Critical behavior of the van der Waals bonded ferromagnet Fe$_{3-x}$GeTe$_2$

The critical properties of the single-crystalline van der Waals bonded ferromagnet Fe$_{3-x}$GeTe$_2$ were investigated by bulk dc magnetization around the paramagnetic (PM) to ferromagnetic (FM) phase transition. The Fe$_{3-x}$GeTe$_2$ single crystals grown by self-flux method with Fe deficiency $x \approx 0.36$ exhibit bulk FM ordering below $T_c = 152$ K. The M\"{o}ssbauer spectroscopy was used to provide information on defects and local atomic environment in such crystals. Critical exponents $\beta = 0.372(4)$ with a critical temperature $T_c = 151.25(5)$ K and $\gamma = 1.265(15)$ with $T_c = 151.17(12)$ K are obtained by the Kouvel-Fisher method whereas $\delta = 4.50(1)$ is obtained by a critical isotherm analysis at $T_c = 151$ K. These critical exponents obey the Widom scaling relation $\delta = 1+\gamma/\beta$, indicating self-consistency of the obtained values. With these critical exponents the isotherm $M(H)$ curves below and above the critical temperatures collapse into two independent universal branches, obeying the single scaling equation $m = f_\pm(h)$, where $m$ and $h$ are renormalized magnetization and field, respectively. The exponents determined in this study are close to those calculated from the results of the renormalization group approach for a heuristic model of three-dimensional Heisenberg ($d = 3, n = 3$) spins coupled with the attractive long-range interactions between spins that decay as $J(r)\approx r^{-(3+\sigma)}$ with $\sigma=1.89$.