Topological insulator (TI) thin films with surface magnetism are expected to exhibit a quantized anomalous Hall effect (QAHE) when the magnetizations on the top and bottom surfaces are parallel, and a quantized topological magnetoelectric effect (QTME) when the magnetizations have opposing orientations (axion-insulator phase) and the films are sufficiently thick. We present a unified picture of both effects that associates deviations from exact quantization of the QTME caused by finite thickness with nonlocality in the sidewall current response function. Using realistic tight-binding model calculations, we show that in Bi2Se3 TI thin films, deviations from quantization in the axion-insulator phase are reduced in size when the exchange coupling of tight-binding model basis states to the local magnetization near the surface is strengthened. Stronger exchange coupling also reduces the effect of potential disorder, which is unimportant for the QAHE but detrimental for the QTME, which requires that the Fermi energy lie inside the gap at all positions.

Breaking time-reversal symmetry in three-dimensional topological insulator thin films can lead to different topological quantum phases, such as the Chern insulator (CI) phase and the axion insulator (AI) phase. Using first-principles density functional theory methods, we investigate the onset of these two topological phases in a trilayer heterostructure consisting of a Bi₂Se₃ (Bi₂Te₃) TI thin film sandwiched between two antiferromagneticMnTe layers. We find that an orthogonal exchange field from the MnTe layers, stabilized by a small anisotropy barrier, opens an energy gap of the order of 10 meV at the Dirac point of the TI film. A topological analysis demonstrates that, depending on the relative orientation of the exchange field at the two interfaces, the total Chern number of the system is either C = 1 or C = 0, characteristic of the CI and AI phases, respectively. Nontopological surface states inside the energy-gap region, caused by the interface potential, complicate this identification. Remarkably though, the calculation of the anomalous Hall conductivity shows that such nontopological surface states do not affect the topology-induced transport properties. Given the size of the exchange gap, we estimate that gapless chiral edge states, leading to the quantum anomalous Hall effect, should emerge on the sidewalls of these heterostructures in the CI phase for widths > 200 nm. We also discuss the possibility of inducing transitions between the CI and the AI phases by means of the spin-orbit torque caused by the spin Hall effect in an adjacent conducting layer.

Over the past two decades, the discovery of topological Dirac materials, such as topological insulators and semimetals, has defined a new paradigm in condensed matter physics and established new platforms for realizing spin electronic devices and quantum computing. The novel quantum properties of these materials, which are mainly due to the presence of strong spin-orbit coupling, are invariant under topological transformations, making them more resilient to structural disorder, electromagnetic and thermal fluctuations in comparison to the conventional silicon-based materials. Their topological nature, characterized by topological invariants, depends crucially on the global symmetries of the system, with time-reversal symmetry and inversion symmetry being the most important ones. If these symmetries are broken by external fields, it is possible to cause transitions between different topological phases. One of the main challenges in this field is to determine the physical conditions under which different non-trivial topological phases can be realized, detected, and manipulated.

The present thesis aims at investigating the onset of different topological phases and associated quantum phenomena in topological material nanostructures such as thin films and nanoribbons where time-reversal symmetry is broken by the presence of magnetism. Specifically, using theoretical and computational methods based on atomistic tight-binding models and density functional theory, we have addressed four problems: (i) the possibility of realizing both the Chern insulator and the axion insulator phase in the same heterostructure consisting of a TI thin film sandwiched by two antiferromagnetic layers; (ii)  the origin of the deviations from exact quantization of the elusive topological magneto-electric effect in TI thin films in the axion insulator phase. For this purpose we have introduced a novel approach based on a non-local side-wall response that treats the quantum anomalous Hall effect and the topological magnetoelectric effect on the same footing; (iii) signatures in quantum transport of different topological phases in two-dimensional (graphene) and three-dimensional (Bi₂Se₃) TI nanoribbons under different configurations of the exchange field; (iv) non-magnetic- and magnetic-impurity-induced topological phase transitions in topological semimetals, with the prediction of a coexisting Dirac-Weyl mixed-phase under some given conditions. The results of this theoretical work in part elucidate some fundamental issues of magnetic topological materials, but also give indications for the experimental realization of quantum phenomena which have important technological applications.

Using first-principles density functional theory calculations, combined with a topological analysis, we have investigated the electronic properties of Cd₃As₂ and Na₃Bi Dirac topological semimetals doped with nonmagnetic and magnetic impurities. Our systematic analysis shows that the selective breaking of the inversion, rotational, and time-reversal symmetry, controlled by specific choices of the impurity doping, induces phase transitions from the original Dirac semimetal to a variety of topological phases such as topological insulator, trivial semimetal, nonmagnetic and magnetic Weyl semimetal, and Chern insulator. The Dirac semimetal phase can exist only if the rotational symmetry Cn with n>2 is maintained. One particularly interesting phase emerging in doped Cd₃As₂ is a coexisting Dirac-Weyl phase, which occurs when only inversion symmetry is broken while time-reversal symmetry and rotational symmetry are both preserved. To further characterize the low-energy excitations of this phase, we have complemented our density functional results with a continuum four-band k⋅p model, which indeed displays nodal points of both Dirac and Weyl types. The coexisting phase appears as a transition point between two topologically distinct Dirac phases but may also survive in a small region of parameter space controlled by external strain.

Using the non-equilibrium Green’s function method and the Keldysh formalism, we study the effects of spin–orbit interactions and time-reversal symmetry breaking exchange fields on non-equilibrium quantum transport in graphene armchair nanoribbons. We identify signatures of the quantum spin Hall (QSH) and the quantum anomalous Hall (QAH) phases in nonequilibrium edge transport by calculating the spin-resolved real space charge density and local currents at the nanoribbon edges. We find that the QSH phase, which is realized in a system with intrinsic spin–orbit coupling, is characterized by chiral counter-propagating local spin currents summing up to a net charge flow with opposite spin polarization at the edges. In the QAH phase, emerging in the presence of Rashba spin–orbit coupling and a ferromagnetic exchange field, two chiral edge channels with opposite spins propagate in the same direction at each edge, generating an unpolarized charge current and a quantized Hall conductance  . Increasing the intrinsic spin–orbit coupling causes a transition from the QAH to the QSH phase, evinced by characteristic changes in the non-equilibrium edge transport. In contrast, an antiferromagnetic exchange field can coexist with a QSH phase, but can never induce a QAH phase due to a symmetry that combines time-reversal and sublattice translational symmetry.

We study thermally induced spin resolved current in a zigzag silicene nanoribbon when the left and right leads are respectively affected by ferromagnetic (FM) and anti-ferromagnetic (AFM) exchange fields (FM/normal/AFM junction). We show that pure spin current is generated due to the leads temperature difference and the junction can work as a spin Seebeck diode. The pure spin current can be easily controlled by a perpendicular electric field and the junction, in this case, can work as a spin current switch. In addition, we study the effect of a single vacancy and show that the vacancy can slightly destroy the pure spin current property which leads to induce a weak spin polarized current. In the presence of both vacancy and electric field, current with high and tunable spin polarization can be achieved.

Regarding the spin field effect transistor (spin FET) challenges such as mismatch effect in spin injection and insufficient spin life time, we propose a silicene based device which can be a promising candidate to overcome some of those problems. Using non-equilibrium Green's function method, we investigate the spin-dependent conductance in a zigzag silicene nanoribbon connected to two magnetized leads which are supposed to be either in parallel or anti-parallel configurations. For both configurations, a controllable spin current can be obtained when the Rashba effect is present; thus, we can have a spin filter device. In addition, for anti-parallel configuration, in the absence of Rashba effect, there is an intrinsic energy gap in the system (OFF-state); while, in the presence of Rashba effect, electrons with flipped spin can pass through the channel and make the ON-state. The current voltage (I–V) characteristics which can be tuned by changing the gate voltage or Rashba strength, are studied. More importantly, reducing the mismatch conductivity as well as energy consumption make the silicene based spin FET more efficient relative to the spin FET based on two-dimensional electron gas proposed by Datta and Das. Also, we show that, at the same conditions, the current and   ratio of silicene based spin FET are significantly greater than that of the graphene based one.

We study spin-dependent electron transport properties of zigzag silicene nanoribbons in the presence of anti-ferromagnetic exchange field using a nonequilibrium Green׳s function method. Applying a transverse electric field, spin splitting can take place and the silicene nanoribbon can work as a spin filter. The spin polarization is calculated and it is shown that the spin filtering is perfect and the spin states of electrons are fully coherent. The spin direction of transmitted electrons through the silicene filter can be easily controlled by changing the transverse electric field direction. Using Hubbard model, we take into account the electron–electron interaction and we find that although this interaction causes some changes in the electron conductance, it has no destructive effect on spin filtering properties. The effect of a single vacancy on electron transport is also investigated and it is found that, the vacancy causes to decrease the electron conductance; however, the spin-dependent properties remain the same. The vacancy in the near of the edges of nanoribbon has less destructive effect on electron conductance than that in the middle.

Using non-equilibrium Green's function, we study the spin-dependent electron transport properties in a zigzag silicene nanoribbon. To produce and control spin polarization, it is assumed that two ferromagnetic strips are deposited on the both edges of the silicene nanoribbon and an electric field is perpendicularly applied to the nanoribbon plane. The spin polarization is studied for both parallel and anti-parallel configurations of exchange magnetic fields induced by the ferromagnetic strips. We find that complete spin polarization can take place in the presence of perpendicular electric field for anti-parallel configuration and the nanoribbon can work as a perfect spin filter. The spin direction of transmitted electrons can be easily changed from up to down and vice versa by reversing the electric field direction. For parallel configuration, perfect spin filtering can occur even in the absence of electric field. In this case, the spin direction can be changed by changing the electron energy. Finally, we investigate the effects of nonmagnetic Anderson disorder on spin dependent conductance and find that the perfect spin filtering properties of nanoribbon are destroyed by strong disorder, but the nanoribbon retains these properties in the presence of weak disorder.