The complex interplay of spin, charge, orbital and lattice degrees of freedom provides a plethora of exotic phases and physical phenomena1,2,3,4,5. In recent years, complex spin topologies have emerged as a consequence of the electronic band structure and the interplay between spin and spin–orbit coupling in materials6,7. Here we produce complex topologies of electrical polarization—namely, nanometre-scale vortex–antivortex (that is, clockwise–anticlockwise) arrays that are reminiscent of rotational spin topologies6—by making use of the competition between charge, orbital and lattice degrees of freedom in superlattices of alternating lead titanate and strontium titanate layers. Atomic-scale mapping of the polar atomic displacements by scanning transmission electron microscopy reveals the presence of long-range ordered vortex–antivortex arrays that exhibit nearly continuous polarization rotation. Phase-field modelling confirms that the vortex array is the low-energy state for a range of superlattice periods. Within this range, the large gradient energy from the vortex structure is counterbalanced by the corresponding large reduction in overall electrostatic energy (which would otherwise arise from polar discontinuities at the lead titanate/strontium titanate interfaces) and the elastic energy associated with epitaxial constraints and domain formation. These observations have implications for the creation of new states of matter (such as dipolar skyrmions, hedgehog states) and associated phenomena in ferroic materials, such as electrically controllable chirality.
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Signatures of enhanced out-of-plane polarization in asymmetric BaTiO3 superlattices integrated on silicon
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A.K.Y. and R.R. acknowledge support from the Office of Basic Energy Sciences, US Department of Energy (DE-AC02-05CH11231), for the synthesis and characterization of superlattices. R.R. and C.T.N. acknowledge support from the Office of Basic Energy Sciences, US Department of Energy (DE-AC02-05CH11231), for TEM characterization of superlattices. S.L.H. acknowledges support from the National Science Foundation under the MRSEC program (DMR-1420620). Z.H. acknowledges support from the National Science Foundation (DMR-1210588). J.D.C. acknowledges support from the Office of Basic Energy Sciences, US Department of Energy (DE-AC02-05CH11231). C.M.S. acknowledges use of the Advanced Photon Source, which was supported by the US Department of Energy, Office of Science, Office of Basic Energy Science (DE-AC02-06CH11357), for the synchrotron-based reciprocal space map studies of superlattice structures at Sector 33-BM-C beamline. A.R.D. acknowledges support from the Army Research Office (W911NF-14-1-0104). P.S. and E.A. acknowledge support from the Director, Office of Science, Office of Basic Energy Sciences, US Department of Energy (DE-AC02-05CH11231), for the synchrotron-based studies of the superlattice structures. L.R.D. acknowledges support from the US Department of Energy, Office of Basic Energy Sciences (DE-SC0012375) for the chemical analysis of the superlattice structures. D.C. acknowledges support from the National Science Foundation under the MRSEC program (DMR-1420620). A.V. and A.M.M. acknowledge support from Office of Basic Energy Sciences, US Department of Energy (DE-AC02-05CH11231). L.Q.C. acknowledges support from the National Science Foundation (DMR-1210588). L.W.M. acknowledges support from the National Science Foundation (DMR-1451219). Electron microscopy of superlattice structures was performed at the Molecular Foundry, LBNL, supported by the Office of Science, Office of Basic Energy Sciences, US Department of Energy (DE-AC02-05CH11231). Z.H. thanks J. Britson, F. Xue and J. Wang for help and discussions.
The authors declare no competing financial interests.
Extended data figures and tables
RHEED oscillations for five periods of a (SrTiO3)6/(PbTiO3)6 superlattice grown using a 20% excess Pb target are shown. The oscillations were present throughout the growth of the 100-nm-thick superlattice. a, RHEED oscillations for six unit cells of PbTiO3, followed by oscillations for six unit cells of SrTiO3, that is, one 6 × 6 superlattice period. b, RHEED intensity variation for growth of PbTiO3 from a stoichiometric PbTiO3 or 0% Pb excess target. Inset, RHEED diffraction pattern obtained at the end of growth, highlighting the rapid transition to 3D growth mode when a stoichiometric target is used (Methods).
An RBS spectrum of a 200-nm PbTiO3 film grown on a single-crystalline GdScO3(001)pc substrate is shown, along with the simulated spectrum (indicated by ‘Best fit’ in the figure) for the same structure, revealing an effective Pb:Ti ratio of 1.02:1, which is very close to ideal stoichiometry.
a, An HR-STEM HAADF image of a (SrTiO3)10/(PbTiO3)10 superlattice. Polar displacement vectors (PPD) are calculated by first finding atom positions by fits of 2D Gaussians to the image intensity. b, An example subregion from an HR-STEM image (unrotated). c, The corresponding Gaussian fit of b. d, An example of the B-site-centred five-atom clusters which are fitted simultaneously.
a, The Gaussian fit of a region of an HAADF STEM image. b, A magnified view of the boxed subregion in a, showing two overlapping unit cells, namely, the A-site-centred perovskite unit cell, and the B-site-centred perovskite unit cell. c, The mean position of each atom’s four nearest cation neighbours (MNP, mean neighbour position) is used to determine a relative sublattice offset. d, A polar displacement vector (yellow arrow for A-site and red arrow for B-site), taken as the difference between each atom (filled circle) and the MNP (‘×’), is given opposite sign for A- and B-sites to maintain a consistent direction.
Extended Data Figure 5 Polar displacement vector map and phase-field simulation of a (SrTiO3)10/(PbTiO3)10 superlattice.
a, The polar displacement vector map (black arrows) corresponding to the (SrTiO3)10/(PbTiO3)10 superlattice in Fig. 2a is shown overlaid on the colourized (∇ × PPD). b, A magnified view of the highlighted polar vortex shows the smooth rotation increasing towards a vortex core. The extremes of the colour scale of the curl vector are −16.75 pm nm−1 and 16.75 pm nm−1, with blue and red colours indicating negative and positive values of curl (units for polar displacement are in pm). c, A cross-section of the phase-field simulation for the (SrTiO3)10/(PbTiO3)10 superlattice showing the polarization vectors (red arrows) with interlayer alignment. This sample exhibits preference for alignment of matched rotations (indicated by blue arrows), but a phase shift of anti-alignment is also found (indicated by ‘Anti-rotation boundary’ in the figure). Such defects are found occasionally. d, A subregion of the simulation, highlighting a vortex–antivortex pair with the curl of the polarization overlaid.
a, Cross-sectional DF-TEM of a (SrTiO3)10/(PbTiO3)10 superlattice taken along the pc zone axis (inset shows the SAED pattern and the g-vector for image formation). b, Cross-sectional DF-TEM of the same (SrTiO3)10/(PbTiO3)10 superlattice sample taken along the orthogonal pc zone axis (inset as a). In both cases, clear vortex–antivortex structures are observed.
Extended Data Figure 7 Differentiation of vortex structure from a flux-closure domain structure using phase-field simulations and HR-STEM polar displacement mapping.
a, HR-STEM polar displacement vector map of a single vortex in a (SrTiO3)10/(PbTiO3)10 superlattice. b, Phase-field-calculated polarization vector map of a single vortex in a (SrTiO3)10/(PbTiO3)10 superlattice. c, Total energy density variation within a single vortex in a (SrTiO3)10/(PbTiO3)10 superlattice calculated from phase-field simulations. d, HR-STEM polar displacement vector map of a 20-nm PbTiO3 layer in a heterostructure having the same layer stacking as in ref. 19. e, Phase-field-calculated polarization vector map of a flux-closure domain in a 20-nm PbTiO3 heterostructure (red, blue, green and pink arrows in d and e indicate the direction of net polarization within their respective domains). f, Total energy density variation within a flux-closure domain calculated using phase-field simulations.
The phase-field-calculated total energy for various phases competing with a vortex phase is depicted. See Methods for details.
a, Polar displacement vector map of a (SrTiO3)10/(PbTiO3)10 superlattice obtained from the same HR-STEM data set as Fig. 2a, showing the polar displacement vectors at all atom positions. b, A magnified image of a single vortex. c, Polar displacement vector map of the same region as b, but plotting only Pb (A-site) centred displacements (like Fig. 2a). d, Polar displacement vector map of a single vortex showing the raw displacement vectors.
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Yadav, A., Nelson, C., Hsu, S. et al. Observation of polar vortices in oxide superlattices. Nature 530, 198–201 (2016). https://doi.org/10.1038/nature16463
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