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We study the statistical properties of charge and energy transport inelectron conducting junctions with electron-phonon interactions, specifically,the thermoelectric efficiency and its fluctuations. The system comprises donorand acceptor electronic states, representing a two-site molecule or a doublequantum dot system. Electron transfer between metals through the two molecularsites is coupled to a particular vibrational mode which is taken to be eitherharmonic or anharmonic- a truncated (two-state) spectrum. Considering thesemodels we derive the cumulant generating function in steady state for chargeand energy transfer, correct to second-order in the electron-phononinteraction, but exact to all orders in the metal-molecule coupling strength.This is achieved by using the non-equilibrium 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 itsnoise and the large deviation function for the thermoelectric efficiency. Wedemonstrate that at large bias the charge current, differential conductance,and the current noise can identify energetic and structural properties of thejunction. We further examine the operation of the junction as a thermoelectricengine and show that while the macroscopic thermoelectric efficiency isindifferent to the nature of the mode (harmonic or anharmonic), efficiencyfluctuations do reflect this property.

Full counting statistics of vibrationally-assisted electronic conduction: transport and fluctuations of the thermoelectric efficiency