Resonant electron transport in single-molecule junctions: Vibrational excitation, rectification, negative differential resistance, and local cooling

Vibronic effects in resonant electron transport through single-molecule junctions are analyzed. The study is based on generic models for molecular junctions, which include electronic states on the molecular bridge that are vibrationally coupled and exhibit Coulomb interaction. The transport calculations employ a master equation approach. The results, obtained for a series of models with increasing complexity, show a multitude of interesting transport phenomena, including vibrational excitation, rectification, negative differential resistance, as well as local cooling. While some of these phenomena have been observed or proposed before, the present analysis extends previous studies and allows a more detailed understanding of the underlying transport mechanisms. In particular, it is shown that many of the observed phenomena can only be explained if electron-hole pair creation processes at the molecule-lead interface are taken into account. Furthermore, vibronic effects in systems with multiple electronic states and their role for the stability of molecular junctions are analyzed.

Figure 1

Basic schemes of vibrationally coupled electron transport processes for a single electronic state. Panels (a) and (c) depict examples for emission processes, where an electron sequentially tunnels from the left lead onto the molecule and further to the right lead, thereby singly exciting the vibrational mode of the molecular bridge (red wiggly line). Such emission processes are effectively “heating” the junction (local heating). An example for a respective absorption process is shown in panel (b), where an electron tunnels from the left to the right lead by absorbing a quantum of vibrational energy (blue wiggly line). These processes result in local cooling of the junction.

Figure 2

Example processes for electron-hole pair creation in a molecular junction. Panel (a) depicts an electron-hole pair creation process with respect to the left lead by absorption of a single vibrational quantum. Panel (b) represents an electron-hole pair creation process with respect to the right lead by absorption of two vibrational quanta. The absorption of two vibrational quanta normally occurs with lower probability.

Figure 3

Current and vibrational excitation for a generic model system with a single electronic state at the molecular bridge that is symmetrically coupled to the left and the right lead, and moderately coupled to a single vibrational mode. The inset shows the corresponding population $n_1$ of the electronic level. The dashed line refers to a calculation where the electronic-vibrational coupling is set to zero. The solid gray and black line are obtained for an electronic-vibrational coupling strength $\lambda =0.06$ eV. Thereby, the gray line is calculated employing the thermal equilibrium state of the vibrational mode (at 10K), and the black line is obtained with its full current-induced nonequilibrium state. The solid red line depicts the vibrational excitation for this model system with a reduced electronic-vibrational coupling $\lambda =0.03$ eV.

Figure 4

Current-voltage characteristics for a generic model system with a single electronic state that is asymmetrically coupled to the leads, and moderately coupled to a single vibrational mode with different coupling strengths $\lambda$. The current-voltage characteristics of the upper panel have been obtained with the vibrational mode kept in thermal equilibrium, while for the ones in the lower panel, the full nonequilibrium state of the vibrational mode is taken into account. The insets represent the corresponding populations of the electronic level.

Figure 5

Average vibrational excitation corresponding to the currents shown in Fig. 4b.

Figure 6

Example of transport processes that become active at a bias voltage $2(\Omega -{\stackrel{̲}{\epsilon }}_1)$, as the electronic state of our model system is located close to the Fermi level of the system, that is, $|{\stackrel{̲}{\epsilon }}_1-{\stackrel{̲}{\epsilon }}_F|<\Omega /2$. If ${\stackrel{̲}{\epsilon }}_1-{\stackrel{̲}{\epsilon }}_F>\Omega /2$, the hopping process from the left lead onto the molecular bridge requires a higher bias voltage, $\Phi =2{\stackrel{̲}{\epsilon }}_1>2(\Omega -{\stackrel{̲}{\epsilon }}_1)$, and in that case, these transport processes become active at the same bias voltage as, for example, elastic transport processes that do not change the vibrational state.

Figure 7

Current and vibrational excitation for a generic model system with a single electronic state close to the Fermi energy, $|{\stackrel{̲}{\epsilon }}_1-{\epsilon }_F|<\Omega /2$, and a single vibrational mode. The dashed lines refer to a calculation with the vibrational mode in thermal equilibrium, while for the solid lines the vibrational mode is treated in its full current-induced nonequilibrium state. The current-voltage characteristics for positive (black and blue line) and negative bias voltages (gray and green line) are overlayed to highlight areas, where $I(\Phi )=-I(-\Phi )$ holds.

Figure 8

Current-voltage characteristics for a single electronic state, which is located close to the Fermi energy, $|{\stackrel{̲}{\epsilon }}_1-{\epsilon }_F|<\Omega /2$, and coupled to a single vibrational mode by different coupling strengths $\lambda$. Features of NDR disappear one by one for increasing electronic-vibrational coupling. The inset shows the difference of the transition probabilities $\left|X_{00}\right|{}^2-(|X_{11}|{}^2+|X_{01}|{}^2)$ (red line) and $(|X_{00}|{}^2+|X_{01}|{}^2)-(|X_{11}|{}^2+|X_{01}|{}^2+|X_{12}|{}^2)$ (dashed green line) as functions of the electronic-vibrational coupling strength $\lambda$. The zero crossings mark the value of the electronic-vibrational coupling, where the respective NDR feature vanishes.

Figure 9

Basic processes for vibrationally coupled electron transport involving two electronic states, including resonant emission (a), resonant absorption (b), and electron-hole pair creation (c) processes. Due to resonant emission processes with respect to the lower-lying electronic state [panel (a)], resonant absorption and electron-hole pair creation processes with respect to the second electronic state become active even before this state enters the conduction window set by the applied bias voltage.

Figure 10

Current-voltage characteristics for a model molecular junction comprising two electronic states, which are coupled to a single vibrational mode. The dashed line corresponds to a calculation without electronic-vibrational coupling ${\lambda }_{1/2}=0$. The solid black and gray line show results with a moderate coupling of the vibrational mode to both electronic states. For the solid gray line, the vibrational mode is assumed to be in its thermal equilibrium state, while for the solid black line the full current-induced nonequilibrium state of the vibrational mode is taken into account.

Figure 11

Vibrational excitation for the two-state model molecular junction employed for the current-voltage characteristics of Fig. 10 (black line). The different results correspond to different energies ${\epsilon }_2$ for the higher-lying electronic state. The solid black line shows the vibrational excitation that corresponds to the respective current-voltage characteristics of Fig. 10. The solid blue and red line depict results where the higher-lying electronic state is located closer to the lower-lying electronic level.

Figure 12

Current-voltage characteristics and vibrational excitation for a molecular junction with two electronic states that are moderately coupled to a single vibrational mode. The solid red line refers to a calculation with asymmetric molecule-lead coupling, where both electronic states are strongly coupled to the left lead and weakly coupled to the right lead: ${\nu }_{\mathrm{L},1/2}=10{\nu }_{\mathrm{R},1/2}$. The solid black lines represent the result for the corresponding symmetric junction, which are the same as the solid black lines in Figs. 10 and 11. The corresponding current-voltage characteristics is thereby rescaled by a factor of $1/100$ for a better comparison with the red line. The inset in the upper panel shows the population of the electronic states $n_{1/2}$ for the asymmetric junction.

Figure 13

Upper panel: Vibrational excitation of a model molecular junction with two electronic states, both of which are moderately coupled to a single vibrational mode and symmetrically to the leads. In addition, a repulsive electron-electron interaction $U=0.5$ eV is taken into account. Lower panel: Difference between the vibrational excitation shown in Fig. 11 and the upper panel, that is, with and without Coulomb interaction, $\langle a^{†}a\rangle {}_{H,U=0}-\langle a^{†}a\rangle {}_{H,U=0.5\mathrm{eV}}$.

Figure 14

Current-voltage characteristics for a two-state molecular junction with a single vibrational mode, where a repulsive electron-electron interaction of $U=0.5$ eV is taken into account. The solid black line represents the current for a symmetrically coupled junction, and the solid red line represents the current for a junction with a blocking state,[47, 94, 95] that is, a higher-lying state that is weakly coupled to the right lead. The populations of the electronic states $n_{1/2}$, which correspond to the current-voltage characteristics depicted by the solid red line, are shown in the inset. Therein, the solid black line displays the population of the blocking state.

Figure 15

Current-voltage characteristics for a symmetrically coupled molecular junction, where two electronic states are moderately coupled to a single vibrational mode and exhibit repulsive Coulomb interaction, $U=0.5$ eV. The solid black line, as in Fig. 14, represents the result where both states are coupled to the leads with the same coupling strengths ${\nu }_{K,i}$. For the solid red line, the coupling of the higher-lying electronic state to both leads was decreased to ${\nu }_{\mathrm{L}/\mathrm{R},2}=0.1{\nu }_{\mathrm{L}/\mathrm{R},1}$.

Figure 16

Current-voltage characteristics and vibrational excitation for a model system similar to the one of Fig. 12. Here, the energy of the second electronic state is chosen such that the $\nu$th vibrational level of the electronically excited state of the anion is degenerate with respect to the $(\nu +1)$th vibrational state of the anionic ground state: ${\epsilon }_2={\epsilon }_1+\Omega$. The solid red line is obtained disregarding all coherences of the reduced density matrix, while the solid green line is obtained taking all coherences of $\rho$ into account. The inset shows the respective population of the electronic levels, where the dashed lines refer to the population of state 1 ($n_1$) and the solid lines to the population of state 2 ($n_2$).