Giant magneto-optical responses in magnetic Weyl semimetal Co3Sn2S2

The Weyl semimetal (WSM), which hosts pairs of Weyl points and accompanying Berry curvature in momentum space near Fermi level, is expected to exhibit novel electromagnetic phenomena. Although the large optical/electronic responses such as nonlinear optical effects and intrinsic anomalous Hall effect (AHE) have recently been demonstrated indeed, the conclusive evidence for their topological origins has remained elusive. Here, we report the gigantic magneto-optical (MO) response arising from the topological electronic structure with intense Berry curvature in magnetic WSM Co3Sn2S2. The low-energy MO spectroscopy and the first-principles calculation reveal that the interband transitions on the nodal rings connected to the Weyl points show the resonance of the optical Hall conductivity and give rise to the giant intrinsic AHE in dc limit. The terahertz Faraday and infrared Kerr rotations are found to be remarkably enhanced by these resonances with topological electronic structures, demonstrating the novel low-energy optical response inherent to the magnetic WSM.

2 Supplementary Note 1: Hall conductivity spectra for the single anti-crossing point.
The single anti-crossing point in the two-dimensional electronic structure was employed as the simplest model for the anomalous Hall effect (Fig. 1a in the main text), which is expressed by the following Hamiltonian H(k), where 0 and i (i=x, y, z) are the identity and Pauli matrices, respectively, and  is the chemical potential. Assuming that the band dispersion is two-dimensional and (hx, hy, hz) = (kx, ky, m), the corresponding band dispersion has the level splitting of 2|m| at the Weyl point, which is schematically illustrated in the inset of Fig. 1a in the main text.
We calculated the optical Hall conductivity xy() with use of the general expression given by the Kubo formula; where the Jx(y) is the current operator given by is the Fermi distribution function, and | ⟩ are the energy and the Bloch wave function of the n-th band, respectively, and  is the damping constant. The energy-dependent Hall conductivity xy() is thus given by; For the calculations shown in the main text, we assumed  = 1 meV.
3 Supplementary Note 2: Quantitative discussion of the Faraday rotations.
The Faraday rotation for the free-standing bulk crystal is given by, where c and d are the speed of light and thickness of the sample, respectively. n represents the difference in the refractive indices for the right and left circularly polarized light. Therefore, the F/d is a good measure to evaluate the magnitude of the Faraday rotation quantitatively for the bulk case, i.e. d/c>>1. However, in case of the thin film on top of the substrate, the terahertz Faraday rotation is given by, F+iF = Z0xyd/(1+ns+ Z0xxd), as described in Method section; thus, the F is the nonlinear function of d and does not necessarily represent the intrinsic material parameters. Therefore, to discuss the magnitude of the Faraday rotation in a unified manner, the n and the figure of merit defined by ndp/2c, where dp is the penetration depth, are more appropriate. Here, for the Co3Sn2S2 thin film, |n| and ndp/2c are estimated to be 10.6 and 451 mrad, respectively at 7.5 meV, both of which are much larger than those of YIG at 4.16 eV S1,2 (Supplementary Table 1). We also calculate the F/d for the free-standing film (F calc. /d); it decreases from the experimentally observed F obs. /d, but is still larger than that of YIG. Re xy() and Im xx() show the sharp resonance peaks, while Im xy() and Re xx() have the step-function like structure at the interband transition energy (2) (Fig. 1a). respectively. The zero-field Hall conductivity disappears above 120 K because the single domain state is unstable in zero field near TC. Overall characteristics of transport properties including the magnitude are comparable between the bulk and thin film. We note that the TC of the thin film (~ 184 K) is higher than that of the bulk (~ 172 K), which 7 is possibly due to slight composition discrepancy because the TC tends to scatter in sample to sample in this system. One other possible cause is the strain effect. Since the reduction of the TC by applying isotropic pressure has been reported for the bulk system S6 , the uniaxial pressure effect for the thin film may play a role. negligibly small compared with that of the Co3Sn2S2 film, and therefore the possible modification of the terahertz spectra owing to the SiO2 capping layer can be neglected within the accuracy of our measurement.