Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices. Nature 556, 43–50 (2018).

Article 
CAS 
PubMed 

Google Scholar
 

Andrei, E. Y. & MacDonald, A. H. Graphene bilayers with a twist. Nat. Mater. 19, 1265–1275 (2020).

Article 
CAS 
PubMed 

Google Scholar
 

Pryds, N., Park, D.-S., Jespersen, T. S. & Yun, S. Twisted oxide membranes: a perspective. APL Mater. 12, 010901 (2024).

Article 
CAS 

Google Scholar
 

Nuckolls, K. P. & Yazdani, A. A microscopic perspective on moiré materials. Nat. Rev. Mater. 9, 460–480 (2024).

Article 

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).

Article 
CAS 
PubMed 

Google Scholar
 

Pesquera, D., Fernández, A., Khestanova, E. & Martin, L. W. Freestanding complex-oxide membranes. J. Phys. Condens. Matter 34, 383001 (2022).

Article 
CAS 

Google Scholar
 

Zhang, J. et al. Super-tetragonal Sr4Al2O7 as a sacrificial layer for high-integrity freestanding oxide membranes. Science 383, 388–394 (2024).

Article 
CAS 
PubMed 

Google Scholar
 

Sánchez-Santolino, G. et al. A 2D ferroelectric vortex pattern in twisted BaTiO3 freestanding layers. Nature 626, 529–534 (2024).

Article 
PubMed 
PubMed Central 

Google Scholar
 

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).

Article 
CAS 
PubMed 
PubMed Central 

Google Scholar
 

Kim, M.-S. et al. Charge disproportionation at twisted SrTiO3 bilayer interface driven by local atomic registry. ACS Nano 19, 39714–39724 (2025).

Article 
CAS 
PubMed 
PubMed Central 

Google Scholar
 

Zubko, P., Catalan, G. & Tagantsev, A. K. Flexoelectric effect in solids. Annu. Rev. Mater. Res. 43, 387–421 (2013).

Article 
CAS 

Google Scholar
 

Wang, B., Gu, Y., Zhang, S. & Chen, L.-Q. Flexoelectricity in solids: progress, challenges, and perspectives. Prog. Mater Sci. 106, 100570 (2019).

Article 
CAS 

Google Scholar
 

Sha, H. et al. Polar vortex hidden in twisted bilayers of paraelectric SrTiO3. Nat. Commun. 15, 10915 (2024).

Article 
PubMed 
PubMed Central 

Google Scholar
 

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).

Article 
CAS 

Google Scholar
 

Haeni, J. H. et al. Room-temperature ferroelectricity in strained SrTiO3. Nature 430, 4 (2004).

Article 

Google Scholar
 

Yadav, A. K. et al. Observation of polar vortices in oxide superlattices. Nature 530, 198–201 (2016).

Article 
CAS 
PubMed 

Google Scholar
 

Das, S. et al. Observation of room-temperature polar skyrmions. Nature 568, 368–372 (2019).

Article 
CAS 
PubMed 

Google Scholar
 

Zabalo, A. & Stengel, M. Switching a polar metal via strain gradients. Phys. Rev. Lett. 126, 127601 (2021).

Article 
CAS 
PubMed 

Google Scholar
 

Peng, W. et al. Flexoelectric polarizing and control of a ferromagnetic metal. Nat. Phys. 20, 450–455 (2024).

Article 
CAS 

Google Scholar
 

Catalan, G. Metal poles around the bend. Nat. Phys. 20, 358–359 (2024).

Article 
CAS 

Google Scholar
 

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).

Article 
CAS 

Google Scholar
 

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).

Article 
CAS 
PubMed 

Google Scholar
 

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).

Article 
PubMed 
PubMed Central 

Google Scholar
 

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).

Article 
CAS 
PubMed 

Google Scholar
 

Kp, H. et al. Mind the gap—imaging buried interfaces in twisted oxide moirés. Adv. Mater. 38, e21189 (2026).

Article 
CAS 
PubMed 

Google Scholar
 

Chen, Z. et al. Electron ptychography achieves atomic-resolution limits set by lattice vibrations. Science 372, 826–831 (2021).

Article 
CAS 
PubMed 

Google Scholar
 

Wakonig, K. et al. PtychoShelves, a versatile high-level framework for high-performance analysis of ptychographic data. J. Appl. Crystallogr. 53, 574–586 (2020).

Article 
CAS 
PubMed 
PubMed Central 

Google Scholar
 

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).

Article 
CAS 
PubMed 

Google Scholar
 

Allen, P. B. et al. Transport properties, thermodynamic properties, and electronic structure of SrRuO3. Phys. Rev. B 53, 4393–4398 (1996).

Article 
CAS 

Google Scholar
 

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).

Article 
CAS 
PubMed 

Google Scholar
 

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).

Article 
CAS 

Google Scholar
 

Klein, L. et al. Transport and magnetization in the badly metallic itinerant ferromagnet. J. Phys. Condens. Matter 8, 10111–10126 (1996).

Article 
CAS 

Google Scholar
 

Hill, N. A. Why are there so few magnetic ferroelectrics? J. Phys. Chem. B 104, 6694–6709 (2000).

Article 
CAS 

Google Scholar
 

Spaldin, N. A. & Ramesh, R. Advances in magnetoelectric multiferroics. Nat. Mater. 18, 203–212 (2019).

Article 
CAS 
PubMed 

Google Scholar
 

Junquera, J. et al. Topological phases in polar oxide nanostructures. Rev. Mod. Phys. 95, 025001 (2023).

Article 
CAS 

Google Scholar
 

Han, L. et al. High-density switchable skyrmion-like polar nanodomains integrated on silicon. Nature 603, 63–67 (2022).

Article 
CAS 
PubMed 

Google Scholar
 

Ngai, J. H., Walker, F. J. & Ahn, C. H. Correlated oxide physics and electronics. Annu. Rev. Mater. Res. 44, 1–17 (2014).

Article 
CAS 

Google Scholar
 

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).

Article 
CAS 

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).

Article 
PubMed 
PubMed Central 

Google Scholar
 

Hÿtch, M. J., Snoeck, E. & Kilaas, R. Quantitative measurement of displacement and strain fields from HREM micrographs. Ultramicroscopy 74, 131–146 (1998).

Article 

Google Scholar
 

Zhang, C. et al. Bayesian optimization for multi-dimensional alignment: tuning aberration correctors and ptychographic reconstructions. Microsc. Microanal. 28, 3146–3148 (2022).

Article 

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).

Article 
CAS 

Google Scholar
 

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).

Article 
CAS 

Google Scholar
 

Blöchl, P. E. Projector augmented-wave method. Phys. Rev. B 50, 17953–17979 (1994).

Article 

Google Scholar
 

Perdew, J. P., Burke, K. & Ernzerhof, M. Perdew, Burke, and Ernzerhof reply. Phys. Rev. Lett. 80, 891 (1998).

Article 
CAS 

Google Scholar
 

Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).

Article 
CAS 
PubMed 

Google Scholar
 

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).

Article 
PubMed 

Google Scholar
 

Garanin, D. A. Self-consistent Gaussian approximation for classical spin systems: thermodynamics. Phys. Rev. B 53, 11593–11605 (1996).

Article 
CAS 

Google Scholar
 

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).

Article 
CAS 

Google Scholar
 

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).

Article 
CAS 

Google Scholar
 

Lun, Y. et al. Ptychography dataset for ‘Polarization vortices in a ferromagnetic metal via twistronics’. Zenodo https://doi.org/10.5281/zenodo.21917406 (2026).