Abstract: Novel quantum phases of matter and developing practical control over their characteristics is one of the primary aims of current condensed matter physics. It offers the potential for a new generation of energy, electronic and photonic technologies. Among all the newly found phases of matter, topological insulators are novel phases of quantum matter with fascinating bulk band topology and surface states protected by specific symmetries. For example, at the boundary of a strong topological insulator and a trivial insulator, metallic surface states appear that are protected by time-reversal symmetry. As a result, the bulk continues to be insulating, while the surface can support exotic high-mobility spin-polarized electronic states. Since there is no such thing as a clean system, impurities and other disorders are always present in materials. Even while impurities appear to be unfavorable to a system at first look, doping the host system with impurities allows us to engineer different electronic properties of systems, such as the Fermi level or electron density. Because of the symmetry protected metallic states in topological insulators, charge transport responds distinctively to magnetic and non-magnetic impurities. This doctoral dissertation explores how the longitudinal charge transport in magnetic topological thin films and the anomalous Hall effect on the surface of 3D magnetic topological insulators is influenced by point-like and randomly distributed dilute magnetic impurities. We are interested in how charge transport in these systems responds to the orientation of the magnetization orientation and how this response evolves based on the system's main characteristics, such as the magnitude of the Fermi level or gate voltage. Because topological insulators have a strong spin-orbit coupling, the interaction between conducting electrons and local magnetic impurities is very anisotropic. We will show that this anisotropy even enhances when magnetic topological thin films are exposed to a substrate or gate voltage. Therefore, to properly capture this anisotropy in charge transport calculations, we rely on a generalized Boltzmann formalism together with a modified relaxation time scheme. We show that magnetic impurities affect the charge transport in topological insulators by inducing a transition selection rule that governs scatterings of electrons between various electronic states. We see that this selection rule is highly sensitive to the spin direction of the magnetic impurities as well as the position of the Fermi level. According to this selection rule and depending on the position of the Fermi level, two different transport regimes are realized in magnetic topological thin films. In one of these regimes, our findings show that a dissipation less charge current can be generated. In other words, even if there are many magnetic impurities in the system, electrons do not notice them and, remarkably, conduct charge without dissipation. Outside this regime, the charge transport is always dissipative and its sensitivity to the spatial orientation of the magnetic impurities can be effectively modulated by a substrate or gate voltage. In this doctoral thesis, we also explore the anomalous Hall effect (AHE) on the surface of 3D magnetic topological insulators. The AHE is generated by three mechanisms: the intrinsic effect (owing to a nonzero Berry curvature), the side jump effect, and the skew scattering effect. They compete to dominate the AHE in distinct regimes. Analytically, we calculate the contributions of all three mechanisms to the scattering of massive Dirac fermions by magnetic impurities. Our results reveal three transport regimes based on the relative importance of the engaged mechanisms. The identification of these three distinctive transport regimes can assist experimentalists in achieving a regime in which each contribution is dominant over the others, allowing them to measure them separately. Typically, this is not feasible empirically since the total value of the experimentally observed AHE conceals the specific information of each of the three contributions. Based on our analytical calculations, we prove that the AHE can change sign by varying the orientation of the surface magnetization, the concentration of impurities, and the location of the Fermi level, which is consistent with previous experimental findings. In addition, we show that by suitably adjusting the given parameters, any contribution to the AHE, or even the entire AHE, can be turned off. For example, in a system with in-plane magnetization, one can turn off the AHE by pushing the system into the completely metallic regime. Furthermore, we demonstrate that any contribution to the AHE, or even the whole AHE, can be turned off by appropriately changing the provided parameters. For example, in a system with in-plane magnetization, the AHE can be turned off by pushing the system into the fully metallic regime.

Abstract: This thesis presents, in its first part, an investigation on the trembling motion of wave packets known as zitterbewegung (ZBW), in multilayer graphene, as well as in moiré excitons in twisted MoS2/WSe2 hetero-bilayers. In the last few decades, the dynamics of wave packets has been subject of many theoretical and experimental studies in various types of systems such as semiconductors, superconductors, crystalline solids and cold atoms. The discovery of graphene and moiré excitons in twisted hetero-bilayers, brought two new platforms for the investigation on time evolution of wave packets and possible observation of ZBW. This trembling motion was first theoretically predicted by Schrödinger for wave packets describing particles that obey the Dirac equation. This is exactly the case of low energy electrons in graphene, as well as of moiré exciton in twisted MoS2/WSe2 under an external applied electromagnetic field. ZBW in multilayer graphene was studied both analytically and computationally, respectively, through the Green's function and split-operator methods. In this system, it is found that ZBW depends not only on the wave packet width and initial pseudospin polarization, but also on the number of layers. Furthermore, the analytical and numerical methods proposed here allow to investigate wave packet dynamics in graphene systems with an arbitrary number of layers and arbitrary potential landscapes. For moiré excitons in twisted MoS2/WSe2 hetero-bilayers, it is shown that, analogously to other Dirac-like particles, this system also exhibits ZBW when under a perpendicular applied field. In this case, the ZBW presents long timescales that are compatible with current experimental techniques for exciton dynamics. This promotes the study of the dynamics of moiré excitons in van der Waals heterostructures as an advantageous solidstate platform to probe zitterbewegung, broadly tunable by gating and inter-layer twist angle. In the second part of this thesis, a study into graphene plasmonic in van der Waals heterostructure (vdWhs) are treated in a linear response framework within the Random Phase Approximation and with support of the quantum electrostatic heterostructure (QEH), a DFT-based method. Since Dirac plasmons in graphene are very sensitive to the dielectric properties of the environment, it is possible to explore this property to probe the structure and composition of van der Waals heterostructures (vdWh) placed underneath a single graphene layer. In this way, one can achieve a layer sensitivity of a single layer and differentiate between different TMDs for heterostructures thicker than 2 layers. As a consequence of this, study, the hybridization of Dirac plasmons in graphene with phonons of transition metal dichalcogenides (TMDs), when the materials are combined in so-called van der Waals heterostructures (vdWh) forming surface plasmon-phonon polaritons (SPPPs) are also investigated. It was found that it is possible to realize both strong and ultrastrong coupling regimes by tuning graphene’s Fermi energy and changing TMD layer number.