In this chapter, we have computed the analytical expressions for the edge modes using a tight-binding Hamiltonian for 2D Dirac (graphene) and semi-Dirac systems. We solved these equations numerically to observe the nature of the edge states of the ribbon. For the Dirac case (graphene), we see the exponentially decaying amplitude of the wave functions which indicates that the edge states are localized at the edges and decay into the bulk. We also
compute the band structure, where we observe the flat bands between a finite range of mo- mentum that corresponds to the two edge modes and are gapless in bulk. The transport prop- erties are explored by calculating the charge conductance. The conductance spectra show a plateau at a non-zero value 2e2/h near the zero of the Fermi energy due to the presence of the edge modes. To continue the study in graphene, we have also looked into the bilayer graphene nanoribbon. The analytic results show that the behavior of the edge states of a bilayer graphene is quite different than that of a monolayer graphene. For the band structure, the four bands correspond to four edge states which implies that there exist two edge modes per edge. Moreover, the charge conductance spectrum shows plateaus at 4e2/h for bilayer graphene near the zero of the Fermi energy. For the semi-Dirac case, the edge states are more localized at both edges and decay faster inside the ribbon than the Dirac one. It can also be seen from the band dispersion that the edge modes are completely separated from the bulk ones. Also, the flat band gets enhanced in the case of semi-Dirac one. The conductance plateau is quantized with the same value (2e2/h) as observed for the Dirac case, but the width of the plateau has diminished. For all the cases, we have shown LDOS results that support well our edge states derived analytically.
3 Transport properties of a Kane-Mele Dirac and semi-Dirac nanoribbon
3.1 Kane-Mele model in a Dirac system . . . . 55
3.1.1 Edge states: Analytical expressions 56
3.1.2 Results and discussion 59
3.1.3 Intrinsic SOC 60
3.1.4 Rashba SOC 62
3.1.5 Intrinsic and Rashba SOC 63
3.1.6 Bilayer system 66
3.2 Kane-Mele model in semi-Dirac system . . . . 79
3.2.1 Edge states: Analytical expressions 80
3.2.2 Results and discussion 82
3.2.3 Intrinsic SOC 82
3.2.4 Intrinsic and Rashba SOC 83
3.3 Summary . . . . 86
Recently, the study of the effects of spin-orbit coupling (SOC) has become one of the most important topics, especially in systems that do not have surface or bulk inversion symmetry. Some of these systems assumably have exciting prospects of spintronic appli- cations [118–120] where spin current can be used to transmit dissipationless information.
On the other hand, it has been realized that SOC can lead to a new quantum state of matter that supports gapless edge (or surface) states protected by the TRS, while the bulk remains insulating. It is named as a topological insulator (TI) [22,23] or more specifically, as the QSH insulator. There may be different kinds of SOC present in the system due to different physical origins. Mainly, two kinds of SOCs are thought to be relevant in the context of graphene, namely the intrinsic SOC and the Rashba SOC [15,29]. Kane and Mele [15,29]
predicted that a QSH state could be observed in the presence of a complex next-nearest neighbor hopping, the so-called intrinsic SOC, which triggered an enormous study on topo- logically nontrivial electronic materials [18,22,23,121]. Unfortunately, the QSH phase in pristine graphene is still not observed experimentally owing to its vanishingly small intrinsic SOC strength (typically ∼ 0.01−0.05 meV) [122,123], whereas, in strong SOC materials, such as CdTe/HgTe quantum wells, the QSH phase has been observed . From differ- ent first-principles studies [122,123], the strength of the intrinsic SOC emerges that is of the order of 10−3 meV. This value is much weaker than the value predicted by Kane-Mele compared to what is needed to realize the topological phase. Nevertheless, owing to its vast and potential applications as spintronic devices, several experimental studies could yield en- hanced SOC values which are realized by doping with heavy adatoms, such as Indium or Thallium , using the proximity to a three-dimensional topological insulator, such as Bi2Se3[125,126], by functionalization with methyl  etc. Recently, many other 2D ma- terials have been found with prospects of a tunable SOC, such as silicene, germanene, and stanene [128–132] etc. From the first-principles calculation, it is reported that Rashba SOC can be enhanced via doping with 3d or 5d transition-metal atoms [133,134]. Recent obser- vations showed that the strength of the Rashba SOC can also be enhanced up to 100 meV from Gold (Au) intercalation at the graphene-Ni interface . A Rashba splitting of about 225 meV in epitaxial graphene layers grown on the surface of Ni  and a giant Rashba SOC (∼600 meV) from Pb intercalation at the graphene-Ir surface  are also observed in experiments.
There are only very few studies on the effects of SOC in a bilayer graphene so far. In- trinsically, the magnitude of the SOC in a bilayer graphene is about one order of magnitude larger than that in monolayer graphene due to mixing of the π andσ bands via interlayer hoppings (typically∼0.01−0.1 meV) . A bulk energy gap can be opened by breaking the inversion symmetry via the staggered sublattice potential term, and it plays a similar role in a monolayer graphene as that played by the gate bias in a bilayer graphene [139,140].
It has also been studied that the bias voltage may reduce the bulk energy gap induced by
the intrinsic SOC . The topological phases of a bilayer Kane-Mele model have been studied in detail in the presence of both the SOCs [142,143]. The main findings are that a Z2-metallic phase can be achieved with nontrivialZ2invariant, which gives rise to spin heli- cal edge states in the presence of the TRS, whereas a Chern metallic phase can be achieved with nontrivial Chern invariant. The latter gives rise to chiral edge states due to breaking of the TRS by a Zeeman-like coupling term . A stable topological insulator phase can be achieved in gated bilayer graphene in the presence of large Rashba SOC . Further, the study of the band structure reveals that aMexican-hat feature appears in the vicinity of the Dirac points in the presence of SOC and without any bias voltage . Moreover, the conventional charge transport in bilayer graphene has been studied earlier , but a sys- tematic study of charge and spin transport in a spin-orbit coupled bilayer graphene is still new and hence needs to be explored.
In this chapter, we explore the roles of different SOCs in a Kane-Mele Dirac and semi- Dirac nanoribbon and emphasize its various physical properties. We show the effects of SOC on the edge states and the band structure. To see the interplay between these two types of SOC on the band structure and the edge states with a view to understand the data on charge conductances, we consider a Kane-Mele model. We write the Kane-Mele model for a nanoribbon and perform an analytical investigation of the edge states for a few choices of intrinsic SOC and Rashba SOC. We derive the fundamental eigenvalue equations that form the backbone of our results for the discussion on the edge states and the band structure.
On the other hand, the transport properties are investigated in order to understand charge conductance. For our numerical calculation on the LDOS and conductance, we have used Kwant .