Misfit strain-misfit strain phase diagram of (110)-oriented ferroelectric PbTiO3 films: a phase-field study

Ferroelectric thin films with high index orientations are found to possess unique structures and properties. In this work, we constructed the misfit strain-misfit strain phase diagram of (110)-oriented PbTiO 3 thin films by phase-field simulations. The evolutions of ferroelectric phase structures, domain morphologies, volume fraction and polarization components with the anisotropic strains were analyzed in detail. There exist large anisotropic strains between the orthorhombic scandate substrates and (110)- oriented PbTiO 3 films, which makes it possible to engineer the structures and properties by anisotropic strain. These results deepen the understanding of ferroelectric domain structures of (110)-oriented PbTiO 3 films under the anisotropic strain

How to regulate the excellent properties of ferroelectric materials is the focus of attention.In this process, researchers have tried many methods, such as strain [6][7][8][9][10] , film thickness [11][12][13] , electrical boundary condition [14,15] , growth orientation [16][17][18][19][20] , etc.Both experiments and theoretical simulations have proved that the ferroelectric thin films with high index orientations, such as (110)-and (111)-orientations, have unique structures and properties different from those with low index orientation, such as the (001)-orientation [21][22][23][24][25][26] .PbTiO3 (PTO) is a prototypical ferroelectric material, which undergoes a cubic-to-tetragonal ferroelectric transition at about 765 K [27,28] .For (110)oriented PTO films, the temperature-misfit strain phase diagram [29] were constructed by the phenomenological theory, which indicates that various low-symmetry phases could emerge at different strain states.However, the phenomenological theory only considers single-domain states and prescribed multi-domain states.In contrast, the phase field simulations could predict the optimal multi-domain structures under certain external conditions and their evolutions with the external field.In our previous work, the temperature-misfit strain phase diagram of the (110)-oriented PTO film was constructed by phase field simulations, and the effect of epitaxial strain on the structures and properties were systematically investigated [30] .
Due to the anisotropy of the crystal, the (110)-oriented ferroelectric films can exhibit unique properties.Experimentally, there are many orthogonal substrates, which exert asymmetric in-plane strain.For example, the orthogonal NdGaO3 substrate applies the asymmetric strain to the (110)-oriented Ba1-xSrxTiO3 film, resulting in strong in-plane dielectric anisotropy [31,32] .For ferroelectric PTO, the anisotropic misfit strain phase diagram [33] were constructed by the phenomenological theory.However, there are no phase-field studies, which could provide predictive anisotropic misfit strain multi-domain phase diagrams for experimentalists.
In this paper, the effect of asymmetric misfit strain on ferroelectric phase (domain) structures of (110)-orientated PTO films is constructed by analyzing the phase-field data via the stereographic projection (SP) method [30,34,35] .Then, the typical phase (domain) structures under asymmetric misfit strain state and their evolution with strain are analyzed in detail.Finally, it is pointed out that a series of orthogonal substrates can apply large asymmetric misfit strains to achieve the regulation of (110)-oriented PTO films.These results help to deepen the understanding of ferroelectric domain structures under high index asymmetric misfit strain, and provide theoretical guidance for the design of ferroelectric devices based on asymmetric misfit strain regulation.

Phase field model
The phase-field model suitable for the (110)-oriented ferroelectric films was constructed in our previous work [30] .Here, the main formulae were briefly outlined.A where the transformation matrix T ij can be written as follows: The temporal evolution of P i ' is modelled via numerically solving the timedependent Ginzburg-Landau (TDGL) equation: where t is the time step, L is the kinetic coefficient related to the domain wall mobility and the total free energy F ' consists of the following contributions: represent the bulk, gradient, elastic and electrostatic energy contributions, respectively.The detailed numerical expressions of these energy contributions in the common and new coordinate systems, as well as the method of solving the phase-field equations can be found in the previous work [30] .
The thickness of the PTO thin film and the deformable substrate are 20 nm and 4 nm, respectively.The in-plane misfit strains ε11 and ε22 range from −4% to 4%, and the temperature is chosen as the room temperature (25 ℃).Periodic boundary conditions were applied along the in-plane x' and y' directions.The mixed mechanical boundary condition was applied that the top surface of the film is in a traction-free state, while the bottom of the simulation region in the substrate is fixed.The short-circuit electric boundary condition is considered where the electric potential at the top film surface and the film/substrate interface is fixed to zero.Random noise is used to simulate the annealing process as the initial set-up.All related coefficients of PTO are adopted from the previous literature [7,36] .

Misfit strain-misfit strain phase diagram
A series of equilibrium structures under different misfit strains ε11 and ε22 are calculated by phase field simulations.Firstly, the structures of (110)-oriented PTO films are analyzed via the SP method and the type of phase at each position of misfit strainmisfit strain space is determined, as shown in Figure 1.There are several phases,  misfit strain phase diagram of the (110)-oriented PTO film is asymmetric [37] and the symmetries of the phases in the (110)-oriented PTO are also generally lower than those under the (001) orientation.Compared with the temperature-symmetric strain phase diagram of the (110)-oriented PTO film [30] , it is found that the high-temperature Ta phase can be stabilized at room temperature under the asymmetric strain.previous work [30] .

The effect of asymmetric strain on domain structures
In order to further reveal the effect of asymmetric strain on the domain structure of (110)-oriented PTO films, the domain structure and domain morphology of the Ta/Mc phase under different strains are analyzed in this section, as shown in Figure 5. Figure

Asymmetric strain between the orthogonal substrates and the film
In recent years, with the development of a series of commercial orthogonal substrates, it is possible to control the domain structure in (110)-oriented PTO films by the asymmetric strain.The lattice constants of the commonly used orthorhombic scandate and gallate substrates and the mismatch strain between them and the (110)oriented PTO film are listed in Table 1.In the phase field simulation, the lattice constants of the ( 110 first-principles calculations [38] , which indicates that the PTO film grown on gallate substrates might exhibit enhanced piezoelectricity.

CONCLUSIONS
In this work, the misfit strain-misfit strain multi-domain phase diagram of (110)oriented PTO thin films at the room temperature was constructed by phase field simulations.The typical domain structures and domain wall structures in the misfit strain-misfit strain phase diagram were systematically analyzed.The effect of asymmetric strain on the phase (domain) structures of (110)-oriented PTO thin films was studied.By analyzing the phase (domain) structures under different strains, the evolution of phase (domain) structures, domain morphology, phase volume fraction, polarization component and polarization angle of (110)-oriented PTO thin film with asymmetric strain is revealed.This paper points out that there is a large in-plane anisotropic strain between the (110)-oriented PTO film and the orthorhombic substrate such as scandate, which makes it possible to experimentally control the asymmetric strain of the (110)-oriented film.

DECLARATIONS
new coordinate system (x ~1, x ~2, x ~3) with axes along the [100], [01 ̅ 1] and [011] directions is introduced along with the common one (x1, x2, x3) with axes along the [100], [010], and [001] directions of a perovskite unit cell.The polarization components P i ' in the coordinate system (x ~1, x ~2, x ~3) are related to that in the coordinate system (x1, x2, x3) via the transformation matrix Tij: including the tetragonal phase (Ta: P1 ≠ 0, P2 = P3 = 0, Tc: P2 = P3 ≠ 0, P1 = 0), the orthogonal phase (Oc: P3 ≠ 0, P1 = P2 = 0), the monoclinic phase (Mc: P2 ≠ P3 ≠ 0, P1 = 0) and the rhombohedral phase (R: P1 ≠ P2 ≠ 0, P3 = 0).The ranges of tetragonal-like, orthogonal and rhombohedral-like phases are marked by green, red and orange solid circles whose radiuses are 10°, respectively.The polarization vector of the monoclinicMcphase rotates in the x2-x3 plane.When the angle between the polarization vector of the Mc phase and the original [001] axis is less than 10°, it is defined as the Tc phase.In fact, the Tc phase is a tetragonal-like monoclinic Mc phase.Comparing the SPs under each misfit strain with the ideal position schematic diagram of the (110)-oriented PTO ferroelectric phase shown in the center of Figure 1, the phase and polarization variants under different misfit strains can be determined accurately.When the strain state of the PTO film is the symmetrical strain of ε11 = ε22 = −4%, the peak of the projection point is located at the center of the SPs as shown in Figure1A, which indicates that the polarization vectors are parallel to the x3 axis, that is, the Oc phase.Keeping ε11 at compressive strain of -4% and changing ε22 toward the tensile strain gradually, the peak of the projection point splits into two parts along the x2 axis, corresponding to the two polarized variants of the Mc phase, as shown in Figure1B.When ε22 continues to increase to 4% of the tensile strain, the peaks of the Mc phase enter the range of the Tc phase, the green circles, as shown in Figure1C.When ε22 is maintained at 4%, and ε11 is gradually changed toward the tensile strain, it is found that the peak of Mc phase continues to deflect in the in-plane direction.When ε11 = 1%, the peaks of Ta phase emerge at both ends of the x1 axis and the phase structure is Ta/Mc mixed phase, as shown in Figure1D.Further increasing ε11, the peaks of the R phase emerge.When the strain is ε11 = ε22 = 4%, the peaks of the Mc phase disappear, resulting in the Ta/R phase, as shown in Figure1E.Keeping ε11 at 4% and decreasing ε22, it is found that the peak of the R phase gradually weakens and the peak of the Mc phase begins to appear on the x2 axis.When the strain state is ε11 = 4% and ε22 = 1%, the phase structure evolves into the Ta/Mc phase again, as shown in Figure1F.With the decrease of ε22, the peaks of the Mc phase gradually disappears and the phase structure transforms into the pure Ta phase, as shown in Figure1G.Finally, the compressive strain of ε22 is maintained at − 4%, and then ε11 is gradually reduced.Under the asymmetric strain of ε11 = 0% and ε22 = − 4%, the peak of the Oc phase appears in the central region of the SPs, forming the Ta/Oc phase.As ε11 continues to change toward the compressive strain, the peaks of the Ta phase gradually weaken and the phase structure transforms into a pure Oc phase, as shown in Figure1A.

Figure 1 .
Figure 1.The stereographic projections of ferroelectric phases for (110)-oriented PTO thin films under anisotropic misfit strains at room temperature.A-H: The stereographic projections of typical ferroelectric phases under different strain states.The schematic diagram in the center gives the ideal locations of different ferroelectric phases for (110)oriented PTO films.The ranges of tetragonal-like, orthorhombic and rhombohedral-like phases are marked by green, red and orange solid circles.The magenta dashed circle indicates the position where the polarization vector deviates from the x3 axis by 45°, and the cyan dashed lines mark the four in-plane <111> directions.The color reflects the intensity of the projection scatters.

Figure 3 .
Figure 3.Typical domain structures of (110)-oriented PTO thin films under different misfit strains.A: The Oc phase; B: The Mc phase; C: The Ta phase; D: The Ta/Oc phase; E: The Ta/Mc phase; F: The Ta/R phase.The polarization variants are denoted by different colors as labeled on the right.

Figure 4 .
Figure 4. Domain structures of the Ta/Oc phase in (110)-oriented PTO thin films under the anisotropy strain of ε11 = 0% and ε22 = −4%.A: The 3D domain structure; B: The vertical cross section; C: The horizontal cross section of the white dashed boxes in A. The bars in B and C indicate 10 nm.

5A
shows the Ta/Mc phase structure under the symmetric strain of ε11 = ε22 = 1%.The strip-like Ta phase and the Mc phase (the volume fraction is about 66.1%) coexist and the domain wall density is high.When increasing the strain in the x2 direction to ε22 = 4%, as shown in Figure 5B, the volume fraction of the Mc phase increase (about 88.3%) at the expanse of the strip Ta phase.When increasing the strain in the x1 direction to ε11 = 4%, however, the volume fraction of the Ta phase increases and that of the Mc phase decreases (about 9.3%), as shown in Figure 5C.

Figure 6 .
Figure 6.Phase structure evolutions of (110)-oriented PTO films with respective to misfit strain ε11 at various constant strain ε22.A-C: ε22 = −2%; D-F: ε22 = 0%; G-I: ε22 = 2%.(A, D, G) are the volume fractions.(B, E, H) are the polarization components of various phases.(C, F, I) are the angle θ between the polarizations of the Mc(Oc) phase and the x3 axis, in which the bars represent the standard deviations of the angle.The schematic of the angle θ is shown as an insert in (I).

Figure 6
Figure 6 is the evolution of the phase structure with the strain ε11 under fixed ε22.When ε22 = −2%, it can be seen from Figure 6A that with the increase of ε11, the phase structure undergoes the evolution path of Oc → Mc → Ta/Mc → Ta.In the Ta/Mc phase region, with the increase of ε11, the volume fraction of the Ta phase increases and the volume fraction of the Mc phase decreases.Figure 6B is the corresponding polarization

Figure
Figure 6C is the angle evolution diagram between the polarization of Mc(Oc) phase and the x3 axis.The polarization angle increases from 4° to 25°, indicating that the polarization vector gradually rotates to the in-plane direction with the increase of ε11.In addition, we also analyzed the evolution of phase volume fraction, polarization component and polarization angle with the misfit strain ε11 when ε22 = 0% and ε22 = 2%, as shown in Figure 6D-F and 6G-I, respectively.At ε22 = 0%, with the increase of ε11, the evolution path of the phase is Mc → Ta/Mc → Ta, as shown in Figure 6D.At ε22 = 2%, with the increase of ε11, the evolution path is Mc → Ta/Mc, as shown in Figure 6G.The corresponding polarization component diagrams (Figure 6E and 6H) and polarization angle diagrams (Figure 6F and 6I) have similar evolution trend as those of ε22 = −2%.With the increase of ε11, the out-of-plane Pz component corresponding to Mc gradually decreases, and the in-plane Py component gradually increases.The Px component corresponding to Ta increases gradually.Similarly, the angle between the Mc phase and the x3 axis is also increasing, indicating that the polarization of the Mc phase rotates to the in-plane direction.

Figure 7
Figure 7 is the evolution of the phase structure with the misfit strain ε22 when the misfit strain ε11 is fixed.At ε11 = −2%, with the increase of ε22, the phase structure evolves from the Oc phase to the Mc phase, as shown in Figure 7A. Figure 7B reflects the evolution of the polarization component.When ε22 = −3%, the in-plane Py component emerges and gradually increases with ε22, while the out-of-plane Pz component increases first and then decreases.Figure 7C shows the change of the angle between the polarization of the Mc(Oc) phase and the x3 axis.In the Mc phase region, the angle increases from 0° to 51.5°, indicating that the polarization of the Mc phase gradually rotates to the in-plane direction with the increase of ε22.At ε11 = 0%, it can be seen from the phase volume fraction in Figure 7D that with the increase of ε22, the evolution path is Ta/Oc → Ta/Mc → Mc.In the Ta/Mc phase region, with the increase of

Figure 7 .
Figure 7. Phase structure evolutions of (110)-oriented PTO films with respective to misfit strain ε22 at various constant strain ε11.A-C: ε11 = −2%; D-F: ε11 = 0%; G-I: ε11 = 2%.(A, D, G) are the volume fractions.(B, E, H) are the polarization components of various phases.(C, F, I) are the angle θ between the polarizations of the Mc(Oc) phase and the x3 axis, in which the bars represent the standard deviations of the angle.
)-oriented PTO film in the in-plane x1[100] and x2[011] directions are ac = 3.957 Å and √2ac = 5.596Å, respectively.In order to make the (110)-oriented film and the orthogonal substrate match the lattice in two directions in the interface, there are two growth orientations.One is grown on the (100)O plane, which satisfies [100]PC//[001]O, [011]PC//[010]O.The other one is grown on the (010)O plane, which satisfies [100]PC//[001]O, [011]PC//[100]O, where the subscripts PC and O represent the pseudo-cubic index of the film and the orthogonal index of the substrate, respectively.The misfit strains corresponding to the orthogonal (100)O and (010)O scandate and gallate substrates are plotted in the phase diagram, as shown in Figure 8.It can be seen that the strains corresponding to the (100)O and (010)O oriented scandate substrates are located in the Ta/Mc, Ta/Oc, and Mc phase regions.The strains corresponding to the (100)O and (010)O oriented gallate substrates are located near the phase boundary between the Oc phase and the Mc phase.In our recent work, we predicted that giant piezoelectricity could emerge at this phase boundary by phase-field simulation and

Figure 8 .
Figure 8.The strain positions of orthorhombic scandate and gallate substrates in the phase diagram.

Table 1 .
The lattice constants of commonly used orthorhombic scandate and gallate substrates and the misfit strains between these substrates and (110)oriented PTO films.In this work, two orthorhombic orientations (100)O and (010)O of these substrates are considered, where the subscript O denotes orthorhombic indices.The in-plane lattice constants in the [100] and [011] directions of the (110)-oriented PTO film are 3.957Å and 5.596Å, respectively.