Yadav, A. K. et al. Observation of polar vortices in oxide superlattices. Nature 530, 198–201 (2016).
Google Scholar
Hsu, S.-L. et al. Emergence of the vortex state in confined ferroelectric heterostructures. Adv. Mater. 31, 1901014 (2019).
Google Scholar
Shafer, P. et al. Emergent chirality in the electric polarization texture of titanate superlattices. Proc. Natl Acad. Sci. USA 115, 915–920 (2018).
Google Scholar
Gruverman, A. et al. Vortex ferroelectric domains. J. Phys. Condens. Matter 20, 342201 (2008).
Google Scholar
Nelson, C. et al. Spontaneous vortex nanodomain arrays at ferroelectric heterointerfaces. Nano Lett. 11, 828–834 (2011).
Google Scholar
Das, S. et al. Observation of room-temperature polar skyrmions. Nature 568, 368–372 (2019).
Google Scholar
Caretta, L. et al. Non-volatile electric-field control of inversion symmetry. Nat. Mater. 22, 207–215 (2023).
Google Scholar
Yuan, S. et al. Hexagonal close-packed polar-skyrmion lattice in ultrathin ferroelectric PbTiO3 films. Phys. Rev. Lett. 130, 226801 (2023).
Google Scholar
Chen, S. et al. Recent progress on topological structures in ferroic thin films and heterostructures. Adv. Mater. 33, 2000857 (2021).
Google Scholar
Rusu, D. et al. Ferroelectric incommensurate spin crystals. Nature 602, 240–244 (2022).
Google Scholar
Jia, C.-L., Urban, K. W., Alexe, M., Hesse, D. & Vrejoiu, I. Direct observation of continuous electric dipole rotation in flux-closure domains in ferroelectric Pb(Zr,Ti)O3. Science 331, 1420–1423 (2011).
Google Scholar
Peters, J. J. P., Apachitei, G., Beanland, R., Alexe, M. & Sanchez, A. M. Polarization curling and flux closures in multiferroic tunnel junctions. Nat. Commun. 7, 13484 (2016).
Google Scholar
Schilling, A. et al. Domains in ferroelectric nanodots. Nano Lett. 9, 3359–3364 (2009).
Google Scholar
Tang, Y. L. et al. Observation of a periodic array of flux-closure quadrants in strained ferroelectric PbTiO3 films. Science 348, 547–551 (2015).
Google Scholar
Naumov, I. I., Bellaiche, L. & Fu, H. Unusual phase transitions in ferroelectric nanodisks and nanorods. Nature 432, 737–740 (2004).
Google Scholar
Kornev, I., Fu, H. & Bellaiche, L. Ultrathin films of ferroelectric solid solutions under a residual depolarizing field. Phys. Rev. Lett. 93, 196104 (2004).
Google Scholar
Naumov, I. & Bratkovsky, A. M. Unusual polarization patterns in flat epitaxial ferroelectric nanoparticles. Phys. Rev. Lett. 101, 107601 (2008).
Google Scholar
Choi, K. J. et al. Enhancement of ferroelectricity in strained BaTiO3 thin films. Science 306, 1005–1009 (2004).
Google Scholar
Catalan, G. et al. Flexoelectric rotation of polarization in ferroelectric thin films. Nat. Mater. 10, 963–967 (2011).
Google Scholar
Zubko, P., Catalan, G. & Tagantsev, A. K. Flexoelectric effect in solids. Annu. Rev. Mater. Res. 43, 387–421 (2013).
Google Scholar
Pertsev, N. A., Zembilgotov, A. G. & Tagantsev, A. K. Effect of mechanical boundary conditions on phase diagrams of epitaxial ferroelectric thin films. Phys. Rev. Lett. 80, 1988–1991 (1998).
Google Scholar
Aguado-Puente, P. & Junquera, J. Ferromagneticlike closure domains in ferroelectric ultrathin films: first-principles simulations. Phys. Rev. Lett. 100, 177601 (2008).
Google Scholar
Hong, J., Catalan, G., Fang, D. N., Artacho, E. & Scott, J. F. Topology of the polarization field in ferroelectric nanowires from first principles. Phys. Rev. B 81, 172101 (2010).
Google Scholar
Lu, H. et al. Mechanical writing of ferroelectric polarization. Science 336, 59–61 (2012).
Google Scholar
Cao, Y. et al. Tunable correlated states and spin-polarized phases in twisted bilayer–bilayer graphene. Nature 583, 215–220 (2020).
Google Scholar
Andrei, E. Y. et al. The marvels of moiré materials. Nat. Rev. Mater. 6, 201–206 (2021).
Google Scholar
Kazmierczak, N. P. et al. Strain fields in twisted bilayer graphene. Nat. Mater. 20, 956–963 (2021).
Google Scholar
Lu, D. et al. Synthesis of freestanding single-crystal perovskite films and heterostructures by etching of sacrificial water-soluble layers. Nat. Mater. 15, 1255–1260 (2016).
Google Scholar
Hong, S. S. et al. Extreme tensile strain states in La0.7Ca0.3MnO3 membranes. Science 368, 71–76 (2020).
Google Scholar
Dong, G. et al. Super-elastic ferroelectric single-crystal membrane with continuous electric dipole rotation. Science 366, 475–479 (2019).
Google Scholar
Han, L. et al. High-density switchable skyrmion-like polar nanodomains integrated on silicon. Nature 603, 63–67 (2022).
Google Scholar
Shao, Y.-T. et al. Emergent chirality in a polar meron to skyrmion phase transition. Nat. Commun. 14, 1355 (2023).
Google Scholar
Puebla, S. et al. Combining freestanding ferroelectric perovskite oxides with two-dimensional semiconductors for high performance transistors. Nano Lett. 22, 7457–7466 (2022).
Google Scholar
Shen, J. et al. Observation of moiré patterns in twisted stacks of bilayer perovskite oxide nanomembranes with various lattice symmetries. ACS Appl. Mater. Interfaces 14, 50386–50392 (2022).
Google Scholar
Li, Y. et al. Stacking and twisting of freestanding complex oxide thin filmsAdv. Mater. 34, e2203187 (2022).
Google Scholar
Devonshire, A. F. XCVI. Theory of barium titanate. London, Edinburgh, Dublin Philos. Mag. J. Sci. 40, 1040–1063 (1949).
Google Scholar
Íñiguez, J., Ivantchev, S., Perez-Mato, J. M. & García, A. Devonshire-Landau free energy of BaTiO3 from first principles. Phys. Rev. B 63, 144103 (2001).
Google Scholar
King-Smith, R. D. & Vanderbilt, D. First-principles investigation of ferroelectricity in perovskite compounds. Phys. Rev. B 49, 5828–5844 (1994).
Google Scholar
López-Pérez, J. & Íñiguez, J. Ab initio study of proper topological ferroelectricity in layered perovskite La2Ti2O7. Phys. Rev. B 84, 075121 (2011).
Google Scholar
Zheng, Z. et al. Unconventional ferroelectricity in moiré heterostructures. Nature 588, 71–76 (2020).
Google Scholar
Yasuda, K., Wang, X., Watanabe, K., Taniguchi, T. & Jarillo-Herrero, P. Stacking-engineered ferroelectricity in bilayer boron nitride. Science 372, 1458–1462 (2021).
Google Scholar
Woods, C. R. et al. Charge-polarized interfacial superlattices in marginally twisted hexagonal boron nitride. Nat. Commun. 12, 347 (2021).
Google Scholar
Vizner Stern, M. et al. Interfacial ferroelectricity by van der Waals sliding. Science 372, 1462–1466 (2021).
Google Scholar
Bennett, D. Theory of polar domains in moiré heterostructures. Phys. Rev. B 105, 235445 (2022).
Google Scholar
Wang, J. et al. Polar Solomon rings in ferroelectric nanocrystals. Nat. Commun. 14, 3941 (2023).
Google Scholar
Yu, X. Z. et al. Transformation between meron and skyrmion topological spin textures in a chiral magnet. Nature 564, 95–98 (2018).
Google Scholar
Sanchez-Santolino, G. et al. Resonant electron tunnelling assisted by charged domain walls in multiferroic tunnel junctions. Nat. Nanotechnol. 12, 655–662 (2017).
Google Scholar
Pesquera, D. et al. Beyond substrates: strain engineering of ferroelectric membranes. Adv. Mater. 32, 2003780 (2020).
Google Scholar
Borisevich, A. Y., Lupini, A. R. & Pennycook, S. J. Depth sectioning with the aberration-corrected scanning transmission electron microscope. Proc. Natl Acad. Sci. USA 103, 3044–3048 (2006).
Google Scholar
Ishikawa, R., Lupini, A. R., Hinuma, Y. & Pennycook, S. J. Large-angle illumination STEM: Toward three-dimensional atom-by-atom imaging. Ultramicroscopy 151, 122–129 (2015).
Google Scholar
Verbeeck, J. & Van Aert, S. Model based quantification of EELS spectra. Ultramicroscopy 101, 207–224 (2004).
Google Scholar
Thomas, P. J. & Twesten, R. D. A simple, model based approach for robust quantification of EELS spectra and spectrum-images. Microsc. Microanal. 18, 968–969 (2012).
Google Scholar
Galindo, P. L. et al. The Peak Pairs algorithm for strain mapping from HRTEM images. Ultramicroscopy 107, 1186–1193 (2007).
Google Scholar
Nord, M., Vullum, P. E., MacLaren, I., Tybell, T. & Holmestad, R. Atomap: a new software tool for the automated analysis of atomic resolution images using two-dimensional Gaussian fitting. Adv. Struct. Chem. Imaging 3, 9 (2017).
Google Scholar
Ghosez, Ph., Michenaud, J. & Gonze, X. Dynamical atomic charges: The case of ABO3 compounds. Phys. Rev. B 58, 6224–6240 (1998).
Google Scholar
Smeaton, M. A., Schnitzer, N., Zheng, H., Mitchell, J. F. & Kourkoutis, L. F. Channeling-Induced Artifacts in Atom Tracking of Cations in Distorted Perovskites Imaged by HAADF-STEM. Microsc. Microanal. 28, 1736–1738 (2022).
Google Scholar
Oveisi, E., Spadaro, M. C., Rotunno, E., Grillo, V. & Hébert, C. Insights into image contrast from dislocations in ADF-STEM. Ultramicroscopy 200, 139–148 (2019).
Google Scholar
Kim, Y. et al. Remote epitaxy through graphene enables two-dimensional material-based layer transfer. Nature 544, 340–343 (2017).
Google Scholar
Kum, H. S. et al. Heterogeneous integration of single-crystalline complex-oxide membranes. Nature 578, 75–81 (2020).
Google Scholar
Kong, W. et al. Polarity governs atomic interaction through two-dimensional materials. Nat. Mater. 17, 999–1004 (2018).
Google Scholar
Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54, 11169–11186 (1996).
Google Scholar
Kresse, G. & Joubert, D. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B 59, 1758–1775 (1999).
Google Scholar
Perdew, J. P. et al. Restoring the density-gradient expansion for exchange in solids and surfaces. Phys. Rev. Lett. 100, 136406 (2008).
Google Scholar
Blöchl, P. E. Projector augmented-wave method. Phys. Rev. B 50, 17953–17979 (1994).
Google Scholar
Monkhorst, H. J. & Pack, J. D. Special points for Brillouin-zone integrations. Phys. Rev. B 13, 5188–5192 (1976).
Google Scholar
Harada, J., Axe, J. D. & Shirane, G. Neutron-scattering study of soft modes in cubic BaTiO3. Phys. Rev. B 4, 155–162 (1971).
Google Scholar
Yudin, P. V., Ahluwalia, R. & Tagantsev, A. K. Upper bounds for flexoelectric coefficients in ferroelectrics. Appl. Phys. Lett. 104, 082913 (2014).
Google Scholar
Wang, B., Gu, Y., Zhang, S. & Chen, L.-Q. Flexoelectricity in solids: Progress, challenges, and perspectives. Prog. Mater. Sci. 106, 100570 (2019).
Google Scholar
Ma, W. & Cross, L. E. Strain-gradient-induced electric polarization in lead zirconate titanate ceramics. Appl. Phys. Lett. 82, 3293–3295 (2003).
Google Scholar
Stengel, M. Surface control of flexoelectricity. Phys. Rev. B 90, 201112 (2014).
Google Scholar
Dreyer, C. E., Stengel, M. & Vanderbilt, D. Current-density implementation for calculating flexoelectric coefficients. Phys. Rev. B 98, 075153 (2018).
Google Scholar