A topological meron features a non-coplanar structure, whose order parameters in the core region are perpendicular to those near the perimeter. A meron is half of a skyrmion, and both have potential applications for information carrying and storage. Although merons and skyrmions in ferromagnetic materials can be readily obtained via inter-spin interactions, their behaviour and even existence in ferroelectric materials are still elusive. Here we observe using electron microscopy not only the atomic morphology of merons with a topological charge of 1/2, but also a periodic meron lattice in ultrathin PbTiO3 films under tensile epitaxial strain on a SmScO3 substrate. Phase-field simulations rationalize the formation of merons for which an epitaxial strain, as a single alterable parameter, plays a critical role in the coupling of lattice and charge. This study suggests that by engineering strain at the nanoscale it should be possible to fabricate topological polar textures, which in turn could facilitate the development of nanoscale ferroelectric devices.
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We are grateful to D. S. Ma, at Nankai University (now at Cornell University), for participation in film growth by PLD and TEM specimen preparation, and C. J. Li at Shenyang National Laboratory for Materials Science for experimental assistance with XRD and RSM. This work is supported by the Key Research Program of Frontier Sciences CAS (QYZDJ-SSW-JSC010), the National Natural Science Foundation of China (no. 51671194, no. 51971223, no. 51922100) and Shenyang National Laboratory for Materials Science (L2019R06, L2019R08, L2019F01, L2019F13). Y.L.T. acknowledges the Youth Innovation Promotion Association CAS (no. 2016177).
The authors declare no competing interests.
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Extended Data Fig. 1 The in-plane strain and lattice rotation of polar convergent meron in the PTO/SSO film.
(a) The atomic-resolved cross-sectional HAADF-STEM image of the trapezoidal domain. Yellow and red circles denote the Pb and Ti atom columns, respectively. Yellow arrows denote the directions of −δTi vectors. (b) The 2D mapping of the in-plane strain (εxx). (c) The 2D mapping of the lattice rotation (Rx). Source data
Extended Data Fig. 2 The statistical spaces of two adjacent convergent merons at the direction along stripe domain walls.
Note the most probable space is about 8 nm. Source data
Extended Data Fig. 3 Reversed Ti-displacement vector map based on atomic-resolved planar-view HAADF-STEM images of stripe domains, showing the divergent meron arrays (black dashed circles) at “tail-to-tail” domain walls (black dashed lines).
The inset is a magnified polarization map of the divergent meron corresponding to the area labeled as “1”. Source data
(a) Reversed Ti-displacement vector map, showing the relationship between convergent merons, divergent merons and antimerons. (b) and (c) The polarization maps of two antimerons corresponding to the areas labeled as “1-2” in (a). Source data
(a) The 3D domain structure of a 5 nm film. (b) and (c) Vertical cross sections corresponding to black boxes in a. A convergent IP polarization distribution is found in b and the ordinary a/c domain is found in c. (d and e) The horizontal cross-sectional slice (marked by black boxes in a) and the zoom-in images. (f) The corresponding topological density distribution. (g) The schematic diagram of the process of a meron and an antimeron to form a MAC. The bars in these figures indicate 2 nm, except that in (d). The topological density is expressed as a value per square nanometre. Source data
(a) The classification of merons (M), antimerons (A) and MACs according to their topological charges and volumes. (b) The relationship between the coercive field and the volume of the central c domain for merons (M), antimerons (A) and MACs. It is clear that the coercive fields of merons and antimerons are generally larger than those of MACs. Source data
(a-c) The Pz component, the bulk energy density and the elastic energy density of the lattice model. (d-f) Those quantities of the random model. Source data
(a) A cross-sectional low-magnification HAADF-STEM image viewed along the  direction of PTO. (b) GPA of (a) reveals the in-plane strain (εxx). (c) An atomic-resolution image corresponding to the white dashed rectangular box in (a). Yellow and red circles denote the Pb and Ti atom columns, respectively. Yellow arrows denote the directions of −δTi vectors. (d) The mapping of −δTi vectors, which are consistent with the spontaneous polarization directions of PTO. Source data
(a) A cross-sectional low-magnification HAADF-STEM image viewed along the  direction of PTO. (b) GPA of (a) reveals the in-plane strain (εxx). (c) and (e) The atomic-resolution images corresponding to two white dashed rectangular boxes in (a), respectively. Yellow and red circles denote the Pb and Ti atom columns, respectively. Yellow arrows denote the directions of −δTi vectors. (d) and (f) The mappings of −δTi vectors, which are consistent with the spontaneous polarization directions of PTO. Source data
(a) The variation of domain structure with respective to the epitaxial strain. Only c domains exist at very small tensile strain, such as 0.6%. When the strain is 1.3% (corresponding to DSO), a/c domains emerge. When the strain is very large, such as >= 3.1% (corresponding to PSO), only a domains exist. Merons were observed at intermediated strain, such as 1.8% (corresponding to GSO) and 2.3% (corresponding to SSO). (b) and (c) The densities of merons (b) and antimerons (c) as the function of the epitaxial strain. Source data
Experimental RSM data of Fig. 1b.
Off-centre shift data of Ti atoms in Fig. 2b,c.
Polarization vector matrix of the lattice model for Fig. 3.
Polarization vector and topological density matrices of the random model for Fig. 4.
IP strain and lattice rotation data in Extended Data Fig. 1b,c.
Statistical source data of Extended Data Fig. 2.
Off-centre shift data of Ti atoms in Extended Data Fig. 3.
Off-centre shift data of Ti atoms in Extended Data Fig. 4a.
Polarization vector and topological density matrices of the random model for Extended Data Fig. 5.
Source data of Extended Data Fig. 6.
Polarization vector and energy density matrices of the lattice (Extended Data Fig. 7a–c) and random (Extended Data Fig. 7d–f) models.
Off-centre shift data of Ti atoms in Extended Data Fig. 8d.
Off-centre shift data of Ti atoms in Extended Data Fig. 9d,f.
Polarization vector matrices for Extended Data Fig. 10a (*.dat files) and the statistical densities of merons and antimerons for Extended Data Fig. 10b,c (the *.xlsx file).
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Wang, Y.J., Feng, Y.P., Zhu, Y.L. et al. Polar meron lattice in strained oxide ferroelectrics. Nat. Mater. (2020). https://doi.org/10.1038/s41563-020-0694-8