Two-dimensional ferromagnetism in few-layer van der Waals crystals: Renormalized spin-wave theory and calculations

Dimensionality shapes the behavior of magnetism. Recent experiments on quasi two-dimensional (2D) van der Waals crystals have demonstrated unusual ferromagnetic orders and phase transitions distinct from their bulk counterparts. In this work, we present a method using renormalized spin-wave theory to study the ferromagnetic behavior in Heisenberg-type 2D ferromagnets, and apply our method to few-layer Cr2Ge2Te6 crystals. We present the details of the method, which includes a self-consistent updating scheme of the spin-wave spectra at the Hartree-Fock level. The theoretical results obtained quantitatively explain the experimental measurements, showing strong dimensionality effects and sensitive external magnetic-field dependence which leads to easily tunable magnetism in this class of 2D materials. This method can be generalized to study various magnetic phenomena in other systems.