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

The critical properties of the single-crystalline van der Waals bonded ferromagnet ${\mathrm{Fe}}_{3-x}{\mathrm{GeTe}}_2$ were investigated by bulk dc magnetization around the paramagnetic to ferromagnetic (FM) phase transition. The ${\mathrm{Fe}}_{3-x}{\mathrm{GeTe}}_2$ single crystals grown by self-flux method with Fe deficiency $x\approx 0.36$ exhibit bulk FM ordering below $T_c=152$ K. The Mössbauer spectroscopy was used to provide information on defects and local atomic environment in such crystals. Critical exponents $\beta =0.372\left(4\right)$ with a critical temperature $T_c=151.25\left(5\right)$ K and $\gamma =1.265\left(15\right)$ with $T_c=151.17\left(12\right)$ K are obtained by the Kouvel-Fisher method, whereas $\delta =4.50\left(1\right)$ 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 $M\left(H\right)$ curves below and above the critical temperatures collapse into two independent universal branches, obeying the single scaling equation $m=f_{\pm }\left(h\right)$, 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 $J\left(r\right)\approx r^{-(3+\sigma )}$ with $\sigma =1.89$.

Figure 1

Crystal structure of ${\mathrm{Fe}}_{3-x}{\mathrm{GeTe}}_2$ from (a) side and (b) top views. The unit cell is enclosed by blue solid lines. Inequivalent Fe sites are labeled as Fe1 and Fe2, respectively. (c) Powder x-ray diffraction (XRD) and (d) single-crystal XRD patterns of ${\mathrm{Fe}}_{3-x}{\mathrm{GeTe}}_2$. The vertical tick marks represent Bragg reflections of the $P{6}_3/mmc$ space group.

Figure 2

(a) Temperature dependence of magnetization for ${\mathrm{Fe}}_{3-x}{\mathrm{GeTe}}_2$ measured with the external magnetic field $H=1$ kOe applied along the $c$ axis and in the $ab$ plane under zero-field-cooling (ZFC) and field-cooling (FC) modes. The yellow solid lines are fitted by the modified Curie-Weiss law $\chi =\frac{C}{T-\theta }+{\chi }_0$, where ${\chi }_0$ is the temperature-independent susceptibility, $C$ is the Curie-Weiss constant, and $\theta$ is the Weiss temperature. (Inset) The derivative magnetization $dM/dT$ vs $T$ in different applied fields along the $c$ axis. (b) Field dependence of magnetization for ${\mathrm{Fe}}_{3-x}{\mathrm{GeTe}}_2$ measured at $T=2$ K. (Inset) The magnification in the low-field region. (c) Mössbauer spectrum at $T=300$ K of the ${\mathrm{Fe}}_{3-x}{\mathrm{GeTe}}_2$ crushed single crystal. The experimental data are presented by solid circles, and the fit is given by the red solid line. Vertical arrow denotes relative position of the lowermost peak with respect to the basal line (relative transmission). The fitted lines of subspectra are plotted above the main spectrum fit: The Fe1 doublet is blue, the ${\mathrm{Fe}1}^{*}$ doublet is violet, and the Fe2 doublet is olive. Error is depicted as the absolute of difference; the largest value is 0.176%.

Figure 3

(a) Typical initial isothermal magnetization curves measured along the $c$ axis around $T_c=152$ K (in an orange symbol and line) for ${\mathrm{Fe}}_{3-x}{\mathrm{GeTe}}_2$. (b) Arrott plots of $M^2$ vs $H/M$ around $T_c$ for ${\mathrm{Fe}}_{3-x}{\mathrm{GeTe}}_2$.

Figure 4

The isotherms of $M^{1/\beta }$ vs ${(H/M)}^{1/\gamma }$ with parameters of (a) 3D Heisenberg model, (b) 3D $XY$ model, (c) 3D Ising model, and (d) tricritical mean-field model. The straight lines are the linear fit of isotherms at different temperatures.

Figure 5

Temperature dependence of the normalized slopes $NS=S\left(T\right)/S\left(T_c\right)$.

Figure 6

(a) Temperature dependence of the spontaneous magnetization $M_s$ (left) and the inverse initial susceptibility ${\chi }_0^{-1}$ (right) with solid fitting curves for ${\mathrm{Fe}}_{3-x}{\mathrm{GeTe}}_2$. (b) Kouvel-Fisher plots of $M_s{(d{M}_s/dT)}^{-1}$ (left) and ${\chi }_0^{-1}{(d{\chi }_0^{-1}/dT)}^{-1}$ (right) with solid fitting curves for ${\mathrm{Fe}}_{3-x}{\mathrm{GeTe}}_2$.

Figure 7

Isotherm $M$ vs $H$ plot collected at $T_c=151$ K for ${\mathrm{Fe}}_{3-x}{\mathrm{GeTe}}_2$. (Inset) The same plot in log-log scale with a solid fitting curve.

Figure 8

(a) Scaling plots of renormalized magnetization $m$ vs renormalized field $h$ below and above $T_c$ for ${\mathrm{Fe}}_{3-x}{\mathrm{GeTe}}_2$. (Inset) The same plots in log-log scale. (b) The renormalized magnetization and field replotted in the form of $m^2$ vs $h/m$ for ${\mathrm{Fe}}_{3-x}{\mathrm{GeTe}}_2$. (Inset) The rescaling of the $M\left(H\right)$ curves by $M{H}^{-1/\delta }$ vs $\varepsilon H^{-1/\left(\beta \delta \right)}$.