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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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.
The authors declare no competing financial interests.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
a, Planar-view dark-field STEM imaging shows the widespread occurrence of nanometer-size (~8 nm) skyrmion formation along the  and  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.
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).
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.
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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