  The incorporation of bismuth (Bi) in GaAs results in a large reduction of the
band gap energy (E$_g$) accompanied with a large increase in the spin-orbit
splitting energy ($\bigtriangleup_{SO}$), leading to the condition that
$\bigtriangleup_{SO} > E_g$ which is anticipated to reduce so-called CHSH Auger
recombination losses whereby the energy and momentum of a recombining
electron-hole pair is given to a second hole which is excited into the
spin-orbit band. We theoretically investigate the electronic structure of
experimentally grown GaBi$_x$As$_{1-x}$ samples on (100) GaAs substrates by
directly comparing our data with room temperature photo-modulated reflectance
(PR) measurements. Our atomistic theoretical calculations, in agreement with
the PR measurements, confirm that E$_g$ is equal to $\bigtriangleup_{SO}$ for
$\textit{x} \approx$ 9$%$. We then theoretically probe the inhomogeneous
broadening of the interband transition energies as a function of the alloy
disorder. The broadening associated with spin-split-off transitions arises from
conventional alloy effects, while the behaviour of the heavy-hole transitions
can be well described using a valence band-anticrossing model. We show that for
the samples containing 8.5% and 10.4% Bi the difficulty in identifying a clear
light-hole-related transition energy from the measured PR data is due to the
significant broadening of the host matrix light-hole states as a result of the
presence of a large number of Bi resonant states in the same energy range and
disorder in the alloy. We further provide quantitative estimates of the impact
of supercell size and the assumed random distribution of Bi atoms on the
interband transition energies in GaBi$_{x}$As$_{1-x}$. Our calculations support
a type-I band alignment at the GaBi$_x$As$_{1-x}$/GaAs interface, consistent
with recent experimental findings.
