*3.2. The Role of Disorder*

As we have now established general principles for the traversal times, we concentrate our discussion on disordered GNRs of fixed length and fixed applied (time-independent) voltage. In Figure 3, we introduce disorder into the GNR without breaking chiral symmetry. This is done by drawing a random number from a uniform distribution of width *w* around the average hopping matrix value *tC*. We exclude second and third nearest neighbour hoppings, and we consider randomness only in the hoppings, for we wish to preserve particle–hole symmetry [49,69]. We see that the disorder appears to increase the traversal time and also has the effect of decreasing the quality of the signal, so that multiple side-peaks are visible. These are caused by internal reflections induced by the disorder. In addition, the intrinsic resonant frequencies for the disordered GNRs (shown by the Fourier spectra) are red-shifted due to the hopping disorder. This finding is consistent with the idea of reduced operational frequencies for the GNR devices due to disorder.

**Figure 3.** Current cross-correlations for fixed length disordered GNRs (*L* = 5 nm) with fixed voltage *VL* = − *VR* = *tC*/2. A uniformly-distributed disorder *w* is included in the intramolecular hopping energies, which preserves the chiral symmetry.

In Figure 4, we break the chiral symmetry by adding a random term to the on-site energy levels of the GNR. This is also done by drawing a random number from a uniform distribution of width *f* around the zero on-site energy for the pristine GNR. In Figure 4a, we look at the AGNR case, and we observe in this case the deterioration of a clear traversal time signal as the Anderson localisation increases the average dwell time in the interior of the GNR. In Figure 4b, the average traversal time is once again seen to be larger in the ZGNR case. Interestingly, there is a crossover in both GNR configurations as the disorder destroys the coherence of the propagating wave packet around *f* = *tC*/2. In contrast to the case of hopping disorder, here the operational frequencies of the GNR device (shown by the Fourier spectra) remain roughly unchanged for the on-site disorder. This finding could be related to the character of the disorder: While both types of disorder may introduce an effective tunnel barrier around the disordered GNR that the propagating electrons must overcome, the hopping disorder case corresponds to deformation of the lattice geometry, whereas the on-site disorder corresponds to a change in charge neutrality or chemical potential.

**Figure 4.** Current cross-correlations for fixed length disordered GNRs (*L* = 5 nm) with fixed voltage *VL* = −*VR* = *tC*/2. A uniformly-distributed chiral symmetry-breaking disorder *f* is included in the on-site energy.
