Topological Weyl semimetals in the chiral antiferromagnetic materials Mn3Ge and Mn3Sn

Recent experiments revealed that Mn3Sn and Mn3Ge exhibit a strong anomalous Hall effect at room temperature, provoking us to explore their electronic structures for topological properties. By ab initio band structure calculations, we have observed the existence of multiple Weyl points in the bulk and corresponding Fermi arcs on the surface, predicting antiferromagnetic Weyl semimetals in Mn3Ge and Mn3Sn. Here the chiral antiferromagnetism in the Kagome-type lattice structure is essential to determine the positions and numbers of Weyl points. Our work further reveals a new guiding principle to search for magnetic Weyl semimetals among materials that exhibit a strong anomalous Hall effect.

In a WSM, the conduction and valence bands cross each other linearly through nodes called Weyl points. Between a pair of Weyl points with opposite chiralities (sink or source of the Berry curvature) 4 , the emerging Berry flux can lead to the anomalous Hall effect (AHE) 29 , as observed in GdPtBi 26,27 , and an intrinsic spin Hall effect (SHE), as predicted in TaAstype materials 30 , for systems without and with time-reversal symmetry, respectively. Herein, we raise a simple recipe to search for WSM candidates among materials that host strong AHE or SHE.
Recently, Mn 3 X (where X = Sn, Ge, and Ir), which exhibit noncollinear antiferromagetic (AFM) phases at room temperature, have been found to show large AHE [31][32][33][34] and SHE 35 , provoking our interest to investigate their band structures. In this work, we report the existence of Weyl fermions for Mn 3 Ge and Mn 3 Sn compounds and the resultant Fermi arcs on the surface by ab initio calculations, awaiting experimental verifications. Dozens of Weyl points exist near the Fermi energy in their band structure, and these can be well under-stood with the assistance of lattice symmetry.

II. METHODS
The electronic ground states of Mn 3 Ge and Mn 3 Sn were calculated by using density-functional theory (DFT) within the Perdew-Burke-Ernzerhof-type generalized-gradient approximation (GGA) 36 using the Vienna ab initio Simulation Package (vasp) 37 . The 3d 6 4s 1 , 4s 2 4p 2 , and 5s 2 5p 2 electrons were considered as valance electrons for Mn, Ge, and Sn atoms, respectively. The primitive cell with experimental crystal parameters a = b = 5.352 and c = 4.312 Å for Mn 3 Ge and a = b = 5.67 and c = 4.53 Å for Mn 3 Sn were adopted. Spin-orbit coupling (SOC) was included in all calculations.
To identify the Weyl points with the monopole feature, we calculated the Berry curvature distribution in momentum space. The Berry curvature was calculated based on a tightbinding Hamiltonian based on localized Wannier functions 38 projected from the DFT Bloch wave functions. Chosen were atomic-orbital-like Wannier functions, which include Mn-spd and Ge-sp/Sn-p orbitals, so that the tight-binding Hamiltonian is consistent with the symmetry of ab initio calculations. From such a Hamiltonian, the Berry curvature can be calculated using the Kubo-formula approach 39 , where Ω γ n ( k) is the Berry curvature in momentum space for a given band n,v α(β,γ) = 1 ∂Ĥ ∂k α(β,γ) is the velocity operator with α, β, γ = x, y, z, and |u n ( k) and E n ( k) are the eigenvector and eigenvalue of the HamiltonianĤ( k), respectively. The summation of Ω γ n ( k) over all valence bands gives the Berry curvature vector Ω (Ω x , Ω y , Ω z ).
In addition, the surface states that demonstrate the Fermi arcs were calculated on a semi-infinite surface, where the momentum-resolved local density of states (LDOS) on the surface layer was evaluated based on the Green's function method. We note that the current surface band structure corresponds to the bottom surface of a half-infinite system. arXiv:1608.03404v2 [cond-mat.mtrl-sci] 17 Aug 2016

III. RESULTS AND DISCUSSION
A. Symmetry analysis of the antiferromagnetic structure Mn 3 Ge and Mn 3 Sn share the same layered hexagonal lattice (space group P6 3 /mmc, No. 193). Inside a layer, Mn atoms form a Kagome-type lattice with mixed triangles and hexagons and Ge/Sn atoms are located at the centers of these hexagons. Each Mn atom carries a magnetic moment of 3.2 µB in Mn 3 Sn and 2.7 µB in Mn 3 Ge. As revealed in a previous study 40 , the ground magnetic state is a noncollinear AFM state, where Mn moments align inside the ab plane and form 120-degree angles with neighboring moment vectors, as shown in Fig.1b. Along the c axis, stacking two layers leads to the primitive unit cell. Given the magnetic lattice, these two layers can be transformed into each other by inversion symmetry or with a mirror reflection (M y ) adding a half-lattice (c/2) translation, i.e., a nonsymmorphic symmetry {M y |τ = c/2}. In addition, two other mirror reflections (M x and M z ) adding time reversal (T), M x T and M z T , exist.
In momentum space, we can utilize three important symmetries, M x T , M z T , and M y , to understand the electronic structure and locate the Weyl points. Suppose a Weyl point with chirality χ (+ or −) exists at a generic position k (k x , k y , k z ). Mirror reflection reverses χ while time reversal does not and both of them act on k. The transformation is as follows: Each of the above three operations doubles the number of Weyl points. Thus, eight nonequivalent Weyl points can be generated at (±k x , +k y , ±k z ) with chirality χ and (±k x , −k y , ±k z ) with chirality −χ (see Fig. 1d). We note that the k x = 0/π or k z = 0/π plane can host Weyl points. However, the k y = 0/π plane cannot host Weyl points, because M y simply reverses the chirality and annihilates the Weyl point with its mirror image if it exists. Similarly the M y mirror reflection requires that a nonzero anomalous Hall conductivity can only exist in the xz plane (i.e., σ xz ), as already shown in Ref. 34.
In addition, the symmetry of the 120-degree AFM state is slightly broken in the materials, owing to the existence of a tiny net moment (∼0.003 µB per unit cell) 33,34,40 . Such weak symmetry breaking seems to induce negligible effects in the transport measurement. However, it gives rise to a perturbation of the band structure, for example, shifting slightly the mirror image of a Weyl point from its position expected, as we will see in the surface states of Mn 3 Ge.

B. Weyl points in the bulk band structure
The bulk band structures are shown along high-symmetry lines in Fig. 2 for Mn 3 Ge and Mn 3 Sn. It is not surprising that the two materials exhibit similar band dispersions. At first glance, one can find two seemingly band degenerate points at Energies are relative to the Fermi energy E F . Each of W 1,2,7 has four copies whose coordinates can be generated from the symmetry as (±k x , ±k y , k z = 0). W 4 has four copies at (k x ≈ 0, ±k y , ±k z ) and W 9 has two copies at (k x ≈ 0, ±k y , k z = 0). Each of the other Weyl points has four copies whose coordinates can be generated from the symmetry as (±k x , ±k y , ±k z Z and K points, which are below the Fermi energy. Because of M z T and the nonsymmorphic symmetry {M y |τ = c/2}, the bands are supposed to be quadruply degenerate at the Brillouin zone boundary Z, forming a Dirac point protected by the nonsymmorphic space group [41][42][43] . Given the slight mirror symmetry breaking by the residual net magnetic moment, this Dirac point is gapped at Z (as shown in the enlarged panel) and splits into four Weyl points, which are very close to each other in k space. A tiny gap also appears at the K point. Nearby, two additional Weyl points appear, too. Since the Weyl point separations are too small near both Z and K points, these Weyl points may generate little observable consequence in experiments such as those for studying Fermi arcs. Therefore, we will not focus on them in the following investigation. Mn 3 Sn and Mn 3 Ge are actually metallic, as seen from the band structures. However, we retain the terminology of Weyl semimetal for simplicity and consistency. The valence and conduction bands cross each many times near the Fermi energy, generating multiple pairs of Weyl points. We first investigate the Sn compound. Supposing that the total valence electron number is N v , we search for the crossing points between the N th v and (N v + 1) th bands. As shown in Fig. 3a, there are six pairs of Weyl points in the first Brillouin zone; these can be classified into three groups according to their positions, noted as W 1 , W 2 , and W 3 . These Weyl points lie in the M z plane (with W 2 points being only slightly off this plane owing to the residual-momentinduced symmetry breaking) and slightly above the Fermi energy. Therefore, there are four copies for each of them according to the symmetry analysis in Eq. 2. Their representative coordinates and energies are listed in Table I and also indicated in Fig. 3a. A Weyl point (e.g., W 1 in Figs. 3b and 3c) acts as a source or sink of the Berry curvature Ω, clearly showing the monopole feature with a definite chirality.
In contrast to Mn 3 Sn, Mn 3 Ge displays many more Weyl points. As shown in Fig. 4a and listed in Table II, there are nine groups of Weyl points. Here W 1,2,7,9 lie in the M z plane with W 9 on the k y axis, W 4 appears in the M x plane, and the others are in generic positions. Therefore, there are four copies of W 1,2,7,4 , two copies of W 9 , and eight copies of other Weyl points. Although there are many other Weyl points in higher energies owing to different band crossings, we mainly focus on the current Weyl points that are close to the Fermi energy. The monopole-like distribution of the Berry curvature near these Weyl points is verified; see W 1 in Fig. 4 as an example. Without including SOC, we observed a nodalring-like band crossing in the band structures of both Mn 3 Sn and Mn 3 Ge. SOC gaps the nodal rings but leaves isolating band-touching points, i.e., Weyl points. Since Mn 3 Sn exhibits stronger SOC than Mn 3 Ge, many Weyl points with opposite chirality may annihilate each other by being pushed by the strong SOC in Mn 3 Sn. This might be why Mn 3 Sn exhibits fewer Weyl points than Mn 3 Ge.

C. Fermi arcs on the surface
The existence of Fermi arcs on the surface is one of the most significant consequences of Weyl points inside the threedimensional (3D) bulk. We first investigate the surface states of Mn 3 Sn that have a simple bulk band structure with fewer Weyl points. When projecting W 2,3 Weyl points to the (001) surface, they overlap with other bulk bands that overwhelm the surface states. Luckily, W 1 Weyl points are visible on the Fermi surface. When the Fermi energy crosses them, W 1 Weyl points appear as the touching points of neighboring hole and electron pockets. Therefore, they are typical type-II Weyl points 20 . Indeed, their energy dispersions demonstrate strongly tilted Weyl cones.
The Fermi surface of the surface band structure is shown in Fig. 3d for the Sn compound. In each corner of the surface Brillouin zone, a pair of W 1 Weyl points exists with opposite chirality. Connecting such a pair of Weyl points, a long Fermi arc appears in both the Fermi surface (Fig. 3d) and the band structure (Fig. 3e). Although the projection of bulk bands exhibit pseudo-symmetry of a hexagonal lattice, the surface Fermi arcs do not. It is clear that the Fermi arcs originating from two neighboring Weyl pairs (see Fig. 3d) do not exhibit M x reflection, because the chirality of Weyl points apparently violates M x symmetry. For a generic k x -k z plane between each pair of W 1 Weyl points, the net Berry flux points in the −k y direction. As a consequence, the Fermi velocities of both Fermi arcs point in the +k x direction on the bottom surface (see Fig. 3f). These two right movers coincide with the nonzero net Berry flux, i.e., Chern number = 2.
For Mn 3 Ge, we also focus on the W 1 -type Weyl points at the corners of the hexagonal Brillouin zone. In contrast to Mn 3 Sn, Mn 3 Ge exhibits a more complicated Fermi surface. Fermi arcs exist to connect a pair of W 1 -type Weyl points with opposite chirality, but they are divided into three pieces as shown in Fig. 4d. In the band structures (see Figs. 4e and f), these three pieces are indeed connected together as a single surface state. Crossing a line between two pairs of W 1 points, one can find two right movers in the band structure, which are indicated as p1 and p2 in Fig. 4f. The existence of two chiral surface bands is consistent with a nontrivial Chern number between these two pairs of Weyl points.

IV. SUMMARY
In summary, we have discovered the Weyl semimetal state in the chiral AFM compounds Mn 3 Sn and Mn 3 Ge by ab initio band structure calculations. Multiple Weyl points were observed in the bulk band structures, most of which are type II. The positions and chirality of Weyl points are in accordance with the symmetry of the magnetic lattice. For both compounds, Fermi arcs were found on the surface, each of which connects a pair of Weyl points with opposite chirality, calling for further experimental investigations such as angleresolved photoemission spectroscopy. The discovery of Weyl points verifies the large anomalous Hall conductivity observed recently in titled compounds. Our work further reveals a guiding principle to search for Weyl semimetals among materials that exhibit a strong anomalous Hall effect.