Circularly-Polarized TO Phonons

• Circularly-polarized TO phonons are quantized lattice vibrations whose atomic displacements follow a circular trajectory, endowing them with intrinsic angular momentum and handedness.
• They are characterized by polarization-dependent band splitting revealed through Raman, infrared, and inelastic X-ray scattering techniques, linking symmetry-breaking mechanisms with measurable optical effects.
• Their interaction with light and electron–phonon coupling facilitates innovative applications in magnetization control, topological transport, and spintronic device engineering.

Circularly-polarized transverse-optical (TO) phonons are quantized lattice vibrations whose atomic displacements trace a circular trajectory in the plane transverse to the direction of propagation or symmetry axis. These excitations, often called “chiral phonons,” possess well-defined angular momentum and can exhibit handedness (left- or right-circular polarization), typically arising when lattice, symmetry, or coupling conditions lift the degeneracy of otherwise linearly polarized TO modes. The paper of circularly-polarized TO phonons has advanced through both experimental probes (such as Raman, infrared, and inelastic X-ray scattering) and theoretical developments invoking Berry curvature, symmetry analysis, and nonlocal geometric forces. Their relevance extends from fundamental lattice dynamics to optically-controlled magnetism, topological transport, and magneto-phononic device engineering.

1. Symmetry, Micropolar Elasticity, and the Conditions for TO Phonon Chirality

Micropolar elasticity theory provides a foundational framework for understanding chiral phonons. In this continuum description, each point in the crystal is endowed with both a displacement vector $u(\mathbf{r})$ and a microrotation vector $\phi(\mathbf{r})$. In chiral (noncentrosymmetric) crystals, the quadratic strain energy density includes a chiral coupling term:

$$U = \tfrac{1}{2}A_{klmn} \varepsilon_{kl} \varepsilon_{mn} + \tfrac{1}{2} B_{klmn} \gamma_{kl} \gamma_{mn} + C_{klmn} \varepsilon_{kl} \gamma_{mn},$$

where $C_{klmn}$ encodes chirality (odd under parity) while time-reversal symmetry may remain preserved (Kishine et al., 2020). This structural term yields equations of motion wherein the transverse modes, written in a circular basis $u_+ = u_1 + i u_2$, $u_- = u_1 - i u_2$, split into left- and right-circularly polarized eigenmodes.

The eigenfrequencies depend on polarization:

$$[\omega_{(\alpha)}^{(O/A)}]^2 = \frac{1}{2\rho j}[b_{(\alpha)} + j a_{(\alpha)} \pm \sqrt{(b_{(\alpha)} - j a_{(\alpha)})^2 + 4j\Delta_{(\alpha)}^2}],$$

with $\alpha = \pm1$ for circular polarizations, and $a_{(\pm)}, b_{(\pm)}, \Delta_{(\pm)}$ determined by elastic coefficients and chiral terms. The presence of linear-in-gradient terms ($\partial_3$ multiplied by chiral coefficients) leads to a polarization-dependent band splitting even in the absence of external fields or explicit time-reversal symmetry breaking (Kishine et al., 2020). This splitting is a direct result of parity violation, and the two circular modes propagate with unequal group velocities, yielding true phononic chirality.

2. Berry Curvature, Nonlocal Geometric Forces, and the Emergence of Chiral Optical Phonons

An alternative microscopic mechanism, dominant in systems with broken time-reversal symmetry (for example, magnetic or topological insulators), is the emergence of a molecular Berry curvature in the Born–Oppenheimer framework. Here, the adiabatic evolution of the electronic ground state $\Phi_0(\{R\})$ with ionic positions $\{R\}$ assigns a geometric gauge field $A_{l,\kappa}$ to each ion (Saparov et al., 2021, Chen et al., 4 Jun 2025):

$$A_{l,\kappa} = i\langle \Phi_0 | \nabla_{l,\kappa} \Phi_0 \rangle,$$

which, upon quantization, gives a gauge-invariant curvature $G$ appearing as a nonlocal geometric force in the lattice Hamiltonian (Chen et al., 4 Jun 2025). This “magnetic” term modifies the equations of motion, such that:

$$\frac{d}{dt} \begin{pmatrix} u_k \ p_k \end{pmatrix} = \begin{pmatrix} \tilde{G}_k & I \ -D_k & \tilde{G}_k \end{pmatrix} \begin{pmatrix} u_k \ p_k \end{pmatrix}.$$

The Berry curvature $\tilde{G}_k$ typically lifts the degeneracy of the optical modes at the $\Gamma$ point, leading to split, circularly-polarized phonon branches:

$$\delta \omega \approx \pm \frac{\hbar}{M}\, \mathrm{Re}[G(\mathbf{k}=0)]$$

(Saparov et al., 2021, Chen et al., 4 Jun 2025). The phonon eigenmodes become $\psi = (1/\sqrt{2})(\gamma_1 \pm i \gamma_2)$, combining basis vectors with phase difference to yield left- and right-circular polarization and nonzero intrinsic angular momentum per phonon.

This framework has been rigorously connected to real material systems: for example, monolayer CoCl$_2$ displays a splitting of $\sim 3\times10^{-2}$ THz in optical branches due to such a nonlocal Berry curvature-derived force, transforming them into circularly-polarized (chiral) modes characterized by quantized pseudoangular momentum (Chen et al., 4 Jun 2025).

3. Experimental Probes and Symmetry Constraints: Raman, Infrared, and X-ray Spectroscopy

The identification and analysis of circularly-polarized TO phonons rely on a suite of polarization-sensitive spectroscopies, including Raman, infrared (IR), and inelastic X-ray scattering (IXS):

• Raman Spectroscopy: In chiral crystals (e.g. tellurium, point group 32), selection rules determined by the conservation of pseudo-angular momentum (PAM) dictate that only certain circularly-polarized TO modes (e.g. $\Gamma_3$ in Te) are active under specific incident/scattered photon configurations (Ishito et al., 2022). The splitting observed between RL and LR polarization channels, interpreted via the conservation law

$$\sigma_i - \sigma_s \equiv \pm m_\mathrm{PAM}^{(s)}\quad (\mathrm{mod}\ 3)$$

(with photon spin PAM $\sigma$ and phonon spin PAM $m_\mathrm{PAM}^{(s)}$), is a direct marker of phonon chirality.

• Infrared and Kerr/Ellipticity Spectroscopy: In magnetic Weyl semimetals such as Co$_3$Sn$_2$S$_2$, polar Kerr rotation and ellipticity spectroscopy record the frequency-resolved response of the sample to right- and left-circularly polarized IR light (Yang et al., 29 Oct 2024). Circular dichroism in the phonon response indicates lifting of degeneracy between E$_u$ optical phonon modes via electron–phonon coupling with topological electronic states.
• IXS: In tungsten carbide (WC), IXS probes circularly-polarized acoustic phonons near the $K, K'$ points (Cai et al., 2021). Discrepancies between in-plane and out-of-plane scattering geometries reveal the necessity to revise conventional scattering functions to properly account for the angular momentum transfer between x-rays and chiral phonon modes.

Notably, a comprehensive symmetry-based framework shows that nonzero PAM alone is insufficient to guarantee nonzero real angular momentum or measurable chirality (Zhang et al., 28 Mar 2025). Depending on crystal/magnetic point group symmetry, even a mode with “apparent” circular polarization may have vanishing net AM if sublattice or mirror-related contributions cancel.

4. Coupling to Light and Electron–Phonon Interactions: Bandgap Tuning and Exciton Manipulation

Circularly-polarized TO phonons not only couple efficiently to circularly-polarized light but also can mediate indirect optical transitions that are forbidden in the absence of chiral phononic degrees of freedom. In systems such as monolayer $\beta$-borophene under circularly-polarized electromagnetic radiation, the interplay between Floquet-induced bandgap opening and strong electron–phonon coupling via surface optical phonons (described by Fröhlich interaction and Lee–Low–Pines transformation) enables dynamic control over the electronic spectrum (Akay et al., 2022). The effective bandgap

$$E_\mathrm{gap} = 2(E_{cv} - \Sigma_{e-\mathrm{ph}})$$

depends both on the photoinduced mass gap ($\Delta$) from light and the polaronic self-energy shift arising from dressing by TO phonons.

In magnetically ordered or topological materials with narrow band gaps, circularly-polarized optical phonons can “light up” optically dark excitons via the conservation of pseudoangular momentum (Chen et al., 4 Jun 2025):

$$\ell_e^C - \ell_e^V = \ell_\mathrm{photon} - \ell_\mathrm{ph}\pmod{4}$$

where $\ell_e^{C,V}$ are the electronic PAM quantum numbers, and $\ell_\mathrm{ph}, \ell_\mathrm{photon}$ the phonon and photon PAM. This enables novel selection rules for optical transitions, expanding the toolkit for valley-selective optoelectronics.

5. Angular Momentum Transport, Hall Viscosity, and Spintronic Applications

Chiral TO phonons carry intrinsic angular momentum,

$$J_z^{ph} = \sum_{l,\kappa} \left(u_{l,\kappa}^x \dot{u}_{l,\kappa}^y - u_{l,\kappa}^y \dot{u}_{l,\kappa}^x\right),$$

and can transport angular momentum even in the acoustic regime (Saparov et al., 2021, Chen et al., 4 Jun 2025). Under temperature gradients, the out-of-equilibrium occupation of right- and left-moving chiral phonons results in a net angular momentum density:

$$J^{(\mathrm{ph},\alpha)} = -\frac{\hbar \tau}{V} \sum_{q,\sigma, \beta} s^\alpha_{q,\sigma} v^\beta_{q,\sigma} \frac{\partial f_0(\omega_{q,\sigma})}{\partial T}\, \frac{\partial T}{\partial x^\beta},$$

with $s^\alpha_{q,\sigma}$ the phonon “spin,” $v^{\beta}$ the group velocity and $\tau$ the relaxation time (Pols et al., 26 Nov 2024). This angular momentum can, via the Einstein–de Haas effect or coupling to electron spins, generate measurable currents or torques, with applications in spintronic and “orbitronic” devices.

In 2D halide perovskites, the CISS effect and spin Seebeck phenomena may be mediated by such chiral phonons, whose net angular momentum under thermal gradients can inject spin-polarized carriers or magnetization into adjacent materials (Pols et al., 26 Nov 2024).

Moreover, the presence of phonon Berry curvature and the splitting of TO branches in magnetic systems contributes to Hall-like effects, such as phonon angular momentum Hall currents and finite nondissipative Hall viscosity in the elastic tensor (Chen et al., 4 Jun 2025). The dynamic matrix for coupled LA/TA modes in this context includes off-diagonal terms arising from the Berry curvature, resulting in transverse flows of momentum and angular momentum.

6. Magnetization Control with Circularly-Polarized Phonons

The robust transfer of angular momentum from circularly-polarized TO phonons to a magnetic layer enables helicity-dependent magnetization reversal—achieved experimentally via resonant excitation of specific IR-active TO phonon modes in a substrate (Fennema et al., 1 Sep 2025). In such systems, a polarization-modulated transient grating—composed of two perpendicularly polarized IR pulses—creates spatially varying circular polarization across the sample, driving localized magnetization switching in a magnetic overlayer.

The mechanism’s efficiency is maximized at phonon resonance, and is resilient to moderate deviations from perfect ellipticity, while being highly sensitive off-resonance. This demonstrates a route for ultrafast, energy-efficient, non-thermal magnetization control, scalable to a variety of device architectures.

7. Quantitative and Symmetry-Based Criteria: Angular Momentum, PAM, and Helicity

The distinction between angular momentum (AM), pseudo-angular momentum (PAM), and helicity is crucial for the classification and detection of circularly-polarized TO phonons (Zhang et al., 28 Mar 2025, Ishito et al., 2022). These quantities are defined by:

• AM: $$l = \hbar\, \mathbf{\epsilon}^\dagger\, M\, \mathbf{\epsilon}$$, with $\mathbf{\epsilon}$ the phonon eigenvector and $M$ the angular momentum generator.
• PAM: determined via the action of the $n$-fold rotation operator, $$\hat{C}_n\, \mathbf{\epsilon} = e^{-i \frac{2\pi l_{ph}}{n}}\, \mathbf{\epsilon}$$.
• Helicity: projection of AM onto the phonon propagation direction, $h = \hat{\mathbf{q}} \cdot \mathbf{l}$.

Analyses based solely on PAM (e.g., the flipping of CPL polarization in Raman) may not uniquely reveal true chiral phonon character unless the point group and magnetic symmetries are properly accounted for. This has been validated by both symmetry analysis and experiments across a range of crystals and magnetic phases (Zhang et al., 28 Mar 2025).

Summary Table: Mechanisms and Observations

Mechanism	Material Examples	Key Observables
Micropolar elasticity, parity breaking	Chiral micropolar media	Polarization-dependent band splitting, "roton" minimum
Molecular Berry curvature, nonlocal geometric force	CoCl$_2$, Haldane model	Splitting of optical branches, circular polarization
Electron–phonon coupling with electronic topology	Co$_3$Sn$_2$S$_2$	Phonon circular dichroism, Fano lineshape asymmetry
Circularly-polarized light, Floquet effect	$\beta$-borophene	Tunable bandgap, manipulation of optical transitions
Phonon-magnon hybridization	FePSe$_3$	Nondegenerate elliptically polarized phonons (magnon polarons)

The paper of circularly-polarized transverse-optical phonons thus forms a nexus at which symmetry, Berry phase phenomena, lattice dynamics, and magneto-optical interactions intersect, providing both a refined framework for understanding fundamental phonon properties and opportunities for leveraging chiral excitations in informational and functional materials design.