We observed significant deviations in the transport properties due to the anisotropic band distribution of the semi-Dirac system compared to the results known for Dirac materials. 1.1 (Color online) The longitudinal and the Hall resistances (ρxx and ρxy) are shown as a function of the magnetic field.

General perspectives

## Topological phases of matter: quantum Hall and quantum spin Hall phases . 6

Dirac system (graphene) 10

### Landauer-Büttiker formalism 14

After the discovery of the quantum Hall effect (QH), Haldane showed that the necessary condition for observing the QH effect is not the presence of a magnetic field, but the breaking of the time-reversal symmetry (TRS), which leads to the quantum anomalous Hall effect (QAH). The forward Green's function can be calculated with respect to the backward Green's function via GA=(GR)†.

Landauer formula used for computing transport

## Dirac and semi-Dirac systems

When t2=t or 2t, we have a 3D tight binding band structure for Dirac or semi-Dirac, as shown in Figure 2.2.3), shows that the dispersion is linear (Dirac-like) along the y-direction, while the dispersion along of the x direction is quadratic (non-relativistic), the combination of which results in half-Dirac dispersion.

## Tight-binding model of Dirac nanoribbon

### Edge states and band structure 35

To confirm the behavior of the edge states, we calculated the local density of states (LDOS) near the Fermi energy. Now we can use a momentum representation of the electron operator due to the periodicity in the x-direction, which is, . To ascertain the effects of the edge modes on the conductivity properties of a ZGNR, we further calculated the conductivity of zigzag graphene as shown in Fig.

The value of the conductance depends on the number of transverse modes in the system. Using the above boundary condition mentioned in Eq. induction method, we finally obtain the following matrix equations for the amplitudes of the wave function at A and B sublattices as, To have the essence of the edge mode on the band dispersion, we have also plotted the band structure for N = 100 in Fig.

## Summary

The conductance spectra show a plateau with a non-zero value of 2e2/h near the zero Fermi energy due to the presence of edge modes. Analytical results show that the behavior of the edge states of a bilayer graphene is quite different from that of a single layer graphene. Additionally, the charge conduction spectrum shows plateaus at 4e2/h for bilayer graphene near zero Fermi energy.

The latter gives rise to chiral edge states due to the breaking of the TRS by a Zeeman-like coupling term [143]. Furthermore, the study of the band structure reveals that a Mexican-hat feature appears in the vicinity of the Dirac points in the presence of SOC and without any bias voltage [145]. We write the Kane-Mele model for a nanoribbon and perform an analytical investigation of the boundary conditions for some choices of intrinsic SOC and Rashba SOC.

## Kane-Mele model in a Dirac system

Edge states: Analytical expressions 56

### Edge states: Analytical expressions 80

All the atoms in the zigzag edges belong to the same sublattice as shown in Fig. We have plotted the probability density of the edge states with the strength of the intrinsic coupling, λS O=0.1 as shown in Fig. To explore further, we have plotted the charge conductance, G as a function of Fermi energy, E in the presence of the intrinsic SOC as shown in Fig.

This indicates that the topological properties of the QSH phase are destroyed in the presence of intra- and interlayer Rashba coupling together with intrinsic intralayer SOC. It is interesting to note that the edge modes can only be seen in the presence of the intrinsic SOC, while the inclusion of Rashba interlayer coupling together with the Rashba intrinsic SOC destroys the edge modes. For λR=0.1 and λ⊥R =0, the band crossing in the kxrange as previously observed in the presence of only internal SOC.

The behavior of the plateau confirms that it is separated from the bulk even in the presence of internal SOC. We show the existence of topologically protected edge states due to the presence of parity and the TRS Hamiltonian.

## Hofstadter butterfly spectra in semi-Dirac and Dirac system

Thus the tight-binding Hamiltonian in the presence of the perpendicular magnetic field can be written in terms of mandn (wherein increases along the x-direction and increases along the negative s-direction) (see Fig. 4.1.2) where the summni is hmni over the nearest neighbors. In the following we solve the above equation (Eq. 4.1.3)) to observe the influence of the magnetic field on the fractal spectrum as well as the LL spectrum. The consequences of the different LLs properties for the Dirac and the semi-Dirac systems are described below.

We can see that the gap closes completely around zero energy with the flat band continuing strongly2=t. The central LLs (around zero energy) have the characteristic square-root behavior as a function of the magnetic field in graphene, while the upper and lower LLs at low field values vary linearly with the magnetic field. The reason is clearly that electrons far from zero energy no longer remain Dirac fermions.

Landau levels

## Summary

The clear LL behavior for the semi-Dirac dispersion leads to a different quantization than a Dirac-like case. As mentioned earlier, semi-Dirac dispersion has been observed in a few layers of black phosphorene through the in situ deposition of potassium atoms in experiments [182]. Since phosphorene can be considered as a realistic material possessing semi-Dirac properties, it is necessary to continue QH studies on the semi-Dirac systems.

As discussed above, the energy dispersion of the phosphorus is similar to that of the semi-Dirac systems, and it is therefore likely that other properties also show similar characteristics. In this chapter, we have investigated the transport properties in the presence of a magnetic field for a semi-Dirac system using a tight-binding Hamiltonian on a honeycomb lattice. The Hall conductance in a semi-Dirac system shows the standard quantization, namely σxy∝nas compared to the previously observed anomalous quantization, that is, σxy∝2(n+1/2) for a Dirac system.

## Transport properties

### Methodology 101

The components of the dc conductivity tensor (ω→ 0 limit of the ac conductivity) for non-interacting electrons are given by the Kubo-Bastin formula [70,187] which can be written as whereT is the temperature, µ is the chemical potential, vαis The component of the velocity operator , A is the surface area of the sample, f(ε) is the Fermi-Dirac distribution and G±(ε,H)=. To get a feel for the evolution of single-particle properties between the Dirac and semi-Dirac limits, we have plotted the DOS in the absence of a magnetic field for various This gives a faster convergence of the Hall conductance in the limit of large magnetic fields.

In the presence of the magnetic field, the DOS consists of peaks of discrete energy levels (LLs) as shown in Fig. However, we get broad DOS peaks at lower values of the magnetic field (B = 50T), which is especially visible for the semi-Dirac case due to the small energy separation between the LLs (less than 3 meV). Below we present a systematic exploration of the MO transport for a semi-Dirac system in the visible frequency range.

## Keldysh formalism

The effect is characterized by the angle by which the plane of polarization is rotated and is called the Faraday angle, which is calculated for the semi-Dirac and Dirac cases. 6.1.5) The following vector potential can be used to introduce a static magnetic field and a uniform electric field. In real space, this commutator gives the matrix element of the Hamiltonian connecting the two locations i and j multiplied by the distance vector di j between them. di j will be a well-defined quantity in the case of a periodic boundary condition if we define this quantity as the distance between neighbors, rather than the difference between positions .

Therefore, the ˆhoperators can be defined in the position space by multiplying the elements of the Hamiltonian matrix by the required product of the difference vectors. The current operator can be calculated from the Hamiltonian, via ˆJα=−Ω1∂H. where Ω denotes the volume of the sample). Backward and forward Green's functions, Dirac deltas, and generalized rate operators are written in a position basis expanded in a truncated set of Chebyshev polynomials [237] (see Appendix A).

## Results

### Zero magnetic field 115

Graphs with a more moderate value of the magnetic field (say 100 T) are shown in the inset of the figure. In particular, the peaks for the real and imaginary parts of the MO conductivity are modified compared to the Dirac case. The other peaks (third, fourth and so on) are similar to the second peak; come from a pair of transitions from −nto n+ 1 and −(n+ 1) to us shown in fig.

In the lower panel of Fig. 6.5, we have plotted the real part of the MO Hall conductance, Re(σxy) as a function of photon energy for both the half-Dirac (t2=2t) and Dirac (t2= t) cases at B=400T (denoted by red curve) as shown in the main frame. In the case of the Dirac system (t2=t), Im(σxy) shows a single sharp positive peak at the same energy corresponding to the peak of Re(σxy) as shown in Fig. Due to the different properties of the LL, the MO conductances show some distinct features for the semi-Dirac case compared to the Dirac one.

## Summary of the work done

In the third chapter, we have calculated analytical expressions for zero-energy edge modes for a Dirac and semi-Dirac nanoribbon in the presence of intrinsic and Rashba SOC within the framework of the Kane-Mele model. We have shown the analytical expressions for the zero-energy edge modes for a half-Dirac nanoribbon in the presence of both SOCs. We observe two identical spectra with gaps in the Hofstadter butterfly spectrum for the semi-Dirac case, which is absent for the Dirac case.

In the fifth chapter, we investigated the quantum Hall properties of a semi-Dirac nanoribbon in the presence of an external magnetic field using a close-fitting model on a honeycomb lattice and compared it with the Dirac case. In the sixth chapter, we investigated the magneto-optical transport properties of a semi-Dirac system in the presence of an external magnetic field using Keldysh formalism. The MO conductivity shows different characteristics for the semi-Dirac case compared to the Dirac case.

## Future prospects

Shi, Strong electron-hole symmetric rashba spin-orbit coupling in graphene/monolayer transition metal dichalcogenide heterostructures, Physical Review B96nr. Basu, Adatoms in graphene nanoribbons: spintronic properties and the quantum spin half-phase, Materials Research Express4no. Mokrousov, Electrically tunable quantum anomalous hall effect in graphene decorated with 5d transition metal adatoms, Phys.

Miranda, Spatial variation of the giant spin-orbit effect induces electron confinement in graphene on Pb islands, Nature Physics 11 no. Guinea, Spin-orbit coupling in bilayer graphene and in graphite, New Journal of Physics 12 no. Sen, Electronic dynamics in graphene with spin-orbit couplings and periodic potentials, Journal of Physics: Condensed Matter 29 no.