Magnon spectrum of Bloch hopfion beyond ferromagnetic resonance

Abstract

With the development of new nanofabrication technologies and measurement techniques, the interest of researchers is moving toward 3D structures and 3D magnetization textures. Special attention is paid to the topological magnetization textures, particularly hopfions. In this paper, we investigate the magnetization dynamics of the hopfion through the numerical solution of the eigenvalue problem. We show that the spectrum of spin-wave modes of the hopfion is much richer than those attainable in ferromagnetic resonance experiments or time-domain simulations reported so far. We identified four groups of modes that differ in the character of oscillations (clockwise or counter-clockwise rotation sense), the position of an average amplitude localization along the radial direction, and different oscillations in the vertical cross section. The knowledge of the full spin-wave spectrum shall help in hopfion identification, understanding of the interaction between spin waves and hopfion dynamics as well as the development of the potential of hopfion in spintronic and magnonic applications.