In the past two decades theoretical and experimental efforts have brought forth numerous advances in the field of topological matter. Notably, theoretical predictions like the Quantum Spin Hall phase in mercury/cadmium telluride quantum well structures have subsequently been confirmed experimentally. Another example is the prediction of topological superconductivity in systems combining Zeeman splitting, spin-orbit coupling and conventional superconductivity which has been tentatively confirmed in various experiments. These efforts are motivating further research on how to realize systems hosting topological phases and are bringing forward the possibility of engineering physical devices using topological states of matter.
One of the first examples of the engineering of a topological phase out of topological domain-wall states was recently reported in two experiments. Also following a theoretical prediction, the experiments used graphene nanoribbons of alternating width to create an effective Su-Schrieffer-Heeger chain composed of the topological domain-wall states arising at the interfaces of nanoribbon regions of different width. In this thesis we provide a theoretical toy model that captures the basic physics of these systems. We show the possible end state configurations of the model and how the end states of the engineered topological phase can interact with those of the underlying topological system.
With most of the theoretical foundations already in place, the research has focused on finding strategies to realize topological phases in a more reliable, stable manner and provide unequivocal signatures of the sought-for topological states. Within this context, magnetic adatom chains on superconducting substrates have attracted attention as they are predicted to realize topological superconductivity and host Majorana quasiparticles at their ends. Due to an enhanced lateral extension of the Yu-Shiba-Rusinov wave functions of the magnetic adatoms in two dimensional superconductors, substrates such as NbSe$_2$ have been proposed. This enhancement increases the coupling between the adatoms leading to a more robust topological phase. In this doctoral thesis we show that the effects of the charge-density modulation present in NbSe$_2$ must be taken into account for an optimal engineering of topological superconductivity.
The signatures of the presence of Majorana quasiparticles at the ends of a topological superconducting phase are sometimes compatible with those of other (conventional) states. These states can have near-zero energies and thus be confused with the desired zero-energy Majorana quasiparticles. In this dissertation, we develop a theory of photon-assisted Andreev reflections resonantly enhanced by subgap states. Our theory is in excellent agreement with a recent experiment which shows deviations from the Tien-Gordon model, widely successful in explaining photon-assisted processes. Of relevance in its own right, our theory also provides a technique to measure near-zero energies of the subgap states. Indeed, separate threshold conditions for electron and hole tunneling processes lead to a higher sideband multiplicity in the case of small but non-zero subgap state energies. Experiments with tunable photon energies can benefit from this fact to measure the subgap state energy.