In this chapter, we conclude with the main outcome of this thesis and give a brief idea that could be the possible extension of the work done by us. In our thesis, we have largely explored the transport properties of 2D semi-Dirac materials within the framework of a tight- binding model on a honeycomb lattice in the presence of spin-orbit couplings and magnetic fields. The results are compared and contrasted with those for the Dirac systems. Magnetic field plays a significant role in studying the quantum Hall effect, whereas the spin-orbit coupling, which does not violate the time-reversal symmetry, helps us to understand the spin transport and hence should be applicable to spintronic systems. Also, the topological phases of matter of these systems open up new directions for better understanding and reveal many possible applications. We found several exotic quantum phenomena shown by the 2D semi-Dirac systems due to their anisotropic behavior as compared to the Dirac ones.

As a starting point, we have presented the band structure of a 2D Dirac and a semi-
Dirac system using the tight-binding Hamiltonian. The anisotropic behavior of the band
structure reveals that 2D semi-Dirac system is Dirac-like in one direction, whereas in the
other direction it is like that of normal metals. We have shown the evolution of the edge
states as we tune the band structure from being Dirac to semi-Dirac systems. To study the
edge state properties, we derive the eigenvalue equations for the zero-energy edge modes of
Dirac and semi-Dirac systems using nanoribbons. For the Dirac case (such as graphene), we
see that the amplitude of the wave functions decays exponentially indicating the edge states
are highly localized at the edges of the ribbon and there is an exponential decay as one moves
into the bulk. We compute the band structure where the flat bands exist within a certain finite
momentum range that corresponds to the two edge modes. We observe a quantized plateau at
2e^{2}/haround zero of the Fermi energy in the conductance spectra due to the presence of the
edge modes. In the case of bilayer nanoribbon, the behavior of the edge states of a bilayer
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graphene is different than that of a monolayer graphene as seen from the probability density
plot. We calculated the band structure and found that there are four flat bands corresponding
to four edge states. This implies that there exist two edge modes per edge. Instead of a
2e^{2}/hplateau, we observe a plateau at 4e^{2}/hfor bilayer graphene near the zero of the Fermi
energy as compared to a single layer. For the semi-Dirac nanoribbon, the edge states are
more localized at both the edges of the ribbon and decay faster inside the ribbon than its
Dirac counterpart. The band dispersion shows that the edge modes are completely separated
from the bulk one. Also, the flat bands exist and are much more extended in the case of semi-
Dirac ribbons. The conductance plateaus are quantized with the same value, namely 2e^{2}/h,
as observed for the Dirac case. However, the width of the plateau diminishes as compared to
the Dirac case. The LDOS results provide robust support for our results on the edge states
derived analytically for all these cases.

In thethird chapter, we have computed analytical expressions for the zero-energy edge
modes for a Dirac and a semi-Dirac nanoribbon in presence of the intrinsic and the Rashba
SOC within the framework of the Kane-Mele model. We have calculated the band struc-
ture and the conductance for graphene with intrinsic SOC, graphene with Rashba SOC, and
graphene with both SOCs. We re-establish the existence of topologically protected edge
states owing to the presence of parity and time-reversal symmetry of the Hamiltonian. The
system acquires edge states in presence of spin-orbit couplings as observed from band struc-
ture and both analytic and numeric calculation of electron probability densities. The conduc-
tance spectra further show a plateau at a non-zero value (=2e^{2}/h) near the zero of the Fermi
energy. In addition, we derive analytical expressions for the edge modes for a bilayer Kane-
Mele model in presence of both SOCs. An asymmetry in the finite-size ribbon is observed in
presence of intrinsic SOC, which otherwise is absent for a tight-binding model in the band
structure. Moreover, the band structure plots show that the QSH phase can be destroyed
with the inclusion of interlayer RSOC. Further, we observe a plateau at=4e^{2}/hfor pristine
bilayer graphene near the zero of the Fermi energy in the charge conductance spectra, while
it decreases in presence of both intrinsic and Rashba SOCs. Studies on spin transport re-
veal that bilayer graphene should be an appropriate material for spintronic applications. To
make a connection with experiments, we have computed the effective mass and have shown
that it can be tuned by the inclusion of Rashba SOC. Next, we have explored the similar
characteristics for a semi-Dirac nanoribbon as described earlier. We have shown the analytic
expressions for the zero-energy edge modes for a semi-Dirac nanoribbon in presence of both
the SOCs. With the inclusion of intrinsic SOC (Rashba SOC is turned off), we find that the

edge modes are completely separated from the bulk modes, similar to the tight-binding case.

The charge conductance shows the same feature as observed from the band structure. When
we turn on the Rashba SOC, the 2e^{2}/hplateau vanishes at some critical value when both the
parameter values become equal, and the charge conductance becomes zero eventually, which
is also true for higher values of Rashba SOC.

In thefourth chapter, we include a magnetic field and study the different physical prop- erties of a semi-Dirac nanoribbon within the framework of a tight-binding model of a hon- eycomb lattice. We also present our results for the Dirac cases. We consider a semi-infinite semi-Dirac nanoribbon and study the Hofstadter butterfly and the properties of the Landau level spectra. We observe two identical gapped spectra in the Hofstadter butterfly spectrum for the semi-Dirac case, which is absent for the Dirac case. A zero-energy mode is seen in the spectrum in contrast to a Dirac system. The Landau levels become fully dispersive in bulk for moderate values of the magnetic flux, which is not true for the Dirac case.

In the fifth chapter, we have explored the quantum Hall properties of a semi-Dirac
nanoribbon in presence of an external magnetic field using a tight-binding model on a hon-
eycomb lattice and compared it with the Dirac case. We numerically explore the magneto-
transport properties using the Kubo formula based on Kernel Polynomial Method (KPM) via
calculating the longitudinal and Hall conductivities. We observe that the transverse or Hall
conductivity shows standard quantization similar to that of a conventional semiconductor
two-dimensional electron gas. This is sharply in contrast with respect to a Dirac system. The
density of states shows the absence of a zero Landau level peak in the case of a semi-Dirac
system. Owing to the anisotropic dispersion, the longitudinal conductivities in thexxandyy
directions, that is,σ_{xx}andσ_{yy}show distinct behavior.

In the sixth chapter, we have explored the magneto-optical transport properties of a semi-Dirac system in presence of an external magnetic field using Keldysh formalism. For comparisons, we present the results for the Dirac systems. The MO conductivities show various distinct features for the semi-Dirac case as compared to the Dirac one. The real parts of the longitudinal conductivities show a series of absorption peaks owing to the transition between the Landau levels with the semi-Dirac case having additional features owing to an asymmetric distribution of the Landau levels and their densities of states. Further, the MO Hall conductivity shows a pair of extra absorption peaks for the real, as well as the imaginary parts, owing to the additional optical transitions for the semi-Dirac case. The effect of electron filling on the absorption spectra has been studied by tuning the chemical potential between the consecutive Landau levels. The polarization effect of the incident

radiation with MO transport has been explored. Finally, we have studied Faraday rotation which should be possible to be realized in the experiments.