Differential Geometry Applied to Rings and Möbius Nanostructures

Nanostructure shape effects have become a topic of increasing interest due to advancements in fabrication technology. In order to pursue novel physics and better devices by tailoring the shape and size of nanostructures, effective analytical and computational tools are indispensable. In this chapter, we present
analytical and computational differential geometry methods to examine particle quantum eigenstates and eigenenergies in curved and strained nanostructures. Example studies are carried out for a set of ring structures with different radii and it is shown that eigenstate and eigenenergy changes due to curvature are
most significant for the groundstate eventually leading to qualitative and quantitative changes in physical properties. In particular, the groundstate in-plane symmetry characteristics are broken by curvature effects, however, curvature contributions can be discarded at bending radii above 50 nm. A more complicated topological structure, the Möbius nanostructure, is analyzed and geometry effects for eigenstate properties are discussed including dependencies on the Möbius nanostructure width, length, thickness, and strain. The second edition contains a derivation of phonon equations-of-motion of thin shells applied to 2D graphene using a differential geometry formulation. The third editions adds a presentation of eigenstate and energy properties of open and closed nanorings for a general normal plane geometry. In the case of closed
nanorings it is demonstrated that eigenstate energies depend on a holonomy angle which is given by the integral of the torsion around the centerline modulo 2*pi.