JIANG Zhexin，WANG Jie*

1.School of Aeronautics and Astronautics，Zhejiang University，Hangzhou 310027，P.R.China；2.Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province，Zhejiang University，Hangzhou 310027，P.R.China

Abstract: The ferroelectric superlattices have been widely studied due to their distinguished electromechanical coupling properties. Under different biaxial mismatch strains，ferroelectric superlattices exhibit different domain structures and electromechanical coupling properties. A three-dimensional phase field model is employed to investigate the detailed domain evolution and electromechanical properties of the PbTiO3/SrTiO3（PTO/STO）superlattices with different biaxial mismatch strains. The phase field simulations show that the ferroelectric superlattice exhibits large electrostrain in the stacking direction when an external field is applied. Under a large compressive mismatch strain，vortex domains appear in ferroelectric layers with the thickness of 4 nm. The vortex domains become stable cdomain under a large external electric field，which remains when the electric field is removed. When the initial compressive mismatch strain decreases gradually，the waved or a1/a2 domains replaces the initial vortex domains in the absence of electric field. The fully polarized c-domain by a large electric field switches to diagonal direction domain or a/c domain when the electric field is small. Furthermore，when a biaxial tensile strain is applied to the superlattice，ferroelectric domains switch back to the initial a1/a2 twin-like domain structure，resulting in the recoverable and large electrostrain. This provides an effective way to obtain the large and recoverable electrostrain for the engineering application.

Key words：large electrostrain；ferroelectric superlattices；domain switching；recoverable domains

0 Introduction

Ferroelectric materials have been widely used in many electronic and electromechanical devices such as memories and actuators due to their unique dielectric，ferroelectric and piezoelectric properties.The properties of ferroelectric materials are closely related to the polarization distribution and domain structures. Below the Curie temperature，ferroelec?tric materials with different compositions usually possess various kinds of domain structures. These domain structures are often dependent not only on the compositions but also on the geometries. Due to the dependence of domain structures on composi?tions and geometries，different structures［1］and compositions［2-3］have been designed to tune the do?main structures and the electromechanical coupling properties of ferroelectric materials. Meanwhile，the influence of temperature［4］，stress［5］and electric field［6-7］on the electromechanical coupling properties of ferroelectric materials have also been studied by many researchers. Due to the controllability of elec?tric field，controlling domain switching by external electric field to achieve the optimal performance has attracted much attention.

In addition to ferroelectric single crystals and ceramics，the interesting properties of ferroelectric superlattices have gradually become a research hot?spot in recent years［8］. The improvement of experi?mental techniques about epitaxial strain and layerby-layer growth led scientists to produce high quali?ty nanometer ferroelectric superlattices. For exam?ple，Yadav et al. used pulsed-laser deposition meth?od to synthesize ferroelectric superlattice films with different period-thicknesses［9］. Due to the different thicknesses of ferroelectric and dielectric layers in superlattices，domain structures are known to in?clude a1/a2 domain，a/c domain，flux-closure do?main，vortex domain［10］and polar skyrmions［11］.Hong et al. predicted that vortex structures appear in the superlattice grown on the DyScO3substrate when the thickness of the ferroelectric layer is 4 nm.If the external fields such as stress，strain and elec?tric field are considered，more domains with special morphologies could be generated and mixed.

In this work，the influence of the biaxial misfit strain on the domain structures and electromechani?cal response of the three-dimensional ferroelectric superlattice（PbTiO3（PTO）/SrTiO3（STO））is in?vestigated by using the phase field model. In order to get the domain evolution and electromechanical response，a quasi-static cyclic electric field is ap?plied in the stacking direction of the ferroelectric su?perlattice. The large range of electrostrain induced by the switching of domain structure and the change of polarization magnitude is the focus of the work.The recoverable and large electrostrain due to the domain switching from a1/a2 domain to c-domain under a biaxial tension strain is predicted.

1 Phase Field Modeling

In order to study the polarization evolution in ferroelectric superlattices，phase field method is fre?quently used. Phase field method has great advan?tage in investigating the influence of electric field and stress/strain on the evolution of ferroelectric do?main structure. It can not only qualitatively reveal the detailed shapes of domain and domain wall，but also quantitatively calculate the polarization and electric field. In this study，the polarization vector P =（P1，P2，P3）is chosen as the order parameter，and the polarization spatial distribution represents ferroelectric domain structure. Various properties of superlattice can be derived from domain structure.The temporal evolution of the domain structure can be obtained by solving the time-dependent Ginzburg-Landau（TDGL）equations［12］

where L，r and t denote the kinetic coefficient，spa?tial position vectors and time，respectively.

The total free energy of ferroelectric materialsincludes the Landau free energy，the domain wall energy，the elastic energy and the elec?tric energy. For the PTO and STO in the superlat?tice，their energy forms can be expressed as the fol?lowing functions related to polarization Pi，strain εijand electric field Ei，shown as

where αi，αij，αijkare the Landau energy coeffi?cients；gijklis the gradient energy coefficient，cijklthe elastic constant，and qijklthe electrostrictive coeffi?cient；ε0and εrrepresent the dielectric constant of the vacuum and relative dielectric constant of the background material （εr=66 for PTO and εr=300 for STO），respectively.

In addition，both the mechanical equilibrium equations

and the Maxwell’s equations

should be satisfied at the same time，where σijis the stress displacement component，Dithe electric dis?placement component and f the free-energy density.

Following previous work［12］，the spontaneous strains are related to spontaneous polarizations，which are given by［13］

where Qijare the electrostrictive coefficients.

In this work，the semi-implicit Fourier-spectral method is employed to numerically solve the related equations［14］. All of the related material parameters of PTO and STO at room temperature are taken from previous works［15］. Table 1 lists all the related material parameters of PTO and STO with SI units，and the unit of temperature T is K. Small ran?dom fluctuation（＜0.01P0，where P0=0.757 C/m2is the spontaneous polarization of PTO at room tem?perature）is used as the initial values to initiate the domain evolution.

Table 1 Material parameters of PTO and STO

Fig.1 shows the three-dimensional model of fer?roelectric superlattices. The x，y and z axes are par?allel to the［1 0 0］，［0 1 0］and［0 0 1］crystal?lographic directions， respectively. The structure consists of six layers，including three layers of ferro?electric film（PTO）and three layers of dielectric film （STO）. Monolayer of PTO and STO are stacked alternately along the z-axis direction. In or?der to generate vortex domain in the ferroelectric layer，the thickness of each layer is set to be 4 nm.Periodic boundary conditions are used in the stack?ing direction and in-plane direction. A three-dimen?sional mesh of 100 × 100 × 60 is used. Each grid is set as 0.4 nm to simulate the lattice size of the ma?terial approximately. The four arrows in Fig.1 repre?sent uniformly distributed biaxial strains（εxxand εyy）in the x and y directions. A vertical quasi-static electric field is applied in the z direction. Here，to meet the periodic boundary conditions in the stack?ing direction，uniform electric field is applied in?stead of the electric potential.

Fig.1 Schematic diagram of 3-D ferroelectric superlattices under given biaxial strains and vertical electric field

2 Results and Discussion

Under the large biaxial compressive strain of εxx=εyy=-1%，F(xiàn)ig.2 shows the polarization di?rection and the polarization component in the z direc?tion denoted by black arrows and color contour，re?spectively. The panels in Fig.2 are the domain states in the whole simulated model，while the be?low panels are the detailed polarization distribution in the white squares of the panels. The colors in Fig.2 denote the magnitudes of polarization compo?nents in the z direction. It can be seen that the stripe vortex domain structure in the ferroelectric layer in Fig.2（a）is a stable initial state under large biaxial compression strain when the layer thickness is set as 4 nm. Under a large compression strain，vortex and anti-vortex domains appear in the ferroelectric and dielectric layers of superlattice due to the competi?tion among electric field energy，strain energy and domain wall energy［8-9］. When the vertical electric field increases，the vortex domain structures are de?stroyed gradually. When superlattice finally reaches the state of single domain，the domain structure in superlattice is transformed into single c-domain and has the same polarization direction in the ferroelec?tric and the dielectric layer，as shown in Fig.2（b）.As a major characteristic of ferroelectric materials，residual polarization also exists in ferroelectric super?lattices. When the external electric field is removed，the domain structure in the superlattice remains，but the polarization magnitude decreases significantly.Because the in-plane domains are unfavorite under the large compressive strain，the out-of-plane polar?ization is induced by the vertical electric field，which are shown by the color contour in Fig.2（b）and Fig.2（c）.

Fig.2 Polarization distributions of the superlattice with different states

Based on the polarization along the stacking di?rection in the ferroelectric and dielectric layers，the average polarization of the whole superlattice is cal?culated. The hysteresis loop curves under different biaxial strains are plotted in Fig.3，in which the numbers 1，2，3 and their nearest solid lines repre?sent the three domains in Figs.3（d—f），respective?ly. The blue lines and numbers represent the transi?tion of the electric field from positive to negative，while the red ones represent the opposite. The ar?rows of solid line represent the order in which the polarization and domain structure change in a cyclic electric field.

The domain structures in Fig.2（b）and Fig.2（c）correspond to the two points where the electric field are equal to 60 and 0 kV/mm，respectively，in the hysteresis loop of Fig.3（a）. Under large com?pressive stain，the c-domain remains up or down in the vertical direction. Even if the electric field is re?moved，the initial vortex state cannot be recovered.However，when the strain decreases，the situation begins to change. Fig.3（b）shows that when the bi?axial strain is -0.5%，the shape of hysteresis loop begins to change，and the numbers of domain types begin to increase. When the positive electric field turns to the opposite direction，the c-domain in fer?roelectric layer becomes unstable. Without the re?strain of large biaxial compressive strain，the do?main structure begins to switch to the in-plane direc?tion，thus the diagonal direction domain and a/c do?main appear. Moreover，with the further decrease of the compressive strain，the range of electric field generating the c-domain further shrinks，and the electric field required for generating the single do?main is larger. When the electric field is small，the a/c domain and the diagonal direction domain be?come the dominant domain structure in the superlat?tice.

It should be noted that during the quasi-static loading and unloading of electric field，the variation of domain structure is not the same. The hysteresis loop is symmetric with respect to the center. With the decrease of biaxial misfit compression strain，the enclosed area of the hysteresis loop gradually shrinks. This suggests that it is possible to close the hysteresis loop by adjusting the biaxial misfit strain.The closing of the hysteresis loop means that the co?ercive field is eliminated，so the recoverable elec?trostrain is possible.

Fig.3 Hysteresis loops and typical domain structures of the superlattice under different biaxial compression strains

According to spontaneous strain equations of Eq.（5），the electrostrain of the superlattice can be obtained by the average polarization and the related electrostrictive coefficient. Under three different bi?axial compression strains， the curves of elec?trostrain versus electric field are presented in Fig.4.Lines and numbers in Fig.3 and Fig.4 correspond one to one. The strain calculated refers to the elec?trostrain along the stacking direction of the superlat?tice. Fig.4（a）is a common butterfly-shaped loop of ferroelectric strain versus electric field. Due to the change of domain structure，the corresponding loop of strain versus electric field deforms，as shown in Fig.4（b）and Fig.4（c）. For the biaxial misfit strains of -1%，-0.5% and 0.3%，the electric fields for inducing single domain are 40，50 and 80 kV/mm，respectively. Although the magnitudes of the elec?tric field leading to the single domain are not the same，the electrostrains induced by the same elec?tric field are basically equal in the saturated states for the three cases. Therefore，to find the largest change of electrostrain within a limited range of elec?tric field，we focus on the lowest point of the elec?trostrain curve rather than the highest point. By comparing the three curves in Fig.4，the decline of the lowest points is obvious. Although the decrease of the biaxial compression strain leads to the in?crease of the saturated electric fields，the largest variation of electrostrain increases. The largest vari?ation of electrostrain mainly results from the domain switching from out-of-plane to the in-plane direc?tion. When c-domain has only the 180° switching，the magnitude of polarization component Pzalong the stacking direction is still relatively large. Howev?er，when c-domain turns into diagonal direction do?main or a/c domain，Pzdecreases significantly，which causes the lowest point of electrostrain curves to go down. Under the biaxial compression strain of-0.3%，considering the state just reaches the sin?gle domain，the superlattice has the electrostrain range of over 3%，which is more than twice as much as that of-1%.

Fig.4 Electrostrain curves of the superlattice under different biaxial compression strains

It has been shown that the decreases of biaxial compressive strain increase the variation of elec?trostrain. However，the vortex or waved domain structures in the initial state do not have good recov?erability under a small compressive strain. When the electric field is removed，the ferroelectric layer of the superlattice is more inclined to form a/c domain to minimize the total energy，which causes the elec?trostrain in the stacking direction fail to return back to its original state. In order to solve this problem，a regulated way of applying tensile strain is consid?ered. It is found that in the initial state，a1/a2 do?main is stable in the superlattices under biaxial mis?fit tensile strain. Similar domain structures have been reported in the work of Hong et al.［10］，but the a1/a2 domain is realized by limiting the thickness of ferroelectric layers. In our simulation，a1/a2 do?main structure can be realized under larger thickness size by applying the tensile misfit strain. More sur?prisingly，the domain structure is recoverable under the biaxial tensile strain. As shown in Fig.5（a），when the external electric field is removed，the do?main structure of the superlattice will eventually switch back to the a1/a2 domain，and the color con?tour indicates the size and direction of Px.Figs.5（b—d）represent the detailed polarization dis?tribution of the corresponding area in the white box.It must be stated that for the initial state，the direc?tion of a1/a2 domain is same at the same position in each layer，but the domain structure recovered from c-domain is not. As can be clearly seen from Fig.5，the a1/a2 domains in the adjacent ferroelectric lay?ers are in opposite directions，while the domains in the ferroelectric layers separated by one layer are in the same direction. Due to the polarization along the stacking direction of a1/a2 domain is almost negligi?ble and the contribution of in-plane polarization to electrostrain in the stacking direction is the same，the differences in domain directions have no influ?ence and the recoverability of electrostrain is ob?tained.

Fig.6 Polarization versus electric field and electrostrain versus electric field under different tensile misfit strains

Using the same method as Fig.3 and Fig.4，we plot curves of the polarization versus electric field and the electrostrain versus electric field under two biaxial misfit tensile strains in Fig.6. The tensile strains of 0.3% and 0.5% correspond to Figs.6（a，b），respectively. The solid lines，colors，and num?bers have the same meaning as in Figs.3，4，except?ing that the new number 4 represents the a1/a2 do?main. It is clear that a/c domain has been replaced by a1/a2 domain under the biaxial tensile strain.This change is responsible for the recoverability of the electrostrain. It can be seen from Fig.6（a）that when the electric field is fully removed，the average polarization in the stacking direction returns to zero，which also means the absence of coercive fields.When the electric field changes from -50 kV/mm to 50 kV/mm，electrostrain curve has no hysteresis loop. When the electric field extends the range from-50 kV/mm to 50 kV/mm，two small hysteresis loops appear in the curve，which is caused by the different evolution of domain structures during the loading and unloading of electric field. Since what we seek is the largest range between the maximum and minimum electrostrain induced by electric field and the ability of turning back to the initial deforma?tion state after the electric field is removed，the models and outcomes are all acceptable.

As mentioned above，the smaller the compres?sive strain is，the lower the value of the minimum electrostrain will be. However，by comparing the in?fluence of different biaxial tensile strains on elec?trostrain in Figs.6（a，b），we find the above effect is almost negligible，because the contribution of a1/a2 domain to Pzis already very small. The further in?crease of biaxial tensile misfit strain has little influ?ence on the lowest point of electrostrain. Therefore，a small biaxial tensile misfit strain is enough for the superlattice to achieve the largest electrostrain range. When the single domain is formed，the polar?ization and electrostrain are always equal under the same electric field，no matter tensile strain or com?pressive strain. For the superlattice with the biaxial strain of 0.3%，an electric field of 250 kV/mm is re?quired to achieve the single domain state. Based on the domain switching from initial a1/a2 domain to single c-domain，5% electrostrain along the stack?ing direction is obtained.

3 Conclusions

In summary，the influence of biaxial misfit strains on the evolution of domain structures in the PTO/STO superlattices has been studied. The hys?teresis loops and electrostrain curves are obtained for the superlattices under different electric fields and misfit strains. The different domain structures and evolution processes are found under different misfit strains and electric fields. It is found that the 5% electrostrain along the stacking direction is ob?tained under a small biaxial tensile misfit strain. In addition，when the external electric field is re?moved，the fully polarized c-domain switches back to the initial a1/a2 domain and the electrostrain in the stacking direction recovers to the initial state.The present work provides an effective way to real?ize large and recoverable electromechanical response of ferroelectric nanomaterials by using domain engi?neering and strain engineering.