Single-molecule electron tunnelling through multiple redox levels with environmental relaxation

We present an analytical theory for electron flow through a molecule with several redox levels, enclosed between a pair of metallic electrodes. The levels can be electronic or electronic-vibrational levels. Vibrational energy spacing characterises the latter sets. The levels are further coupled to environmental nuclear motion. Cryogenic temperatures are considered. This implies that thermal activation is disregarded but vibrational relaxation in the molecular charge states is central. The electrochemical potentials of the electrodes are controlled relative to a reference electrode. The electrodes represent the substrate and tip in electrochemical in situ scanning tunnelling microscopy. An equivalent three-electrode configuration represents a molecular single-electron transistor in which the enclosing electrodes constitute source and drain, and the reference electrode the gate. Current-bias voltage relations at fixed electrochemical overpotential or gate voltage, and current–overpotential or current-gate voltage relations at fixed bias voltage are equivalent in the two systems. Due to the activation-less nature of the processes, electron flow between the electrodes through the molecular redox levels can be only achieved when the latter are located between the Fermi levels of the substrate and tip or source and drain electrodes. The redox levels can be brought into this “energy window”, either by the overpotential or bias and gate voltages, or by vibrational relaxation of (a) given (set of) redox level(s) subsequent to electron transfer. Several physical mechanisms can be distinguished and distinctive current–overpotential/gate voltage or current-bias voltage relations obtained. These reflect electronic level separation, environmental nuclear reorganisation, and coherent or incoherent multi-electron flow. The models and formalism have bearings on construction of single-molecule devices, illustrated by a short discussion of single-electron tunnelling in semiconductor quantum dots and reported low-temperature single-molecular transistor effects.