Labyrinthine domains in ferroelectric nanoparticles: Manifestation of a gradient-induced morphological transition

In the framework of the Landau-Ginzburg-Devonshire (LGD) approach we studied finite-size effects of the phase diagram and domain structure evolution in spherical nanoparticles of a uniaxial ferroelectric. The particle surface is covered by a layer of screening charge characterized by finite screening length. The phase diagram, calculated in coordinates particle radius and screening length has a wide region of versatile polydomain structures separating single-domain ferroelectric and nonpolar paraelectric phases. Unexpectedly, we revealed a region of irregular labyrinthine domains in the nanoparticles of uniaxial ferroelectric CuInP₂S₆ with the first-order paraelectric-ferroelectric phase transition. We established that the origin of labyrinthine domains is the mutual balance of LGD, polarization gradient, and electrostatic energies. The branching of the domain walls appears and increases rapidly when the polarization gradient energy decreases below the critical value. Allowing for the generality of the LGD approach, we expect that the gradient-induced morphological transition can be the source of labyrinthine domain appearance in many spatially confined ferroics with long-range order parameter, including relaxors, ferromagnetics, antiferrodistortive materials, and materials with incommensurate ferroic phases.