Electronic structure of periodic curved surfaces -- continuous surface versus graphitic sponge

TL;DR

This paper explores the electronic band structure of electrons on periodic curved surfaces, using Schrödinger's equation with Weierstrass representation and tight-binding models, revealing how surface connectivity influences electronic properties.

Contribution

It formulates a Schrödinger equation approach for minimal surfaces and demonstrates the equivalence of continuous surfaces and atomic network models in their electronic spectra.

Findings

Band structures depend on surface connectivity and Bonnet transformations.

Low-energy spectra of atomic networks match those of continuous curved surfaces.

The approach provides insights into electronic properties of complex curved nanostructures.

Abstract

We investigate the band structure of electrons bound on periodic curved surfaces. We have formulated Schr\"{o}dinger's equation with the Weierstrass representation when the surface is minimal, which is numerically solved. Bands and the Bloch wavefunctions are basically determined by the way in which the ``pipes'' are connected into a network, where the Bonnet(conformal)-transformed surfaces have related electronic strucutres. We then examine, as a realisation of periodic surfaces, the tight-binding model for atomic networks (``sponges''), where the low-energy spectrum coincides with those for continuous curved surfaces.