Time-resolved charge detection with cross-correlation techniques

We present time-resolved charge-sensing measurements on a GaAs double quantum dot with two proximal quantum point-contact (QPC) detectors. The QPC currents are analyzed with cross-correlation techniques, which enable us to measure dot charging and discharging rates for significantly smaller signal-to-noise ratios than required for charge detection with a single QPC. This allows us to reduce the current level in the detector and therefore the invasiveness of the detection process and may help to increase the available measurement bandwidth in noise-limited setups.

Figure 1

(Color online) (a) Inset: AFM micrograph of the sample which consists of two QDs in series (QD1 and QD2) with two charge readout QPCs, denoted PC1 and PC2. The source, drain, and center barriers can be tuned with in-plane gates S, D, and C. Main graph: part of the DQD charge stability diagram obtained by counting the number of switching events in $I_2$. (b) Detector currents recorded at two different gate configurations indicated in (a). Dot-lead tunneling processes cause (I) identical switching directions in both QPCs whereas interdot processes cause (II) opposite switching. (c) Current correlator $C_0=⟨I_1{I}_2⟩/{\left(⟨I_1^2⟩⟨I_2^2⟩\right)}^{1/2}$ extracted from the raw data used in (a), revealing the correlation-anticorrelation pattern in the $V_{\text{S}}\text{−}{V}_{\text{D}}$ plane. Numbers $\left(n,m\right)$ indicate the electron occupancy of the dots relative to the state (0,0).

Figure 2

(Color online) (a) Evolution of the current cross-correlation function $C\left(\tau \right)=⟨I_1\left(t+\tau \right)I_2\left(t\right)⟩$ when crossing the $\left(1,0\right)\to \left(1,1\right)$ charging line in the DQD stability diagram [outside the scan range of Fig. 1b]. The curves are offset for clarity; going from the lowest to the uppermost curve corresponds to adding one electron to QD2. $C\left(\tau \right)$ exhibits an exponential decay $\sim \text{exp}\left(-\left|\tau \right|/{\tau }_0\right)$ characteristic for the random telegraph signal present in both QPC currents. The center curves, which feature the largest amplitude $C\left(0\right)$, belong to QPC signals with rather balanced occupation of the high- and low-current states. (b) Noise reduction due to cross correlation in time traces measured on a Coulomb peak. Plotted are the Fourier transform $S_C$ of the cross-correlation function and the geometric mean of the two QPC current spectra $S_{\text{PC}1}$ and $S_{\text{PC}2}$. The dashed line represents the ideal RTS spectrum $S_{\text{RTS}}\left(f\right)=2⟨I_{\text{RTS}}^2⟩{\tau }_0/\left[1+{\tau }_0^2{\left(2\pi f\right)}^2\right]$ with parameters $⟨I_{\text{RTS}}^2⟩=0.22\times {10}^{-3}{\text{nA}}^2$ and ${\tau }_0=0.8\text{ms}$. (c) Standard deviation $\sigma \left(⟨I_1{I}_2⟩\right)$ of the average $⟨I_1{I}_2⟩=T^{-1}{\int }_0^T{I}_1\left(t\right)I_2\left(t\right)dt$ as a function of the integration time $T$. The solid line marks the expected $\sim T^{-1/2}$ behavior.

Figure 3

(Color online) Plots of the rates ${\Gamma }_{\text{in}}$, ${\Gamma }_{\text{out}}$, and ${\Gamma }_{\text{tot}}={\Gamma }_{\text{in}}{\Gamma }_{\text{out}}/\left({\Gamma }_{\text{in}}+{\Gamma }_{\text{out}}\right)$ as determined by electron counting (marked with the letter “C”) and by current cross-correlation (“X”). The sweep range is indicated by a line in the top graph of Fig. 4a.

Figure 4

(Color online) (a) Top: tunneling rate ${\Gamma }_{\text{tot}}^X$ between QD2 and lead as a function of QPC bias (applied to both PC1 and PC2) and S gate voltage determined by cross-correlation analysis. Bottom: SNR for the counting analysis and expected false count rate as a function of QPC bias. The false count rate was calculated from the SNR data assuming Gaussian amplifier noise with an autocorrelation time of ${\tau }_n=0.1\text{ms}$. A SNR of 6 will result in a false count rate on the order of 10 Hz and can therefore be considered as the minimum requirement for the counting analysis. It is reached for bias voltages above approximately $150\mu \text{V}$. (b) Examples of the time dependence (left column) and current distribution function $P\left(I\right)$ (right column) of $I_2$ recorded at three different QPC bias voltages indicated in the top graph in (a). The RTS as a component of the current is recognized by the naked eye in all three cases, but only the trace “I” allows for a determination of the transition rates without significant error when analyzed with a counting algorithm.