The piezoelectric effect is a cross-correlation phenomenon governed by crystal symmetry. It is active in 20 point groups, excluding the point group O, and is permitted in all inversion symmetry breaking (ISB) point groups. Studies have shown that electric dipoles, octupoles, and toroidal quadrupoles (ETQs) can function as active multipoles for the piezoelectric effect.

Cd₂Re₂O₇ is a fascinating material known as a spin–orbit-coupled metal. At room temperature, Cd₂Re₂O₇ exists in a centrosymmetric cubic pyrochlore structure (Phase I, point group O_h). However, upon cooling, Cd₂Re₂O₇ undergoes an ISB phase transition to Phase II (point group D_2d) at 205 K, followed by successive transitions to Phase XI (point group D₂) at 120 K and Phase III (point group D₄) at 100 K before eventually exhibiting superconductivity at 1 K. Unlike standard phase transitions driven by atomic shifts, the ISB in this metal is believed to be primarily electronic in origin. The phenomenon is driven by an odd-parity multipolar order parameter known as an ETQ and thus provides an example of an electronic nematic state.

In this study, a laser Doppler vibrometer with high sensitivity was used to detect the extremely subtle structural changes associated with this electronic order by measuring nanometer‑scale surface displacements induced by an AC electric field. The results revealed that a finite piezoelectric response emerges only below the ISB transition. This confirms that the ETQ order breaks the crystal’s inversion symmetry.

However, the pronounced directional anisotropy of the piezoelectric response was also a striking discovery. Two configurations were compared, including an electric field E applied in parallel (E //) and perpendicular (E ⊥) to the axis [111]. Specifically, for E // [111], the response peaked in Phases II and XI but vanished in Phase III. Conversely, for E ⊥ [111], the response was suppressed in Phase II, emerged in Phase XI, and became substantial in Phase III. This behavior is explained by a reconfiguration of the ETQ order parameter within the two‑component E_u representation. Specifically, Phase II is dominated by an x²–y²-type ETQ, while Phase III involves a shift to a 3z²–r²-type ETQ. This internal “rotation” of the electronic order parameter changes the components of the piezoelectric tensor and thus causes the observed enhancement or cancellation of the signal depending on the field geometry.

Overall, these results establish piezoelectricity as a direct and symmetry-sensitive tool for mapping complex electronic orders in quantum materials.

(Written by Kazuyuki Matsuhira on behalf of all authors.)

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