  We perform a systematic theoretical analysis of the nature and importance of
alloy disorder effects on the electronic and optical properties of
GaN$_{y}$As$_{1-x-y}$Bi$_{x}$ alloys and quantum wells (QWs), using large-scale
atomistic supercell electronic structure calculations based on the
tight-binding method. Using ordered alloy supercell calculations we also derive
and parametrise an extended basis 14-band \textbf{k}$\cdot$\textbf{p}
Hamiltonian for GaN$_{y}$As$_{1-x-y}$Bi$_{x}$. Comparison of the results of
these models highlights the role played by short-range alloy disorder --
associated with substitutional nitrogen (N) and bismuth (Bi) incorporation --
in determining the details of the electronic and optical properties. Systematic
analysis of large alloy supercells reveals that the respective impact of N and
Bi on the band structure remain largely independent, a robust conclusion we
find to be valid even in the presence of significant alloy disorder where N and
Bi atoms share common Ga nearest neighbours. Our calculations reveal that N-
(Bi-) related alloy disorder strongly influences the conduction (valence) band
edge states, leading in QWs to strong carrier localisation, as well as
inhomogeneous broadening and modification of the conventional selection rules
for optical transitions. Our analysis provides detailed insight into key
properties and trends in this unusual material system, and enables quantitative
evaluation of the potential of GaN$_{y}$As$_{1-x-y}$Bi$_{x}$ alloys for
applications in photonic and photovoltaic devices.
