Analytical description of domain morphology and phase diagrams of ferroelectric nanoparticles

Analytical description of domain structure morphology and phase diagrams of ferroelectric nanoparticles is developed in the framework of Landau-Ginzburg-Devonshire approach. To model realistic conditions of incomplete screening of spontaneous polarization at the particle surface, it was considered covered by an ultra-thin layer of screening charge with effective screening length. The phase diagrams, calculated for spherical Sn₂P₂S₆ nanoparticles in coordinates “temperature” and “effective screening length” by finite element modelling, demonstrate the emergence of poly-domain region at the tricritical point and its broadening with increasing the screening length for the particle radius over a critical value. Metastable and stable labyrinthine domain structures exist in Sn₂P₂S₆ nanoparticles with radius (8–10) nm and more. We derived simple analytical expressions for the boundaries between paraelectric, single-domain and poly-domain ferroelectric phases, tricritical point and the necessary condition for the appearance of labyrinthine domains, and demonstrated their high accuracy in comparison with finite element modelling results. Analytical expressions for the dependence of the ferroelectric-paraelectric transition temperature on the particle radius in the single-domain and poly-domain states of the particle were compared with analogous dependences experimentally measured for SrBi₂Ta₂O₉ nanoparticles and simulated for Sn₂P₂S₆ nanocrystals by Monte Carlo method within the framework of axial next-nearest-neighbours Ising model. The analytical expression for the ferroelectric-paraelectric transition temperature in the poly-domain state quantitatively agrees with experimental and simulated results, and it accurately reproduces empirical Ishikawa equation at all temperatures, justifying it theoretically. Analytical description shows that phase diagrams and domain morphologies, which are qualitatively similar to the ones calculated in this work, can be expected in other ferroelectric nanoparticles covered by the screening charges, being rather different for the ferroelectrics with the first and second order ferroelectric-paraelectric transitions respectively.