Ferroelectric Domain Walls Approaching Morphotropic Phase

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Ferroelectric Domain Walls Approaching Morphotropic Phase Boundary Jinghui Gao, Xinghao Hu, Yongbin Liu, Yan Wang, Xiaoqin Ke, Dong Wang, Lisheng Zhong, and Xiaobing Ren J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b11595 • Publication Date (Web): 09 Jan 2017 Downloaded from http://pubs.acs.org on January 11, 2017

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The Journal of Physical Chemistry

Ferroelectric Domain Walls Approaching Morphotropic Phase Boundary Jinghui Gao1, Xinghao Hu1, Yongbin Liu1, Yan Wang1, Xiaoqin Ke2, Dong Wang2*, Lisheng Zhong1* , Xiaobing Ren1,3 *

1. State Key Laboratory of Electrical Insulation and Power Equipment and Multidisciplinary Materials Research Center, Frontier Institute of Science and Technology, Xi’an Jiaotong University, Xi’an, 710049, China 2. Center of Microstructure Science, Frontier Institute of Science and Technology, Xi’an Jiaotong University, Xi’an, 710049, China 3. Ferroic Physics Group, National Institute for Materials Science, Tsukuba, 3050047, Ibaraki, Japan

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ACS Paragon Plus Environment


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Abstract Domain walls play an important role on tailoring the properties of ferroic materials. We employ a Landau-Ginzburg thermodynamic model to investigate the ferroelectric domain walls around the morphotropic phase boundary (MPB) which contribute to the large piezoelectric response, including 90o domain wall for tetragonal(T) symmetry and 109o and 71o domain walls for rhombohedral(R) symmetry. The domain wall energy for each type of domain wall has been analytically calculated by quasi-one-dimensional solution of the spatial evolution for polarization. Our results suggest that the domain wall energy for both T-T and R-R domains decrease as the composition approaches MPB accompanied by the vanishing of polarization anisotropy, which can be viewed as the origin for the experimental-observed miniaturized domains. And the reduced domain wall energy caused by the polarization anisotropy vanishing is responsible for the large piezoelectric response due to the displacement of domain walls.

Corresponding Authors: *E-mail: [email protected] Tel: +86-029-83395072. *E- mail: [email protected]. Tel: +86-029-82668268 *E- mail: [email protected]. Tel: +81-29-859-2731 J. G. and X. H equally contribute to this paper.

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1. Introduction Owing to the high efficiency on the conversion between mechanical energy and electric energy, the piezoelectric materials have been broadly employed in the manufacture of electronic devices, such as sensor, actuator, transducers and so on1. Most of the high performance pieozoelectric materials are inevitably achieved on a so-called morphotropic phase boundary (MPB), which is regarded as the compositioninduced phase boundary connecting the tetragonal (T) and rhombohedral (R) ferroelectric phases in piezoelectric systems, such as Pb(Zr,Ti)O3 (PZT), Pb(Mg,Nb)O3-PbTiO3 (PMN-PT), Pb(Zn,Nb)O3-PbTiO3 (PZN-PT)

2-9

. It is crucial to

understand the mechanism for large piezoelectricity in MPB region, which enable the advanced design of high-performance piezoelectric materials. Early crystallographic investigation suggests that the vertical MPB for example in PZT system can be considered as the sluggish transition from T to R with temperature10.

Later, an intermediate phase with monoclinic symmetry (M) was

reported to exists in a narrow composition region interleafing the R and T phases for PZT, PMNPT, PZNPT by high-resolution X-ray diffraction method, which is believed to facilitate the polarization rotation thus responsible for large piezoelectric

11-14

.

Furthermore, Jin et. al. pointed out that M symmetry can be mimicked by an adaptive phase showing a hierarchical domain structure of nano T twin or nano R twin with low domain wall energy

15-19.

There are intensive investigations on the interpretation

of large piezoelectricity on MPB from the theoretical modeling point of view. First principle calculations pointed out the importance of polarization rotation on facilitating large electromechanical response for piezoelectric systems20-22. And the phenomenological Landau-Devonshire model suggest that the reduction of 3



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polarization anisotropy was responsible for enhanced piezoelectric response approaching to MPB 23-25. Further phase field simulation based on the time-dependent Ginzburg-Landau model suggests that the domain walls, including their density, configuration, and distribution are crucial on understanding the origin of MPB

26-29

. It

is generally believed that the vanishing or low domain wall energy is the reason for miniaturized domain size, broadened domain wall and large extrinsic piezoelectric response on MPB. However, we notice that although the importance of domain wall has been recognized, there is still lack of the analytical solution or approximation for the domain wall energy close to MPB, especially its quantitative relationship with the polarization anisotropy is missing. In this paper, we construct a Landau-Ginzburg modeling to describe the domain walls between T-T domains on one side of MPB and R-R domains on the other side regarding the polarization anisotropy. The spatial evolution of polarization across the domain wall has been analytically calculated by using a quasi-one-dimensional solution. And the corresponding domain wall energy and wall thickness have been described as a function of the coefficient of polarization anisotropy term in the landau free energy polynomial. And the domain miniaturization, domain wall broadening as well as enhanced extrinsic piezoelectric activity can be predicted from such a theoretical modeling. Our findings of thermodynamic principles also give general implications for ferroic materials with displacive transitions with domain walls

30, 31

,

and may help design and development of new functional materials with advanced properties. 2. Theory

4


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The Landau-Ginzburg thermodynamic model for ferroelectric twins (90o T-T domains and 109o, 71o R-R domains) have been considered. The free energy for the system includes chemical free energy (fL) describing the local phase stability and gradient energy (fG) describing the interface or domain wall between different phases. To simplify the calculations without influencing the static results, long range order interaction of elastic energy and electromechanical coupling are not considered in our model. And the free energy can be written as follows:

F = ∫ f L + fG

(2.1)

The first term in eq. 2.1 is Landau-Devonshire chemical free energy, which can be expanded with respect to the polarization vector P=(P1 P2 P3) 32, FL (x,T, Pi ) =α1(P12 + P22 + P32 ) +α110 (P14 + P24 + P34 ) +α120 (P12P22 + P22P32 + P12P32 ) +α1110 (P16 + P26 + P36 ) +α1120[P12 (P24 + P34 ) + P22 (P14 + P34 ) + P32 (P14 + P24 )]

(2.2)

+α1230P12P22P32

α1, α110, α120, α1110, α1120, α1230 are the polynomial coefficients which depend on the systems and vary with temperature T and composition x. The second term in Eq. 2.1 is the gradient energy which describes the interface energy between two domains. FG = +

g11 2 2 (P1,1 + P2,2 + P3,32 ) + g12 (P1,1P1,2 + P1,1P3,3 + P2,2 P3,3 ) 2

g44 [(P1,2 + P2,1)2 + (P1,3 + P3,1)2 + (P2,3 + P3,2 )2 ] 2

(2.3)

Here Pi,,j are the directional derivatives of Pi with the direction j. And gij are gradient coefficients. The equilibrium configuration of spontaneous polarization across domain wall can be obtained by solving the following equation: ∂F =0 ∂P

(2.4)

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To solve the above-mentioned Eq. 2.4 on static equilibrium conditions, we used the Euler equation in Eq. 2.5 33: ∂ ∂F ∂F ( )− = 0, (i, j = 1, 2,3) ∂x j ∂Pi , j ∂Pi

(2.5)

To further simplify the solution for Eq. 2.4, quasi-one-dimensional analytical solution has been used in this work. Such a method has been used by Cao et. al. to study the interface of tetragonal twin in ferroelectric perovskites successfully

33

. To

avoid volume charges, a domain wall usually aligns along the bisector between two polarizations of the adjacent ferroelectric domains

19

. Further charge neutrality

condition of domain wall guarantees that only one polarization component varies with space

33

. Thereby the 3D spatial polarization evolution across the domain wall has

been simplified into 1D partial differential equation. Three kinds of twinning systems (including 90o T-T domains and 71 o, 109 o R-R domains) are considered since we notice that not all types of domain configuration contribute to the piezoelectric response. For example, in tetragonal phase there are two types of domain configuration, i.e. 90 o domains and 180 o domains. It is known that 180

o

domain switching does not induce any strain change, and thus cannot

contribute to the piezoelectric response. Besides, the arrangements for polarization alignment between two polarization domains can be head-to-head, tail-to-tail, and head-to-tail. The former two arrangements will result in the accumulation of net charge on the twin boundaries, and thus is not energetically stable. On the other hand, the head-to-tail twin configuration can neutralize the charge accumulating on the domain wall, which can be considered as the prevailing domain configuration in piezoelectric system. 3. Results & Discussion 6


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3.1. The composition dependent anisotropy coefficient approaching MPB region The Landau-Devonshire free energy in Eq.2.2 which depends on composition and temperature by changing coefficients reflects the crystal symmetry, phase transition and the physical properties relevant for ferroelectric systems. The coefficient of the fourth order term of Eq. 2.2 influences the polarization anisotropy23,

24

. Figure 1

shows the composition dependence of phase stability and anisotropy in PbTiO3PbZrO3 systems. At PbTiO3-rich region, T phase is the stable phase. So, we can transform Eq. 2.2 into the following form to obtain equilibrium T-T domains and its interface: FL (x,T, Pi ) =α1(P12 + P22 + P32 ) +α11(P12 + P22 + P32 )2 +α12 (P12P22 + P22P32 + P12P32 ) +α111(P12 + P22 + P32 )3

(3.1)

where, the fourth order terms have been transformed into a combination of isotropic term (with coefficient α11) as well as anisotropic term (with coefficient α12). The anisotropic term is minimized for T phase with P=(P0 0 0) or (0 P0 0) or (0 0 P0) when α12>0. And these coefficients have the following relationship with the 4th order term coefficients in Eq. 2.2: α11 = α110

(3.2)

α12 =α120 − 2α110

(3.3)

On the other hand, with the decreasing of PbTiO3 content to a PbZrO3-rich region (see Figure 1), we can transform Eq. 2.2 into the following form to obtain equilibrium R-R domains and its interface: FL (x,T, Pi ) = α1(P12 + P22 + P32 ) +α11(P12 + P22 + P32 )2 +α12 (P14 + P24 + P34 ) +α111(P12 + P22 + P32 )3 +α123P12P22P32

7


(3.4)


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In such a form, the isotropic term (with the coefficient α11) and anisotropic term (with the coefficient α12) guarantee a stable R phase with P=(P0 P0 P0), and have the following relationship with the coefficient in Eq. 2.2: 1 2

α11 = α120

(3.5) 1 2

α12 =α110 − α120

(3.6)

The composition dependence of the anisotropic term α12 has been obtained by experimental measurements from references 34-38. As shown in Figure 1, the α12 varies with the PbTiO3 content x. Such a parameter reduces when approaching to MPB from both T and R sides. In MPB, the anisotropic term α12 tends to decrease to zero suggested by the Landau model proposed by Rossetti et al. Although it cannot strictly vanish when calculating from the analytical expression for the Landau polynomial coefficients, the role of vanishing polarization anisotropy has been recognized for the piezoelectricity enhancement of MPB, which will be discussed in the latter part. It should be noticed that the domain configuration is quite complicated for the specimen in MPB region39-49, especially for the microstructure of phase coexistence. Therefore, we only discuss the modeling of ferroelectric domain walls for single phases close to MPB, including the 90o domain walls for T phase and 109o, 71o domain walls for R phase. 3.2. 90o domain walls of T phase The 90o domains, which can contribute to piezoelectric response, consist of two adjacent domains with spontaneous polarization directions perpendicular with each other. Figure 2 shows the schematic head-to-tail domain wall separating two tetragonal domains, the spontaneous polarizations of which can be described by P1 8


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and P2. And the twin plane or domain wall plane is (110). For convenience, the coordinate system has been rotated by 45o around [001] axis, which results in a new coordinate with r= [110], s= [110], z= [001]. r axis is inside the twin plane and s axis is along twin plane normal direction (see Figure 2). And the free energy in Eq. 2.1 can be described as a function of new coordinate with Ps and Pr after subscribing Eq. 3.1: 1 1 F = FL + FG = α1(Ps2 + Pr2 ) + (α11+ α12 )Ps4 + (α11+ α12 )Pr 4 4 4 +(2α11 −

α12

)Ps2Pr 2 +α111Pr6 +α111Ps6 + 3α111Pr4Ps2 + 3α111Pr 2Ps4

(3.7)

2 Gss 2 Grs 2 + Ps,s + Pr,s 2 2

Here Gss =(g11 + g12 + 2 g 44 ) / 2

(3.8)

Grs =(g11 -g12 ) / 2

(3.9)

We solved the differential equation ∂F = 0 by using the Euler equation in Eq. 2.5. ∂P

It should be noted that in order to keep the charge neutrality, we assume that the polarization component Ps along s direction keeps unchanged across the domain wall. And the solution is as follows: Ps =

2 P0 2

(3.10)

sinh( s / ξ )

Pr = P0

[ A + sinh 2 ( s / ξ )]

(3.11) 1 2

Where 1

ξ=

1 P0

A=

 2 Grs  2 +   6α111 P0 + 2α11 

(3.12)

3α111 P02 + α11+ 2α111 P02 + α11+

(3.13)

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(3.14)

α1+ =1/4(4α11 − α12 )P0 2 + α1 + 3 / 4α111P04

α11+ =3 / 2α111P02 + (α11 +

α12 ) 4

(3.15)

The spatial evolution of polarization across the interface can thus be described by the kink solution in Eq. 3.11. And the solution is depicted in Figure 3(a) as the change of Pr with s. It can be seen that Pr evolves from negative value to positive value through a polarization variation region, i.e. domain wall.

We then obtained the

changes of P1 (T domain 1) and P2 (T domain 2) with s. It can be seen from Figure 3(b) that the curve for P1 and P2 are symmetric indicating a twinning nature around the 90o domain wall. To study the composition dependence of domain wall configuration, we further calculated the spatial evolution of Pr with polarization anisotropy coefficient α12 which depends on the composition (see Figure 1). As shown in Figure 3(a) and (b), although all the curves show the same tendency, the one with large polarization anisotropy coefficient exhibits a very steep curve, while zero polarization anisotropy shows a relatively gradual change with distance. Furthermore, the illustrations for polarization variation have been shown in Figure 4(a). The illusion of polarization variation suggests that, with the decrease of α12 the width for domain wall region is enlarged. More precise result is further shown in Figure 4(b), and the coefficient ζ, known as the parameter for domain wall width, becomes larger. Moreover, the energy density stored in 90o domain wall E can be calculated from the polarization variation relationship of Eq. 3.11 as follows: +∞

E=

2Grs

∫ ( F − F )dx = [ α 0

−∞

]1/2 [

α11+ P0 2

111

α+ −( 11 − α1+ ) arcsin h 4α111

(3P02 +

P0 (2 P02 +

α11+ 1/2 ) 2α111

α11+ 1/2 ) 2α111

]

10


(3.16)

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, which also changes with the polarization anisotropy. As shown in Figure 4(c), the domain wall energy decreases with the reduction of α12, and it reaches to the minimum value when polarization anisotropy is vanishing with α12→0.

3.3. 109o and 71o domain walls of R phase 109o and 71o domain walls are two typical head-to-tail polarization alignments for the adjacent R-phase domains with polar axis along orientations. The schematic drawings for these two types of domain walls have been depicted in Figure 5(a) and Figure 5(b) respectively. As shown in Figure 5(a), 109o domain wall has the (110) twin plane bordering two domains with the 109o angle between the adjacent polarization vectors. We assume that the polarization only varies on (110) plane. For charge neutrality interface consideration, the polarization along the domain wall normal direction (PS∥[110]) keeps unchanged, and the polarization variation across the domain wall can be described by the spatial evolution of Pr∥ [001] along the domain wall plane. In the same way, for 71o domain wall (Figure 5(b)), the polarization evolution is only calculated on (110) plane, and the polarization along the domain wall Pr∥[110] varies with distance across the domain wall. Therefore, these two types of domain walls can be turned into quasi-one-dimensional models. Accordingly, the coordinate transformations are required for both domain walls as demonstrated in Figure5(a) and Figure5(b), and the free energy for the system can be expressed as: For 109o domain wall,

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F =FL +FG =α111 Pr 6 + 8α111 Ps 6 + 6α111 Pr 4 Ps 2 + 12α111 Pr 2 Ps 4 +(4α11 +4α12 ) Ps 4 + (α11 +α12 ) Pr 4 -2α12 P12 P2 2 + 4α11 Ps 2 Pr 2 +α1 (2 Ps 2 + Pr 2 )+α123 Ps 4 Pr 2 +

Gss 2 Grs 2 Ps , s + Pr ,s 2 2

(3.17)

Here, Gss =(g11 + g12 + 2 g 44 ) / 2 and Grs =g 44 . For 71o domain wall, F = FL + FG =8α111 Pr 6 + α111 Ps 6 + 12α111 Pr 4 Ps 2 + 6α111 Pr 2 Ps 4 + (α11 +α12 ) Ps 4 + (4α11 +2α12 +α123 Ps 2 ) Pr 4 + 4α11 Ps 2 Pr 2 +α1 ( Ps 2 + 2 Pr 2 )+

(3.18)

Gss 2 Grs 2 Ps , s + Pr , s 2 2

Here, Gss =g11 / 2 and Grs =g 44 / 2 . The solutions of differential equation ∂F = 0 for these two types of domain walls ∂P

have the similar expression form as follows: sinh( s / ξ )

Pr = P0

1

[ A + sinh 2 (s / ξ )]2

(3.19)

1

1 ξ= P0

A=

 2 Grs  + 2 +  α α 6 P + 2 11   111 0

(3.20)

+ 3α111 P02 + α11+ + 2α111 P02 + α11+

(3.21)

However, 109o and 71o domain walls have different coefficients forms and Ps expressions as follows: For 109o domain wall, Ps =

8 3

2 P0 3

(3.22) 2 9

8 3

α1+ =( α11 + α123 ) P02 + 2α1 + α111P0 4

(3.23) α11+ =8α111 P0 2 + 4α11 +4α12

(3.24) 12


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+ α111 =α111

(3.25)

For 71o domain wall, 1 P0 3

Ps =

(3.26)

2 4 α1+ = α111 P0 4 + α11 P0 2 + 2α1 3 3

(3.27)

1 α11+ =4α111P02 +4α11 +2α12 + α123 P02 3

(3.28)

+ α111 =8α111

(3.29)

The spatial evolution of polarization for 109o and 71o domain walls can thus be described by the kink solution in Eq. 3.19 and 3.22. The changes of Pr with s from these solutions are depicted in Figure 6(a) and Figure 6(b) for 109o and 71o domain walls. Further space profiles for different α12 (Figure 6(a) and Figure 6(b)) and the associated illustrations (Figure 8(a) and Figure 8(b)) demonstrate the domain wall broadening phenomenon approaching MPB. And the domain wall width coefficient ζ in Figure 7(a) becomes larger when α12 approaches zero, which is similar with 90o domain wall in T phase. Furthermore, the domain wall energy for 109o and 71o domain walls can be calculated subscribing the kink solutions, and they have the same expression, but with different coefficient values: +∞

E=

2Grs

∫ ( F − F )dx = [ α 0

−∞

−(

α

+ 11 + 111

4α

+ 111

]1/2 [

α11+ P0 2

(3P02 +

P0

− α1+ ) arcsin h

(2 P + 2 0

α11+ 1/ 2 ) + 2α111

α11+ 1/ 2 ) + 2α111

]

(3.30)

The domain wall energy for both types vary with the polarization anisotropy coefficient. As shown in Figure 7(b), the domain wall energy decreases with the reduction of α12, and it reaches to the minimum value when polarization anisotropy is 13



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vanishing with α12→0. But it seems that the 109o domain wall energy decreases much larger compared with 71o domain wall. Also the 109 o domain wall exhibits lower energy, and is thus more stable than 71o domain wall in the low polarization anisotropy region of 0

Ferroelectric Domain Walls Approaching Morphotropic Phase

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