  Newly discovered Bi-O-S compounds remain an enigma in attempts to understand
their electronic properties. A recent study of Bi$_{4}$O$_{4}$S$_{3}$ has shown
it to be a mixture of two phases, Bi$_{2}$OS$_{2}$ and Bi$_{3}$O$_{2}$S$_{3}$,
the latter being superconducting [W. A. Phelan et al., J. Am. Chem. Soc. 135,
5372 (2013)]. Using density functional theory, we explore the electronic
structure of both the phases and the effect of the introduction of stacking
faults. Our results demonstrate that the S$_{2}$ layers dope the
bismuth-sulphur bands. The bands at the Fermi level are of clear
two-dimensional character. One band manifold is confined to the two adjacent,
square-lattice bismuth-sulphur planes, a second manifold is confined to the
square lattice of sulphur dimers. We show that the introduction of defects in
the stacking does not influence the electronic structure. Finally, we also show
that spin-orbit coupling does not have any significant effect on the states
close to the Fermi level at the energy scale considered.
