Modeling time-dependent current through electronic open channels using a mixed time-frequency solution to the electronic equations of motion

A nonequilibrium Green’s function model based on time-dependent perturbation theory is developed to propagate electronic structure and molecular conductance of extended electrode-molecule-electrode nanostructures. In this model, we use the two-time variable nature of the Kadanoff-Baym equations of motion to formulate a mixed time-frequency representation for the electronic density expressed by the appropriate Green’s function $\left(G^{<}\right)$. This allows for the dynamical treatment of open systems. Furthermore, highly informative time-dependent Wigner distributions are used to shed light on the features of dynamical observables, such as electron current. Calculations, performed on model systems, resolve the dynamic current into direct and alternating components. The direct current is due to electronic open channels near the Fermi level and the alternating response is due to interference fringes from a superposition of extended states. We analyze the transient conductance with respect to the fundamental system’s parameters, the effect of bound states, and the conductance driven by laser-induced coherence affected by detuning due to an applied dc bias. The amplitude of the alternating transient current can be adjusted by reshaping the bias pulse or by controlling the electronic coupling terms. Bound states may yield a persisting oscillating response depending on their relative electronic densities. In the analysis we utilize the calculated highly informative time-dependent current distributions.

Figure 1

The Keldysh contour.

Figure 2

(Color online) One-dimensional wire composed of an array of hopping sites. The central region corresponds to the device, which can be set to different site to model doping. The coupling terms within the device are denoted as ${\beta }_d$, coupling of the device bulk ${\beta }_l$ and $\beta$ for the bulk region. Corresponding $s$ terms for the overlap are defined as well.

Figure 3

(Color online) TD conductance $\left[I\left(t\right)\right]$ due to three different pulses was studied. All pulses correspond to an analytical TD potential bias function, where the same rate is used for the turn on (elapsing 15 fs). They differ in the rate of the turn-off and the length of the pulse at the target bias. An $I\left(t\right)$ for a case with bound states is also provided: (i) Black dotted line: slow rate of turn off (15 fs). (ii) Red line with plus symbols: shorter pulse at target. (iii) Green line with circles: fast turn-off pulse (elapsing 1 fs). (iv) Brown fine and solid line: like (i) with two additional bound states. (Bottom) The potential ramping profiles are illustrated.

Figure 4

(Color online) Time span at the target bias study: (i) short (50 fs), (ii) medium (100 fs), and (iii) long (200 fs) pulse. Left panel: $I\left(t\right)$ curves. Three right panels: Current distribution with differing color maps between the lower and upper panels.

Figure 5

(Color online) Switch off rate study: (i) slow switching (30 fs), (ii) medium (15 fs), and (iii) quick (1 fs) rates. Left panel: $I\left(t\right)$ curves. Three right panels: Current distributions with differing color maps between the lower and upper panels.

Figure 6

(Color online) The two-site system with three different broadening (base) factors. (Left) The electronic DOS. (Right) The associated $I\left(t\right)$ curves.

Figure 7

(Color online) The two-site system including two additional bound states. The interference involving the bound-state results with a persistent current oscillation: (a) the electronic density of states. (b) The $I\left(t\right)$ curve. Wigner plot of the current distribution: (c) with two stable bound states, (d) with one stable bound state, and with one unoccupied.

Figure 8

(Color online) The effect of the strength of electronic coupling $\left({\beta }_d\right)$ on TD conductance: (a) the electronic density of states. (b) The $I\left(t\right)$ curve. (c) Three Wigner plots of the current distribution for the system including the two additional bound states with the different ${\beta }_d$ each with a different color scaling scheme.

Figure 9

(Color online) Coherence-driven current of a two-state system with and without a steady dc bias. (Left) The DOS and the $I\text{−}t$ of the two-state system. Current distribution of: (up) tuned system and (low) dc detuned, and (center) one state occupied and (right) two states occupied