Image charge dynamics in time-dependent quantum transport

In this work, we investigate the effects of the electron-electron interaction between a molecular junction and the metallic leads in time-dependent quantum transport. We employ the recently developed embedded Kadanoff-Baym method [Phys. Rev. B 80, 115107 (2009)] and show that the molecule-lead interaction changes substantially the transient and steady-state transport properties. We first show that the mean-field Hartree-Fock (HF) approximation does not capture the polarization effects responsible for the renormalization of the molecular levels neither in nor out of equilibrium. Furthermore, due to the time-local nature of the HF self-energy, there exists a region in parameter space for which the system does not relax after the switch-on of a bias voltage. These and other artifacts of the HF approximation disappear when including correlations at the second-Born or GW levels. Both these approximations contain polarization diagrams, which correctly account for the screening of the charged molecule. We find that by changing the molecule-lead interaction, the ratio between the screening and relaxation time changes, an effect which must be properly taken into account in any realistic time-dependent simulation. Another important finding is that while in equilibrium the molecule-lead interaction is responsible for a reduction of the highest occupied molecular orbital-lowest unoccupied molecular orbital (HOMO-LUMO) gap and for a substantial redistribution of the spectral weight between the main spectral peaks and the induced satellite spectrum, in the biased system it can have the opposite effect, i.e., it sharpens the spectral peaks and opens the HOMO-LUMO gap.

Figure 1

Image charge model for quantum transport.

Figure 2

Top left and right panels: The real and imaginary parts of the dynamical response function. Bottom panel: The electron density at the terminal site of the TB chain as a function of time when the impurity potential $U=0.5$ is suddenly switched on. The different curves correspond to different values of the hopping parameter $b=-0.5,-1.0,-1.5,-2.0$.

Figure 3

Self-energy diagrams for ${\lambda }^l={\lambda }^r=0$. (a) The only nonzero diagrams of the GW self-energy. (b) Second-order self-energy diagrams that are zero. All higher-order diagrams are also zero.

Figure 4

Green's function $G_{LL}^{>}(t,0)$ for (a) HF and 2B with $b=-0.6$, $U=1$, and $\lambda =0$, (b) HF and 2B with $b=-0.6$, $U=1.0$, and $\lambda =-0.2$.

Figure 5

Green's function $G_{LL}^{>}(t,0)$ for HF and 2B with $b=-1.0$ [(a) and (b)] and $b=-0.6$ [(c) and (d)] and $\lambda =-0.2$ and different values of $U$.

Figure 6

LUMO spectral function in the 2B approximation for $b=-0.6$, $U=1.0$, and $\lambda =0$ (left panel), $b=-0.6$, $\lambda =-0.2$, and $U=0,1.0,1.3$ (right panel).

Figure 7

HF and GW equilibrium spectral functions $A_{\mathrm{mol}}\left(\omega \right)$ for $\lambda =0$, $b=-1.0$, and $U=0,0.5,1.0,1.5$. The vertical lines indicate the exact and 2Bs(HF) peak positions.

Figure 8

HF, 2B, and GW equilibrium spectral functions $A_{\mathrm{mol}}\left(\omega \right)$ for $\lambda =-0.2$, $b=-1.0$, and $U=0,0.5,1.0,1.5$.

Figure 9

The real and imaginary components of the 2B(HF) self-energy for the molecular HOMO level with $U=0.5$ (left panel) and $U=1.5$ (right panel). The rest of the parameters are $U_{HL}=U_0=-b=1$ and $\lambda =0$.

Figure 10

(a) Time-dependent right current for $U=0.0,0.5,1.0$. (b) Ground-state (GS) and steady-state spectral functions of Eq. (20). (c) Time-dependent densities $n_{1r}\left(t\right)$ and $n_{1l}\left(t\right)$ at the terminal sites. (d) Time-dependent HOMO and LUMO densities $n_H\left(t\right)$ and $n_L\left(t\right)$. In all the plots, the simulations have been performed within the HF approximation with bias $W^l=-W^r=0.8$.

Figure 11

(a) Time-dependent right current for $U=0.0,0.5,1.0$. (b) Ground-state (GS) and steady-state spectral functions of Eq. (20). (c) Time-dependent densities $n_{1r}\left(t\right)$ and $n_{1l}\left(t\right)$ at the terminal sites. (d) Time-dependent HOMO and LUMO densities $n_H\left(t\right)$ and $n_L\left(t\right)$. In all the plots, the simulations have been performed within the HF approximation with bias $W^l=-W^r=1.2$.

Figure 12

(a) Fourier transform of the transient current of Fig. 11 with $U=1.0$. (b) Ground-state and steady-state spectral functions $A_{\mathrm{mol}}\left(\omega \right)$ for $U=1.0$. (c) Time-dependent density matrix components $G_{L,1l}(t,t^{+})$ and $G_{1l,L}(t,t^{+})$ for $U=0.0$ and $U=1.0$ (d) HF time-dependent density matrix components $G_{H,1l}(t,t^{+})$ and $G_{1l,H}(t,t^{+})$ for $U=0.0$ and $1.0$. In all the plots, the simulations have been performed within the HF approximation with bias $W^l=-W^r=1.2$.

Figure 13

Time-dependent density in the noninteracting part of the right lead for $U=0.0$ (top panel) and $U=1.0$ (bottom panel). The simulations have been performed within the HF approximation with bias $W^l=-W^r=1.2$. Site number 2 corresponds to the first noninteracting site in the right lead.

Figure 14

(a) Time-dependent right current for $U=0.0,0.5,1.0$. (b) Ground-state (GS) and steady-state spectral functions of Eq. (20). (c) Time-dependent densities $n_{1r}\left(t\right)$ and $n_{1l}\left(t\right)$ at the terminal sites. (d) Time-dependent HOMO and LUMO densities $n_H\left(t\right)$ and $n_L\left(t\right)$. In all the plots, the simulations have been performed within the 2B approximation with bias $W^l=-W^r=1.2$.

Figure 15

(a) Time-dependent right current for $U=0.0,0.5,1.0$. (b) Ground-state (GS) and steady-state spectral functions of Eq. (20). (c) Time-dependent densities $n_{1r}\left(t\right)$ and $n_{1l}\left(t\right)$ at the terminal sites. (d) Time-dependent HOMO and LUMO densities $n_H\left(t\right)$ and $n_L\left(t\right)$. In all the plots, the simulations have been performed within the GW approximation with bias $W^l=-W^r=1.2$.

Figure 16

Time-dependent right current (top panel) and total number of particles in the molecule (bottom panel) for bias voltages in the range $[0.7,1.2]$ and molecule-lead interaction $U=1.0$.

Figure 17

(a) Time-dependent left, right, and total currents. (b) Fourier transform of the total current. (c) Time-dependent terminal site densities. (d) Time-dependent HOMO and LUMO densities and the total number of particles in the molecule. In all panels, $U=1.0$ and the bias voltage is $W^l=-W^r=0.8$.

Figure 18

HF time-dependent right current for different interaction strengths $U=0,0.5$ (top left and right panels) and $U=1.0$ (bottom left panel). The $I-V$ curves extracted from the long-time limit are displayed in the bottom right panel.

Figure 19

Time-dependent number of electrons on the molecule $N_{\mathrm{mol}}\left(t\right)$ versus the applied bias voltage. HF approximation with $U=1.0$.

Figure 20

Ground-state (GS) and steady-state spectral functions in the HF approximation for $U=1.0$ and bias $W^l=-W^r=0.55$. (a) Full spectral function of the interacting region. (b) Spectral function on the terminal site of the left lead. (c) HOMO and LUMO spectral functions. (d) Spectral function on the terminal site of the right lead.

Figure 21

2B time-dependent right current for different interaction strengths $U=0,0.5$ (top left and right panels) and $U=1.0$ (bottom left panel). The $I-V$ curves extracted from the long-time limit are displayed in the bottom right panel.

Figure 22

GW time-dependent right current for different interaction strengths $U=0,0.5$ (top left and right panels) and $U=1.0$ (bottom left panel). The $I-V$ curves extracted from the long-time limit are displayed in the bottom right panel.

Figure 23

HF, 2B, and GW spectral functions for different bias voltage and lead-molecule interaction $U=0.0$ and $1.0$.

Figure 24

Electronic configuration for a two-site lead. For an extra electron on the molecule, there are three energy eigenspaces for the lead. Note that the parallel-spin electron states do not contribute in the response properties since the bias preserves the spin orientation.

Figure 25

Formation of the split-off state (sharp peak in the bottom right panel) as $U$ increases. In all plots, $b=-1.0$.

Figure 26

The value of the renormalization constant as a function of $U$ for $b=-1.0$ and $\lambda =-0.2$. The inset shows the dependence of the factor $C$ as a function of $U$.