A charge parity ammeter

A metallic double-dot is measured with radio frequency reflectometry. Changes in the total electron number of the double-dot are determined via single electron tunnelling contributions to the complex electrical impedance. Electron counting experiments are performed by monitoring the impedance, demonstrating operation of a single electron ammeter without the need for external charge detection.

electrometer. Specifically, we demonstrate real-time measurements of the relative charge parity of the metallic double dot by probing the complex impedance of the device itself.
When any of the capacitors in a single electron device is driven by an alternating potential, periodic tunnelling of electrons can occur, resulting in an alternating current. For example, if an alternating potential is applied to the gate of a single electron box, a current is periodically driven across the tunnel barrier. If the drive frequency is sufficiently high that electron tunnelling does not happen adiabatically, energy is dissipated each half cycle leading to an effective resistance 17 -the 'Sisyphus resistance'. In general the driven tunnel current is not in phase with the applied alternating potential and both real and imaginary components of the Sisyphus effect need to be considered. [18][19][20] In this Letter we refer to the real and imaginary Sisyphus impedances as Z Re S and Z Im S respectively. They are measured here by their dissipative and dispersive effects on a radiofrequency resonant circuit. They can be observed in any single electron system; in this sense they are dissimilar to the 'quantum capacitance' due to the change in bandstructure curvature close to an anti-crossing between two levels, which is present in, for example, Cooper pair boxes 21   We now describe measurements on the HR device. In fig. 2a we show phase as a function of V L and V R , and highlight three charge state cells. We note that the sign of the phase shift induced by Z Im S is dependent upon the junction resistance and the r.f. drive frequency, 18 and here is opposite to that for the LR sample. From the stability diagram and bias dependency measurements, we estimate the island charging energies to be E c1 ≈ E c2 ∼ 230 µeV, and the electrostatic coupling energy to be E cm ∼ 140 µeV.
To probe the tunnelling dynamics of the device, we fix the gate voltages at a point close to the triple point between the (m,n), (m+1,n) and (m,n+1) charge regions, and concentrate on the phase response. We then take a long (50 s) time trace. In fig. 3a a typical trace segment is shown, for V ds = 0 mV and T = 35 mK. We see a stochastic switching between two phases, separated by 1.4 • . From the S 11 of the resonant circuit, we deduce a corresponding change in 1/ f 0 Z Im S of 33 aF between the two states.
We attribute this impedance change to the thermally driven tunnelling of a single charge through the highly resistive tunnel junctions of the leads. When the device is in the (m,n) state (or other state with even parity relative to this state), the r.f. stimulus is unable to drive an electron through the middle junction, and so no Sisyphus impedance is observed ( fig. 3b, left panel). The addition of an electron to the double island, however, places the device in either the (m+1,n) or (m,n+1) state (or another odd relative parity state), and the Sisyphus impedance is now present due to the extra electron ( fig. 3b, right panel).
To determine the tunnel rates of electrons on to and off the device, a time trace is divided into 'high' (where there is no Sisyphus impedance) and 'low' (where the Sisyphus impedance is present) capacitance periods. The dwell times of these are determined, and they are then histogrammed ( fig. 3c). We fit separate exponential decays, and extract transition rates between the even and odd parity states, Γ EO and Γ OE . In general, the analysis of Poissonian transition rates requires the finite bandwidth of the measurement setup (in this case ∼ 15 kHz) to be considered, 26 but here we note that the measured rates are more than two orders of magnitudes lower than the bandwidth, and so this correction is negligible.
The average rate of parity change is given by Γ = 1 2 (Γ EO + Γ OE ). In fig. 3d we show Γ as a function of the temperature of the dilution fridge mixing chamber for two different r.f. carrier powers, with V ds = 0. We observe a constant cycle rate for temperatures up to 250 mK (P r f = −105 dBm) and 325 mK (P r f = −95 dBm). At these temperatures the electron temperature begins to increase and Γ, which is thermally driven, begins to rise. At lower r.f. powers and higher temperatures the signal to noise ratio (SNR) degrades such that the two phase levels cannot be reliably distinguished.  We now quantify the behaviour of the charge parity ammeter by measuring Γ as a function of V ds . We show data at T = 35 mK and for two carrier powers (Fig 4). Two regimes can be identified.
At low bias (V ds < 0.1 mV) Γ is dominated by tunnelling events driven both thermally and by the r.f. carrier signal. The net charge flow in this regime is significantly less than 1 2 e. At higher bias (V ds > 0.2 mV) events in which electrons are transferred from source to the device, or the device to the drain, dominate. This regime extends to lower bias for reduced carrier power.
We model this behaviour by considering individual tunnel events onto and off the island pair.
The tunnel rate through a single NIN junction is given by 27 where ∆E is the change in chemical potential, R k = h e 2 and T e is the electron temperature. There are two contributions to ∆E for each junction; a d.c. contribution, which includes V ds and the chemical potential of the charge state involved, and an a.c. contribution from the r.f. drive. We include the r.f. drive by integrating the tunnel rate over one drive period and renormalising. For forward transport with ∆E(t) = (V r. f . sin(ωt) +V d.c. )e, and V r. f . determined from the carrier power and Q factor of the resonant circuit.
To determine the rate of parity change for a given V ds we solve the master equation for the double dot charge states to determine populations in each charge state. The total charge on the island can be increased (decreased) by the tunnelling of an electron from (to) either the source or drain. The rates for these processes are determined by solving eqn 1 numerically for each state.
The total rate of parity change is then given by the rate for each charge state weighted by the population of that state.
This description remains valid if a large source-drain bias such that (eV ds > Ec) is applied to the device, and it is therefore no longer in the Coulomb blockade regime. In this regime charge states other than (m,n), (m+1,n) and (m,n+1) can be occupied and the presence of the Sisyphus impedance depends upon the charge state being odd parity relative to (m,n). By measuring the phase shift, we therefore monitor changes in the charge parity of the device in real time, and at large V ds each switch of parity is due to a net charge transfer of 1 2 e from drain to source. In fig. 4 we show a fit (solid lines) to the measured rates. The electron temperature is determined as above, and we find a good fit for resistances of 54 TΩ for each lead-island tunnel junction, for both r.f. powers.
We measured Γ EO,OE tunnel rates in the range 10-100 Hz, corresponding to electrical currents of order aA. This demonstrates proof-of-principle of the charge parity ammeter, and we now discuss the prospects of measuring larger currents, as required for example in current metrology. In our experiment, the rates were limited by the high lead-dot tunnel resistances and we could simply decrease these resistances to achieve higher currents. However, we cannot do this indefinitely as the signal to noise ratio will start to limit measurable currents: as an example we were not able to measure the discrete switching in the 5 MΩ LR device. From the data in fig 2c we find an SNR of 7.3 at a measurement bandwidth of 15 kHz. A sensitivity of 5.3 × 10 −3 e/ √ Hz is therefore implied for the present measurement. There are ways to increase bandwidth: by using higher charging energy devices allowing a higher amplitude r.f. carrier signal; by optimising the signal from the Sisyphus impedance; by using a lower noise temperature r.f. amplifier (at present T N = 10 K); and by using a low-loss superconducting resonant circuit. With these modifications it seems feasible to measure Γ in the MHz range, corresponding to pA electrical currents.
In conclusion, we have performed r.f. reflectometry measurements on a high resistance aluminium double dot. In measuring the electrical impedance of the device itself we avoided the need for external charge detection in single electron ammetery, and could directly determine the relative charge parity of a metal double dot. This configuration benefits from simplicity in the design, and also is well suited to the use of high bandwidth electrical techniques, in principle enabling relatively large currents to be measured. A related charge parity measurement has application in measuring the spin-state of semiconductor quantum dots. 23 A.J.F. acknowledges support from the Hitachi Research Fellowship and Hitachi Cambridge Laboratory. This work was funded by EPSRC grant EP/H016872/1.