Dirac fermions on wires confined to the graphene Moebius strip

TL;DR

This paper studies how the unique geometry of a graphene Moebius strip influences Dirac fermions, revealing geometric phases, symmetry breaking effects, and topological modifications to electronic states and energy levels.

Contribution

It provides exact solutions to the Dirac equation on a Moebius-shaped graphene strip, highlighting the impact of geometry and topology on electronic properties.

Findings

Geometric potential induces a geometric phase in wave functions.

Electronic states are localized due to parity symmetry breaking.

Energy levels exhibit half-integer multiples and altered periodicity.

Abstract

We investigate the effects of the curved geometry on a massless relativistic electron constrained to a graphene strip with a Moebius strip shape. The anisotropic and parity-violating geometry of the Moebius band produces a geometric potential that inherits these features. By considering wires along the strip width and the strip length, we find exact solutions for the Dirac equation and the effects of the geometric potential on the electron were explored. In both cases, the geometric potential yields to a geometric phase on the wave function. Along the strip width, the density of states depends on the direction chosen for the wire, a consequence of the lack of axial symmetry. Moreover, the breaking of the parity symmetry enables the electronic states to be concentrated on the inner or on the outer portion of the strip. For wires along the strip length, the nontrivial topology influences the eigenfunctions by modifying their periodicity. It turns out that the ground state has a period of $4\pi$ whereas the first excited state is a $2\pi$ periodic function. Moreover, we found that the energy levels are half-integer multiples of the energy of the ground state.