Group theory analysis of phonons in monolayer chromium trihalides and their Janus structures

A contrastive investigation of the symmetry aspects of phonons in monolayer chromium trihalides and their Janus structures Y₃-Cr₂-X₃ (X, Y = F, Cl, Br, I) was presented within group theory. We classified all phonon vibration modes at the Brillouin-zone center (Γ) into irreducible representations. The infrared and Raman activity of optic phonons, Raman tensors, and the possible polarization assignments of Raman active phonons are further predicted. We clarify the discrepancy concerning the Raman and infrared activity of optic modes in monolayer CrI₃. It is also found that the Raman and infrared spectra are exclusive for X₃-Cr₂-X₃ but coincident for Janus Y₃-Cr₂-X₃. This distinction plays a vital role in optic spectra identification of the Janus Y₃-Cr₂-X₃ monolayer from the X₃-Cr₂-X₃ monolayer. To demonstrate the origin of phonon chirality and magnetism intuitively, we derive the symmetry-matched phonon eigenfunctions and the corresponding schematic representations of the eigenvectors for both the F₃-Cr₂-I₃ and I₃-Cr₂-I₃ monolayers. Our analysis indicates that the spin-phonon coupling, the magneto-optical effect of infrared and Raman active phonons, and the phonon chirality could be observed in the Janus Y₃-Cr₂-X₃ monolayer more easily than in the X₃-Cr₂-X₃ monolayer. Our results thus provide a detailed guiding map for experimental identification and characterization of the Janus Y₃-Cr₂-X₃ monolayer.