Design of super-elastic freestanding ferroelectric thin films guided by phase-field simulations

Abstract

Understanding the dynamic behavior of domain structures is critical to the design and application of super-elastic freestanding ferroelectric thin films. Phase-field simulations represent a powerful tool for observing, exploring and revealing the domain-switching behavior and phase transitions in ferroelectric materials at the mesoscopic scale. This review summarizes the recent theoretical progress regarding phase-field methods in freestanding ferroelectric thin films and novel buckling-induced wrinkled and helical structures. Furthermore, the strong coupling relationship between strain and ferroelectric polarization in super-elastic ferroelectric nanostructures is confirmed and discussed, resulting in new design strategies for the strain engineering of freestanding ferroelectric thin film systems. Finally, to further promote the innovative development and application of freestanding ferroelectric thin film systems, this review provides a summary and outlook on the theoretical modeling of freestanding ferroelectric thin films.

Keywords

Freestanding ferroelectric thin films / super-elastic / mechanical structure / topological domain structure / phase-field simulations

References

[1] Hussain AM.CMOS-technology-enabled flexible and stretchable electronics for internet of everything applications.Adv Mater2016;28:4219-49

[2] Vilouras A,Gupta S.Modeling of CMOS devices and circuits on flexible ultrathin chips.IEEE Trans Electron Devices2017;64:2038-46

[3] Zhang H,Yang Y.High-performance carbon nanotube complementary electronics and integrated sensor systems on ultrathin plastic foil.ACS Nano2018;12:2773-9

[4] Comiskey B,Yoshizawa H.An electrophoretic ink for all-printed reflective electronic displays.Nature1998;394:253-5

[5] Rogers JA,Baldwin K.Paper-like electronic displays: large-area rubber-stamped plastic sheets of electronics and microencapsulated electrophoretic inks.Proc Natl Acad Sci USA2001;98:4835-40 PMCID:PMC33123

[6] Gelinck GH,van Veenendaal E.Flexible active-matrix displays and shift registers based on solution-processed organic transistors.Nat Mater2004;3:106-10

[7] McAlpine MC,Wang D.Highly ordered nanowire arrays on plastic substrates for ultrasensitive flexible chemical sensors.Nat Mater2007;6:379-84 PMCID:PMC3695594

[8] Segev-Bar M.Flexible sensors based on nanoparticles.ACS Nano2013;7:8366-78

[9] Lee HS,Hwang G.Flexible inorganic piezoelectric acoustic nanosensors for biomimetic artificial hair cells.Adv Funct Mater2014;24:6914-21

[10] Yamamoto Y,Yamamoto D.Printed multifunctional flexible device with an integrated motion sensor for health care monitoring.Sci Adv2016;2:e1601473 PMCID:PMC5262446

[11] Wang X,Zhang T.Flexible sensing electronics for wearable/attachable health monitoring.Small2017;13:1602790

[12] Chen Y,Zhang S.Skin-like biosensor system via electrochemical channels for noninvasive blood glucose monitoring.Sci Adv2017;3:e1701629 PMCID:PMC5738229

[13] Tee BC,Dunn RR,Eason E.Tunable flexible pressure sensors using microstructured elastomer geometries for intuitive electronics.Adv Funct Mater2014;24:5427-34

[14] Wang Y,Pfattner R.A highly stretchable, transparent, and conductive polymer.Sci Adv2017;3:e1602076 PMCID:PMC5345924

[15] Lu L,Liu J.Flexible PVDF based piezoelectric nanogenerators.Nano Energy2020;78:105251

[16] Pan L,Zhai D.Hierarchical nanostructured conducting polymer hydrogel with high electrochemical activity.Proc Natl Acad Sci USA2012;109:9287-92 PMCID:PMC3386113

[17] Sun JY,Illeperuma WR.Highly stretchable and tough hydrogels.Nature2012;489:133-6 PMCID:PMC3642868

[18] Kubo M,Kim C.Stretchable microfluidic radiofrequency antennas.Adv Mater2010;22:2749-52

[19] Gao Y,Schaler EW.Wearable microfluidic diaphragm pressure sensor for health and tactile touch monitoring.Adv Mater2017;29:1701985

[20] Yan J,Maleski K.Flexible MXene/graphene films for ultrafast supercapacitors with outstanding volumetric capacitance.Adv Funct Mater2017;27:1701264

[21] Gao W,Wang Y,Liu J.A review of flexible perovskite oxide ferroelectric films and their application.J Mater2020;6:1-16

[22] Bertoldi K,Christensen J.Flexible mechanical metamaterials.Nat Rev Mater2017;2:1-11

[23] Xue Z,Rogers JA,Huang Y.Mechanically-guided structural designs in stretchable inorganic electronics.Adv Mater2020;32:e1902254

[24] Kim DH,Choi WM.Stretchable and foldable silicon integrated circuits.Science2008;320:507-11

[25] Bae HJ,Yoon J.Self-organization of maze-like structures via guided wrinkling.Sci Adv2017;3:e1700071 PMCID:PMC5493415

[26] Peraza-hernandez EA,Malak Jr RJ.Origami-inspired active structures: a synthesis and review.Smart Mater Struct2014;23:094001

[27] Song Z,Tang R.Origami lithium-ion batteries.Nat Commun2014;5:3140

[28] Shyu TC,Dodd PM.A kirigami approach to engineering elasticity in nanocomposites through patterned defects.Nat Mater2015;14:785-9

[29] Callens SJ.From flat sheets to curved geometries: origami and kirigami approaches.Materials Today2018;21:241-64

[30] Meng Y,Hu C.All-graphene core-sheath microfibers for all-solid-state, stretchable fibriform supercapacitors and wearable electronic textiles.Adv Mater2013;25:2326-31

[31] Ghosh T.Stretch, wrap, and relax to smartness.Science2015;349:382-3

[32] Scott JF.Ferroelectric memories.Science1989;246:1400-5

[33] Auciello O,Ramesh R.The physics of ferroelectric memories.Phys Today1998;51:22-7

[34] Wang J,Huang H.Ferroelectric domain-wall logic units.Nat Commun2022;13:3255 PMCID:PMC9170692

[35] Sun H,Wang Y.Nonvolatile ferroelectric domain wall memory integrated on silicon.Nat Commun2022;13:4332 PMCID:PMC9325887

[36] Muralt P.Ferroelectric thin films for micro-sensors and actuators: a review.J Micromecha Microeng2000;10:136-46

[37] Damjanovic D,Setter N.Ferroelectric sensors.IEEE Sensors J2001;1:191-206

[38] Kirby P,Imura M,Su Q.High frequency thin film ferroelectric acoustic resonators and filters.Integr Ferroelectr2001;41:91-100

[39] Dragoman M,Modreanu M.Extraordinary tunability of high-frequency devices using Hf_0.3Zr_0.7O₂ ferroelectric at very low applied voltages.Appl Phys Lett2017;110:103104

[40] Bowen CR,Weaver PM.Piezoelectric and ferroelectric materials and structures for energy harvesting applications.Energy Environ Sci2014;7:25-44

[41] Zhang Y,Roake E.Thermal energy harvesting using pyroelectric-electrochemical coupling in ferroelectric materials.Joule2020;4:301-9

[42] Li Q,Gadinski MR,Wang Q.High energy and power density capacitors from solution-processed ternary ferroelectric polymer nanocomposites.Adv Mater2014;26:6244-9

[43] Thakur VK, Gupta RK. Recent progress on ferroelectric polymer-based nanocomposites for high energy density capacitors: synthesis, dielectric properties, and future aspects.Chem Rev2016;116:4260-317

[44] Xu K,Dong S,Huang H.Antiferroelectric phase diagram enhancing energy-storage performance by phase-field simulations.ACS Appl Mater Interfaces2022;14:25770-80

[45] Xu S,Pan H.Strain engineering of energy storage performance in relaxor ferroelectric thin film capacitors.Adv Theory Simul2022;5:2100324

[46] Ohigashi H,Suzuki M,Kimura K.Piezoelectric and ferroelectric properties of P (VDF-TrFE) copolymers and their application to ultrasonic transducers.Ferroelectrics1984;60:263-76

[47] Zhang S,Jiang X,Luo J.Advantages and challenges of relaxor-PbTiO₃ Ferroelectric crystals for electroacoustic transducers-a review.Prog Mater Sci2015;68:1-66 PMCID:PMC4267134

[48] Zhang G,Huang H.Toward wearable cooling devices: highly flexible electrocaloric Ba_0.67Sr_0.33TiO₃ nanowire arrays.Adv Mater2016;28:4811-6

[49] Gao R,Wang J,Huang H.Designed giant room-temperature electrocaloric effects in metal-free organic perovskite [MDABCO](NH₄)I₃ by phase-field simulations.Adv Funct Mater2021;31:2104393

[50] Qian X,Zheng L.High-entropy polymer produces a giant electrocaloric effect at low fields.Nature2021;600:664-9

[51] Gao R,Wang J.Understanding electrocaloric cooling of ferroelectrics guided by phase-field modeling.J Am Ceram Soc2022;105:3689-714

[52] Ge JF,Liu C.Superconductivity above 100 K in single-layer FeSe films on doped SrTiO₃.Nat Mater2015;14:285-9

[53] Bégon-lours L,Sander A.High-temperature-superconducting weak link defined by the ferroelectric field effect.Phys Rev Appl2017;7

[54] Lynch CS,Suo Z,Yang W.Crack growth in ferroelectric ceramics driven by cyclic polarization switching.J Intell Mater Syst Struct1995;6:191-8

[55] Arias I,Ortiz M.A phenomenological cohesive model of ferroelectric fatigue.Acta Mater2006;54:975-84

[56] Horiuchi S.Organic ferroelectrics.Nat Mater2008;7:357-66

[57] Bhansali US,Alshareef H.Organic ferroelectric memory devices with inkjet-printed polymer electrodes on flexible substrates.Microelect Eng2013;105:68-73

[58] Zabek D,Boulbar EL.Micropatterning of flexible and free standing polyvinylidene difluoride (PVDF) films for enhanced pyroelectric energy transformation.Adv Energy Mater2015;5:1401891

[59] Owczarek M,Ferris DP.Flexible ferroelectric organic crystals.Nat Commun2016;7:13108 PMCID:PMC5065626

[60] Guo M,Qian J.Flexible robust and high-density FeRAM from array of organic ferroelectric nano-lamellae by self-assembly.Adv Sci2019;6:1801931 PMCID:PMC6425439

[61] Dong G,Yao M.Super-elastic ferroelectric single-crystal membrane with continuous electric dipole rotation.Science2019;366:475-9

[62] Guo C,Zhou Z.Domain evolution in bended freestanding BaTiO₃ ultrathin films: a phase-field simulation.Appl Phys Lett2020;116:152903

[63] Dong G,Li T.Periodic wrinkle-patterned single-crystalline ferroelectric oxide membranes with enhanced piezoelectricity.Adv Mater2020;32:e2004477

[64] Zhou Y,Dong G.Tip-induced in-plane ferroelectric superstructure in zigzag-wrinkled BaTiO₃ thin films.Nano Lett2022;22:2859-66

[65] Guo M,Han J.Toroidal polar topology in strained ferroelectric polymer.Science2021;371:1050-6

[66] Dong G,Guo C.Self-assembled epitaxial ferroelectric oxide nanospring with super-scalability.Adv Mater2022;34:e2108419

[67] Chen X,Chen X.Nano-imprinted ferroelectric polymer nanodot arrays for high density data storage.Adv Funct Mater2013;23:3124-9

[68] Fujikake H,Murashige T.Polymer-stabilized ferroelectric liquid crystal for flexible displays.Displays2004;25:3-8

[69] Sekine T,Tashiro T.Fully printed wearable vital sensor for human pulse rate monitoring using ferroelectric polymer.Sci Rep2018;8:4442

[70] Han X,Tang X,Liu J.Flexible polymer transducers for dynamic recognizing physiological signals.Adv Funct Mater2016;26:3640-8

[71] Liu Z,Zheng Q.Human motion driven self-powered photodynamic system for long-term autonomous cancer therapy.ACS Nano2020;14:8074-83

[72] Shi Q,Lee C.MEMS based broadband piezoelectric ultrasonic energy harvester (PUEH) for enabling self-powered implantable biomedical devices.Sci Rep2016;6:24946 PMCID:PMC4844957

[73] Ryu J,Park C.Enhanced domain contribution to ferroelectric properties in freestanding thick films.J Appl Phys2009;106:024108

[74] Zuo Z,Zhan Q.Preparation and ferroelectric properties of freestanding Pb(Zr,Ti)O₃ thin membranes.J Phys D Appl Phys2012;45:185302

[75] Pesquera D,Qualls A.Beyond substrates: strain engineering of ferroelectric membranes.Adv Mater2020;32:e2003780

[76] Shi Q,Cheng X.The role of lattice dynamics in ferroelectric switching.Nat Commun2022;13:1110 PMCID:PMC8891289

[77] Tian M,Yang Y.Perovskite oxide ferroelectric thin films.Adv Elect Mater2022;8:2101409

[78] Jin C,Han W.Exchange bias in flexible freestanding La_0.7Sr_0.3MnO₃/BiFeO₃ membranes.Appl Phys Lett2020;117:252902

[79] Xu R,Barnard ES.Strain-induced room-temperature ferroelectricity in SrTiO₃ membranes.Nat Commun2020;11:3141 PMCID:PMC7305178

[80] Chang L,Wang J.The path to flexible ferroelectrics: approaches and progress.Jpn J Appl Phys2018;57:0902A3

[81] Yao M,Zhou Z.Recent progress on the fabrication and applications of flexible ferroelectric devices.J Mater Chem C2020;8:14-27

[82] Chiabrera FM,Li Y.Freestanding perovskite oxide films: synthesis, challenges, and properties.Annalen Physik2022;534:2200084

[83] Li S,Yang M.Ferroelectric thin films: performance modulation and application.Mater Adv2022;3:5735-52

[84] Won SS,Kawahara M.Flexible vibrational energy harvesting devices using strain-engineered perovskite piezoelectric thin films.Nano Energy2019;55:182-92

[85] De Dobbelaere C,Jiménez R.Aqueous solutions for low-temperature photoannealing of functional oxide films: reaching the 400 °C Si-technology integration barrier.J Am Chem Soc2011;133:12922-5

[86] Bretos I,Ricote J.Low-temperature crystallization of solution-derived metal oxide thin films assisted by chemical processes.Chem Soc Rev2018;47:291-308

[87] Bretos I,Ricote J.Low-temperature solution approaches for the potential integration of ferroelectric oxide films in flexible electronics.IEEE Trans Ultrason Ferroelectr Freq Control2020;67:1967-79

[88] Bretos I,Ricote J,Calzada ML.Photoferroelectric thin films for flexible systems by a three-in-one solution-based approach.Adv Funct Mater2020;30:2001897

[89] Barrios Ó,Ricote J,Calzada ML.A sustainable self-induced solution seeding approach for multipurpose BiFeO₃ active layers in flexible electronic devices.Adv Funct Mater2022;32:2112944

[90] Jiang J,Huang CW.Flexible ferroelectric element based on van der Waals heteroepitaxy.Sci Adv2017;3:e1700121

[91] Zheng M,Ni H,Gao J.van der Waals epitaxy for highly tunable all-inorganic transparent flexible ferroelectric luminescent films.J Mater Chem C2019;7:8310-5

[92] Bitla Y.van der Waals oxide heteroepitaxy for soft transparent electronics.Nanoscale2020;12:18523-44

[93] Lee SA,Kim ES,Choi WS.Highly oriented SrTiO₃ Thin film on graphene substrate.ACS Appl Mater Inter2017;9:3246-50

[94] Kum HS,Kim S.Heterogeneous integration of single-crystalline complex-oxide membranes.Nature2020;578:75-81

[95] Wong WS,Cheung NW.Damage-free separation of GaN thin films from sapphire substrates.Appl Phys Lett1998;72:599-601

[96] Xu J,Wang Y.Preparation of large area freestanding GaN by laser lift-off technology.Mater Lett2002;56:43-6

[97] Lin I,Lee K.Preparation of ferroelectric Pb(Zr1-xTix)O₃/Si films by laser lift-off technique.J Eur Ceram Soc2004;24:975-8

[98] Lee CH,Oh Y,Yoon Y.Use of laser lift-off for flexible device applications.J Appl Phys2010;108:102814

[99] Zhang Y,Lu X.Recent progress on flexible inorganic single-crystalline functional oxide films for advanced electronics.Mater Horiz2019;6:911-30

[100] Bakaul SR,Lee M.Single crystal functional oxides on silicon.Nat Commun2016;7:10547 PMCID:PMC4748113

[101] Bakaul SR,Zhang Q.Freestanding ferroelectric bubble domains.Adv Mater2021;33:e2105432

[102] Lu D,Hong SS,Hikita Y.Synthesis of freestanding single-crystal perovskite films and heterostructures by etching of sacrificial water-soluble layers.Nat Mater2016;15:1255-60

[103] Baek DJ,Hikita Y,Kourkoutis LF.Ultrathin epitaxial barrier layer to avoid thermally induced phase transformation in oxide heterostructures.ACS Appl Mater Inter2017;9:54-9

[104] Hong SS,Lu D.Two-dimensional limit of crystalline order in perovskite membrane films.Sci Adv2017;3:eaao5173 PMCID:PMC5696264

[105] Ji D,Paudel TR.Freestanding crystalline oxide perovskites down to the monolayer limit.Nature2019;570:87-90

[106] Han L,Zhao Y.Giant uniaxial strain ferroelectric domain tuning in freestanding PbTiO₃ films.Adv Mater Inter2020;7:1901604

[107] Takahashi R.Sacrificial water-soluble BaO layer for fabricating free-standing piezoelectric membranes.ACS Appl Mater Inter2020;12:25042-9

[108] Zhong H,Zhang Q.Large-scale Hf_0.5Zr_0.5O₂ membranes with robust ferroelectricity.Adv Mater2022;34:e2109889

[109] Guo R,Lin W.Continuously controllable photoconductance in freestanding BiFeO₃ by the macroscopic flexoelectric effect.Nat Commun2020;11:2571 PMCID:PMC7244550

[110] Peng B,Zhang YQ.Phase transition enhanced superior elasticity in freestanding single-crystalline multiferroic BiFeO₃ membranes.Sci Adv2020;6:eaba5847 PMCID:PMC7442355

[111] Jin C,Li X.Super-flexible freestanding BiMnO₃ membranes with stable ferroelectricity and ferromagnetism.Adv Sci2021;8:e2102178 PMCID:PMC8693045

[112] Han L,Prokhorenko S.High-density switchable skyrmion-like polar nanodomains integrated on silicon.Nature2022;603:63-7

[113] Elangovan H,Seremi S.Giant superelastic piezoelectricity in flexible ferroelectric BaTiO₃ membranes.ACS Nano2020;14:5053-60

[114] Cai S,Ji D.Enhanced polarization and abnormal flexural deformation in bent freestanding perovskite oxides.Nat Commun2022;13:5116 PMCID:PMC9433432

[115] Chen L.Phase-field models for microstructure evolution.Annu Rev Mater Res2002;32:113-40

[116] Artyukhin S,Spaldin NA.Landau theory of topological defects in multiferroic hexagonal manganites.Nat Mater2014;13:42-9

[117] Xue F,Shi Y,Chen L.Strain-induced incommensurate phases in hexagonal manganites.Phys Rev B2017;96:104109

[118] Wang J,Chen L,Zhang T.Phase-field simulations of ferroelectric/ferroelastic polarization switching.Acta Materialia2004;52:749-64

[119] Cao W.Constructing landau-ginzburg-devonshire type models for ferroelectric systems based on symmetry.Ferroelectrics2008;375:28-39

[120] Chen L.Phase-field method of phase transitions/domain structures in ferroelectric thin films: a review.J Am Ceram Soc2008;91:1835-44

[121] Chen HT,Ni Y.Phase field modeling of flexoelectric effects in ferroelectric epitaxial thin films.Acta Mech2014;225:1323-33

[122] Wang J,Chen L.Understanding, predicting, and designing ferroelectric domain structures and switching guided by the phase-field method.Annu Rev Mater Res2019;49:127-52

[123] Peng R,Peng B,Chen L.Domain patterns and super-elasticity of freestanding BiFeO₃ membranes via phase-field simulations.Acta Materialia2021;208:116689

[124] Peng R,Peng B,Chen L.Boundary conditions manipulation of polar vortex domains in BiFeO₃ membranes via phase-field simulations.J Phys D Appl Phys2021;54:495301

[125] Chen WJ,Xiong WM,Wang B.Effect of mechanical loads on stability of nanodomains in ferroelectric ultrathin films: towards flexible erasing of the non-volatile memories.Sci Rep2014;4:5339 PMCID:PMC4061556

[126] Lacour S,Suo Z.Design and performance of thin metal film interconnects for skin-like electronic circuits.IEEE Electron Device Lett2004;25:179-81

[127] Cheng H,Hwang K,Huang Y.Buckling of a stiff thin film on a pre-strained bi-layer substrate.Int J Solids Struct2014;51:3113-8

[128] Pan K,He L.Nonlinear analysis of compressed elastic thin films on elastic substrates: from wrinkling to buckle-delamination.Int J Solids Struct2014;51:3715-26

[129] Xu F,Belouettar S.3D finite element modeling for instabilities in thin films on soft substrates.Int J Solids Struct2014;51:3619-32

[130] Yan D,Hu G.Wrinkling of structured thin films via contrasted materials.Soft Matter2016;12:3937-42

[131] Park HG,Jung YH.Control of the wrinkle structure on surface-reformed poly(dimethylsiloxane) via ion-beam bombardment.Sci Rep2015;5:12356 PMCID:PMC4508831

[132] Zhu W,Perebeinos V.Structure and electronic transport in graphene wrinkles.Nano Lett2012;12:3431-6

[133] Chung JY,Stafford CM.Diffusion-controlled, self-organized growth of symmetric wrinkling patterns.Adv Mater2009;21:1358-62

[134] Guvendiren M,Burdick JA.Swelling-induced surface patterns in hydrogels with gradient crosslinking density.Adv Funct Mater2009;19:3038-45

[135] Jiang H,Song J,Huang Y.Finite deformation mechanics in buckled thin films on compliant supports.Proc Natl Acad Sci USA2007;104:15607-12 PMCID:PMC2000418

[136] Hendricks TR,Lee I.Buckling in nanomechanical films.Soft Matter2010;6:3701

[137] Huang Z,Suo Z.Nonlinear analyses of wrinkles in a film bonded to a compliant substrate.J Mech Phys Solids2005;53:2101-18

[138] Genzer J.Soft matter with hard skin: From skin wrinkles to templating and material characterization.Soft Matter2006;2:310-23

[139] Audoly B.Buckling of a stiff film bound to a compliant substrate-part I: formulation, linear stability of cylindrical patterns, secondary bifurcations.J Mech Phys Solids2008;56:2401-21

[140] Zhang Y,Yan Z.Printing, folding and assembly methods for forming 3D mesostructures in advanced materials.Nat Rev Mater2017;2

[141] Rogers JA,Huang Y.Materials and mechanics for stretchable electronics.Science2010;327:1603-7

[142] Kim JB,Pégard NC.Wrinkles and deep folds as photonic structures in photovoltaics.Nat Photon2012;6:327-32

[143] Zhang W,Qiu J,Liu N.Topological structures of transition metal dichalcogenides: a review on fabrication, effects, applications, and potential.InfoMat2021;3:133-54

[144] Stafford CM,Beers KL.A buckling-based metrology for measuring the elastic moduli of polymeric thin films.Nat Mater2004;3:545-50

[145] Chung JY,Stafford CM.Surface wrinkling: a versatile platform for measuring thin-film properties.Adv Mater2011;23:349-68

[146] Dervaux J,Guedeau-Boudeville MA.Shape transition in artificial tumors: from smooth buckles to singular creases.Phys Rev Lett2011;107:018103

[147] Guvendiren M,Yang S.Solvent induced transition from wrinkles to creases in thin film gels with depth-wise crosslinking gradients.Soft Matter2010;6:5795

[148] Tan Y,Song J,Wu W.Bioinspired multiscale wrinkling patterns on curved substrates: an overview.Nanomicro Lett2020;12:101 PMCID:PMC7770713

[149] Naumov II,Fu H.Unusual phase transitions in ferroelectric nanodisks and nanorods.Nature2004;432:737-40

[150] Tang YL,Ma XL.Observation of a periodic array of flux-closure quadrants in strained ferroelectric PbTiO₃ films.Science2015;348:547-51

[151] Hadjimichael M,Zatterin E.Metal-ferroelectric supercrystals with periodically curved metallic layers.Nat Mater2021;20:495-502

[152] Yadav AK,Hsu SL.Observation of polar vortices in oxide superlattices.Nature2016;530:198-201

[153] Hong Z,Xue F.Stability of polar vortex lattice in ferroelectric superlattices.Nano Lett2017;17:2246-52

[154] Liu D,Wang J,Huang H.Phase-field simulations of surface charge-induced ferroelectric vortex.J Phys D Appl Phys2021;54:405302

[155] Liu D,Jafri HM.Phase-field simulations of vortex chirality manipulation in ferroelectric thin films.NPJ Quantum Mater2022;7

[156] Das S,Hong Z.Observation of room-temperature polar skyrmions.Nature2019;568:368-72

[157] Zhang Y,Huang H,Wang X.Strain manipulation of ferroelectric skyrmion bubbles in a freestanding PbTiO₃ film: a phase field simulation.Phys Rev B2022;105

[158] Wang YJ,Zhu YL.Polar meron lattice in strained oxide ferroelectrics.Nat Mater2020;19:881-6

[159] Vasudevan RK,Tai HH.Exploring topological defects in epitaxial BiFeO3 thin films.ACS Nano2011;5:879-87

[160] Wang X,Han MG.Unfolding of vortices into topological stripes in a multiferroic material.Phys Rev Lett2014;112:247601

[161] Shimada T,Nagano K,Kitamura T.Hierarchical ferroelectric and ferrotoroidic polarizations coexistent in nano-metamaterials.Sci Rep2015;5:14653 PMCID:PMC4589792

[162] Cavallo F.Semiconductors turn soft: inorganic nanomembranes.Soft Matter2010;6:439-55

[163] Chen Z,Trase I,Mei Y.Mechanical self-assembly of a strain-engineered flexible layer: wrinkling, rolling, and twisting.Phys Rev Applied2016;5

[164] Yang M.Nanoscale helices from inorganic materials.J Mater Chem2011;21:6775

[165] Guo Q,Grover MA,Lynn DG.Shape selection and multi-stability in helical ribbons.Appl Phys Lett2014;104:211901

[166] Guo Q,Li W.Mechanics of tunable helices and geometric frustration in biomimetic seashells.EPL Europhys Lett2014;105:64005

[167] Yu X,Hu N.Shape formation of helical ribbons induced by material anisotropy.Appl Phys Lett2017;110:091901

[168] Wang B,Zhang S.Flexoelectricity in solids: progress, challenges, and perspectives.Prog Mater Sci2019;106:100570