Nonequilibrium resonant spectroscopy of molecular vibrons

Quantum transport through single molecules is essentially affected by molecular vibrations. We investigate the behavior of the prototype single-level model with weak to intermediate electron-vibron interactions and arbitrary couplings to the leads. For this, we have developed a nonequilibrium self-consistent theory which allows us to explore the nonperturbative regime via the nonequilibrium Green function formalism. We show that the nonequilibrium resonant spectroscopy is able to determine the energies of molecular orbitals and the spectrum of molecular vibrations. Our results are relevant to scanning tunneling spectroscopy experiments and demonstrate the importance of the systematic and self-consistent investigation of the effects of the vibronic dynamics onto the transport through single molecules.

Figure 1

(Color online) Schematic picture of the considered electron-vibron single-level model, coupled to the left and right leads.

Figure 2

(Color online) Spectral function at different electron-vibron couplings: $\lambda ∕{\omega }_0=0.4$ (black), $\lambda ∕{\omega }_0=1.2$ (blue/dark gray, dashed), and $\lambda ∕{\omega }_0=2$ (red/gray); at ${ϵ}_0∕{\omega }_0=5$, ${\Gamma }_L∕{\omega }_0={\Gamma }_R∕{\omega }_0=0.1$. In the inset, the spectral function at $\lambda ∕{\omega }_0=1.2$ is shown at a finite voltage when the level is partially filled. Energies are in units of $\hslash {\omega }_0$.

Figure 3

Differential conductance of a symmetric junction $(\eta =0.5,{\Gamma }_R={\Gamma }_L)$ at different molecule-to-lead couplings, from ${\Gamma }_L∕{\omega }_0=0.1$ (lower curve) to ${\Gamma }_L∕{\omega }_0=10$ (upper curve), $\lambda ∕{\omega }_0=1$ and ${ϵ}_0∕{\omega }_0=2$. Voltage is in units of $\hslash {\omega }_0∕e$.

Figure 4

Differential conductance of an asymmetric junction $(\eta =0,{\Gamma }_R=20{\Gamma }_L)$ at different molecule-to-lead couplings, from ${\Gamma }_R∕{\omega }_0=0.2$ (lower curve) to ${\Gamma }_R∕{\omega }_0=4$ (upper curve), $\lambda ∕{\omega }_0=2$ and ${ϵ}_0∕{\omega }_0=5$. The voltage is in units of $\hslash {\omega }_0∕e$.

Figure 5

(Color online) Differential conductance at different molecule-to-STM couplings (see the text)—from asymmetric junction with ${\Gamma }_L∕{\omega }_0=0.025$, ${\Gamma }_R∕{\omega }_0=0.5$, and $\eta =0.2$ (lower curve, blue/dark gray thick line) to symmetric junction with ${\Gamma }_L∕{\omega }_0={\Gamma }_R∕{\omega }_0=0.5$, and $\eta =0.5$ (upper curve, red/gray thick line)—$\lambda ∕{\omega }_0=1$ and ${ϵ}_0∕{\omega }_0=2$. Voltage is in units of $\hslash {\omega }_0∕e$.