Phase‐Field Method of Phase Transitions/Domain Structures in Ferroelectric Thin Films: A Review

Abstract

This article briefly reviews recent applications of phase‐field method to ferroelectric phase transitions and domain structures in thin films. It starts with a brief introduction to the thermodynamics of coupled electromechanical systems and the Landau description of ferroelectric transitions in homogeneous ferroelectric single crystals. The thermodynamic potentials of a homogeneous crystal under different mechanical boundary conditions are presented, including the thin‐film boundary conditions. The phase‐field approach to inhomogeneous systems containing domain structures is then outlined. It describes a domain structure using the spatial distribution of spontaneous polarization. The evolution of a domain structure towards equilibrium is driven by the reduction in the total‐free energy of an inhomogeneous domain structure including the chemical driving force, domain wall energy, electrostatic energy as well as elastic energy. A number of examples are discussed, including phase transitions and domain stability in ferroelectric thin films and superlattices. It is demonstrated that using a set of independently measured thermodynamic parameters for the corresponding bulk single crystals, the phase‐field approach is able to quantitatively predict not only the strain effect on phase transition temperatures but also the correct ferroelectric domain structures for a given strain and temperature.

Ferroelastic domain switching dynamics under electrical and mechanical excitations

“…Rather, switching occurs because of the stabilization of the c-domain by the strain state caused by both the applied strain and in-plane clamping of the film. Application of strain, e yy , along the viewing direction results in an out-of-plane elastic strain, e zz , of 33 e zz ¼ À c 12 ðe xx þ e yy Þ=c 11 where e xx is the in-plane coherency strain that remains unchanged in the clamped film with application of strain along the viewing direction, and c 11 and c 12 are the elastic stiffness constants of the cubic parent material used as reference in the model. At higher compressive e yy strains the strain e zz is increased, which results in increased tetragonality of the system and stability of the c-domain 33,34 .…”

Atomic-scale mechanisms of ferroelastic domain-wall-mediated ferroelectric switching

“…Details about solving equations (7)(8)(9) for the film/substrate system and the required boundary conditions are presented in depth elsewhere 23,39 . Domain wall energy in the system was included through the polarization gradient energy.…”

“…Phase-field modelling is a mesoscale-modelling approach that evolves the distribution of the three polarization components, P i , in time towards a minimum energy state by solving the time-dependent Ginzburg-Landau equation [22][23][39][40][41] …”

“…The thickness of the film was taken to be 56D while the substrate was assumed to have a thickness of 14D (ref. 39). Each cubic voxel was assumed to have an edge length of 1 nm.…”

Film size-dependent voltage-modulated magnetism in multiferroic heterostructures

“…These domain structures would lead to different voltage-induced magnetizationswitching characteristics accordingly [57,58], which is critical to the potential device application [59,60]. Here, we use a phase-field model [81][82][83] to investigate such lateral size-dependent voltage manipulation of magnetization. This computational model is capable of simulating the domain structure evolution process without any a priori assumptions on the possible domain structures, and thus shows better accuracy than thermodynamic calculations [18,56].…”