Hierarchical quantum master equation approach to electronic-vibrational coupling in nonequilibrium transport through nanosystems

Within the hierarchical quantum master equation (HQME) framework, an approach is presented, which allows a numerically exact description of nonequilibrium charge transport in nanosystems with strong electronic-vibrational coupling. The method is applied to a generic model of vibrationally coupled transport considering a broad spectrum of parameters ranging from the nonadiabatic to the adiabatic regime and including both resonant and off-resonant transport. We show that nonequilibrium effects are important in all these regimes. In particular, in the off-resonant transport regime, the inelastic cotunneling signal is analyzed for a vibrational mode in full nonequilibrium, revealing a complex interplay of different transport processes and deviations from the commonly used $G_0/2$ rule of thumb. In addition, the HQME approach is used to benchmark approximate master equation and nonequilibrium Green's function methods.

Figure 1

$I\text{−}V$'s obtained by the HQME approach and different approximate methods. The results are shown for model 1 [(a), (c), (e)] as well as for model 2 [(b), (d), (f)].

Figure 2

IETS for model 3 and $\mathrm{\Gamma }/\mathrm{\Omega }=1/15$. The purely electronic contribution $I_{\mathrm{el}}$ has been subtracted for a better resolution of inelastic effects. The fourth-order ME as well as the NEGF-SCBA approach are compared to the HQME approach in (a). The inset shows the peak-dip structure at $\mathrm{\Phi }=0.328\mathrm{V}$ for $T=50\mathrm{K}$. (b) depicts a comparison with the approximate version of the HQME approach.

Figure 3

(a) IETS and (b) differential conductance for model 4 and different molecule-lead couplings $\mathrm{\Gamma }$. The graphs depict HQME results obtained with a truncation after the third (solid lines) and fourth (solid circles) level of the hierarchy. The dashed and dotted lines in (a) represent results of a NEGF calculation within the SCB and FSCB approximation. In (b), the conductance-voltage characteristics is also depicted for the noninteracting system by dashed-dotted lines.