Theoretical and experimental investigation of the metal-insulator transition in disordered anti-dot graphene

We study the transport behavior of anti-dot graphene both theoretically and experimentally, where the term 'anti-dot' denotes the graphene layer to be nanostructured with a periodic array of holes. It has been shown that the electronic band structure of the anti-dot graphene can be described by a 4 by 4 effective Hamiltonian (Pan J et al 2017 Phys. Rev. X. 7 031043) with a gap around the Dirac point, attendant with a 0 to pi variation of the Berry phase as a function of energy, measured from the band edge. Based on the diagrammatic method analysis and experiments, we identify an energy-dependent metal-to-insulator transition (MIT) in this two-dimensional (2D) system at a critical Fermi energy e (c), characterized by the divergence of the localization length in the Anderson localization phase to a de-localized metallic phase with diffusive transport. By measuring the conductance of square samples with varying dimension and at different Fermi energies, experiments were carried out to verify the theory predictions. While both theory and experiment indicate the existence of a 2D MIT with similar localization length divergence exponent, the values of the critical energy e (c) and that of the localization length do not show quantitative agreement. Given the robust agreement in the appearance of a 2D MIT, we attribute the lack of quantitative agreement to the shortcomings in the theoretical model. The difficulties in addressing such shortcomings are discussed.