Fig. 1

(a) The interlinking of contours corresponding to m_x = + 1 (blue) and m_x = − 1 (red), demonstrating the topological nature of the hopfion with a linking number of unity. (b) Magnetic hopfion profile at the mid-layer in the computational domain, with the coordinate axes used in this study indicated as x, y, and z. F_STT and F_b represent the effective forces acting on the hopfion, originating from spin-transfer torque and boundary repulsion, respectively.