  Flat bands form in a 3D Hopf-linked graphene crystal or a 3D carbon allotrope
named Hopfene, which qualitatively differ from bands of only graphenes. This
paper discusses carbon-hexagon deformation on the level shift of a flat band
via density-functional-theoretical (DFT) analysis to set the flat-band level to
the Fermi level, viz., to utilize its large density of states for magnetic- and
electronic-property researches. Tight-binding (TB) analysis is also performed
for a comparison with the DFT analysis; here, a qualitative agreement between
TB and DFT bands is obtained. The DFT analysis shows an almost linear flat-band
level shift to the lattice-deformation rate, where electron-interaction effects
are included within the Kohn-Sham method. To tune the flat-band level so that
it fits the Fermi level, a double-hetero-like structure is also proposed as a
way of hexagon-deformation control.
