  Exciton dynamics can be strongly affected by lattice vibrations through
electron-phonon coupling. This is rarely explored in two-dimensional magnetic
semiconductors. Focusing on bilayer CrI3, we first show the presence of strong
electron-phonon coupling through temperature-dependent photoluminescence and
absorption spectroscopy. We then report the observation of periodic broad modes
up to the 8th order in Raman spectra, attributed to the polaronic character of
excitons. We establish that this polaronic character is dominated by the
coupling between the charge-transfer exciton at 1.96 eV and a longitudinal
optical phonon at 120.6 cm-1. We further show that the emergence of long-range
magnetic order enhances the electron-phonon coupling strength by about 50$\%$
and that the transition from layered antiferromagnetic to ferromagnetic order
tunes the spectral intensity of the periodic broad modes, suggesting a strong
coupling among the lattice, charge and spin in two-dimensional CrI3. Our study
opens opportunities for tailoring light-matter interactions in two-dimensional
magnetic semiconductors.
