Fundamental Spin Interactions Underlying the Magnetic Anisotropy in the Kitaev Ferromagnet CrI3

We lay the foundation for determining the microscopic spin interactions in two-dimensional (2D) ferromagnets by combining angle-dependent ferromagnetic resonance (FMR) experiments on high quality ${\mathrm{CrI}}_{3}$ single crystals with theoretical modeling based on symmetries. We discover that the Kitaev interaction is the strongest in this material with $K\ensuremath{\sim}\ensuremath{-}5.2\text{ }\text{ }\mathrm{meV}$, 25 times larger than the Heisenberg exchange $J\ensuremath{\sim}\ensuremath{-}0.2\text{ }\text{ }\mathrm{meV}$, and responsible for opening the $\ensuremath{\sim}5\text{ }\text{ }\mathrm{meV}$ gap at the Dirac points in the spin-wave dispersion. Furthermore, we find that the symmetric off-diagonal anisotropy $\mathrm{\ensuremath{\Gamma}}\ensuremath{\sim}\ensuremath{-}67.5\text{ }\text{ }\ensuremath{\mu}\mathrm{eV}$, though small, is crucial for opening a $\ensuremath{\sim}0.3\text{ }\text{ }\mathrm{meV}$ gap in the magnon spectrum at the zone center and stabilizing ferromagnetism in the 2D limit. The high resolution of the FMR data further reveals a $\ensuremath{\mu}\mathrm{eV}$-scale quadrupolar contribution to the $S=3/2$ magnetism. Our identification of the underlying exchange anisotropies opens paths toward 2D ferromagnets with higher ${T}_{C}$ as well as magnetically frustrated quantum spin liquids based on Kitaev physics.