Nodal-line resonance generating the giant anomalous Hall effect of ${\mathrm{Co}}_3{\mathrm{Sn}}_2{\mathrm{S}}_2$

Giant anomalous Hall effect (AHE) and magneto-optical activity can emerge in magnets with topologically nontrivial degeneracies. However, identifying the specific band-structure features such as Weyl points, nodal lines, or planes which generate the anomalous response is a challenging issue. Since the low-energy interband transitions can govern the static AHE, we addressed this question in the prototypical magnetic Weyl semimetal ${\mathrm{Co}}_3{\mathrm{Sn}}_2{\mathrm{S}}_2$ also hosting nodal lines by broadband polarized reflectivity and magneto-optical Kerr effect spectroscopy with a focus on the far-infrared range. In the linear dichroism spectrum we observe a strong resonance at 40 meV, which also appears in the optical Hall conductivity and primarily determines the static AHE, and thus confirms its intrinsic origin. Our material-specific theory reproduces the experimental data remarkably well and shows that strongly tilted nodal-line segments around the Fermi energy generate the resonance. While the Weyl points only give vanishing contributions, these segments of the nodal lines gapped by the spin-orbit coupling dominate the low-energy optical response and generate the giant AHE.

Figure 1

Optical spectra measured for several temperatures between 10 and 200 K of (a) the reflectivity on the kagome plane and (b) along the stacking direction. The insets highlight the orientation of the electric field. (c) Kerr-rotation and (d) ellipticity on the $\mathit{ab}$-plane in the energy range up to 3 eV.

Figure 2

Comparison of the experimental conductivity spectra measured between 10 and 200 K (colored lines) and the theoretical DFT spectra (black lines) calculated as described in the text. (a), (b), (c), and (d) respectively show the real parts of the diagonal, $Re{\sigma }_{xx}$ and $Re{\sigma }_{zz}$, as well as the imaginary and real parts of the off-diagonal conductivity spectra, $Im{\sigma }_{xy}$ and $Re{\sigma }_{xy}$. For comparison, the static conductivity values are plotted as colored squares at zero energy. The respective insets show a broad energy range.

Figure 3

(a) Optical anisotropy spectra $Re{\sigma }_{zz}/Re{\sigma }_{xx}$. The inset shows a broader energy range. (b) Hall angle spectra ${\mathrm{\Theta }}_{\text{H}}=\text{arctan}Re({\sigma }_{xy}/{\sigma }_{xx})$ with a maximum of 42.7°.

Figure 4

(a) Location of the nodal lines and the Weyl nodes in the Brillouin zone. The color scale encodes the position relative to the Fermi energy. (b) The Hall spectral weight of the calculated 34.3-meV peak on the mirror plane containing the nodal line and Weyl points. (c) Band structure along the main contributing areas of the real part of the off-diagonal optical conductivity. The gray shading highlights the energy range below 50 meV where we expect contributions to the peak in $Re{\sigma }_{xy}$. The non-high-symmetry points $A$ and $B$ are (0.0, 0.4002, 0.301) and (0.0, 0.7373, 0.0) in reciprocal lattice units.