Unraveling topological magnetic structure and its complex electronic polarization in Janus structures CrInX₃ (X=Se, Te)

In this work, by using first-principles calculations combined with atomistic spin dynamics, we provide a complete picture of the magnetic interactions and magnetic excitations in Janus structure CrInX₃ (X=Se, Te). Specifically, the rank-2 exchange matrix is obtained, and the exchange mechanism for both the isotropic Heisenberg exchange and the antisymmetric Dzyaloshinzkii-Moriya (DM) exchange is analyzed. The calculated orbitally-resolved exchange coupling distinctly demonstrates that the ferromagnetic coupling between Cr-Cr atoms results from a competition between ferromagnetic t_2g-e_g coupling and antiferromagnetic e_g-e_g coupling. This highlights the significance of e_g electrons, which originate from the robust hybridization between the Cr d orbital and the ligand p orbital. Furthermore, a direct connection between the three-dimensional DM vectors and the ground state spin helical configuration is effectively established. Generation conditions of magnetic skyrmions are analyzed based on the energy decomposition of effective Hamiltonian. In addition, the magnetic excited states and magnetic equilibrium states are studied thoroughly by obtaining an adiabatic magnon spectrum and a detailed phase diagram. At last, instead of considering only the ionic and electronic contribution of the ferroelectric polarization using modern polarization theory, spin-spiral-induced polarization is also obtained by employing the microscopic approach proposed by Katsura, Nagaosa, and Balatsky.