Localisation and finite-size effects in graphene flakes

TL;DR

This paper demonstrates that disordered graphene exhibits electron localization at all energies for sufficiently large samples, clarifying the role of finite-size effects and sample dimensions in observing localization phenomena.

Contribution

The study reveals that graphene flakes of about 200 nm or larger are necessary to observe Anderson localization, highlighting the importance of finite-size effects in experimental and theoretical analyses.

Findings

Electron states are localized at all energies for disorder as small as 1/6 of the band width.

Samples smaller than 200 nm show extended or critical states due to large wave function extent.

Finite-size effects explain divergent results in previous studies of graphene localization.

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 1/6 of the band width of clean graphene. We clarify that, in order for this Anderson-type localisation to be manifested, graphene flakes of size of approximately 200 x 200 nm^2 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.