  Electronic and structural properties of a 3D carbon allotrope made of
Hopf-linked graphenes, which we call a Hopfene - a type of topological crystal,
are examined by semi-empirical molecular-orbital and
density-functional-theoretical methods, where band-structure analyses reveal
very different properties from those of 2D graphenes. Furthermore, the analyses
give an interesting finding that, depending on graphene-sheet spacings,
Hopfenes exhibit different band features between primary-type Hopfene with a
finite minimum sheet spacing and secondary type with its double-sized spacing.
The primary type shows semi-metallic nature and the secondary type exhibits
semi-metallic or semiconducting nature at different bands and also has flat
bands; these conducting features can be utilised by Fermi-level control. A
device application of Hopfenes is also provided.
