[Paper Review] Full Counting Statistics of a charge shuttle

F. Pistolesi|HAL (Le Centre pour la Communication Scientifique Directe)|Jan 20, 2004

Advanced Statistical Process Monitoring4 citations

TL;DR

This paper develops a full counting statistics (FCS) framework for charge transfer in a nanoscale charge shuttle, where a small grain oscillates between two leads under Coulomb blockade. By generalizing the master equation approach to include time-dependent tunneling rates due to oscillation, the study reveals that for large amplitudes, the FCS becomes sharply peaked at one electron per cycle, with asymmetric tails controlled by distinct parameters for n<1 and n>1, and identifies conditions where the effective charge appears as half the elementary charge due to dynamical correlations.

ABSTRACT

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 $ ilde n$ electrons per cycle is strongly peaked around one. The peak is asymmetric since its form is controlled by different parameters for $ ilde n&gt;1$ and $ ilde n &lt; 1$. Under certain conditions the systems behaves as if the effective charge is 1/2 of the elementary one. Knowledge of the counting statistics gives a new insight on the mechanism of charge transfer.

Motivation & Objective

• To develop a theoretical framework for full counting statistics (FCS) of charge transfer in a nanomechanical charge shuttle.
• To understand how oscillation amplitude and tunneling rate affect the probability distribution of electrons transferred per cycle.
• To investigate the role of dynamical correlations in modifying the effective charge and reducing current noise.
• To provide a quantitative description of electron transfer statistics beyond average current and noise, including higher-order moments.
• To clarify the transition from static to shuttling regimes and identify conditions for well-defined single-electron transfer.

Proposed method

• Generalizes the FCS technique of Bagrets and Nazarov to time-dependent tunneling rates arising from oscillating grain position.
• Uses a master equation approach with time-dependent transition rates ΓL(t) and ΓR(t) that depend on the grain's position in a harmonic potential.
• Derives the generating function for the FCS using a cumulant expansion and solves it numerically for arbitrary oscillation amplitudes.
• Performs analytical approximations in the limits of small and large oscillation amplitudes (a ≪ 1 and a ≫ 1), identifying distinct physical regimes.
• Extracts the probability distribution P(ñ) for transferring ñ electrons per cycle from the generating function via inverse Laplace transform.
• Analyzes the first two moments (current and noise) and the Fano factor, linking them to the underlying dynamics and asymmetry in the distribution.

Experimental results

Research questions

• RQ1 How does the full counting statistics of electron transfer in a charge shuttle depend on oscillation amplitude and tunneling rate?
• RQ2 What causes the asymmetry in the FCS distribution around ñ = 1, and which parameters control the behavior for ñ < 1 versus ñ > 1?
• RQ3 Under what conditions does the system exhibit an effective elementary charge of e/2 instead of e?
• RQ4 How does the Fano factor evolve with increasing oscillation amplitude, and what does this imply about current fluctuations?
• RQ5 In what regime does the system transition from stochastic tunneling to coherent shuttling with well-defined one-electron transfer per cycle?

Key findings

• For large oscillation amplitudes (a ≫ 1), the FCS probability distribution P(ñ) becomes sharply peaked at ñ = 1, indicating highly deterministic one-electron transfer per cycle.
• The peak at ñ = 1 is asymmetric: for ñ < 1, the decay is governed by α = Γ/ω, while for ñ > 1, it is controlled by τ = 2Γ/a, leading to different functional forms on either side.
• In the limit of large a, the Fano factor F = ⟨(ΔI)²⟩ / ⟨I⟩ decreases significantly, indicating strong suppression of current fluctuations due to Coulomb blockade and coherent shuttling.
• When the tunneling rate Γ > 1, the contribution from the central region (x ≈ 0) remains significant, requiring very large amplitudes to achieve the well-defined shuttling regime with ⟨ñ⟩ ≈ 1.
• The effective charge can appear as e/2 due to dynamical correlations in the transfer process, particularly in the symmetric case where both leads contribute equally.
• Numerical results confirm analytical predictions: a discontinuity in the slope of ln P(ñ) at ñ = 1 emerges for large a, signaling a qualitative change in the transfer mechanism.