Domain splitting and enhanced piezoelectricity in ferroelectric thin films with square grid structure

doi: 10.1209/0295-5075/108/27009

Domain splitting and enhanced piezoelectricity in ferroelectric thin films with square grid structure

Y. Zhang¹, P. Chu^1(a), Y. L. Xie¹, D. P. Chen¹, Z. B. Yan¹ andJ.-M. Liu^1,2(b)

1 Laboratory of Solid State Microstructures and Innovation Center for Advanced Microstructures, Nanjing University Nanjing 210093, China

2 Institute for Quantum Materials, Hubei Polytechnic University - Huangshi 435000, China received 30 April 2014; accepted in final form 2 October 2014

published online 29 October 2014

PACS 77.80.Dj– Domain structure; hysteresis

PACS 77.65.Bn– Piezoelectric and electrostrictive constants

PACS 75.40.Mg– Numerical simulation studies

Abstract– The pattern evolution of the stripe-like 90^◦domain structure in tetragonal ferroelectric thin films upon introduction of square void arrays is investigated using Monte Carlo simulation based on the Landau-Devonshire phenomenological theory. The regular splitting of the stripe- like domains near the voids and substantial release of both the dipole-dipole interaction and the electromechanical energy are revealed. Consequently, a significant enhancement of piezoelectric coefficientd33is predicted. It is suggested that the delicate balance between the two energy terms modulated by the void array periodicity and void size is responsible for the domain splitting. The present work proposes an alternative promising approach to tune the domain structure via simple micro-fabrication techniques.

Copyright cEPLA, 2014

The performances of ferroelectric (FE) thin films and nano-structures have been receiving considerable interest not only due to the significance in fundamental researches but also for their diverse applications in piezoelectric micro-systems, FE memories, high-frequency electronic components, etc. [1–3]. Those applications rely substan- tially on the FE polarization and electromechanical (piezo- electric) behaviors, which are related to the FE domain structures and their responses to different environmental stimuli such as electrical field, stress, spatial confinement (dimensionality), etc. Along this line, extensive investiga- tions on the FE domain structure and its control in order to optimize these properties have been reported [4–6]. The so-called domain engineering on ferroelectrics has become of high concern in recent years since those application per- formances can be then remarkably improved [7,8].

The most representative class of FE materials target- ing such applications includes FE perovskite oxides like BaTiO₃ and PbTiO₃ which have tetragonal FE phase originating from cubic paraelectric phase upon decreas- ing temperatureT [9,10]. For a FE system like BaTiO₃, the well-aligned stripe-like 90^◦domain structure is one of

(a)E-mail:playerc308@gmail.com

(b)E-mail:liujm@nju.edu.cn

the core characteristics [9]. For the sake of simplification, we discuss a two-dimensional (2D) FE lattice as an ap- proach to FE thin film, where the FE polarizations are confined onto the xy-plane along the [01] and [10] easy axes. This 90^◦ domain structure is the consequence of the delicate balance mainly between the electric dipole- dipole interactions (depolarization effect) associated with the FE polarization and the electromechanical (elastic plus electrostrictive) energy associated with the piezoelectricity and electrostriction effect, while the domain wall energy may not be negligible, too [11]. This delicate balance is the reason for sensitive responses of the domain structure to either external electric field or stress/pressure, so as to exhibit excellent electromechanical performances in addi- tion to the favored FE properties [12].

It is noted that both the dipole-dipole interactions and the electromechanical energy in a FE lattice are long- ranged and they can be strongly modulated by imposing various spatial constraints, in particular the dimension- ality and geometric constraints. They are responsible for very scattered FE and piezoelectric properties of finite-size FE materials such as nano- or micro-scale dots and islands as well as substrates confined thin films [13–15]. Con- sequently, the regular stripe-like or twinned 90^◦ domain structure in bulk lattice may be replaced by vortex-like

domain pattern and others. Quite a number of works re- ported observations of various fascinating domain patterns by modulating these spatial constraints [16,17]. Taking a naked FE island (in-plane sizeL and thicknessD, and D/L1 for thin films) with free boundary as an example, one can easily predict the evolution of domain structure, with decreasingL from stripe-like domain pattern into a coexistence of the stripe-like pattern and the vertex-like one, eventually into a single vertex pattern [18]. For such small-size island, the dipole-dipole interaction in compet- ing with the electromechanical energy dominates the for- mation of vertex structure. It was reported that such FE islands in some cases do exhibit better electromechanical performances than their bulk counterpart.

Given the rapid progress in micro-fabrication technol- ogy of materials, one may have additional approaches to engineer the FE domain structure. While a fabrication of well-aligned FE island lattices (arrays) faces technical challenging, it represents one end of releasing the spatial constraints. In this case, the dipole-dipole interaction and electromechanical energy are seriously released. The other end would be an alternative scheme in which an infinite FE thin film can be patterned by removing finite-size units, so that a grid thin film structure with periodically aligned voids (void array) can be fabricated. This end scheme seems quite less touched in the literature so far but offers several particulars. Starting from a stripe-like 90^◦domain structure, fabricating such voids would result in several consequences. First, the thin film as a whole remains continuous, but the voids will partially release the elec- tromechanical energy of the neighboring film region while the dipole-dipole interaction enforces the electric polar- izations surrounding the voids to re-align. By properly designing the spatial distribution, shape, and size of the voids, one is allowed to modulate substantially the stripe- like domain pattern for specific purposes. Second, due to the high mobility of the 90^◦domain walls, such voids may benefit to the enhancement of the FE and electromechan- ical responses. Third, with respect to the FE island array, this grid thin film structure can be easily and efficiently fabricated by up-bottom fabrication techniques, which is technologically favored and cost-competitive.

In this work, we investigate the spatial evolution of the 90^◦ domain structure in a FE thin film with periodically aligned square void array, based on the well-known Landau-Devonshire phenomenological theory [19–22].

Certainly, the shape of voids can be other than square but the underlying physics should be similar. We start from a 2D L ×L square lattice with periodic bound- ary conditions as an approach to an infinite FE thin film. On each site i two order parameters are imposed:

polarization vector P(r) = (Px, Py) and displacement vectoru(r) = (ux,uy), wherexandyare the coordinate axes. Following earlier literature, the free energy of the FE lattice includes the Landau energy, gradient energy, dipole-dipole interaction, elastic energy, electrostrictive interaction, and electrostatic energy [18]. For such a

lattice, the total free energy density can be written as f_total=fld+fg+f_dip+f_ela+fes+fE, (1) where the termsfld,fg,f_dip,f_ela,fes,fE are the Landau energy, gradient energy, dipole-dipole interaction, elastic energy, electrostrictive energy, and electrostatic energy, respectively. The electromechanical energy includes f_ela plusfes. The Landau energy is a polynomial extended to the sixth order:

fld =A₁

P_x²+P_y²

+A₁₁

P_x⁴+P_y⁴

+A₁₂ P_x²P_y² +A₁₁₁

P_x⁶+P_y⁶

+A₁₁₂

P_x⁴P_y²+P_y⁴P_x² , (2) where A with various subscripts are the Landau coeffi- cients. Usually,A₁ isT-dependent and can be written as A₁=A₁₀(T−T0), whereT₀is the critical point for the FE transition, andA₁₀= (2ε₀C)⁻¹with vacuum permittivity ε₀ and Curie constantC. Positive A₁ and negative A₁₁ are required for the FE transitions. The domain wall energy is expressed by the gradient energy form:

fg = 1 2

G₁₁

P_x,x² +P_y,y²

+G₁₂Px,xPy,y

+G₄₄(Px,y−Py,x)²+G₄₄(Px,y+Py,x)²

, (3)

where Pi,j = ∂Pi/∂rj, and Gij > 0 are the gradient energy coefficients. The long-range electric dipole-dipole interaction is written as

f_dip = 1 4πε

j

Pi·Pj

|ri−rj|³

−3Pi·(ri−rj) [Pj·(ri−rj)]

|ri−rj|⁵

, (4)

where the pre-factor (1/4πε) = 1 is taken for normalizing purpose. We can re-write eq. (4) as

f_dip =−1

2E₁·Pi, (5)

where the internal electric fieldE₁ is E₁=− 1

2πε

j

Pj

|ri−rj|³ −3 (ri−rj) [Pj·(ri−rj)]

|ri−rj|⁵

. (6) Due to the existence of voids in the lattice, big lattice size L has to be taken. A precise account of this long- range term using the Ewald summation scheme becomes practically overtedious [23]. Fortunately, for 2D lattice, one can choose a preset truncating distanceR₀ from the central site in order make the computation tractable. This scheme is sufficient for a reliable account of this interaction (R₀= 8 in our simulation) [24,25].

Table 1: Parameters chosen for the simulation (1/4πε= 1, Boltzmann constantkB = 1).

Parameter (unit) Value Parameter (unit) Value Parameter (unit) Value A₁₀ (α₁₀) 0.10 A₁₁(α₁₁P₀²/|α₀|) −0.24 A₁₂ (α₁₂P₀²/|α₀|) 4.50 A₁₁₁ (α₁₁₁P₀⁴/|α₀|) 0.49 A₁₁₂(α₁₁₂P₀⁴/|α₀|) 1.20 G₁₁ (g₁₁/a²₀|α₀|) 1.60 G₁₂ (g₁₂/a²₀|α₀|) 0.00 G₄₄ (g₄₄/a²₀|α₀|) 0.80 G₄₄ (g₄₄ /a²₀|α₀|) 0.80 C₁₁ (c₁₁/|α₀|P₀²) 2.75 C₁₂ (c₁₂/|α₀|P₀²) 1.79 C₄₄ (c₄₄/|α₀|P₀²) 0.54 Q₁₁ (q₁₁/|α0|) 0.142 Q₁₂(q₁₂/|α0|) −0.0074 Q₄₄(q₄₄/|α0|) 0.0157 L(a₀) 64 T₀(Tc/(Tc-T₁)) 5.00

Subsequently, the elastic energy reads f_ela=1

2C₁₁

u²_x,x+u²_y,y

+C₁₂ux,xuy,y+1

2C₄₄u²_x,y, (7) where ui,i =∂ui/∂ri, ui,j =∂ui/∂rj+∂uj/∂ri, and Cij

are the elastic coefficients. The electrostrictive energy ac- counting the coupling between the polarization field and strain field is written as

fes=−ux,xηxx−uy,yηyy−ux,yηxy, (8) with strainηxx=Q₁₁P_x²+Q₁₂P_y²,ηyy=Q₁₁P_y²+Q₁₂P_x² and ηxy = Q₄₄PxPy, where Q₄₄ are the electrostrictive coefficients. Finally, the electrostatic energy is

fE =−E·Pi, (9) with applied electric fieldE= (Ex, Ey).

The void array is simply introduced into the FE lattice by imposingP(r)≡0 andui,j(r)≡0 (i, j =x, y) in the pre-designed regions. Here we only consider square voids with the two sides aligning along the x-axis and y-axis, respectively. It is noted that these voids in the lattice makes it mathematically challenging to properly handle the mechanical balance conditions on the boundaries be- tween thin film matrix and voids. This difficulty becomes particularly clear if the time-dependent Ginzburg-Landau equation (TDGL, Langevin-type) is used to solve this is- sue numerically [16]. However, no such difficulty appears if one employs the Monte Carlo (MC) simulation to track the evolution of the FE lattice [18]. A reasonable assump- tion here is that the response time of elastic strain is far shorter than that for dipole relaxation, such that the me- chanical equilibrium is instantly established when an elec- tric dipole flip event occurs [15,26]. The object lattice of our simulation is BaTiO₃thin film. The relevant materials parameters are taken from the literature [18]. We assume that the lattice constant a₀, Landau coefficient α₀, and polarizationP₀ at T =T₁ far below T₀are known as ref- erence, whereα₀=α₁₀(T₁−T₀) andα₁₀>0. All the free energy terms are normalized by term|α₀|P₀²which has an energy unit [27]. Table 1 lists the chosen parameters for simulation.

For practical simulation, we submit a FE lattice free of any void to a high T T₀ for annealing. Then the lattice is gradually cooled with sufficient relaxation time down to a low T T₀. The details of the simulation

Fig. 1: (Color online) Simulated in-plane domain patterns for Ls = 0 (a), 4 (b), 6 (c), and 12 (d), respectively. The arrows indicate the domain polarization orientation which is defined by angleθ.

were reported in our earlier work [24,25] and no descrip- tion is given here. Figure 1(a) shows the simulated 90^◦ domain structure with L = 64. The well-aligned stripe- like 90^◦domain walls along the [1¯1] direction are observed.

The results are well consistent with phase-field simulations using the TDGL equation [15,26,28]. Subsequently, a pe- riodically distributed square void array of given size Ls

is introduced into the lattice and a sufficiently long-time relaxation of the domain structure is allowed. The even- tually reached domain structure as the energy lowest state (the ground state) is checked carefully by choosing various annealing paths.

We perform extensive simulations via the standard Metropolis algorithm on the domain structure evolution by imposing different spatial distribution and size of the voids. It is shown that the evolutions share qualitatively similar characteristics. As an example, we present in figs. 1(b)–(d) the simulated domain structures with a 4×4 void array, forming a grid FE thin film. The void sizeLs

takes 0, 4, 6, and 12, respectively. At the first glance, the existence of voids in the FE thin film has marked impact on the domain structure. However, differently from FE islands where the vertex pattern is preferred, we observe an interesting pattern evolution when Ls takes different values. AtLs≤4, the well-aligned 90^◦ domain structure

with straight walls (fig. 1(a)) can be maintained roughly, whereas only local deformation in the domain walls neigh- boring the void boundaries is seen, as shown in fig. 1(b).

This local deformation becomes complicated as Ls > 4 and the pattern atLs = 6 is shown in fig. 1(c). Surpris- ingly, regular domain splitting sequence is clearly observed due to the introduction of the void array. By such split- ting, one can halve the domain width and periodicity. This domain splitting can sustain untilLsis big enough, such as Ls = 12, as shown in fig. 1(d), where the FE lattice consists of crossing thin grids along thex-axis andy-axis.

Due to the strong depolarization effect, all the domains in the horizontal grid edges align along thex-axis and those in the vertical grid edges align along they-axis. However, every edge-crossing region is occupied by local 90^◦domain structure (two 90^◦ walls).

It seems that for the grid FE lattice, the domain walls favor the self-adaptive evolution so that the electric polar- izations neighboring the voids can align in parallel to the boundary rather than any other choice. This is a common feature almost applicable for all the cases. It is found that the domain splitting sequence can be realized by properly assigning the void array periodicity (λ) and void sizeLsin a broad range, thus a flexible and easily accessed approach to control the domain structure is obtained. In compari- son with FE islands, the grid FE lattice seems to offer rich possibilities of domain patterns [17].

In order to see more clearly the domain splitting, we choose the case of the 4×4 void array and Ls = 6 for an illustration. The snapshoted domain patterns at four annealing times (T = 0.05) are shown in figs. 2(a)–(d).

Starting from the initial state in fig. 2(a), each stripe- like domain with vertical polarization begins to split at positions near the voids and generates two branches, as shown in fig. 2(b). The splitting develops continuously with time along theθ∼135^◦direction and ends near the neighboring void. This sequence thus generates the final well-aligned and regularly split domain structure shown in fig. 2(d) and fig. 1(d).

Considering the well-organized split domain structures, we also consult the Fourier transform spectrum of the dipole correlation function in the wave vector (k = (kx, ky)) space. The Fourier spectrum of the spatial cor- relation function is defined as

S(k) = 1 L⁴

ij

Pi·Pjexp[ik·(ri−rj)], (10) where subscripts i and j denote the lattice sites.

Figures 2(e)–(g) show, respectively, the Fourier spectra of the FE lattice free of void (fig. 1(a)), the FE lat- tices with 4×4 void array and Ls = 6 without any an- nealing and with sufficient annealing. A comparison of figs. 2(a) and (b) helps us to identify the indications of the spots. The bright spot atk= (4,4), as indicated by the ar- row in fig. 2(e), reflects the stripe domain pattern whereas the weaker spot atk∼(12,12) comes from the high-order effect from the Fourier transform. The additional dim

Fig. 2: (Color online) Instantly snapshoted domain patterns upon thermal annealing atT = 0.05 for various timest= 0 (a), 1×10⁵mcs (b), 3×10⁵mcs (c), and 5×10⁶mcs (d) for a FE lattice with 4×4 void array andLs= 6. The Fourier spectra of the spatial correlation function for the FE lattices in (a) and (d) are plotted in (f) and (g), respectively. The Fourier spectrum of the lattice free of any void is shown in (e) for comparison. The arrows in (e)–(g) indicate the spots from the domain structures.

spots in fig. 2(f) obviously come from the 4×4 void array of Ls= 6, noting that the brightest spot is atk= (4,4) from the stripe domain pattern. Figure 2(g) plots the Fourier spectrum of the well-developed domain pattern after the sufficient annealing. The brightest spot at k = (8,8) characterizes the well-split stripe domain pattern. This spot originates from the shifting of the brightest spot in fig. 2(f), due to the splitting of the stripe domains.

To understand the physics underlying such domain splitting behaviors, one may evaluate the spatial contours of the dipole-dipole interaction (f_dip) and electromechan- ical energy (f_ela +fes). For convenience of compari- son, figs. 3(a) and (b) show the spatial contours of f_dip around one void of Ls = 6 at annealing time t = 0 and t = 5×10⁶mcs, respectively. It is clearly shown that those polarizations perpendicular to the void sur- faces at t = 0 re-align themselves to be parallel to the surfaces during the annealing, resulting in a significant reduction of the local f_dip, which is one of the driving forces for the domain splitting. This process can also be illustrated by the spatial contours of the internal electric field. Figures 3(c) and (d) present the spatial contours of the magnitude of the internal electric field|E1| at t = 0

Fig. 3: (Color online) Simulated energy contours for the dipole- dipole interaction fdip ((a), (b)), magnitude contours of the internal electric field|E1|((c), (d)), and the electromechanical energy (fela+fes) ((e), (f)), at two annealing timest= 0 and 5×10⁶mcs, respectively.

and 5×10⁶mcs, with the arrows indicating the distri- butions of E1. At the up and left boundaries of voids, the dipole-dipole fields are initially offset by the strong depolarization field. The total internal field in this re- gion is relatively small, as shown in fig. 3(c). The anneal- ing process drives the dipoles to realign in parallel to the voids boundaries, leading to the modulated internal elec- tric field, as shown in fig. 3(d) in comparison with fig. 3(c), focusing on the void-field boundaries. Figures 3(e) and (f) present the spatial contours of (f_ela+fes). Similar be- haviors are observed and the existence of voids releases remarkably the local electromechanical energy. In partic- ular, the high (f_ela+fes) energy on those domain walls (red color) in fig. 3(e) is fully released, and the whole re- gion has much lower (f_ela+fes) energy via the polariza- tion re-alignment. It is noted that the total gain in the electromechanical energy and dipole-dipole interaction is much greater than the cost for additional domain walls arising from the domain splitting. Therefore, the domain splitting is thermodynamically spontaneous. It should be mentioned that those energy terms in eq. (1) other than f_dip and (f_ela+fes) may change too upon the domain splitting. However, these changes are much less remark- able thanf_dip and (f_ela+fes).

One of the favored consequences of the domain split- ting is the expected enhancement of the electromechanical performance. To confirm this prediction, we evaluate the piezoelectric coefficientd₃₃ defined asd₃₃= dη^[10]/dE^[10]

where η^[10] = ηxx = Q11P_x²+Q₁₂P_y², E^[10] denotes the electric field along the x-axis, and represents the

Fig. 4: (Color online) (a) Simulated longitudinal piezoelectric coefficientd33 and its effective valued^eff₃₃, as a function ofLs

respectively for the lattice with 4×4 void array. The d33

contours for Ls = 0 and Ls = 8 are shown in (b) and (c), respectively.

configuration average [28]. When the averaging is made over the whole lattice including the voids, the evaluated piezoelectric coefficient is denoted byd^eff₃₃, while the coef- ficient is denoted byd₃₃if the averaging is made over the lattice excluding the voids. We take−0.2< E^[10] <0.2, within which term η^[10] exhibits good linear dependence onE^[10]. The evaluated d₃₃ andd^eff₃₃ as a function ofLs, respectively, are plotted in fig. 4(a). One observes that bothd₃₃ andd^eff₃₃ increase rapidly with increasingLs and reach the maximal asLs= 4–8. Afterwards, bothd₃₃and d^eff₃₃ begin to fall and become low as Ls is over 12. It is noted that more than 20 times enhancement ind₃₃ is ob- tained asLs increases from zero toLs= 4. This implies that the introduction of the void array allows a remarkably enhanced piezoelectric coefficient, a surprising but highly favored effect.

To find the correlation of this piezoelectric enhance- ment with the domain structure, one plots the spatial contours of the local d₃₃ for the lattices of Ls = 0 and Ls= 8 in figs. 4(b) and (c), respectively, where the shad- owed stripes roughly mark the domain walls. As shown in figs. 3(a) and (c), the stripe-like 90^◦ domain structure free of any void has both the high dipole-dipole interaction (f_dip) and the high electromechanical energy (f_ela+fes).

In particular, the high (f_ela+fes) implies that the whole lattice is in the seriously strained state. Therefore, ad- ditional lattice deformation becomes highly resistive,i.e.

d₃₃ is very small. As expected, thed₃₃contour forLs= 0 is roughly homogeneous over the whole lattice with small magnitude, and only the domain walls exhibit slightly higher d₃₃. Differently from this case, the lattice with the Ls = 8 shows very high local d₃₃ not only on the domain walls, as illustrated in fig. 4(c). The physics is

simple. On the one hand, the sufficiently released dipole- dipole interaction and electromechanical energy by the voids introduction benefit substantially to the high mo- bility and deformation capability of the domains under electric field. On the other hand, the domain splitting al- lows the doubling of the domain walls, contributing to the electromechanical response.

It should be mentioned that the present simulations are far from perfect. We only deal with the effects of the square void array on the 90^◦ domain structure and elec- tromechanical behaviors. Also, only the 2D thin film lat- tice with in-plane FE domains is considered here. One may consider various arrays of voids of different shapes and deal with more realistic domain structures in FE thin films, where in-plane domains and out-of-plane domains coex- ist. The simulations may be extended to 3D cases with much more complicated domain structures. Nevertheless, for these extended cases, the underlying physics would be similar and the core issue here is how to efficiently release and even modulate the concerned free energy terms by im- planting various geometric structures, so that the favored FE and electromechanical performances can be enhanced.

In this sense, the present work, in addition to the FE is- lands, disks, dots, or composite structures, may shed light on alternative approaches to engineer the domain struc- ture of FE materials for specific purposes. In fact, a recent experiment revealed that introduction of low permittivity defects into KTiOPO₄ lamellae can dramatically modu- late the domain switching dynamics [29]. An anisotropic electric field pattern near the air holes is believed to be responsible for the observed results. In the present work, the internal electric field also shows a spatial distribution due to the depolarization field near the film-void interface.

This experiment together with our findings here allows us to believe that modification of the FE domain structure by introducing voids-like defects is effective.

In summary, we have investigated the domain patterns and piezoelectricity of the tetragonal FE thin films with square grid structure. It is found that by properly choos- ing the size and array of the square voids, the stripe-like domains near the voids may spilt regularly. The com- petition between the dipole-dipole interaction and the electromechanical energy is responsible for the domain splitting. The calculations on the longitudinal piezo- electric coefficient predict a remarkable enhancement of the piezoelectric coefficient, caused by the sufficiently re- leased dipole-dipole interaction and electromechanical en- ergy due to the voids introduction and thus the doubling of domain walls.

∗ ∗ ∗

This work was supported by the National Key Projects for Basic Researches of China (2015CB654602), the Nat- ural Science Foundation of China (51431006, 51332006, 11374147), and the Priority Academic Program Develop- ment of Jiangsu Higher Education Institutions, China.

REFERENCES

[1] Setter N., Damjanovic D., Eng L., Fox G., Gevor- gian S., Hong S., Kingon A., Kohlstedt H., Park N. Y., Stephenson G. B., Stolitchnov I., Taganstev A. K., Taylor D. V., Yamada T.and Streiffer S., J. Appl. Phys.,100(2006) 051606.

[2] Scott J. F.,Science,315(2007) 954.

[3] Bondurant D. W. and Gnadiger F. P., IEEE Spectrum,26(1989) 30.

[4] Jin L., Li F. and Zhang S., J. Am. Ceram. Soc., 97 (2014) 1.

[5] Sluka T., Tagantsev A. K., Damjanovic D., Gureev M.andSetter N.,Nat. Commun.,3(2012) 748.

[6] Dawber M., Rabe K. M.andScott J. F.,Rev. Mod.

Phys.,77(2005) 1083.

[7] Jang H. W., Ortiz D., Baek S.-H., Folkman C. M., Das R. R., Shafer P., Chen Y., Nelson C. T., Pan X., Ramesh R.andEom C.-B.,Adv. Mater.,21(2009) 817.

[8] Davis M., Damjanovic D., Hayem D.andSetter N., J. Appl. Phys.,98(2005) 014102.

[9] Arlt G.andSasko P.,J. Appl. Phys.,51(1980) 4956.

[10] Goo E. K. W., Mishra R. K.andThomas G.,J. Appl.

Phys.,52(1981) 2940.

[11] Hu H.-L. and Chen L.-Q., J. Am. Ceram. Soc., 81 (1998) 492.

[12] Wang J., Shi S.-Q., Chen L.-Q., Li Y. and Zhang T.-Y.,Acta Mater.,52(2004) 749.

[13] Naumov I. I., Bellaiche L. and Fu H., Nature, 432 (2004) 737.

[14] Ponomareva I., Naumov I. I.andBellaiche L.,Phys.

Rev. B,72(2005) 214118.

[15] Hu H.-L. and Chen L.-Q., Mater. Sci. Eng. A, 238 (1997) 182.

[16] Li Y. L., Hu S. Y., Liu Z. K. andChen L. Q.,Acta Mater.,50(2002) 395.

[17] Zhang J. X., Wu R., Choudhury S., Li Y. L., Hu S.

Y.andChen L. Q.,Appl. Phys. Lett.,92(2008) 122906.

[18] Xue F., Gao X. S.andLiu J.-M.,J. Appl. Phys.,106 (2009) 114103.

[19] Ahn C. H., Rabe K. M.andTriscone J. M.,Science, 303(2004) 488.

[20] Wang Y. G., Zhong W. L. and Zhang P. L., Phys.

Rev. B,51(1995) 5311.

[21] Li S., Eastman J. A., Li Z., Foster C. M., Newnham R. E.andCross L. E.,Phys. Lett. A,212(1996) 341.

[22] Wang J., Kamlah M., Zhang T.-Y., Li Y.andChen L.-Q.,Appl. Phys. Lett.,92(2008) 162905.

[23] De’Bell K., MacIsaac A. B.and Whitehead J. P., Rev. Mod. Phys.,72(2000) 225.

[24] Wang X., Liu J. M., Chan H. L. W.andChoy C. L., J. Appl. Phys.,95(2004) 4282.

[25] Li B. L., Liu X. P., Fang F., Zhu J. L.andLiu J. M., Phys. Rev. B,73(2006) 014107.

[26] Wang L. F.andLiu J.-M.,Appl. Phys. Lett.,89(2006) 092909.

[27] Chen D. P.andLiu J.-M.,Appl. Phys. Lett.,100(2012) 062904.

[28] Wang L.-F.andLiu J.-M.,Appl. Phys. Lett.,91(2007) 092908.

[29] Whyte J. R., McQuaid R. G. P., Sharma P., Canalias C., Scott J. F., Gruverman A.andGregg J. M.,Adv. Mater.,26(2014) 293.