Anomalous Hall effect in the van der Waals bonded ferromagnet ${\mathrm{Fe}}_{3-x}{\mathrm{GeTe}}_2$

We report the anomalous Hall effect (AHE) in single crystals of a quasi-two-dimensional ${\mathrm{Fe}}_{3-x}{\mathrm{GeTe}}_2$ ($x\approx 0.36$) ferromagnet grown by the flux method which induces defects on the Fe site and bad metallic resistivity. Fe $K$-edge x-ray absorption spectroscopy was measured to provide information on the local atomic environment in such crystals. The dc and ac magnetic susceptibility measurements indicate a second-stage transition below 119 K in addition to the paramagnetic to ferromagnetic transition at 153 K. A linear scaling behavior between the modified anomalous Hall resistivity ${\rho }_{xy}/{\mu }_0{H}_{\mathrm{eff}}$ and longitudinal resistivity ${\rho }_{xx}^{2}M/{\mu }_0{H}_{\mathrm{eff}}$ implies that the AHE in ${\mathrm{Fe}}_{3-x}{\mathrm{GeTe}}_2$ should be dominated by the intrinsic Karplus-Luttinger mechanism rather than the extrinsic skew-scattering and side-jump mechanisms. The observed deviation in the linear-$M$ Hall conductivity ${\sigma }_{xy}^A$ below 30 K is in line with its transport characteristic at low temperatures, implying the scattering of conduction electrons due to magnetic disorder and the evolution of the Fermi surface induced by a possible spin-reorientation transition.

Figure 1

(a) Normalized Fe $K$-edge XANES spectra and (b) Fourier-transformed magnitudes of EXAFS data of ${\mathrm{Fe}}_{3-x}{\mathrm{GeTe}}_2$ measured at room temperature. The experimental data are shown as blue symbols alongside the model fit plotted as a red line. The inset in (b) shows the corresponding EXAFS oscillation with the model fit.

Figure 2

(a) Temperature dependence of dc magnetization $M\left(T\right)$ with zero-field cooling (ZFC) and field cooling (FC) taken at ${\mu }_{0}H=0.01$ T and ${\mu }_{0}H=5$ T (inset) for ${\mu }_{0}H\parallel ab$ and ${\mu }_{0}H\parallel c$, respectively. (b) Field dependence of magnetization $M\left({\mu }_{0}H\right)$ taken at $T=2$ K.

Figure 3

(a) Temperature dependence of ac susceptibility real part ${\chi }^{\prime }\left(T\right)$ measured in zero external field. Oscillated ac field of 3.8 Oe is applied in the $ab$ plane and the $c$ axis, respectively. (b) Temperature dependence of heat capacity $C_p\left(T\right)$. Insets: The low-temperature $C_p\left(T\right)/T$ vs $T^2$ curve fitted by $C_p\left(T\right)/T=\gamma +\beta T^2$ and the enlargement of the $\lambda$-type anomaly around $T_c$ = 153 K. The red curve in the right inset represents the phonon contribution fitted by a polynomial. The right axis and its associated blue curve denote the magnetic entropy calculated by $S_{\mathrm{mag}}\left(T\right)={\int }_0^T{C}_{\mathrm{mag}}/TdT$.

Figure 4

(a) Temperature-dependent in-plane resistivity ${\rho }_{xx}\left(T\right)$ for ${\mathrm{Fe}}_{3-x}{\mathrm{GeTe}}_2$. Inset: Magnetoresistance of ${\rho }_{xx}\left({\mu }_{0}H\right)$ at different temperatures. (b) Temperature-dependent Seebeck coefficient $S\left(T\right)$ for ${\mathrm{Fe}}_{3-x}{\mathrm{GeTe}}_2$.

Figure 5

(a), (b) Effective field dependence of magnetization $M\left({\mu }_0{H}_{\mathrm{eff}}\right)$ and (c),(d) Hall resistivity ${\rho }_{xy}\left(B\right)$ for ${\mathrm{Fe}}_{3-x}{\mathrm{GeTe}}_2$ at various temperatures with ${\mu }_{0}H\parallel c$. The red dashed lines in (a) and (c) are linear fits of $M\left({\mu }_0{H}_{\mathrm{eff}}\right)$ and ${\rho }_{xy}\left(B\right)$ at the high-field region.

Figure 6

(a),(b) Temperature-dependent $R_0\left(T\right)$ and $R_s\left(T\right)$ fitted from ${\rho }_{xy}(B,T)$ using ${\rho }_{xy}=R_{0}B+R_s{\mu }_{0}M$. (c) Anomalous Hall conductivity ${\sigma }_{xy}^A$ vs $M_s$. (d) The ${\rho }_{xy}/{\mu }_0{H}_{\mathrm{eff}}$ vs ${\rho }_{xx}^{2}M/{\mu }_0{H}_{\mathrm{eff}}$ curves at indicated temperatures with subsequent offset of 0.02 ${\mathrm{cm}}^3{\mathrm{C}}^{-1}$. The solid lines in (d) represent linear fits of the data at different temperatures.