Critical behavior of quasi-two-dimensional semiconducting ferromagnet CrGeTe$_3$

The critical properties of the single-crystalline semiconducting ferromagnet CrGeTe$_3$ were investigated by bulk dc magnetization around the paramagnetic to ferromagnetic phase transition. Critical exponents $\beta = 0.200\pm0.003$ with critical temperature $T_c = 62.65\pm0.07$ K and $\gamma = 1.28\pm0.03$ with $T_c = 62.75\pm0.06$ K are obtained by the Kouvel-Fisher method whereas $\delta = 7.96\pm0.01$ is obtained by the critical isotherm analysis at $T_c = 62.7$ 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 determined exponents match well with those calculated from the results of renormalization group approach for a two-dimensional Ising system coupled with long-range interaction between spins decaying as $J(r)\approx r^{-(d+\sigma)}$ with $\sigma=1.52$.