Model of impurity potential

As shown in figure 1, the impurities are randomly distributed with a density n _imp in the nanoribbons. In our model, we assume that the each impurity potential has a Gaussian form

Figure 7. L -dependence of the averaged dimensionless conductance h g i for a zigzag nanoribbon with N = 10, (a) d / a = 1.5 (no inter-valley scattering), (b) d / a = 0.1 (inter-valley scattering). Here u ₀ = 1.0, and n _imp = 0.1. More than 9000 samples with different impurity configurations are included in the ensemble average.

2.3. PCC: absence of Anderson localization

We focus first on the case of LRI using a potential with d / a = 1.5, which is already sufficient to avoid inter-valley scattering. Figure 7(a) shows the averaged dimensionless conductance as a function of L for different incident energies (Fermi energies), averaging over an ensemble of 40000 samples with different impurity configurations for ribbons of width N = 10. The potential strength and impurity density are chosen to be u ₀ = 1.0 and n _imp = 0.1, respectively. As a typical localization effect, we observe that h g i gradually decreases with increasing length L (figure 7). However, h g i converges to h g i = 1 for LRIs (figure 7(a)), indicating the presence of a single PCC. It can be seen that h g i(L) has an exponential behavior as

h g i−1 ∼ exp(− L /ξ) (34)

with ξ as the localization length.

Figure 8. (a) Average conductance h g i as a function of the ribbon length L in the presence of LRIs for several different Fermi energies E. Conductance is almost unaffected by impurities for single-channel transport (E = 0.1,0.2 and 0.3), while it shows a conventional exponential decay for multi-channel transport (E > 0.4). Here, N = 14, n _imp = 0.1 and d / a = 1.5. Ensemble average is taken over 10⁴ samples. (b) The Fermi energy dependence of h g i for LRI. (c) The same as (a) for SRIs. Here, N = 14, n _imp = 0.1 and d / a = 0.1.

We performed a number of tests to confirm the presence of this PCC. First of all, it exists up to L = 3000 a for various ribbon widths up to N = 40 for the potential range (d / a = 1.5). Moreover the PCC remains for LRI with d / a = 2.0,4.0,6.0,8.0, and u ₀ = 1.0, n _imp = 0.1 and N = 10. As the effect is connected with the subtle feature of an excess mode in the band structure, it is natural that the result can only be valid for sufficiently weak potentials. For potential strengths comparable to the energy scale of the band structure, e.g. the energy difference between the transverse modes, the result should be qualitatively altered [81]. Deviations from the limit h g i → 1 also occur, if the incident energy lies at a value close to the change between g = 2 n −1 and 2 n +1 for the ribbon without disorder. This is for example visible in the above calculations for E = 0.4 where the limiting value h g i < 1

(figure 7(a)).

Turning to the case of SRI the inter-valley scattering becomes sizable enough to ensure TRS, such that the perfect transport supported by the effective chiral mode in a single valley ceases to exist. In figure 7(b), the nanoribbon length dependence of the averaged conductance for SRIs is shown. Since SRI causes inter-valley scattering for any incident energy, the electrons tend to be localized and the averaged conductance decays exponentially, h g i ∼ exp(− L /ξ), without developing a perfect conduction channel.

In this subsection, we have completely neglected the effect of electron–electron interaction, which may acquire the energy gap for non-doped zigzag nanoribbons at very low temperatures accompanying edge spin polarization [7, 21, 45]. In such a situation, a small transport gap will appear near E = 0. Since the edge states have less Fermi instability for the doped regime, the spin polarized states might be less important for the doped system.