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We study the statistical properties of charge and energy transport in electron conducting junctions with electron-phonon interactions, specifically, the thermoelectric efficiency and its fluctuations. The system comprises donor and acceptor electronic states, representing a two-site molecule or a double-quantum-dot system. Electron transfer between metals through the two molecular sites is coupled to a particular vibrational mode which is taken to be either harmonic or anharmonic, a truncated (two-state) spectrum. Considering these models we derive the cumulant generating function in steady state for charge and energy transfer, correct to second order in the electron-phonon interaction, but exact to all orders in the metal-molecule coupling strength. This is achieved by using the nonequilibrium Green's function approach (harmonic mode) and a kinetic quantum master-equation method (anharmonic mode). From the cumulant generating function we calculate the charge current and its noise and the large-deviation function for the thermoelectric efficiency. We demonstrate that at large bias the charge current, differential conductance, and the current noise can identify energetic and structural properties of the junction. We further examine the operation of the junction as a thermoelectric engine and show that while the macroscopic thermoelectric efficiency is indifferent to the nature of the mode (harmonic or anharmonic), efficiency fluctuations do reflect this property.
Agarwalla et al. (Mon,) studied this question.