Kadanoff-Baym approach to quantum transport through interacting nanoscale systems: From the transient to the steady-state regime

We propose a time-dependent many-body approach to study the short-time dynamics of correlated electrons in quantum transport through nanoscale systems contacted to metallic leads. This approach is based on the time propagation of the Kadanoff-Baym equations for the nonequilibrium many-body Green’s function of open and interacting systems out of equilibrium. An important feature of the method is that it takes full account of electronic correlations and embedding effects in the presence of time-dependent external fields, while at the same time satisfying the charge conservation law. The method further extends the Meir-Wingreen formula to the time domain for initially correlated states. We study the electron dynamics of a correlated quantum wire attached to two-dimensional leads exposed to a sudden switch on of a bias voltage using conserving many-body approximations at Hartree-Fock, second Born and GW level. We obtain detailed results for the transient currents, dipole moments, spectral functions, charging times, and the many-body screening of the quantum wire as well as for the time-dependent density pattern in the leads, and we show how the time dependence of these observables provides a wealth of information on the energy level structure of the quantum wire out of equilibrium. For moderate interaction strengths the second Born and GW results are in excellent agreement at all times. We find that many-body effects beyond the Hartree-Fock approximation have a large effect on the qualitative behavior of the system and lead to a bias-dependent gap closing and quasiparticle broadening, shortening of the transient times and washing out of the step features in the current-voltage curves.

Figure 1

(Color online) Sketch of the transport setup. The correlated central region (C) is coupled to semi-infinite left (L) and right (R) tight-binding leads via tunneling Hamiltonians ${\mathbf{H}}_{\alpha \text{C}}$ and ${\mathbf{H}}_{\text{C}\alpha }$, $\alpha =\text{L},\text{R}$.

Figure 2

(Color online) Keldysh contour $\gamma$. Times on the upper/lower branch are specified with the subscript $\mp$.

Figure 3

Diagrammatic representation of the many-body approximations for ${\mathbf{\Sigma }}_{\text{CC}}^{\text{MB}}$.

Figure 4

(Color online) The imaginary part of the lesser Green’s function ${\mathcal{G}}_{\text{CC},HH}^{<}\left(t_1,t_2\right)$ of the central region in molecular orbital basis corresponding to the HOMO level of the central chain. Bias voltage $U=1.2$, HF approximation.

Figure 5

(Color online) The imaginary part of the mixed Green’s function ${\mathcal{G}}_{\text{CC},HH}^{\rceil }\left(t,\tau \right)$ of the central region in molecular orbital basis. Bias voltage $U=1.2$, HF approximation.

Figure 6

(Color online) Transient currents flowing into the right lead for the HF, 2B, and GW approximations with the applied bias $U=0.8$ (three lowest curves) and $U=1.2$.

Figure 7

(Color online) Spectral functions $A\left(\omega \right)$ for HF (uppermost plot), 2B (middle plot), and GW (bottom plot) approximation with the applied bias $U=0.8$ (solid line) and $U=1.2$ (dashed line). A vertical dashed line that intersects the $\omega$ axis at the chemical potential $\mu =2.26$ is also displayed.

Figure 8

(Color online) Real-time evolution of the spectral function $A\left(T/2,\omega \right)$ for the HF (left panel) and the 2B approximation (right panel) for an applied bias of $U=1.2$. On the horizontal axis the time $T$ and on the vertical axis the energy $\omega$.

Figure 9

(Color online) Transient right current $I_{\text{R}}\left(U,t\right)$ as a function of applied bias voltage and time in the HF (left panel) and 2B (right panel) approximations.

Figure 10

(Color online) Spectral function $A\left(\omega \right)$ for the HF (left panel) and 2B (right panel) approximation, as a function of the bias voltage. For the 2B approximation the spectral functions for bias voltages until $U=0.6$ were divided by a factor 30 (blue lines in the figure)

Figure 11

(Color online) Dipole moment of the central region as a function of time for bias $U=1.2$. The inset shows the absolute value of the Fourier transform of the dipole moment.

Figure 12

(Color online) Imaginary part of the trace of the screened interaction $W^{<}\left(t_1,t_2\right)$ in the GW approximation.

Figure 13

(Color online) Snapshots of the density in left lead within the HF approximation after the switch on of a bias of $U=1.2$. On the horizontal axes the transverse dimension of the lead (nine rows wide, with the site connected to the chain in the center). The vertical axis extends to ten layers deep. Upper panel left: Initial density, Upper panel right: density at time $t=1.7$, Lower left panel: density at time $t=3.6$, Lower right panel: density at time $t=10$. The upper color bar refers to the initial density in the upper left panel. The lower color bar refers to the remaining panels.

Figure 14

(Color online) Tight-binding system for finite 2D leads connected to a central scattering region.