Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556, 43–50 (2018).
Andrei, E. Y. & MacDonald, A. H. Graphene bilayers with a twist. Nat. Mater. 19, 1265–1275 (2020).
Pryds, N., Park, D.-S., Jespersen, T. S. & Yun, S. Twisted oxide membranes: a perspective. APL Mater. 12, 010901 (2024).
Nuckolls, K. P. & Yazdani, A. A microscopic perspective on moiré materials. Nat. Rev. Mater. 9, 460–480 (2024).
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).
Pesquera, D., Fernández, A., Khestanova, E. & Martin, L. W. Freestanding complex-oxide membranes. J. Phys. Condens. Matter 34, 383001 (2022).
Zhang, J. et al. Super-tetragonal Sr4Al2O7 as a sacrificial layer for high-integrity freestanding oxide membranes. Science 383, 388–394 (2024).
Sánchez-Santolino, G. et al. A 2D ferroelectric vortex pattern in twisted BaTiO3 freestanding layers. Nature 626, 529–534 (2024).
Lee, S., De Sousa, D. J. P., Jalan, B. & Low, T. Moiré polar vortex, flat bands, and Lieb lattice in twisted bilayer BaTiO3. Sci. Adv. 10, eadq0293 (2024).
Kim, M.-S. et al. Charge disproportionation at twisted SrTiO3 bilayer interface driven by local atomic registry. ACS Nano 19, 39714–39724 (2025).
Zubko, P., Catalan, G. & Tagantsev, A. K. Flexoelectric effect in solids. Annu. Rev. Mater. Res. 43, 387–421 (2013).
Wang, B., Gu, Y., Zhang, S. & Chen, L.-Q. Flexoelectricity in solids: progress, challenges, and perspectives. Prog. Mater Sci. 106, 100570 (2019).
Sha, H. et al. Polar vortex hidden in twisted bilayers of paraelectric SrTiO3. Nat. Commun. 15, 10915 (2024).
Shahed, N. A. et al. Prediction of polarization vortices, charge modulation, flat bands, and moiré magnetism in twisted oxide bilayers. Phys. Rev. B 111, 195420 (2025).
Haeni, J. H. et al. Room-temperature ferroelectricity in strained SrTiO3. Nature 430, 4 (2004).
Yadav, A. K. et al. Observation of polar vortices in oxide superlattices. Nature 530, 198–201 (2016).
Das, S. et al. Observation of room-temperature polar skyrmions. Nature 568, 368–372 (2019).
Zabalo, A. & Stengel, M. Switching a polar metal via strain gradients. Phys. Rev. Lett. 126, 127601 (2021).
Peng, W. et al. Flexoelectric polarizing and control of a ferromagnetic metal. Nat. Phys. 20, 450–455 (2024).
Catalan, G. Metal poles around the bend. Nat. Phys. 20, 358–359 (2024).
Saeed, U. et al. Beware of the water: hidden hydrogenation of perovskite membranes made by the water-soluble sacrificial layer method. Phys. Rev. Mater. 10, 024408 (2026).
Sandholt, W. et al. Designing dislocation-driven polar vortex networks in twisted perovskites. Preprint at https://arxiv.org/abs/2603.27272 (2026).
Choi, K. J. et al. Phase-transition temperatures of strained single-crystal SrRuO3 thin films. Adv. Mater. 22, 759–762 (2010).
Haria, R. et al. Polar topologies in a ferroelastic metal membrane. Preprint at https://arxiv.org/abs/2604.28120 (2026).
Meng, M. et al. Interface-induced magnetic polar metal phase in complex oxides. Nat. Commun. 10, 5248 (2019).
Gu, Y. et al. Interfacial control of ferromagnetism in ultrathin SrRuO3 films sandwiched between ferroelectric BaTiO3 layers. ACS Appl. Mater. Interfaces 12, 6707–6715 (2020).
Kp, H. et al. Mind the gap—imaging buried interfaces in twisted oxide moirés. Adv. Mater. 38, e21189 (2026).
Chen, Z. et al. Electron ptychography achieves atomic-resolution limits set by lattice vibrations. Science 372, 826–831 (2021).
Wakonig, K. et al. PtychoShelves, a versatile high-level framework for high-performance analysis of ptychographic data. J. Appl. Crystallogr. 53, 574–586 (2020).
Lee, C.-H., Zeltmann, S. E., Yoon, D., Ma, D. & Muller, D. A. PtyRAD: a high-performance and flexible ptychographic reconstruction framework with automatic differentiation. Microsc. Microanal. 31, ozaf070 (2025).
Allen, P. B. et al. Transport properties, thermodynamic properties, and electronic structure of SrRuO3. Phys. Rev. B 53, 4393–4398 (1996).
Eom, C. B. et al. Single-crystal epitaxial thin films of the isotropic metallic oxides Sr1−xCaxRuO3 (0 ≤ x ≤ 1). Science 258, 1766–1769 (1992).
Gan, Q., Rao, R. A., Eom, C. B., Garrett, J. L. & Lee, M. Direct measurement of strain effects on magnetic and electrical properties of epitaxial SrRuO3 thin films. Appl. Phys. Lett. 72, 978–980 (1998).
Klein, L. et al. Transport and magnetization in the badly metallic itinerant ferromagnet. J. Phys. Condens. Matter 8, 10111–10126 (1996).
Hill, N. A. Why are there so few magnetic ferroelectrics? J. Phys. Chem. B 104, 6694–6709 (2000).
Spaldin, N. A. & Ramesh, R. Advances in magnetoelectric multiferroics. Nat. Mater. 18, 203–212 (2019).
Junquera, J. et al. Topological phases in polar oxide nanostructures. Rev. Mod. Phys. 95, 025001 (2023).
Han, L. et al. High-density switchable skyrmion-like polar nanodomains integrated on silicon. Nature 603, 63–67 (2022).
Ngai, J. H., Walker, F. J. & Ahn, C. H. Correlated oxide physics and electronics. Annu. Rev. Mater. Res. 44, 1–17 (2014).
Balents, L., Dean, C. R., Efetov, D. K. & Young, A. F. Superconductivity and strong correlations in moiré flat bands. Nat. Phys. 16, 725–733 (2020).
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).
Hÿtch, M. J., Snoeck, E. & Kilaas, R. Quantitative measurement of displacement and strain fields from HREM micrographs. Ultramicroscopy 74, 131–146 (1998).
Zhang, C. et al. Bayesian optimization for multi-dimensional alignment: tuning aberration correctors and ptychographic reconstructions. Microsc. Microanal. 28, 3146–3148 (2022).
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).
Kresse, G. & Furthmüller, J. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mater. Sci. 6, 15–50 (1996).
Blöchl, P. E. Projector augmented-wave method. Phys. Rev. B 50, 17953–17979 (1994).
Perdew, J. P., Burke, K. & Ernzerhof, M. Perdew, Burke, and Ernzerhof reply. Phys. Rev. Lett. 80, 891 (1998).
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).
Grimme, S., Antony, J., Ehrlich, S. & Krieg, H. A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H–Pu. J. Chem. Phys. 132, 154104 (2010).
Garanin, D. A. Self-consistent Gaussian approximation for classical spin systems: thermodynamics. Phys. Rev. B 53, 11593–11605 (1996).
Huang, A., Hung, S.-H. & Jeng, H.-T. Strain induced metal–insulator transition of magnetic SrRuO3 single layer in SrRuO3/SrTiO3 superlattice. Appl. Sci. 8, 2151 (2018).
Jiao, P., Liu, Y., Wang, X. & Chen, J. First principles investigation of Na doping effects on the structural, magnetic, and electronic properties in SrRuO3. Comput. Mater. Sci. 69, 284–288 (2013).
Lun, Y. et al. Ptychography dataset for ‘Polarization vortices in a ferromagnetic metal via twistronics’. Zenodo https://doi.org/10.5281/zenodo.21917406 (2026).