Electronic structure of an electron on the gyroid surface: A helical labyrinth

A previously reported formulation for electrons on curved periodic surfaces is used to analyze the band structure of an electron bound on the gyroid surface (the only triply periodic minimal surface that has screw axes). We find that an effect of the helical structure appears as the multiple band sticking on the Brillouin zone boundaries. We elaborate how the band sticking is lifted when the helical and inversion symmetries of the structure are degraded. We find from this that the symmetries give rise to prominent peaks in the density of states.