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柔性铁电材料在可穿戴电子领域具有重要应用前景, 然而其动态弯曲过程中应变梯度与极化翻转的力-电耦合物理机制仍缺乏系统性的研究. 本研究基于相场模拟, 系统地探讨了(SrTiO3)10/(PbTiO3)10/(SrTiO3)10三层异质膜在U型和N型弯曲下的畴结构演化及其宏观电学响应. 研究表明, 通过改变弯曲变形方向可以产生方向相反的挠曲电场, 导致电滞回线发生相应方向的偏移. 此外, 弯曲应变和应变梯度可驱动极性涡旋态与单畴态之间的拓扑相变, 其中界面静电能、弹性约束及梯度能的协同作用对拓扑结构的稳定性起关键作用. 本研究揭示了弯曲变形通过力-电耦合效应实现畴构型与电响应的定向调控机制, 为高密度柔性存储器和能量收集器件的跨尺度设计奠定了理论基础, 并进一步拓展了拓扑态工程在柔性电子领域的应用前景.Flexible ferroelectric materials possess considerable potentials for wearable electronics and bio-inspired devices, yet their mechano-electric coupling mechanisms under dynamic bending conditions remain incompletely understood. In his work, the effects of bending deformation on domain structures and macroscopic ferroelectric responses in (SrTiO3)10/(PbTiO3)10/(SrTiO3)10 flexible ferroelectric trilayer films are systematically investigated using phase-field simulations. By constructing computational models for upward-concave (U-shaped) and downward-concave (N-shaped) bending configurations, the strain distribution and its regulation mechanism on polarization patterns under different curvature radii are analyzed. The results reveal distinct strain gradients across bending modes: U-shaped bending induces compressive strain in the upper layer and tensile strain in the lower layer, generating a negative out-of-plane strain gradient. Conversely, N-shaped bending reverses this strain distribution. Such inhomogeneous strains drive significant polarization reconfiguration within the PTO layer. At a moderate curvature (large R), the system retains stable vortex-antivortex pairs. Reducing bending radius (smaller R) promotes divergent topological transitions—U-shaped bending facilitates vortex pair transformation into zigzag-like domains, while N-shaped bending drives vortex-to-out-of-plane c-domain evolution. Notably, bending-induced strain gradients impose transverse flexoelectric fields that markedly change trilayer hysteresis loops. U-shaped bending introduces a negative flexoelectric field, shifting loops rightward with maximum polarization (Pmax) decreasing. In contrast, N-shaped bending generates a positive field, enhancing Pmax via leftward loop shifting. The polarization switching analysis under electric field further demonstrates bending-mediated control of domain evolution pathway and reversal dynamics. These findings not only elucidate profound bending effects on flexible ferroelectrics’ domain architectures and functional properties but also provide theoretical guidance for designing strain-programmable ferroelectric memories, adaptive sensors, and neuromorphic electronics.
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Keywords:
- phase-field simulation /
- bending /
- strain gradient /
- ferroelectric vortex
[1] Setter N, Damjanovic D, Eng L, Fox G, Gevorgian S, Hong S, Kingon A, Kohlstedt H, Park N Y, Stephenson G B, Stolitchnov I, Taganstev A K, Taylor D V, Yamada T, Streiffer S 2006 J. Appl. Phys. 100 051606
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图 1 (a) 铁电三层膜的两种弯曲示意图, 这里R是弯曲曲率的半径; (b) 铁电三层膜在U型弯曲和N型弯曲下的面内应变εxx分布; (c) 铁电三层膜在U型弯曲和N型弯曲下的应变梯度εxx,z分布
Fig. 1. (a) Schematic diagrams of the two bending configurations of the ferroelectric trilayer film, where R is the radius of curvature; (b) in-plane strain εxx of the ferroelectric trilayer film under U-shaped and N-shaped bending; (c) strain gradient εxx,z of the ferroelectric trilayer film under U-shaped and N-shaped bending.
图 2 STO/PTO/STO三层膜在不同U型弯曲半径下畴结构演化行为
Fig. 2. Domain evolution behavior in STO/PTO/STO trilayer films under different U-shaped bending radii.
图 3 STO/PTO/STO三层膜在不同N型弯曲半径下畴结构演化行为
Fig. 3. Domain evolution behavior in STO/PTO/STO trilayer films under different N-shaped bending radii.
图 4 STO/PTO/STO三层膜在不同U型弯曲半径下的电滞回线
Fig. 4. Hysteresis loops of STO/PTO/STO trilayer films under different U-shaped bending radii.
图 5 STO/PTO/STO异质膜在U型弯曲变形下的电场调控极化分布特性
Fig. 5. Electric field-modulated polarization distribution in STO/PTO/STO trilayers under U-shaped bending deformation.
图 6 STO/PTO/STO三层膜在不同N型弯曲半径下的电滞回线
Fig. 6. Hysteresis loops of STO/PTO/STO trilayer films under different N-shaped bending radii.
图 7 STO/PTO/STO异质膜在N型弯曲变形下的电场调控极化分布特性
Fig. 7. Electric field-modulated polarization distribution in STO/PTO/STO trilayers under N-shaped bending deformation.
表 1 相场模拟所用的材料参数取值[33,34] (SI单位制, 温度为300 K)
Table 1. Material parameter values in the phase-field simulations (SI unit, T=300 K).
变量 数值 变量 数值 PTO $ {\alpha _1}/({10^8}{\text{ }}{\mathrm{J}} {\cdot} m {\cdot} {{\mathrm{C}}^{ - 2}}) $ $ - 1.706 $ $ {{{Q}}_{11}}/({{\mathrm{m}}^4} {\cdot} {{\mathrm{C}}^{ - 2}}) $ 0.089 $ {\alpha _{11}}/({10^7}{\text{ }}{\mathrm{J}} {\cdot} {{\mathrm{m}}^5} {\cdot} {{\mathrm{C}}^{ - 4}}) $ $ - 7.3 $ $ {{{Q}}_{12}}/({{\mathrm{m}}^4} {\cdot} {{\mathrm{C}}^{ - 2}}) $ –0.026 $ {\alpha _{12}}/({10^8}{\text{ }}{\mathrm{J}} {\cdot} {{\mathrm{m}}^5} {\cdot} {{\mathrm{C}}^{ - 4}}) $ $ 7.5 $ $ {{{Q}}_{44}}/({{\mathrm{m}}^4} {\cdot} {{\mathrm{C}}^{ - 2}}) $ 0.0675 $ {\alpha _{111}}/({10^8}{\text{ }}{\mathrm{J}} {\cdot} {{\mathrm{m}}^9} {\cdot} {{\mathrm{C}}^{ - 6}}) $ $ 2.6 $ $ {{{G}}_{11}}/({10^{ - 10}}{\text{ }}{\mathrm{N}} {\cdot} {{\mathrm{m}}^4} {\cdot} {{\mathrm{C}}^{ - 2}}) $ $ 1.44 $ $ {\alpha _{112}}/({10^8}{\text{ }}{\mathrm{J}} {\cdot} {{\mathrm{m}}^9} {\cdot} {{\mathrm{C}}^{ - 6}}) $ $ 6.1 $ $ {{{G}}_{12}}/({\mathrm{N}} {\cdot} {{\mathrm{m}}^4} {\cdot} {{\mathrm{C}}^{ - 2}}) $ 0 $ {\alpha _{123}}/({10^8}{\text{ }}{\mathrm{J}} {\cdot} {{\mathrm{m}}^9} {\cdot} {{\mathrm{C}}^{ - 6}}) $ $ - 3.7 $ $ G_{44},G_{44}'/(10^{-11}\text{ }\mathrm{N}{\cdot}\mathrm{m}^4{\cdot}\mathrm{C}^{-2}) $ $ 7.2 $ $ {{c}_{11}}/({10^{11}}{\text{ }}{\mathrm{J}} {\cdot} {{\mathrm{m}}^{ - 3}}) $ $ 2.3 $ $ {{f}_{11}}/{\mathrm{V}} $ 1.6 $ {{c}_{12}}/({10^{11}}{\text{ }}{\mathrm{J}} {\cdot} {{\mathrm{m}}^{ - 3}}) $ $ 1 $ $ {{f}_{12}}/{\mathrm{V}} $ –0.8 $ {{c}_{44}}/({10^{10}}{\text{ }}{\mathrm{J}} {\cdot} {{\mathrm{m}}^{ - 3}}) $ $ 7 $ $ {{f}_{44}}/{\mathrm{V}} $ 0.15 STO $ {\alpha _1}/({10^8}{\text{ }}{\mathrm{J}} {\cdot} {\mathrm{m}} {\cdot} {{\mathrm{C}}^{ - 2}}) $ $ 2.017 $ $ {{{Q}}_{44}}/({{\mathrm{m}}^4} {\cdot} {{\mathrm{C}}^{ - 2}}) $ 0.00957 $ {\alpha _{11}}/({10^9}{\text{ }}{\mathrm{J}} {\cdot} {{\mathrm{m}}^5} {\cdot} {{\mathrm{C}}^{ - 4}}) $ $ 1.7 $ $ {{{G}}_{11}}/({10^{ - 10}}{\text{ }}{\mathrm{N}} {\cdot} {{\mathrm{m}}^4} {\cdot} {{\mathrm{C}}^{ - 2}}) $ $ 1.44 $ $ {\alpha _{12}}/({10^9}{\text{ }}{\mathrm{J}} {\cdot} {{\mathrm{m}}^5} {\cdot} {{\mathrm{C}}^{ - 4}}) $ $ 4.45 $ $ {{{G}}_{12}}/({\mathrm{N}} {\cdot} {{\mathrm{m}}^4} {\cdot} {{\mathrm{C}}^{ - 2}}) $ 0 $ {{c}_{11}}/({10^{11}}{\text{ }}{\mathrm{J}} {\cdot} {{\mathrm{m}}^{ - 3}}) $ $ 3.3 $ $ {{{G}}_{44}}, {{G}}_{44}'/({10^{ - 11}}{\text{ }}{\mathrm{N}} {\cdot} {{\mathrm{m}}^4} {\cdot} {{\mathrm{C}}^{ - 2}}) $ $ 7.2 $ $ {{c}_{12}}/({10^{11}}{\text{ }}{\mathrm{J}} {\cdot} {{\mathrm{m}}^{ - 3}}) $ $ 1 $ $ {{f}_{11}}/{\mathrm{V}} $ –3.21 $ {{c}_{44}}/({10^{11}}{\text{ }}{\mathrm{J}} {\cdot} {{\mathrm{m}}^{ - 3}}) $ $ 1.25 $ $ {{f}_{12}}/{\mathrm{V}} $ 1.47 $ {{{Q}}_{11}}/({{\mathrm{m}}^4} {\cdot} {{\mathrm{C}}^{ - 2}}) $ 0.0457 $ {{f}_{44}}/{\mathrm{V}} $ 1.07 $ {{{Q}}_{12}}/({{\mathrm{m}}^4} {\cdot} {{\mathrm{C}}^{ - 2}}) $ –0.0135 εr (PTO/STO) 20 
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表 2 STO/PTO/STO三层膜在U型弯曲变形下电滞回线的矫顽电场、最大极化强度和剩余极化强度
Table 2. Coercive electric field, maximum polarization, and remnant polarization of the ferroelectric hysteresis loop in STO/PTO/STO trilayer films under U-shaped bending deformation.
U型弯曲-
R/nmεxx,z/
(106 m–1)Ec/
(kV·cm–1)Pmax/
(μC·cm–2)Pr/
(μC·cm–2)未弯曲 0 0 43.90 0 1200 –0.61 2.78 43.72 –0.29 937 –0.77 2.78 43.64 –0.31 600 –1.14 5.56 43.52 –0.46 400 –1.82 11.11 43.30 –0.57 300 –2.42 13.89 43.05 –0.64 240 –3.03 16.67 42.86 –0.69
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表 3 STO/PTO/STO三层膜在N型弯曲变形下电滞回线的矫顽电场、最大极化强度和剩余极化强度
Table 3. Coercive electric field, maximum polarization, and remnant polarization of the ferroelectric hysteresis loop in STO/PTO/STO trilayer films under N-shaped bending deformation.
N型弯曲-
R/nmεxx,z/
(106 m–1)Ec/
(kV·cm–1)Pmax/
(μC·cm–2)Pr/
(μC·cm–2)未弯曲 0 0 43.90 0 1200 0.61 –2.78 44.13 0.21 600 1.14 –5.56 44.33 0.43 400 1.82 –8.33 44.53 0.67 300 2.42 –13.89 44.78 0.93 240 3.03 –16.67 44.88 1.18 200 3.61 –19.44 45.15 1.42
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[1] Setter N, Damjanovic D, Eng L, Fox G, Gevorgian S, Hong S, Kingon A, Kohlstedt H, Park N Y, Stephenson G B, Stolitchnov I, Taganstev A K, Taylor D V, Yamada T, Streiffer S 2006 J. Appl. Phys. 100 051606
Google Scholar
[2] Martin L W, Rappe A M 2017 Nat. Rev. Mater. 2 16087
Google Scholar
[3] Scott J F, Paz de Araujo C A 1989 Science 246 1400
Google Scholar
[4] Ramesh R, Aggarwal S, Auciello O 2001 Mater. Sci. Eng., R 32 191
Google Scholar
[5] R. Bowen C, A. Kim H M, Weaver P, Dunn S 2014 Energy Environ. Sci. 7 25
Google Scholar
[6] Silva J P B, Silva J M B, Oliveira M J S, Weingärtner T, Sekhar K C, Pereira M, Gomes M J M 2019 Adv. Funct. Mater. 29 1807196
Google Scholar
[7] Singh A, Monga S, Sharma N, Sreenivas K, Katiyar R S 2022 J. Asian Ceram. Soc. 10 275
Google Scholar
[8] Lancaster M J, Powell J, Porch A 1998 Supercond. Sci. Technol. 11 1323
Google Scholar
[9] Wang W, Li J, Liu H, Ge S 2021 Adv. Sci. 8 2003074
Google Scholar
[10] Han X, Ji Y, Yang Y 2022 Adv. Funct. Mater. 32 2109625
Google Scholar
[11] Park J S, Jung S Y, Kim D H, Park J H, Jang H W, Kim T G, Baek S H, Lee B C 2023 Microsyst. Nanoeng. 9 1
Google Scholar
[12] Yu H, Chung C C, Shewmon N, Ho S, Carpenter J H, Larrabee R, Sun T, Jones J L, Ade H, O’Connor B T, So F 2017 Adv. Funct. Mater. 27 1700461
Google Scholar
[13] Yao M, Cheng Y, Zhou Z, Liu M 2020 J. Mater. Chem. C 8 14
Google Scholar
[14] Gao W, Zhu Y, Wang Y, Yuan G, Liu J M 2020 J. Materiomics 6 1
Google Scholar
[15] Jia X, Guo R, Tay B K, Yan X 2022 Adv. Funct. Mater. 32 2205933
Google Scholar
[16] Ryu J, Priya S, Park C S, Kim K Y, Choi J J, Hahn B D, Yoon W H, Lee B K, Park D S, Park C 2009 J. Appl. Phys. 106 024108
Google Scholar
[17] Shi Q, Parsonnet E, Cheng X, Fedorova N, Peng R C, Fernandez A, Qualls A, Huang X, Chang X, Zhang H, Pesquera D, Das S, Nikonov D, Young I, Chen L Q, Martin L W, Huang Y L, Íñiguez J, Ramesh R 2022 Nat. Commun. 13 1110
Google Scholar
[18] Ji D, Cai S, Paudel T R, Sun H, Zhang C, Han L, Wei Y, Zang Y, Gu M, Zhang Y, Gao W, Huyan H, Guo W, Wu D, Gu Z, Tsymbal E Y, Wang P, Nie Y, Pan X 2019 Nature 570 87
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[24] Tanwani M, Gupta P, Powar S, Das S 2025 Small 21 2405688
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[25] Yadav A K, Nelson C T, Hsu S L, et al. 2016 Nature 530 198
Google Scholar
[26] Hsu S L, McCarter M R, Dai C, Hong Z, Chen L Q, Nelson C T, Martin L W, Ramesh R 2019 Adv. Mater. 31 1901014
Google Scholar
[27] Hong Z, Damodaran A R, Xue F, Hsu S L, Britson J, Yadav A K, Nelson C T, Wang J J, Scott J F, Martin L W, Ramesh R, Chen L Q 2017 Nano Lett. 17 2246
Google Scholar
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Google Scholar
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Liu D, Wang J, Wang J S, Huang H B 2020 Acta Phys. Sin. 69 127801
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Google Scholar
Gao R Z, Wang J, Wang J S, Huang H B 2020 Acta Phys. Sin. 69 217801
Google Scholar
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Google Scholar
Liang D S, Huang H B, Zhao Y N, Liu Z H, Wang H Y, Ma X Q 2021 Acta Phys. Sin. 70 044202
Google Scholar
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Google Scholar
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