Rudge, Samuel L., and Kosov, Daniel S. (2019) Nonrenewal statistics in quantum transport from the perspective of first-passage and waiting time distributions. Physical Review B, 99 (11). 115426.
The waiting time distribution has, in recent years, proven to be a useful statistical tool for characterizing transport in nanoscale quantum transport. In particular, as opposed to moments of the distribution of transferred charge, which have historically been calculated in the long-time limit, waiting times are able to detect nonrenewal behavior in mesoscopic systems. They have failed, however, to correctly incorporate backtunneling events. Recently, a method has been developed that can describe unidirectional and bidirectional transport on an equal footing: the distribution of first-passage times. Rather than the time between successive electron tunnelings, the first passage refers to the first time the number of extra electrons in the drain reaches +1. Here, we demonstrate the differences between first-passage time statistics and waiting time statistics in transport scenarios where the waiting time either cannot correctly reproduce the higher-order current cumulants or cannot be calculated at all. To this end, we examine electron transport through a molecule coupled to two macroscopic metal electrodes. We model the molecule with strong electron-electron and electron-phonon interactions in three regimes: (i) sequential tunneling and cotunneling for a finite bias voltage through the Anderson model, (ii) sequential tunneling with no temperature gradient and a bias voltage through the Holstein model, and (iii) sequential tunneling at zero bias voltage and a temperature gradient through the Holstein model. We show that for each transport scenario, backtunneling events play a significant role; consequently, the waiting time statistics do not correctly predict the renewal and nonrenewal behavior, whereas the first-passage time distribution does.