Full-counting statistics for molecular junctions: Fluctuation theorem and singularities

TL;DR

This paper derives an analytic full-counting statistics for charge transport through a quantum dot with electron-phonon interactions, revealing how phonons influence electron transfer and satisfy the fluctuation theorem, with implications for understanding quantum transport phenomena.

Contribution

It provides a second-order analytic expression for the cumulant generating function accounting for nonequilibrium phonons, and explores singularities and phase-transition-like features in electron transport.

Findings

Identifies regimes where phonons affect electron transfer differently.

Discovers a kink in the probability distribution indicating a phase transition.

Confirms that current flows against bias only at finite temperature, consistent with the fluctuation theorem.

Abstract

We study the full-counting statistics of charges transmitted through a single-level quantum dot weakly coupled to a local Einstein phonon which causes fluctuations in the dot energy. An analytic expression for the cumulant generating function, accurate up to second order in the electron-phonon coupling and valid for finite voltages and temperatures, is obtained in the extended wide-band limit. The result accounts for nonequilibrium phonon distributions induced by the source-drain bias voltage, and concomitantly satisfies the fluctuation theorem. Extending the counting field to the complex plane, we investigate the locations of possible singularities of the cumulant generating function, and exploit them to identify regimes in which the electron transfer is affected differently by the coupling to the phonons. Within a large-deviation analysis, we find a kink in the probability distribution, analogous to a first-order phase transition in thermodynamics, which would be a unique hallmark of the electron-phonon correlations. This kink reflects the fact that although inelastic scattering by the phonons once the voltage exceeds their frequency can scatter electrons opposite to the bias, this will never generate current flowing against the bias at zero temperature, in accordance with the fluctuation theorem.