1 Topology and applications of 2D Dirac and semi-Dirac materials

Abstract

Two dimensional (2D) Dirac materials, such as graphene, hold promise of being useful in energy storage, and thus have merged as candidates that are worth exploring through the last couple of decades. In this chapter, we mainly focus on three aspects of these materials, namely, the electronic properties, via computing the band structure, the topological properties through the topological invariants, and the prospects of these 2D materials for spintronic applications, via studying the spin polarized transport. All of these properties are correlated, and hence warrant a thorough discussion. Further, in order to ascertain whether a band deformation induces noticeable effects on the electronic, topological and spintronic properties, we have considered a 2D semi-Dirac system, that does not have Dirac cones, however the conduction and the valence bands touch at an intermediate to the Dirac points in the Brillouin zone. From our studies, we infer that the behaviour of these semi-Dirac systems is quite distinct from their Dirac counterpart. Finally, in order to have noticeable spin polarized transport, we use heavy adatoms (such as, Au) on the graphene matrix which enhances the spin–orbit coupling, and thereby propose a mechanism that will ramify on the spintronic applications.

1.1 Introduction

Since the discovery of graphene in 2004 [1] by exfoliation technique, the world has realized a perfect two dimensional (2D) material consisting of C atoms placed on the vertices of a honeycomb lattice. Later on C has been replaced by P (phosphorene) [2, 3], Ge (germanium) [4], Si (silicon) [5, 6], transition metal dichalcogenide (TMDC), boron nitride (BN) etc., and similar 2D structures have been realized in experiments. During the last couple of decades, scientists and engineers have discovered a few thousands of atomically thin materials. Some of these actually have buckled structures and have added to the richness of their properties, along with enhanced interest into the research of 2D materials. Several other 2D materials, for example, transition metal oxides, Carbides/Nitrides (which have a special name Mxene), dichalcogenides etc. have been discovered and studied with renewed intensity. The astonishing features of all these materials are extremely high mobility, electrical conductivity, elastic properties, high packing density, surface properties etc. The latter is particularly significant because of the electrocatalytic and photocatalytic utilities, especially in the infrared region. However, the property that stands out especially from the societal perspective is their utility in electrochemical energy storage applications [7]. Also the flexibility of developing various kinds of heterostructure with 2D materials makes them feasible candidates for the design of integrated circuit based devices.

However, these 2D materials suffer from several shortcomings, because of which their applicabilities are limited. For example, in experiments, the limitations of using graphene as electrodes arise owing to its ion-accessible surface area due to stacking one layer over the other. Thus a large quantity of electrolytes is often required to compensate for this. To take remedial measures, 2D materials with porous structures have emerged as a game changer with increased exposure of their surface area to electrolytes and mesoscale sized pores for enhanced charge transfer. The versatility of such technical developments has led to improved stabilities of energy storage devices, including supercapacitors, lithium-ion batteries etc. Currently, more effort is being given to acquire precise control on the number of porous layers and their thickness to suit the intended applications. Moreover, Janus1 materials in the form of nanosheets, with distinct properties of each face, have led to formation of completely new composites [8].

A very large volume of literature has been devoted to energy storage and conversion, such as design and fabrication of Li/Na-ion batteries, supercapacitors for their large power density and long life cycle, electrocatalytic applications and so on. Simultaneously, the condensed matter physicists across the globe have contributed to the understanding of the electronic and the topological properties of graphene, and its derivatives2. Surprisingly, and interestingly, the low energy dispersion of graphene is relativistic (massless Dirac like) in nature. However, the relativistic feature is restricted to the energy varying linearly with the wave vector; the velocity of the electrons is still limited by the Fermi energy. A further dichotomy arises with the effective mass of the electrons, which according to standard definitions of solid state physics can be written as, m*=1ℏ2(∂2ϵ∂k2)−1 becomes infinity, while the corresponding Hamiltonian is that of a massless particle. However, that is reconciled with a modified definition, namely, m*=ℏ2k(∂ϵ∂k)−1 , which yields sensible results. Further, the low energy dispersion shows band touching3 at six locations of the Brillouin zone (BZ) of which two are distinct (four others can be obtained by adding and subtracting proper reciprocal lattice vectors), that are called as the Dirac points. In standard language, these two Dirac points are called as valleys, and scattering of electrons from one valley to another is strictly prohibited.

With quantum Hall systems unveiling the first realization of a topological insulators, 2D Dirac materials, and in particular graphene was conceptualized as a prospective candidate for showing topological properties. A Berry phase of ±π at the Dirac points added fuel to the speculations. However, the minimum requirement for obtaining a topologically trivial state is that a spectral gap needs to open up at the Dirac points. The question that follows immediately afterwards is whether such a gap is accompanied by conducting edge states. If the answer is yes, then we have a topological insulator.

In a completely different context, Haldane [9] envisioned realizing quantum Hall effect in absence of an external magnetic field, and hence without the requirement of formation of Landau levels. Breaking the time reversal symmetry (TRS) was thought to be supremely important for realizing the quantum Hall effect, which was done by Haldane by introducing a complex direction dependent second neighbour hopping. The scenario is equivalent to a staggered field produced at the vertices and at the centre of the honeycomb structure. This, of course, gives graphene the states of a topological insulator, which is also known as the Chern insulator because of its non-zero Chern number, which is the topological invariant in this case.

A further development took place with the intervention of Kane and Mele [10, 11], where they have repaired the loss of time reversal invariance by explicitly considering the spin degrees of freedom of the electrons, such that spin-↑ electrons experience a flux +π, while that by the spin-↓ is −π. The situation does not support a quantum Hall state, since the TRS is intact, however, gives rise to an important, and the second type of topological insulators, called as the quantum spin Hall insulator. The realization of such insulators, along with spin–orbit coupling (SOC) raised the possibility of using graphene as spintronic devices. Quite frustratingly, the Rashba SOC, which is the main ingredient for such an application is too low and hence impedes graphene’s application as a spintronic device. While HgTe-CdTe quantum well structures [12] have conveniently demonstrated realization of the quantum spin Hall phase, we still work on the prospects of enhancing the Rashba SOC in graphene via inclusion of heavy adatoms.

Before we embark on the Hall effect in graphene, let us review the electronic properties. We give a pedagogical description of the tight binding bandstructure and compute the location of the Dirac points in the BZ, where the low energy dispersion looks relativistic. We further give definitions of all the quantities, such as, Berry phase, Berry curvature, Chern number and the ℤ2 invariants (the last two being the topological invariants that characterize the topological properties).

1.2 Basic electronic properties of...