Completely flat bands and fully localized states on surfaces of anisotropic diamond-lattice models

We discuss flat-band surface states on the (111) surface in the tight-binding model with nearest-neighbor hopping on the diamond lattice, in analogy to the flat-band edge states in graphene with a zigzag edge. The bulk band is gapless, and the gap closes along a loop in the Brillouin zone. The verge of the flat-band surface states is identical with this gap-closing loop projected onto the surface plane. When anisotropies in the hopping integrals increase, the bulk gap-closing points move and the distribution of the flat-band states expands in the Brillouin zone. Then when the anisotropy is sufficiently large, the surface flat bands cover the whole Brillouin zone. Because of the completely flat bands, we can construct surface-state wavefunctions which are localized in all the three directions.

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