Magnetic anisotropy in ferromagnetic CrI3

We use neutron scattering to show that ferromagnetic (FM) phase transition in the two-dimensional (2D) honeycomb lattice ${\mathrm{CrI}}_{3}$ is a weakly first order transition and controlled by spin-orbit coupling (SOC) induced magnetic anisotropy, instead of magnetic exchange coupling as in a conventional ferromagnet. With increasing temperature, the magnitude of magnetic anisotropy, seen as a spin gap at the Brillouin zone center, decreases in a power law fashion and vanishes at ${T}_{C}$, while the in-plane and $c$-axis spin-wave stiffnesses associated with magnetic exchange couplings remain robust at ${T}_{C}$. We also compare parameter regimes where spin waves in ${\mathrm{CrI}}_{3}$ can be described by a Heisenberg Hamiltonian with Dzyaloshinskii-Moriya interaction or a Heisenberg-Kitaev Hamiltonian. These results suggest that the SOC induced magnetic anisotropy plays a dominant role in stabilizing the FM order in single layer 2D van der Waals ferromagnets.