From Vortex Labyrinths to Polar Bubbles—A Mean-Field Perspective

Partially screened ultra-thin PZT films are known to host a variety of non-trivial domain topologies ranging from vortex stripes^1-2 and labyrinth patterns³ to bi-meron arrays and bubble lattices^1,3-5. The stability of any of these polar textures critically depends on numerous external parameters^1-5 (e.g. misfit strain, temperature, built-in or applied bias field, external stress, screening conditions), and even a slight change of the setting can trigger a topological transition. At the same time, the complexity of the phase canvas is significantly reduced since all of the aforementioned states are, in fact, related via universal mechanisms of phase separation kinetics³.<br/><br/>Here, we further explore such a relationship using a simple, symmetry-based, mean-field model³. While being analytically traceable, our model allows not only to reproduce the topological temperature-electric field phase diagram obtained from effective Hamiltonian simulations but also to correctly predict the behavior of the system under changing screening or mechanical boundary conditions. Thereby, this approach aids in navigating the multi-dimensional space of polar modulated phases and allows to reveal simple rules governing the stability of various polar topological defects. Lastly, we discuss the relation of the developed model to interatomic interactions and the possible bridges between spin and polar topologies it entails.<br/><br/>This work is supported by the DARPA Grant No. HR0011727183-D18AP00010 (TEE Program), the DARPA Grant No. HR0011-15-2-0038 (MATRIX program) and the Vannevar Bush Faculty Fellowship (VBFF) Grant No. N00014-20-1-2834 from the Department of Defense. Computations were made possible thanks to the use of the Arkansas High-Performance Computing Center and the Arkansas Economic Development Commission. The research at the University of New South Wales (UNSW) was partially supported by an Australian Research Council (ARC) Discovery Project and supported by the Australian Research Council Centre of Excellence in Future Low- Energy Electronics Technologies (project number CE170100039). We also thank the support from a FLEET Seed Grant and the Women in FLEET Fellowship.<br/><br/>[1] Kornev, I., Fu, H., Bellaiche, L. Phys. Rev. Lett. <b>93</b>, 196104 (2004)<br/>[2] Yadav, A., Nelson, C., Hsu, S. <i>et al.</i> <i>Nature</i> <b>530, </b>198–201 (2016)<br/>[3] Nahas, Y., Prokhorenko, S., Zhang, Q. <i>et al.</i> <i>Nat. Commun.</i> <b>11, </b>5779 (2020)<br/>[4] Zhang, Q., Xie, L., Liu, G. <i>et al.</i> <i>Adv. Mater.</i> 29, 1702375 (2017)<br/>[5] Zhang, Q., Prokhorenko, S., Nahas, Y. <i>et al</i>. <i>Adv. Funct. Mater.</i> 2019, 29, 1808573