Optical phonons as a testing ground for spin group symmetries

Lattice vibrations are highly sensitive to crystal symmetries and their changes across phase transitions. The latter can modify irreducible (co)representations and corresponding infrared and Raman selection rules of phonons. This concept is established for relativistic magnetic point groups, simultaneously transforming spatial and spin coordinates. However, in altermagnets described by non-relativistic spin groups with disjunct symmetry operations for both vector spaces, the phonon selection rules have remained unexplored. Here, we present a detailed study of the infrared- and Raman-active modes in the collinear antiferromagnet and altermagnet candidate Co$_2$Mo$_3$O$_8$. Comparing to ab initio calculations accurately capturing the eigenfrequencies, we identify all expected phonon modes at room temperature and deduce their selection rules using both symmetry approaches. Importantly, we observe the change of selection rules upon antiferromagnetic ordering, agreeing with the relativistic symmetry approach, while the spin group formalism predicts no changes. Therefore, optical phonons can reveal the appropriate symmetry treatment.

💡 Research Summary

In this work the authors use infrared (IR) and Raman spectroscopy combined with first‑principles phonon calculations to test two competing symmetry frameworks for the polar hexagonal oxide Co₂Mo₃O₈. The material crystallizes in the non‑centrosymmetric space group P6₃mc (point group 6 mm) and undergoes a collinear antiferromagnetic (AFM) transition at T_N ≈ 39 K, with all spins aligned along the crystallographic c‑axis.

Two symmetry descriptions are considered. (i) The relativistic magnetic point‑group approach, which assumes a finite spin‑orbit coupling and therefore treats spatial and spin coordinates as a single entity. For Co₂Mo₃O₈ the magnetic point group is 6′m′m, obtained by halving the crystallographic group to the unitary subgroup 3 m and adding anti‑unitary time‑reversal operations. (ii) The non‑relativistic spin‑group approach, appropriate for altermagnets, where spin‑orbit coupling is neglected and spatial and spin symmetries are treated independently. The corresponding spin group is ¯1 6 ¯1 m ¯1 m.

Group‑theoretical analysis shows that in the paramagnetic phase the optical phonon representation decomposes into 9 A₁ + 12 E₁ + 13 E₂ modes (IR+Raman active) plus acoustic and silent modes. In the relativistic magnetic phase the unitary part is reduced to 3 m, so the original irreducible representations are promoted to corepresentations. This promotion predicts that several modes, previously IR‑only or Raman‑only, become simultaneously active in both channels after magnetic ordering. By contrast, the spin‑group analysis keeps the spatial part unchanged; therefore it predicts no change in selection rules across the AFM transition.

The authors measured IR reflectivity for electric field parallel to c (Eω∥c) and to a (Eω∥a) and Raman spectra in three polarization configurations (y(zz)¯y, y(xz)¯y, and z(xx)¯z) at 295 K (paramagnetic) and 5–10 K (AFM). Density‑functional perturbation theory (DFPT) calculations provide the phonon frequencies for all symmetry‑allowed modes. The experimental spectra at 295 K display exactly the nine A₁, twelve E₁ and thirteen E₂ modes predicted by the crystallographic 6 mm group, with deviations of only 1–3 cm⁻¹ from the calculated values.

Upon cooling below T_N, the IR spectra reveal that several A₁ and E₁ modes acquire Raman intensity in the y(zz)¯y or y(xz)¯y configurations, and vice‑versa, confirming the emergence of mixed IR/Raman activity. For example, the A₁ mode near 264 cm⁻¹, which is IR‑active only in the paramagnetic phase, becomes Raman‑active in the AFM phase. These observations exactly match the predictions of the 6′m′m magnetic point group and contradict the spin‑group prediction of unchanged selection rules.

The discrepancy is interpreted as evidence that the spin‑phonon coupling in Co₂Mo₃O₈ involves terms that require simultaneous transformations of spin and real‑space coordinates, i.e., the coupling respects the full relativistic magnetic symmetry rather than the decoupled spin‑group symmetry. Consequently, optical phonons act as a sensitive probe of whether spin‑orbit effects are relevant in a given material.

The study concludes that, for Co₂Mo₃O₈ and likely for other altermagnet candidates, the relativistic magnetic point‑group description remains the appropriate framework for describing phonon selection rules, while the non‑relativistic spin‑group approach fails to capture the observed changes. This result highlights the importance of considering spin‑orbit coupling and spin‑phonon interactions when using symmetry analysis to predict spectroscopic signatures in magnetic materials.