Phys. Rev. B 111, 094436 (2025)
Hopfions in chiral magnetic trilayers
X. C. Hu1, X. S. Wang2, C. Wang3, and X. R. Wang4,1,*
1 Physics Department, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong.
2 School of Physics and Electronics, Hunan University, Changsha 410082, China.
3 Center for Joint Quantum Studies and Department of Physics, School of Science, Tianjin University, Tianjin 300350, China.
4 School of Science and Engineering, Chinese University of Hong Kong (Shenzhen), Shenzhen 51817, China.
* phxwan@ust.hk
Abstract
Magnetic hopfions are three-dimensional localized topological spin textures characterized by the Hopf invariant. Stable hopfions exist in chiral magnetic trilayers consisting of a middle layer with a weaker perpendicular magnetic anisotropy (PMA) and two capping layers with a stronger PMA. The middle layer holds a hopfion, and the capping layers prevent the hopfion from dilating. However, the existence criteria for a stable hopfion are not adequately known. Here, we numerically obtain the phase diagram for stable hopfions in the plane of the middle layer thickness and dimensionless parameter κ which measures the relative strength of the Dzyaloshinskii-Moriya interaction (a chiral exchange interaction) compared with the interactions preferring collinear spin alignment (due to the ferromagnetic exchange interaction and the anisotropy). κ is also the parameter determining the locality of skyrmions in chiral magnetic films. Our results deepen our understanding of hopfions and provide practical guidance to experimentalists for seeking materials and nanostructures for stable hopfions.
