Charge dynamics in molecular junctions: Nonequilibrium Green's function approach made fast

Real-time Green's function simulations of molecular junctions (open quantum systems) are typically performed by solving the Kadanoff-Baym equations (KBE). The KBE, however, impose a serious limitation on the maximum propagation time due to the large memory storage needed. In this work we propose a simplified Green's function approach based on the generalized Kadanoff-Baym ansatz (GKBA) to overcome the KBE limitation on time, significantly speed up the calculations, and yet stay close to the KBE results. This is achieved through a twofold advance: First, we show how to make the GKBA work in open systems and then construct a suitable quasiparticle propagator that includes correlation effects in a diagrammatic fashion. We also provide evidence that our GKBA scheme, although already in good agreement with the KBE approach, can be further improved without increasing the computational cost.

Figure 1

Diagrams for the 2B correlation self-energy.

Figure 2

Example of an open system as described in the main text with $N_{\tau }=9$ transverse channels and a chain of four sites.

Figure 3

Top panels: Density $n_1={\rho }_{11}$ of a one-site chain connected to leads with $N_{\tau }=1$ after the sudden switch-on of a bias $V_L=2$ for different $T_{\lambda }=-9,-5,-2$. Bottom left panel: HF density of site 1 of a two-site chain connected to leads with $N_{\tau }=1$ after the sudden switch-on of a bias $V_L=-V_R=1$. Bottom right panel: HF current at the right interface of a four-site chain connected to leads with $N_{\tau }=9$ after the sudden switch-on of a bias $V_L=-V_R=0.8,1.2$.

Figure 4

Results for the density $n_1\left(t\right)$ of the one-site chain with $T_{\lambda }=-9$ and same parameters as in Fig. 3. In the left panel the system is unperturbed and $n_1\left(t_{\mathrm{in}}\right)$ is varied. In the right panel $n_1\left(t_{\mathrm{in}}\right)=1/2$, a bias $V_L=2$ is switched on at $t=0$ and $t_{\mathrm{in}}$ is varied. For clarity the curves with $t_{\mathrm{in}}=-2n$, $n=0,1,2,3$ are shifted upward by $n/10$.

Figure 5

Time-dependent occupation of the one-site chain for different $\Gamma$. Left panel: $t_{\mathrm{in}}=-8$ and unperturbed system. Middle panel: $t_{\mathrm{in}}=-8$ and bias $V_L=2$ switched on at $t=0$. Right panel: $t_{\mathrm{in}}=-12$ and bias $V_L=2$ switched on at $t=0$.

Figure 6

Time-dependent current at the right interface of the four-site junction with $V_L-V_R=1.6$ (left panel) and 2.4 (right panel); same parameters as in Fig. 4.

Figure 7

Numerical evidence of the fulfillment of the continuity equation in the GKBA+QP and GKBA + SC schemes.

Figure 8

Schematic illustration of the photovoltaic junction described in the main text.

Figure 9

Time-dependent occupations in the HF approximation using GKBA (solid) and KBE (circles). The junction is perturbed by a monochromatic pulse.

Figure 10

Thermalization of the occupations in the correlated case. The correlated KBE value is represented by a dotted horizontal line.

Figure 11

Time-dependent occupations after a pulse in KBE and GKBA + SC. For $n_l$ and $n_1$ we also show the results of the GKBA0 scheme (dotted-dashed).

Figure 12

Time-dependent KBE (top panel) and GKBA + SC (bottom panel) spectral functions for the photovoltaic junction subject to a light pulse. The light pulse is switched on at time $t=40$ for the KBE and $t=75$ for the GKBA + SC. The parameters are the same as in Fig. 11.

Figure 13

Time-dependent occupations in the presence of continuous radiation in KBE and GKBA + SC. For $n_1$ we also show the results of the GKBA0 scheme as well as of the hybrid scheme (thermalization with GKBA + SC, positive-time propagation with GKBA+QP).