Effect of nonadiabatic electronic-vibrational interactions on the transport properties of single-molecule junctions

The interaction between electronic and vibrational degrees of freedom in single-molecule junctions may result from the dependence of the electronic energies or the electronic states of the molecular bridge on the nuclear displacement. The latter mechanism leads to a direct coupling between different electronic states and is referred to as nonadiabatic electronic-vibrational coupling. Employing a perturbative nonequilibrium Green's function approach, we study the influence of nonadiabatic electronic-vibrational coupling in model molecular junctions. Thereby, we distinguish between systems with well-separated and quasidegenerate electronic levels. The results show that the nonadiabatic electronic-vibrational interaction can have a significant influence on the transport properties. The underlying mechanisms, in particular the difference between nonadiabatic and adiabatic electronic-vibrational couplings, are analyzed in some detail.

Figure 1

Current-voltage characteristics of two molecular junctions described by the models AD and NONAD for two different vibrational energies. The onset of resonant transport through the electronic levels in the absence of vibrations is marked by vertical dashed lines. The inset provides a close-up of a region relevant for nonadiabatic vibrational effects.

Figure 2

Conductance map for the model systems (top) AD and (bottom) NONAD. In both plots, the conductance $\frac{\partial I}{\partial V}$ is plotted as a function of voltage $V$ and and vibrational energy $\hslash \mathrm{\Omega }$. The dashed lines indicate the positions of the unperturbed electronic resonances ${\epsilon }_{1/2}$, the electronic resonances renormalized by the coupling to the vibrations ${\overline{\epsilon }}_{1/2}$ as well as their respective vibrational satellites.

Figure 3

Schematic representation of example processes for (a) purely electronic and (b)–(e) vibronic transport involving the emission of one vibrational quantum and two broadened electronic levels. The abbreviations L and R denote the left and the right leads, respectively, and M stands for the molecule. The arrows illustrate tunneling electrons, whereas the color of the arrow denotes the molecular level involved in the tunneling process.

Figure 4

Current-voltage characteristics for the system ASYMM. The pictures given in this plot visualize the most important vibrational transport processes. The vertical dashed lines mark the voltages at which relevant unperturbed electronic energy levels enter the voltage window.

Figure 5

Current-voltage characteristics for the model system INTPLY for $M_{11},M_{22}=0.05$ eV and $M_{12}=0,\pm 0.02$ eV. The vertical dashed lines denote the voltages at which resonant transport through the molecular states, renormalized by the adiabatic electronic-vibrational coupling, becomes possible.

Figure 6

Interference (dashed lines) and incoherent (solid lines) contributions to the currents of the system INTPLY for $M_{11},M_{22}=0.05$ eV and $M_{12}=0,\pm 0.02$ eV depicted in Fig. 5.

Figure 7

Current-voltage characteristics for the model system INTPLY. The electronic-vibrational coupling parameters are specified in the legend.

Figure 8

Current-voltage characteristics of the system DESNONAD for ${\epsilon }_2=0.505$ eV with and without nonadiabatic electronic-vibrational coupling.

Figure 9

Conductance map of the system DESNONAD for varying electronic energy ${\epsilon }_2$. The purely nonadiabatic electronic-vibrational coupling strength is $M_{12}=0.05$ eV. The white dashed lines indicate the position of the electronic resonances. The positions of the corresponding two-level system with empirically determined constant coupling strength $\mathrm{\Delta }=0.0075$ eV are marked by black dashed lines in the relevant region. The light colors on the right-hand side of the plot close to the line ${\epsilon }_2={\epsilon }_1=0.5$ eV indicate regions of negative differential resistance.

Figure 10

Current-voltage characteristics of the system DESVIB for varying nonadiabatic electronic-vibrational coupling. The gray line gives the current without electronic-vibrational interaction, the black line is the current for the system with purely adiabatic electronic-vibrational coupling, and the blue and green lines corresponds to the current of the system with different nonadiabatic electronic-vibrational coupling ranging from $M_{12}=0.005$ eV to $M_{12}=0.02$ eV.

Figure 11

Interference (dashed lines) and incoherent (solid lines) contributions to the currents of the system DESVIB for varying nonadiabatic electronic-vibrational coupling strength $M_{12}$.