The conductive helical edge states in topological insulators (TIs) have been lauded over the last decade as a means towards the development of low-energy electronic and spintronic devices. Unfortunately, development of TI devices have been hampered by issues such as impurities and difficulty in fabrication, meaning that they are no closer to replacing traditional semiconductor platforms than when they were discovered over a decade ago. However, while the low carrier mobility of edge states in TI devices may always preclude their use in the electronics industry, the non-trivial topological nature of these edge states mean they are also a novel playground for the development and observation of exotic physics and emergent phenomena. Indeed, in recent years focus has transitioned from finding utility in bare TI devices to investigating combination of topological protection with other effects, such as magnetism and superconductivity. In particular, the introduction of magnetism into TIs leads to a variety of unique phenomena and exotic quasiparticles not observed in conventional material systems, such as the quantum anomalous Hall effect (QAHE) and Majorana fermions. The study and development of devices based on such phenomena are not only interesting from the perspective of fundamental physics, but also propose practical applications in spintronics and quantum computing. This thesis presents work on the interplay between magnetism and topology and discusses the technological significance of such an interaction.

Early stage research into the introduction of magnetism into TIs focused on doping TIs with magnetic adatoms, or engineering magnetic insulator/TI interfaces to induce magnetism into the TI edge states through a proximity effect. However, as was the case with bare TI devices, fabricating devices based on such platforms is not without difficulty. Inhomogeneities in the concentration of magnetic adatoms in doped samples, and the very weak interaction between the TI and magnetic insulator in proximity based devices mean that interesting effects and phenomena are only observable at temperatures on the range of 10's of millikelvins. In recent years, the discovery of intrinsic magnetic TIs has attracted an intense amount of research activity, as they provide a novel platform to investigate both topology and magnetism in van der Waals materials without suffering from many of the shortcomings of magnetically doped TIs or TI/magnetic insulator heterostructures. Furthermore, the antiferromagnetic coupling between layers gives rise to interesting layer dependent phenomena, where the electronic structure of samples is dependent on the parity of the number of layers, i.e. whether there are an even or odd number of layers. Literature has mainly focused on (\text{MnBi}{2}\text{Te}{4}), whose order along the (\hat{z}) axis means that it can host the QAHE and axionic insulator state. However, the family of intrinsic magnetic TIs is extensive and can host different magnetic configurations.

In particular, in this thesis we have investigated in-plane magnetisation as a means to engineer flat-bands in the energy dispersion relation of topologically non-trivial materials. When considering antiferromagnetic interlayer coupling, we uncover an interesting dependence of the electronic dispersion and the local density of states on the parity of the number of layers. Furthermore, we demonstrate that magnetic textures at the surface of magnetic topological materials lead to spin-polarised flat-bands. In addition, the infinite mass quasiparticles occupying these flat-band states are strongly localised around magnetic domain walls.

The means of engineering flat-bands developed in this thesis may have great technological significance in electronic and spintronic applications. For instance, we propose that the system discussed in chapter \ref{chap:AFMTI_dws} could be used in re-configurable magnetic memory, however they may also prove useful in the investigation of exotic physics and emergent phenomena. It is well known that the high density of electronic states in flat-bands leads to many-body interactions gaining greater importance in the overall dynamics of a system. As a result, flat-band systems can exhibit strong electronic correlations, which can result in the emergence of interesting phenomena such as non-BCS superconductivity, and other non-Fermi liquid phases, and charge or spin fractionalization. While the investigation of strongly correlated physics is beyond the scope of this thesis, the results presented here nevertheless demonstrate that magnetic TI systems are novel playgrounds for the investigation of emergent phenomena that could advance our understanding of fundamental physics in condensed matter systems.