Impedance of the single-electron transistor at radio-frequencies

We experimentally characterize the impedance of a single-electron transistor (SET) at an excitation frequency comparable to the electron tunnel rate. In contrast to usual radio-frequency-SET operations, the excitation signal is applied to the gate of the device. At zero source–drain bias, the SET displays both resistive (Sisyphus resistance) and reactive (tunnelling capacitance) components to its impedance. We study the bias dependence of the complex impedance, investigating its response as the electron tunnel rate becomes large with respect to the driving frequency. The experimental data are compared with values calculated from a master equation model.

capacitance has been investigated in gallium arsenide quantum dots [8,9] and, similar to the Sisyphus resistance, allows measurements of single-electron charging with just a single-tunnel junction. This tunnelling capacitance is distinct from the quantum capacitance that arises from bandstructure curvature in single Cooper pair devices [10,11] and double quantum dots [12]. Aluminium SETs were fabricated by a standard double-angle evaporation technique using bilayer resist and controlled oxidation [13]. They were measured in a dilution fridge at an electron temperature lower than 200 mK. A magnetic field (B = 600 mT) was applied to suppress superconductivity in the aluminium. An rf-resonant circuit, consisting of a chip inductor, was connected to the gate of the SET (figures 1(a) and (b)). This is different from the usual configuration of the rf-SET [2], where the resonant circuit is connected to the source-drain of the SET and where the largest contribution to the reflection coefficient is from modulation of the differential conductance.
The circuit was driven at resonance ( f 0 = 347 MHz). The amplitude of the rf-signal sent to the device was dn g = C g dV g ∼ 0.09 e (2 × Q × δV g × C g = 2 × 60 × 2 µV × 0.1 fF, where δV g is the amplitude of the rf-drive and Q is the resonator's Q-factor, estimated from the 3 dB part of the resonance curve shown in figure 1(c)). After amplification by a low-temperature and room-temperature amplifier, the reflected signal from the resonant circuit was mixed with the reference signal to provide phase-sensitive detection. The output of the mixer was amplified, low pass filtered and digitized with an oscilloscope. Using a line stretcher, we varied the phase of the incident signal to the device with respect to the reference signal. As a result, we obtained the in-phase and quadrature components of the reflected signal, allowing us to calculate the amplitude and phase response of the SET.
We briefly compare the differential conductance of the SET, measured using a standard low-frequency lock-in technique, to the amplitude and phase response of the reflected signal (figure 2). In the phase response, we can clearly observe the Coulomb diamonds that appear in the differential conductance graph, these corresponding to the blockade regime in which transport through the island is forbidden. Additionally, there is a peak in the phase shift around the zero bias degeneracy points (V sd = 0 V, n g = (m + 1/2) e). In the amplitude response, there is a dissipative signal at the degeneracy points that decreases with increasing source drain bias. Strikingly, by comparison with the phase response, the Coulomb diamonds are not observed.
We now analyse the phase response, starting with observation of the Coulomb diamonds. When the device is in Coulomb blockade (CB), the gate capacitance (C g ) and the junction This includes the resistance and geometrical capacitance of the junction. The top right image is a snapshot of the graph of the reflected signal phase as a function of the dc voltage applied to the gate and of the source-to-drain bias. On the graph, three points are shown: point A indicates the point of zero source-to-drain bias, point B marks the peak in the phase shift at low source-to-drain biases and point C indicates the region of high source-to-drain biases. The three diagrams at the bottom show the tunnelling events on the timescale of the rf drive for each of these three points. I net is the net current to ground that results from the tunnelling of the electrons. We see that, when V sd is increased (B), the tunnelling events happen 'early' in the rf cycle and I net develops a negative phase shift with respect to the drive. When V sd is further increased (C), however, I net is generated by stochastic tunnelling events no longer correlated to the drive. (b, c) Graph of the measured and calculated variations of the reflected signal phase shift as a function of the dc gate voltage (b) and of the source-to-drain voltage (c). capacitances (C 1 + C 2 ) are in series (figure 3(a)), with a very little contribution to the impedance from the tunnel junction resistance. Outside the blockaded region, the capacitance becomes shunted by the junction resistance. This results in an increase in capacitance ( C g +C 1 +C 2 ) shown by the resonator, lowering the resonant frequency and explaining the phase shift.

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Near the zero bias degeneracy points, a description of the electron tunnelling dynamics is necessary. With the rf signal superimposed on the gate, the island cyclically transits between the nearly degenerate n and n + 1 charge states. If the tunnel rates at zero instantaneous rf bias (we estimate at the zero bias degeneracies = kT e 2 R ∼ 2 GHz, where R is the resistance of one tunnelling junction) are greater than the drive frequency, the electrons on average tunnel early in the rf cycle. This phase shifts the current with respect to the rf voltage, leading to an effective capacitance C eff in parallel with the SET. The total capacitance C can then be written as C = dV g , where en is the average charge on the island. This expression can be found for the expression of the polarization charge induced on the island by the gate where V i is the island potential. The first term represents the dc limit of the capacitance, obtained by considering C g in series with the capacitances of the source and drain tunnelling junctions. The second term represents the contribution from C eff and can be written as − , where (1/C g ) dn g is the amplitude of the rf drive and P n is the probability of having n electrons on the island. When k B T E c and dn g e, tunnelling to higher-energy states can be neglected and P n is found by solving the master equation, which involves the states n and n + 1 [1]: P n = n+1,n P n+1 − n,n+1 P n , P n+1 = n,n+1 P n − n+1,n P n+1 .
Here n+1,n and n,n+1 are the tunnelling rates of the island between the two states n and n + 1 [14]. An increase in V sd leads to an increase in the tunnelling rate of the electrons at the drain lead. We therefore expect the tunnelling events to happen 'earlier' with respect to the case of zero bias and the current to be more phase shifted with respect to the rf signal ( figure 3(a)). Accordingly, a maximum in phase (maximum C eff = 0.03 fF) occurs at V sd = 30 µV ( figure 3(c)). When V sd is increased above this value, the contribution of the source to drain potential becomes dominant, the tunnelling events are no longer sensitive to the rf drive, therefore leading to a decrease in the phase shift. In figures 3(b) and (c), we report the calculated phase shift as a function of both n g and V sd . The simulations have been performed at a finite temperature of 120 mK and with dn g = 0.09 e. By comparing the simulations with the experimental results, we observe that there is good agreement in the n g dependence. The V sd dependence agrees qualitatively with the experimental observations, but presents some discrepancies that could be fitted by using a higher value for the temperature or dn g . We could not explain, however, the origin of this disagreement. We next consider the behaviour of the amplitude. From figures 4(b) and (c), we observe that the reflected (dissipated) signal reaches a minimum (maximum) at n g = 0.5e and V sd = 0 V. When V sd = 0 V, the SET effectively behaves like an SEB and its dissipative response to an rf signal can be understood in terms of the Sisyphus resistance [7]. On average electrons tunnel after the degeneracy point between the lead and island chemical potentials is passed; therefore energy dissipation occurs in each half of the rf cycle, dissipating energy from the resonator. From the average powerP dissipated by the SET (figure 4), we deduce the value of the effective resistance R eff , R eff = ((1/C g ) dn g ) 2 /2P. Our resonator is lossy (Q ∼ 60), so we findP by measuring the relative variation of the reflected power. At V sd = 0 V, the effective resistance reaches a minimum value R eff = 16 M .
As V sd becomes greater than the rf amplitude, the tunnelling events onto and off the island are driven largely by the dc bias rather than by the rf excitation ( figure 4(a)) and the contribution from the Sisyphus dissipation becomes smaller. The absence of the CB diamonds in figure 2(c) Figure 4. (a) Reflected signal amplitude as a function of V sd and n g . The decrease at V sd = 0 V and n g = 0.5e is explained by an increase in energy dissipation by the SET due to the Sisyphus tunnelling processes. The top right diagrams show the different tunnelling processes involved at zero and positive V sd . In the first case, the Sisyphus tunnelling events lead to the absorption of extra energy from the resonator and thus to a decrease in the reflected power amplitude. In the second case, the energy needed for the tunnelling is provided by the battery and no decrease in amplitude is observed. (b, c) Graphs of the measured and calculated variation of the reflected power amplitude as a function of (b) the dc gate voltage and (c) the source-to-drain voltage.
shows that the amplitude response is completely dominated by the Sisyphus dissipation, which overshadows the effects of the SET resistance variation at higher values of V sd . The average power dissipated in one period of the rf drive can be calculated as where E n+1 − E n is the energy difference between n + 1 and n states. The Heaviside step function (θ ) ensures that the transitions between the states n and n + 1 are energetically allowed. There is close agreement between the theory and the experimentally measured amplitude response as a function of n g and V sd (figures 4(b) and (c)). We expect that the reactive and dissipative components of the SET response to an rf drive will strongly depend on the frequency of the excitation. In our experiment, we do not have the flexibility to make a frequency-dependent study of this response; however, we show in figure 5 the simulated variation of the reflected signal phase shift and of the dissipated power,P, as a function of the excitation frequency. As the frequency of the drive becomes much higher than the electrons' tunnelling rate, the tunnelling events are random and no longer correlated to it. On the graph of the phase shift, three points are shown: point A indicates the region in which the electrons' tunnelling rate is comparable to the frequency of the rf drive, point B marks the minimum value of the phase shift, which is reached at higher excitation frequencies, and point C indicates the region in which the frequency of the excitation is much higher than the electrons' tunnelling rate. The three diagrams on the right show the tunnelling events involved in each of these three cases on the timescale of the rf drive.
However, the probability for them to occur is higher at the maximum amplitude of the rf drive as the tunnelling rate is higher. Accordingly, the power dissipated in the device saturates towards a maximum value at higher frequencies. C eff , on the other hand, tends to zero and the phase shift of the reflected signal is only determined by (C 1 +C 2 )C g C . It is interesting to note that for frequencies higher than our working frequency (347 MHz), a negative phase shift is expected, opposite to what we observe. In diagram (B) of figure 5, this is explained by the tunnelling events happening in the second half of the rf cycle, resulting in a positive shift of the net current with respect to the drive.
By connecting the gate, rather than the source or drain, to the rf circuit, we are able to determine the Sisyphus resistance and tunnelling capacitance contributions to the SET impedance. These effects will also be present in a conventionally measured rf-SET (resonator on source-drain) and should be included if the impedance needs to be accurately known.