Effects of charge-dependent vibrational frequencies and anharmonicities in transport through molecules

As a step towards a more realistic modeling of vibrations in single-molecule devices, we investigate the effects of charge-dependent vibrational frequencies and anharmonic potentials on electronic transport. For weak phonon relaxation, we find that in both cases vibrational steps split into a multitude of substeps. This effectively leads to a bias-dependent broadening of vibrational features in current-voltage and conductance characteristics, which provides a fingerprint of nonequilibrium vibrations whenever other broadening mechanisms are secondary. In the case of an asymmetric molecule-lead coupling, we observe that frequency differences can also cause negative differential conductance.

Figure 1

(Color online) $IV$ and conductance for a symmetric device ($t_L=t_R$, symmetric voltage splitting) with slightly different vibrational frequencies for the neutral and charged molecule $({\omega }_1=1.1{\omega }_0)$, and $\lambda =1.2$, $U\to \infty$. (a) $IV$ at $\epsilon =0$ and $k_{B}T⪡\hslash {\omega }_0$ for strong and weak vibrational relaxation. While for equilibrated phonons, step locations are specified by Eq. (7), for unequilibrated phonons a multitude of steps is generated, leading to an effective step-broadening even at low temperatures. For comparison, the inset shows results with identical frequencies ${\omega }_0={\omega }_1$. (b),(c) Conductance plots at $k_{B}T=0.02\hslash {\omega }_0$ showing the subsplitting and washing out of phonon features for unequilibrated phonons.

Figure 2

(Color online) Current-voltage characteristics for an asymmetric device ($t_R^2=0.3t_L^2$, symmetric voltage splitting) in the unequilibrated regime with $\lambda =1.1$, $k_{B}T⪡\hslash \omega$, $U\to \infty$. The combination of asymmetric coupling and differing vibrational frequencies is observed to lead to weak peaklike structures at the onsets of phonon steps with negative differential conductance regions (marked by arrows).

Figure 3

(Color online) $IV$ and conductance plots for a symmetric device ($t_L=t_R$, symmetric voltage splitting) with a Morse potential $(j=30)$, for $\lambda =1.5$, $U\to \infty$. (a) $IV$ at $\epsilon =0$ and $k_{B}T⪡\hslash {\omega }_e$ for strong and weak vibrational relaxation. For equilibrated phonons steps along $\epsilon =0$ are located at voltages specified by Eq. (7), for unequilibrated phonons the step-splitting leads to an effective step-broadening even at low temperatures. (b),(c) Conductance plots at $k_{B}T=0.02\hslash {\omega }_e$ showing the subsplitting and broadening of phonon features for unequilibrated phonons.