Elucidating the origin of long-range ferromagnetic order in Fe$_3$GeTe$_2$ by low-energy magnon excitation studies

Abstract: We report a detailed high-field/high-frequency ferromagnetic resonance (HF-FMR) study of low-energy magnon excitations in the van der Waals ferromagnet Fe$3$GeTe$_2$. At 2 K, the field dependence of the magnon branches is well described by a semiclassical domain-based model, from which we extract key microscopic parameters including the anisotropy gap $\Delta = 170\pm 4$ GHz, the anisotropy field $B{\rm A} = 5.85\pm 0.08$ T, and the effective $g$-factor $g_{\rm ab}\simeq g_{\rm c} = 2.07(4)$. Furthermore the uniaxial anisotropy constant was determined to be $K = (10.5\pm 0.23) \times 10^{-6}$ erg/cm$^3$. Anisotropic short-range magnetic order persists above $T_{\rm C}$ up to approximately 270 K, as evidenced by a finite anisotropy gap and anisotropic shifts in the FMR resonance fields. Both results clearly show the presence of anisotropic local magnetic fields well above $T_{\rm C}$. Our findings underscore the crucial role of magneto-crystalline anisotropy in driving long-range magnetic order in Fe$_3$GeTe$_2$.