Electronic properties of graphene superlattices : transport and superconductivity

In this thesis, we undertake a theoretical investigation of the properties of graphene-based heterostructures. It has long been understood that substrates, such as hexagonal boron nitride (hBN), can affect the electronic properties of graphene [1]. This can be understood by considering what is known as the moiré superlattice: periodic structures with long wavelengths compared to that of graphene, formed by differences in lattice constants or relative twists between graphene and a substrate. We introduce and derive models describing superlattices formed by placing graphene on a variety of substrates, and use these to explain and predict the electronic behaviour of such systems. We first consider the superlattice produced by placing monolayer graphene on a hBN substrate, which is known to produce a spectrum of subbands at low energies [2-4]. By examining the Chern numbers of these subbands we find that the system is topologically nontrivial. We then introduce a real-space model of the superlattice to study the band spectra in finite-width superlattice nanoribbons, and find that in the presence of the superlattice flat bands localised at the edges of the nanoribbons become dispersive. This results in formerly gapped graphene spectra becoming gapless, offering an explanation for the low resistivity commonly measured in graphene/hBN devices [5]. Combined with the topological nature of the subbands, we attribute this behaviour to a valley Hall effect. These results were published in Ref. [6]. We then study the electronic properties of bilayer graphene on a hBN substrate by extending the model used previously. Superlattices formed by bilayer graphene with small twist angles between the layers have attracted much attention for their unusual superconducting properties [7, 8]. We focus on the case of aligned bilayer graphene which has also been found to exhibit superconductivity [9], the superlattice forming due to the substrate instead. We find that the electronic spectrum of this system contains isolated flat bands, the positions of which can be tuned using an external bias across the bilayer. These flat bands are expected to be susceptible to strong interactions, which may be responsible for inducing the measured superconductivity. We also analyse the behaviour of graphene in contact with a type-II superconductor. Here the superlattice is formed by the Abrikosov lattice of magnetic vortices present in type-II superconductors. We derive a model describing the effect of the Abrikosov vortex lattice on charge carriers in graphene, and find that this lifts the degeneracies of the Landau levels found in graphene in the presence of a magnetic field, producing an electronic spectrum with many flat bands in close proximity. On including the superconductivity induced in graphene by proximity to the type-II superconductor, we find indications that these bands become superconducting. As a result, we propose that graphene on a type-II superconductor is a potential platform to readily study superconductivity in graphene systems.