**4. Conclusions**

We have employed the recently developed TD-LB formalism to compute the two-time current correlation functions for disordered GNRs. This methodology is a fast and accurate way of addressing mesoscopic quantum transport phenomena out of equilibrium as it is well-supported by the underlying nonequilibrium Green's function theory [70]. By our analysis, we confirm that the current cross-correlation is a good measure of electron traversal time. We find that the traversal time scales roughly linearly with the length of the GNRs, and that the traversal time also depends strongly on the GNR orientation. In general, the results presented in our work indicate that the traversal time can be most sensibly connected to the group velocity of noninteracting electrons passing through the graphene nanoribbon. The distance between the first maxima in the cross-correlations corresponds to the time for an electron with velocity *v* ≈ 1 nm/fs to cross the ribbon. This velocity is consistent with the value of the Fermi velocity in graphene *v*F = 3*tCa*/2*h*¯ where *a* = 1.42 Å is the carbon–carbon distance [49]. This can be related to the Büttiker–Landauer time for tunnelling [4] through a rectangular barrier as the time it would take a particle with velocity *v*F to traverse the barrier.

We found that disorder in GNRs increases the traversal times, in general, and ultimately destroys the whole picture of coherent information transfer over the GNR junction when the disorder-induced scattering is strong. The "rule-of-thumb" character of our findings is summarised in Figure 6. We considered two types of disorder, one that preserves (hopping disorder) and one that breaks (on-site disorder) the chiral symmetry of the GNR. In Figure 6b, we find that the intrinsic operational frequency of the GNR is redshifted for the hopping disorder while, in Figure 6c, we see that it remains roughly unchanged for the on-site disorder. However, in the latter case, the statistical spread of *τ*tr is significantly enhanced as the on-site disorder is increased. To measure the current cross-correlation and extract experimental values for *τ*tr, there exist a range of spectroscopic techniques which relate the field strength of photons emitted from each lead to the current. The zero- and finite-frequency current cross-correlations can then be extracted from these current measurements [66,71].

**Figure 6.** Electron traversal times estimated from the distance between the first maxima in the current cross-correlation (cf. Equation (4) and Figure 2): (**a**) no disorder, fixed voltage *VL* = −*VR* = *tC*/2, varying length *L*; (**b**) fixed length *L* = 5 nm, fixed voltage *VL* = −*VR* = *tC*/2, varying hopping disorder *w*; and (**c**) fixed length *L* = 5 nm, fixed voltage *VL* = −*VR* = *tC*/2, varying on-site disorder *f* . The error bars are empirically estimated from the cross-correlation peaks as the full width at half maximum.

Our noninteracting approach is sufficient for graphene structures since monolayer graphene devices have been experimentally shown to have ballistic transfer lengths in the range from 100 nm at room temperatures to 1 μm at sub-Kelvin temperatures [2,72]. Even though our approach is limited to noninteracting electrons, we expect the current correlations and traversal times to be similarly related even when dealing with, e.g., electron–electron or electron–phonon interactions [47,73–75]. If perturbation theory could be applied, i.e., when the interaction is weak, current correlation or noise simulations are still feasible to perform in terms of the one-particle Green's function [76–81]. Disordered interacting systems have also been studied using mean-field or density-functional theories [82–84]. Here, out-of-equilibrium dynamics only due to voltage bias was considered but also thermal gradients could be included similarly [46,85–88]. At present, for the case of strong interaction, these approaches cannot ye<sup>t</sup> be extended to realistic device structures since considerably more complicated and numerically expensive methods are required [89].

We also confirmed that the overall picture of electron traversal times is not qualitatively changed by introducing an ac driving voltage compared to the response to a dc drive. Possible quantitative differences in the response signals to an ac drive could be related to signatures of photon assisted tunnelling on traversal time [37,65] but, for now, will be left for future work. On the other hand, it would also be interesting to consider, e.g., a short laser pulse for exciting the system out of equilibrium [90–92] instead of the quench of the voltage bias employed in the present work. These topics will also be addressed more thoroughly in a forthcoming paper.

**Author Contributions:** Conceptualization, M.R. and R.T.; Formal Analysis, M.R., M.A.S. and R.T.; Writing—Original Draft, M.R. and R.T.; Writing—Review & Editing, M.R., M.A.S. and R.T.

**Funding:** This research was funded by the Raymond and Beverly Sackler Center for Computational Molecular and Materials Science, Tel Aviv University (M.R.) and by the DFG Grant No. SE 2558/2-1 through the Emmy Noether program (M.A.S. and R.T.).

**Acknowledgments:** We wish to thank Robert van Leeuwen for productive discussions.

**Conflicts of Interest:** The authors declare no conflict of interest.
