Time-dependent Landauer-Büttiker formula: Application to transient dynamics in graphene nanoribbons

In this work, we develop a time-dependent extension of the Landauer-Büttiker approach to study transient dynamics in time-dependent quantum transport through molecular junctions. A key feature of the approach is that it provides a closed integral expression for the time dependence of the density matrix of the molecular junction after switch-on of a bias in the leads or a perturbation in the junction, which in turn can be evaluated without the necessity of propagating individual single-particle orbitals or Green's functions. This allows for the study of time-dependent transport in large molecular systems coupled to wide-band leads. As an application of the formalism, we study the transient dynamics of zigzag and armchair graphene nanoribbons of different symmetries. We find that the transient times can exceed several hundreds of femtoseconds while displaying a long-time oscillatory motion related to multiple reflections of the density wave in the nanoribbons at the ribbon-lead interface. This temporal profile has a shape that scales with the length of the ribbons and is modulated by fast oscillations described by intraribbon and ribbon-lead transitions. Especially in the armchair nanoribbons there exists a sequence of quasistationary states related to reflections at the edge state located at the ribbon-lead interface. In the case of zigzag nanoribbons, there is a predominant oscillation frequency associated with virtual transitions between the edge states and the Fermi levels of the electrode. We further study the local bond currents in the nanoribbons and find that the parity of the edges strongly affects the path of the electrons in the nanoribbons. We finally study the behavior of the transients for various added potential profiles in the nanoribbons.

Figure 1

Schematic of the quantum transport setup: a noninteracting central region $C$ is coupled to an arbitrary number of leads.

Figure 2

Transport setup of a (zigzag) graphene nanoribbon connected to metallic leads: contacts to leads are between doubly colored bonds; bridge (explained in text) is shown by the green cutting line. The structure of the leads is shown for illustrative purposes. Voltage profile is shown below the structure.

Figure 3

Time-dependent bond currents through ribbons of varying length: (a) aGNR [fixed width $W=1.5$ nm (13)], (b) zGNR [fixed width $W=1.6$ nm (8)], and (c) the corresponding Fourier transforms (zGNR is offset for clarity); the inset shows the long-time behavior of the currents for $L=10.5$ nm in (a) and (b). [The line colors and styles correspond to those in (a) and (b).]

Figure 4

Time-dependent bond currents through ribbons of varying width: (a) aGNR (fixed length $L=4.1$ nm), (b) zGNR (fixed length $L=4.1$ nm), and (c) the corresponding Fourier transforms (zGNR is offset for clarity); the inset shows the long-time behavior of the currents for $W=3.6$ nm in (a) and $W=3.7$ nm in (b), respectively. [The line colors and styles correspond to those in (a) and (b).]

Figure 5

Time-dependent bond currents through fixed-size ribbons with varying bias voltage (a) aGNR [$W=1.5$ nm (13), $L=4.1$ nm], (b) zGNR [$W=4.1$ nm (8), $L=4.1$ nm], and (c) the corresponding Fourier transforms (zGNR is offset for clarity); the inset shows the long-time behavior of the currents for $V_{\mathrm{sd}}=10.6$ eV in (a) and (b).

Figure 6

First transients of the time-dependent current through ribbons of varying length divided by the number of bonds in the bridge. The horizontal axis is scaled by the length of the corresponding ribbon.

Figure 7

Temporal snapshots of spatial charge densities and bond currents along aGNRs. Upper panel shows the fully symmetric aGNR and lower panel transversally asymmetric aGNR. Left panel shows the snapshots corresponding to the first maximum in the transient current and the right panel shows the ones corresponding to the first minimum. The charge densities are calculated as the difference from the ground-state density (color map). The bond currents are drawn as solid arrows where the width of the arrow indicates the relative strength of the current.

Figure 8

Temporal snapshots as in Fig. 7 but for zGNRs and at different times.

Figure 9

Time-dependent bond currents through a 4-zGNR (length $0.7$ nm and width $0.9$ nm) with fixed bias voltage $V_{\text{sd}}/2=3.5$ eV and with varying potentials: (a) linear potential profile, (b) sinusoidal potential profile, (c) the corresponding Fourier transforms (sinusoidal is offset for clarity).

Figure 10

Nonequilibrium spectral functions of the studied zGNR with varying potential: (a) linear potential profile, and (b) sinusoidal potential profile.