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### Abstract

The topology of the electron wavefunctions in certain band insulators can give rise to novel topological phases. Materials harbouring such topological phases are termed topological insulators (TI). A gapped bulk electronic spectrum, described by a topological invariant, and gapless boundary modes, tend to characterize the non-trivial topology. This work describes a theoretical investigation of the ${Z}_{2}$ topological insulator phase in ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$ and ${\mathrm{Bi}}_{2}{\mathrm{Te}}_{3}$, and the topological crystalline insulator (TCI) phase in SnTe, subject to nanoscale confinement. Specifically, it details the electronic structure, and properties of low-dimensional nanostructures derived from the bulk topological phase. For the bismuth chalcogenides, a first principles methodology is applied to compute the energetics of high-index surfaces, followed by an analysis of the electronic properties of corresponding topological surface state charge carriers. Our calculations find several stable terminations of high-index surfaces, which can be realized at different values of the chemical potential of one of constituent elements. For the uniquely defined stoichiometric termination, the Dirac fermion surface states exhibit a strong anisotropy, with a clear dependence of Fermi velocities and spin polarization on the surface orientation. Non-stoichiometric surfaces undergo self-doping effects, which results in the presence of topologically trivial mid-gap states. These findings guide the construction of ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$ nanostructures of a nanowire (NW) and nanoribbon (NR) morphology. A tight-binding formalism is utilised to study, firstly, the impact of finite-size effects on the electronic spectrum of each nanostructure. Secondly, the effects of confinement on the topological properties of two-dimensional (2D) Dirac fermion surface states. Quantum confinement around each nanostructure perimeter entails the formation of a series of discrete one-dimensional (1D) sub-bands in the bulk gap. An analysis of how the band gap varies as a function of nanostructure dimensions finds that the dependence is highly sensitive to nanostructure morphology. We reveal a clear correspondence between the spin helicity of the 2D surface Dirac cone and the spin properties of the 1D sub-bands. This is exemplified in the real space spin textures of each nanostructure. For the NW morphology, this correspondence gives rise to an energy dependent spin polarization density. Whereas for the NR morphology the presence of two separate surface types results in a more complex relationship. Finally, via a similar tight-binding formalism, we establish how the crystal-symmetry-dependent topological phases of SnTe (001) thin films are exhibited in lower dimensional nanowires. SnTe (001) thin films, defined by either mirror or glide symmetry, realise distinct 2D TCI phases. As the band dispersion of NWs are characterised by which of these symmetry classes they belong to, we subsequently connect the distinctive NW surface states to the respective parent 2D TCI phase. Lastly, we show that the robust topological protection offered by the mirror symmetry protected 2D TCI phase is manifested in robust surface states of NWs of equivalent symmetry.