Geometric effects on the electronic structure of curved nanotubes and curved graphene: the case of the helix, catenary, helicoid, and catenoid

TL;DR

This study investigates how the geometry of curved nanostructures like helices and catenoids influences the quantum behavior of electrons, revealing quantized angular momentum and bound states related to surface topology.

Contribution

It provides a detailed analysis of quantum particles constrained to various curved surfaces using differential geometry and numerical methods, highlighting geometry-induced effects on electronic states.

Findings

Quantum particles on a helix exhibit quantized angular momentum.

Particles on catenary, helicoid, and catenoid surfaces can have bound or excited states.

Proposes measurements to distinguish surface topologies via topological metrology.

Abstract

Since electrons in a ballistic regime perceive a carbon nanotube or a graphene layer structure as a continuous medium, we can use the study of the quantum dynamics of one electron constrained to a curve or surface to obtain a qualitative description of the conduction electrons' behavior. The confinement process of a quantum particle to a curve or surface leads us, in the so-called "confining potential formalism" (CPF), to a geometry-induced potential (GIP) in the effective Schr\"odinger equation. With these considerations, this work aims to study in detail the consequences of constraining a quantum particle to a helix, catenary, helicoid, or catenoid, exploring the relations between these curves and surfaces using differential geometry. Initially, we use the variational method to estimate the energy of the particle in its ground state, and thus, we obtain better approximations with the use of the confluent Heun function through numerical calculations. Thus, we conclude that a quantum particle constrained to an infinite helix has its angular momentum quantized due to the geometry of the curve, while in the cases of the catenary, helicoid, and catenoid the particle can be found either in a single bound state or in excited states which constitute a continuous energy band. Additionally, we propose measurements of physical observables capable of discriminating the topologies of the studied surfaces, in the context of topological metrology.