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Large cross-polarized Raman signal in CrI$_3$: A first-principles study

We find unusually large cross-polarized (and anti-symmetric) Raman signature of A$_{\rm g}$ phonon mode in CrI$_3$, in agreement with experiments. The signal is present only when the following three effects are considered in concert: ferromagnetism on Cr atoms, spin-orbit interaction, and resonant effects. Somewhat surprisingly, we find that the relevant spin-orbit interaction potential originates from iodine atoms, despite magnetism being mostly on chromium atoms. We analyze the Raman signature as a function of magnetic order, the direction of the magnetic moment, energy and polarization of light used for Raman scattering, as well as carrier lifetime. In addition to a strong cross-polarized Raman signal, we also find unusually strong phonon modulated magneto-optical Kerr effect (MOKE) in CrI$_3$.