Modelling topological surface states in rhombohedral graphene

这个理论项目的目的是模拟菱面体堆积的多层石墨烯中表面态的电子性质，这种表面态的不同寻常的拓扑性质导致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菱面体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相似之处来理解。这意味着，菱面体石墨烯中的两个表面态(每个表面态都位于外部两个石墨烯层中的一个附近)几乎是简并的，为大范围的面内波矢提供了固定在主体带隙中间的平带。该项目的目的是在考虑对称破缺效应的情况下对它们的性质进行建模，同时发展量子力学的分析技能和数值建模技能。科学主管由爱德华·麦肯(主管)和尼尔·德拉蒙德(联合主管)组成。单层和双层石墨烯的手性导致了迷人的电子传输性质，包括通过消逝模式的传输，克莱因隧穿和有效的负折射率聚焦。我们的目标是模拟RG中的输运，并了解是否发生这些效应的一般化，即确定有限大小样品的电导，并确定势垒和n-p结处的传输特性。就像在石墨烯中常见的那样，我们将考虑平面内方向的输运，但我们也将考虑垂直输运(即垂直于平面)，其中由有效的绝缘体在空间上分隔的两个简并表面态将起到隧道晶体管的作用(这种输运以前在石墨烯异质结构中实现，但其他材料，如氮化硼作为势垒)。我们预计，由于近平的表面态，输运性质将非常敏感地依赖于费米能级的位置。我们将考虑外部扰动(如电场、面内磁场和应变)对输运性质的影响。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，并使用波匹配技术。对于数值计算，我们将使用实空间紧束缚模型和使用Landauer-Buttiker形式的模型输运。这种方法的优点是它是一种基于实空间晶格的方法，因此不依赖于平移不变性：它可以对具有无序或有限尺寸效应的系统进行建模。在该项目的后期阶段，我们将研究使用凯尔德什-格林函数方法来发展一种形式主义来描述超越线性响应的非平衡效应。

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.