1 Introduction

The advances in film-growth methods including the pulsed laser deposition (PLD) (Ramesh et al. 1990; Hubler 1992; Frey et al. 1994; Koster et al. 1999) and molecular beam epitaxy (MBE) (Chao and Arthur 1975; Spah et al. 1988; Schlom et al. 1988; Kwo et al. 1988; Eckstein and Bozovic 1995; Theis et al. 1998) have allowed the growth of high-quality epitaxial ferroelectric thin films. An in-plane biaxial strain is typically generated and imposed on the film via the epitaxial lattice mismatch. Compared with the ferroelectric bulk counterparts which would normally fracture under moderate strains such as ∼±0.1%, epitaxial thin films can withstand much larger in-plane biaxial strains up to ∼±3% (Schlom et al. 2007, 2008, 2014; Martin and Rappe 2017). Owing to the strong coupling between the polarization and the lattice degrees of freedom in ferroelectrics, novel phases, physics, and properties may emerge in heavily strained thin films. For example, a stress-free SrTiO3 crystal is paraelectric down to 0 K. However, when a (001)-oriented SrTiO3 was epitaxially grown on a (110)-oriented DyScO3 substrate which can impose a tensile biaxial strain of ∼0.8% on SrTiO3 film, a room-temperature ferroelectricity was observed in the SrTiO3 thin film (Haeni et al. 2004). The similar effect was observed in strained BaTiO3 thin films which show enhanced room-temperature remnant polarization and ferroelectricity at temperatures much higher than the bulk Curie point (Choi et al. 2004). While a stress-free BiFeO3 bulk crystal has a stable rhombohedral symmetry below the Curie temperature of 830 °C (Catalan and Scott 2009), a BiFeO3 thin film on a LaAlO3 substrate is subjected to a huge compressive strain (∼−4.5%) and was shown to exhibit a highly distorted tetragonal symmetry. As thickness of a BiFeO3 film increases, it gradually loses the coherency strains, leading to the formation of a rhombohedral-tetragonal alternating “stripe-like” mixed-phase structure (Zeches et al. 2009; Li et al. 2015).

The thermodynamic analysis was normally performed to understand the strain effect on the phase transition of ferroelectric thin films (Pertsev et al. 1998; Roytburd 1998; Sheng et al. 2008; Shirokov et al. 2009; Qiu et al. 2010). It starts from the Landau phenomenological theory of a ferroelectric bulk crystal, wherein the Gibbs energy density is used to describe the total free energy of the bulk system with temperature, electric field, and the stress as independent variables. For studying the effect of strain on transition temperatures, it is actually more convenient to use strain rather than stress as the independent variable. Therefore, a Legendre transformation can be performed to obtain the Helmholtz free energy density, in which strain is the independent variable in the mechanical energy term. The stress components can be solved from the combination of thermodynamic property equations (first-order derivatives of the Gibbs free energy) and the mechanical boundary conditions of the epitaxial thin film. More specifically, the first-order derivative of the Gibbs free energy with respect to a stress component is equal to the corresponding negative strain component. The in-plane strain components of the thin film are fixed by the film/substrate lattice mismatch due to interfacial coherency, and the out-of-plane stress components of the thin film are zero due to the traction-free film surface. Eventually, the equilibrium state and the thermodynamic properties of the thin film can be established by minimizing the total thermodynamic potential of the thin film with respect to the thermodynamic variables.

The thermodynamic analysis has been extensively employed to predict the strain-temperature phase diagrams of various ferroelectric thin films including PbTiO3 (Pertsev et al. 1998; Zembilgotov et al. 2005), PbZr1-xTixO3 (Pertsev et al. 2003; Li et al. 2003a; Qiu et al. 2010), BaTiO3 (Emelyanov et al. 2002; Shirokov et al. 2007; Sheng et al. 2008), BiFeO3 (Zhang et al. 2007; Ma et al. 2008; Karpinsky et al. 2017), etc. One advantage of the thermodynamic analysis is the easy implementation of the thermodynamic potential into MATHEMATICA or MATLAB programs. Using the “minimization function” built in the software, the thermodynamic potential can be conveniently minimized to obtain the equilibrium state and the polarization-related properties such as the polarization, spontaneous strain, dielectric permittivity, and piezoelectric coefficients. However, when considering polydomain structures, the thermodynamic analysis becomes much more complicated (Kouhar et al. 2001; Bratkovsky and Levanyuk 2002; Xu et al. 2015). Moreover, there is no direct access or even impossible to get the kinetic behavior of the polarization and properties on the domain walls through the thermodynamic analysis. In order to achieve this, the interfacial energy or domain wall energy and the time-evolving behavior of the polarization need to be considered.

2 Phase–Field Method

Phase-field method combines the material thermodynamics and kinetics. It has become an important tool for studying microstructure evolution (Chen 2002; Boettinger et al. 2002; Gránásy et al. 2006; Chen 2008; Steinbach 2009). The phase-field method for ferroelectric epitaxial thin films was firstly developed by Li et al., and it has been employed to study the strain-temperature phase diagrams, domain structures, domain switching, and other domain-related properties for various ferroelectric thin films (Li et al. 2001, 2002a, 2002b, and 2003b, 2006; Li and Chen 2006; Zhang et al. 2008; Kontsos and Landis 2010; Winchester et al. 2011; Britson et al. 2014; Hong et al. 2014; Chen et al. 2014; Wang et al. 2016). In the phase-field method of ferroelectric thin films, the polarization vector is selected as the order parameter. The spatial distribution of the polarization represents the domain structures of a thin film. The evolution of polarization can be described by the time-dependent Landau-Ginzburg (TDGL) equations, i.e.:

$$ \frac{\partial \mathbf{P}\left(\mathbf{x},t\right)}{\partial t}=-L\frac{\delta F}{\delta \mathbf{P}\left(\mathbf{x},t\right)}, $$
(1)

where P, x, t, F, and L represent the polarization vector, spatial coordinate, time, the total free energy of the thin-film system, and the kinetic coefficient that is related to the domain wall mobility, respectively.

In the phase-field method of ferroelectric thin films (Li et al. 2001; Chen 2008), only polarization-related energy contributions are considered in the total free energy, which normally includes the Landau energy (bulk chemical energy), elastic energy, electric energy, and gradient energy, i.e.:

$$ F=\underset{V}{\iiint}\left[{f}_{\mathrm{bulk}}\left(\mathbf{P},T\right)+{f}_{\mathrm{grad}}\left(\nabla \mathbf{P}\right)+{f}_{\mathrm{electric}}\left(\mathbf{P},\mathbf{E}\right)+{f}_{\mathrm{elastic}}\left(\mathbf{P},\boldsymbol{\upvarepsilon} \right)\right] dV, $$
(2)

where fbulk, fgrad, felectric, felastic, ε, E, and ∇P represent the bulk, gradient, electric, elastic energy densities, strain tensor, electric field vector, and polarization gradient, respectively. The bulk energy density can be expressed by a Landau polynomial as function of polarization, i.e.:

$$ {\displaystyle \begin{array}{l}{f}_{\mathrm{bulk}}={\alpha}_1\left({P}_1^2+{P}_2^2+{P}_3^2\right)+{\alpha}_{11}\left({P}_1^4+{P}_2^4+{P}_3^4\right)\\ {}\ +{\alpha}_{12}\left({P}_1^2{P}_2^2+{P}_1^2{P}_3^2+{P}_2^2{P}_3^2\right)+{\alpha}_{111}\left({P}_1^6+{P}_2^6+{P}_3^6\right)\\ {}\ +{\alpha}_{112}\left[{P}_1^4\left({P}_2^2+{P}_3^2\right)+{P}_2^4\left({P}_1^2+{P}_3^2\right)+{P}_3^4\left({P}_1^2+{P}_2^2\right)\right]\\ {}\ +{\alpha}_{123}{P}_1^2{P}_2^2{P}_3^2+{\alpha}_{1111}\left({P}_1^8+{P}_2^8+{P}_3^8\right)\\{}\ +{\alpha}_{1122}\left({P}_1^4{P}_2^4+{P}_1^4{P}_3^4+{P}_2^4{P}_3^4\right)\\ {}\ +{\alpha}_{1112}\left[{P}_1^6\left({P}_2^2+{P}_3^2\right)+{P}_2^6\left({P}_1^2+{P}_3^2\right)+{P}_3^6\left({P}_1^2+{P}_2^2\right)\right]\\ {}\ +{\alpha}_{1123}\left({P}_1^4{P}_2^2{P}_3^2+{P}_1^2{P}_2^4{P}_3^2+{P}_1^2{P}_2^2{P}_3^4\right),\end{array}} $$
(3)

where αi, αij, αijk, and αijkl are called Landau coefficients relating to the different-order dielectric stiffness. They can be fitted to the bulk properties at zero stress (Haun et al. 1989a; 1989b; 1989c; Li et al. 2005; Wang et al. 2007; Liang et al. 2009; Pohlmann et al. 2017).

The gradient energy density is defined as

$$ {f}_{\mathrm{grad}}=\frac{1}{2}{\gamma}_{ijkl}{P}_{i,j}{P}_{k,l}, $$
(4)

where γijkl are the components of the gradient energy coefficient tensor and Pi,j=∂Pi/∂xj.

The electric energy density is given by:

$$ {f}_{\mathrm{electric}}=-{P}_i\left(\mathbf{x}\right){E}_i\left(\mathbf{x}\right)-\frac{1}{2}{\varepsilon}_0{\kappa}_{ij}^{\mathrm{b}}{E}_i\left(\mathbf{x}\right){E}_j\left(\mathbf{x}\right), $$
(5)

where \( {\kappa}_{ij}^{\mathrm{b}} \) are components of the background dielectric constant tensor (Rupprecht and Bell 1964; Tagantsev 2008). The electric field distribution can be solved from the electrostatic equilibrium equation, which can be dramatically affected by the electric boundary condition. For example, under open-circuit boundary conditions, if the bound charges at the film surfaces are not compensated, a depolarization field will be generated and may significantly suppress the polarization. This effect will be dominant when the thickness of the thin film is down to several unit cells, leading to the disappearance of the ferroelectricity, which is also called the size effect of the ferroelectric thin films (Scott 1988, 1991; Junquera and Ghosez 2003).

The elastic energy density can be calculated according to the Hooke’s law:

$$ {f}_{\mathrm{elastic}}=\frac{1}{2}{c}_{ij kl}{e}_{ij}\left(\mathbf{x}\right){e}_{kl}\left(\mathbf{x}\right)=\frac{1}{2}{c}_{ij kl}\left[{\varepsilon}_{ij}\left(\mathbf{x}\right)-{\varepsilon}_{ij}^0\left(\mathbf{x}\right)\right]\left[{\varepsilon}_{kl}\left(\mathbf{x}\right)-{\varepsilon}_{kl}^0\left(\mathbf{x}\right)\right], $$
(6)

where cijkl are components of the elastic stiffness tensor and eij(x), εij(x), and \( {\varepsilon}_{ij}^0\left(\mathbf{x}\right) \) are components of the elastic strain tensor, total strain tensor, and eigenstrain tensor, respectively. The eigenstrain tensor \( {\varepsilon}_{ij}^0\left(\mathbf{x}\right) \), or the stress-free strain tensor, arises from the coupling between the polarization and the lattice degrees of freedom in ferroelectrics. This coupling is also named the electrostrictive effect (Cross et al. 1980; Cross 1996; Sundar and Newnham 1992). It is a universal effect in insulators, and in ferroelectrics the quadratic coupling can be especially strong. The strength of the quadratic coupling between polarization and the lattice degrees of freedom is described by the electrostrictive coefficient Qijkl, which is a fourth-order tensor. Its values can be obtained by experimental measurements (Yamada 1972) or first-principle calculations (Wang et al. 2010). The eigenstrain \( {\varepsilon}_{ij}^0\left(\mathbf{x}\right) \) can be calculated from \( {\varepsilon}_{ij}^0\left(\mathbf{x}\right) \) = QijklPk(x)Pl(x). The total strain εij(x) can be solved from the mechanical equilibrium equation using Khachaturyan’s microelasticity theory (Khachaturyan 1983), in combination with the thin-film mechanical boundary conditions (Li et al. 2002).

In the phase-field model of ferroelectric thin film, materials parameters needed include the Landau coefficients, gradient energy coefficients, background dielectric constant tensor, elastic compliance tensor, and electrostrictive coefficient tensor. With these materials parameters available, the domain structures under different temperatures and in-plane biaxial strain conditions can be obtained starting from a random distribution of polarization or from an existing domain structure.

3 Phase Transitions and Domain Structures

In the phase-field simulation of ferroelectric thin films, the equilibrium domain structures can be predicted by quenching an annealed paraelectric state with a fixed substrate constraint to lower temperatures. The main advantage of the phase-field method is the fact that one does not have to make a priori assumptions on the possible ferroelectric phases and domain structures that might appear under a given temperature and an epitaxial strain.

3.1 Misfit Strain–Temperature Phase Diagrams

The phase-field method has been employed to calculate the strain-temperature phase diagrams for several ferroelectric thin-film oxides such as PbTiO3 (Li et al. 2001), BaTiO3 (Choi et al. 2004), SrTiO3 (Haeni et al. 2004), and PbZr1-xTixO3 (Choudhury et al. 2005) (Fig. 1a–d). A common feature of these phase diagrams is the enhanced Curie temperature by the epitaxial strain. This effect can be dramatic as the Curie temperature can be increased by several hundred degrees by only 1% compressive or tensile strain. This effect originates from the strong coupling between the polarization and the lattice degrees of freedom. Another common feature among these phase diagrams is the existence of transition regions along the ferroelectric/paraelectric phase boundaries. For the stress-free bulk ferroelectrics including PbTiO3, BaTiO3, SrTiO3, and PbZr1-xTixO3, the phase transition is normally first-order at which the polarization changes sharply at the transition temperature. However, under thin-film mechanical boundary conditions with constraint strains in-plane and stress-free out-of-plane, the paraelectric to ferroelectric phase transition may become second-order (Pertsev et al. 1999). This is caused by the renormalization of the fourth-order Landau coefficients due to the coupling between the polarization and the strain. On the strain-temperature phase diagram of PbTiO3 thin film (Fig. 1a), there is a large region wherein the in-plane a domain and out-of-plane c domain can coexist, which has been verified by numerous experimental observations (Seifert et al. 1995; Kiguchi et al. 2011; Tang et al. 2014). For PbZr0.53Ti0.47O3 thin film shown in Fig. 1d, there are also some mixed-phase regions. For example, under moderate strains and low temperatures, the tetragonal phase, orthorhombic phase, and distorted rhombohedral phase may coexist. These mixed-phase regions are normally not predicted from thermodynamic analysis without considering the multi-domain structures.

Fig. 1
figure 1

Misfit strain-temperature phase diagrams of (a) PbTiO3 (adapted from Li et al. 2001), (b) BaTiO3 (adapted from Choi et al. 2004), (c) SrTiO3 (adapted from Haeni et al. 2004), and (d) PbZr0.53Ti0.47O3 (adapted from Choudhury et al. 2005) thin films predicted using the phase-field method. (Copyright 2001, American Institute of Physics, Copyright 2004 American Association for the Advancement of Science, Copyright 2004, Springer Nature, Copyright 2001, American Institute of Physics)

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3.2 Domain Structures in Single–Layered Thin Films

A common feature of ferroelectric materials is the formation of domain structures when the material is cooled down from a paraelectric phase to a ferroelectric phase. A ferroelectric domain is defined by a region with uniform electrical polarization. The domain size may range from several nanometers to hundreds of nanometers depending on many factors such as the film thickness, the epitaxial strain, and the electric boundary conditions. The type of domain structures is determined by the total free energy, which can be altered by external stimuli including the mechanical, electric, thermal, and chemical stimuli. One advantage of phase-field modeling of the domain structures in ferroelectric thin films is the theoretically feasible incorporation of arbitrary boundary conditions, although only some of the possible boundary conditions can be realized in experimental configurations.

By incorporating materials parameters and incorporating boundary conditions of the experimental thin films into the phase-field model, the equilibrium domain structures can be obtained and compared with available experimental observations. For example, Fig. 2a shows the a/c domain structures predicted in a (001)-oriented PbTiO3 thin film under a moderate compressive strain, while Fig. 2b gives the domain structures observed in a PbTiO3 thin film epitaxially grown on a (001)-oriented SrTiO3 substrate which can impose a compressive strain on the PbTiO3 thin film. The a domain represents the domains with the polarization along either the x1 or x2 axis of the pseudocubic lattice, and the c domain represents the domains with the polarization along the x3 axis of the pseudocubic lattice. The predicted domain structures (Li et al. 2002) show amazing similarity to the domain structures observed (Seifert et al. 1995) using the transmission electron microscopy (TEM). The formation of the a/c domain structures can be understood from the competition between the elastic energy, the electrostatic energy, and the gradient energy. For the moderately compressed thin films under short-circuit electric boundary conditions, a single c domain can minimize the electrostatic energy and the gradient energy. In this case however the macroscopic deformation along the out-of-plane direction will be maximized, leading to a high elastic energy. Therefore, some a domains are generated to decrease the macroscopic deformation and the elastic energy but comprise to slightly increase the electrostatic energy and the gradient energy.

Fig. 2
figure 2

(a) Predicted a/c domain structures in a (001)-oriented PbTiO3 thin film under moderate compressive strains (adapted from Li et al. 2002), compared with (b) the experimental observation in a PbTiO3 thin film epitaxially grown on a (001)-oriented SrTiO3 substrate (adapted from Seifert et al. 1995). (c) Predicted a1/a2 twinning domain structures in a PbZr0.2Ti0.8O3 thin film under equally biaxial tensile strains (adapted from Wang et al. 2016), compared with (d) the experimental observation in a PbTiO3 thin film epitaxially grown on a (001)-oriented KTaO3 substrate. (Adapted from Lee et al. 2001. Copyright 2002, 2016, Elsevier, Copyright 2001, American Institute of Physics)

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3.3 Domain Structures in Bilayer Thin Films

The phase-field method was also employed by Artemev et al. to study the domain structures in ferroelectric bilayer thin films (Artemev et al. 2008). They found that a poled state can be achieved by a low electric field in the bilayer film with one layer in a single-domain state and the other layer in a polydomain state. In contrast to the bilayer film with both layers in single-domain states, the presence of a polydomain layer in the bilayer film is responsible for a higher dielectric constant and lower coercive field. The increase in the applied electric field can lead to a transition to single-domain states in both layers of the bilayer film, and the critical field depends on the thermodynamic potentials of the two layers (Artemev et al. 2008). Using the thin-film phase-field method, Xue et al. investigated the effect of interfacial coherency between the two layers in a Pb(Zr0.3Ti0.7)O3/Pb(Zr0.7Ti0.3)O3 bilayer thin films on the domain structures (Xue et al. 2013). If the lattice parameters of the two layers end up with the same average lattice parameter, the two layers can be regarded as coherent. For example, the cubic lattice parameters for Pb(Zr0.3Ti0.7)O3 and Pb(Zr0.7Ti0.3)O3 are 4.0185 Å and 4.1032 Å, respectively, giving rise to an average lattice parameter of 4.0609 Å when the two layers have the same thickness. With a coherent interface, the strained bilayer film has a lattice parameter of 4.0609 Å. Consequently, the Pb(Zr0.3Ti0.7)O3 layer tolerates a tensile strain of (4.0609−4.0185)/4.0185 = 1.05%, and the Pb(Zr0.7Ti0.3)O3 layer tolerates a compressive strain of (4.0609−4.1032)/4.1032=−1.03%. For a totally incoherent bilayer film, each layer can be regarded to have similar lattice parameters to their bulk counterparts, unless the clamping effect from the substrate is considered.

The interfacial coherency has significant effects on the domain structures in each layer (Fig. 3a and b). For Pb(Zr0.3Ti0.7)O3/Pb(Zr0.7Ti0.3)O3 bilayer thin films with coherent interface, tetragonal a1/a2 twinning domain structures (all in-plane polarizations) are predicted in the Pb(Zr0.3Ti0.7)O3 layer (Fig. 3a), while in its single-layer thin film, the domain structures are tetragonal a/c types (Fig. 3c, mixed in-plane and out-of-plane polarizations). The Pb(Zr0.7Ti0.3)O3 layer in the bilayer film shows tetragonal c domains (all out-of-plane polarizations), compared with rhombohedral domains in its single-layer thin-film counterpart (Fig. 3d). For the bilayer film with incoherent interface, the domain structures in each layer are similar to their single-layer thin-film counterparts, i.e., a/c domains in Pb(Zr0.3Ti0.7)O3 layer and rhombohedral domains in Pb(Zr0.7Ti0.3)O3 layer (Fig. 3b). The interfacial coherency effects on the domain morphologies can be understood from the strain configuration in the bilayer film. In Pb(Zr0.3Ti0.7)O3 layer, the tensile strain 1.05% tends to align all polarizations in-plane, leading to the formation of a1/a2 twinning domain structures under the synergistic effect from the bulk chemical energy and elastic energy. In Pb(Zr0.7Ti0.3)O3 layer, the compressive strain −1.03% tends to switch all polarizations out-of-plane, leading to the formation of c domains (Xue et al. 2013; Liu et al. 2016).

Fig. 3
figure 3

Effects of interfacial coherency on the domain structures in a Pb(Zr0.3Ti0.7)O3/Pb(Zr0.7Ti0.3)O3 bilayer thin film (Xue et al. 2013). Domain structures of the bilayer film when the interface between the two layers is (a) coherent and (b) incoherent, compared with the domain structures in single-layer (c) Pb(Zr0.3Ti0.7)O3 thin film and (d) Pb(Zr0.7Ti0.3)O3 thin film under a moderate strain of 0.2% due to thermal expansion. (Copyright 2013, Elsevier)

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3.4 Strain Phase Separation

The thin-film phase-field model was also used to understand the formation of mixed phases and domains in strained thin films from the perspective of strain phase separation. In a constrained system, domains or phases can form and coexist with each other to minimize the overall elastic energy, given that each domain or phase has different local strains. This phenomenon is similar to the phase decomposition process from a chemically homogeneous phase to a two-phase mixture wherein each phase has a different chemical composition. The volume fraction of each phase with a specific composition can be obtained from the lever rule. This idea was used to understand the observed rhombohedral-tetragonal mixed phases in severely compressed BiFeO3 thin films (Xue et al. 2016, 2017). BiFeO3 has a pseudocubic lattice parameter of ∼4.0 Å. When it is epitaxially grown on a LaAlO3 substrate with a lattice parameter of ∼3.79 Å, a large compressive strain up to ∼−5.25% will be imposed on the thin film, leading to a transition from the rhombohedral phase to a tetragonal-like monoclinic phase in BiFeO3 thin film (Zeches et al. 2009; Li et al. 2015; Zhang et al. 2012). With the film thickness increasing to be larger than 100 nm, the large epitaxial strain is partially relaxed, and the rhombohedral phase emerges, forming a tetragonal-like/rhombohedral mixed domain structure (Fig. 4a and b). The phase-field model was employed to predict domain structures (Fig. 4c and d) which agree well with the experimental observations and give insights to the underlying physics. With the relaxation of the mismatch strain, the overall strain will be smaller than the critical strain, under which the transition from rhombohedral phase to tetragonal-like phase occurs. Therefore, the phase separation happens to decrease the total elastic energy, analogous to the chemical decomposition process (Xue et al. 2016, 2017).

Fig. 4
figure 4

Experimentally observed (a) topography and (b) the corresponding in-plane piezoresponse force microscopy result of a mixed-phase BiFeO3 thin film grown on LaAlO3 substrate, compared with the simulated (a) three-dimensional domain structures and (b) in-plane domain morphology from thin-film phase-field method. (Adapted from Xue et al. 2016. Copyright 2016, American Physics Society)

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3.5 Understanding the Polarization Switching in Ferroelectric Thin Films

One advantage of the phase-field method over the thermodynamic analysis is the capability of studying the polarization switching behavior in ferroelectric thin films. The polarization switching driven by externally applied electric fields was simulated by Li et al. for epitaxial BaTiO3 thin films (Li et al. 2008). They found that the coercive field and saturated polarization in the hysteresis loop strongly depend on the type and density of the interfacial dislocation. For example, the coercive field can be decreased by increasing the density of interfacial misfit dislocations on a (100) slip plane with a Burger vector along [−100] direction. With an optimal combination of different misfit dislocations, the remnant polarization can be greatly enhanced and the coercive field can be greatly reduced. The polarization versus electric field hysteresis loops can be distorted to be asymmetric by threading dislocations. Wu et al. investigated the effects of background dielectric constant and interfacial coherency on the hysteresis loops in BaTiO3/SrTiO3 bilayer films (Wu et al. 2012, 2015). It was found that the hysteresis loops can be constricted when the background dielectric constant is decreased. The decrease in the background dielectric constant has a similar effect to the decrease in the space charge density. The phase-field simulation predicted the domain structure evolution during the switching process in the ferroelectric/paraelectric bilayer film, revealing a dipole-dipole interaction similar to the exchange-spring interaction in magnetic multilayer systems (Wu et al. 2012). The interfacial coherency also has a remarkable impact on the polarization switching behavior. In the bilayer film with fully coherent interfaces, the hysteresis loops are square-like, exhibiting a large coercive field and a large remnant polarization. For the bilayer films with partially relaxed heterointerfaces, the remnant polarization is relatively high but the coercive field is dramatically reduced. For the bilayer film with fully relaxed heterointerfaces, the hysteresis loops are very slim, showing both small remnant polarization and coercive field.

The polarization switching is normally achieved by the externally applied electric field. However, it can also be achieved by a mechanical loading via the flexoelectric effects in epitaxial thin films. The flexoelectric effect is defined as the coupling between the polarization and the strain gradient. This effect is much stronger in ferroelectric thin films than in their bulk counterparts. It has been implemented into the thin-film phase-field model by several research groups (Chen et al. 2014; Ahluwalia et al. 2014; Gu et al. 2014; Chen et al. 2015). In these works, the flexoelectric effects on the domain structures and switching were studied. For example, Chen et al. studied the stability and switching patterns of domain structures in ferroelectric thin films subjected to mechanical loads and related flexoelectric fields (Chen et al. 2015). As shown in Fig. 5a and b, a mixed a/c domain structure can be switched to a single-domain structure by cylindrical bending with a large surface flexoelectric field. For the upward bending case shown in Fig. 5a, the out-of-plane c− domains are gradually switched to in-plane at first and then are switched to the out-of-plane c+, while for the downward bending case shown in Fig. 5b, all in-plane domains are switched to the out-of-plane c− domain. These two switching patterns can be understood from the different flexoelectric fields due to the different bending directions. For the film with an upward bending, the induced flexoelectric field is upward, leading to the switching of polarization to c+. For the film with a downward bending, the induced flexoelectric field is downward, leading to the switching of polarization to c−.

Fig. 5
figure 5

Switching behavior of the domain structure from a/c mixed structures to (a) single out-of-plane c+ domain under upward bending and to (b) single out-of-plane c− domain under downward bending. (Adapted from Chen et al. 2015. Copyright 2015, Elsevier)

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4 Toward a Multiscale Simulation

Although the thin-film phase-field method has gained great success in understanding the experimental domain structures and phase transitions, and in predicting the domain structures and switching in thin films under conditions beyond the experiment, it has only been applied to limited material systems due to the lack of the thermodynamic parameters including the Landau coefficients, electrostrictive coefficients, elastic constants, and gradient energy coefficients. For example, up to now, there are very limited materials that have available Landau potentials such as BaTiO3, PbTiO3, PbZr1-xTixO3, BiFeO3, CaTiO3, SrTiO3, KNa1-xNbxO3, LiNbO3, LiTaO3, SrBi2Nb2O9, and Sr0.8Bi2.2Ta2O9. However, the ferroelectrics that have been discovered are far more beyond these. In order to obtain the Landau coefficients, the polarization and dielectric responses including both the first-order and higher-order dielectric stiffness as function of temperature have to be measured, which is experimentally challenging especially when the required temperature range is broad. An alternative approach to obtaining the Landau coefficients is the first-principle calculation, which has been employed to obtain the Landau coefficients for BaTiO3 (Íñiguez et al. 2001; Geneste 2009), CaTiO3 (Eklund 2010; Gu et al. 2012), and BiFeO3 (Marton et al. 2017). In addition to the Landau coefficients, the first-principle method has also been employed to calculate the phase transitions (Zhong et al. 1994), elastic constants (Wu et al. 2005), electrostrictive coefficients (Wang et al. 2010), and polarization domain wall energies (Wang et al. 2013; Diéguez et al. 2013; Li et al. 2014). For the domain wall motion speed which is related to the kinetic coefficient L in the TDGL equation, it can be studied using the atomistic molecular dynamics and coarse-grained Monte Carlo simulations (Shin et al. 2007). All these efforts point toward the multiscale simulation-assisted phase-field modeling for ferroelectric thin films (Völker et al. 2011a, 2011b), which can provide principal inputs for future high-throughput calculation (Shen et al. 2017) and optimization of properties and microstructure design for ferroelectric thin films using machine learning (Li et al. 2017).