相场模拟应变调控PbZr(1–x)TixO3薄膜微观畴结构和宏观铁电性能

刘迪; 王静; 王俊升; 黄厚兵

doi:10.7498/aps.69.20200310

摘要

外延生长铁电薄膜中基底失配应变能够调控微观铁电畴结构和宏观铁电性能. 本文选择了三种相结构(四方相、四方和菱方混合相、菱方相) PbZr_(1–x)Ti_xO₃ (x = 0.8, 0.48, 0.2)铁电薄膜, 利用相场模拟研究了在不同基底失配应变(ε_sub)作用下, 三种成分铁电薄膜中微观畴结构的演化以及宏观极化-电场回线. 随着应变从–1.0%变化到1.0%, 三种相结构铁电薄膜的矫顽场、饱和极化值以及剩余极化值全都降低, 其中PbZr_0.52Ti_0.48O₃薄膜的饱和极化值和剩余极化值比另外两种薄膜降低更快. 模拟结果表明拉应变能提高铁电薄膜储能效率, 其中准同型相界处应变提升储能效率最快. 本工作揭示了应变对PbZr_(1–x)Ti_xO₃铁电薄膜中畴结构、电滞回线以及储能等方面的影响, 为铁电功能薄膜材料的实验设计提供了理论基础.

关键词:

Abstract

Ferroelectric domain structures and ferroelectric properties in the hetero-epitaxially constrained ferroelectric thin films can be manipulated by substrate misfit strain. In this work, three kinds of phase structures of PbZr_(1–x)Ti_xO₃ thin films, including tetragonal, tetragonal- rhombohedral-mixed and rhombohedral phases, are investigated. Firstly, the ferroelectric domain structures at different substrate misfit biaxial strains are obtained by the phase-field simulation. Then we calculate the polarization-electric field hysteresis loops at different misfit strains, and obtain the coercive field, saturation polarization, and remnant polarization. In the tetragonal PbZr_(1–x)Ti_xO₃ (x = 0.8) thin film, compressive strain contributes to the formation of out-of-plane c1/c2 domain, and tensile strain favors in-plane a1/a2 domain formation. With the increase of compressive strain, the tetragonal phase and the rhombohedral phase coexist in PbZr_(1–x)Ti_xO₃ (x = 0.48) film near the morphotropic phase boundary, while the tensile strain reduces the rhombohedral domain size. In the rhombohedral PbZr_(1–x)Ti_xO₃ (x = 0.2) film, the rhombohedral domains are steady states under compressive strain and tensile strain. As the misfit strain changes from –1.0% to 1.0%, the value of the coercive field, saturation polarization and remnant polarization decrease. Among them, for tetragonal-rhombohedral mixed phase, the reductions of saturation field and remnant polarization are larger than for tetragonal phase and rhombohedral phase. The coercive field of mixed phase decreases rapidly under the compressive strain, but deceases slowly under the tensile strain. It is worth noting that the remnant polarization decreases faster than the saturation polarization in three components of ferroelectric thin film. Due to the electromechanical coupling, when x = 0.48 at the morphotropic phase boundary it is shown that the remnant polarization reduction is faster than those of the other two types of ferroelectric thin films, and the small coercive field is obtained in the case of large tensile strain. Therefore, tensile strain can effectively improve the energy storage efficiency in ferroelectric thin films, and the efficiency of x = 0.48 thin film increases significantly compared with that of x = 0.8 or 0.2 thin film. Both the ratio of rhombohedral/tetragonal phase and the domain size will play a significant role in ferroelectric performance. Therefore, our results contribute to the understanding of the electromechanical coupling mechanism of PbZr_(1–x)Ti_xO₃, and provide guidance for the experimental design of ferroelectric functional thin film materials.

Keywords:

PACS:

作者及机构信息

1.
北京理工大学材料学院, 北京　100081

2.
北京理工大学前沿交叉科学研究院, 北京　100081

通信作者: 黄厚兵, hbhuang@bit.edu.cn

基金项目: 国家级-国家重点基础研究发展计划(2019YFA0307900)

Authors and contacts

1.
School of Materials Science and Engineering, Beijing Institute of Technology, Beijing 100081, China

2.
Advanced Research Institute of Multidisciplinary Science, Beijing Institute of Technology, Beijing 100081, China

Corresponding author: Huang Hou-Bing, hbhuang@bit.edu.cn

文章全文

1. 引　言

锆钛酸铅PbZr_(1–_x₎Ti_xO₃ (PZT)因其优异的铁电、压电、热电和介电性能^[1-5], 在国防和工业中得到广泛应用, 包括非易失性存储器^[6,7]、压电传感器^[8,9]、光电器件^[10,11]以及铁电储能^[12-14]等. 其中铁电随机存储器因非易失性存储和低功耗等优势, 具有潜在的商业化应用前景. 铁电存储器主要通过外加电场翻转铁电极化来实现“0”和“1”存储状态的切换, 进而实现信息的快速写入过程. 目前, 铁电材料的极化翻转稳定性是实现铁电存储器商业化应用的重要指标^[6,7], 而其中铁电薄膜的力电耦合机制可以影响铁电极化翻转的稳定性.

随着外延薄膜生长技术的不断发展^[15,16], 可以选择不同的基底(衬底), 通过界面晶格参数差异产生的失配应变来调控铁电薄膜性能^[17]. 前期已有相关实验和理论工作证明铁电材料的畴结构和宏观铁电、压电性能可以受外延应变的调控, 包括BaTiO₃ (BTO)^[18-20], SrTiO₃ (STO)^[21,22], BiFeO₃ (BFO)^[23-25], PZT^[26-29]等. 例如, 实验上, 通过调节基底失配应变可以提升BTO薄膜的居里温度以及剩余极化值^[18]、提升BFO薄膜的压电性能^[25]和调控PZT薄膜的极化翻转以及畴结构演变^[30]. 理论上, Xue等^[31]和Lin等^[32]通过相场模拟研究发现基底失配应变对四方相PZT薄膜中畴结构演化产生影响, 结果表明拉应变促进面内a1/a2畴长大, 而压应变促进面外c畴长大. 在PbTiO₃ (PTO)薄膜中, 基底拉应变触发铁弹畴翻转, 可提升PTO铁电薄膜的压电性能^[29]. 虽然基底失配应变在畴结构演变方面已被广泛研究, 但是基底失配应变对铁电薄膜不同铁电相的宏观铁电性能(即电滞回线)的影响还需深入探讨.

因此, 本文选择三种不同成分的PZT (x = 0.8, 0.48, 0.2)薄膜作为研究对象, 通过相场方法模拟基底失配应变对其微观畴结构及宏观铁电性能的影响. 研究不同基底失配应变对PZT薄膜四方相、四方和菱方混合相、以及菱方相的铁电畴结构演变影响. 在此基础上分析基底失配应变对三种相结构电滞回线和铁电薄膜储能效率的影响, 为铁电功能薄膜材料的实验设计提供理论基础.

2. 相场模型

相场方法是基于Ginzburg-Landau-Devonshire理论^[33], 通过微分方程表征具有特定物理机制的扩散、有序化势和热力学驱动的综合作用, 通过求解包含序参量的相场方程, 获取研究体系在时间和空间上的瞬时微观形貌. 针对铁电材料, 选用极化强度作为序参量, 根据能量最小化原理, 求解时间相关的Ginzburg-Landau方程, 获得铁电材料畴结构演化过程^[34,35].

$\frac{{\partial {P_{i}}\left( {{{r}},t} \right)}}{{\partial t}} = - L\frac{{\delta {F_{\rm{P}}}}}{{\delta {{{P}}_i}\left( {{{r}},t} \right)}}~~\left( {i = x,y,z} \right),$

(1)

其中t表示时间, r表示空间矢量位置, ${{P}}_{i}\left({{r}}, t\right)$ 表示某一时刻及某一位置处极化强度分量, L为动力学系数, ${F}_{\rm{P}}$ 为体系总能量. 体系总能量由体自由能、梯度能、弹性能以及静电场能组成:

$\begin{split} {F_{\rm{P}}} =\; & \iiint_{V}\left[ {f_{{\rm{bulk}}}}\left( {{{{P}}_i}} \right) + {f_{{\rm{grad}}}}\left( {{{{P}}_{i,j}}} \right)\right. \\ & \left. + {f_{{\rm{elas}}}}\left( {{{{P}}_{i,\;}}{\varepsilon _{ij}}} \right)+ {f_{{\rm{elec}}}}\left( {{{{P}}_{i,}}{{{E}}_i}} \right) \right]{\rm{d}}V.\end{split}$

(2)

体自由能密度f_bulk展开到六阶多项式, 具体表达式如下:

$\begin{split} {f_{{\rm{bulk}}}} =\; & {a_1}\left( {P_1^2 + P_2^2 + P_3^2} \right) + {a_{11}}\left( {P_1^4 + P_2^4 + P_3^4} \right)\\ & + {a_{12}}\left( {P_1^2P_2^2 + P_2^2P_3^2 + P_3^2P_1^2} \right)\\ & + {\rm{}}{a_{111}}\left( {P_1^6 + P_2^6 + P_3^6} \right) \\ & + {a_{112}}[P_1^4(P_2^2 + P_3^2) + P_2^4\left( {P_3^2 + P_1^2} \right)\\ & + P_3^4\left( {P_1^2 + P_2^2} \right)] + {a_{123}}P_1^2P_2^2P_3^2,\\[-12pt] \end{split}$

(3)

其中 ${a}_{1}$ 是温度相关的介电刚度参数, 根据居里-外斯定律, ${a}_{1}$ = (T – T_C)/(2ε₀C₀), T_C为材料居里温度, ε₀是真空介电常数, C₀是居里常数; ${a}_{11}$ , ${a}_{12}$ , ${a}_{111}$ , ${a}_{112}$ , ${a}_{123}$ 是高阶介电刚度.

梯度能密度 ${f_{{\rm{grad}}}}$ 的具体表达式如下:

${f_{{\rm{grad}}}} = \frac{1}{2}{G_{ijkl}}{{{P}}_{i,j}}{{{P}}_{k,l}},$

(4)

其中 ${{P}}_{i, j} = \partial {P}_{i}/\partial {x}_{j}$ , ${G}_{ijkl}$ 称作梯度能系数, 通过Voigt标记法表示, ${g}_{ij}$ 和 ${G}_{ijkl}$ 可以相互转换, 例如 ${g}_{11} = {G}_{1111}$ . 梯度能密度展开式如下:

$\begin{split} & {f_{{\rm{grad}}}} = \frac{1}{2}{g_{11}}( {P_{1,1}^2 + P_{2,2}^2 + P_{3,3}^2} ) \\ &+ {g_{12}}( {{P_{1,1}}{P_{2,2}} + {P_{3,3}}{P_{2,2}} + {P_{3,3}}{P_{1,1}}} )\\ &+ \frac{1}{2}{g_{44}}\big[{{( {{P_{1,2}}\! + \!{P_{2,1}}} )}^2}\! + \!{{( {{P_{2,3}}\! + \!{P_{3,2}}} )}^2}\! + \!{{( {{P_{1,3}}\! + \!{P_{3,1}}} )}^2} \big]\\ &\! + \!\frac{1}{2}g_{44}'\big[{{( {{P_{1,2}}\! - \!{P_{2,1}}} )}^2}\! + \!{{( {{P_{2,3}}\! - \! {P_{3,2}}} )}^2}\! + \!{{( {{P_{1,3}}\! - \!{P_{3,1}}} )}^2} \big]. \end{split}$

(5)

对各向同性体系, 梯度能密度可以简化为

$\begin{split}{f_{{\rm{grad}}}} =\;& \frac{1}{2}{g_{11}}\left(P_{1,1}^2 + P_{1,2}^2 + P_{1,3}^2 + P_{2,1}^2 + P_{2,2}^2\right. \\ & \left.+ P_{2,3}^2 + P_{3,1}^2 + P_{3,2}^2 + P_{3,3}^2 \right).\end{split}$

(6)

弹性能密度 ${f_{{\rm{elas}}}}$ 具体表达如下:

${f_{{\rm{elas}}}} = \frac{1}{2}{C_{ijkl}}\left( {{\varepsilon _{ij}} - \varepsilon _{ij}^0} \right)\left( {{\varepsilon _{kl}} - \varepsilon _{kl}^0} \right),$

(7)

式中 ${C}_{ijkl }$ 表示弹性刚度张量; ${\varepsilon }_{ij}$ 表示体系总应变; ${\varepsilon }_{ij}^{0}$ 表示本征应变, 其中 ${\varepsilon }_{ij}^{0}={Q}_{ijkl}{P}_{k}{P}_{l}$ , ${Q}_{ijkl}$ (i, j, k, l = 1, 2, 3)为电致伸缩常数. 求解力学平衡方程^[36]: ${\sigma }_{ij, j}\!=\!0$ , 其中 ${\sigma }_{ij}\!=\!{C}_{ijkl}{e}_{kl}\!=\!{C}_{ijkl}({\varepsilon }_{kl}- {\varepsilon }_{kl}^{0})$ . 薄膜上表面设为无应力边界条件, 即为 ${\sigma }_{i3}{|}_{x3=hf}= 0$ . ${\varepsilon _{11}} = {\varepsilon _{22}} = {\varepsilon _{{\rm{sub}}}}$ , ${\varepsilon _{{\rm{sub}}}}$ 指基底对薄膜的约束值, 即基底失配应变值^[37]. 本征应变和电致伸缩常数以及极化强度之间的关系之间如下:

$\begin{split} & \varepsilon _{11}^0 = {Q_{11}}P_1^2 + {Q_{12}}(P_2^2 + P_3^2),\\ & \varepsilon _{22}^0 = {Q_{11}}P_2^2 + {Q_{12}}(P_1^2 + P_3^2),\\ & \varepsilon _{33}^0 = {Q_{11}}P_3^2 + {Q_{12}}(P_1^2 + P_2^2),\\ & \varepsilon _{23}^0 = {Q_{44}}{P_2}{P_3}, ~~ \varepsilon _{13}^0 = {Q_{44}}{P_1}{P_3},\\ & \varepsilon _{12}^0 = {Q_{44}}{P_1}{P_2}, \end{split}$

(8)

其中, ${\varepsilon }_{12}^{0}={\varepsilon }_{21}^{0}={Q}_{44}{P}_{1}{P}_{2}$ , ${\varepsilon }_{23}^{0}={\varepsilon }_{32}^{0}=0$ , ${\varepsilon }_{13}^{0}= {\varepsilon }_{31}^{0}= 0$ . 其中铁电薄膜在外加电场和应变作用下产生的力电耦合效应和材料的电致伸缩常数 ${Q}_{ijkl}$ 密切相关^[38,39].

体系静电场自由能密度由退极化场和外电场两项组成, 因此静电场能密度 ${f_{{\rm{elec}}}}$ 具体表达如下:

${f_{{\rm{elec}}}} = - \frac{{{\varepsilon _0}{\varepsilon _r}}}{2}{E_i}{E_j} - {E_i}{P_i},$

(9)

其中真空介电常数 ${\varepsilon }_{0}= 8.85 \times 10^{-12}$ F/m, 电势可通过求解静电平衡方程得^[40].

本文通过半隐性傅里叶变换谱方法求解Ginzburg-Landau方程^[41], PZT薄膜的参数来源于文献[42], 表1列出了三种PZT薄膜介电刚度系数和电致伸缩常数. 模拟尺寸 $128 \Delta x\times 128 \Delta y\times 16 \Delta z$ , 其中 $\Delta x=\Delta y=\Delta z=1$ nm, 选择10个格点 $\Delta z$ 来模拟PZT薄膜, 其厚度为10 nm. 力学和电学边界条件设为薄膜边界条件^[34,36]. 模拟过程中, 初始畴结构为随机畴. 为了便于计算, 在模拟过程中使用无量纲变量(右上角标记星号)^[27,43], 为此选定当体系的温度T = 25 ℃时, 定义 ${a}_{0}={a}_{1}$ , 选定自发极化强度 ${P}_{0}=\left|{{{P}}_{0}}\right|= 0.757$ C/m². 因此本文中模拟电场和极化强度无量纲变量的转化如下: ${E}^{*}=E/\left({a}_{0}{P}_{0}\right)$ , ${P}^{*}=P/{P}_{0}$ .

表 1 三种成分PZT铁电薄膜介电刚度系数和电致伸缩常数

Table 1. Corresponding material constants for the Landau free energy, the electrostrictive coefficients of three components PZT thin films.

Coefficients	PbZr_0.2Ti_0.8O₃	PbZr_0.52Ti_0.48O₃	PbZr_0.8Ti_0.2O₃
${a}_{1}$ /C^–2·m²·N	3.44 × 10⁵(T – 456.38)	1.45 × 10⁵(T – 387.06)	2.71 × 10⁵(T – 300.57)
${a}_{11}$ /C^–4·m⁶·N	–3.05 × 10⁷	5.83 × 10⁷	3.13 × 10⁸
${a}_{12}$ /C^–4·m⁶·N	6.32 × 10⁸	1.82 × 10⁸	–3.45 × 10⁶
${a}_{111}$ /C^–6·m¹⁰·N	2.47 × 10⁸	1.50 × 10⁸	4.29 × 10⁸
${a}_{112}$ /C^–6·m¹⁰·N	9.68 × 10⁸	6.88 × 10⁸	1.81 × 10⁹
${a}_{123}$ /C^–6·m¹⁰·N	–4.90 × 10⁹	–3.24 × 10⁹	–7.54 × 10⁹
${Q}_{11}$ /C^–2·m⁴	0.081	0.094	0.056
${Q}_{12}$ /C^–2·m⁴	–0.024	–0.044	–0.017
${Q}_{44}$ /C^–2·m⁴	0.032	0.040	0.026

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3. 结果与讨论

PZT是典型的ABO₃钙钛矿结构氧化物. 当温度高于T_C时, PZT表现为顺电相, 其晶体结构为立方相(cubic phase, C相), 如图1(a)所示. 其中立方六面体的顶点A位被金属铅(Pb)原子占据, 中心B位被金属钛(Ti)或锆(Zr)原子占据. 氧原子位于立方体的6个面心, 构成氧八面体. 当温度低于T_C时, 顺电相晶胞中的原子发生位移, 导致正负离子中心不重合而产生电偶极矩, 此时PZT转变为铁电相. 图1(b)—(d)分别为PZT铁电相的三种典型的晶体结构示意图, 四方相(tetragonal phase, T相)、正交相(orthorhombic phase, O相)以及菱方相(rhombohedral phase, R相).

$图 1 PZT铁电材料的晶体结构示意图　(a)立方顺电相结构; (b)四方铁电相结构; (c)正交铁电相结构; (d)菱方铁电相结构\r\nFig. 1. Schematic of PZT ferroelectric structure: (a) Paraelectric cubic phase; (b) ferroelectric tetragonal phase; (c) ferroelectric orthorhombic phase; (d) ferroelectric rhombohedral phase.$

图 1 PZT铁电材料的晶体结构示意图　(a)立方顺电相结构; (b)四方铁电相结构; (c)正交铁电相结构; (d)菱方铁电相结构

Fig. 1. Schematic of PZT ferroelectric structure: (a) Paraelectric cubic phase; (b) ferroelectric tetragonal phase; (c) ferroelectric orthorhombic phase; (d) ferroelectric rhombohedral phase.

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在考虑基底应变效应之前, 首先通过相场模拟确定不同成分PZT (x = 0.8, 0.48, 0.2)处于铁电相时对应的稳态晶体结构和极化取向. 在室温T = 25 ℃ (T < T_C^[44])下, 通过(3)式对体自由能密度求最小值, 可以获得PZT材料的三维自由能曲面. 图2(a)—(f)分别展示了随Ti原子百分比降低PZT所对应的三维自由能曲面. 当x = 0.8时, 三维自由能曲面中的极小点沿着 $\langle 001\rangle$ 晶体轴方向, 因此PZT (x = 0.8)的稳态相为T相(图2(a)). 随着Ti成分降低, 当x = 0.48时(图2(d)), 三维自由能曲面中的极小点转到 $\langle 111\rangle$ 晶体轴方向, 但 $\langle 001\rangle$ 晶体轴方向存在局域极小值点, 因此, x = 0.48时PZT为T相、R相共存, 但R相结构相对稳定. 随着Ti成分进一步降低(x = 0.3或0.2), 三维自由能曲面的极小点仍然沿 $\langle 111\rangle$ 晶体轴方向 (图2(e)和图2(f)), 因此PZT (x = 0.3或0.2)的稳态相为R相.

$图 2 (a)−(f)室温下随Ti成分降低(x = 0.8—0.2) PZT的三维自由能曲面, 蓝色代表最小值, 红色代表最大值; (g)−(i) T相、R/T混合相及R相二维自由能双势阱示意图\r\nFig. 2. (a)−(f) Free energy surface of PZT with the decrease of Ti composition (x = 0.8–0.2) at room temperature. Blue and red color represents the minimum and maximum value respectively; (g)−(i) Schematic of double well potential of tetragonal phase (g), mixed phase (h) and rhombohedral phase (i).$

图 2 (a)−(f)室温下随Ti成分降低(x = 0.8—0.2) PZT的三维自由能曲面, 蓝色代表最小值, 红色代表最大值; (g)−(i) T相、R/T混合相及R相二维自由能双势阱示意图

Fig. 2. (a)−(f) Free energy surface of PZT with the decrease of Ti composition (x = 0.8–0.2) at room temperature. Blue and red color represents the minimum and maximum value respectively; (g)−(i) Schematic of double well potential of tetragonal phase (g), mixed phase (h) and rhombohedral phase (i).

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根据朗道唯象理论, T相结构的铁电极化沿 $\langle 001\rangle$ 晶体轴方向, 而R相结构的铁电极化沿 $\langle 111\rangle$ 方向. 图2(g)和图2(i)分别给出了T相和R相的自由能二维双势阱示意图. 当PZT位于准同型相界附近时(x = 0.48)^[44], 表现为T相与R相结构共存的状态, 势垒高度降低^[45,46], 如图2(h)所示. 在后续讨论中, 本文将 $\langle 001\rangle$ 取向的T相畴结构分别记为a1 + [100], a1–[ ${\bar {1}} 00$ ], a2 + [010], a2–[ $0 {\bar {1}}0$ ], c1[001], c2[ $00 {\bar {1}}$ ]; $\langle 111\rangle$ 取向的R相畴结构分别记为R1 $\left[111\right]$ , R2 $\left[{\bar {1}}11\right]$ , R3 $\left[{\bar {1}}{\bar {1}}1\right]$ , R4 $\left[1{\bar {1}}1\right]$ , R5 $\left[{\bar {1}}{\bar {1}}{\bar {1}}\right]$ , R6 $\left[1{\bar {1}}{\bar {1}}\right]$ , R7 $\left[11{\bar {1}}\right]$ , R8 $\left[{\bar {1}}1{\bar {1}}\right]$ ; 将 $\langle 011\rangle$ 取向的O相畴结构记为O畴.

下面将详细探讨基底失配应变对PZT薄膜微观畴结构的影响. 图3为PZT (x = 0.8, 0.48, 0.2)薄膜在不同基底应变(ε_sub = 0, –0.5%, 0.5%)作用下畴结构的演化. 对于T相PZT (x = 0.8)薄膜, 当不施加基底失配应变时, 薄膜的畴结构最终演变为a1/a2/c1/c2畴, 且相邻畴区域之间形成90°畴壁, 具体如图3(a)所示. 当基底应变为–0.5%时(图3(b)), a1/a2畴消失, 整个薄膜最终演化为c1/c2畴, 且相邻畴区域之间形成180°畴壁, 这是由于基底面内压应变促进面内a1/a2畴翻转形成面外c1/c2畴. 当基底应变为0.5%时(图3(c)), PZT (x = 0.8)薄膜中a1/a2畴的比例增加, 同时伴随着c1/c2畴的比例减小, 这是由于面内拉应变使得面外c1/c2畴向面内翻转形成a1/a2畴. 而对于准同型相界附近的PZT (x = 0.48)薄膜, 在无应变时(图3(d)), R畴为稳定相, 同时伴随少量的T畴, 这与图2中所示的热力学计算结果一致. 当施加面内压应变(ε_sub = –0.5%)时(图3(e)), 除了R畴, 薄膜中新形成a畴和c畴, 说明压应变使得PZT (x = 0.48)薄膜形成R/T混合相. 其中a畴产生原因是相邻R畴在压应变下铁电极化分量产生叠加或抵消. 以a1–[ ${\bar {1}} 00$ ]畴为例, 其相邻R畴分别是R5 $\left[{\bar {1}}{\bar {1}}{\bar {1}}\right]$ 和R8 $\left[{\bar {1}}1{\bar {1}}\right]$ , y方向铁电极化分量相互抵消, x和z方向铁电极化分量相互叠加, 因此在压应变下R畴紧邻畴是a1–和c2畴. 当施加面内拉应变(ε_sub = 0.5%)时(图3(f)), R畴的尺寸减小, 畴壁密度增加, 且有少量O畴出现. 而对于R相的PZT (x = 0.2)薄膜, 在无应变时(图3(g)), R畴为稳定相, 这与图2中的热力学计算结果相符. 当施加面内压应变(ε_sub = –0.5%)时(图3(h)), 与PZT (x = 0.48)薄膜类似, 也形成了R/T混合相, 但是T相的比例远小于PZT (x = 0.48)薄膜. 而当施加面内拉应变(ε_sub = 0.5%)时(图3(i)), 与PZT (x = 0.48)薄膜的情况不同, R畴的平均尺寸只有很小程度的减小, 畴壁密度变化也不大. 因此基底失配应变对四方相和混合相畴类型与畴尺寸的影响比菱方相更加显著. 从自由能角度分析(见图2)准同型相界处PZT (x = 0.48)自由能势垒较低, 铁电畴翻转对应变比较敏感, 因此应变对畴结构尺寸改变明显. 菱方PZT (x = 0.48)薄膜自由能势垒较高, 所以应变对R畴尺寸影响较小.

$图 3 相场模拟PZT薄膜在不同基底失配应变下的畴结构　(a)—(c)分别对应于PZT (x = 0.8)薄膜εsub = 0, εsub = –0.5%, εsub = 0.5%; (d)—(f)分别对应于PZT (x = 0.48)薄膜εsub = 0, εsub = –0. 5%, εsub = 0.5%; (g)—(h)分别对应于PZT (x = 0.2)薄膜εsub = 0, εsub = –0.5%, εsub = 0.5%\r\nFig. 3. Domain structures of PZT (x = 0.8, x = 0.48, x = 0.2) thin film with different substrate biaxial misfit strain (εsub = 0, εsub = –0.5%, εsub = 0.5%): (a)−(c) Domain structures of PZT (x = 0.8) thin films at εsub = 0, εsub = –0.5%, εsub = 0.5%; (d)−(f) domain structures of PZT (x = 0.48) thin films at εsub = 0, εsub = –0.5%, εsub = 0.5%; (g)−(h) domain structures of PZT (x = 0.2) thin films at εsub = 0, εsub = –0.5%, εsub = 0.5%.$

图 3 相场模拟PZT薄膜在不同基底失配应变下的畴结构　(a)—(c)分别对应于PZT (x = 0.8)薄膜ε_sub = 0, ε_sub = –0.5%, ε_sub = 0.5%; (d)—(f)分别对应于PZT (x = 0.48)薄膜ε_sub = 0, ε_sub = –0. 5%, ε_sub = 0.5%; (g)—(h)分别对应于PZT (x = 0.2)薄膜ε_sub = 0, ε_sub = –0.5%, ε_sub = 0.5%

Fig. 3. Domain structures of PZT (x = 0.8, x = 0.48, x = 0.2) thin film with different substrate biaxial misfit strain (ε_sub = 0, ε_sub = –0.5%, ε_sub = 0.5%): (a)−(c) Domain structures of PZT (x = 0.8) thin films at ε_sub = 0, ε_sub = –0.5%, ε_sub = 0.5%; (d)−(f) domain structures of PZT (x = 0.48) thin films at ε_sub = 0, ε_sub = –0.5%, ε_sub = 0.5%; (g)−(h) domain structures of PZT (x = 0.2) thin films at ε_sub = 0, ε_sub = –0.5%, ε_sub = 0.5%.

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在充分了解基底失配应变对PZT薄膜中微观畴结构的影响之后, 接下来利用相场模拟研究其对PZT薄膜宏观铁电性能的影响. 图4展示了不同基底应变(ε_sub = ± 0.1%, ± 0.5%, ± 1.0%)下PZT (x = 0.8, 0.48, 0.2)薄膜的电滞回线. 在压应变情况下(如图4(a)—(c)所示), 随着Ti比例的降低, PZT薄膜的矫顽场、饱和极化值以及剩余极化值都相应减小. 对于PZT (x = 0.8)和PZT (x = 0.48)薄膜, 矫顽场受压应变的调控比PZT (x = 0.2)薄膜更敏感, 这主要是因为在这两种成分的PZT薄膜中, 基底面内压应变使得c畴比例明显增加. 然而相比于PZT (x = 0.8)和PZT (x = 0.2)薄膜, PZT (x = 0.48)薄膜的饱和极化和剩余极化值对压应变要更敏感, 这是由于PZT (x = 0.48)薄膜在准同型相界处的强力电耦合效应, 畴结构对应变响应较为敏感^[38,47,48]. 这主要体现在面内压应变使得PZT (x = 0.48)薄膜中T相比例增加, 而R相比例减小. 而在施加基底拉应变情况下, 与PZT (x = 0.48)和PZT (x = 0.2)薄膜相比, PZT (x = 0.8)薄膜展现出更大的饱和极化和剩余极化值, 这是因为其主要由T畴构成, 而前两者主要由R畴构成. 随着拉应变增加, PZT (x = 0.8)薄膜中面内a畴比例增加, 同时面外c畴比例减小. 而PZT (x = 0.48)薄膜R畴尺寸减小, 同时伴随畴壁密度增大, 所以这两个成分的PZT薄膜矫顽场对拉应变都非常敏感. 而PZT (x = 0.2)薄膜中, R相畴尺寸随拉应变变化不是很明显, 所以其对应的矫顽场的变化幅度也最小.

$图 4 室温下PZT铁电薄膜四方相(x = 0.8), 混合相(x = 0.48)以及菱方相(x = 0.2)在不同的基底失配应变下(εsub = ± 0.1%, ± 0.5%, ± 1.0%)的电滞回线, 其中P *和E *表示归一化后的极化强度和电场强度值　(a)−(c)分别表示压应变下四方相、混合相和菱方相的电滞回线; (d)−(f)分别表示拉应变下四方相、混合相和菱方相的电滞回线\r\nFig. 4. Hysteresis loops of PZT thin films with three Ti components at different substrate biaxial misfit strains (εsub = ± 0.1%, ± 0.5%, ± 1.0%), and P * and E * are normalized polarization and electric field: (a)−(c) The case of compressive strains; (d)−(f) the case of tensile strains.$

图 4 室温下PZT铁电薄膜四方相(x = 0.8), 混合相(x = 0.48)以及菱方相(x = 0.2)在不同的基底失配应变下(ε_sub = ± 0.1%, ± 0.5%, ± 1.0%)的电滞回线, 其中P^*和E^*表示归一化后的极化强度和电场强度值　(a)−(c)分别表示压应变下四方相、混合相和菱方相的电滞回线; (d)−(f)分别表示拉应变下四方相、混合相和菱方相的电滞回线

Fig. 4. Hysteresis loops of PZT thin films with three Ti components at different substrate biaxial misfit strains (ε_sub = ± 0.1%, ± 0.5%, ± 1.0%), and P^* and E^* are normalized polarization and electric field: (a)−(c) The case of compressive strains; (d)−(f) the case of tensile strains.

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依据上述模拟结果, 进一步统计了基底失配应变(ε_sub = ± 0.1%, ± 0.5%, ± 1.0%)对不同成分的PZT (x = 0.8, 0.48, 0.2)薄膜矫顽场、饱和极化值以及剩余极化值的影响. 如图5所示, 随基底失配应变从压应变逐渐过渡到拉应变, PZT (x = 0.8, 0.48, 0.2)薄膜中矫顽场、饱和极化值以及剩余极化值都呈现出减小的趋势. 结合上面对图3与图4的讨论可知, 这主要与PZT (x = 0.8, 0.48, 0.2)薄膜中T相与R相的相对比例以及R相畴的尺寸随应变的变化是密切相关的. 对于PZT (x = 0.48, 0.2)薄膜来讲, 所有随应变变化的函数曲线(包含矫顽场、饱和极化与剩余极化)都存在一个交点(图5中蓝色和紫色曲线). 在交点左侧, PZT (x = 0.48)薄膜的相关铁电性能(矫顽场、饱和极化和剩余极化)都优于PZT (x = 0.2)薄膜. 在压应变情况下, 虽然这两个R相薄膜中都有T相畴的形成, 但PZT (x = 0.48)薄膜中T相畴的比例更高, 所以造成上面的现象. 而在交点右侧, PZT (x = 0.2)薄膜的相关铁电性能要优于PZT (x = 0.48)薄膜. 这是因为随着拉应变的增加, PZT (x = 0.2)薄膜中R相畴的尺寸变化不大, 而PZT (x = 0.48)薄膜中R相畴的尺寸急剧减小, 畴壁密度迅速增加所导致的. PZT (x = 0.2)薄膜中, 随着应变从–1.0%变化到1.0%时, 矫顽场、剩余极化值和饱和极化值都缓慢降低, 而对于PZT (x = 0.48)薄膜随应变增加, 矫顽场、剩余极化值以及饱和极化值显著降低. 从自由能角度分析, 准同型相界处PZT (x = 0.48)双势阱能垒小于PZT (x = 0.8)和PZT (x = 0.2), 在能量双势阱中能垒被拉平(图2(h)), 准同型相界处PZT薄膜对应变响应更为敏感, 其铁电极化强度也更容易翻转. 因此, x = 0.48时PZT薄膜随基底失配应变从–1.0%变化到1.0%, 矫顽场、饱和极化和剩余极化等值的变化速率大于另外两种PZT薄膜.

$图 5 三种相PZT铁电薄膜的矫顽场、饱和极化和剩余极化值与基底应变的关系　(a) 矫顽场Ec*; (b) 饱和极化值Ps*; (c) 剩余极化值Pr*\r\nFig. 5. Normalized coercive field (Ec*), saturation polarization (Pr*), and remnant polarization (Ps*) as a function of substrate misfit strain (εsub), where three PZT ferroelectric thin films with x = 0.8, 0.48 and 0.2 Ti component are considered: (a) Coercive field vs. strain; (b) saturation polarization vs. strain; (c) remnant polarization vs. strain.$

图 5 三种相PZT铁电薄膜的矫顽场、饱和极化和剩余极化值与基底应变的关系　(a) 矫顽场E_c^*; (b) 饱和极化值P_s^*; (c) 剩余极化值P_r^*

Fig. 5. Normalized coercive field (E_c^*), saturation polarization (P_r^*), and remnant polarization (P_s^*) as a function of substrate misfit strain (ε_sub), where three PZT ferroelectric thin films with x = 0.8, 0.48 and 0.2 Ti component are considered: (a) Coercive field vs. strain; (b) saturation polarization vs. strain; (c) remnant polarization vs. strain.

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我们继续探讨应变调控PZT铁电薄膜在储能方面的应用. 储能密度和储能效率之间的计算公式为

$\eta = \frac{{w1}}{{w1 + w2}} \times 100\% ,$

(10)

其中η表示储能效率, w1为可放电能量密度, w2为损失能量密度, 具体定义可见文献[49]. 图6(a)表示电滞回线中对应的能量储存示意图, 其中绿色面积表示可放电能量密度, 黄色区域表示放电过程中损失能量密度.

$图 6 (a)电滞回线中充放电过程中储能示意图; (b) 三种PZT薄膜材料能量存储效率与基底应变之间的关系\r\nFig. 6. (a) Schematic of P-E loop used for energy storage; (b) the energy storage efficiency as a function of substrate misfit strain.$

图 6 (a)电滞回线中充放电过程中储能示意图; (b) 三种PZT薄膜材料能量存储效率与基底应变之间的关系

Fig. 6. (a) Schematic of P-E loop used for energy storage; (b) the energy storage efficiency as a function of substrate misfit strain.

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基底失配应变能够有效地调控PZT薄膜电滞回线, 计算了三种PZT薄膜中应变对储能效率的影响(具体数值列于表2). 计算结果表明随着基底失配应变从压应变过渡到拉应变, 材料储能效率值逐渐增加. 图6(b)表示PZT薄膜储能效率和应变之间的关系. 其中相比于另外两种成分PZT, PZT (x = 0.48)薄膜从压应变到拉应变其储能效率提升最快. 而R相PZT (x = 0.2)薄膜储能效率提升速率高于T相PZT (x = 0.8)薄膜. 因此, 拉应变能够有效地提高PZT薄膜储能效率, 其中准同型相界处PZT (x = 0.48)薄膜随应变增加其储能效率提升最快.

表 2 三种PZT薄膜材料在不同应变下的储能效率值η

Table 2. Energy storage efficiency values of the PZT thin films under different strains.

Strain/%	Energy storage efficienc η/%
Strain/%	PbZr_0.2Ti_0.8O₃	PbZr_0.52Ti_0.48O₃	PbZr_0.8Ti_0.2O₃
–1.0	8.0	7.8	11.2
–0.5	12.0	17.4	18.7
–0.1	16.8	31.0	23.4
0.1	20.0	40.6	27.4
0.5	30.7	61.0	34.7
1.0	43.9	73.6	55.8

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4. 结　论

本文通过相场方法系统研究了基底失配应变对三种结构PZT (x = 0.8, 0.48, 0.2)薄膜中铁电畴结构演变以及宏观铁电性能的影响. 在应变调控畴结构演变方面, 四方相中应变可以实现面内a畴和面外c畴的翻转; 混合相中压应变可以诱导从R畴到T畴的相变, 而拉应变则诱导R畴尺寸减小; 菱方相中, 压和拉应变对于R畴尺寸产生差异较小. 随着应变从–1.0%变化到1.0%, 三种相结构铁电薄膜中的矫顽场、饱和极化值和剩余极化值全部都降低, 其中准同型相界处PZT (x = 0.48)薄膜的矫顽场、饱和极化和剩余极化等值变化速率大于另外两种PZT薄膜. 模拟结果表明拉应变能够有效地提高铁电薄膜储能效率, 在准同型相界处铁电薄膜随应变增加其储能效率提升最快. 本工作揭示了应变能够有效地调控铁电畴结构、电滞回线和铁电薄膜储能效率, 这为铁电功能薄膜材料的实验设计提供理论基础.

参考文献

[1]	Xu F, Trolier-McKinstry S, Ren W, Xu B, Xie Z L, Hemker K J 2001 J. Appl. Phys. 89 1336 Google Scholar
[2]	Kim D J, Maria J P, Kingon A I, Streiffer S K 2003 J. Appl. Phys. 93 5568 Google Scholar
[3]	Karthik J, Martin L 2011 Phys. Rev. B 84 024102 Google Scholar
[4]	Karthik J, Damodaran A R, Martin L W 2012 Phys. Rev. Lett. 108 167601 Google Scholar
[5]	赵晓英, 刘世建, 褚君浩, 戴宁, 胡古今 2008 物理学报 57 5968 Google Scholar Zhao X Y, Liu S J, Chu J H, Dai N, Hu G J 2008 Acta Phys. Sin. 57 5968 Google Scholar
[6]	Arimoto Y, Ishiwara H 2004 MRS Bull. 29 823 Google Scholar
[7]	Ganapathi K L, Rath M, Rao M S R 2019 Semicond. Sci. Technol. 34 055016 Google Scholar
[8]	Won S S, Seo H, Kawahara M, Glinsek S, Lee J, Kim Y, Jeong C K, Kingon A I, Kim S H 2019 Nano Energy 55 182 Google Scholar
[9]	Hoshyarmanesh H, Ebrahimi N, Jafari A, Hoshyarmanesh P, Kim M, Park H H 2019 Sensors 19 13 Google Scholar
[10]	Gupta R, Gupta V, Tomar M 2020 Mater. Sci. Semicond. Process. 105 104723 Google Scholar
[11]	Rath M, Varadarajan E, Premkumar S, Shinde S, Natarajan V, Rao M S R 2019 Ferroelectrics 551 17 Google Scholar
[12]	Yao Z H, Song Z, Hao H, Yu Z Y, Cao M H, Zhang S J, Lanagan M T, Liu H X 2017 Adv. Mater. 29 1601727 Google Scholar
[13]	Pan H, Li F, Liu Y, Zhang Q, Wang M, Lan S, Zheng Y, Ma J, Gu L, Shen Y, Yu P, Zhang S, Chen L Q, Lin Y H, Nan C W 2019 Science 365 578 Google Scholar
[14]	Wang J J, Su Y J, Wang B, Ouyang J, Ren Y, Chen L Q 2020 Nano Energy 72 104665 Google Scholar
[15]	Li A D, Mak C L, Wong K H, Shao Q Y, Wang Y J, Wu D, Ming N B 2002 J. Cryst. Growth 235 307 Google Scholar
[16]	Ehara Y, Shimizu T, Yasui S, Oikawa T, Shiraishi T, Tanaka H, Kanenko N, Maran R, Yamada T, Imai Y Sakata O, Valanoor N, Funakubo H 2019 Phys. Rev. B 100 104116 Google Scholar
[17]	Izyumskaya N, Alivov Y I, Cho S J, Morkoç H, Lee H, Kang Y S 2007 Crit. Rev. Solid. State Mater. Sci. 32 111 Google Scholar
[18]	Choi K J, Biegalski M, Li Y L, Sharan A, Schubert J, Uecker R, Reiche P, Chen Y B, Pan X Q, Gopalan V, Chen L Q, Schlom D G, Eom C B 2004 Science 306 1005 Google Scholar
[19]	Noguchi Y, Maki H, Kitanaka Y, Matsuo H, Miyayama M 2018 Appl. Phys. Lett. 113 012903 Google Scholar
[20]	Li Y L, Chen L Q 2006 Appl. Phys. Lett. 88 072905 Google Scholar
[21]	Pertsev N A, Tagantsev A K, Setter N 2000 Phys. Rev. B 61 R825 Google Scholar
[22]	Haeni J H, Irvin P, Chang W, Uecker R, Reiche P, Li Y L, Choudhury S, Tian W, Hawley M E, Craigo B, Tagantsev A K, Pan X Q, Streiffer S K, Chen L Q, Kirchoefer S W, Levy J, Schlom D G 2004 Nature 430 758 Google Scholar
[23]	Zhang J X, Li Y L, Choudhury S, Chen L Q, Chu Y H, Zavaliche F, Cruz M P, Ramesh R, Jia Q X 2008 J. Appl. Phys. 103 094111 Google Scholar
[24]	Ren W, Yang Y, Diéguez O, Íñiguez J, Choudhury N, Bellaiche L 2013 Phys. Rev. Lett. 110 187601 Google Scholar
[25]	Zhang Y, Xue F, Chen Z H, Liu J M, Chen L Q 2020 Acta Mater. 183 110 Google Scholar
[26]	Li Y L, Hu S Y, Liu Z K, Chen L Q 2002 Acta Mater. 50 395 Google Scholar
[27]	Li Y L, Hu S Y, Chen L Q 2005 J. Appl. Phys. 97 034112 Google Scholar
[28]	Yu Q, Li J, Zhu F, Li J 2014 J. Mater. Chem. C 2 5836 Google Scholar
[29]	Lu X Y, Chen Z H, Cao Y, Tang Y L, Xu R J, Saremi S, Zhang Z, You L, Dong Y Q, Das S, Zhang H B, Zheng L M, Wu H P, Lv W M, Xie G Q, Liu X J, Li J Y, Chen L, Chen L Q, Cao W W, Martin L W 2019 Nat. Commun. 10 3951 Google Scholar
[30]	Nguyen M D, Dekkers M, Houwman E, Steenwelle R, Wan X, Roelofs A, Schmitz-Kempen T, Rijnders G 2011 Appl. Phys. Lett. 99 252904 Google Scholar
[31]	Xue F, Wang J J, Sheng G, Huang E, Cao Y, Huang H-H, Munroe P, Mahjoub R, Li Y, Valanoor N, Chen L 2013 Acta Mater. 61 2909 Google Scholar
[32]	Lin F Y, Cheng X, Chen L Q, Sinnott S B 2018 J. Am. Ceram. Soc. 101 4783 Google Scholar
[33]	Devonshire A F 1949 The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 40 1040 Google Scholar
[34]	Chen L Q 2008 J. Am. Ceram. Soc. 91 1835 Google Scholar
[35]	Chen L Q 2002 Ann. Rev. Mater. Res. 32 113 Google Scholar
[36]	Li Y L, Hu S Y, Liu Z K, Chen L Q 2001 Appl. Phys. Lett. 78 3878 Google Scholar
[37]	Li Y L, Choudhury S, Liu Z K, Chen L Q 2003 Appl. Phys. Lett. 83 1608 Google Scholar
[38]	Shu W L, Wang J, Zhang T Y 2012 J. Appl. Phys. 112 064108 Google Scholar
[39]	Wang J J, Wang B, Chen L Q 2019 Ann. Rev. Mater. Res. 49 127 Google Scholar
[40]	Li Y L, Chen L Q, Asayama G, Schlom D G, Zurbuchen M A, Streiffer S K 2004 J. Appl. Phys. 95 6332 Google Scholar
[41]	Chen L Q, Shen J 1998 Comput. Phys. Commun. 108 147 Google Scholar
[42]	Haun M J, Zhuang Z Q, Furman E, Jang S J, Cross L E 1989 Ferroelectrics 99 45 Google Scholar
[43]	Hu H L, Chen L Q 1998 J. Am. Ceram. Soc. 81 492
[44]	Damjanovic D 2005 J. Am. Ceram. Soc. 88 2663 Google Scholar
[45]	Liu W F, Ren X B 2009 Phys. Rev. Lett. 103 257602 Google Scholar
[46]	Li F, Lin D B, Chen Z B, Cheng Z X, Wang J L, Li C C, Xu Z, Huang Q W, Liao X Z, Chen L Q, Shrout T R, Zhang S J 2018 Nat. Mater. 17 349 Google Scholar
[47]	Liao Z Y, Xue F, Sun W, Song D S, Zhang Q Q, Li J F, Chen L Q, Zhu J 2017 Phys. Rev. B 95 214101 Google Scholar
[48]	Liu H, Chen J, Huang H B, Fan L L, Ren Y, Pan Z, Deng J X, Chen L Q, Xing X R 2018 Phys. Rev. Lett. 120 055501 Google Scholar
[49]	Ma Z, Ma Y, Chen Z, Zheng F, Gao H, Liu H, Chen H 2018 Ceram. Int. 44 4338 Google Scholar

施引文献

期刊类型引用(4)

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2.	李昊晴，苏煜. BaTiO_3纳米单晶薄膜在外加电场作用下畴结构演化的相场研究. 人工晶体学报. 2024(07): 1136-1149 . 百度学术
3.	白刚，韩宇航，高存法. (111)取向无铅K_(0.5)Na_(0.5)NbO_3外延薄膜的相变和电卡效应:外应力与错配应变效应. 物理学报. 2022(09): 347-354 . 百度学术
4.	高荣贞，王静，王俊升，黄厚兵. Landau-Devonshire理论探究不同类型铁电材料的电卡效应. 物理学报. 2020(21): 177-186 . 百度学术

其他类型引用(5)

图 1 PZT铁电材料的晶体结构示意图　(a)立方顺电相结构; (b)四方铁电相结构; (c)正交铁电相结构; (d)菱方铁电相结构

Fig. 1. Schematic of PZT ferroelectric structure: (a) Paraelectric cubic phase; (b) ferroelectric tetragonal phase; (c) ferroelectric orthorhombic phase; (d) ferroelectric rhombohedral phase.

下载: 全尺寸图片幻灯片

图 5 三种相PZT铁电薄膜的矫顽场、饱和极化和剩余极化值与基底应变的关系　(a) 矫顽场E_c^*; (b) 饱和极化值P_s^*; (c) 剩余极化值P_r^*

下载: 全尺寸图片幻灯片

图 6 (a)电滞回线中充放电过程中储能示意图; (b) 三种PZT薄膜材料能量存储效率与基底应变之间的关系

Fig. 6. (a) Schematic of P-E loop used for energy storage; (b) the energy storage efficiency as a function of substrate misfit strain.

下载: 全尺寸图片幻灯片

表 1 三种成分PZT铁电薄膜介电刚度系数和电致伸缩常数

Table 1. Corresponding material constants for the Landau free energy, the electrostrictive coefficients of three components PZT thin films.

Coefficients	PbZr_0.2Ti_0.8O₃	PbZr_0.52Ti_0.48O₃	PbZr_0.8Ti_0.2O₃
${a}_{1}$ /C^–2·m²·N	3.44 × 10⁵(T – 456.38)	1.45 × 10⁵(T – 387.06)	2.71 × 10⁵(T – 300.57)
${a}_{11}$ /C^–4·m⁶·N	–3.05 × 10⁷	5.83 × 10⁷	3.13 × 10⁸
${a}_{12}$ /C^–4·m⁶·N	6.32 × 10⁸	1.82 × 10⁸	–3.45 × 10⁶
${a}_{111}$ /C^–6·m¹⁰·N	2.47 × 10⁸	1.50 × 10⁸	4.29 × 10⁸
${a}_{112}$ /C^–6·m¹⁰·N	9.68 × 10⁸	6.88 × 10⁸	1.81 × 10⁹
${a}_{123}$ /C^–6·m¹⁰·N	–4.90 × 10⁹	–3.24 × 10⁹	–7.54 × 10⁹
${Q}_{11}$ /C^–2·m⁴	0.081	0.094	0.056
${Q}_{12}$ /C^–2·m⁴	–0.024	–0.044	–0.017
${Q}_{44}$ /C^–2·m⁴	0.032	0.040	0.026

下载: 导出CSV

表 2 三种PZT薄膜材料在不同应变下的储能效率值η

Table 2. Energy storage efficiency values of the PZT thin films under different strains.

Strain/%	Energy storage efficienc η/%
Strain/%	PbZr_0.2Ti_0.8O₃	PbZr_0.52Ti_0.48O₃	PbZr_0.8Ti_0.2O₃
–1.0	8.0	7.8	11.2
–0.5	12.0	17.4	18.7
–0.1	16.8	31.0	23.4
0.1	20.0	40.6	27.4
0.5	30.7	61.0	34.7
1.0	43.9	73.6	55.8

下载: 导出CSV

[1]	Xu F, Trolier-McKinstry S, Ren W, Xu B, Xie Z L, Hemker K J 2001 J. Appl. Phys. 89 1336 Google Scholar
[2]	Kim D J, Maria J P, Kingon A I, Streiffer S K 2003 J. Appl. Phys. 93 5568 Google Scholar
[3]	Karthik J, Martin L 2011 Phys. Rev. B 84 024102 Google Scholar
[4]	Karthik J, Damodaran A R, Martin L W 2012 Phys. Rev. Lett. 108 167601 Google Scholar
[5]	赵晓英, 刘世建, 褚君浩, 戴宁, 胡古今 2008 物理学报 57 5968 Google Scholar Zhao X Y, Liu S J, Chu J H, Dai N, Hu G J 2008 Acta Phys. Sin. 57 5968 Google Scholar
[6]	Arimoto Y, Ishiwara H 2004 MRS Bull. 29 823 Google Scholar
[7]	Ganapathi K L, Rath M, Rao M S R 2019 Semicond. Sci. Technol. 34 055016 Google Scholar
[8]	Won S S, Seo H, Kawahara M, Glinsek S, Lee J, Kim Y, Jeong C K, Kingon A I, Kim S H 2019 Nano Energy 55 182 Google Scholar
[9]	Hoshyarmanesh H, Ebrahimi N, Jafari A, Hoshyarmanesh P, Kim M, Park H H 2019 Sensors 19 13 Google Scholar
[10]	Gupta R, Gupta V, Tomar M 2020 Mater. Sci. Semicond. Process. 105 104723 Google Scholar
[11]	Rath M, Varadarajan E, Premkumar S, Shinde S, Natarajan V, Rao M S R 2019 Ferroelectrics 551 17 Google Scholar
[12]	Yao Z H, Song Z, Hao H, Yu Z Y, Cao M H, Zhang S J, Lanagan M T, Liu H X 2017 Adv. Mater. 29 1601727 Google Scholar
[13]	Pan H, Li F, Liu Y, Zhang Q, Wang M, Lan S, Zheng Y, Ma J, Gu L, Shen Y, Yu P, Zhang S, Chen L Q, Lin Y H, Nan C W 2019 Science 365 578 Google Scholar
[14]	Wang J J, Su Y J, Wang B, Ouyang J, Ren Y, Chen L Q 2020 Nano Energy 72 104665 Google Scholar
[15]	Li A D, Mak C L, Wong K H, Shao Q Y, Wang Y J, Wu D, Ming N B 2002 J. Cryst. Growth 235 307 Google Scholar
[16]	Ehara Y, Shimizu T, Yasui S, Oikawa T, Shiraishi T, Tanaka H, Kanenko N, Maran R, Yamada T, Imai Y Sakata O, Valanoor N, Funakubo H 2019 Phys. Rev. B 100 104116 Google Scholar
[17]	Izyumskaya N, Alivov Y I, Cho S J, Morkoç H, Lee H, Kang Y S 2007 Crit. Rev. Solid. State Mater. Sci. 32 111 Google Scholar
[18]	Choi K J, Biegalski M, Li Y L, Sharan A, Schubert J, Uecker R, Reiche P, Chen Y B, Pan X Q, Gopalan V, Chen L Q, Schlom D G, Eom C B 2004 Science 306 1005 Google Scholar
[19]	Noguchi Y, Maki H, Kitanaka Y, Matsuo H, Miyayama M 2018 Appl. Phys. Lett. 113 012903 Google Scholar
[20]	Li Y L, Chen L Q 2006 Appl. Phys. Lett. 88 072905 Google Scholar
[21]	Pertsev N A, Tagantsev A K, Setter N 2000 Phys. Rev. B 61 R825 Google Scholar
[22]	Haeni J H, Irvin P, Chang W, Uecker R, Reiche P, Li Y L, Choudhury S, Tian W, Hawley M E, Craigo B, Tagantsev A K, Pan X Q, Streiffer S K, Chen L Q, Kirchoefer S W, Levy J, Schlom D G 2004 Nature 430 758 Google Scholar
[23]	Zhang J X, Li Y L, Choudhury S, Chen L Q, Chu Y H, Zavaliche F, Cruz M P, Ramesh R, Jia Q X 2008 J. Appl. Phys. 103 094111 Google Scholar
[24]	Ren W, Yang Y, Diéguez O, Íñiguez J, Choudhury N, Bellaiche L 2013 Phys. Rev. Lett. 110 187601 Google Scholar
[25]	Zhang Y, Xue F, Chen Z H, Liu J M, Chen L Q 2020 Acta Mater. 183 110 Google Scholar
[26]	Li Y L, Hu S Y, Liu Z K, Chen L Q 2002 Acta Mater. 50 395 Google Scholar
[27]	Li Y L, Hu S Y, Chen L Q 2005 J. Appl. Phys. 97 034112 Google Scholar
[28]	Yu Q, Li J, Zhu F, Li J 2014 J. Mater. Chem. C 2 5836 Google Scholar
[29]	Lu X Y, Chen Z H, Cao Y, Tang Y L, Xu R J, Saremi S, Zhang Z, You L, Dong Y Q, Das S, Zhang H B, Zheng L M, Wu H P, Lv W M, Xie G Q, Liu X J, Li J Y, Chen L, Chen L Q, Cao W W, Martin L W 2019 Nat. Commun. 10 3951 Google Scholar
[30]	Nguyen M D, Dekkers M, Houwman E, Steenwelle R, Wan X, Roelofs A, Schmitz-Kempen T, Rijnders G 2011 Appl. Phys. Lett. 99 252904 Google Scholar
[31]	Xue F, Wang J J, Sheng G, Huang E, Cao Y, Huang H-H, Munroe P, Mahjoub R, Li Y, Valanoor N, Chen L 2013 Acta Mater. 61 2909 Google Scholar
[32]	Lin F Y, Cheng X, Chen L Q, Sinnott S B 2018 J. Am. Ceram. Soc. 101 4783 Google Scholar
[33]	Devonshire A F 1949 The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 40 1040 Google Scholar
[34]	Chen L Q 2008 J. Am. Ceram. Soc. 91 1835 Google Scholar
[35]	Chen L Q 2002 Ann. Rev. Mater. Res. 32 113 Google Scholar
[36]	Li Y L, Hu S Y, Liu Z K, Chen L Q 2001 Appl. Phys. Lett. 78 3878 Google Scholar
[37]	Li Y L, Choudhury S, Liu Z K, Chen L Q 2003 Appl. Phys. Lett. 83 1608 Google Scholar
[38]	Shu W L, Wang J, Zhang T Y 2012 J. Appl. Phys. 112 064108 Google Scholar
[39]	Wang J J, Wang B, Chen L Q 2019 Ann. Rev. Mater. Res. 49 127 Google Scholar
[40]	Li Y L, Chen L Q, Asayama G, Schlom D G, Zurbuchen M A, Streiffer S K 2004 J. Appl. Phys. 95 6332 Google Scholar
[41]	Chen L Q, Shen J 1998 Comput. Phys. Commun. 108 147 Google Scholar
[42]	Haun M J, Zhuang Z Q, Furman E, Jang S J, Cross L E 1989 Ferroelectrics 99 45 Google Scholar
[43]	Hu H L, Chen L Q 1998 J. Am. Ceram. Soc. 81 492
[44]	Damjanovic D 2005 J. Am. Ceram. Soc. 88 2663 Google Scholar
[45]	Liu W F, Ren X B 2009 Phys. Rev. Lett. 103 257602 Google Scholar
[46]	Li F, Lin D B, Chen Z B, Cheng Z X, Wang J L, Li C C, Xu Z, Huang Q W, Liao X Z, Chen L Q, Shrout T R, Zhang S J 2018 Nat. Mater. 17 349 Google Scholar
[47]	Liao Z Y, Xue F, Sun W, Song D S, Zhang Q Q, Li J F, Chen L Q, Zhu J 2017 Phys. Rev. B 95 214101 Google Scholar
[48]	Liu H, Chen J, Huang H B, Fan L L, Ren Y, Pan Z, Deng J X, Chen L Q, Xing X R 2018 Phys. Rev. Lett. 120 055501 Google Scholar
[49]	Ma Z, Ma Y, Chen Z, Zheng F, Gao H, Liu H, Chen H 2018 Ceram. Int. 44 4338 Google Scholar

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相场模拟应变调控PbZr_(1–x)Ti_xO₃薄膜微观畴结构和宏观铁电性能

Phase field simulation of misfit strain manipulating domain structure and ferroelectric properties in PbZr_(1–x)Ti_xO₃ thin films

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1.
北京理工大学材料学院, 北京　100081

2.
北京理工大学前沿交叉科学研究院, 北京　100081

通信作者: 黄厚兵, hbhuang@bit.edu.cn

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1.
School of Materials Science and Engineering, Beijing Institute of Technology, Beijing 100081, China

2.
Advanced Research Institute of Multidisciplinary Science, Beijing Institute of Technology, Beijing 100081, China

Corresponding author: Huang Hou-Bing, hbhuang@bit.edu.cn

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