Magnetic, transport and topological properties of Co-based shandite thin films

After the prediction of the Berezinskii-Kosterlitz-Thouless transition and Haldane’s seminal work for a spontaneous quantum anomalous Hall effect (AHE) in two spatial dimensions, it has been widely recognised that condensed matters can be characterised using the concept of topology Nobel . The Weyl semimetals are a representative in three dimensions that have attracted much attention because of the variety of interesting physical properties arising from their unique topology AMV . The energy spectrum of the Weyl semimetals possesses Weyl nodes, which can be regarded as “monopoles” with chirality $+1$ or $-1$ in momentum space. The Weyl nodes always appear in pairs with positive and negative chirality NN , and this feature generates the Fermi arc on the surface through the bulk-edge correspondence Murakami , as indeed observed by the angle-resolved photoemission spectroscopy Xu ; Souma . The peculiar topology of the Weyl semimetals is also detected in quantum transport phenomena, such as the large AHE Murakami ; Burkov ; Suzuki , the anomalous Nernst effect (ANE) Sakai and the negative magnetoresistance associated with the chiral anomaly due to the monopoles Burkov ; XHuang ; Zhang .

A prescription for the Weyl semimetals is to lift the degeneracy in the Dirac nodes by breaking symmetry. The Weyl semimetals with broken spatial inversion symmetry have been confirmed in several materials, e.g., transition metal pnictides Xu ; Souma ; WFFBD and dichalcogenides Soluyanov ; Feng , whereas those with broken time-reversal symmetry have been rather scarce, with only a few theoretical proposals WTVS ; XWWDF ; KF and experiments under the magnetic field Suzuki ; Sakai . Recently, however, a kagome ferromagnet Co-based shandite, Co₃Sn₂S₂, has been discovered to possess the Weyl nodes by spontaneous breaking of time-reversal symmetry, in the vicinity of $60$ meV from the Fermi level QXu ; Wang ; Liu . Indeed, theoretical and experimental studies revealed that the giant AHE and ANE observed in this compound are of the intrinsic contribution from the Weyl nodes QXu ; Wang ; Liu ; Guin ; Yang ; Minami ; Yanagi .

Co₃Sn₂S₂ has a quasi-two-dimensional lattice structure with Co kagome layers, as shown in Fig. 1a, b. Recently, two types of Co kagome monolayers with different surface terminations, Co₃Sn₃S₂ and Co₃SnS₂, were investigated theoretically Muechler . As a remnant of the Weyl nodes in the bulk, both monolayers are predicted to show a large AHE; especially in Co₃Sn₃S₂ with Sn-end surfaces, the Weyl gap opens at the Fermi level, which leads to the quantum AHE. Stimulated by this theoretical prediction, experimental attempts to fabricate monolayer systems have been carried out, and the magnetism, transport phenomena and surface states have been discussed for atomically thin films of Co₃Sn₂S₂ Fujiwara ; Shiogai ; Ikeda1 ; Ikeda2 . In the theoretical study, however, the lattice parameters were fixed at the bulk values for both monolayers, despite the fact that the lattice structures are expected to be relaxed in a different manner depending on the surface termination, sensitively affecting the magnetism and band topology of the atomically thin films GG ; BHuang . More importantly, since the Weyl points are the topological singularities predicted for the three-dimensional bulk systems, it remains elusive how they appear while changing the film thickness.

In this study, we perform comprehensive ab initio calculations of the band topology and transport properties for thin films of Co₃Sn₂S₂, with special attention to the optimal lattice structures and stable magnetic states. We study monolayers, bilayers and trilayers of Co kagome planes with two different surface terminations by Sn and S atoms (Fig. 1c–h). From the obtained magnetic and electronic states, we compute the anomalous Hall conductivity and the anomalous Nernst conductivity. Through the systematic study in comparison with the bulk, we discuss the evolution of the topological properties with the number of kagome layers from one, two, three to infinity.

Next, for the optimised structures, we compare the ab initio energy for various magnetic states (see Supplementary Note 2). The lowest-energy states and the magnitudes of Co moments are summarised in Table 1 (see Methods); see also Fig. 2a, e, i, m and Fig. 3a, e, i. For the bulk, a ferromagnetic state is obtained with Co moment of $0.350$ $\mu_{\rm B}$ along the out-of-plane direction, where $\mu_{\rm B}$ is the Bohr magneton; the total magnetic moment per unit cell is $1.00$ $\mu_{\rm B}$ because of small but nonzero antiparallel magnetic moments at Sn and S atoms. The result agrees well with the previous studies Liu ; Wang ; Yanagi . For the Sn-end thin films, similar out-of-plane ferromagnetic states are obtained for all the monolayer, bilayer and trilayer systems. The Co moment changes with the number of Co layers: $0.415$ $\mu_{\rm B}$ for monolayer, $0.372$ $\mu_{\rm B}$ for bilayer, and $0.365$ $\mu_{\rm B}$ (top and bottom layers) and $0.362$ $\mu_{\rm B}$ (middle layer) for trilayer; the value is monotonically reduced to the bulk one as the number of Co layers increases. We note that the net moment per primitive unit cell is $1.00$, $2.01$ and $3.00$ $\mu_{\rm B}$ for monolayer, bilayer and trilayer, respectively, indicating that the value per Co layer is the same as in the bulk. In contrast, for the S-end thin films, we obtain different magnetic states depending on the layer number: an out-of-plane ferromagnetic state for monolayer, an intralayer in-plane ferromagnetic and interlayer antiferromagnetic state for bilayer, and an intralayer out-of-plane ferromagnetic and interlayer ferrimagnetic state for trilayer, as shown in Fig. 3a, e, i, whose net magnetic moments per unit cell are $3.01$, $0.00$ and $1.00$ $\mu_{\rm B}$, respectively. The Co moment is also largely different from the Sn-end cases: $1.01$ $\mu_{\rm B}$ for monolayer, $0.339$ $\mu_{\rm B}$ for bilayer, and $0.207$ $\mu_{\rm B}$ (top and bottom layers) and $0.080$ $\mu_{\rm B}$ (middle layer) for trilayer. The contrasting behaviours between the Sn- and S-end series are summarised in Fig. 4a.

Our results indicate that the ferromagnetic order within each Co kagome layer is stable, regardless of the number of Co layers and the surface termination. In addition, the magnetic order has out-of-plane anisotropy, except for the S-end bilayer system. These suggest that the out-of-plane ferromagnetic state of the Co layer, which is a building block for the bulk system, is robust in most cases of thin films. Meanwhile, our results reveal that the surface termination matters to the interlayer magnetism as well as the magnetic anisotropy. Analysing the maximally-localised Wannier functions, we indeed find that the spatial extensions of the Co $d$ orbitals are substantially different for different surface terminations, which would be relevant to the interlayer magnetic interactions (see Supplementary Note 3).

Let us first discuss the Sn-end films in Fig. 2. In the case of the monolayer, the band structure has a gap of $\sim 50$ meV at the Fermi level between the bands $M_{1}$ and $M_{2}$, and there is no Fermi surface. The bands $M_{1}$ and $M_{2}$ have the Chern number $6$ and $-3$, respectively. The gap is opened by the spin-orbit coupling at the Weyl nodes on the $\Gamma$-$\rm K$ line in momentum space, resulting in the large values of the Berry curvatures near the band top and bottom. Thus, the system is an insulator with the Weyl gap, as pointed out in the previous study without structural optimisation Muechler . This is reasonable, given the little changes in the lattice structure as well as the magnetism after the optimisation in our calculations. In the bilayer, however, the system becomes metallic with the electron Fermi surfaces around the $\rm K$ point in the band $B_{1}$ and the pairs of the hole pockets on both sides of each $\Gamma$-$\rm M$ line in the band $B_{2}$. The Chern numbers of the bands $B_{1}$ and $B_{2}$ are $9$ and $-1$, respectively. By closely looking at the band structure, we can identify the Weyl gap of $\sim 30$ meV between $B_{1}$ and $B_{2}$ slightly above the Fermi level, which appears to correspond to that in the monolayer case but split into two, leading to the hole pocket pair; see also the band structure in Fig. 2h. In the trilayer system, the band structure becomes more complicated, but still retains the characteristic features in the monolayer and bilayer: One of the Weyl gaps opens between the bands $T_{2}$ and $T_{3}$ on the $\Gamma$-K line above the Fermi level, while the other two are along the $\Gamma$-M line, as shown in the band structure in Fig. 2l. The Chern numbers of the bands $T_{1}$, $T_{2}$ and $T_{3}$ are $-3$, $6$ and $3$, respectively. These overall systematic changes of the electronic states appear to smoothly connect to the bulk. In fact, as shown in Fig. 2n, o, the $k_{z}=0$ cut of the bulk band structure shows similar distribution of the Weyl gap and the Berry curvature to the above Sn-end films. In particular, the large Berry curvature in the Sn-end films can be regarded as a remnant of the intersections of the nodal gap in the bulk system QXu ; Minami . Thus, in the Sn-end series, where the out-of-plane ferromagnetic state persists, the electronic structure and the topological properties change systematically as the number of Co layers increases.

Next, we discuss the S-end cases. In the monolayer, the system is metallic with two electron pockets around the $\Gamma$ point as shown in Fig. 3b, in contrast to the Sn-end case. The result is also different from the previous study without structural optimisation Muechler . The difference is ascribed to the substantial shrinkage of the lattice constants shown in Table 1. We find considerable values of the Berry curvature in the bands close to the Fermi level, which affect the transport properties as discussed later. In the case of the bilayer, the system is also metallic with one electron pocket around the $\Gamma$ point; all the bands are doubly degenerate due to the interlayer antiferromagnetic ordering, cancelling out the Berry curvature. In the trilayer, there are three bands forming two electron pockets around the $\Gamma$ point, one electron pocket around the K point, and one hole pocket around the M point. In each band, the Berry curvature takes nonzero values, similar to the monolayer case. In addition, we find two small Weyl gaps between the bands $t_{1}$ and $t_{2}$ along the $\rm M$-$\Gamma$ line and another one between $t_{2}$ and $t_{3}$ along $\Gamma$-$\rm K$ line close to the Fermi level. Although this apparently resembles the case of Sn-end trilayer and bulk systems, these bands show qualitatively different Berry curvatures due to the different magnetic state. Thus, in stark contrast to the Sn-end cases, the S-end series shows no systematic change not only in the magnetic orders but also in the electronic structure and the band topology.

In contrast, $\sigma_{\rm H}$ does not show systematic changes in the S-end cases, as shown in Fig. 3d, h and l: $\sigma_{\rm H}/(e^{2}/h)\simeq-0.34$, $0.00$ and $-2.71$ for the monolayer, bilayer and trilayer, respectively. The absence of the AHE in the bilayer case is due to the cancellation of the Berry curvature by the band degeneracy discussed above, and the negative values of $\sigma_{\rm H}$ for the monolayer and trilayer originate from the different band topology from the Sn-end series. Note that in the monolayer case, $\sigma_{\rm H}$ is largely different from that of the previous study Muechler because of the structural optimisation.

We also compute the anomalous Nernst conductivity $\alpha_{\rm N}$ for both series of the films. According to the generalised Mott formula $\alpha_{\rm N}=-\frac{k_{\rm B}}{e}\int d{\varepsilon}s({\varepsilon})\frac{\partial\sigma_{\rm H}({\varepsilon})}{\partial{\varepsilon}}$ XYFN ; Yanagi , $\alpha_{\rm N}$ is related with the energy derivative of $\sigma_{\rm H}$. This holds for our results, as plotted in Fig. 2d, h, l, p and Fig. 3d, h, l. In the Sn-end series, $\alpha_{\rm N}$ is almost zero at the Fermi level for the monolayer because of the quantisation of $\sigma_{\rm H}$, while it takes a negative value for the bilayer and trilayer cases corresponding to the positive derivative of $\sigma_{\rm H}$. Interestingly, $\alpha_{\rm N}$ takes a large negative value for the bilayer since the Fermi level is on the verge of the plateau-like feature of $\sigma_{\rm H}$ where $\sigma_{\rm H}$ increases rapidly. Since $\alpha_{\rm N}$ is positive in the bulk, our results suggest a sign change while increasing the number of Co layers, in contrast to $\sigma_{\rm H}$. Meanwhile, in the S-end cases, $\alpha_{\rm N}$ vanishes for the bilayer, while it takes small negative values for the monolayer and trilayer. The layer number dependences for both cases are summarised in Fig. 4b.

Our systematic ab initio study for the thin films of Co-based shandite revealed that the number of Co kagome layers and the surface termination are both relevant to the lattice structure, the magnetic state, the electronic band structure and its topology, and transport properties. We found that the Sn-end series shows similar behaviours to the bulk, and retains the large AHE down to the monolayer limit where the quantum AHE is realised, reflecting the systematic change of the gapped Weyl nodes in the band structure. In contrast, the S-end series exhibits nonmonotonic behaviours with respect to the layer number, owing to the dominantly antiferromagnetic interlayer coupling. Our results would stimulate further experimental studies of the thin films of Co-based shandite to realise the quantum AHE in the monolayer form. The important implication of our results is that the measurements of the AHE and ANE as well as the magnetism will distinguish the film thickness and the surface termination, both of which are usually hard to directly identify in experiments. In particular, the sign of $\sigma_{\rm H}$, which is positive in the Sn-end films while negative in the S-end films, can be relevant measurement. Furthermore, the different sign of $\alpha_{\rm N}$ from the bulk will also be useful to measure the film thickness. Furthermore, our prediction of the systematic evolution of the topological band structure would stimulate optical measurements, which have been useful to detect the chirality and gap structure of the Weyl nodes away from the Fermi level Ma ; YXu ; Okamura .

The authors thank M. H. N. Assadi, K. Fujiwara, K. Nomura, S. Okumura, A. Ozawa and A. Tsukazaki for fruitful discussions. The parts of the calculation have been done using the facilities of the Supercomputer Center, the Institute for Solid State Physics, the University of Tokyo. This work is supported by JST CREST (Grant No. JPMJCR18T2) and JSPS KAKENHI (Grant No. JP21K13875 and JP22K03509).

We present additional information of the optimised lattice structures in Supplementary Table 1: the averaged angles for neighbouring Co-Sn-Co and Co-S-Co bonds, $\theta_{\rm Co{{\mathchar 45\relax}}Sn{{\mathchar 45\relax}}Co}$ and $\theta_{\rm Co{{\mathchar 45\relax}}S{{\mathchar 45\relax}}Co}$, respectively, and the averaged bond lengths of the nearest-neighbour Co ions in different layers, $d_{\rm inter{{\mathchar 45\relax}}Co}$. In most cases, the bond angles and lengths do not change so much from the bulk values, except for $\theta_{\rm Co{{\mathchar 45\relax}}S{{\mathchar 45\relax}}Co}$ at the surfaces.

We show in Supplementary Table 2 the energy for different magnetic states obtained by the fully relativistic ab initio calculations for the optimised lattice structures. As listed in Supplementary Figure 2, we took the initial states as ferromagnetic (FM) states, 120 degree noncollinear antiferromagnetic (120AFM) states, and intralayer FM and interlayer collinear antiferromagnetic (IAFM) states [interlayer collinear ferrimagnetic (IFRM) state for trilayer]. For the FM states, we considered three types whose magnetic moments are along the out-of-plane ($c$) and two in-plane ($a$ and $b^{\prime}$) directions; for the 120AFM states, we considered three types with different chirality and helicity; for the IAFM states, we considered three types similar to the FM cases. For the Sn-end systems, the FM states with the out-of-plane magnetic moments are the most stable, similar to the bulk, with the energy differences from the in-plane FM states being about 1$\sim$2 meV. In contrast, for the S-end systems, the stable magnetic states and the directions of the magnetic moments depend on the layer number as described in the main text. The energy differences between the most stable states and the other competing states are less than $1$ meV, which are smaller than the Sn-end cases. This implies that the magnetic anisotropy is smaller for the S-end films than the Sn-end ones.

We display the maximally-localised Wannier functions with Co $d_{z^{2}}$ and $d_{x^{2}-y^{2}}$ orbital characters in Supplementary Figure 3, which are mostly different between the Sn- and S-end films among the $d$ orbitals. The results are obtained for the paramagnetic state by the relativistic calculations. In all cases, the Wannier functions in the S-end systems, especially of the Co ions near the surfaces, have wider spatial extensions to the interlayer directions. This difference leads to different overlaps of the wave functions between the Sn- and S-end systems, possibly being relevant to the difference in the interlayer magnetic interactions.

Supplementary Figure 4 shows the results of the GGA+$U$ calculations for the S-end monolayer under different $U$ on the Co $d$ electrons within the fully relativistic framework. We find that the band gap is not opened up at least to $U=4$ eV, indicating that the S-end monolayer remains metallic even when including the Coulomb interactions at the level of GGA+$U$. However, the transport properties will be altered because of the large modification of the band structure. We also performed GGA+$U$ calculations for the S-end bilayer, and find that the system turns into a FM state for $U\gtrsim 2$ eV, where the double band degeneracy is lifted and the anomalous Hall and Nernst effects are expected to occur.