Phonon Magnetochiral Effect (PMCE)

• PMCE is a nonreciprocal phonon phenomenon in chiral systems where propagation velocity differs based on the relative orientation of phonon wave vector and magnetic field.
• Experimental studies in chiral magnets and Weyl semimetals have quantified PMCE using metrics like Δv/v₀ and the magnetochiral coefficient, demonstrating frequency and field-dependent effects.
• Theoretical models emphasize symmetry breaking and Berry curvature, offering material design insights to optimize magnon–phonon hybridization for advanced phononic applications.

The phonon magnetochiral effect (PMCE) is a bulk nonreciprocal phenomenon in which the propagation velocity and attenuation of phonons differ depending on the relative orientation between the phonon wave vector and an applied magnetic field in a medium lacking inversion symmetry. This effect is rooted in fundamental symmetry requirements: it emerges only when both spatial inversion and time-reversal symmetries are simultaneously broken, typically realized in chiral crystals under external magnetic fields. PMCE is observable as a linear-in-field difference in acoustic or thermal transport, with the nonreciprocal velocity quantified by $\Delta v/v_0$ or a dimensionless magnetochiral coefficient $g_{\rm mCh}$, and arises from mechanisms ranging from magnon-phonon hybridization in chiral magnets to Berry curvature and orbital moment effects in topological semimetals. Experimental and theoretical studies have delineated the physical origin, symmetry selection rules, and material-dependent tunability of PMCE in both insulating and metallic systems, as well as in Weyl semimetals where Berry-phase phenomena dominate (Nomura et al., 2022, Nomura et al., 2018, Sengupta, 27 Nov 2025, Sengupta et al., 2020).

1. Symmetry Principles and Selection Rules

PMCE is fundamentally a nonreciprocal phonon transport phenomenon. Its observation necessitates two symmetry conditions:

• Broken spatial inversion (chirality): Chiral crystal structures, such as those in space groups $P4_1 32$ or $P4_3 32$, ensure that the phonon propagation along $+\mathbf{k}$ is not symmetrically equivalent to $-\mathbf{k}$.
• Broken time-reversal symmetry (magnetic field): Application of a magnetic field $\mathbf{H}$ establishes a time-reversal-odd axis.

The allowed free-energy contributions then contain terms proportional to $(\mathbf{k} \cdot \mathbf{H})$, whose sign reverses under inversion of crystal chirality ($\sigma \to -\sigma$), wave vector ($\mathbf{k} \to -\mathbf{k}$), or magnetic field ($\mathbf{H} \to -\mathbf{H}$) (Nomura et al., 2018). This symmetry directly leads to the selection rule: the sign of the nonreciprocal effect reverses upon inversion of any one of $\sigma$, $\mathbf{k}$, or $\mathbf{H}$. In the Voigt geometry ($\mathbf{k} \perp \mathbf{H}$), PMCE is forbidden by symmetry, while it is maximal for the Faraday configuration ($\mathbf{k} \parallel \mathbf{H}$).

2. Experimental Realizations and Quantification

Two principal material classes have been investigated for PMCE:

• Chiral magnets (insulators): Cu$_2$OSeO$_3$ exhibits PMCE below its Curie temperature ($T_C \approx 58$ K), detected via phase-sensitive ultrasound velocity measurements. The dimensionless nonreciprocal coefficient $g_{\rm mCh}$ is extracted from velocity differences in the field-reversed configuration (Nomura et al., 2018).
• Chiral magnets (metals): Co$_9$Zn$_9$Mn$_2$, with a $\beta$-Mn-type chiral structure, exhibits robust PMCE up to 250 K, exceeding the temperature range of Cu$_2$OSeO$_3$. Nonreciprocity in velocity and attenuation are mapped against field, frequency, and temperature. The PMCE is most prominent in the conical magnetic phase and peaks near the conical–collinear transition (Nomura et al., 2022).
• Chiral Weyl semimetals: PMCE is predicted in Weyl materials via electronic mechanisms. Here, the effect is intrinsically linked to Berry-curvature and orbital magnetic moment corrections in the presence of a magnetic field, without requiring magnetic order or hybridization with magnons (Sengupta, 27 Nov 2025, Sengupta et al., 2020).

The effect is quantified by: $$\Delta v/v_0(H) \equiv \frac{v(\mathbf{k},H) - v(\mathbf{k},-H)}{v_0},$$ or equivalently,

$$g_{\rm mCh}(H) \equiv \frac{\Delta v(+H)}{v_0} - \frac{\Delta v(-H)}{v_0}.$$

For magnon–phonon hybridization scenarios, the nonreciprocal coefficient is: $$g_{\rm mCh} = \frac{4 \gamma^2 \langle S^z \rangle^2 S^2 D^3 k^3}{c \Delta_0^2} \propto f^3,$$ where parameters specify the magnetoelastic coupling ($\gamma$), DM interaction ($D$), relevant spins, phonon wave vector $k=2\pi f/v$, elastic constant $c$, and magnon gap $\Delta_0$ (Nomura et al., 2022, Nomura et al., 2018).

3. Microscopic Mechanisms

A. Magnon–Phonon Hybridization (Chiral Magnets)

In chiral ferromagnets, the Dzyaloshinskii–Moriya (DM) interaction gives rise to an asymmetric magnon dispersion,

$\omega_m(k) = \Delta_0 + Ak^2 + Dk,$

which, when hybridized with circularly polarized acoustic phonons, leads to an anticrossing. The band repulsion is $k$-asymmetric, resulting in $v(+k) \ne v(-k)$. Only the phonon mode with polarization matching the magnetization direction hybridizes, maximizing the PMCE under the resonance condition ($v k^* \simeq \omega_m(k^*)$). The hybridization gap and nonreciprocal shift scale with $\Delta_0^{-2}$, giving a sensitive dependence on magnon gap and bandwidth (Nomura et al., 2022, Nomura et al., 2018).

B. Berry Curvature and Orbital Moment Mechanism (Weyl Semimetals)

In chiral Weyl semimetals, electronic band-geometry effects provide an alternate PMCE mechanism. Absence of inversion symmetry leads to inequivalent Fermi velocities $v_F^{(\pm)}$ and energies $\varepsilon_F^{(\pm)}$ at Weyl nodes. Deformation-potential electron–phonon coupling induces a dynamical chiral imbalance among these nodes. Berry curvature $\Omega(\mathbf{p})$ and orbital magnetic moment $m(\mathbf{p})$ lead to $B$-linear corrections to the electronic distribution, producing a feedback force on the phonon lattice and nonreciprocal corrections to phonon dispersion. The resulting frequency shifts for acoustic ($\Delta \omega^{\rm ac}$) and optical ($\Delta \omega^{\rm op}$) phonons scale as (Sengupta, 27 Nov 2025, Sengupta et al., 2020): $$\Delta \omega^{\rm ac} \propto q^3 B,\quad \Delta \omega^{\rm op} \propto qB,$$ with amplitudes controlled by Berry curvature, electron–phonon couplings, and slow inter-node relaxation rate $\Gamma_E$ (chiral anomaly enhancement).

4. Frequency, Field, and Temperature Dependence

• Frequency: In both magnon–phonon hybridization and Weyl semimetal models, PMCE magnitude increases rapidly with frequency. Specifically, $g_{\rm mCh} \propto f^3$ (acoustic modes), a dependence confirmed experimentally from 200–950 MHz (Nomura et al., 2022, Nomura et al., 2018).
• Field: PMCE peaks at field-induced phase boundaries—particularly the conical–collinear boundary in chiral magnets—where the anticrossing condition optimizes magnon–phonon hybridization. In Weyl semimetals, the field dependence is strictly $B$-odd and linear in $B$ for small fields, as expected from symmetry (Sengupta, 27 Nov 2025, Sengupta et al., 2020).
• Temperature: In insulating magnets (Cu$_2$OSeO$_3$), PMCE collapses above $T_C$ due to vanishing ordered moment and increased magnon damping. In metallic magnets (Co$_9$Zn$_9$Mn$_2$), the effect is enhanced up to 250 K; the temperature dependence is governed by competing effects: magnon–phonon coupling increases as electron–magnon scattering (Gilbert damping) is suppressed with warming, even as the ordered moment $\langle S^z\rangle$ diminishes. In Weyl semimetals, the main constraint is the intervalley relaxation rate $\Gamma_E$ (Nomura et al., 2022, Sengupta et al., 2020).

5. Theoretical Descriptions and Key Formulas

In both classes of materials, the PMCE has been formulated using semiclassical and field-theoretical approaches:

• Hybridized magnon–phonon Hamiltonian: Coupling induced by strain-modulated Dzyaloshinskii–Moriya interaction, with chiral Lagrangian terms of the form

$$\mathcal{L}_{\rm me} = \gamma D \langle S^z \rangle [ -\partial_z u_x \partial_z S^y + \partial_z u_y \partial_z S^x ],$$

quantifies the interaction in collinear/chiral magnets (Nomura et al., 2018).

• Berry-phase corrected Boltzmann-kinetic theory: This approach incorporates nodal Berry curvature and orbital magnetic moment contributions to the electron distribution, leading to field- and geometry-dependent drag forces on the phonon lattice (Sengupta, 27 Nov 2025, Sengupta et al., 2020). The predicted fractional velocity shift and attenuation difference are

$$v_{\rm MC} \simeq \frac{e |C|}{\pi^2 \hbar^2 \rho c_s} \frac{1}{\Gamma_E^2} \frac{\langle 1^{(+)} \rangle - \langle 1^{(-)} \rangle}{\langle 1^{(+)} \rangle + \langle 1^{(-)} \rangle} (q_z |q_z| B_z) (\delta \lambda)^2,$$

$$r_{\rm MC} \sim \frac{L e |C|}{\pi^2 \hbar^2 \rho c_s^2} \frac{1}{\Gamma_E} (q_z |q_z| B_z) \frac{1}{\frac{\langle \lambda_{zz}\rangle}{\langle 1\rangle} \delta \lambda},$$

where $C$ is the monopole charge of the Weyl node, and $\delta \lambda$ is the difference in deformation-potential couplings between nodes.

PMCE is therefore sensitive to frequency, field, material disorder, magnon and electron lifetimes, and band-geometry parameters.

6. Experimental Signatures and Material Design

Measurement techniques include ultrasound pulse-echo velocity determinations, phase-sensitive detection (MHz–GHz range), and Brillouin/Raman spectroscopy for optical phonon branches (Nomura et al., 2022, Nomura et al., 2018, Sengupta, 27 Nov 2025). Key observable signatures are as follows:

Observable	Typical Magnitude	Materials
$\Delta v/v_0$	$10^{-7}$–$10^{-5}$	Chiral magnets, WSMs
$g_{\rm mCh}$	$0.2 \times 10^{-7}$–$0.8 \times 10^{-7}$ (Co$_9$Zn$_9$Mn$_2$)	(Nomura et al., 2022)
Attenuation $\Delta r$	$0.1$–$1$ (fractional)	Chiral WSMs
Upper temperature limit	58 K (insulator), 250 K (metal)	Cu$_2$OSeO$_3$, Co$_9$Zn$_9$Mn$_2$

Material engineering for enhanced PMCE involves reducing magnon or electron damping (minimize disorder, select materials with low Gilbert damping, utilize end-member alloys), increasing DM coupling or band-geometry effects, tuning elastic constants or magnon-phonon overlap, and nanostructuring to create minibands with strong hybridization (Nomura et al., 2022).

7. Outlook and Applications

PMCE research continues to advance in exploring new symmetry platforms and operational regimes:

• High-temperature and room-temperature PMCE is now realized in metallic chiral magnets, providing new possibilities for nonreciprocal phonon devices (Nomura et al., 2022).
• Berry-phase–driven PMCE in WSMs opens a pathway to nonreciprocal acoustic and optical devices without requiring magnetic order, with enhanced effects via intervalley relaxation and chiral anomaly physics (Sengupta, 27 Nov 2025, Sengupta et al., 2020).
• Experimental detection thresholds are now within reach for velocity shifts down to $10^{-6}$ and attenuation nonreciprocity of $0.1$–$1$, enabling the exploration of PMCE signatures across diverse materials and probes.

The design principle emerging from comparative studies is clear: maximize magnon or electron–phonon hybridization, minimize linewidths (Gilbert or intervalley relaxation), and optimize chiral lattice characteristics to achieve large and tunable PMCE for applications in phononic information transport, magnetoacoustic transduction, and topological phononics.