Magnetic critical behavior and anomalous Hall effect in $\mathrm{2H}\text{−}{\mathrm{Co}}_{0.22}{\mathrm{TaS}}_2$ single crystals

We report ferromagnetism in $2\mathrm{H}\text{−}{\mathrm{Co}}_{0.22}\mathrm{Ta}{\mathrm{S}}_2$ single crystals where Co atoms are intercalated in the van der Waals gap, and a systematic study of its magnetic critical behavior in the vicinity of $T_c\sim 28$ K. The obtained critical exponents $\beta =0.43\left(2\right)$, $\gamma =1.15\left(1\right)$, and $\delta =3.54\left(1\right)$ fulfill the Widom scaling relation $\delta =1+\gamma /\beta$ and follow the scaling equation. This indicates that the spin coupling in $2\mathrm{H}\text{−}{\mathrm{Co}}_{0.22}\mathrm{Ta}{\mathrm{S}}_2$ is of three-dimensional Heisenberg type coupled with long-range magnetic interaction, and that the exchange interaction decays with distance as $J\left(r\right)\approx r^{-4.69}$. $2\mathrm{H}\text{−}{\mathrm{Co}}_{0.22}\mathrm{Ta}{\mathrm{S}}_2$ exhibits a weak temperature-dependent metallic behavior in resistivity and negative values of thermopower with dominant electron-type carriers, in which obvious anomalies were observed below $T_c$ as well as the anomalous Hall effect (AHE). The linear scaling behavior between the modified anomalous Hall resistivity ${\rho }_{xy}/{\mu }_{0}H$ and longitudinal resistivity ${\rho }_{xx}^{2}M/{\mu }_{0}H$ implies that the origin of the AHE in $2\mathrm{H}\text{−}{\mathrm{Co}}_{0.22}\mathrm{Ta}{\mathrm{S}}_2$ should be dominated by the extrinsic side-jump mechanism.

Figure 1

(a) Single crystal and (b) powder x-ray diffraction (XRD) patterns of $2\mathrm{H}\text{−}{\mathrm{Co}}_{0.22}\mathrm{Ta}{\mathrm{S}}_2$. The inset in (a) shows the crystal structure of $2\mathrm{H}\text{−}{\mathrm{Co}}_{0.22}\mathrm{Ta}{\mathrm{S}}_2$. The tiny extra peaks [labeled as an asterisk in (b)] at $32.2^{\circ }$ and $38.9^{\circ }$ are from pure iodine and sulfur, respectively. (c) Normalized Co $K$-edge x-ray absorption near edge structure (XANES) spectra. The inset shows the derivative $\mu \left(E\right)$ curve. (d) Fourier transform magnitudes of the extended x-ray absorption fine structure (EXAFS) oscillations (symbols) for Co $K$-edge with the phase-shift correction. The model fits are shown as solid lines. The inset shows the corresponding filtered EXAFS (symbols) with $k$-space model fits (solid lines). Temperature-dependent magnetization $M\left(T\right)$ with both zero-field-cooling (ZFC) and field-cooling (FC) modes for (e) $\mathbf{H}\parallel \mathbf{ab}$-plane and (f) $\mathbf{H}\parallel \mathbf{c}$-axis at various magnetic fields. The insets show the corresponding inverse susceptibility with $H=1$ kOe fitted by the Curie-Weiss law (solid lines). Field-dependent magnetization for (g) $\mathbf{H}\parallel \mathbf{ab}$-plane and (h) $\mathbf{H}\parallel \mathbf{c}$-axis at various temperatures.

Figure 2

(a) Typical initial isothermals measured with $\mathbf{H}\parallel \mathbf{c}$-axis from 18 to 38 K with a temperature step of 1 K. (b) The Arrott plots of $M^2$ vs $H/M$. (c) Temperature-dependent spontaneous magnetization $M_s$ (left) and inverse initial susceptibility ${\chi }_0^{-1}$ (right) with solid fitting curves for $2\mathrm{H}\text{−}{\mathrm{Co}}_{0.22}\mathrm{Ta}{\mathrm{S}}_2$. The inset shows the $M\left(H\right)$ curve in ${\log }_{10}\text{−}{\log }_{10}$ scale collected at 28 K with a linear fitting curve. (d) The 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. (e) Scaling plots of renormalized $m$ vs $h$ with $m\equiv {\varepsilon }^{-\beta }M(H,\varepsilon )$ and $h\equiv {\varepsilon }^{-(\beta +\gamma )}H$. (f) Rescaling plots of $M\left(H\right)$ isothermals by $M{H}^{-1/\delta }$ vs $\varepsilon H^{-1/\left(\beta \delta \right)}$.

Figure 3

Temperature dependence of in-plane (a) thermal conductivity $\kappa \left(T\right)$, (b) thermopower $S\left(T\right)$, and (c) resistivity ${\rho }_{xx}\left(T\right)$ in zero field for $2\mathrm{H}\text{−}{\mathrm{Co}}_{0.22}\mathrm{Ta}{\mathrm{S}}_2$ single crystal. Out-of-plane field dependence of (d) Hall resistivity ${\rho }_{xy}\left({\mu }_{0}H\right)$ and (e) initial magnetization $M\left({\mu }_{0}H\right)$ for $2\mathrm{H}\text{−}{\mathrm{Co}}_{0.22}\mathrm{Ta}{\mathrm{S}}_2$ at indicated temperatures. Temperature dependence of (f) ordinary Hall coefficient $R_0$ and (g) anomalous Hall coefficient $R_s$, fitted from the ${\rho }_{xy}\left({\mu }_{0}H\right)$ curves using ${\rho }_{xy}=R_0{\mu }_{0}H+R_{s}M$. (h) Anomalous Hall conductivity ${\sigma }_{xy}^A$ vs $M_s$ with ${\sigma }_{xy}^A\approx {\rho }_{xy}^A/{\rho }_{xx}^2$. (i) The ${\rho }_{xy}/{\mu }_{0}H$ vs ${\rho }_{xx}^{2}M/{\mu }_{0}H$ curves at indicated temperatures with subsequent offset of 0.05 ${\mathrm{cm}}^3{\mathrm{C}}^{-1}$. (j) Temperature dependence of coefficient $S_H={\mu }_0{R}_s/{\rho }_{xx}^2$.