Full counting statistics of a charge shuttle

We study the charge transfer in a small grain oscillating between two leads. Coulomb blockade restricts the charge fluctuations in such a way that only zero or one additional electrons can sit on the grain. The system thus acts as a charge shuttle. We obtain the full counting statistics of charge transfer and discuss its behavior. For large oscillation amplitude the probability of transferring electrons per cycle is strongly peaked around one. The peak is asymmetric since its form is controlled by different parameters for and . Under certain conditions the systems behaves as if the effective charge is of the elementary one. Knowledge of the counting statistics gives a new insight on the mechanism of charge transfer.

Figure 1

(Color online) Simple schematic of the system. A small grain oscillating in an harmonic potential between two leads kept at a constant voltage bias.

Figure 2

(Color online) The two equivalent systems for the statistics of charge transferred in the long measurement time limit: (a) single electron transistor with equal probability $\mathrm{\Gamma }$ of hopping from the left lead to the grain and from the grain to the right lead; (b) a fictitious system of charges $1∕2$ that tunnel with probability $\mathrm{\Gamma }$.

Figure 3

(Color online) Exact and approximate dependence on $\varphi$ of ${\mathrm{\Gamma }}_L$ used to obtain analytically the FCS in the large oscillation limit. ${\mathrm{\Gamma }}_R\left(\varphi \right)$ has the same form shifted by $\pi$ in the $\varphi$ axis.

Figure 4

(Color online) Current (plain line) and noise (dashed line) as a function of the oscillation amplitude for different values of $\mathrm{\Gamma }$. From the top-left pannel $\mathrm{\Gamma }=0.001$, 0.1, 1, 5. The dashed light lines are obtained with the analytical approximation (29). The short lines at $a=0$ are the static results given by (15).

Figure 5

(Color online) $\ln \left[P_N\left(\tilde{n}\right)\right]∕N$ as a function of the number of electrons transferred per cycle $\tilde{n}$ for $\mathrm{\Gamma }=0.1$, and oscillation amplitude $a=0$, 1, 2, 3, 4, and 5 (from left to right).

Figure 6

(Color online) The same as Fig. 5 for $\mathrm{\Gamma }=1$.