Three-dimensional magnetic critical behavior in ${\mathrm{CrI}}_3$

${\mathrm{CrI}}_3$ is a promising candidate for van der Waals bonded ferromagnetic devices since its ferromagnetism can be maintained upon exfoliating bulk crystals down to a single layer. In this work we studied critical properties of bulk ${\mathrm{CrI}}_3$ single crystals around the paramagnetic to ferromagnetic phase transition. Critical exponents $\beta$ = 0.260(4) with a critical temperature $T_c$ = 60.05(13) K and $\gamma$ = 1.136(6) with $T_c$ = 60.43(4) K are obtained by the Kouvel-Fisher method, whereas $\delta$ = 5.32(2) is obtained by a critical isotherm analysis at $T_c$ = 60 K. The critical exponents determined in bulk ${\mathrm{CrI}}_3$ single crystals suggest a three-dimensional long-range magnetic coupling with the exchange distance decaying as $J\left(r\right)\approx r^{-4.69}$.

Figure 1

(a) Crystal structure of ${\mathrm{CrI}}_3$ at room temperature. (b) Image of representative single crystals. (c) Single-crystal x-ray diffraction (XRD) and (d) powder XRD patterns of ${\mathrm{CrI}}_3$ at room temperature. The vertical tick marks represent Bragg reflections of the $C2/m$ space group.

Figure 2

(a) Temperature dependence of magnetization for ${\mathrm{CrI}}_3$ measured in the magnetic field $H$ = 1 kOe applied in the $ab$ plane and along the $c$ axis. The solid lines are fitted by the modified Curie-Weiss law $\chi ={\chi }_0+\frac{C}{T-\theta }$, where ${\chi }_0$ is the temperature-independent susceptibility, $C$ is the Curie-Weiss constant, and $\theta$ is the Weiss temperature. Inset: $dM/dT$ vs $T$. (b) Field dependence of magnetization for ${\mathrm{CrI}}_3$ measured at $T$ = 2 K. Inset: the magnification of the low-field region.

Figure 3

(a) Typical initial isothermal magnetization curves measured along the $c$ axis from $T$ = 45 K to $T$ = 75 K for ${\mathrm{CrI}}_3$. (b) Arrott plots of $M^2$ vs $H/M$. Plots of $M^{1/\beta }$ vs ${(H/M)}^{1/\gamma }$ with parameters from (c) 3D Heisenberg model, (d) 3D Ising model, (e) 3D XY model, and (f) tricritical mean-field model. The straight lines are the linear fit of isothermals at different temperatures.

Figure 4

(a) Temperature dependence of the spontaneous magnetization $M_s$ (left) and the inverse initial susceptibility ${\chi }_0^{-1}$ (right) with solid fitting curves for ${\mathrm{CrI}}_3$. (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 ${\mathrm{CrI}}_3$.

Figure 5

(a) Temperature dependence of the normalized slopes $\text{NS}=S\left(T\right)/S\left(T_c\right)$ for different models. (b) Isotherm $M\left(H\right)$ collected at $T_c$ = 60 K. Inset: the same plot in log-log scale with a solid fitting curve.

Figure 6

(a) Scaling plots of renormalized magnetization $m$ vs renormalized field $h$ below and above $T_c$ for ${\mathrm{CrI}}_3$. Inset: the same plots in log-log scale. (b) Plots of $m^2$ vs $h/m$ for ${\mathrm{CrI}}_3$. Inset: the rescaling of the $M\left(H\right)$ curves by $M{H}^{-1/\delta }$ vs $\varepsilon H^{-1/\left(\beta \delta \right)}$.