Critical behavior of intercalated quasi-van der Waals ferromagnet $\mathrm{F}{\mathrm{e}}_{0.26}\mathrm{Ta}{\mathrm{S}}_2$

In the present work, single-crystalline quasi-van der Waals ferromagnet $\mathrm{F}{\mathrm{e}}_{0.26}\mathrm{Ta}{\mathrm{S}}_2$ was successfully synthesized with Fe atoms intercalated at ordered positions between $\mathrm{Ta}{\mathrm{S}}_2$ layers. Its critical behavior was systematically studied by measuring the magnetization around ferromagnetic to paramagnetic phase transition temperature, $T_C\sim 100.7\mathrm{K}$, under different magnetic fields. The critical exponent β for the spontaneous magnetization below $T_C$, γ for the inverse initial susceptibility above $T_C$, and δ for the magnetic isotherm at $T_C$ were determined with modified Arrott plots, the Kouvel-Fisher method, the Widom scaling law, and critical isotherm analysis, and found to be the following values: $\beta =0.459\left(6\right),\gamma =1.205\left(11\right)$, and $\delta =3.69\left(1\right)$. The obtained critical exponents are self-consistent and follow the scaling equation, indicating the reliability and intrinsicality of these parameters. A close analysis within the framework of renormalization group theory reveals that the spin coupling inside $\mathrm{F}{\mathrm{e}}_{0.26}\mathrm{Ta}{\mathrm{S}}_2$ crystal is of the three-dimensional Heisenberg $\left(\left\{d:n\right\}=\left\{3:3\right\}\right)$ type with long-range magnetic interaction and that the exchange interaction decays with distance as $J\left(r\right)\sim r^{-4.71}$.

Figure 1

(a) Schematic diagram of $\mathrm{F}{\mathrm{e}}_{1/4}\mathrm{Ta}{\mathrm{S}}_2$ crystal structure. Ta, S, and Fe atoms are denoted by green, orange, and purple balls, respectively. The dashed lines in the top view indicate the crystal unit cell. (b) Optical image of a typical as-grown FTS crystal. The size of the grid is $1\mathrm{m}{\mathrm{m}}^2$. (c) XRD pattern of FTS measured with the slab surface lying flat on the sample holder.

Figure 2

(a) Experimental random (black) and channeling (blue) RBS spectra of the single-crystalline $\mathrm{F}{\mathrm{e}}_{0.26}\mathrm{Ta}{\mathrm{S}}_2$ sample with the $\mathrm{H}{\mathrm{e}}^{+}$ beam aligned along $c$ axis. The red line is the fitted RBS random spectra for $\mathrm{F}{\mathrm{e}}_{0.26}\mathrm{Ta}{\mathrm{S}}_2$ using the SIMNRA code. (b) Experimental two-dimensional backscattered-yield mapping patterns of the Fe in the crystal obtained from RBS/C spectra. θ and ϕ are the scanning angles when performing the two-dimensional mapping, and the color scale is the integral backscattering signals (counts).

Figure 3

(a) Magnetic-field-dependent magnetic moment of a $\mathrm{F}{\mathrm{e}}_{0.26}\mathrm{Ta}{\mathrm{S}}_2$ sample measured at different temperatures with the field being applied along the $c$ axis and in the $ab$ plane, respectively. (b) Temperature-dependent magnetization curves after zero-field-cooling (solid) and field-cooling (open) processes followed by a 100 Oe magnetic field being applied along the $c$ axis (red squares) and in the $ab$ plane (blue triangles), respectively. The lines for the $H\left|\right|ab$ plane case are enlarged 100 times for clarity. Inset: the dM/dT vs $T$ of field-cooling $M\left(T\right)$ curves. (c) Temperature dependence of inverse susceptibility with a 70 kOe field applied along the $c$ axis. The red dashed line describes the fitting from the Curie-Weiss equation in which the temperature region is chosen from 180 to 300 K as highlighted in blue.

Figure 4

(a) Magnetic-field-dependent initial isothermal magnetization curves measured from 96 to 106 K with $H\left|\right|c$, and (b) is the corresponding Arrott plot. The orange-highlighted solid curve represents the measurement taking place at the Weiss temperature of 100 K estimated by the Curie-Weiss law. In the range of 98–102 K, the temperature step is 0.5 K. Otherwise, it is 1 K.

Figure 5

The modified Arrott plots of $M^{1/\beta }$ vs ${\left(H/M\right)}^{1/\gamma }$ in the description of (a) 3D Heisenberg model $\left(\beta =0.365,\gamma =1.386\right)$, (b) 3D Ising model $\left(\beta =0.325,\gamma =1.241\right)$, (c) 3D XY model $\left(\beta =0.345,\gamma =1.316\right)$, and (d) tricritical mean-field model $\left(\beta =0.25,\gamma =1.0\right)$.

Figure 6

(a) Temperature-dependent spontaneous magnetization $M_S$ (left) and inverse initial susceptibility ${\chi }_0^{-1}$ (right) with red fitting curves for FTS. (b) The modified Arrott plot of $M^{1/\beta }$ vs ${\left(H/M\right)}^{1/\gamma }$ with parameters of $\beta =0.460$ and $\gamma =1.216$.

Figure 7

(a) The KF plot of $M_S\left(T\right)/\left[d{M}_S\left(T\right)dT\right]$ (left) ${\chi }_0^{-1}\left(T\right)/\left[d{\chi }_0^{-1}\left(T\right)/dT\right]$ and (right) with red fitting curves for FTS. (b) The isotherm magnetic-field-dependent magnetization measured at $T_C=100.7\mathrm{K}$ with a red fitting curve. Inset: the corresponding log-log scale plot.

Figure 8

(a) Renormalized magnetization $m$ as a function of renormalized magnetic field $h$ below and above $T_C$ for FTS. Inset: the corresponding log-log scale plot. (b) The $m^2$ vs $h/m$ plot. Inset: the plot of $M{H}^{-1/\delta }$ vs $\varepsilon H^{-1/\left(\beta \delta \right)}$.