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The magneto-optical (MO) effects not only are a powerful probe of magnetismand electronic structure of magnetic solids but also have valuable applicationsin 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-opticaleffects and have also attracted the attention for fundamental physics studies.In particular, YIG has been routinely used as a spin current injector. In thispaper, we present a thorough theoretical investigation on magnetism,electronic, optical and MO properties of YIG and BIG, based on the densityfunctional theory with the generalized gradient approximation plus onsiteCoulomb repulsion. We find that both semiconductors exhibit large MO effectswith their Kerr and Faraday rotation angles being comparable to that ofbest-known MO materials such as MnBi. Especially, the MO Kerr rotation anglefor bulk BIG reaches -1.2$ ^{\circ}$ at photon energy $\sim2.4$ eV, and the MOFaraday rotation angle for BIG film reaches -74.6 $ ^{\circ}/\mu m$ at photonenergy $\sim2.7$ eV. Furthermore, we also find that both valence and conductionbands across the MO band gap in BIG are purely spin-down states, i.e., BIG is asingle spin semiconductor. These interesting findings suggest that the irongarnets will find valuable applications in semiconductor MO and spintronicnanodevices. The calculated optical conductivity spectra, MO Kerr and Faradayrotation angles agree well with the available experimental data. The mainfeatures in the optical and MO spectra of both systems are analyzed in terms ofthe calculated band structures especially by determining the band statesymmetries and the main optical transitions at the $\Gamma$ point in theBrillouin zone.

A First Principle Study on Magneto-Optical Effects and Magnetism in Ferromagnetic Semiconductors Y$_3$Fe$_5$O$_{12}$ and Bi$_3$Fe$_5$O$_{12}$