Large anomalous Hall effect induced by weak ferromagnetism in the noncentrosymmetric antiferromagnet ${\mathrm{CoNb}}_3{\mathrm{S}}_6$

We study the mechanism of the exceptionally large anomalous Hall effect (AHE) in the noncentrosymmetric antiferromagnet $\mathrm{Co}{\mathrm{Nb}}_3{\mathrm{S}}_6$ by angle-resolved photoemission spectroscopy (ARPES) and magnetotransport measurements. From ARPES measurements of $\mathrm{Co}{\mathrm{Nb}}_3{\mathrm{S}}_6$ and its family compounds ($\mathrm{Fe}{\mathrm{Nb}}_3{\mathrm{S}}_6$ and $\mathrm{Ni}{\mathrm{Nb}}_3{\mathrm{S}}_6$), we find a band dispersion unique to the Co intercalation existing near the Fermi level. We further demonstrate that a slight deficiency of sulfur in $\mathrm{Co}{\mathrm{Nb}}_3{\mathrm{S}}_6$ eliminates the ferromagnetism and the AHE simultaneously while hardly changing the band structure, indicating that the weak ferromagnetism is responsible for the emergence of the large AHE. Based on our results, we propose Weyl points near the Fermi level to cause the large AHE.

Figure 1

Crystal structure and band dispersion of $\mathrm{Co}{\mathrm{Nb}}_3{\mathrm{S}}_6$ obtained by VUV-ARPES ($h\nu =120\mathrm{eV}$). (a) Crystal structure of $\mathrm{Co}{\mathrm{Nb}}_3{\mathrm{S}}_6$. Two $\mathrm{Nb}{\mathrm{S}}_2$ layers of different orientations are represented by orange and yellow. In the bottom panels, the crystallographic unit cells of $\mathrm{Nb}{\mathrm{S}}_2$ and $\mathrm{Co}{\mathrm{Nb}}_3{\mathrm{S}}_6$ are represented by the dashed and solid rhomboids respectively. The red arrows represent antiferromagnetic moments of Co atoms. (b1) and (c1) Fermi surfaces of $\mathrm{Nb}{\mathrm{S}}_2$ and $\mathrm{Co}{\mathrm{Nb}}_3{\mathrm{S}}_6$. (b2) and (c2) Band dispersions of $\mathrm{Nb}{\mathrm{S}}_2$ and $\mathrm{Co}{\mathrm{Nb}}_3{\mathrm{S}}_6$ along the $k_y$ direction [vertical arrows in (b1) and (c1)]. (b3) and (c3) Schematics of the band dispersions of $\mathrm{Nb}{\mathrm{S}}_2$ and $\mathrm{Co}{\mathrm{Nb}}_3{\mathrm{S}}_6$. The black filled and open dots represent peak positions extracted from energy and momentum distribution curves (Figs. S7 and S8 [10]), respectively. Band dispersions originating from $\mathrm{Nb}{\mathrm{S}}_2$ are drawn by pink and additional dispersions in (c3) are marked by blue or light blue. (d1)–(e3) Fermi surfaces and band dispersions of $\mathrm{Fe}{\mathrm{Nb}}_3{\mathrm{S}}_6$ and $\mathrm{Ni}{\mathrm{Nb}}_3{\mathrm{S}}_6$. In the Fermi surface maps, the solid and dashed hexagons correspond to the Brillouin zones of $\mathrm{Nb}{\mathrm{S}}_2$ and $M{\mathrm{Nb}}_3{\mathrm{S}}_6$ respectively.

Figure 2

Evaluation of the broken inversion symmetry in $\mathrm{Co}{\mathrm{Nb}}_3{\mathrm{S}}_6$ by the $k_z$ dispersion of $2H\text{−}\mathrm{Nb}{\mathrm{S}}_2$. (a)–(c) Schematics of layered materials and estimated ARPES spectra along the $k_z$ direction. (d) $k_z$ dispersions of $\mathrm{Nb}{\mathrm{S}}_2$ along the $M\text{−}L$ path by SX-ARPES and DFT calculations. In the left panel, peak positions extracted from energy distribution curves (Fig. S15(a) [10]) are shown by the blue and light blue points. In the right panel, band dispersions corresponding to those observed by SX-ARPES are highlighted by the same colors. (e) Brillouin zone of $\mathrm{Nb}{\mathrm{S}}_2$. (f) Band dispersions of SX-ARPES ($h\nu =525\mathrm{eV}$) and DFT calculations along the $k_y$ direction. Blue curves in the right panel correspond to those in (d). Filled and open circles in the left panel represent peak positions extracted from energy distribution curves (EDCs) [Figs. S14(b) and S14(c) 10], while the open circles are those determined from EDC spectra in $k_y>0$ region and symmetrized with respect to the $\mathrm{\Gamma }$ point. (g) Maximally localized Wannier functions of two bands highlighted by blue in (d) and (f). (h) Schematic of the electronic structure near the Fermi level. Left and right schematics are exchanged by the inversion for the black point, which keeps Co atoms unchanged.

Figure 3

Magnetic and magnetotransport properties of $\mathrm{Co}{\mathrm{Nb}}_3{\mathrm{S}}_6$. (a1) Magnetic susceptibility $\chi$ of $\mathrm{Co}{\mathrm{Nb}}_3{\mathrm{S}}_6$ (light blue and blue) and $\mathrm{Co}{\mathrm{Nb}}_3{\mathrm{S}}_{6-x}$ (pink and red), obtained by field-cooled (FC) and zero-field-cooled (ZFC) processes. (a2) Temperature dependence of the longitudinal resistivity ${\rho }_{xx}$ in $\mathrm{Co}{\mathrm{Nb}}_3{\mathrm{S}}_6$ (light blue) and $\mathrm{Co}{\mathrm{Nb}}_3{\mathrm{S}}_{6-x}$ (pink). (b1) Out-of-plane ferromagnetic moment extracted from $M\text{−}H$ curves like Fig. S16(b) [10]. (b2) Anomalous Hall resistivity ${\rho }_{\mathrm{AHE}}$ extracted from ${\rho }_{xy}\text{−}H$ curves like Fig. S16(c) [10]. (c) Band dispersion of $\mathrm{Co}{\mathrm{Nb}}_3{\mathrm{S}}_{6-x}$ along the $k_y$ direction taken by the 120 eV VUV light. (d) Energy distribution curves of $\mathrm{Co}{\mathrm{Nb}}_3{\mathrm{S}}_6$ and $\mathrm{Co}{\mathrm{Nb}}_3{\mathrm{S}}_{6-x}$ in the red rectangular area in (c). The dashed ovals and solid rectangles in (c) and (d) correspond to each other.

Figure 4

Possibility of Kramers-Weyl nodes near the Fermi level in $\mathrm{Co}{\mathrm{Nb}}_3{\mathrm{S}}_6$. (a) Band dispersions of $M{\mathrm{Nb}}_3{\mathrm{S}}_6(M=\mathrm{Fe},\mathrm{Co},\mathrm{Ni})$ along the blue arrow in the right schematic of the Brillouin zones. (b) Schematic of band dispersions near the Fermi level for $\mathrm{Co}{\mathrm{Nb}}_3{\mathrm{S}}_6$. The Kramers-Weyl node is marked by the blue circle and labeled as “KW.”