Topological Weyl semimetals in the chiral antiferromagnetic materials Mn₃Ge and Mn₃Sn

Mn₃Ge and Mn₃Sn share the same layered hexagonal lattice (space group $P{6}_{3}/{mmc}$, No. 194). 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 Mn₃Sn and 2.7 μB in Mn₃Ge. As revealed in a previous study [41], the ground magnetic state is a noncollinear AFM state, where Mn moments align inside the ab plane and form 120° angles with neighboring moment vectors, as shown in figure 1(b). 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 $\{{M}_{y}| \tau =c/2\}$. In addition, two other mirror reflections (M_x and M_z) adding time reversal (T), ${M}_{x}T$ and ${M}_{z}T$, exist.

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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 ${\bf{k}}\,({k}_{x},{k}_{y},{k}_{z})$. Mirror reflection reverses χ while time reversal does not and both of them act on ${\bf{k}}$. Further, mirror reflection ${M}_{\gamma }$ preserves the Berry curvature ${{\rm{\Omega }}}^{\gamma }$ while time reversal reserves it. The transformation is as follows:

$$\begin{eqnarray}{M}_{x}T\,:({k}_{x},{k}_{y},{k}_{z}) & \to & ({k}_{x},-{k}_{y},-{k}_{z});\chi \to -\chi ;{{\rm{\Omega }}}^{x}\to -{{\rm{\Omega }}}^{x}\\ {M}_{z}T\,:({k}_{x},{k}_{y},{k}_{z}) & \to & (-{k}_{x},-{k}_{y},{k}_{z});\chi \to -\chi ;{{\rm{\Omega }}}^{z}\to -{{\rm{\Omega }}}^{z}\\ {M}_{y}\,:({k}_{x},{k}_{y},{k}_{z}) & \to & ({k}_{x},-{k}_{y},{k}_{z});\chi \to -\chi ;{{\rm{\Omega }}}^{y}\to +{{\rm{\Omega }}}^{y}.\end{eqnarray} \tag{ 2 }$$

Each of the above three operations doubles the number of Weyl points. Thus, eight nonequivalent Weyl points can be generated at $(\pm {k}_{x},+{k}_{y},\pm {k}_{z})$ with chirality χ and $(\pm {k}_{x},-{k}_{y},\pm {k}_{z})$ with chirality $-\chi$ (see figure 1(c)). We note that the ${k}_{x}=0/\pi$ or ${k}_{z}=0/\pi$ plane can host Weyl points. However, the ${k}_{y}=0/\pi$ plane cannot host Weyl points, because M_y simply reverses the chirality and annihilates the Weyl point with its mirror image if it exists.

In addition, the symmetry of the 120° AFM state is slightly broken in the materials, owing to the existence of a tiny net moment (∼0.003 μB per unit cell) [41–43]. 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 Mn₃Ge.

The intrinsic anomalous Hall conductivity ${\sigma }_{\gamma }(\gamma =x,y,z)$ can be calculated by integrating the Berry curvature ${{\rm{\Omega }}}^{\gamma }$ over the whole Brillouin zone [40, 44]

$$\begin{eqnarray}&&{\sigma }^{\gamma }=-\displaystyle \frac{{e}^{2}}{{\hslash }}{\int }_{{BZ}}\displaystyle \frac{{{\rm{d}}}^{3}\vec{k}}{{(2\pi )}^{3}}{f}_{n}(\vec{k}){{\rm{\Omega }}}^{\gamma }(\vec{k}).\end{eqnarray} \tag{ 3 }$$

According to equation (2), ${{\rm{\Omega }}}^{x}$ and ${{\rm{\Omega }}}^{z}$ are odd with respect to ${M}_{x}T$ and ${M}_{y}T$, respectively. Thus, corresponding ${\sigma }^{x}$ and ${\sigma }^{z}$ are zero. Since ${{\rm{\Omega }}}^{y}$ is even with respect to the M_y mirror plane, corresponding ${\sigma }^{y}$ is nonzero. This is also consistent with the distribution of Weyl points in the k-space. As shown in figure 1(c), only '+' Weyl points appear on one side of the M_y plane and only '–' Weyl points locate on the other side of M_y plane. Then there are net Berry flux (starting from '+' to '–' Weyl points) ${{\rm{\Omega }}}^{y}$ crossing the M_y plane, resulting in the nonzero anomalous Hall conductivity ${\sigma }^{y}$. In contrast, an equal number of '+' and '–' Weyl points appear on each side of M_x (M_z) planes. Consequently, the net Berry flux of ${{\rm{\Omega }}}^{x}$ (${{\rm{\Omega }}}^{z}$) should be zero, giving rise to vanishing ${\sigma }^{x}$ (${\sigma }^{z}$). According to recent numerical calculations [36], ${\sigma }^{y}=330(133)\,{{\rm{S}}}^{-1}\,{\mathrm{cm}}^{-1}$ for Mn₃Ge (Mn₃Sn).

In the measurement of AHE, an external magnetic field is usually applied to uniform different magnetic domains. Further, for Mn₃Ge and Mn₃Sn, the triangular spins can be rotated inside the xy plane even by a very weak magnetic field due to the residual magnetic moment [43]. The rotation of an arbitrary angle can break the M_y and ${M}_{x}T$ symmetry, showing nonzero ${\sigma }^{y}$ and ${\sigma }^{x}$. However, ${\sigma }^{z}$ is still zero due to the ${M}_{z}T$ symmetry. As observed for both compounds in experiment [34, 35], ${\sigma }^{x,y}$ are indeed very large and ${\sigma }^{z}$ is negligible. The in-plane anomalous Hall conductivity is about 500 (100) ${{\rm{S}}}^{-1}\,{\mathrm{cm}}^{-1}$ for Mn₃Ge (Mn₃Sn) at low temperature, which are in the same order of magnitude as the calculations [36].

The bulk band structures are shown along high-symmetry lines in figure 2 for Mn₃Ge and Mn₃Sn. It is not surprising that the two materials exhibit similar band dispersions. At first glance, one can find two seemingly band degenerate points at Z and K points, which are below the Fermi energy. Because of ${M}_{z}T$ and the nonsymmorphic symmetry {${M}_{y}| \tau =c/2$}, the bands are supposed to be quadruply degenerate at the Brillouin zone boundary Z, forming a Dirac point protected by the nonsymmorphic space group [45–47]. 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.

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Mn₃Sn and Mn₃Ge 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 ${N}_{{\rm{v}}}^{\mathrm{th}}$ and ${({N}_{{\rm{v}}}+1)}^{\mathrm{th}}$ bands.

As shown in figure 3(a), 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 W₁, W₂, and W₃. These Weyl points lie in the M_z plane (with W₂ points being only slightly off this plane owing to the residual-moment-induced symmetry breaking) and slightly above the Fermi energy. Therefore, there are four copies for each of them according to the symmetry analysis in equation (2). Their representative coordinates and energies are listed in table 1 and also indicated in figure 3(a). A Weyl point (e.g., W₁ in figures 3(b) and (c)) acts as a source or sink of the Berry curvature ${\boldsymbol{\Omega }}$, clearly showing the monopole feature with a definite chirality.

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In contrast to Mn₃Sn, Mn₃Ge displays many more Weyl points. As shown in figure 4(a) and listed in table 2, there are nine groups of Weyl points. Here ${{\rm{W}}}_{\mathrm{1,2},\mathrm{7,9}}$ lie in the M_z plane with W₉ on the k_y axis, W₄ appears in the M_x plane, and the others are in generic positions. Therefore, there are four copies of ${{\rm{W}}}_{\mathrm{1,2},\mathrm{7,4}}$, two copies of W₉, 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 W₁ in figure 4 as an example. Without including SOC, we observed a nodal-ring-like band crossing in the band structures of both Mn₃Sn and Mn₃Ge. SOC gaps the nodal rings but leaves isolating band-touching points, i.e., Weyl points. Since Mn₃Sn exhibits stronger SOC than Mn₃Ge, many Weyl points with opposite chirality may annihilate each other by being pushed by the strong SOC in Mn₃Sn. This might be why Mn₃Sn exhibits fewer Weyl points than Mn₃Ge.

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The existence of Fermi arcs on the surface is one of the most significant consequences of Weyl points inside the three-dimensional (3D) bulk. We first investigate the surface states of Mn₃Sn that have a simple bulk band structure with fewer Weyl points. When projecting ${{\rm{W}}}_{\mathrm{2,3}}$ Weyl points to the (001) surface, they overlap with other bulk bands that overwhelm the surface states. Luckily, W₁ Weyl points are visible on the Fermi surface. When the Fermi energy crosses them, W₁ Weyl points appear as the touching points of neighboring hole and electron pockets. Therefore, they are typical type-II Weyl points [20, 48]. Indeed, their energy dispersions demonstrate strongly tilted Weyl cones.

The Fermi surface of the surface band structure is shown in figure 3(d) for the Sn compound. In each corner of the surface Brillouin zone, a pair of W₁ Weyl points exists with opposite chirality. Connecting such a pair of Weyl points, a long Fermi arc appears in both the Fermi surface (figure 3(d) and the band structure (figure 3(e)). 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 figure 3(d)) 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 W₁ 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 figure 3(f)). These two right movers coincide with the nonzero net Berry flux, i.e., Chern number = 2.

For Mn₃Ge, we also focus on the W₁-type Weyl points at the corners of the hexagonal Brillouin zone. In contrast to Mn₃Sn, Mn₃Ge exhibits a more complicated Fermi surface. Fermi arcs exist to connect a pair of W₁-type Weyl points with opposite chirality, but they are divided into three pieces as shown in figure 4(d). In the band structures (see figures 4(e) and (f)), these three pieces are indeed connected together as a single surface state. Crossing a line between two pairs of W₁ points, one can find two right movers in the band structure, which are indicated as p1 and p2 in figure 4(f). The existence of two chiral surface bands is consistent with a nontrivial Chern number between these two pairs of Weyl points.