The 90° domain splitting and electromechanical behaviors in ferroelectric thin ﬁlms with triangle anti-dot array

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The 90° domain splitting and electromechanical behaviors in ferroelectric thin ﬁlms with triangle anti-dot array

Y. Zhang

^a

, Y.L. Wang

^a

, P. Chu

^a

, Y.L. Xie

^a

, Z.B. Yan

^a

, J.-M. Liu

^a,b,^⇑

aLaboratory of Solid State Microstructures and Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China

bInstitute for Quantum Materials, Hubei Polytechnic University, Huangshi 435000, China

a r t i c l e i n f o

Article history:

Received 14 December 2014 Received in revised form 2 March 2015 Accepted 5 March 2015

Available online 25 March 2015

Keywords:

Ferroelectric thin ﬁlms Anti-dot array Piezoelectricity

a b s t r a c t

The 90°-domain structure in ferroelectrics of tetragonal symmetry is one of the core ingredients determining the ferroelectric and electromechanical performances. We perform Monte Carlo simulations on the splitting of the 90°twin-like domains in ferroelectric thin ﬁlms with differently oriented triangle anti-dot arrays. It is revealed that the domain splitting is substantially dependent of the array orientation, and thus the piezoelectric and dielectric responses are the orientation-dependent too. The mechanism for the domain structure evolution is the competition between the dipole–dipole interaction and electromechanical energy, similar to the cases of square anti-dot arrays. The present work suggests the role of geometry and symmetry of the anti-dot array in modulating the electromechanical behaviors of 90°- domained ferroelectric thin ﬁlms.

1. Introduction

Ferroelectric (FE) materials have piezoelectricity and thus electromechanical coupling is one of the core ingredients of physics for FE domain structure[1]. A full understanding of the physics has been of fundamental interest and useful for guiding domain structure control for diverse applications[2–5]. We consider the most concerned tetragonal ferroelectrics like BaTiO3 and PbTiO3, and discuss the 90°-domain structure which is the most interested characteristic of tetragonal ferroelectrics [6,7], so that the electromechanical and dielectric performances can be improved[8,9].

In the phenomenological Landau–Devonshire framework on tetragonal ferroelectrics, the 90° twin-like domain structure is basically the consequence of competition among the multifold interactions, including the dipole–dipole interactionfdi, gradient energy fgr, elastic energy fel, and electrostrictive energy fes, etc.

[10]. The last two terms can be called the electromechanical energy fem=fel+fes. This competition results in a delicate balance among these energy terms in the 90°-domain structure [11]. The 90°

twin-like domain structure is thus the optimized self-organized pattern, while the competition between the long-range dipole–

dipole interaction and electromechanical energy is dominant in determining the domain structure[12,13].

One understands that the 90°twin-like structure favors an efficient energy release[11], in particular the release of electromechanical energy fem (elastic energy). However, this release is far from sufficient and there remains substantial energy stored inside the domains[14,15]. It is known that the piezoelectricity is deter- mined by the dynamic response of the domains against electric field or vice versa via the electromechanical coupling. Unfortunately, the elastic energy stored inside domains still imposes high resistance against electric field driven domain wall motion. This case is an ana- log with the pre-stressing mechanism usually employed in concrete construction engineering. Therefore, additional efficient energy release scheme is still favored. To our knowledge, a promising approach to overcome this challenge is to downscale the materials into nanostructures such as nano/micro-scale pillars, bubbles, and dots[16–18]. These structures have free boundaries allowing sufficient energy release[19]. However, these small-scale structures have insufficient volume to support mechanical load even though the piezoelectric coefficient itself is large.

A possible and alternative scheme may be to embed mesoscopic anti-dot array into FE lattice, where the anti-dot size or/and array periodicity is on the same order of magnitude as the twin-like domain width. This scheme has recently been addressed in our phase-field simulations (Monte Carlo simulations) on FE thin films with embedded square anti-dot array [20]. By optimizing the anti-dot size and array periodicity, it is predicted that the electromechanical energy can be efficiently released by the 90°-domain splitting process and consequently the piezoelectric coefficient can be significantly enhanced.

⇑ Corresponding author at: Laboratory of Solid State Microstructures and Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China.

E-mail address:[email protected](J.-M. Liu).

Contents lists available atScienceDirect

Computational Materials Science

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m m a t s c i

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Nevertheless, it should be mentioned that a hidden issue for the 90°-domain structure is the lattice symmetry. For a three-dimen- sional (3D) tetragonal FE lattice, there are six FE domain variants stemming from the cubic symmetry of the paraelectric parent lattice[21]. For a FE thin ﬁlm with in-plane polarization as an example, the parent lattice has square symmetry. We thus consider the 2D in-plane FE domains which have four equivalent variants. In this case, the square anti-dot array can induce domain splitting over the whole lattice[20]. It will be of interest to see the domain splitting behavior if an anti-dot array different from the square symmetry, such as triangle anti-dot array, is embedded into the lattice. This is the major motivation of the present work.

In this paper, we will investigate the 90°-domain structures in FE thin films with embedded triangle anti-dot arrays. As a representative case, we choose isosceles right triangle anti-dot array with different orientations. We first perform Monte Carlo simulations on the 2D square FE lattice with periodic boundary conditions as an approximation to infinite tetragonal FE thin films. Then, we compute the piezoelectric and dielectric responses to check how the embedded triangular anti-dot array modulates these responses. It should be mentioned that extension to the 3D lattice is direct.

2. Model and simulation methodology

We start from a 2DLLsquare lattice, and the energy on each sitei(or energy density) can be represented by two order parameters: electric dipole vectorP(r)=(Px,Py) and elastic displacement vector u(r)=(ux,uy), where x and y are the coordinate axes.

Following earlier work, we consider the same energy terms as the Landau–Devonshire phenomenological theory, including the Landau energy fld, gradient energy fgr, dipole–dipole interaction fdi, elastic energy fel, electrostrictive energyfes, and electrostatic energyfse[20,22,23]. For such a lattice, the total energy density can be written as:

f_total¼f_ldþf_grþf_diþf_emþf_se; ð1Þ

where electromechanical energy fem=fel+fes. For the Landau energy, we consider a six-order polynomial:

f_ld¼A1ðP²_xþP²_yÞ þA11ðP⁴_xþP⁴_yÞ þA12ðP²_xP²_yÞ þA111ðP⁶_xþP⁶_yÞ þA112ðP⁴_xP²_yþP⁴_yP²_xÞ; ð2Þ whereA1,A11,A12,A111, andA112are the Landau coefﬁcients.A1> 0 andA11< 0 are required for the ﬁrst-order transitions.A1 is sup- posed to be temperature (T) dependent and can be written as A1=A10(TT0), whereT0is the critical temperature for the FE tran- sition, and A10= (2

e

0C)¹ with vacuum permittivity

e

0and Curie constantC. The gradient energy which expresses the domain wall energy is:

f_gr¼1

2hG11ðP²_x;xþP²_y;yÞ þG12Px;xPy;yþG44ðPx;yPy;xÞ²þG⁰₄₄ðPx;yþPy;xÞ²i

; ð3Þ with positive gradient coefﬁcientG11,G12,G44,G⁰44, andPi,j=@Pi/@rj. The dipole–dipole interaction can be written as:

fdi¼ 1 4

pe

X

hji

PiPj

jrirjj³3Pi ðrirjÞ½Pj ðrirjÞ jrirjj⁵

" #

; ð4Þ

In the strict case, the calculation of this term should take account of whole lattice for the long-range feature. Precise numeri- cal methods like the Ewald summation scheme are used but very tedious[24]. Here, one can choose a preset truncating distanceR0

from the central site to make the computation simple. For a 2D lattice, this scheme is sufﬁciently reliable (R0= 8 in our simulation)

and a pre-checking with a comparison with the Ewald summation scheme was done[15,26]. It should be mentioned that another efﬁ- cient scheme to deal with the computation offdi is the spectral iterative perturbation (SIP) method[25].

Subsequently, the elastic energy is written as:

fel¼1

2C11ðu²_x;xþu²_y;yÞ þC12ux;xuy;yþ1

2C44u²_x;y; ð5Þ with the elastic coefﬁcientsC11,C12,C44, andui,i=@ui/@ri,ui,j=@ui/

@rj+@uj/@ri. Then, the electrostrictive energy reads:

fes¼ ux;x

g

_xxuy;y

g

_yyux;y

g

_xy; ð6Þ

where

g

_xx¼Q₁₁P²_xþQ₁₂P²_y,

g

_yy¼Q₁₁P²_yþQ₁₂P²_x,

g

xy=Q44PxPy, and Q11,Q12,Q44are the electrostrictive coefﬁcients. Finally, the electrostatic energy is

f_se¼ EPi; ð7Þ

with applied electric ﬁeldE= (Ex,Ey).

Given an anti-dot array, we assume P(r)0 and ui,j(r)0 (i,j=x,y) for the sites within anti-dots, which means the anti-dots are empty (full of air). Such kind of model could be dealt with more easily. This scheme corresponds to realistic structure which may be experimentally realized by micro-fabrication [27]. We consider isosceles right triangle anti-dots with four different orientations, where the hypotenuses are parallel to [1 1] or [11] directions and the two legs align along the x-axis and y-axis, respectively.

Mathematically, mechanical balance conditions on the boundaries between anti-dots and matrix can be handled by solving the time- dependent Ginzburg–Landau (TDGL) equation[21,28,29]or using the SIP method[30]. However, following earlier work[20], we still employ Monte Carlo simulation to track the domain patterns for the sake of convenience and contrast[10]. A reasonable assump- tion here is that the response speed of elastic strain is much faster than that for dipole relaxation, such that the mechanical equilib- rium is established just after an electric dipole flip event occurs [13]. In our simulation, we choose BaTiO3thin films as objects of simulation and relevant materials parameters are taken from literature[10]. We choose lattice constanta0, Landau coefficient

a

0with

a

0=

a

10(T₁T₀) and

a

10> 0, and polarizationP₀atT=T₁ far belowT0as reference values to normalize other parameters.

Term |

a

0|P²₀, which has an energy unit, is used to normalize those energy terms [31], and thus the dimensionless parameters are listed inTable 1 [12,32].

3. Results and discussion

We start from a FE lattice free of any anti-dot. The lattice is gradually annealed from a high TT0 to a sufﬁciently low TT0. The details of simulation were described in earlier work [25]and no description is given here.Fig. 1(a) shows the simulated FE domain pattern with lattice sizeL= 64. One can observe the well-aligned stripe-like 90°-domains, while domain walls align along the [11] direction. The results are in accord with those based on phase-ﬁeld simulations using the TDGL equation [13,15,33].

Subsequently, an anti-dot array is embedded into the lattice, and an adequately long-time relaxation of domain pattern is allowed.

We carefully check the total energy by choosing various annealing paths to make sure that the domain structure reaches the energy- lowest state (ground state) eventually.

As a reference, we ﬁrst look at the domain structure in a lattice with a 44 square anti-dot array, as shown inFig. 1(b), where the anti-dot size isLs= 6. It is seen that the stripe-like domains near the anti-dots split regularly[20]. Then we embed isosceles right triangle anti-dot arrays with different orientations into the lattice respectively and perform extensive simulations upon different

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anti-dot sizeLsand arrays, whereLsis the hypotenuse length. For convenience, we choose the anti-dot hypotenuse orientation as signature of the array, and we consider four representative types of arrays: type-(1 1), type-( 11), type-(1 1), and type-(1 1). The

simulated domain structures for the four cases are shown in Fig. 1(c)–(f), given the 44 arrays andLs= 7. It is clear that the anti-dot array orientation has notable impact on the domain pattern. An immediate claim is that local domain splitting will occur Table 1

Parameters chosen for the simulation (1/4pe= 1, Boltzmann constantkB= 1).

Parameter (unit) Value Parameter (unit) Value Parameter (unit) Value

A10(a10) 0.10 A11(a11P²₀/|a0|) 0.24 A12(a12P²₀/|a0|) 4.50

A111(a111P⁴₀/|a0|) 0.49 A112(a112P⁴₀/|a0|) 1.20 G11(g11/a²₀|a0|) 1.60

G12(g12/a²₀|a0|) 0.00 G44(g44/a²₀|a0|) 0.80 G⁰44(g⁰44/a²₀|a0|) 0.80

C11(c11/|a0|P²₀) 2.75 C12(c12/|a0|P²₀) 1.79 C44(c44/|a0|P²₀) 0.54

Q11(q11/|a0|) 0.142 Q12(q12/|a0|) 0.0074 Q44(q44/|a0|) 0.0157

L(a0) 64 T0(Tc/(TcT1)) 5.00

Fig. 1.Simulated in-plane domain patterns for lattice free of anti-dot (a), lattice with square anti-dot array (Ls= 6) (b), lattice with the type-(1 1) anti-dot array (c), lattice with the type-(11) anti-dot array (d), lattice with the type-(11) anti-dot array (e), and lattice with the type-(1 1) anti-dot array (f), respectively. The red arrows indicate the direction of polarization and the anti-dot array periodicity is 16. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

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if domains near the anti-dots have polarizations perpendicular to the legs, so that the polarizations near the anti-dots ﬂip to be parallel to the edges. If the polarization is not perpendicular to one edge, such a splitting will not necessarily occur. It is seen the higher the probability of domain splitting, the closer to ±

p

/2 the anglehPbetween the polarizationPand one nearby triangle edge.

The underlying mechanism is the efficient release of the depolarization field on the anti-dot edges via the domain splitting. This depolarization field is attributed to the dipole–dipole interaction and electromechanical energy.

We take the lattice with the type-(1 1) anti-dot array as an example, as shown inFig. 1(c). It is seen that the original dipoles next to anti-dots are perpendicular to the triangle legs, and thus

the depolarization ﬁeld has to be suppressed by the domain splitting, while the original dipoles near the triangle hypotenuses have hP ±

p

/4. In contrast, for the type-(11) array shown in Fig. 1(e), the domain structure remains roughly unchanged since all the dipoles near the triangle legs are parallel to the edges. For the type-(11) and type-(1 1) arrays, shown inFig. 1(d) and (f), similar analysis can be made and one sees local domain splitting events on those locations at which the original domains are perpendicular to the triangle edges.

It seems that the size and orientation of the anti-dots plus the array periodicity all have inﬂuences on the domain structure splitting. The anti-dot array embedding has two impacts. First, an anti- dot itself allows the release of local elastic energy in surrounding Fig. 2.Simulatedfdi-contours for the lattices (a) free of anti-dot, (b) with the type-(1 1) anti-dot array, (c) with the type-(11) anti-dot array, (d) with the type-(11) anti-dot array, (e) with the type-(1 1) anti-dot array, respectively. The anti-dot array periodicity is 16.

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region. Second, the depolarization ﬁeld near the anti-dot edges has to be suppressed by the domain splitting. The twofold effect can be understood by looking into the energy landscape. We calculate the spatial contours of termsf_diandf_em.Fig. 2(b)–(e) shows the contours of termfdisurrounding one anti-dot withLs= 7 for the four different orientations, while the contour for the lattice free of anti-dot (Ls= 0) is shown inFig. 2(a) as a reference. It is clearly revealed that for Ls= 0, the lattice has higher fdi on the domain walls than that inside domains. With the embedded anti-dot arrays, the regions near the two triangle legs have the lowestfdi, while the regions near the triangle hypotenuses have the highest

f_diwith the angleh_P ±

p

/4 for almost all local dipoles. Obviously, the anti-dot array embedding results in higherfdi averaged over the lattice.

Subsequently, we look at the f_em-contours, as shown in Fig. 3(b)–(e) for the four cases, while the contour atLs= 0 is shown inFig. 3(a) for reference. It is seen that the lattice free of anti-dot accommodates highfemover the whole lattice. In contrast, the lattice with the type-(1 1) array, which experiences domain splitting, has the lowest averagedfemalthough thefem-contour is inhomoge- neous, as shown inFig. 3(b). The lattice with the type-(11) array, which has no domain splitting and thus wide domains, shows high Fig. 3.Simulatedfem-contours for the lattices (a) free of anti-dot, (b) with the type-(1 1) anti-dot array, (c) with the type-(11) anti-dot array, (d) with the type-(1 1) anti-dot array, (e) with the type-(1 1) anti-dot array, respectively. The anti-dot array periodicity is 16.

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fem-contour too, as shown inFig. 3(d). As mentioned above, the lattices inFig. 3(c) and (e) accommodate local domain splitting and thus have lowerfem.

The above results clearly demonstrate that the electromechanical energyfemcan be efﬁciently released by the domain splitting, although the local density of dipole–dipole interaction may be enhanced as compensation. It is noted that energy termsfld, and fgrmay change too upon the domain splitting, but the variations are small and can be neglected. In the overall sense, the depolarization ﬁeld near the anti-dot triangle edges is the key driving force for the splitting initiation, while the electromechanical energy drives the splitting across the whole region between two neighbor anti-dots. As the reduction infemis more than the gain infdi, the domain splitting is thermodynamically spontaneous.

One of the consequences of the domain splitting is the enhanced electromechanical responses in sufﬁciently split domain

structure, i.e. enhanced piezoelectric and dielectric responses. The anti-dot embedding itself also functions the energy release. For the triangle anti-dot arrays, these responses are dependent of the array orientation. We evaluate the effective longitudinal piezoelectric coefﬁcient d^eff₃₃= d

g

^[10]/dE^[10], where

g

^[10]=hQ₁₁P²_xþQ₁₂P²_yi, E^[10]

denotes the electric ﬁeld along thex-axis, andh i represents the spatial averaging over the whole lattice including anti-dots[33].

We choose0.2 <E^[10]< 0.2, within which a good linear depen- dence of term

g

^[10]on fieldE^[10]is shown and the slope is defined asdêff₃₃. For possible comparison with experiments, we have a rough estimation of the realisticd33, following the procedure given in Refs.[12,32]. It is noted thatdêff₃₃ has the unit ofQ₁₁P²_x=E, withQ11

has the unit ofq11/C11. The parametersq11,C11,Px, andEare nor- malized by |

a

0|, |

a

0|P²₀,P0, and |

a

0|P0, respectively. One can obtain the unit of d33 to be 1/|

a

0|P0, where |

a

0| = 10¹¹N m²/C², P0= 1 Fig. 4.(a) Simulated effective longitudinal piezoelectric coefficientdêff33as a function ofLsfor the lattices with four types of triangle anti-dot arrays. Thedêff33contours for the four types of anti-dot arrays are shown in (b)–(e), respectively. The anti-dot array periodicity is 16.

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C/m². By this way, we have the calculatedd^eff₃₃ in the unit of pC/N.

Fig. 4(a) plots the evaluateddêff₃₃as a function ofL_sfor the four types of 44 arrays. The error bars for the data are ±10%, upon averaging over ten simulation cycles. It is seen that thedêff₃₃for all the four cases increases withLsand reaches the peak atLs= 7 for the type- (1 1) and type-(1 1) arrays, and atLs= 9 for the type-(11) and type- (1 1) arrays. It is evidenced that, first, the lattices embedded with the triangle anti-dot arrays do have significantly enhanced piezoelectric performance. The peakeddêff₃₃ can be one order of magnitude larger than that of the lattice free of anti-dot. Second, the piezoelectric performance is indeed orientation-dependent. The lattice with the type-(11) array, which shows nearly no domain splitting, has the lowestdêff₃₃, suggesting the better electromechanical performance of the domain structure with narrower stripe width. In the other words, the domain splitting may benefit to the performance. However, it is not the domain structure with the type-(1 1) anti-dot array that exhibits the highest dêff₃₃. Instead, the lattices with local domain splitting, as the cases with the type-(11) and type-(1 1) arrays, show higher dêff₃₃than that for the lattice with the type-(1 1) array, implying that the type-( 11) and type-(1 1) arrays themselves can release the energy more efficiently than the type-(1 1) and type-(1 1) arrays. InFig. 4(b)–(e) are plotted the latticed33-contours for the four types of anti-dot arrays atLs= 7, while thed33-contour for the lattice free of anti-dot is roughly 10 and homogeneous over the whole lattice. It is shown that embedding the anti-dot array does allow the domain walls to exhibit higherd33for all the four cases. Finally, it should be mentioned that the predicteddêff₃₃ values of 100 pC/N are definitely quite high for BaTiO₃ thin films in such microwave range.

Without the embedded anti-dot array, the reald33would be 10–

20 pC/N, typical values for thin ﬁlms.

Another consequence of the domain splitting associated with the anti-dot array embedded lattice is the enhanced dielectric

response in the microwave frequency[15], because the dielectric response in this frequency range comes from the domain wall vibration driven by ac electric ﬁeldE(t). Therefore, more domain walls and free boundaries contribute higher dielectric permittivity.

The ac dielectric permittivity

e

(x) can then be calculated by P(x) =

e

(x)E(x), where P(x) and E(x) are the Fourier trans- formations of polarizationP(t) and electric ﬁeldE(t), as described in details in Ref.[15]. The Monte Carlo step (mcs) is used to scale time t. Similarly, the dielectric permittivity

e

(x) =P(x)/E(x), where P(x) and E(x) have the unit of P₀mcs and |

a

0|P₀mcs, respectively. The

e

(x) has a unit of 1/|

a

0|.Fig. 5(a) and (b) show the simulated real and imaginary parts of the dielectric permittivity spectra for all the cases. Clearly, the lattices with the embedded anti-dot arrays do show much higher dielectric response than that for the lattice free of anti-dot (Ls= 0), consistent with the fact that the anti-dot array embedded lattices have larger piezoelectric coefﬁcient than the lattice free of anti-dot. It is also noted that the evaluated dielectric constant is also consistent with measured values in the microwave frequency range (GHz).

The present triangle anti-dot array structure can be seen as an alternative to the square anti-dot structure investigated earlier [20], while they both improve the ferroelectric and piezoelectric properties effectively. However, the triangle anti-dot array is different from the square one in symmetry. We consider four different orientations and the performances among them are diverse.

From the analysis of the domain patterns withfdi- andfem-contours of four orientations, one can clearly understand the roles of these two energy terms on the domain splitting. At the same time, the electromechanical and dielectric responses are orientation-dependent, which may be a reference to design more complex anti-dot structures.

Finally, it should be mentioned that the present work only con- siders the effect of triangle anti-dot arrays of four different orientations on the in-plane stripe-like 90°domain structure in FE thin films. We do not deal with the 3D cases where in-plane domains coexist with out-of-plane domains which deserve for future investigation. It is understood that realistic domain structures would have more complicated behaviors than the present results, but the underlying physics should be similar. The main task here is to modulate the 90°-domain structures effectively by embedding different anti-dot arrays. By analyzing the roles of the anti-dot symmetry in controlling ferroelectric, piezoelectric, and dielectric properties of FE thin films, the present simulations pro- vide an approach in practically engineering domain structure in FE materials for specific electromechanical applications.

4. Conclusion

In summary, we have investigated the in-plane 90° domain structure and piezoelectric/dielectric responses of tetragonal FE thin ﬁlms with triangle anti-dot arrays. It has been found that the domain splitting is dependent of the orientation and dimension of triangle anti-dot array. The delicate balance of the dipole–dipole interaction and electromechanical energy results in the domain splitting. Simultaneously, the calculations indicate considerably enhanced electromechanical performance and dielectric response, which are, however, orientation-dependent. The present work offers an alternative approach to control FE domain structures and their features for practical electromechanical applications.

Acknowledgments

This work was supported by the National 973 Projects of China (Grant Nos. 2015CB654602 and 2011CB922101), the Natural Science Foundation of China (Grant Nos. 51431006 and 11234005).

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Fig. 5.Simulatedacdielectric permittivity real part (a) and imaginary part (b) for lattices withLs= 0 (free of anti-dot) and lattices with the four types of triangle anti- dot arrays, respectively. The anti-dot array periodicity is 16.

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