Voltage-induced singularities in transport through molecular junctions

Abstract

The inelastic scattering of electrons which carry current through a single-molecule junction is modeled by a quantum dot, coupled to electron reservoirs via two leads. When the electron is on the dot, it is coupled to a single harmonic oscillator of frequency ${\ensuremath{\omega}}_{0}$. At zero temperature, the resonance peak in the linear-response conductance always narrows down due to the coupling with the vibrational mode. However, this narrowing down is given by the Franck-Condon factor only for narrow resonances. Contrary to some claims in the literature, the linear-response conductance does not exhibit any sidebands at zero temperature. Small sidebands, of order $\text{exp}[\ensuremath{-}\ensuremath{\beta}\ensuremath{\hbar}{\ensuremath{\omega}}_{0}]$, do arise at finite temperatures. The single-particle density of states exhibits discontinuities and logarithmic singularities at the frequencies corresponding to the opening of the inelastic channels, due to the imaginary and real parts of the self-energy. The same singularities also generate discontinuities and logarithmic divergences in the differential conductance at and around the inelastic thresholds. These discontinuities usually involve upwards steps, but these steps become negative within a rather narrow range of the elastic transparency of the junction. This range shrinks further as the excitation energy exceeds the bare resonance width.