Modelling topological surface states in rhombohedral graphene

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The aim of this theoretical project is to model the electronic properties of surface states in rhombohedrally-stacked multilayer graphene which are unusual because their topological nature gives rise to flat, degenerate electronic bands near the Fermi level. Their origin can be understood by noting the similarity between the lattice of rhombohedral graphene and that of a one-dimensional chain with alternating bond strengths as described by the Su-Schrieffer-Heeger (SSH) model which describes a one-dimensional topological insulator. This means that two surface states in rhombohedral graphene (each of which is localised near one of the outer two graphene layers) are almost degenerate, giving flat bands fixed to the middle of the bulk band gap for a broad range of in-plane wavevectors. The aim of the project is to model their properties taking into account symmetry breaking effects, developing both analytical skills in quantum mechanics as well as numerical modelling skills. The scientific supervisors consist of Edward McCann (supervisor) and Neil Drummond (co-supervisor).Chirality in monolayer and bilayer graphene leads to fascinating electronic transport properties including transmission via evanescent modes, Klein tunnelling and focusing with an effective negative refractive index. We aim to model transport in RG and to understand whether generalisations of these effects occur, i.e. to determine the conductance of a finite-size sample and to determine transmission properties at a potential barrier and at an n-p junction. As is usual in graphene, we will consider transport in the in-plane direction, but we will also consider vertical transport (i.e. perpendicular to plane) in which the two degenerate surface states, spatially separated by what is effectively an insulator, will act like a tunnelling transistor (such transport has previously been realised in graphene heterostructures but with other materials such as boron nitride acting as a barrier). We expect the transport properties to depend very sensitively on the position of the Fermi level because of the nearly-flat surface states. We will consider how external perturbations (e.g. electric field, in-plane magnetic field, and strain) affect the transport properties.We will perform analytic and numerical calculations and compare their results. The analytic approach will use effective continuous Hamiltonians and employ wave-matching techniques. For numerical calculations, we will use a real-space tight-binding model and model transport using the Landauer-Buttiker formalism. The strength of this approach is that it is a real-space lattice-based method and, as such, does not rely on translational invariance: it can model systems with disorder or finite-size effects. In the later stages of the project, we will investigate the use of the Keldysh Green's function approach to develop a formalism to describe non-equilibrium effects beyond linear response.