  We report on a Dirac-like Fermi surface in three-dimensional bulk materials
in a distorted spinel structure on the basis of density functional theory (DFT)
as well as tight-binding theory. The four examples we provide in this paper are
BiZnSiO4, BiCaSiO4, BiMgSiO4, and BiAlInO4. A necessary characteristic of these
structures is that they contain a Bi lattice which forms a hierarchy of
chain-like substructures, with consequences for both fundamental understanding
and materials design.
