Polaron effects on the dc- and ac-tunneling characteristics of molecular Josephson junctions

We study the interplay of polaronic effect and superconductivity in transport through molecular Josephson junctions. The tunneling rates of electrons are dominated by vibronic replicas of the superconducting gap, which show up as prominent features in the differential conductance for the dc and ac current. For relatively large molecule-lead coupling, a features that appears when the Josephson frequency matches the vibron frequency can be identified with an over-the-gap structure observed by Marchenkov et al. [Nat. Nanotech. 2, 481 (2007)]. However, we are more concerned with the weak-coupling limit, where resonant tunneling through the molecular level dominates. We find that certain features involving both Andreev reflection and vibron emission show an unusual shift of the bias voltage $V$ at their maximum with the gate voltage $V_g$ as $V\sim (2/3)V_g$. Moreover, due to the polaronic effect, the ac Josephson current shows a phase shift of $\pi$ when the bias $eV$ is increased by one vibronic energy quantum $\hslash {\omega }_v$. This distinctive even-odd effect is explained in terms of the different sign of the coupling to vibrons of electrons and of Andreev-reflected holes.

Figure 1

Imaginary part of the first diagonal element of the retarded self-energy $-\mathrm{Im}{\left({\Sigma }_L^r\right)}_{11}$, which represents scattering of electrons between the molecule and lead $L$, for various electron-vibron coupling strengths $\lambda$. We have taken $\hslash {\omega }_v=8\Delta$ and $\Gamma =0.2\hslash {\omega }_v$. With increasing $\lambda$, multiple vibronic replicas of the superconducting gap edges appear.

Figure 2

Density plot of the dc differential conductance $d{I}_0/dV$ vs on-site energy ${\varepsilon }_0$ and bias voltage $V$ for $\Delta =\hslash {\omega }_v/3$, $\lambda =\hslash {\omega }_v$, and (a) $\Gamma =\hslash {\omega }_v$ and (b) $\Gamma =0.05\hslash {\omega }_v$. Features labeled “1” and “2” do not involve vibron emission or absorption, whereas those labeled “a”–“h” are due to this polaronic effect. All features are discussed in the text.

Figure 3

Schematic representations of the two most important processes in the MJJ that involve neither vibrons nor the molecular level. Left panel: single Andreev reflection at the right molecule-lead contact. Right panel: direct tunneling from the lower gap edge in the left lead to the upper gap edge in the right lead. The filled (empty) shapes represent the energy states of the superconducting leads below (above) the Fermi energy. Energy scales of the superconducting gap $\Delta$ and the bias voltage $eV$ are indicated. The arrows denote possible transitions of the electron or hole.

Figure 4

Schematic representation of various vibron-assisted tunneling and Andreev-reflection processes in the MJJ. The labels are the same as in Fig. 2. The DOS of both leads is modified due to the polaronic effect. The filled (empty) shapes represent the energy states of the superconducting leads below (above) the Fermi energy. The molecular level ${\epsilon }_0$ (blue/gray) is broadened due to the coupling to the leads. Energy scales of the superconducting gap $\Delta$, the vibron energy $\hslash {\omega }_v$, and the bias voltage $eV$ are indicated. The arrows denote possible transitions of the electron or hole.

Figure 5

Density plot of the ac differential conductance of (a) the dissipative Josephson current $I_1^D$ and (b) the nondissipative Josephson current $I_1^S$ vs the molecular energy level ${\varepsilon }_0$ and the bias voltage $V$. The parameters are identical to those used for Fig. 2b.