Observation of acoustic magneto-chiral anisotropy in $α$-quartz

Abstract: We report the experimental observation of magneto-chiral anisotropy in the longitudinal and transverse ultrasound propagation in $α$-quartz. To perform such measurement, we have built an ultrasound spectrometer with unprecedented experimental resolution of the order of $Δv/v \sim 10^{-8}$. We present a simple macroscopic Becquerel-like analytical model that accounts for the magnitude of the observed effect and its frequency dependence.

• The paper demonstrates clear experimental evidence for aMChA in α-quartz using high-resolution ultrasound spectroscopy.
• It employs a Becquerel-type macroscopic model to predict a linear frequency dependence and non-reciprocal acoustic velocities.
• Findings reveal distinct field and polarization dependencies that can inform the design of non-reciprocal phononic devices.

Acoustic Magneto-Chiral Anisotropy in $\alpha$-Quartz: Experimental Observation and Macroscopic Modeling

Introduction

The study presents definitive experimental evidence for acoustic magneto-chiral anisotropy (aMChA) in both longitudinal and transverse ultrasound propagation in diamagnetic $\alpha$-quartz. The investigation leverages a high-resolution ultrasound spectrometer, achieving relative velocity measurement precision of $\Delta v/v \sim 10^{-8}$. The authors complement their measurements with a macroscopic Becquerel-type analytical model capturing both the magnitude and frequency dependence of the aMChA. This work directly extends the paradigm of non-reciprocal transport, previously observed for magnonic and electronic systems, into the domain of phonon propagation in non-magnetic chiral systems.

Background and Motivation

Magneto-chiral anisotropy originates in systems exhibiting both broken time-reversal symmetry (via magnetization or magnetic fields) and broken spatial inversion symmetry (chirality). This combination allows a suite of non-reciprocal phenomena in optical, electronic, and vibrational transport. Optical MChA (oMChA) is well-documented across the EM spectrum, and “electrical” MChA has been seen in a variety of conductors and semiconductors. In phononics, most prior experimental evidence for “acoustic” MChA was limited to materials with strong magnon-phonon coupling near magnetic phase transitions, where ultrasound propagation demonstrates non-reciprocal effects attributed to hybrid modes.

aMChA in purely diamagnetic chiral crystals remained unobserved, and more generally, theoretical understanding of acoustic transport non-reciprocity in such systems was comparatively underdeveloped. The work under review thus asks whether non-reciprocal propagation—free from magnonic origins—can be detected and quantitatively explained in archetypal diamagnetic chiral materials such as $\alpha$-quartz.

Theoretical Modeling

Symmetry analysis provides that, for acoustic wave velocity $v$ propagating parallel to $\mathbf{B}$ in chiral media, the form

$$v_{\pm} = v(\omega) \pm \alpha^{D/L}(\omega) k \pm \beta(\omega) B + \gamma^{D/L}(\omega) (\mathbf{k} \cdot \mathbf{B}),$$

is symmetry-allowed, where $\gamma$ parameterizes the polarization-independent aMChA. $\alpha$, $\beta$ encode acoustic activity and the Faraday effect, respectively.

The authors then formulate a Becquerel-inspired model for the effect of the magnetic field as an effective frequency shift (Larmor frequency), yielding explicit expressions for the expected aMChA:

$\alpha$0

where $\alpha$1 is an effective charge-to-mass ratio of composite degrees of freedom, and $\alpha$2 is a phenomenological parameter for second-order spatial dispersion in acoustic activity.

Remarkably, the model predicts a linear frequency dependence for aMChA in diamagnetic chiral crystals, in contrast with the cubic frequency scaling characteristic of magnon-phonon mixed systems. Using literature data for $\alpha$3-quartz, the model yields $\alpha$4 at 1 T and 1 GHz for transverse waves and a factor of $\alpha$5 lower for longitudinal modes.

Experimental Methodology

The group engineered a dedicated ultrasound interferometric setup utilizing ZnO transducers configured on $\alpha$6-cut $\alpha$7-quartz samples. The configuration simultaneously launches counterpropagating pulses and measures their velocity difference, suppressing symmetric (reciprocal) artefacts. An alternating (AC) magnetic field and phase-sensitive detection further improves signal isolation, delivering a measurement sensitivity two orders of magnitude better than prior ultrasound-based MChA studies.

Both right- and left-handed natural $\alpha$8-quartz samples were used, and both longitudinal and linearly polarized transverse modes were investigated at room temperature over a frequency range spanning several hundreds of MHz.

Results

The experiment demonstrates a robust linear dependence of the non-reciprocal velocity shift ($\alpha$9) on magnetic field amplitude, with the sign of the effect reversing according to crystal handedness. The magnitude and sign characteristics are fully consistent with magneto-chiral symmetry predictions.

Quantitatively, for longitudinal waves at 340 MHz, the observed field slope is $\Delta v/v \sim 10^{-8}$0 T$\Delta v/v \sim 10^{-8}$1, versus a theoretical estimate of $\Delta v/v \sim 10^{-8}$2 T$\Delta v/v \sim 10^{-8}$3. For transverse waves at 352 MHz, the measured slope is $\Delta v/v \sim 10^{-8}$4 T$\Delta v/v \sim 10^{-8}$5, compared to a predicted $\Delta v/v \sim 10^{-8}$6 T$\Delta v/v \sim 10^{-8}$7. The linearity in frequency, as predicted by the model, is confirmed for both polarization modes.

A notable deviation from model expectation is the observation that the sign of aMChA reverses between longitudinal and transverse propagation for a fixed handedness, an effect not explained by the present macroscopic treatment but likely arising from the approximations made regarding longitudinal mode chiral coupling.

Interestingly, the aMChA magnitude in the GHz/1 T regime outpaces the corresponding oMChA in $\Delta v/v \sim 10^{-8}$8-quartz at optical frequencies by three orders of magnitude.

Broader Implications and Outlook

The findings imply that all diamagnetic chiral crystals—by virtue of their structure and elemental response to applied magnetic fields—must display detectable aMChA, provided sufficient experimental sensitivity. This carries implications for the design and engineering of acoustic diodes, non-reciprocal phononic devices, and potentially thermal management, since low-temperature heat transport in dielectric crystals is phonon-dominated and would inherit MChA-originated non-reciprocity.

The model offers guidance on materials optimization for aMChA via selection for higher acoustic activity (strong $\Delta v/v \sim 10^{-8}$9), specific frequency windows, and suitable charge-to-mass coupling. The absence of high-precision data for non-quartz, diamagnetic, and molecular chiral crystals leaves open the systematic exploration of the phenomenon. The observations also suggest that counterpart effects—such as acoustic wave-induced longitudinal magnetization in chiral crystals (inverse MChA)—should exist, providing a pathway to novel non-reciprocal and chiral-photonic functionalities.

Conclusion

The work presents clear experimental validation, supported by analytical modeling, of acoustic magneto-chiral anisotropy in a classic diamagnetic chiral crystal system. The experimental results align with symmetry-based expectations and macroscopic theory up to factors of order unity, with key phenomenological trends (field, frequency, handedness dependence) quantitatively confirmed. The results generalize the class of non-reciprocal materials and invite systematic studies of aMChA across broad families of dielectric chiral crystals, underpinning both fundamental understanding and potential applications in chiral phononics and thermal transport manipulation (2604.01013).