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.

Figure 1

A primitive patch for the G surface. Its center corresponds to the navel, around which the surface is point symmetric.

Figure 2

A cubic unit cell of the G surface in birds-eye (a) and top (b) views.

Figure 3

Band structure of the G surface displayed on the Brillouin zone for the bcc unit cell. The numbers in the parentheses represent the degeneracy of the band sticking.

Figure 4

(Color) The wave functions in the unit cell of the G surface for the sixfold states at the lowest level in the H point (in Fig. 3), with positive (negative) amplitudes color coded in red (blue).

Figure 5

Coincidence of the band energies among G, P, and D surfaces.

Figure 6

(a) Low-energy band structure for the graphitic G sponge, to be compared with Fig. 3, constructed from a graphite fragment shown at the top. We take the hopping parameter $t$ as the unit of energy and the tight-binding band center as the origin. (b) The band structure when we introduce a potential that breaks the helical symmetry, or (c) the inversion symmetry. Extra potential introduced to degrade the symmetry is shown at the top, where an open circle represents the potential for every other patch, while filled one for all the patches. The panels (d) and (e) show the densities of states for the systems (a) and (b), respectively, in units of $1∕\left(Vt\right)$ ($V$: unit cell volume).