Spectroscopic and first principle DFT+eDMFT study of complex structural, electronic, and vibrational properties of $M_2$Mo$_3$O$_8$ ($M$=Fe, Mn) polar magnets

Optical spectroscopy, X-ray diffraction measurements, density functional theory (DFT) and density functional theory + embedded dynamical mean field theory (DFT+eDMFT) have been used to characterize structural and electronic properties of hexagonal $M_2$Mo$_3$O$_8$ ($M$=Fe, Mn) polar magnets. Our experimental data are consistent with the room temperature structure belonging to the space group P6$_3$mc for both compounds. The experimental structural and electronic properties at room temperature are well reproduced within DFT+eDMFT method, thus establishing its predictive power in the paramagnetic phase. With decreasing temperature, both compounds undergo a magnetic phase transition and we argue that this transition is concurrent with a structural phase transition (symmetry change from P6$_3$mc) in the Fe compound and an isostructural transition (no symmetry change from P6$_3$mc) in the Mn compound. In addition, the unusual temperature dependent behavior of electronic d-d transitions in Fe$^{2+}$ ions is discussed.