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The latest discovery of ferromagnetism in atomically thin films ofsemiconductors Cr$_2$Ge$_2$Te$_6$ and CrI$_3$ has unleashed numerousopportunities for fundamental physics of magnetism in two-dimensional (2D)limit and also for technological applications based on 2D magnetic materials.In this paper, we present a comprehensive theoretical study of the magnetic,electronic, optical and magneto-optical(MO) properties of multilayers[monolayer(ML), bilayer and trilayer] and bulk CrI$_3$, based on the densityfunctional theory with the generalized gradient approximation plus on-siteCoulomb repulsion scheme. Interestingly, all the structures are found to besingle-spin ferromagnetic(FM) semiconductors. They all have a largeout-of-plane magnetic anisotropy energy(MAE) of $\sim$0.5 meV/Cr. These largeMAEs suppress transverse spin fluctuations and thus stabilize long-rangemagnetic orders at finite temperatures down to the ML limit. They also exhibitstrong MO effects with their Kerr and Faraday rotation angles being comparableto that of best-known bulk MO materials. The shape and position of the mainfeatures in the optical and MO spectra are found to be nearlythickness-independent although the magnitude of Kerr rotation angles increasesmonotonically with the film thickness. Magnetic transition temperaturesestimated based on calculated exchange coupling parameters, calculated opticalconductivity, MO Kerr and Faraday rotation angles agree quite well withavailable experimental data. The calculated MAE as well as optical and MOproperties are analyzed in terms of the calculated orbital-decomposed densitiesof states, band state symmetries and dipole selection rules. Our findings oflarge out-of-plane MAEs and strong MO effects in these single-spin FMsemiconducting CrI$_3$ ultrathin films suggest that they will find valuableapplications in semiconductor MO and spintronic nanodevices.

Magnetism and Magneto-optical Effects in Bulk and Few-layer CrI$_3$: A Theoretical GGA + U Study