Direct observation of the spin–orbit coupling effect in magnetic Weyl semimetal Co3Sn2S2

The spin–orbit coupling (SOC) lifts the band degeneracy that plays a vital role in the search for different topological states, such as topological insulators (TIs) and topological semimetals (TSMs). In TSMs, the SOC can partially gap a degenerate nodal line, leading to the formation of Dirac/Weyl semimetals (DSMs/WSMs). However, such SOC-induced gap structure along the nodal line in TSMs has not yet been systematically investigated experimentally. Here, we report a direct observation of such gap structure in a magnetic WSM Co3Sn2S2 using high-resolution angle-resolved photoemission spectroscopy. Our results not only reveal the existence and importance of the strong SOC effect in the formation of the WSM phase in Co3Sn2S2, but also provide insights for the understanding of its exotic physical properties.


INTRODUCTION
The spin-orbit coupling (SOC) effect originates from the relativistic interaction between a particle's spin and its orbital motion, which can modify atomic energy levels and split the energy bands in crystalline materials. The SOC effect is important in numerous research fields of physics, such as spintronics 1 , ultracold atoms 2 , high-temperature superconductivity 3,4 , and topological materials [5][6][7] . In solids, it can lift the degeneracy of critical bands and play a vital role in the formation of exotic topological states [5][6][7] . As an example, in some topological materials, depending on how the SOC effect modifies the topological nodal line, different topological states can be formed: (i) the topological nodal line semimetals (TNLSMs) can be formed if the nodal line is not gapped 8,9 ; (ii) the formation of topological insulators (TIs) if the nodal line is fully gapped [10][11][12] ; (iii) if the nodal line is partially gapped with isolated nodal points, topological semimetals (TSMs) can be formed [13][14][15][16][17][18][19][20][21] , such as Dirac semimetals (DSMs) or Weyl semimetals (WSMs).
In TIs and TSMs, the strength of SOC can be characterized by the energy gap size between the inverted bands and is dependent on the atomic mass. In compounds with heavy elements (e.g. bismuth-based TIs), the SOC-induced energy gap up to several hundred millielectron volts has been observed [22][23][24][25] . On the other hand, although many TSMs have been discovered [13][14][15][16][17][18][19][20][21][26][27][28][29] up to date, the SOC-induced gap structure along the nodal line has not yet been systematically investigated. However, the recently discovered magnetic WSM Co 3 Sn 2 S 2 18-21 provides an opportunity for such a study.
The WSM Co 3 Sn 2 S 2 has three pairs of Weyl points formed by partially gapped nodal lines due to the SOC effect; 20 which also give rise to giant anomalous Hall effect 18,19 . The scanning tunneling microscopy (STM) measurements observed a large negative flat band magnetism 30 , spin-orbit polaron 31 and impurity-induced magnetic resonance 32 in Co 3 Sn 2 S 2 again indicate the effect of strong SOC in Co 3 Sn 2 S 2 . Interestingly, a recent photoemission study 33 suggests the SOC effect in Co 3 Sn 2 S 2 is negligible and a degenerate Weyl loop state is formed.
In this report, we systematically investigate the SOC-induced gap structure along the nodal line in Co 3 Sn 2 S 2 using highresolution angle-resolved photoemission spectroscopy (ARPES), and directly observed large SOC-induced energy gap (up tõ 55 meV) distribution in the momentum space, which can be well reproduced by our ab initio calculations. These results clearly support the WSM nature, rather than the Weyl loop state in Co 3 Sn 2 S 2 ; which provide a solid electronic structure foundation for understanding many exotic physical properties in Co 3 Sn 2 S 2 , such as the large anomalous Hall conductivity (AHC) 18,19 , large anomalous Hall angle (AHA) 18 , and anomalous Nernst effect (ANE) 34,35 .

RESULTS
SOC effect in Co 3 Sn 2 S 2 Co 3 Sn 2 S 2 is crystallized in a trigonal rhombohedral structure and composed of stacked…-Sn-[S-(Co 3 -Sn)-S]… layers (Fig. 1a). In each layer, Co atoms form a two-dimensional (2D) kagome lattice. Such a unique crystal structure guarantees the inversion symmetry, C 3z rotation symmetry and three mirror planes in Co 3 Sn 2 S 2 . It undergoes a ferromagnetic (FM) transition at~177 K 18,19 . The ab initio calculations on Co 3 Sn 2 S 2 in its FM state reveal a band inversion occurs near Fermi level (E F ) 20 . When SOC is ignored, the band inversion forms a nodal line as illustrated in Fig. 1c (see Supplementary Note 1 for more discussion) and totally six nodal lines locating in three mirror planes of the Brillouin zone (BZ) are formed (Fig. 1b). Each nodal line will be partially gapped when SOC is considered, leaving two nodal points in the formation of the Weyl points with opposite chiralities as illustrated in Fig. 1b, d.
To quantitatively visualize the position of the nodal line and the SOC effect on the nodal line, we first calculated the band dispersions along the M0 À Γ À M direction with and without SOC as shown in Fig Fig. 2d (ii) forming the Weyl point. We illustrate the SOC-induced energy gap size along the nodal line in Fig. 2b. It shows strong anisotropy ranging from 0 tõ 50 meV. As ARPES probes the occupied states below E F , the SOCinduced gap structure in Fig. 2d (iii, iv) can be observed experimentally. Based on the calculations, the portion of the gapped nodal line lying below E F is marked by the cyan region as illustrated in Fig. 2a.

Observation of the SOC effect
To search for the gapped structure along the nodal line, we carried out detailed photon energy-dependent measurements along the We also carried out high-resolution measurements on the Fermi surface (FS) topology using the photon energy of 125 eV (Fig. 4a) and 134 eV (Fig. 4c)   k y (Å -1 ) k y (Å -1 ) k y (Å -1 ) k y (Å -1 ) k y (Å -1 ) k y (Å -1 )   along the Γ M direction on both FSs in the first BZ. These features originate from the upper branch of the gapped nodal line. In addition to the SOC-induced gap observed along the Γ M direction (Fig. 3), we extracted the band dispersion through the spot-like feature perpendicular to the Γ M direction as shown in Fig. 4b, d.
Apparently, an energy gap developed between the upper and lower branch of the bands is clearly observed as illustrated by the arrows in Fig. 4b (i) and 4d (i). Such gap structure can also be seen from the EDCs, characterized by the two peaks structure with a dip in the middle [ Fig. 4b (ii) and 4d (ii)]. By tracking the band dispersions, the position of the nodal line were determined experimentally as shown in Fig. 4e. It shows excellent agreement with the calculations. To quantitatively extract the SOC-induced gap size along the nodal line, we fit the two peaks of the EDCs by using two Lorentzian curves [see the insets of Fig. 4b (ii) and 4d (ii) for example]. The gap size is extracted by the energy interval between the two peaks and the results are shown in Fig. 4f. The gap size measured in the cyan region is around 55 meV, which is consistent with the calculations (Fig. 2b). The k z independent gap size in Fig. 4f is mainly caused by the large k z broadening effect 21 .

DISCUSSION
The SOC-induced gap structure in Co 3 Sn 2 S 2 can help to understand the experimentally observed large AHC 18,19 , large AHA 18 and the ANE 34,35 . The anomalous Hall effect in Co 3 Sn 2 S 2 is intrinsic that originates from the large Berry curvature 18 , which will also enhance the thermoelectric response 34,35 . The calculations show the large Berry curvature arises mainly from the gapped region of the nodal line 18 . Our observations of the SOC-induced gap structure along the nodal line and the extracted positions of the nodal line show consistence with the calculations, suggesting that the formation of the gapped nodal line is essential for the large AHC 18,19 , large AHA 18 and the ANE 34,35 .
Compared with our results, the E F in the previous work 33 lies at the top of the lower branch of the gapped nodal line, thus the SOC-induced gap structure lies above the E F and can not be observed. Our results not only demonstrate the WSM nature of Co 3 Sn 2 S 2 with isolated Weyl points, rather than the degenerate Weyl loops, but also reveal the existence and importance of the strong SOC effect in the formation of the WSM phase in Co 3 Sn 2 S 2 , as well as provide important insights for the understanding of many exotic physical properties in Co 3 Sn 2 S 2 .

METHODS
High quality Co 3 Sn 2 S 2 single crystals were grown by self-flux method that can be found elsewhere. ARPES measurements were performed at beamline I05 of the Diamond Light Source (DLS) with a Scienta R4000 analyzer 36 . The angle resolution and overall energy resolution were better than 0.2°and 15 meV, respectively. The single crystals were cleaved in situ below 10 K. The pressure was kept below 2×10 −10 Torr during the whole measurement.

DATA AVAILABILITY
The data of this study are available from the corresponding author upon reasonable request. The extracted band structure (i) and its corresponding EDCs (ii) perpendicular to the Γ M direction through the spot-like feature. The momentum path is illustrated by the red line in (a). The SOC-induced gap is clearly observed as illustrated by the arrow. To quantitatively extract the gap size, the EDC in red is fitted by using two Lorentzian curves as illustrated by the green curves in the inset of (ii). c, d The same to (a, b) but the data are taken at 134 eV photon energy. Note that the experimental plot has been symmetrized according to the crystal symmetry.