Counting Statistics of Single Electron Transport in a Quantum Dot

We have measured the full counting statistics of current fluctuations in a semiconductor quantum dot (QD) by real-time detection of single electron tunneling with a quantum point contact. This method gives direct access to the distribution function of current fluctuations. Suppression of the second moment (related to the shot noise) and the third moment (related to the asymmetry of the distribution) in a tunable semiconductor QD is demonstrated experimentally. With this method we demonstrate the ability to measure very low current and noise levels.

Figure 1

(a) AFM micrograph of the sample consisting of a QD connected to two contacts $S$ and $D$ and a nearby QPC. $G1$, $G2$, and $P$ are lateral gates allowing the tuning of the tunnel coupling to the source $S$, the coupling to the drain $D$, and the conductance of the QPC. $G1$ and $G2$ are also used to tune the number of electrons in the QD. A symmetric bias voltage $V$ is applied between the source and the drain on the QD. (b),(c) Scheme of the quantum dot in the case of equilibrium charge fluctuations (b) and nonequilibrium charge fluctuations (c). (d) Time trace of the current measured through the QPC corresponding to fluctuations of the charge of the dot between $N$ and $N+1$ electrons. The arrows indicate transitions where an electron is entering the QD from the source lead. (e) Probability density of the times ${\tau }_{\mathrm{in}}$ and ${\tau }_{\mathrm{out}}$ (see text) obtained from time traces similar to the one in (d). The lines correspond to the expected exponential dependence (see the text), where the tunneling rates are calculated from $1/{\Gamma }_{S(D)}=1/{\Gamma }_{\mathrm{in}(\mathrm{out})}=⟨{\tau }_{\mathrm{in}(\mathrm{out})}⟩$.

Figure 2

Statistical distribution of the number $n$ of electrons entering the QD during a given time $t_0$. The two panels correspond to two different values of the tunneling rates, obtained for different values of the gate voltage $V_{G1}$. The time $t_0$ is chosen in order to have the same mean value of number of events, $⟨n⟩\approx 3$, for both graphs. We have checked that this choice does not affect the results. The line shows the theoretical distribution calculated from Eqs. (1, 2). The tunneling rates are determined experimentally by the method described in Fig. 1e, and no fitting parameters have been used for the theoretical curves.

Figure 3

(a) Average number of electrons entering the QD, $\mu$, measured as a function of the gate voltage $V_{G1}$ and the bias voltage $V$. Far from the edges of the Coulomb blockade region, i.e., for $|\pm eV/2-E_d|\gg k_{B}T$, the fluctuations of $n$ are directly related to current fluctuations. The dashed line corresponds to the cross section shown in panel (b). (b) Three first moments of the fluctuations of $n$ as a function of the bias voltage $V$ and at a given gate voltage $V_{G1}=-44\mathrm{mV}$. The ground state (GS) as well as two excited states (ES) are clearly visible. The moments are scaled so that $\mu$ corresponds to the number of electrons entering the QD per second. In the gray region, the condition $|\pm eV/2-E_d|\gg k_{B}T$ is not valid, and the number of electrons entering the QD cannot be taken as the current flowing through the QD. The width of this region is $9\times k_{B}T/e\approx 300\mu \mathrm{V}$, determined from the width for which the Fermi distribution is between 0.01 and 0.99. (c) Normalized second and third moments as a function of the bias voltage $V$ and at a given gate voltage $V_{G1}=-44\mathrm{mV}$.

Figure 4

(a) Second and (b) third normalized central moments of the fluctuations of $n$ as a function of the asymmetry of the tunneling rates, $a=({\Gamma }_S-{\Gamma }_D)/({\Gamma }_S+{\Gamma }_D)$. To increase the resolution, each point at a given asymmetry is obtained by averaging over about 50 points at a given voltage $V_{G1}$ and in a window of bias voltage $1.5<V<3\mathrm{mV}$. Error bars correspond to the standard error of this averaging process and are of the size of the points if not shown. The dashed lines are the theoretical predictions given by Eq. (3). No fitting parameters have been used, since the tunneling rates are fully determined experimentally [see Fig. 1e and text]. Inset of (b): Variation of the asymmetry of the tunneling rates, $a$, as a function of $V_{G1}$.