Goldstone-mediated polar instability in hexagonal barium titanate

We discover a rare structural manifestation of the Goldstone paradigm in a hexagonal polytype of the archetypal ferroelectric BaTiO3. First-principles calculations confirm the Goldstone character of the order parameter, and high-resolution diffraction measurements link this to a quasi-continuous domain texture in the vicinity of the low-temperature phase transitions. Our findings highlight how changes in structural topology may be exploited to realize rich polar topologies in bulk ferroelectric perovskites.

💡 Research Summary

In this work the authors report the discovery of a Goldstone‑like polar instability in the six‑layer hexagonal polytype of barium titanate (6H‑BaTiO₃). Using group‑theoretical analysis they identify the Γ₅⁻ irreducible representation of the high‑symmetry P6₃/mmc aristotype as the primary order parameter, describing antipolar Ti displacements in the basal plane. The two‑dimensional order‑parameter vector (η₁, η₂) can adopt three symmetry‑related directions, giving rise to the non‑polar C222₁, the polar Cmc2₁, and the intermediate P2₁ structures observed on cooling. A Landau free‑energy expansion containing a fourth‑order coupling term (F ∼ 3η₁²η₂η₃ − η₂³η₃) links the antipolar Γ₅⁻ mode to a polar Γ₂⁻ mode, stabilising the polar component only for certain orientations of the Γ₅⁻ order parameter.

First‑principles density‑functional theory (DFT) calculations map the potential‑energy surface around the aristotype with respect to the Γ₅⁻ mode. The resulting “Mexican‑hat” landscape shows an almost flat brim with energy barriers below 0.02 meV per formula unit, confirming an emergent U(1) continuous symmetry and the presence of a Goldstone boson. High‑resolution synchrotron X‑ray diffraction (S‑XRD) and neutron powder diffraction (NPD) data collected down to 10 K reveal that the ground state is best described by the Cmc2₁ structure, although conventional Rietveld refinements cannot unambiguously discriminate it from the previously proposed P2₁ model. To overcome this limitation the authors treat the accompanying Γ₅⁺ strain as a two‑dimensional proxy (ρ, φ) for the orientation of the Γ₅⁻ order parameter. By analysing the evolution of (ρ, φ) with temperature they demonstrate a clear crossover from C222₁ to Cmc2₁ via the intermediate P2₁ phase between 10 K and 100 K.

Three‑dimensional X‑ray diffraction (3D‑XRD) on an individual grain at 150 K uncovers a complex microstructure: predominantly C222₁ domains interspersed with micron‑scale Cmc2₁ and P2₁ inclusions, separated by low‑energy, curved domain walls. Diffuse scattering observed in S‑XRD below the higher‑temperature transition (T₀ ≈ 220 K) and persisting down to 10 K indicates a quasi‑continuous distribution of the Γ₅⁻ order‑parameter orientations, especially near the ferroelectric transition (T_c ≈ 70 K). Histogram analysis of the strain‑derived φ distributions shows a progressive broadening of the order‑parameter orientation as temperature approaches T_c, consistent with a flattening of the potential‑energy surface and the restoration of the emergent U(1) symmetry.

The authors extend their analysis to other hexagonal polytypes (4H, 8H, 10H) of BaTiO₃ using DFT phonon calculations, finding that the Γ₅⁻ mode remains unstable and the low‑symmetry phases (C222₁, Cmc2₁, P2₁) are energetically nearly degenerate. This suggests that Goldstone‑like modes are a generic feature of hexagonal perovskite polytypes, offering a broad compositional space for engineering exotic polar topologies without relying on extrinsic disorder.

In conclusion, 6H‑BaTiO₃ provides a model system in which a structural topology change converts the conventional second‑order Jahn‑Teller instability into a Goldstone‑mediated polar instability. The resulting continuous symmetry of the order parameter confines dipolar interactions to two‑dimensional planes, giving rise to a quasi‑continuous domain texture observable by high‑resolution diffraction. The work opens pathways for strain‑engineered control of such textures and highlights the potential of hexagonal perovskite polytypes as platforms for bulk ferroelectric devices that exploit topological polar structures.