Intertwined Swirling Polarization States in BaTiO$_3$ with Embedded BaZrO$_3$ Nanoregions

Ferroelectric materials embedded with dielectric inclusions offer a unique platform for exploring novel topological polar textures. Using first-principles-based atomistic simulations, we investigate the polarization behavior of a BaTiO$_3$ matrix containing segregated BaZrO$_3$ nanoregions. We demonstrate that the polar texture in three-dimensionally ordered arrays of dielectric inclusions is governed by their size and spacing, revealing three distinct regimes. At large separations, the nanocomposite exhibits bulk-like BaTiO3 phase transitions, while at smaller spacings, interconnected swirling polarization patterns give rise to vortex supercrystal states. We analyze the stabilization mechanisms of these states and show that each regime is characterized by distinct switching behavior. Furthermore, we find that nanocomposites with randomly distributed dielectric inclusions exhibit swirling polarization textures, giving rise to an amorphous network of entangled vortices. Our findings provide new insights into the physics of relaxor ferroelectrics, are consistent with recent experimental observations, and open up new possibilities for designing materials with emergent topological functionalities.

💡 Research Summary

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This paper investigates the polarization behavior of BaTiO₃ (BT) matrices containing embedded BaZrO₃ (BZ) nanoregions using first‑principles‑based atomistic simulations. By employing an effective‑Hamiltonian molecular‑dynamics approach, the authors systematically vary the lateral size (d) and inter‑inclusion spacing (s) of cubic (and spherical) BZ inclusions arranged in three‑dimensional periodic arrays. Two order parameters are monitored as a function of temperature: the macroscopic ferroelectric polarization (P) and the toroidal moment per inclusion (G). The simulations reveal three distinct regimes.

1. Bulk‑like regime (large s, low Zr concentration). The composite reproduces the classic rhombohedral‑orthorhombic‑tetragonal‑cubic (R‑O‑T‑C) sequence of pure BT. Near each dielectric inclusion the local polarization bends, forming an anti‑hedgehog (converging) texture that resembles laminar flow around a sink‑like obstacle.

2. Vortex‑supercrystal (VSC) regimes (intermediate s). As the spacing decreases, the inclusions act as nucleation centers for intertwined swirling polarization patterns. Depending on d and s, the system stabilizes V₂, V₄, and V₆ phases, characterized by 2, 4, or 6 independent vortex cores converging at each inclusion. In V₂ the vortices rotate around a single axis (y‑axis), producing alternating ±z ferroelectric domains. V₄ adds a second rotation axis (z), while V₆ exhibits a fully three‑dimensional vortex lattice with non‑zero toroidal components along all Cartesian directions. The toroidal moment components become the fingerprints of each phase.

3. Random‑inclusion regime. When BZ nanoregions are placed at random positions, similar swirling textures emerge. At low Zr content (≈5 %) an ordered‑like V₂ network appears; higher concentrations (15–25 %) generate a disordered, amorphous vortex mesh reminiscent of the crystalline V₆ state.

The authors also explore electric‑field‑driven switching. P‑E hysteresis loops differ markedly across regimes: bulk‑like arrays show rectangular loops with sharp switching; V₂ displays rounded loops and a butterfly‑shaped G_y response, indicating that vortex formation mediates the transition; V₆ exhibits pinched loops, reflecting vortex pinning by the dielectric inclusions. In the random‑inclusion systems, hysteresis trends (decreasing remanent polarization, increasing coercive field with Zr concentration) match experimental observations in BaZrₓTi₁₋ₓO₃ relaxors.

Mechanistically, the larger unit‑cell volume of BZ relative to BT induces elastic strain at the interface, distorting neighboring TiO₆ octahedra and generating small local polarizations directed toward the inclusion. These interfacial dipoles seed the vortex cores, demonstrating that compositional heterogeneity alone can drive complex topological polar textures.

The findings bridge atomistic theory with recent experimental reports of vortex supercrystals, conical polarization patterns in BaTiO₃ nano‑islands, and six‑vortex superstructures in KLNT systems. They suggest a design pathway: by tuning inclusion size and spacing, one can engineer ferroelectric composites with controllable vortex lattices, offering new functionalities for nano‑electronics, data storage, and sensing applications.