Chiral phonons in sixfold chiral CrSi$_2$: Raman spectroscopy and first-principles calculations

Chiral phonons have been identified in several chiral crystals, primarily in those with trigonal symmetry and threefold screw axes. In this study, chiral phonons in CrSi$_2$, a chiral crystal with a sixfold helical structure, were investigated. Circularly polarized Raman spectroscopy revealed a subtle splitting of doubly degenerate $E_2$ phonon modes between cross-circular polarization configurations. These observations, supported by first-principles phonon calculations, indicate the presence of chiral phonons in CrSi$_2$, expanding the scope of materials that exhibit chiral vibrational modes beyond the conventional trigonal class.

💡 Research Summary

In this work the authors investigate the existence of chiral phonons in the semiconductor CrSi₂, which crystallizes in a six‑fold helical structure (space groups P6₄22 for the left‑handed enantiomer and P6₂22 for the right‑handed one). While chiral phonons have previously been reported only in crystals with three‑fold screw axes (e.g., α‑HgS, Te, α‑quartz), CrSi₂ provides the first example of a material with a six‑fold screw symmetry that hosts such modes.

Single crystals were grown by two methods: laser‑diode‑heated floating‑zone (LDFZ) and chemical vapor transport (CVT). The handedness of the LDFZ crystals was controlled by using seed rods of known chirality and subsequently confirmed by single‑crystal X‑ray diffraction (Flack parameters) and circularly polarized resonant X‑ray diffraction.

Raman measurements were performed at room temperature with a 785 nm excitation laser in a back‑scattering geometry perpendicular to the c‑axis. The experimental setup allowed full control of incident and scattered polarizations using two linear polarizers and a quarter‑wave plate, enabling the four circular configurations: RR, LL (parallel) and RL, LR (cross‑circular). In the hexagonal CrSi₂ unit cell (3 Cr + 6 Si atoms) there are 27 phonon branches, of which the Raman‑active ones belong to the irreducible representation Γ = A₁ + 4E₁ + 4E₂. Because the measurement geometry is normal to the c‑axis, the E₁ modes are symmetry‑forbidden, leaving the doubly degenerate E₂ modes observable only in the cross‑circular configurations, while the non‑degenerate A₁ mode appears in the parallel configurations.

The Raman spectra of the left‑handed crystal (L‑CrSi₂) show that each of the four E₂ modes splits into two closely spaced peaks when the polarization is switched from LR to RL. The splitting magnitude is ≤ 1 cm⁻¹, comparable to that reported for three‑fold chiral crystals under similar conditions. Notably, the sign of the splitting reverses for the right‑handed crystal (R‑CrSi₂), providing a direct optical fingerprint of crystal handedness. Peak positions were extracted by fitting Voigt profiles (Gaussian width fixed at the instrumental resolution of 2.7 cm⁻¹). The statistical uncertainties are much smaller than the observed splittings, confirming that the effect is genuine.

First‑principles phonon calculations were carried out within density‑functional perturbation theory (ABINIT, LDA exchange‑correlation). A 4 × 4 × 4 Monkhorst–Pack k‑grid and a plane‑wave cutoff of 25 Ha were employed. The calculated phonon frequencies agree well with experiment, and the eigenvectors were used to evaluate the crystal angular momentum (CAM) of each mode. CAM, defined as the eigenvalue of the six‑fold screw operation acting on the phonon displacement pattern, takes integer values m = 0, ±1, ±2, 3. Along the Γ–A line the doubly degenerate E₂ modes split into two non‑degenerate branches with opposite CAM values (m = ±2).

Raman selection rules for circular polarizations can be expressed as σᵢ − σₛ ≡ ± m (mod 6), where σᵢ and σₛ are the CAM of the incident and scattered photons (±1 for right/left circular). Consequently, in the LR configuration (σᵢ = −1, σₛ = +1) only the m = +2 phonon contributes, whereas in the RL configuration (σᵢ = +1, σₛ = −1) the m = −2 phonon is selected. The finite phonon wave vector involved in Raman scattering is about 1 % of the Brillouin‑zone size along the c‑axis; at this q‑value the calculated frequencies of the m = +2 branch are higher for the E₁₂ and E₃₂ modes but lower for E₂₂ and E₄₂, exactly matching the experimental trend. The opposite CAM sign in R‑CrSi₂ leads to the reversed splitting direction, confirming that the observed energy differences stem from CAM conservation.

The authors discuss three factors governing the magnitude of the splitting: (i) the lattice geometry (size of the Brillouin zone along the screw axis), (ii) the excitation wavelength (shorter wavelength yields larger q), and (iii) the phonon dispersion (group velocities along Γ–A). The linear‑in‑q splitting is a generic consequence of lattice chirality and can be described by an effective Hamiltonian containing a pseudoscalar coupling constant, which is non‑zero only in chiral crystals.

In summary, the study demonstrates that chiral phonons are not limited to three‑fold helical crystals; six‑fold helical CrSi₂ exhibits the same CAM‑driven splitting of doubly degenerate modes, observable via circularly polarized Raman spectroscopy. This expands the family of materials where chiral phonons can be probed, opens the possibility of using Raman spectroscopy as a non‑destructive probe of crystal handedness, and suggests that other six‑fold (or higher‑fold) chiral structures, such as transition‑metal disilicides, may host similar phenomena. The work thus provides a solid experimental‑theoretical framework for exploring chiral vibrational dynamics in a broader class of chiral solids.