  We propose a new band engineering scheme on the biphenylene network, a newly
synthesized carbon allotrope. First, we investigate the mechanism for the
appearance of type II Dirac fermion in a pristine biphenylene network. We show
that the essential ingredients are mirror symmetries and the stabilization of
the compact localized eigenstates via destructive interference. While the
former is used for the band-crossing point along high symmetry lines, the
latter makes the obtained Dirac dispersion highly inclined. Then, we
demonstrate that many other different kinds of Dirac fermions, such as type-I
Dirac, gapped type-II Dirac, and nodal line semimetals, can be developed by
fluorinating the biphenylene network periodically in various ways. In this
program, the key role of the fluorine atoms is manipulating the condition of
the destructive interference and mirror symmetries.
