Margaret R. McCarter, Kook Tae Kim, Vladimir A. Stoica, Sujit Das, Christoph Klewe, Elizabeth P. Donoway, David M. Burn, Padraic Shafer, Fanny Rodolakis, Mauro A. P. Gonçalves, Fernando Gómez-Ortiz, Jorge Íñiguez, Pablo García-Fernández, Javier Junquera, Stephen W. Lovesey, Gerrit van der Laan, Se Young Park, John W. Freeland, Lane W. Martin, Dong Ryeol Lee, Ramamoorthy Ramesh, and Universidad de Cantabria
An escalating challenge in condensed-matter research is the characterization of emergent order-parameter nanostructures such as ferroelectric and ferromagnetic skyrmions. Their small length scales coupled with complex, three-dimensional polarization or spin structures makes them demanding to trace out fully. Resonant elastic x-ray scattering (REXS) has emerged as a technique to study chirality in spin textures such as skyrmions and domain walls. It has, however, been used to a considerably lesser extent to study analogous features in ferroelectrics. Here, we present a framework for modeling REXS from an arbitrary arrangement of charge quadrupole moments, which can be applied to nanostructures in materials such as ferroelectrics. With this, we demonstrate how extended reciprocal space scans using REXS with circularly polarized x rays can probe the three-dimensional structure and chirality of polar skyrmions. Measurements, bolstered by quantitative scattering calculations, show that polar skyrmions of mixed chirality coexist, and that REXS allows valuation of relative fractions of right- and left-handed skyrmions. Our quantitative analysis of the structure and chirality of polar skyrmions highlights the capability of REXS for establishing complex topological structures toward future application exploits. M. R. M. and R. R. were supported by the Quantum Materials program from the Office of Basic Energy Sciences, U.S. Department of Energy (DE-AC02-05CH11231). V. A. S., J. W. F., and L. W. M. acknowledge the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award No. DE-SC-0012375 for support to study complex-oxide heterostructure with x-ray scattering. L. W. M. and R. R. acknowledge partial support from the Army Research Office under the ETHOS MURI via cooperative agreement W911NF-21-2-0162. J. Í. acknowledge financial support from the Luxembourg National Research Fund through project FNR/C18/MS/12705883/REFOX. Diamond Light Source, UK, is acknowledged for beam time on beam line I10 under proposal NT24797. K. T. K., S. Y. P., and D. R. L. acknowledge support from the National Research Foundation of Korea, under Grant No. NRF-2020R1A2C1009597, NRF-2019K1A3A7A09033387, and NRF-2021R1C1C1009494. M. A. P. G. acknowledges support by the Czech Science Foundation (Project No. 19-28594X). This research used resources of the Advanced Light Source, a U.S. DOE Office of Science User Facility under Contract No. DE-AC02-05CH11231. This research used resources of the Advanced Photon Source, a U.S. Department of Energy (DOE) Office of Science User Facility at Argonne National Laboratory and is based on research supported by the U.S. DOE Office of Science-Basic Energy Sciences, under Contract No. DE-AC02-06CH11357. S. D. gratefully acknowledges a start-up grant from Indian Institute of Science, Bangalore, India. F. G.-O., P. G.-F., and J. J. acknowledge financial support from Grant No. PGC2018-096955-B-C41 funded by MCIN/AEI/10.13039/501100011033 and by ERDF “A way of making Europe,” by the European Union. F. G.-O. acknowledges financial support from Grant No. FPU18/04661 funded by MCIN/AEI/10.13039/501100011033