## Curvature-induced electron localization in developable
Möbius-like nanostructures

A.P. Korte & G.H.M. van der Heijden

We study curvature effects and localization of non-interacting electrons
confined to developable one-sided elastic sheets motivated by recent
nanostructured origami techniques for creating and folding extremely thin
membrane structures. The most famous one-sided sheet is the Möbius
strip but the theory we develop allows for arbitrary linking number. Unlike
previous work in the literature we do not assume a shape for the elastic
structures. Rather, we find the shape by minimizing the elastic energy,
i.e., solving the EulerLagrange equations for the bending energy functional.
This shape varies with the aspect ratio of the sheet and affects the
potential experienced by the particles. Depending on the link there is a
number of singular points on the edge of the structure where the bending
energy density goes to infinity, leading to deep potential wells. The
inverse participation ratio is used to show that electrons are increasingly
localized to the higher-curvature regions of the higher-width structures,
where sharp creases radiating out from the singular points could form
channels for particle transport. Our geometric formulation could be used to
study transport properties of Möbius strips and other components in
nanoscale devices.

*J. Phys.: Condens. Matter* **21**, 495301 (2009)