Self-Consistently Renormalized Spin-Wave Theory of Magnetic Two-Dimensional van der Waals Materials

We develop a self-consistently renormalized spin-wave theory, within a mean-field approximation, for the two-dimensional Heisenberg ferromagnet with perpendicular easy-axis anisotropy on the honeycomb lattice, as well as its few-layer and bulk extensions [1]. In this method, the magnetization dependence on temperature is found as the solution of the self-consistency equation. Furthermore, we account for the physical difference of surface and bulk layers by treating the layers as separate sublattices. Thus the method can be readily generalized to study various magnetic phenomena in a broad range of systems, including those comprising magnetically inequivalent sublattices. Using our theory, we calculate the temperature-dependent magnetization for two chromium-based layered van der Waals insulating magnets, Cr₂Ge₂Te₆ and CrI₃, employing various sets of Heisenberg exchange and single-ion anisotropy values reported for these materials in the existing literature. As expected, we observe a strong dimensionality effect where the ordering temperature is reduced and its sensitivity on the anisotropy is enhanced with the decrease of dimensionality. Using the exchange parameters calculated with <i>ab initio</i> description of electron correlations beyond density functional theory [2] yields critical temperatures that agree well with the experimental values.<br/><br/>[1] V. V. Mkhitaryan and Liqin Ke; Physical Review B 104, 064435 (2021)<br/>[2] Liqin Ke and Mikhail I Katsnelson; npj Computational Materials, 7(4):1-8, (2021)