; and modulation of ID against the top- and back-gate voltages (VT and VB, respectively) was systematically measured using SEDs with straightforward Coulomb blockade oscillation. It was demonstrated that the straightforward Coulomb blockade oscillations were controlled by VT and VB, despite the metal and randomly distributed multidot structure. It was also found that the Coulomb blockade oscillations were unevenly responsive to both VT and VB. These results indicate the application potential of a complex array SED in functional devices, including logic gates. The unevenness of the gate operations depends on the thickness of the gate insulators, the dot shape asymmetry (hemispheric rather than spheric), and the arrangement of the surrounding nanodots.

Methods

Figure 1 shows a schematic, and scanning electron microscopy images, of the fabricated Fe nanodot device. The source and drain electrodes (Au/Cr) with a gap length L of 50–400 nm were formed on a thermally oxidized SiO2 (200 nm)/Si substrate. Afterward, an MgF2 (45 nm)/Fe (film thickness tFe = 1.8–2.9 nm) layer was formed between the electrodes at room temperature by using electron-beam deposition (base pressure < 10−7 Pa). In this tFe range, dispersed Fe nanodots were formed owing to surface migration and aggregation of Fe atoms on SiO2. Finally, the gate-insulating SiO2 (300 nm) layer was prepared on the MgF2 layer via sputtering, followed by top-gate electrode (Au/Cr) formation. Details of the device fabrication were given in a previous study37. The drain current between the source and drain electrodes (ID) was measured using a semiconductor parameter analyzer (Agilent 4156C) by applying the voltages VT and VB to the Au/Cr top-gate and Si (substrate) back-gate electrodes, respectively, in a closed-cycle cryogenic probe station at a sample stage temperature T = 8 K.

Figure 1
figure 1

(a) Schematic and (b) and (c) scanning electron microscopy images (plan view) of the fabricated Fe nanodot array device.

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Results and discussion

Almost all of the fabricated devices exhibited Coulomb blockade oscillations where highly reproducible back-and-forth ID–VB curves with the peak/valley ratio larger than ~ 1.1 were seen. About 95% of the fabricated devices were in this category. Some of them (ca. 10%) showed ID–VB oscillations that originated from a single dot as in our previous study37. Typical current oscillation characteristics in this category are demonstrated in Fig. 2 as a function of VB for two SEDs, termed Devices A and B in this report. The VD value utilized here was sufficiently small such that the Coulomb blockade of the dot was not lifted. A brief discussion of ID–VD characteristics and the charging energy of the dot can be seen in the supplementary information as well as our previous study37. In both of the graphs shown in Fig. 2, three ID–VB curves at VT = 1.8, 10, and 20 V are superposed. The observed oscillation curves are evident, and they originated from a single dot, as discussed in our previous study37. The oscillation period was approximately 27 (Device A) or 37 V (Device B); corresponding to a back-gate capacitance CB of 5.9 (Device A) or 4.3 × 10−3 aF (Device B), respectively. Device B has a smaller CB compared with Device A, the dot that contributes the current oscillation in this device should be smaller than that of Device A. In both devices, the current oscillations were added to the constant-background current components [~ 520 pA (Device A) and ~ 1.0 nA (Device B)], which were attributable to the parallel conductive paths comprising the dots with the Coulomb blockade lifted. Such dots (attributable to the background current) are relatively large and/or exhibit higher tunneling conductance than the quantized value (2e2/h)37. By comparing the ID–VB curves with different VT, a systematic shift in the current peak was identified. With increasing VT, the peak ca. 20 V for Device A gradually shifted to 14 and 3.3 V (Fig. 2a). A similar tendency was evident for Device B (Fig. 2b), with a peak shift from 22 to 16 V, and then to 9.3 V. These peak shifts toward the negative VB direction demonstrate that the current oscillation characteristics can be controlled by both the top and back gates, even in the metal multidot SED, where the change in the nanodot charging state by VT can be compensated by VB and vice versa.

Figure 2
figure 2

Current oscillations originating from a single dot measured as a function of the back-gate voltage VB for the following: (a) Device A (tFe = 2.4 nm and L = 50 nm) at drain voltage VD = 20 mV and (b) Device B (tFe = 2.4 nm and L = 400 nm) at VD = 5 mV. The top-gate voltage VT was constant at 1.8, 10, and 20 V.

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To clearly understand the details of the peak shift as a function of the two gate voltages (i.e., VT and VB), a contour plot of the drain current was used27. For this purpose, numerous ID–VB curves were measured using various VT values from 0 to 30 V in 300-mV steps, where the VB sweep was conducted in the sequences of 0–30, 30 to − 30, and − 30 to 0 V. The current oscillations were well-reproducible for the back-and-forth VB sweeps. In addition, the current oscillations were stable over a few days against the peak shift due to the charge offset drift, similar to the findings of a previous study38. Figure 3a and b show the data measured from VB = 30 to − 30 V as two-dimensional (2D) contour maps of the drain current, corresponding to the stability diagrams of the device. The current peaks shown in bright contrast were systematically shifted. Thus, the phase of the current oscillation can be controlled by using VT and VB. These contour maps were straightforward and periodic, as indicated by the yellow dotted lines, although there was an irregularity in Fig. 3b at VT = 17 V, which was caused by charge noise that might be attributable to the effect of satellite nanodots acting as single-electron traps37. These characteristics confirm that the major current oscillation originated from a single dot37. Controllability of the charge state of a single dot by a double-gate was confirmed despite the multidot structure.

Figure 3
figure 3

Typical two-dimensional contour-line maps of drain current ID as a function of VB and VT. (a) Device A at VD = 20 mV and (b) Device B at VD = 5 mV. N is the number of electrons in a single dot.

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The results shown in Figs. 2 and 3 are well-known phenomena for multigate SETs with a single dot configuration28,39,40. However, manifestation of such a phenomenon in metal nanodot arrays is unprecedented and important because of the corresponding suggestion that SEDs comprising randomly dispersed metal multidots can operate as, for example, two-input logic-gate devices.

Via careful analysis of these data, characteristic features of the double-gate SED operation of the self-assembled nanodot system were clarified. For the devices investigated in this study, the top and back gates were capacitively coupled to the SED, and the current peak shift in accordance with the VB and VT follows Eq. (1):

$$ C_{{\text{B}}} V_{{\text{B}}} + C_{{\text{T}}} V_{{\text{T}}} = const, $$
(1)

where CB and CT are the capacitances between the single dot and back/top gates, respectively. When VT changes by ΔVT, the peak shift in the current oscillation (ΔVB) is given by Eq. (2):39

$$ \Delta V_{{\text{B}}} = - \left( {C_{{\text{T}}} /C_{{\text{B}}} } \right)\Delta V_{{\text{T}}} . $$
(2)

Therefore, the gate capacitance ratio CB/CT of the dot is − ΔVT/ΔVB, which was evaluated using the slope of the observed current peak line in the contour map40,41,42,43,44. Using Fig. 3a and b, the gate capacitance ratio CB/CT between the single dot and back/top gates was ~ 1.2 (Device A) or ~ 2.0 (Device B). Therefore, using the CB previously described, CT was ~ 4.9 (Device A) or ~ 2.2 × 10−3 aF (Device B). Table 1 and Fig. 4 show these results along with data from two other devices. For the device structure investigated in this study, the back-gate insulator comprised 200-nm-thick SiO2 and the top-gate insulator comprised 45-nm-thick MgF2 and 300-nm-thick SiO2. Assuming a parallel-plate capacitor structure and known bulk dielectric constants (3.8 for SiO2 and 5.2 for MgF2)45, the capacitance ratio CB/CT was ~ 1.7 (Fig. 4). For a total set of devices, the evaluated CB/CT followed this relationship. Furthermore, each CB/CT ratio was between 1.2 and 2.7, indicating a clear discrepancy from this straightforward estimation by using the parallel-plane model (~ 1.7). This discrepancy was not caused by measurement and/or estimation errors, but was an important characteristic of the self-assembled dot array. In the following paragraphs, this is discussed by using a simplified model.

Table 1 Averaged values of CB/CT, CB, and CT for four devices. CB/CT and CB contains the reading error of the slope and period of the current peak line, respectively.
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Figure 4
figure 4

Plots of CB vs. CT of four devices, indicating simple current oscillation characteristics that originated from a single dot. CB of four devices was evaluated by the period of the current peak line. However, CT of four devices was evaluated by CB and the slope of the current peak line. CB includes the fluctuations that were derived from reading errors of the period of the current peak line, which are shown by the vertical error bars. The horizontal error bars represent the fluctuations of CT that originated from the fluctuations of CB. There was a positive correlation between CB and CT. The slope of the dotted line corresponds to the evaluated CB/CT ~ 1.2 (blue) and ~ 2.7 (green). Although the data approximately follow the relation without considering the shape effect of the nanodot system (red line), clear discrepancy from the red line was identified.

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The nanodot array SED comprised numerous dots with various sizes. Because dots were formed on the substrate plane, the planar arrangement of hemispheric nanodots can be assumed as a model for this discussion. In addition, for straightforward calculations, the model was simplified into parallel-arranged, half-columnar dots with an infinitive axis length. Here, a numerical fit to the experimental data is not the purpose of this simulation. Figure 5 shows an example of the cross-sectional schematic, where the column axis of the dot is perpendicular to the paper surface. The dot is half-circular in this diagram (gray), termed the half-circular dot in the following discussion. By adopting this model, only 2D electric field calculations were required, with few parameters. A compact software (EStat provided by Advanced Science Laboratory, Inc.) based on the finite-element method (with an optimized mesh size) was used to solve the Laplace equations for the following: (1) simulating the electric field and (2) evaluating the capacitances (per unit length along the column axis) between the dot and top/back-gate electrodes. Although the model was simplified, the simulation results provided the intrinsic features of the nanodot array. In the next paragraph, the simulation results are explained with details of the model.

Figure 5
figure 5

Simulation results of the electric potential (color) and the lines of the electric force (white lines), in which the top or the back electrodes are biased by 1 V. Results for the LSL (L = large and S = small) dot configuration with biased (a) back- and (b) top-gate. For details of the model, refer to the text.

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Figure 5 shows a typical model for evaluation. Notably, the horizontal and vertical magnifications of this diagram are different from each other. The in-plane diameter of a metallic half-circular dot is 30, 20, or 14 nm; termed L (large), M (medium), or S (small), respectively, throughout this report. The dots were arranged horizontally to form a dot array. In Fig. 5, the dot arrangement is termed LSL in the central part and M in other regions. The distance between the adjacent dot edges was maintained at 10 nm. The dots were sandwiched between two 100-nm-thick SiO2 layers acting as the top- and back-gate insulators. On the surface of these SiO2 layers, two metallic gates (gray) were attached. For simulations, one of the gate electrodes was biased by 1 V, whereas the dots and another electrode were grounded. Figure 5 shows simulated potential distributions as color maps; the lines of the electric force are represented by white lines. As described next, a remarkable difference was observed between Fig. 5a for voltage application to the back gate and Fig. 5b for voltage application to the top gate. The electric force lines shown in Fig. 5a were almost parallel, similar to those in a parallel capacitor, except near the dot edges. Thus, CB of each dot must be almost proportional to the dot size. Conversely, in Fig. 5b, the electric force lines were strongly curved and those near the central S dot were attracted by the adjacent L dots. Some of the electric force lines were absorbed by the L dots; therefore, CT of the central S dot became smaller in this dot arrangement, whereas CT of the L dot became larger. This is because of the geometrical difference between the upper (roundish) and lower (flat) dot surfaces. A dot shape on the nanometer scale can modulate the electric field, changing the charge distribution on the dot surface and the capacitance with the gate electrode. This suggests that the Coulomb blockade oscillation characteristics in a multidot SED comprising a complex dot array can be modulated by the shape and distribution of the dots, including the surrounding dots.

These simulations were performed for various dot arrangements, and the gate capacitance ratios CB/CT of the central dots were evaluated. Figure 6 shows examples of the SSS, MSM, and LSL arrangements. To check the applicability of the calculations, simulation results of circular dots with the same arrangements were superposed in the graph. Because the dots were symmetric in this case, the CB/CT ratio should be 1.0 in all the cases, reflecting the thickness ratio of the top (100-nm-thick) and bottom (100-nm-thick) insulating layers. The simulation results fit well with this value (the error with the theoretical value was ca. 0.1%). This indicates that the simulation was performed with a sufficient accuracy (number of meshes) to evaluate the CB/CT. Considering half-circular dots, the CB/CT value indicates a large change from 0.95 to 1.27, depending on the adjacent dot size. This is because of the nonuniform distribution of the electric field between the top and bottom gates, which was attributable to the geometrical shape effect of the central dot and surrounding dots (Fig. 5) as well as the vertical asymmetric structure of the dots. In the SSS arrangement, for example, the roundish shape of the surface that was facing the top gate gathered the electric force lines, rather than the flat surface that was facing the bottom electrode. Therefore, CT became larger than CB and CB/CT was < 1. For the large adjacent dot, as in the MSM arrangement, some electric force lines were attracted by the M dots, decreasing the CT value; ultimately resulting in CB/CT > 1. Lager adjacent dots in LSL strongly attracted the electric force lines from the S dot, and the deviation of CB/CT from 1 became large. Similar results were also identified for M or L as the central dot. Such fundamental discussions by using 2D simulations must also be valid for a three-dimensional dot shape and arrangement. The shape asymmetry of the dots and its positional arrangement resulted in a variation of the CB/CT (and thus the device properties) during double-gate operation.

Figure 6
figure 6

Gate capacitance ratio CB/CT of the central S dot for three dot configurations: SSS, MSM, and LSL (L = large, M = medium, and S = small). The CB/CT for circular dots (red) and half-circular dots (black) are presented. The CB/CT value of the circular dots was almost constant at 1.0, regardless of the size of the adjacent dots. Conversely, the value changed from 0.95 to a value of 1.27 for half-circular dots with increasing adjacent dot size.

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Conclusion

A double-gate, Fe nanodot-array-based SED comprising an SiO2–Fe–MgF2 system was fabricated by using vacuum deposition, and its electric characteristics were investigated. The fabricated Fe nanodot device exhibited current oscillations that originated from a single dot37. The charge state of a single dot was controlled by both the top and back gates, even in the multidot structure. This result demonstrates that Coulomb blockade oscillation characteristics that originate from a dot in randomly dispersed multidots can be modulated by two gates. Additionally, the variation of the gate capacitance ratio CB/CT (derived from the randomly distributed and vertically asymmetric nanodot array) was demonstrated. This phenomenon might provide a complex response to the input (gate) voltage. It might be useful to produce flexible logic gates as indicated in previous reports25,26,27, although further comprehensive studies are required in the future. The on/off ratio of the Coulomb blockade oscillations was small because of the many conduction paths that provided the background current. Although this problem is unavoidable in multidot SEDs, the on/off ratio can be improved by using a cascode MOSFET22,26,46. The findings of this study will facilitate new applications of metal nanodot arrays.