Electronic structure of periodic curved surfaces: Topological band structure

Abstract

The electronic band structure for electrons bound on periodic minimal surfaces is differential-geometrically formulated and numerically calculated. We focus on minimal surfaces because they are not only mathematically elegant (with the surface characterized completely in terms of ``navels'') but represent the topology of real systems such as zeolites and negative-curvature fullerenes. The band structure turns out to be primarily determined by the topology of the surface, i.e., how the wave function interferes on a multiply connected surface, so that the bands are little affected by the way in which we confine the electrons on the surface (thin-slab limit or zero thickness from the outset). Another curiosity is that different minimal surfaces connected by the Bonnet transformation (such as Schwarz's P and D surfaces) possess one-to-one correspondence in their band energies at Brillouin-zone boundaries.