Electric toroidal octupolar symmetry in pyrite FeS$_2$ probed by Raman optical activity

Abstract

We report Raman optical activity in pyrite FeS$_2$, which hosts an electric toroidal octupolar symmetry. A clear and reproducible sign reversal of the circular intensity difference is observed between neighboring $\{111\}$ faces under cross-circular polarization. The signal appears only for the doubly degenerate $E_g$ phonon mode and is absent for other modes, consistent with symmetry analysis. First-principles calculations reproduce these features, establishing Raman optical activity as a probe of higher-rank axial multipolar symmetry.

Figures (6)

• Figure 1: (a) Schematic illustrations of the cross-circular polarization configurations (LR and RL) in a backscattering setup. (b) Octahedral and cubic pyrite crystals with {111} and {001} facets, respectively. The four {111} faces are symmetry equivalent in the $T_h$ point group and cannot be distinguished by x-ray diffraction; the labels $(111)$, $(\bar{1}11)$, $(\bar{1}\bar{1}1)$, and $(1\bar{1}1)$ indicate the crystal orientation.
• Figure 2: (a) Anti-Stokes and (b) Stokes Raman spectra measured on $\{111\}$ surfaces at an excitation wavelength of 633 nm in the cross-circular polarization configurations.
• Figure 3: Stokes Raman spectra measured on the {001} surface at 633 nm in the cross-circular configurations.
• Figure 4: Stokes Raman spectra measured on {111} surfaces at excitation wavelengths of (a) 532 nm and (b) 785 nm in the cross-circular configurations. The spectra are zoomed in on the $E_{g}$ phonon mode to highlight the circular intensity difference.
• Figure 5: Schematic illustration of the $xyz$-type electric toroidal octupolar order and its transformation under the mirror reflection $m_{\perp}$ containing the $\langle 111 \rangle$ axis. The orange arrows represent the axial vectors perpendicular to the octahedral faces; their components parallel to the mirror plane are reversed. The faces are color-coded according to the penetrating direction of vectors on each surface.