On the confinement of massless Dirac fermions in topological Möbius strips

Abstract

In this work, we investigate some aspects of the confinement problem in Möbius strips in the context of the low-energy approximation of graphene, aiming to provide an analytical prescription for studying graphene-based nanoribbons in response to the peculiar background proper to such objects. Based on a topological treatment, we model the confinement of massless Dirac fermions in such nanostructures by proposing boundary conditions capable of incorporating their typical topological character, and examine how such configuration affects the physical properties of graphene. Energy eigenvalues are found to be sensitive to the transverse confinement and the positions along the transverse direction, exhibiting an unexpected spatial constraint. In addition, a dual electronic behavior associated with this aspect is also suggested under certain limits within our solution, which includes the presence of a length-dependent gap that responds to the longitudinal boundary conditions. At the same time, we look at the problem from a more mathematical perspective and sketch out some insights into the role played by the notion of self-adjointness in such boundary problem. A peculiar position constraint acts on our solution and manifests itself as partial restriction on the proposed set, pointing to the definition of incomplete domain self-adjoint extensions as well as to the nonexistence of conventional Möbius periodicities compatible with infinite mass confinement.