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

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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, to a geometry-induced potential in the effective Schrödinger 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.

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Acknowledgements

This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior–Brasil (CAPES)—Finance Code 001 (F.F.S.F.), Fundação de Amparo à Ciência e Tecnologia do Estado de Pernambuco (FACEPE), Grants No. IBPG-0487-1.05/19 (J.D.M.L.), and BIC-1187-1.05/20 (E.G.), and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Grant No. 307687/2017-1 (F.M.). The authors are indebted to L.C.B. da Silva for invaluable discussions and suggestions.

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Authors and Affiliations

  1. Departamento de Física, Universidade Federal Rural de Pernambuco, Recife, PE, 52171-900, Brazil

    J. D. M. de Lima, E. Gomes, F. F. da Silva Filho & F. Moraes

  2. Departamento de Matemática, Universidade Federal Rural de Pernambuco, Recife, PE, 52171-900, Brazil

    R. Teixeira

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  1. J. D. M. de Lima
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Correspondence to F. Moraes.

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de Lima, J.D.M., Gomes, E., da Silva Filho, F.F. et al. Geometric effects on the electronic structure of curved nanotubes and curved graphene: the case of the helix, catenary, helicoid, and catenoid. Eur. Phys. J. Plus 136, 551 (2021). https://doi.org/10.1140/epjp/s13360-021-01533-6

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