Voltage-induced singularities in transport through molecular junctions

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 ${\omega }_0$. 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 $\text{exp}\left[-\beta \hslash {\omega }_0\right]$, 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.

Figure 1

(Color online) The function $\Delta E\left(\mu ,\mu ,\mu \right)$, Eq. (27), for two representative values of the oscillator frequency, ${\omega }_0=0.5{\Gamma }_0$ (solid line), and ${\omega }_0={\Gamma }_0$ (dashed line). Here $\gamma ={\Gamma }_0$.

Figure 2

(Color online) The dimensionless linear-response conductance, Eq. (31), for two representative values of the ratio ${\Gamma }_0/{\omega }_0$, 0.5 (a) and 4 (b). It can be seen that the conductance (dashed line) is always smaller than that obtained in the absence of the coupling with the oscillator (depicted by the solid line). (Ref. 42).

Figure 3

(Color online) The local density of states on the dot, Eq. (33), as a function of the energy $\omega$ for $\mu =0$ (a) and for $\mu =1.5{\Gamma }_0$ (b). The solid lines correspond to ${\omega }_0={\Gamma }_0$, while ${\omega }_0=3{\Gamma }_0$ for the dashed lines (here, $\gamma ={\Gamma }_0$). All the graphs should go to zero at $\omega =\pm {\omega }_0$.

Figure 4

(Color online) The linear-response conductance resulting from $I_{\text{inco}}$, for $\beta {\Gamma }_0=1$, 3, 6, and 12 (increasing dash sizes). Here ${\omega }_0=2{\Gamma }_0$, and all energies are in units of ${\Gamma }_0$.

Figure 5

(Color online) The total differential conductance (a) and the contributions to the conductance from the coupling to the vibrational mode, $G_{\text{co}}$ [(b), containing the logarithmic divergence] and $G_{\text{inco}}$ [(c), containing the discontinuity; $G_{\text{inco}}=0$ for $eV<\hslash {\omega }_0$, and the jumps in the various conductances at the thresholds are seen as the lapses in the curves] as function of the bias voltage, for ${\omega }_0=3{\Gamma }_0$, and five different values of the zeroth-order transmission $\mathcal{T}=0.2,{\mathcal{T}}_1=4/17,0.3,{\mathcal{T}}_2=4/9,0.9$ [dash sizes increase with $\mathcal{T}$, appearing in increasing order at small $V$ in (a)]. Here $\gamma ={\Gamma }_0$.