Quantum confinement in low-dimensional Dirac materials

This thesis is devoted to quantum confinement effects in low-dimensional Dirac materials. We propose a variety of schemes in which massless Dirac fermions, which are notoriously diffcult to manipulate, can be trapped in a bound state. Primarily we appeal for the use of external electromagnetic fields. As a consequence of this endeavor, we find several interesting condensed matter analogues to effects from relativistic quantum mechanics, as well as entirely new effects and a possible novel state of matter. For example, in our study of the effective Coulomb interaction in one dimension, we demonstrate how atomic collapse may arise in carbon nanotubes or graphene nanoribbons, and describe the critical importance of the size of the band gap. Meanwhile, inspired by groundbreaking experiments investigating the effects of strain, we propose how to confine the elusive charge carriers in so-called velocity barriers, which arise due to a spatially inhomogeneous Fermi velocity triggered by a strained lattice. We also present a new and beautiful quasi-exactly solvable model of quantum mechanics, showing the possibilities for confinement in magnetic quantum dots are not as stringent as previously thought. We also reveal that Klein tunnelling is not as pernicious as widely believed, as we show bound states can arise from purely electrostatic means at the Dirac point energy. Finally, we show from an analytical solution to the quasi-relativistic two-body problem, how an exotic same-particle paring can occur and speculate on its implications if found in the laboratory.