Local negative permittivity and topological phase transition in polar skyrmions


Topological solitons such as magnetic skyrmions have drawn attention as stable quasi-particle-like objects. The recent discovery of polar vortices and skyrmions in ferroelectric oxide superlattices has opened up new vistas to explore topology, emergent phenomena and approaches for manipulating such features with electric fields. Using macroscopic dielectric measurements, coupled with direct scanning convergent beam electron diffraction imaging on the atomic scale, theoretical phase-field simulations and second-principles calculations, we demonstrate that polar skyrmions in (PbTiO3)n/(SrTiO3)n superlattices are distinguished by a sheath of negative permittivity at the periphery of each skyrmion. This enhances the effective dielectric permittivity compared with the individual SrTiO3 and PbTiO3 layers. Moreover, the response of these topologically protected structures to electric field and temperature shows a reversible phase transition from the skyrmion state to a trivial uniform ferroelectric state, accompanied by large tunability of the dielectric permittivity. Pulsed switching measurements show a time-dependent evolution and recovery of the skyrmion state (and macroscopic dielectric response). The interrelationship between topological and dielectric properties presents an opportunity to simultaneously manipulate both by a single, and easily controlled, stimulus, the applied electric field.

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Fig. 1: Macroscopic demonstration of stabilization of the negative capacitance of polar skyrmions.
Fig. 2: Phase-field, second-principles and macroscopic calculations of the negative permittivity of the skyrmion structure at 300 K in a [(SrTiO3)16/(PbTiO3)16]8 superlattice.
Fig. 3: Measurement of local polarization, electric field and local potential energy of the polar skyrmion using SCBED.
Fig. 4: Electric field control of the topological phase transition of the polar skyrmion of the [(SrTiO3)16/(PbTiO3)16]8 superlattice.
Fig. 5: Generation of a transient state with time-dependent capacitance under out-of-plane measurement of the skyrmion.

Data availability

All data supporting the findings of this study are available within the paper and/or are available from the authors upon reasonable request.


  1. 1.

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

    CAS  Article  Google Scholar 

  2. 2.

    Yadav, A. K. et al. Spatially resolved steady state negative capacitance. Nature 565, 468–471 (2019).

    CAS  Article  Google Scholar 

  3. 3.

    Kosterlitz, J. M. & Thouless, D. J. Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C 6, 1181–1203 (1973).

    CAS  Article  Google Scholar 

  4. 4.

    Nagaosa, N. & Tokura, Y. Topological properties and dynamics of magnetic skyrmions. Nat. Nanotechnol. 8, 899–911 (2013).

    CAS  Article  Google Scholar 

  5. 5.

    Rößler, U. K., Bogdanov, A. N. & Pfleiderer, C. Spontaneous skyrmion ground states in magnetic metals. Nature 442, 797–801 (2006).

    Article  CAS  Google Scholar 

  6. 6.

    Mühlbauer, S. et al. Skyrmion lattice in a chiral magnet. Science 323, 915–919 (2009).

    Article  CAS  Google Scholar 

  7. 7.

    Nahas, Y. et al. Discovery of stable skyrmionic states in ferroelectric nanocomposites. Nat. Commun. 6, 8542 (2015).

    CAS  Article  Google Scholar 

  8. 8.

    Zhang, Q. et al. Nanoscale bubble domains and topological transitions in ultrathin ferroelectric films. Adv. Mater. 29, 1702375 (2017).

    Article  CAS  Google Scholar 

  9. 9.

    Gregg, J. M. Exotic domain states in ferroelectrics: searching for vortices and skyrmions. Ferroelectrics 433, 74–87 (2012).

    CAS  Article  Google Scholar 

  10. 10.

    Naumov, I. I., Bellaiche, L. & Fu, H. Unusual phase transitions in ferroelectric nanodisks and nanorods. Nature 432, 737–740 (2004).

    CAS  Article  Google Scholar 

  11. 11.

    Salahuddin, S. & Datta, S. Use of negative capacitance to provide voltage amplification for low power nanoscale devices. Nano Lett. 8, 405–410 (2008).

    CAS  Article  Google Scholar 

  12. 12.

    Salahuddin, S. & Datta, S. Can the subthreshold swing in a classical FET be lowered below 60 mV/decade? In 2008 IEEE International Electron Devices Meeting 1–4 (IEEE, 2008).

  13. 13.

    Theis, T. N. & Solomon, P. M. It’s time to reinvent the transistor! Science 327, 1600–1601 (2010).

    CAS  Article  Google Scholar 

  14. 14.

    Zubko, P. et al. Negative capacitance in multidomain ferroelectric superlattices. Nature 534, 524–528 (2016).

    CAS  Article  Google Scholar 

  15. 15.

    Appleby, D. J. R. et al. Experimental observation of negative capacitance in ferroelectrics at room temperature. Nano Lett. 14, 3864–3868 (2014).

    CAS  Article  Google Scholar 

  16. 16.

    Íñiguez, J., Zubko, P., Luk’yanchuk, I. & Cano, A. Ferroelectric negative capacitance. Nat. Rev. Mater. 4, 243–256 (2019).

    Article  Google Scholar 

  17. 17.

    Hoffman, M. et al. Unveiling the double-well energy landscape in a ferroelectric layer. Nature 565, 464–467 (2019).

    Article  CAS  Google Scholar 

  18. 18.

    Bratkovsky, A. M. & Levanyuk, A. P. Depolarizing field and “real” hysteresis loops in nanometer-scale ferroelectric films. Appl. Phys. Lett. 89, 253108 (2006).

    Article  CAS  Google Scholar 

  19. 19.

    Wang, X. et al. Van der Waals negative capacitance transistors. Nat. Commun. 10, 3037 (2019).

    Article  CAS  Google Scholar 

  20. 20.

    Lines, M. E. & Glass, A. M. Principles and Applications of Ferroelectrics and Related Materials (Clarendon Press, 1977).

  21. 21.

    Wu, X., Yang, E. S. & Evans, H. L. Negative capacitance at metal-semiconductor interfaces. J. Appl. Phys. 68, 2845–2848 (1990).

    CAS  Article  Google Scholar 

  22. 22.

    Hoffman, M. et al. Ferroelectric negative capacitance domain dynamics. J. Appl. Phys. 123, 184101 (2018).

    Article  CAS  Google Scholar 

  23. 23.

    Damodaran, A. R. et al. Large polarization gradients and temperature-stable responses in compositionally-graded ferroelectrics. Nat. Commun. 8, 14961 (2017).

    Article  Google Scholar 

  24. 24.

    O’Neill, D., Bowman, R. M. & Gregg, J. M. Dielectric enhancement and Maxwell–Wagner effects in ferroelectric superlattice structures. Appl. Phys. Lett. 77, 1520–1522 (2000).

    Article  Google Scholar 

  25. 25.

    Catalan, G., O’Neill, D., Bowman, R. M. & Gregg, J. M. Relaxor features in ferroelectric superlattices: a Maxwell–Wagner approach. Appl. Phys. Lett. 77, 3078–3080 (2000).

    CAS  Article  Google Scholar 

  26. 26.

    Arai, H. et al. Fundamental characteristics of superconducting fault current limiter using LC resonance circuit. IEEE Trans. Appl. Supercond. 16, 642–645 (2006).

    CAS  Article  Google Scholar 

  27. 27.

    Luk’yanchuk, I. et al. Electrodynamics of ferroelectric films with negative capacitance. Phys. Rev. B 98, 024107 (2018).

    Article  Google Scholar 

  28. 28.

    Kamba, S. et al. Dielectric dispersion of the relaxor PLZT ceramics in the frequency range 20 Hz-100 THz. J. Phys. Condens. Matter 12, 497–519 (2000).

    CAS  Article  Google Scholar 

  29. 29.

    Bovtun, V. et al. Comparison of the dielectric response of relaxor PbMg1/3Nb2/3O3 ceramics and single crystals. Integr. Ferroelectr. 69, 3–10 (2005).

    CAS  Article  Google Scholar 

  30. 30.

    Kittel, C.Introduction to Solid State Physics (Wiley, 1966).

  31. 31.

    Watanabe, Y. in Ferroelectric Thin Films: Basic Properties and Device Physics for Memory Applications (eds Okuyama, M. & Ishibashi, Y.) 177–199 (Springer, 2005).

  32. 32.

    Zubko, P., Stucki, N., Lichtensteiger, C. & Triscone, J.-M. X-ray diffraction studies of 180° ferroelectric domains in PbTiO3/SrTiO3 superlattices under an applied electric field. Phys. Rev. Lett. 104, 187601 (2010).

    CAS  Article  Google Scholar 

  33. 33.

    Boulle, A., Infante & Lemée, N. Diffuse X-ray scattering from 180° ferroelectric stripe domains: polarization-induced strain, period disorder and wall roughness. J. Appl. Cryst. 49, 845–855 (2016).

    CAS  Article  Google Scholar 

  34. 34.

    Okamura et al. Transition to and from the skyrmion lattice phase by electric fields in a magnetoelectric compound. Nat. Commun. 7, 12669 (2016).

    CAS  Article  Google Scholar 

  35. 35.

    Fujishiro et al. Topological transitions among skyrmion- and hedgehog-lattice states in cubic chiral magnets. Nat. Commun. 10, 1059 (2019).

    CAS  Article  Google Scholar 

  36. 36.

    Yang et al. Electric field driven evolution of topological domain structure in hexagonal manganites. Phys. Rev. B 96, 144103 (2017).

    Article  Google Scholar 

  37. 37.

    Zong et al. Evidence for topological defects in a photoinduced phase transition. Nat. Phys. 15, 27–31 (2019).

    CAS  Article  Google Scholar 

  38. 38.

    Schulz, A. et al. Emergent electrodynamics of skyrmions in a chiral magnet. Nat. Phys. 8, 301–304 (2012).

    CAS  Article  Google Scholar 

  39. 39.

    Marks, R. B. A multiline method of network analyzer calibration. IEEE Trans. Microw. Theory Tech. 39, 1205–1215 (1991).

    Article  Google Scholar 

  40. 40.

    Williams, D. F., Wang, J. C. M. & Arz, U. An optimal vector-network analyzer calibration algorithm. IEEE Trans. Microw. Theory Tech. 51, 2391–2401 (2003).

    Article  Google Scholar 

  41. 41.

    Li, Q. et al. Simultaneous scanning near-field optical and X-ray diffraction microscopy for correlative nanoscale structure–property characterization. J. Synchrotron Radiat. 26, 1790–1796 (2019).

    Article  Google Scholar 

  42. 42.

    Towns, J. et al. XSEDE: accelerating scientific discovery. Comput. Sci. Eng. 16, 62–74 (2014).

    Article  CAS  Google Scholar 

  43. 43.

    Nystrom, N. A., Levine, M. J., Roskies, R. Z. & Scott, J. R. Bridges: a uniquely flexible HPC resource for new communities and data analytics. In Proc. 2015 XSEDE Conference: Scientific Advancements Enabled by Enhanced Cyberinfrastructure Vol. 30, 1–8 (2015).

  44. 44.

    Chen, L.-Q. Phase‐field method of phase transitions/domain structures in ferroelectric thin films: a review. Appl. Phys. Lett. 91, 1835–1844 (2008).

    CAS  Google Scholar 

  45. 45.

    Sheng, G., Li, Y. L. & Zhang, J. X. A modified Landau–Devonshire thermodynamic potential for strontium titanate. Appl. Phys. Lett. 96, 232902 (2010).

    Article  CAS  Google Scholar 

  46. 46.

    Tagantsev, A. et al. Landau expansion for ferroelectrics: which variable to use? Ferroelectrics 375, 19–27 (2008).

    CAS  Article  Google Scholar 

  47. 47.

    Wojdeł, J. C., Hermet, P., Ljungberg, M. P., Ghosez, P. & Íñiguez, J. First-principles model potentials for lattice-dynamical studies: general methodology and example of application to ferroic perovskite oxides. J. Phys. Condens. Matter 25, 305401 (2013).

    Article  CAS  Google Scholar 

  48. 48.

    Wojdeł, J. C. & Íñiguez, J. Ferroelectric transitions at ferroelectric domain walls found from first-principles. Phys. Rev. Lett. 112, 247603 (2014).

    Article  CAS  Google Scholar 

  49. 49.

    Gonçalves, M. A. P. et al. Theoretical guidelines to create and tune electric skyrmion bubbles. Sci. Adv. 5, eaau7023 (2019).

    Article  CAS  Google Scholar 

  50. 50.

    B. Berg, B. & Lüscher, M. Definition and statistical distributions of a topological number in the lattice O(3) σ-model. Nucl. Phys. B. 190, 412–424 (1981).

    Article  Google Scholar 

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This work was supported by the Quantum Materials program of the Office of Basic Energy Sciences, US Department of Energy (DE-AC02-05CH11231). M.A.P.G. and J.Í. were funded by the Luxembourg National Research Fund through the CORE program (Grant FNR/C15/MS/10458889 NEWALLS). J.W.F., V.A.S., H.W. and L.W.M. acknowledge support from the US Department of Energy, Office of Science, Office of Basic Energy Sciences (Award number DE-SC-0012375) for the development and study of ferroic heterostructures. The phase-field simulations at Penn State were supported as part of the Computational Materials Sciences Program funded by the US Department of Energy, Office of Science, Basic Energy Sciences (Award number DE-SC0020145) and the Extreme Science and Engineering Discovery Environment (XSEDE) cluster, which is supported by the National Science Foundation (Grant ACI-1548562), and specifically, it used the Bridges system, which is supported by the NSF (Award number ACI-1445606) at the Pittsburgh Supercomputing Center (PSC), under allocation DMR170006. F.G.O., P.G.F. and J.J. acknowledge financial support from the Spanish Ministry of Economy and Competitiveness (Grants FIS2015-64886-C5-2-P and PGC2018-096955-B-C41), and P.G.F. acknowledges support from Ramón y Cajal Foundation (Grant RyC-2013-12515). V.A.S., M.R.M., S.D., H.W., Z.Z., J.W.F. and H.Z. acknowledge use of the Advanced Photon Source, a US Department of Energy, Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under contract no. DE-AC02-06CH11357. V.A.S. and H.W. thank Q. Li and S. Marks for kind assistance in operating the XNOM station at the 7-ID-C beamline of APS. Y.T.S. and D.A.M. acknowledge support from the AFOSR Hybrid Materials MURI (Award number FA9550-18-1-0480). We acknowledge the electron microscopy facility of the National Science Foundation (Award numbers DMR-1719875 and DMR-1429155). E.J.M., C.J.L. and N.D.O. acknowledge J. C. Booth for establishing the high-frequency testing facility at NIST, funding E.J.M. and developing the original on-wafer techniques.

Author information




S.D. and R.R. designed the experiments. S.D. synthesized and characterized the trilayer and superlattice samples. S.D. facbricated the devices and S.D. and S. Saremi performed all macroscopic electrical measurements. Y.T.S. performed EMPAD characterization of the samples under the supervision of D.A.M. M.A.P.G., F.G.O., P.G.F., J.Í. and J.J. performed the second-principles calculations. Z.H. performed and analysed the phase-field calculations of the samples. V.A.S., M.R.M. and S.D. carried out the reciprocal space map studies of these samples using laboratory X-ray diffraction and synchrotron X-ray diffraction. V.A.S. carried out the electric field-dependent reciprocal space map studies of these samples using synchrotron X-ray diffraction. E.P. and S.D. performed electric field-dependent pulse dynamics studies. E.J.M. designed the layout, fabricated the in-plane devices and performed the microwave measurements, analysed the data and wrote the microwave section. N.D.O. and A.M.H. supervised the microwave research and N.D.O. wrote the algorithms that calibrated and processed the data, and wrote the analysis tools. C.J.L. wrote the algorithms that generated the layout, optimized the device performances and calibrated and processed the data. S.D., Z.H., V.A.S., J.J., L.Q.C. and R.R. analysed the data and co-wrote the manuscript. R.R., L.Q.C., L.W.M., S. Salahuddin and J.J. supervised the research. All authors contributed to the discussion and manuscript preparation.

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Correspondence to S. Das or R. Ramesh.

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Extended data

Extended Data Fig. 1 Polar skyrmion structure in [(SrTiO3)16/(PbTiO3)16]8 superlattice.

a, Planar-view dark-field STEM imaging shows the widespread occurrence of nanometer-size (~8 nm) skyrmion formation along the [100] and [010] directions. b, Second-principles calculations of the skyrmion structure in the superlattice. The hedgehog-like skyrmion (Topological number +1) at the top and bottom interface between SrTiO3 and PbTiO3, and the Bloch-like skyrmion (topological number +1) at the central plane in PbTiO3 are clearly visible. The arrows represent the normalized electric dipole moments.

Extended Data Fig. 2 Synchrotron-based symmetric three-dimensional reciprocal space mapping about the (002) diffraction condition of the SrTiO3 substrate.

Reciprocal space mapping of a, (SrTiO3)16 /(PbTiO3)16/(SrTiO3)16 trilayer, b, [(PbTiO3)16/(SrTiO3)16]3 superlattice, c, [(PbTiO3)16/(SrTiO3)16]5 superlattice and d, [(PbTiO3)16/(SrTiO3)16]8 superlattice. The satellite peaks along (100) direction (side-lobes; white arrow) which correspond to the skyrmions.

Extended Data Fig. 3 Loss Tangent as a function of electric field.

The loss tangent (tanδ) as a function of electric field for a, [(SrTiO3)16/(PbTiO3)16]8 superlattice, b, [(SrTiO3)16/(PbTiO3)16]5 superlattice, c, [(SrTiO3)16/(PbTiO3)16]3 superlattice, d, (SrTiO3)16/(PbTiO3)16/(SrTiO3)16 trilayer which is in the skyrmion state compared to its constituents e, 50nm SrTiO3 (brown) and f, 50 nm-thick PbTiO3 layer.

Extended Data Fig. 4 Measured out-of-plane dielectric permittivity as a function of electric field for the.

a, [(SrTiO3)16/(PbTiO3)16]8 superlattice (skyrmion sample) compared with 50nm SrTiO3 and 50 nm PbTiO3 layers on SrTiO3 (001) and b, [(SrTiO3)16/(PbTiO3)16]8 superlattice (vortex sample) compared with 50nm SrTiO3 and 50 nm PbTiO3 layers on DyScO3 (110) substrate. The enhancement of dielectric permittivity for the skyrmion sample ~ 20X larger compared to its series permittivity whereas it only ~ 9X for vortex samples.

Extended Data Fig. 5 Out-of-plane dielectric permittivity as a function of frequency measured on three different devices.

a, Capacitance vs Frequency of [(SrTiO3)16/(PbTiO3)16]8 superlattice (superlattice total thickness: 100nm, capacitor size of the measurement: 25µm) compared with 50nm PbTiO3 (orange), 50nm SrTiO3 layers (brown) (capacitor size of the measurement: 25 µm) at electric field E=0 kV/cm and E=1000 kV/cm (light green). b, Tangent loss (Tan δ) vs Frequency of [(SrTiO3)16/(PbTiO3)16]8 superlattice compared with 50nm PbTiO3 (orange) and 50nm SrTiO3 (brown) layers. It shows the Tanδ which is approximately an order of magnitude smaller than for samples which show Maxwell-Wagner (MW) effects in the literature (Ref. 24,25). c, Dielectric permittivity vs Frequency of [(SrTiO3)16/(PbTiO3)16]8 superlattice which is in the skyrmion state measured in three different capacitor devices compared with 50nm PbTiO3 (orange) and 50nm SrTiO3 (brown) layers at electric field E=0 kV/cm and E=1000 kV/cm (light green). d, Capacitance (left axis), permittivity (right axis) as a function of electric field of [(SrTiO3)16/(PbTiO3)16]8 superlattice which is in the skyrmion state at 10kHz (blue), 100kHz (red) and 1MHz (green). The capacitance and permittivity change by ~ 4% over the measured frequency range. This reveals that the calculated negative permittivity from this macroscopic measurement is independent of measured frequency in this range.

Extended Data Fig. 6 In-plane dielectric permittivity of [(SrTiO3)16/(PbTiO3)16]8 superlattice at microwave frequencies from 100 MHz to 10 GHz.

a, A schematic of co-planar Waveguide (CPW) transmission lines for microwave-frequency permittivity characterization. b, A cross-sectional view of the CPW illustrating the electric fields between the center conductor and ground planes. c, A circuit model of the CPW transmission line showing how the total measured capacitance of the transmission line (C) is comprised of a portion related to the permittivity of the SrTiO3 substrate (CSUB), and a portion related to the permittivity of the superlattice (CFILM). d, The measured capacitance of the CPW transmission lines on the superlattice (purple), and SrTiO3 substrate (red) are shown. Finite element simulations and the measurements of the bare SrTiO3 substrate are used to analyze and isolate the portion of the total capacitance which is related to the permittivity of the superlattice (blue). e, The real part of the in-plane dielectric permittivity (left-hand y-axis) and the loss tangent (right-hand y-axis) are determined from the capacitance and conductance measurements via a mapping function obtained from 2D finite element modeling of the CPW structures; f, Cole-Cole model of the in-plane complex permittivity of the superlattice from 100 MHz to 10 GHz (Supplementary Note 2).

Extended Data Fig. 7 Phase field simulation of the switching dynamics of skyrmions under different externally applied fields.

The size of the skyrmion decreases progressively until all the skyrmions implode at ~1500 kV/cm. a, positive field. b, negative field.

Extended Data Fig. 8 Second principle simulation of the evolution of the local polarization texture at the central layer of the PbTiO3 under different external fields.

The arrows represent the polarization on the x-y plane, while the colors represent the component of the out-of-plane polarization as indicated in the legend. The snapshots correspond to Monte Carlo simulations at different electric fields, where the initial configuration for a new magnitude of the field was the relaxed structure from the previous field. a, positive field. b, negative field. For the sake of clarity let us assume that the core of the skyrmion has a polarization pointing along the +z direction, while it points down along −z at the periphery, at null external field. Let us assume also that we switch on an external field parallel to the polarization at the center of the skyrmion, as it is done in panel (a) (positive values of the field). Then, as it is clearly visible from the snapshots at 250 kV/cm and 500 kV/cm, the domain with a polarization parallel to the external field increases at the expenses of the other. The domain wall of the skyrmion has moved, but there will be always a domain wall between the core and the periphery of the skyrmion where the out-of-plane polarization vanishes (see the regions plotted in white delimiting the surface of the skyrmion). There is a tendency of the domain wall to polarize with the appearance of a Bloch component to minimize the energy (see the arrows within the white regions of the panels)48. This Bloch components of the polarization are perpendicular to the applied field and do not couple with it. Therefore, they will be always present, independently of the sizes of the domains up and down. These Bloch components closing a loop around the domain wall are precisely at the origin of the non-trivial topological textures (the vortex they form determine why the skyrmion number is +1). The larger the electric field, the larger the size of the domain that aligns with it (with the criterion chosen for this example, the skyrmion expands). For a critical field, this domain will cover all space and the transition to a monodomain state will be completed. Right before the transition, a singularity appears: the stereographic projection of our dipole patterns does not completely wrap the unit sphere. That is exactly the point where the skyrmion number changes from +1 (non-trivial topological structure) to 0 (trivial case in the monodomain configuration).

Extended Data Fig. 9 Tunable skyrmion size under electric field as a basis for dielectric enhancement [(SrTiO3)16/(PbTiO3)16]8 superlattice.

a, Experimental line cuts through skyrmion diffuse scattering showing 1st and 2nd order satellites, including their suppression and enhancement, respectively, under external electric field. b, Electric field dependency of 1st and 2nd order satellites shown in a, determined experimentally. c, Phase field model of vertical polarization distribution versus electric field depicting planar cuts (80 nm × 80 nm) through the center and parallel with the PbTiO3 layer. Insets show FFT averages for all PbTiO3 layers of the superlattice, which indicates the emergence of 2nd order diffuse scattering skyrmion ring in addition to the dominant 1st order diffuse scattering skyrmion ring. The 2nd order ring intensity is enhanced under application of electric field relative to without field. d, Comparison between the experimentally estimated and phase-field calculated skyrmion areal fraction inside PbTiO3 layers of the superlattice. g, Comparison between the experimentally measured (blue) and phase- field calculated (red) dielectric permittivity of the superlattice, which is closely mapped with electric field dependent normalized satellite intensity of skyrmion calculated from reciprocal space (dark gray, right axis).

Extended Data Fig. 10 Temperature control of skyrmion of [(SrTiO3)16/(PbTiO3)16]8 superlattice.

a-c, Temperature dependent three-dimensional, synchrotron-based symmetric reciprocal space mapping about the (002) diffraction condition of the SrTiO3 substrate. The satellite peaks along (100) direction (side-lobes) which correspond to the skyrmions. d, The in-plane satellite peaks intensity (skyrmion peaks) diminishing upon heating and completely disappeared ~600K. e, Comparison between the temperature-dependent experimentally measured and phase- field calculated out-of-plane dielectric permittivity of the superlattice.

Supplementary information

Supplementary Information

Supplementary Notes 1–4, Figs. 1–10, Tables 1 and 2, and refs. 1–7.

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Das, S., Hong, Z., Stoica, V.A. et al. Local negative permittivity and topological phase transition in polar skyrmions. Nat. Mater. (2020). https://doi.org/10.1038/s41563-020-00818-y

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