Vortex structure transformation of BaTiO₃ nanoparticles through the gradient function

Phase field method has been used to simulate the vortex structures in BaTiO₃ (BTO) nanoparticles. Through the modulation of the gradient coefficients, vortices are found to transform in a path of monoclinic M _A →orthorhombic→monoclinic M_C →tetragonal. Although the gradient coefficients vary significantly, the change in gradient energy is remarkably small. The simulation results show that the rotation and magnitude reduction in polarization dipoles increase the bulk energy, which induces the vortex transformation process in BTO nanoparticles. Moreover, the existence of monoclinic phase is a necessity to start the polarization rotation as well as the vortex transformation process.