  The magneto-optical (MO) effects not only are a powerful probe of magnetism
and electronic structure of magnetic solids but also have valuable applications
in high-density data-storage technology. Yttrium iron garnet
(Y$_3$Fe$_5$O$_{12}$) (YIG) and bismuth iron garnet (Bi$_3$Fe$_5$O$_{12}$)
(BIG) are two widely used magnetic semiconductors with strong magneto-optical
effects and have also attracted the attention for fundamental physics studies.
In particular, YIG has been routinely used as a spin current injector. In this
paper, we present a thorough theoretical investigation on magnetism,
electronic, optical and MO properties of YIG and BIG, based on the density
functional theory with the generalized gradient approximation plus onsite
Coulomb repulsion. We find that both semiconductors exhibit large MO effects
with their Kerr and Faraday rotation angles being comparable to that of
best-known MO materials such as MnBi. Especially, the MO Kerr rotation angle
for bulk BIG reaches -1.2$ ^{\circ}$ at photon energy $\sim2.4$ eV, and the MO
Faraday rotation angle for BIG film reaches -74.6 $ ^{\circ}/\mu m$ at photon
energy $\sim2.7$ eV. Furthermore, we also find that both valence and conduction
bands across the MO band gap in BIG are purely spin-down states, i.e., BIG is a
single spin semiconductor. These interesting findings suggest that the iron
garnets will find valuable applications in semiconductor MO and spintronic
nanodevices. The calculated optical conductivity spectra, MO Kerr and Faraday
rotation angles agree well with the available experimental data. The main
features in the optical and MO spectra of both systems are analyzed in terms of
the calculated band structures especially by determining the band state
symmetries and the main optical transitions at the $\Gamma$ point in the
Brillouin zone.
