Counting statistics and super-Poissonian noise in a quantum dot: Time-resolved measurements of electron transport

We present time-resolved measurements of electron transport through a quantum dot. The measurements were performed using a nearby quantum point contact as a charge detector. The rates for tunneling through the two barriers connecting the dot to source and drain contacts could be determined individually. In the high bias regime, the method was used to probe excited states of the dot. Furthermore, we have detected bunching of electrons, leading to super-Poissonian noise. We have used the framework of full counting statistics to model the experimental data. The existence of super-Poissonian noise suggests a long relaxation time for the involved excited state, which could be related to the spin relaxation time.

Figure 1

(Color online) (a) Quantum dot with integrated charge read-out used in the experiment. (b) Current through the QPC as a function of time, showing a few electrons tunneling into and out of the dot. The arrows mark the steps corresponding to an electron entering the dot.

Figure 2

(Color online) (a) Average dot population versus voltage on gate $G2$. The data was fit to a Fermi distribution function with $T=230\mathrm{mK}$. (b) Counts of events per second for the same data as in (a). The data was fit to Eq. (4), giving $\Gamma =2.63\mathrm{kHz}$ and $T=230\mathrm{mK}$. (c) Events per second versus $V_{G1}$ and $V_{G2}$. For low values of $V_{G1}$ and $V_{G2}$, both the source lead and the drain lead are pinched off. For high voltages, the barriers open up so much that the tunneling occurs on a time scale faster than the measurement bandwidth. (d) Temperature (squares) and tunnel coupling (crosses), extracted from data shown within the ellipse in (c). As $V_{G2}$ is increased, $V_{G1}$ is decreased, in order to keep the dot at a constant potential. For low $V_{G2}$, tunneling occurs between the source lead and the dot, for high $V_{G2}$, the electrons tunnel between the drain and the dot. For intermediate gate values, both leads contribute to the tunneling. The electron temperature was found to be the same for both leads, within the accuracy of the data analysis.

Figure 3

(Color online) (a) Coulomb diamonds, measured by counting events per second. For low values of $V_{G1}$, the source lead is pinched off and tunneling can only occur between the dot and the drain lead. As $V_{G1}$ increases, the source lead opens up and a current can flow through the dot. (b) Diagrams depicting the energy levels of the dot at points I, II, and III. In case III, the bias is higher than the charging energy of the dot, meaning that the dot can contain 0, 1 or 2 excess electrons. (c) Time trace taken at point III. The three possible dot populations ($n$, $n+1$ or $n+2$ electrons) are clearly resolvable.

Figure 4

(Color online) (a) and (b) Blow-up of the upper left region of Fig. 3a, showing the rates for electrons tunneling in (a) and out (b) of the dot, respectively. The solid lines mark the positions where the source lead is lining up with the electrochemical potential of the dot’s ground state. The dashed lines mark the lower edge of the region where condition of Eq. (7) in the text is fulfilled. The color scales are different for the two figures, the rate for tunneling out is around 10 times faster than tunneling in. (c) Diagram depicting the energy levels along the dashed lines in (a) and (b). As the source lead is raised [corresponds to going upward along the dashed lines in (a) and (b)], excited states become available for tunneling. (d) Tunneling rate for electrons entering the dot, measured along the dashed line in (a). With increased gate voltage, excited states become available for transport, giving higher tunneling rates.

Figure 5

(Color online) (a) Time trace of the QPC current showing bunching of electrons. (b) Dot states included in the model used to describe the bunching of electrons. The red circles correspond to electron occupation. State $S_A$ is the $n$-electron ground state, state $S_B$ is an excited $n$-electron state, and state $S_{n+1}$ is the ground state when the dot contains $(n+1)$ electrons. (c) Energy diagram for the model. The two dot transitions are both within the thermal broadening of the lead. Electrons enter the dot from the left lead and may leave through either the left or the right lead. (d) Possible transitions between the different states of the model. The rates ${\Gamma }_{\text{in}}^A$, ${\Gamma }_{\text{in}}^B$ refer to electrons entering the dot, thus taking the dot from state $S_{A∕B}$ to state $S_{n+1}$. The rates ${\Gamma }_{\text{out}}^A$, ${\Gamma }_{\text{out}}^B$ describe electrons leaving the dot, giving transitions from state $S_{n+1}$ to $S_{A∕B}$. $W_{AB}$ and $W_{BA}$ are the direct transition rates between states $S_A$ and $S_B$. Finally, the rates ${\Gamma }_{\text{right}}^A$, ${\Gamma }_{\text{right}}^B$ refer to electrons leaving the dot through the right lead.

Figure 6

(Color online) (a) First, second, and third cumulants of the distribution of charge fluctuations. The markers show values extracted from the experimental data, while the solid lines are calculated from the model given in the text. Fitting parameters are ${\Gamma }_A=1.6\mathrm{kHz}$, ${\Gamma }_B=20.5\mathrm{kHz}$, ${\Gamma }_{\text{right}}^A=4.6\mathrm{kHz}$, ${\Gamma }_{\text{right}}^B=310\mathrm{Hz}$, $T_1=8\mathrm{ms}$, and ${\mu }_A-{\mu }_B=13\mu \mathrm{eV}$. The electronic temperature was $400\mathrm{mK}$. (b) Normalized cumulants $C_3∕C_1$ and $C_2∕C_1$ versus bias voltage. The noise is clearly super-Poissonian in the central region of the graph. (c) Calculated maximal value of $C_3∕C_1$ as a function of the relaxation time between the two states. The values are calculated by varying the relaxation time while keeping the other parameters to the values given by the fit shown in (a). The maximum value $C_3∕C_1$ extracted from the experimental data is 15.9.