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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
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
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[7] Singh A, Monga S, Sharma N, Sreenivas K, Katiyar R S 2022 J. Asian Ceram. Soc. 10 275
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[8] Lancaster M J, Powell J, Porch A 1998 Supercond. Sci. Technol. 11 1323
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[9] Wang W, Li J, Liu H, Ge S 2021 Adv. Sci. 8 2003074
Google Scholar
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Google Scholar
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Gao R Z, Wang J, Wang J S, Huang H B 2020 Acta Phys. Sin. 69 217801
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Google Scholar
[38] Li Y, Zatterin E, Conroy M, et al. 2022 Adv. Mater. 34 2106826
Google Scholar
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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
Figure 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
Figure 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
Figure 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
Figure 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
Figure 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
Figure 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
Figure 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
Google Scholar
[19] Dong G, Li S, Yao M, Zhou Z, Zhang Y Q, Han X, Luo Z, Yao J, Peng B, Hu Z, Huang H, Jia T, Li J, Ren W, Ye Z G, Ding X, Sun J, Nan C W, Chen L Q, Li J, Liu M 2019 Science 366 475
Google Scholar
[20] Peng B, Peng R C, Zhang Y Q, Dong G, Zhou Z, Zhou Y, Li T, Liu Z, Luo Z, Wang S, Xia Y, Qiu R, Cheng X, Xue F, Hu Z, Ren W, Ye Z G, Chen L Q, Shan Z, Min T, Liu M 2020 Sci. Adv. 6 eaba5847
Google Scholar
[21] Zhou Y, Guo C, Dong G, Liu H, Zhou Z, Niu B, Wu D, Li T, Huang H, Liu M, Min T 2022 Nano Lett. 22 2859
Google Scholar
[22] Cai S, Lun Y, Ji D, Lv P, Han L, Guo C, Zang Y, Gao S, Wei Y, Gu M, Zhang C, Gu Z, Wang X, Addiego C, Fang D, Nie Y, Hong J, Wang P, Pan X 2022 Nat. Commun. 13 5116
Google Scholar
[23] Wang J, Liu Z, Wang Q, Nie F, Chen Y, Tian G, Fang H, He B, Guo J, Zheng L, Li C, Lü W, Yan S 2024 Adv. Sci. 11 2401657
Google Scholar
[24] Tanwani M, Gupta P, Powar S, Das S 2025 Small 21 2405688
Google Scholar
[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
[28] Das S, Tang Y L, Hong Z, et al. 2019 Nature 568 368
Google Scholar
[29] Lun Y, Wang X, Kang J, Ren Q, Wang T, Han W, Gao Z, Huang H, Chen Y, Chen L Q, Fang D, Hong J 2023 Adv. Mater. 35 2302320
Google Scholar
[30] Guo C, Yang H, Dong S, Tang S, Wang J, Wang X, Huang H 2024 Adv. Electron. Mater. 10 2400001
Google Scholar
[31] Guo C, Wang J, Huang H 2024 Appl. Phys. Lett. 125 062903
Google Scholar
[32] Zhou M J, Peng K, Tan Y Z, Yang T N, Chen L Q, Nan C W 2025 Acta Mater. 288 120805
Google Scholar
[33] Li Q, Nelson C T, Hsu S L, et al. 2017 Nat. Commun. 8 1468
Google Scholar
[34] Xu T, Wu C, Zheng S, Wang Y, Wang J, Hirakata H, Kitamura T, Shimada T 2024 Phys. Rev. Lett. 132 086801
Google Scholar
[35] 刘迪, 王静, 王俊升, 黄厚兵 2020 物理学报 69 127801
Google Scholar
Liu D, Wang J, Wang J S, Huang H B 2020 Acta Phys. Sin. 69 127801
Google Scholar
[36] 高荣贞, 王静, 王俊升, 黄厚兵 2020 物理学报 69 217801
Google Scholar
Gao R Z, Wang J, Wang J S, Huang H B 2020 Acta Phys. Sin. 69 217801
Google Scholar
[37] 梁德山, 黄厚兵, 赵亚楠, 柳祝红, 王浩宇, 马星桥 2021 物理学报 70 044202
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
[38] Li Y, Zatterin E, Conroy M, et al. 2022 Adv. Mater. 34 2106826
Google Scholar
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