Modelling topological surface states in rhombohedral graphene

本理论项目的目的是模拟菱形堆叠多层石墨烯表面态的电子特性，这是不寻常的，因为它们的拓扑性质会在费米能级附近产生平坦的简并电子带。它们的起源可以通过注意到菱形石墨烯的晶格与描述一维拓扑绝缘体的Su-Schrieffer-Heeger （SSH）模型所描述的具有交替键强度的一维链的晶格之间的相似性来理解。这意味着菱形石墨烯的两个表面态（每个表面态都位于最外层的两个石墨烯层中的一个附近）几乎是简并的，这使得固定在块带隙中间的平坦带可以用于广泛的面内波矢量。该项目的目的是在考虑对称性破缺效应的情况下对它们的性质进行建模，发展量子力学的分析技能和数值模拟技能。科学导师由Edward McCann（导师）和Neil Drummond（联合导师）组成。单层和双层石墨烯的手性导致了令人着迷的电子传输特性，包括通过倏逝模式传输、克莱因隧穿和有效负折射率聚焦。我们的目标是模拟RG中的传输，并了解这些效应是否会发生，即确定有限尺寸样品的电导，并确定势垒和n-p结的传输特性。正如石墨烯中通常的那样，我们将考虑平面内方向的传输，但我们也将考虑垂直传输（即垂直于平面），其中两个简并的表面状态在空间上被有效的绝缘体分开，将像隧穿晶体管一样（这种传输先前已在石墨烯异质结构中实现，但与其他材料如氮化硼作为屏障）。我们期望输运性质非常敏感地依赖于费米能级的位置，因为接近平坦的表面状态。我们将考虑外部扰动（如电场、面内磁场和应变）如何影响输运性质。我们将进行解析和数值计算，并比较它们的结果。解析方法将使用有效的连续哈密顿量和波浪匹配技术。对于数值计算，我们将使用实空间紧密绑定模型和使用Landauer-Buttiker形式的模型传输。这种方法的优势在于它是一种基于实空间格的方法，因此，它不依赖于平移不变性：它可以为具有无序或有限大小效应的系统建模。在项目的后期阶段，我们将研究使用Keldysh Green函数方法来开发一种形式主义来描述线性响应之外的非平衡效应。

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.