Due to quantum confinement, nanoscale superconductivity exhibits richer phenomena than bulk superconductivity. This will allow us to artificially design the electronic properties by changing the size and geometry of the superconductor, leading to the desired control and enhancement of superconductivity. It provides us with a great potential for applications in ultrafast, power-saving electronic devices such as superconducting transistors and single-photon detectors. However, the interplay between superconductivity and quantum confinement effect has not been fully understood yet.

In the present thesis, we theoretically investigated several aspects of nanoscale superconductivity by solving self-consistently the Bogoliubov-de Gennes equations. The topics that are covered range from vortex states under the influence of quantum confinement to the electronic structure in various nano-structures. The density of states obtained in this thesis can be compared with results from STM/STS experiments.

In Chapter. 3, we studied vortex states under the influence of quantum confinement effect in nanoscale superconductors. We found that nanoscale superconductors with coherence length comparable to the Fermi wavelength, that the shape resonances of the order parameter results in an additional contribution to quantum topological confinement - leading to unconventional vortex configurations. Our results for a square geometry in the quantum limit reveal a plethora of asymmetric, giant multi-vortex, and vortex-antivortex structures, stable over a wide range of parameters and which are very different from those predicted by the Ginzburg-Landau theory. Experimentally, these states can be accessed through STM measurements. Additional, competing interactions in the quantum limit for the bound states are different from those for the vortices, so that the conventional picture of a vortex bound to the lowest energy states does not hold. Instead we predict that the maxima in LDOS of the lowest energy states will be observed between vortices and near surfaces. These peculiar phenomena are a consequence of the strong quantum confinement, which induces spatial oscillations in the order parameter. Finally, we proposed that the system consisting of a graphene flake in contact to a superconducting film could be a good candidate to observe experimentally these novel vortex states.

In Chapter. 4, we investigated the vortex states in a nanoscale superconducting square for different sizes , parameters , and temperatures. It is an extension of the previous study presented in Chapter. 3. We found that the inhomogeneous pattern of the order parameter as induced by the quantum confinement effect gives an additional contribution to the competing effects that determine the vortex configurations. Because of this reason, some unconventional vortex states such as asymmetric, edge-parallel and vortex-antivortex states are found as the ground state of our nanoscale system. These were never seen in the Ginzburg-Landau approach. Since the inhomogeneous pattern of the order parameter strongly depends on and the size , the vortex ground states and the magnetic field values of the phase transition are very sensitive to changes in these parameters, which is a direct consequence of the quantum size effect. Finally, we found that, in the quantum limit, nano-size superconductors favor vortex-antivortex molecules while disfavoring giant vortex states, which could be used to observe experimentally the antivortex.

In Chapter. 5, we studied the effect of non-magnetic impurities on the local density of states and the transport properties in superconducting nanowires with diameter comparable to the Fermi wavelength (which is less than the superconducting coherence length). Such impurities have very little effect on the bulk properties of conventional superconductors. However, as the dimensionality is reduced, the effect of impurities becomes more significant. By applying the BdG theory to the case of NbSe2, we uncovered several regimes in which the impurity affects the superconducting properties of the nanowire in different ways. First, depending whether the nanowire is in the resonant or off-resonant regime, the order parameter will show slow or fast oscillations away from the impurity, respectively. This is due to the different nature of the quasi-particles involved in the formation of Cooper pairs, i.e. small or large momentum. Additionally, the impurity has a strong position-dependent effect on the Josephson critical current with opposite behavior in the resonant and off-resonance cases. These effects could be used to investigate the nature of the superconducting condensate and the scattering of the various subbands on the impurity.

In Chapter. 6, we investigated the Tomasch effect on the electronic structure in nanoscale superconductors. The Tomasch effect is due to quasiparticle interference as induced by a nonuniform superconducting order parameter, which results in oscillations in the density of states at energies above the superconducting gap. Here, the Tomasch effect is induced by an inhomogeneous order parameter appearing due to quantum confinement. We found that the Tomasch effect couples degenerate electron and hole states above the superconducting gap leading to additional pairs of BCS-like Bogoliubov-quasiparticles. When the energies of the paired states are far from the Fermi level, the pair states show pseudo-gap-like structures in the DOS. When they are close to the Fermi level, the pair states result in modulated wave patterns in the local density of states. All these are due to the inter-subband electron-hole coupling.

The Tomasch effect is strongly related to the geometrical symmetry of the system and the symmetry, parity and spacial variation of the order parameter. For the nanobelt, the Tomasch effect only leads to two-subband quasiparticles interference processes. For a nanowire, it also results in three-subband quasiparticle interference processes leading to a unique energy band structure. The study on the Tomasch effect can help us to understand how the inhomogeneous order parameter affects quasiparticle states.