# Spin fluctuation enhanced phonon magnetic moment

Phonons are the collective excitation of lattice vibrations and are one of the most common and important quasiparticles in solids. The understanding of the fundamental properties of phonons is the basis of much research in condensed matter physics and materials science. Phonons have energy and quasi-momentum, and can also carry angular momentum and have chirality in systems where symmetry is broken in time or space inversion [1,2]. But for a long time, it was thought that the magnetic moment of phonons was negligible. [3] This is because the mass of the ion is much greater than that of the electron, and the orbital magnetic moment of the phonon produced by the ion is extremely weak, usually less than 10-4 Bohr magnetons (μB), which is four orders of magnitude smaller than the magnetic moment of the electron, so there is almost no observable effect.

In the 1970s, researchers discovered that optical phonons located at 109 cm-1 and 197 cm-1 in rare earth paramagnetics CeCl3 undergo Zeeman splitting under magnetic fields. In this system, phonons obtain a huge magnetic moment, in the order of μB, by coupling with Ce ion 4f orbital crystal field excitation [4]. Figure 1 shows a schematic diagram of measuring the magnetic moment of phonons through the Zeeman effect. Recently, giant phonon magnetic moments in the order of μB and 0.01μB have been observed in topologically non-magnetic materials Cd3As2 and PbTe by terahertz spectroscopy [5,6]. Subsequent theoretical studies have found that under the "Born-Oppenheimer approximation", the rotation of ions is accompanied by the movement of the electron wave packet, and the orbital magnetic moment of the latter can be amplified by the Berry curvature of the band, thus contributing considerable phonon magnetic moments [7]. The above studies based on paramagnetic or non-magnetic systems prove that phonons can obtain magnetic moments through the coupling of two degrees of freedom between lattice-charge. However, for magnetically ordered systems, there is no observational evidence of phonon magnetic moments. Phonon magnetic moments have broad research and application prospects in magnetic systems. On the one hand, the magnetic moment can effectively regulate the energy of phonons by magnetic field, magnetic sequence and magnetic domain. On the other hand, phonons, as carriers of magnetic moments, can directly participate in many magnetic processes. This will open up entirely new possibilities for fundamental magnetic research and spin device design. Can spin-lattice interactions confer magnetic moments on phonons in magnetic systems? How do long-range spin correlations and critical spin fluctuations affect the magnetic properties of phonons? These fundamental physics problems need to be clarified. Recently, the exploration of phonon magnetic moments in magnetically ordered systems has made progress, and researchers have observed huge phonon magnetic moments and their critical fluctuation enhancement effects in antiferromagnets

Using low-wave-number Raman spectroscopy, the researchers observed an excitation pattern (P1) at 1.3 THz(42 cm-1) in Fe2Mo3O8. The spectral signal is still visible at room temperature, which rules out the possibility of magnetic excitation. The phonon properties of this model are further confirmed by inelastic neutron scattering measurements. Through spectral analysis and first-principles calculations, the researchers found that the P1 pattern consists of a pair of chiral phonons carrying opposite angular momentum. The displacement of the magnetic atoms in this mode mainly comes from the in-plane rotation of the tetrahedron Fe, as shown in Figure 2(a). Using cross circularly polarized magneto-optical Raman spectroscopy, the researchers found that the pair of degenerate chiral phonons underwent significant Zeeman splitting under a magnetic field (Figure 2(b)). The frequency of chiral phonons with opposite angular momentum shifts linearly with the magnetic field. According to the slope of Zeeman splitting, the magnetic moment of P1 phonon is 0.11μB, which is more than 3 orders of magnitude higher than that of ordinary solid. This experiment gives conclusive evidence of the magnetic moments of phonons in a magnetically ordered system. It is found that the magnetic moment of P1 phonon is derived from the coupling between the lattice and the spin degrees of freedom. In Fe2Mo3O8, the two antiferromagnetic spin waves are located at 2.6 THz and 3.4 THz respectively. It is through the non-resonant coupling with P1 phonons that the magnetic moment is obtained, which also explains the frequency blue shift of P1 above the phase transition temperature as the temperature increases. Because P1 gradually returns to its uncoupled intrinsic frequency above the phase transition temperature as the coupling with the magenta disappears.

The most interesting phenomenon occurs in the region of the transition from the antiferromagnetic phase to the paramagnetic phase. By extracting the magnetic moments of P1 phonons at different temperatures, the researchers found that the magnetic moments of P1 phonons showed up to 600% enhancement near the phase transition temperature (60 K). At 58 K, the magnetic moment of P1 phonon rapidly increases from 0.11μB at low temperature to 0.68μB, as shown in Figure 3(a). Although the non-resonant coupling of phonons and magnetons can give the magnetic moment of phonons at low temperature antiferromagnetic phase of the system, it is not sufficient to explain the 6-fold enhancement near the phase transition point. Through in-depth study, it is found that the quasi-biviya ferromagnetic fluctuation of the system is the main reason for the enhancement of P1 phonon magnetic moment. The antiferromagnetic sequence of Fe2Mo3O8 can be regarded as a quasi-two-dimensional ferromagnetic sequence, which is formed by interlayer coupling. Fe2Mo3O8 has a strong ferrimagnetic spin fluctuation near the phase transition temperature, which is confirmed by magnetic susceptibility measurements. Near the phase transition temperature, the external magnetic field and the induced molecular field act on P1 phonon as the total equivalent magnetic field, in which the molecular field is amplified by the ferrimagnetic spin fluctuations, resulting in the enhancement of the magnetic moment of the phonon. The P1 phonon magnetic moment does not disappear immediately above 60 K, because the phonon and paramagnon are coupled. At higher temperatures, paramagnetic magnetons further lose coherence, while phonon magnetic moments eventually disappear. Using Monte Carlo simulation, the researchers successfully reproduced the phenomenon of critical fluctuation enhancement of phonon magnetic moments, verifying the reliability of the theoretical explanation, as shown in Figure 3(b). This simple mean field image also explains that the huge splitting of P1 chiral phonons in the ferromagnetic phase comes from the molecular field of the ferromagnetic phase.