Abstract
We show that electron states in disordered graphene, with an onsite potential that induces inter-valley scattering, are localised for all energies at disorder as small as of the band width of clean graphene. We clarify that, in order for this Anderson-type localisation to be manifested, graphene flakes of size or larger are needed. For smaller samples, due to the surprisingly large extent of the electronic wave functions, a regime of apparently extended (or even critical) states is identified. Our results complement earlier studies of macroscopically large samples and can explain the divergence of results for finite-size graphene flakes.
Highlights
States in a 2D system are marginally localised even for small disorder and d = 2 is the lower critical dimension of the Anderson transition in time-reversal–invariant systems. While this statement is true in general, it has been shown that the situation is more complex when correlations in the disorder [7,8] or many-body interactions [6,9] have to be taken into account. Even without these additional factors, the 2D situation remains challenging since the extent of the localised states for weak disorder can become much larger than the available system sizes, which might lead to results of a feigned extended behaviour
It was found that strong disorder leads to localisation at E = 0 [13,14], while disorder that does not lead to intervalley mixing does not [15,16]
Many, mainly numerical, results have indicated the existence of localisation that is unusually weak at E = 0 [20,21,22,23,24] or close to E = 0 [25]
Summary
Even without these additional factors, the 2D situation remains challenging since the extent of the localised states for weak disorder can become much larger than the available system sizes, which might lead to results of a feigned extended behaviour. Recent discussions of results at E = 0 [26,27] or for strong disorder at E 0 [28] indicate complete localisation for disorder with inter-valley mixing, in agreement with the earlier studies [14,15,16] and a true metal-insulator transition has only been observed in hydrogenated graphene [29,30]. Increasing the size M 2 of the graphene samples leads to increasing localisation lengths around E ≈ 0.25, with energy
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