Kinetic equations for transport through single-molecule transistors

Abstract

We present explicit kinetic equations for quantum transport through a general molecular quantum dot, accounting for all contributions up to fourth order perturbation theory in the tunneling Hamiltonian and the complete molecular density matrix. Such a full treatment describes not only sequential, cotunneling, and pair tunneling, but also contains terms contributing to renormalization of the molecular resonances as well as their broadening. Due to the latter all terms in the perturbation expansion are automatically well defined for any set of system parameters: no divergences occur and no by-hand regularization is required. Additionally we show that, in contrast to second order perturbation theory, in fourth order it is essential to account for quantum coherence between nondegenerate states, entering the theory through the nondiagonal elements of the density matrix. As a first application, we study a single-molecule transistor coupled to a localized vibrational mode (Anderson-Holstein model). We find that cotunneling-assisted sequential tunneling processes involving the vibration give rise to current peaks, i.e., negative differential conductance in the Coulomb-blockade regime. Such peaks occur in the crossover to strong electron-vibration coupling, where inelastic cotunneling competes with Franck-Condon suppressed sequential tunneling, and thereby indicate the strength of the electron-vibration coupling. The peaks depend sensitively on the coupling to a dissipative bath, thus providing also an experimental probe of the $Q$ factor of the vibrational motion.