Transport through molecular junctions with a nonequilibrium phonon population

The calculation of the nonlinear conductance of a single-molecule junction is revisited. The self-energy on the junction resulting from the electron-phonon interaction has at low-temperatures logarithmic singularities (in the real part) and discontinuities (in the imaginary one) at the frequencies corresponding to the opening of the inelastic channels. These singularities generate discontinuities and logarithmic divergences (as a function of the bias voltage) in the low-temperature differential conductance around the inelastic thresholds. The self-energy also depends on the population of the vibrational modes. The case of a vibrating free junction (not coupled to a thermal bath), where the phonon population is determined by the bias voltage is examined. We compare the resulting zero-temperature differential conductance with the one obtained for equilibrated phonons and find that the difference is larger, the larger is the bare transmission of the junction and the product of the electron dwell time on the junction with the phonon frequency.

Figure 1

(Color online) The population of the oscillator modes, $N_{\text{ho}}$, at zero temperature, as a function of $eV/{\omega }_0$, for various values of the bare transmission: $\mathcal{T}=0.2$ (smallest dash size), 4/17, 0.3, 4/9, and 0.9 (largest dash size). Here ${\omega }_0/{\Gamma }_0=1/3$.

Figure 2

(Color online) The same as in Fig. 1, for ${\omega }_0={\Gamma }_0$.

Figure 3

(Color online) The same as in Fig. 1, for ${\omega }_0/{\Gamma }_0=3$.

Figure 4

(Color online) The zero-temperature differential conductance, as a function of $eV/{\omega }_0$, for two values of the bare transmission, marked on the figure. The full lines show the conductance with the nonequilibrium phonon population, the dashed curves are with the equilibrium one. Here ${\omega }_0/{\Gamma }_0=1/3$.

Figure 5

(Color online) The same as in Fig. 4, for ${\omega }_0={\Gamma }_0$.

Figure 6

(Color online) The same as in Fig. 4, for ${\omega }_0=3{\Gamma }_0$.