Magnetic critical behavior and anomalous Hall effect in 2H-Co$_{0.22}$TaS$_{2}$ single crystals.

We report ferromagnetism in 2H-Co$_{0.22}$TaS$_2$ 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(2), $\gamma$ = 1.15(1), and $\delta = 3.54(1)$ fulfill the Widom scaling relation $\delta = 1+\gamma/\beta$ and follow the scaling equation. This indicates that the spin coupling in 2H-Co$_{0.22}$TaS$_2$ is of three-dimensional Hersenberg type coupled with long-range magnetic interaction, and that the exchange interaction decays with distance as $J(r)\approx r^{-4.69}$. 2H-Co$_{0.22}$TaS$_2$ 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 $\rho_{xy}/\mu_0H$ and longitudinal resistivity $\rho_{xx}^2M/\mu_0H$ implies that the origin of AHE in 2H-Co$_{0.22}$TaS$_2$ should be dominated by the extrinsic side-jump mechanism.