Iterative summation of path integrals for nonequilibrium molecular quantum transport

We formulate and apply a nonperturbative numerical approach to the nonequilibrium current $I\left(V\right)$ through a voltage-biased molecular conductor. We focus on a single electronic level coupled to an unequilibrated vibration mode (Anderson-Holstein model), which can be mapped to an effective three-state problem. Performing an iterative summation of real-time path integral (ISPI) expressions, we accurately reproduce known analytical results in three different limits. We then study the crossover regime between those limits and show that the Franck-Condon blockade persists in the quantum-coherent low-temperature limit, with a nonequilibrium smearing of step features in the $IV$ curve.

Figure 1

Current $I$ (in units of $e\Gamma /h$) vs bias voltage $V$ (in units of $\Gamma /e$) for $\lambda =0.5\Gamma$, $\Omega =\Gamma$, $\epsilon =0$, and $T=\Gamma$. The ISPI data are depicted as filled red circles, where the dotted red curve is only to guide the eye and the error bars are explained in the main text. We also show the results of perturbation theory in $\lambda$ (dashed black curve) and of the rate equation (solid blue curve). The upper (lower) inset shows the corresponding result for $T=3\Gamma$ ($T=\Gamma /3$).

Figure 2

Same as Fig. 1 but for $\Omega =0.5\Gamma$ (main panel) and $\Omega =2\Gamma$ (inset), both for $T=\Gamma$.

Figure 3

Same as Fig. 1 but for $\Omega =0.5\Gamma$ and $\lambda =\Gamma$. The main panel is for $T=\Gamma$ and compares the ISPI results to NEBO predictions. The insets are for $T=3\Gamma$ and $T=\Gamma /3$, respectively, where the rate equation results are also shown. Notice that, in contrast to ISPI, the rate equation predicts an unphysical current blockade for $T=\Gamma /3$.

Figure 4

ISPI data for the $IV$ curves from weak ($\lambda =0.5\Gamma$) to strong ($\lambda =4\Gamma$) electron-phonon coupling, with $\Omega =2\Gamma$. The main panel is for $T=0.2\Gamma$, the inset for $T=\Gamma$. We used a dense voltage grid yielding smooth $IV$ curves. Error bars are not shown but remain small (cp. Fig. 1).