General ab initio framework for chiral phonons induced by electronic order

TL;DR

This work addresses the failure of conventional ab initio lattice dynamics to capture symmetry breaking driven by electronic order in magnetic materials. It introduces an ab initio framework based on molecular Berry curvature ($MBC$) that adds a $G(\bm{q})$ correction to the dynamical matrix, enabling the description of magnetism-induced phonon splitting when spin–orbit coupling is present. In Co$_3$Sn$_2$S$_2$, the $MBC$ term breaks both $\ ext{T}$ and mirror symmetry, with the $E_g$ splitting predominantly governed by $MBC$ and the $E_u$ splitting largely influenced by a Fano resonance, while mode-resolved $MBC$ maps explain the observed anisotropy and provide a route to identify other candidates. The framework yields design principles for large phonon magnetic moments and predicts materials such as FeCo and Co$_2$MnSi to host significant FM-order–driven phonon splitting, offering a predictive path to phonon magnetism and Hall-type lattice responses from first principles.

Abstract

Conventional ab initio methods fail to describe the emergence of chiral phonons driven by electronic ordering. Here, we develop an ab initio framework, grounded in molecular Berry curvature (MBC), that captures electronic-order-driven symmetry breaking in lattice dynamics and is applicable to both insulating and metallic magnets. Using Co$_3$Sn$_2$S$_2$ as a model system, we show that the MBC term simultaneously breaks time-reversal and mirror symmetries, enabling a quantitative reproduction of the experimentally observed phonon splittings. The analysis uncovers distinct microscopic origins for the $E_g$ and $E_u$ modes: the $E_g$ splitting is governed by MBC and is accurately described by our first-principles scheme, whereas the $E_u$ splitting is enhanced by the Fano resonance, consistent with its asymmetric spectral profile. Leveraging this framework, we further predict several candidates with chiral-phonon splitting. Our results establish a predictive route for identifying and understanding phonon magnetism, chiral phonons, and related Hall-type lattice responses from first principles.

Figures (3)

• Figure 1: Schematic illustration of lattice dynamics in magnetic materials under the standard ab initio approach and the MBC-corrected framework. (a) The model lattice preserves $C_3$, inversion ($\mathcal{P}$), vertical mirror ($\sigma_v$), and time-reversal ($\mathcal{T}$) symmetries. Ferromagnetic order along the out-of-plane direction breaks both $\sigma_v$ and $\mathcal{T}$. (b) Phonon spectra obtained by standard ab initio calculations, where the doubly degenerate phonon dispersion is enforced by $\sigma_v$ and $\mathcal{T}$ in the dynamical matrix $K(\bm{q})$. (c) By introducing the MBC-related term $G(\bm{q})$ into the lattice dynamics, our algorithm accurately captures the phonon spectra of magnetic materials. The MBC term generates opposite phonon magnetic moments $M_{ph}$ for right- and left-handed rotational vibrations, thereby breaking $\sigma_v$ and $\mathcal{T}$ symmetries. (d) Schematic illustration of the molecular Berry curvature arising from the adiabatic evolution of the electronic ground state $\Phi_{0}$ under different phonon modes. Phonon modes with opposite circular polarizations acquire MBC corrections of opposite sign.
• Figure 2: (a) Crystal structure of Co$_3$Sn$_2$S$_2$. Brown arrows indicate the local magnetic moments on Co atoms. This magnetic order breaks the vertical mirror and time-reversal symmetry, but keeps the $C_3$ and inversion symmetry. (b) Brillouin zone of Co$_3$Sn$_2$S$_2$. (c) Phonon spectrum of FM-ordered Co$_3$Sn$_2$S$_2$ with SOC, excluding the molecular Berry curvature contribution. The coordinate of $K$ is ($\frac{1}{3}$, 0, -$\frac{1}{3}$)/ ($\frac{1}{3}$, $\frac{1}{3}$, 0) in the basis of the primitive/conventional lattice vectors. (d1) Phonon dispersion of the $E_g$ mode along the $C_3$-invariant path $-\text{T} \leftrightarrow \Gamma \leftrightarrow \text{T}$, calculated without (black dashed line) and with molecular Berry curvature (blue and red solid lines). The colored stars mark the energy positions of the $E_g$ mode observed experimentally. (d2) Energy splitting of the $E_g$ mode along the $C_3$-invariant path. The insert figure shows the vibrational modes of the S atoms at the $\Gamma$ and T points. (d3) Evolution of the pseudo-angular momentum ($l_{ph}$) and the $z$-component of the angular momentum ($l_{z,\nu \bm{q}}$) along the $C_3$-invariant path. (d4) Atomic displacement patterns of the two $E_g$ modes at the $\Gamma$ point. (e1)-(e4) are the corresponding contents for the $E_u$ mode.
• Figure 3: Distribution of the MBC in the electronic BZ. (a-c) are MBC distributions for the $E_g$ phonon mode with $l_{ph} = +1$ on three distinct $k_z$ planes. The MBC distribution for the $E_g$ mode with $l_{ph} = -1$ is of opposite sign. (d-f) are MBC distributions for the $E_u$ mode with $l_{ph} = +1$. The $E_u$ mode with $l_{ph} = -1$ also exhibits a sign reversal. The MBC magnitude is significantly smaller for the $E_u$ modes, which consequently results in a smaller calculated phonon splitting.