Giant anomalous Hall effect in quasi-two-dimensional layered antiferromagnet ${\mathrm{Co}}_{1/3}{\mathrm{NbS}}_2$

The discovery of the anomalous Hall effect (AHE) in bulk metallic antiferromagnets (AFMs) motivates the search of the same phenomenon in two-dimensional (2D) systems, where a quantized anomalous Hall conductance can, in principle, be observed. Here we present experiments on microfabricated devices based on ${\mathrm{Co}}_{1/3}{\mathrm{NbS}}_2$, a layered AFM that was recently found to exhibit AHE in bulk crystals below the Néel temperature $T_N=29$ K. Transport measurements reveal a pronounced resistivity anisotropy, indicating that upon lowering temperature the electronic coupling between individual atomic layers is increasingly suppressed. The experiments also show an extremely large anomalous Hall conductivity of approximately 400 S/cm, more than one order of magnitude larger than in the bulk, which demonstrates the importance of studying the AHE in small exfoliated crystals, less affected by crystalline defects. Interestingly, the corresponding anomalous Hall conductance, when normalized to the number of contributing atomic planes, is $\sim 0.6e^2/h$ per layer, approaching the value expected for the quantized anomalous Hall effect. The observed strong anisotropy of transport and the very large anomalous Hall conductance per layer make the properties of ${\mathrm{Co}}_{1/3}{\mathrm{NbS}}_2$ compatible with the presence of partially filled topologically nontrivial 2D bands originating from the magnetic superstructure of the antiferromagnetic state. Isolating atomically thin layers of this material and controlling their charge density may therefore provide a viable route to reveal the occurrence of the quantized AHE in a 2D AFM.

Figure 1

Properties of ${\mathrm{Co}}_{1/3}{\mathrm{NbS}}_2$. (a) Crystallographic unit cell of ${\mathrm{Co}}_{1/3}{\mathrm{NbS}}_2$. (b) Top-view of the lattice displaying the regular triangular arrangement of the Co atoms which gives rise to a $\sqrt{3}{a}_0\times \sqrt{3}{a}_0$ superstructure—where $a_0$ is the lattice constant of the parent compound (2H-${\mathrm{NbS}}_2)$—typical of 1/3 fractional intercalation [29]. The crystallographic unit cell of ${\mathrm{Co}}_{1/3}{\mathrm{NbS}}_2$ is indicated by the yellow shaded area. (c) Temperature dependence of the in-plane resistivity ${\rho }_{xx}$ (blue) and of the magnetic susceptibility $\chi$ (green)—acquired in an applied field of 0.1 T along the $c$-axis—both measured in a bulk crystal. The two curves show a clear kink in correspondence of the antiferromagnetic transition temperature at $T_N\simeq 29$ K. (d) Magnetization $M$ as a function of magnetic field $\left(H\parallel c\right)$ for different temperatures above and below $T_N$ measured in a bulk crystal; $M$ is negligible at $H=0$. (e) Optical micrographs of two devices investigated in the present work: a device sculpted using a focused ion beam (top left; the scale bar is $20\mu \mathrm{m})$ and a device based on an exfoliated crystal approximately 50 nm thick (top right; the scale bar is $5\mu \mathrm{m})$. Temperature dependence of the ratio between the resistivity out-of-plane $\left({\rho }_z\right)$ and the longitudinal resistivity in-plane $\left({\rho }_{xx}\right)$, showing a pronounced anisotropy below $T_N$. The inset shows the resistivity along the $c$-axis, ${\rho }_z$, as a function of temperature.

Figure 2

AHE in ${\mathrm{Co}}_{1/3}{\mathrm{NbS}}_2$. (a) Hall resistivity ${\rho }_{xy}$ and (b) anomalous Hall resistivity ${\rho }_{xy}^A$, measured in a representative device upon cycling the magnetic field back and forth from $-14$ T to $+14$ T $\left(H\parallel c\right)$, for selected temperatures above and below $T_N$. The anomalous Hall resistivity in (b) is obtained by subtracting from the curves shown in (a) the linear contribution due to the ordinary Hall effect. ${\rho }_{xy}^A$ exhibits sharp jumps at values of coercive fields $H_c$ that rapidly grow upon lowering $T$. (c) Longitudinal resistivity ${\rho }_{xx}$ as function of magnetic field, acquired simultaneously to ${\rho }_{xy}$; a small magnetoresistance (MR $\sim 2$%) is observed below $T_N$. The small jumps visible in the curves in correspondence of $H_c$ (less than 10% of the absolute change in ${\rho }_{xy}^A)$ can be attributed to small material inhomogeneities mixing the longitudinal and transverse resistivity.

Figure 3

Temperature evolution of AHE in an exfoliated ${\mathrm{Co}}_{1/3}{\mathrm{NbS}}_2$ device. (a) Anomalous Hall resistivity ${\rho }_{xy}^A$ and (b) longitudinal resistivity ${\rho }_{xx}$ as a function of temperature measured in a representative device, after field-cooling (FC) in ${\mu }_0{H}_{FC}=\pm 8$ T (red), $\pm 14$ T (blue), and 0 T (ZFC, grey curve) with $H\parallel c$. The yellow diamonds in (a) represent the values of ${\rho }_{xy}^A$ extracted at $H=0$ from measurements of the hysteresis loops done at fixed $T$ [see Fig. 2]. ${\rho }_{xx}$ shows no dependence on $H_{FC}$: the curves in (b) fall exactly on top of each others and an identical $T$-dependence is observed at ZFC. (c) Remnant magnetization $M$ as a function of temperature for $H_{FC}$ of 8 T (red) and 14 T (blue) with $H\parallel c$. (d) Temperature dependence of the corresponding anomalous Hall conductivity ${\sigma }_{xy}^A$, for FC in 8 T (red) and 14 T (blue). The blue diamond represents the maximum value of the anomalous Hall conductivity reported previously in bulk crystals (27 S/cm at 23 K, see [36]). The right axis indicates the anomalous Hall conductance per atomic layer $G_{xy}^A/N$, obtained normalizing the measured ${\sigma }_{xy}^A$ (left axis) to the total number of contributing crystalline planes [39]. (e) Anomalous Hall conductance per atomic layer $G_{xy}^A/N$, measured at 5 K for different samples as function of their thickness. The pink dashed line and the shaded area represent the mean value and the standard deviation of all measured values.

Figure 4

Detecting structural domain switching in the AHE. Temperature dependence of the (a) anomalous Hall resistivity ${\rho }_{xy}^A$ and (b) longitudinal resistivity ${\rho }_{xx}$ after FC in $\pm 8$ T (red) and $\pm 14$ T (blue) with $H_{FC}\parallel c$, measured in one of our devices. In (a) the red curves display a trend similar to the one observed in Fig. 3, whereas the blue curves measured at ${\mu }_0{H}_{FC}=14$ T exhibit a larger value and a jump at 17 K with a drop to the value measured at ${\mu }_0{H}_{FC}=8$ T. In analogy with Fig. 3, the yellow (light-blue) diamonds correspond to the values of ${\rho }_{xy}^A$ extracted at $H=0$ (14 T) in measurements of the hysteresis loops done at fixed $T$ [see Supplemental Material Fig. S2(b) 38]. The light-blue dashed line is a guide to the eye. In (b) ${\rho }_{xx}$ shows no sizable variation in correspondence of the jump observed in ${\rho }_{xy}^A$, indicating that at the microscopic level the magnetic state of the system remains the same across the jump. (c) Temperature dependence of the anomalous Hall conductance per atomic layer $G_{xy}^A/N$ for FC in 8 T (red) and 14 T (blue). The inset shows a sketch of the envisioned magnetic configuration of the exfoliated crystal, with a structural defect that splits the crystal in two regions (colored in red and blue) with thickness $t_1$ and $t_2$ (the arrow indicates the direction of the $c$-axis). The two regions have opposite chirality upon FC in 8 T, whereas they have the same chirality upon FC in 14 T. (d) $G_{xy}^A/N$ as a function of $H_{FC}$ measured at 5 K with $H\parallel c$; the inset shows a zoom-in at small $H_{FC}$. (e) $G_{xy}^A/N$ as a function of temperature for different $H_{FC}$ applied in the ${\mathrm{NbS}}_2$ planes $\left(H\perp c\right)$. Note that the value of $G_{xy}^A/N$ upon $H_{FC}$ in 8 T corresponds approximately to the value obtained for FC in 0.05 T when $H\parallel c$.