Counting statistics of hole transfer in a $p$-type GaAs quantum dot with dense excitation spectrum

Low-temperature transport experiments on a $p$-type GaAs quantum dot capacitively coupled to a quantum point contact are presented. The time-averaged as well as time-resolved detection of charging events of the dot are demonstrated and they are used to extract the tunneling rates into and out of the quantum dot. The extracted rates exhibit a super-linear enhancement with the bias applied across the dot, which is interpreted in terms of a dense spectrum of excited states contributing to the transport, characteristic for heavy hole systems. The full counting statistics of charge transfer events and the effect of back action is studied. The normal cumulants as well as the recently proposed factorial cumulants are calculated and discussed in view of their importance for interacting systems.

Figure 1

(a) AFM micrograph of the sample consisting of a QD with an integrated charge read-out QPC. The dark regions are trenches created on the surface of the heterostructure by chemical etching. The applied voltages are shown on the same figure. (b) QD current (blue) and QPC current (green) as a function of $V_{\mathrm{G}2}$ measured at the temperature of $T\approx$ 1.2 K. (c) QPC current as a function of time, showing a few holes tunneling into and out of the QD over a timescale of 200 $\mu$s. The lower current level corresponds to a state when the dot holds one excess hole. The QPC current was filtered with a 3 kHz software filter and resampled at a frequency of 14 kHz. The random variables ${\tau }_{\mathrm{in}}$ and ${\tau }_{\mathrm{out}}$ quantify the times it takes for a hole to tunnel into and out of the dot, respectively. (d) The probability density function (PDF) obtained from normalized histogram of the detector current showing two distinct current levels corresponding to the two charge states of the QD. (e) The histogram of times a hole needs to tunnel into the QD (blue), tunnel out of the QD (green), and the event time defined as the sum of two consecutive tunneling in and out events (red) for a symmetric configuration (${\Gamma }_{\mathrm{in}}={\Gamma }_{\mathrm{out}}$). The green curve exactly overlaps with the blue curve. (f) The occupation of the QD and the number of charge events as the gate voltage is swept over a Coulomb blockade peak with fits to the Fermi-Dirac distribution (green curve) and its derivative (red curve), respectively.

Figure 2

(a) Event rate ${\Gamma }_{\mathrm{event}}$ and tunneling rates (b) ${\Gamma }_{\mathrm{in}}$ and (c) ${\Gamma }_{\mathrm{out}}$ as a function of the gate voltage and the applied QD bias, demonstrating half of a Coulomb diamond. All the rates increase with the dot bias. White arrows indicate the relative position of the QD electrochemical potential ${\mu }_N^0$ with respect to those of source and drain. (d) In the case of a moderate energy-dependence of barriers, tunneling into and out of the dot both involves many excited states and relaxation does not change the qualitative picture.

Figure 3

Rate equation simulation of a dot with a dense spectrum of 100 $\mu$eV equidistant energy levels. ${\Gamma }_{\mathrm{in}}$ in (a) and ${\Gamma }_{\mathrm{out}}$ in (b) both increase with the dot bias. The equi-${\Gamma }_{\mathrm{in}}$ lines and equi-${\Gamma }_{\mathrm{out}}$ lines are parallel to source and drain lines, respectively. (c) ${\Gamma }_{\mathrm{event}}$. (d) Horizontal cut through ${\Gamma }_{\mathrm{in}}$, ${\Gamma }_{\mathrm{out}}$, and ${\Gamma }_{\mathrm{event}}$ in Figs. 2a, 2b, 2c at resonance. (e) Same as described in the legend for Fig. 2d but assuming an exponential energy-dependence of barriers. (f) Horizontal cut through the measurement data in Figs. 2a–2c at resonance as described in the legend of Fig. 2d.

Figure 4

(a) Histogram of the number of events during $T=50$ ms at the point A in Fig. 2a with symmetric barriers $(a=0)$. The blue and green curves show the Poisson and Gaussian distributions, respectively, calculated using the mean and variance of the measured data (all these distributions are discrete and the connecting lines are just guides to the eye). While the Gaussian distribution fits reasonably well to the histogram, the data is best described by the Bagrets-Nazarov distribution with the input parameters ${\Gamma }_{\mathrm{in}}$ and ${\Gamma }_{\mathrm{out}}$. (b) The first two normalized cumulants of the statistical distribution of events as a function of tunneling barrier asymmetry $a$, calculated from 100 time traces, each 10 s long. The blue points show $C2/C1$ or the Fano factor and the red points show $C3/C1$, which is the skewness. The solid lines are the model predictions. While the Fano factor agrees quite well with the model the skewness points are scattered much more due to limited statistics. (c) ${\Gamma }_{\mathrm{in}}$, ${\Gamma }_{\mathrm{out}}$, and ${\Gamma }_{\mathrm{event}}$ as a function of the bias voltage on the QPC. The decrease in ${\Gamma }_{\mathrm{in}}$ and increase in ${\Gamma }_{\mathrm{out}}$ is most probably due to a gating effect. (d) Fano factor and skewness as a function of QPC bias showing that the statistics of electron transport is not influenced by the emission of energy quanta by the QPC. The dashed lines show the model predictions discussed in the text augmented by finite-bandwidth corrections (B.W.C.).[31] The red shaded area shows the onset of the detector signal degradation due to charge fluctuators in the QPC (Counting was not possible for $V_{\mathrm{QPC}}^{\mathrm{bias}}<50$ $\mu$V).

Figure 5

(a, b) First 12 normalized cumulants of charge transfer as a function of time (mean number of events in time traces with a given length cut from a very long time trace) calculated from the data collected at point A in Fig. 2a along with the predictions of BN model, which fits remarkably well to the data. The amount of statistics decreases for longer time traces causing the size of error bars to increase. (c, d) First 12 factorial cumulants (FCs)[34] calculated from the normal cumulants. FCs of different order alternate sign and grow exponentially with time and, therefore, it is more convenient to follow their trend in logarithmic scale as shown in Figs. 5e and 5f. No zero-crossing oscillations in FCs are observed, which is consistent with a two-level Markovian model.[34] The white arrows point to the development of faint non-zero-crossing oscillations appearing in higher-order FCs, probably due to finite statistics.