Band structure of the P, D, and G surfaces

A theoretical study on the band structures for electrons constrained to move along triply periodic minimal surfaces is presented. The P, D, and G surfaces, which belong to the same Bonnet family, are considered, and their band structures are calculated numerically. The interrelations between the band structures of these surfaces are discussed in terms of the six-dimensional algebra of the Bonnet transformations [C. Oguey and J.-F. Sadoc, J. Phys. I France 3, 839 (1993)]. Specifically, the coincidence of energy eigenvalues between the surfaces, observed at several wave vectors, are explained in a systematic way. We also analyze in detail the symmetry properties of the band structures. It turns out that there is a close connection between the symmetry properties and the occurrence of nodal lines in the eigenstates. This analysis proves to be useful in understanding the overall characteristics of the band structures. The global connectivity of the band structures is considerably different between the surfaces because of their topological differences. The present study may provide a basis for understanding further the connection between the topology and the band structures.