Effects of coupling to vibrational modes on the ac conductance of molecular junctions

We theoretically examine the effect of the coupling of the transport electrons to a vibrational mode of the molecule on the ac linear-response conductance of molecular junctions. Representing the molecule by a single electronic state, we find that at very low temperatures the frequency-dependent conductance is mainly enhanced (suppressed) by the electron-vibration interaction when the chemical potential is below (above) the energy of that state. The vertex corrections of the electron-vibration interaction induce an additional peak structure in the conductance, which can be observed by tuning the tunnel couplings with the leads.

Figure 1

The diagrams of the ac conductance: (a) without the e-v interaction, (b) the Hartree term, (c) the exchange term, (d) the vertex correction for the Hartree term, and (e) the vertex correction for the exchange term. The solid line denotes the Green’s function of the electrons while the dotted line indicates the Green’s function of the vibrational mode. The wavy line indicates the frequency of the external ac field.

Figure 2

The ac conductance of a fully symmetric bridge, as a function of ${\varepsilon }_0-\mu$ at $\omega =\Gamma$. The coupling strength of the e-v interaction is $\gamma =0.3\Gamma$, and the frequency of the vibrational mode is ${\omega }_0=\Gamma$ (dashed line) and ${\omega }_0=0.5\Gamma$ (dotted line). The solid line is the “bare” conductance $G_{\mathrm{nint}}$, obtained in the absence of the e-v interaction.

Figure 3

(a) The ac conductance of the fully symmetric bridge, as a function of the frequency of the bias voltage $\omega$, at ${\varepsilon }_0-\mu =\Gamma$ and $\gamma =0.3\Gamma$. The frequency of the vibrational mode is ${\omega }_0=\Gamma$ (dashed line) and ${\omega }_0=0.5\Gamma$ (dotted line). The solid line is the “bare” conductance $G_{\mathrm{nint}}$, in the absence of the e-v interaction. (b) The additional conductance due to the Hartree term of the e-v interaction. (c) The additional conductance due to the exchange term.

Figure 4

The same as Fig. 3, but with ${\varepsilon }_0-\mu =-\Gamma$.

Figure 5

The ac conductance of an asymmetric junction, as a function of $\omega$ at (a) ${\varepsilon }_0-\mu =\Gamma$ and (b) ${\varepsilon }_0-\mu =-\Gamma$ when ${\Gamma }_L=\Gamma$. The energy of the vibrational mode is ${\omega }_0=\Gamma$ (dashed line) or ${\omega }_0=0.5\Gamma$ (dotted line). The solid line is the “bare” conductance $G_{\mathrm{nint}}$, in the absence of the e-v interaction. The additional conductances resulting from the vertex corrections of the Hartree and exchange terms are shown in panels (c) and (d).

Figure 6

Top: The thick dotted line denotes the Green’s function of the vibrational mode with the RPA term, see Eqs. (B1) and (B2). Bottom: The diagrams required when the RPA term is included besides the diagrams in Fig. 1 (with the replacement of the dotted line in Fig. 1 by the thick dotted line, as shown in the top part).