Weyl fermions in antiferromagnetic Mn₃Sn and Mn₃Ge

Recently we predicted the occurrence of an anomalous Hall effect (AHE) in the antiferromagnetic compounds Mn₃Sn and Mn₃Ge [1] and, indeed, this effect was subsequently discovered in Mn₃Sn by Nakatsuji et al. [2] as well as in Mn₃Ge by Nayak et al. [3] and independently by Kiyohara and Nakatsuji [4].

While the AHE commonly occurs in ferromagnets [5], it is expected to be absent in antiferromagnets, but Chen et al. [6] found that under certain symmetry conditions it is observable in non-collinear antiferromagnetic compounds. Extending the class of compounds possessing an AHE is of fundamental interest, but it might also be of importance for spintronics applications [7].

The compounds Mn₃Sn and Mn₃Ge can be grown in the hexagonal crystal structure having the symmetry $P6_3/mmc$ and a non-symmorphic spüace group. They have been shown to possess a variety of non-collinear antiferromagnetic orders [8–12], of which one kind is sketched in fig. 1(a), another one in fig. 6(a). Of all the possibilities these two cases have inverse chirality [11].

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We present results of improved calculations of the anomalous Hall conductivity (AHC), changing the previously used outdated hexagonal coordinate system [1], so that we are now conform with ref. [3] and with the International Tables of Crystallography. Special attention is given to the electronic structure at general points in the Brillouin zone, in order to understand the physical origin of the large AHC. Here we show that topological objects, called Weyl points, might supply the answer. These points are singularities in momentum space, originating from band-crossings. They are understood as magnetic monopoles that are sources of Berry curvature. They come in pairs and give rise to characteristic surface properties, the so-called Fermi arcs, which have been predicted theoretically and verified experimentally [13–17]. The ensuing non-trivial topology has been studied in semimetals, but recently the topological approach has been extended to α-iron [18] and to the ferromagnetic Heusler compound Co₂MnAl [19], which possesses a truly gigantic AHC. Recently, Mn₃Sn and Mn₃Ge were also seen to host Weyl points that were all found to be below the Fermi energy [20]. However, the electrical transport properties and especially the thermoelectric properties studied recently [21,22] are quasiparticle properties that reside at the Fermi surface as stressed by Haldane [23]. The purpose of this letter is to show under which conditions the effect of low-lying Weyl points might appear at the Fermi surface. Reference [20] contains a thorough study of the surface arcs of Mn₃Sn and Mn₃Ge but no attempt was made to relate the Weyl nodes to the bulk Fermi surface. Instead of using Haldane's mathematical theory [23], we here present descriptive arguments to show that the Weyl nodes cause the large anomalous Hall conductivity and the distinct thermoelectric effects through the non-zero Chern number of the bands at the Fermi surface. When we say large we have in mind values comparable to those of ferromagnetic α-iron, i.e., of order of 700S/cm, which is almost achieved in Mn₃Ge.

The anomalous Hall effect is obtained by computing the Berry curvature (BC) in momentum space. The quantities responsible for the thermoelectric effects are related to the BC as well [23]. This is a vector, writing its p-component as $\Omega_p(\mathbf{k})$, and is obtained from the curl of the Berry connection, given by ${\cal A}(\mathbf{k})=-i\sum_{n \in occ} \langle{u_{n\textbf{k}}}|\nabla_k|{u_{n\textbf{k}}}\rangle$, where $u_{n\bf k}(\mathbf{r})$ is the crystal-periodic eigenfunction having wave vector k and band index n, $\vec{\Omega}(\bf k)=\nabla \times{\cal A(\bf k)}$. The sum extends over the occupied states. For its evaluation we use the wave functions from density functional calculations [24] following ref. [25], where the numerical work is based on $\Im \ln\det[\langle u_{n \bf k}|u_{m {\textbf{k}^{\prime}}}\rangle]$, which is directly related to the Berry connection [26]. Spin-orbit coupling (SOC) is treated in the second variation and is an essential ingredient. The anomalous Hall conductivity (AHC) follows from the BC by means of

$$\begin{equation} \sigma_{\ell m}=\frac{e^2}{\hbar} \,\int_{\text{BZ}}\frac{\text{d}\mathbf{k}}{(2\pi)^3} \Omega_p(\mathbf{k})f(\mathbf{k}), \end{equation} \tag{ 1 }$$

where $f(\mathbf{k})$ is the Fermi distribution function, the integral extends over the Brillouin zone, $\Omega_p(\mathbf{k})$ is the p-component of the BC for the wave vector k and the components $\ell, m,p$ (for x, y,z) are to be chosen cyclic [27]. While this equation is commonly used for the computation of the AHC, it does not reveal that the transport takes place at the Fermi surface. Haldane [23], as said before, derived another version of eq. (1) that does just that, but will not be used here. We also need the definition of the Chern number [23], which is given by

$$\begin{equation} \frac{1}{2\pi}\int_\text{BZ}\text{d}\mathbf{k} \Omega^n_p(\mathbf{k})=C_n G^n_p. \end{equation} \tag{ 2 }$$

The extra index n appearing here is the state number, which appears in the BC if the sum over states is omitted. We state that the BZ-integral devided by $2\pi$ gives the Chern number of band n, C_n, times $G^n_p$, the p-component of a reciprocal lattice vector. The topology of a crystal is non-trivial if the integer $C_n \ne 0$.

We begin with Mn₃Sn and show the band structure along symmetry lines in fig. 2. It is in good agreement with the band structure shown in ref. [28], where the symmetry conditions are given for the anomalous Hall conductivity and the spin Hall conductivity. The integrated Berry curvature vector for the order shown in fig. 1(a) is in the x-direction and the calculated AHC is $\sigma_{yz}=310\mathrm{ S/cm}$. The experimental value is obtained at 100 K and is $\sigma_{yz}=100\mathrm{ S/cm}$ [2]. The value calculated in ref. [28] is $\sigma_{zx}=133\mathrm{ S/cm}$ which is obtained for the magnetic order sketched in fig. 6(a). The difference is not due to the other magnetic order, but is accounted for by our numerical method. The BC together with contours of the Fermi surface are displayed in fig. 3(a) in the $(k_z=0)$-plane. The BC is very large where the green spots appear. To appreciate the size of the local BC one should multiply the values given in the figure by about 1100 to convert to units S/cm of the Hall conductivity. The dominating values appear at locations of the Fermi surface where the curvature of the Fermi contours shown by the blue dots is large. Figure 3(b) is a two-dimensional rendering of the band structure and band No. 50 is seen to cut the Fermi energy (gray) thus defining the Fermi contours. The lower band No. 49 touches the higher band in two points in the Brillouin zone. Zooming in at one of the points we see the Weyl point that is graphed in part (c) of the figure. The band crossing is at the point K in the Brillouin zone where no signal of the Berry curvature is seen in fig. 3(a). Another Weyl point seen in fig. 3(b) is located at $\textbf{k}=(2/3,0,0)$. There are all together 6 Weyl points of chirality $\pm 1$ in the extended zone, which is displayed in fig. 4. We did not try to verify more points described in ref. [20] because they are too far below the Fermi energy. In fig. 4 we look at the Fermi contours in the extended zone to visualize the development of the BC in the $(k_z=0.4)$-plane from the BC in the $(k_z=0)$-plane. In the $(k_z=0.4)$-plane a second contour appears (yellow dots) due to band No. 51. The contours due to band No. 50 (blue dots) deform and carry the BC spots at the location of large curvature of the Fermi contours. In fig. 5(a) the Berry curvature together with the Fermi contours is displayed in the $(k_x=1/3)$-plane which is sketched schematically in part (b) of the figure. The strong signal at the Fermi energy is a reflection of the Weyl point at K. The Fermi contour that shows up in the upper part of fig. 5(a) is due to band No. 51. There is no BC emanating from this contour.

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We continue with Mn₃Ge. The magnetic structure, shown in fig. 6(a), is different from that of Mn₃Sn. One is obtained from the other by rotating all spins by 90° about the z-axis. Computationally the total energies of the two structures in either one of the systems are very close so that we rely on experimental information for the present assignment. The integrated Berry curvature vector in this case is in the y-direction and the calculated AHC amounts to $\sigma_{zx}=560\mathrm{ S/cm}$. This is to be compared with low-temperature values of $\sigma_{zx}=500\mathrm{ S/cm}$ measured by both groups [3,4]. The calculated value given in ref. [28] is $\sigma_{zx}=330\mathrm{ S/cm}$. The band structure along symmetry lines is given in fig. 6(b) and is in good agreement with ref. [28]. It is nearly identical to fig. 2, except for minute differences near the Fermi energy at K and M. These matter as can be seen in fig. 7(a), where Fermi contours are displayed together with the BC in the $(k_z=0)$-plane. Now the Fermi surface is formed by two bands, viz. bands Nos. 70 and 71. A spot of large value of the BC appears as green and black contours near a Fermi contour that curve strongly below the line $M\text{-}K$. Figure 7(b) gives the band structure near the Fermi energy in a two-dimensional plot. In fig. 8(a) we locate a band crossing between band Nos. 69 and 70. Zooming in near K we see the Weyl point shown in fig. 8(b). Another Weyl point appears at $K^\prime = (2/3, 0, 0)$. As for Mn₃Sn there are 6 Weyl points of chirality $\pm 1$ in the extended zone. The other Weyl points described in ref. [20] we ignored since they are too far below the Fermi energy. In fig. 9 as before we look at the Fermi contours in the extended zone to visualize the development of the BC in the $(k_z=0.4)$-plane from the BC in the $(k_z=0)$-plane. The contours due to band No. 70 (blue dots) deform and carry the BC spots at the location of large curvature of the Fermi contours, whereas the contours due to band No. 71 contract to the yellow circle. Figure 10(a), finally, shows Fermi contours together with the BC in the $(k_y = \frac{1}{2 \sqrt{3}})$-plane, which is shown in part (b) of the figure. The green BC spot in (a) is a reflection of the Weyl points at K and $K^{\prime}$. The yellow Fermi surface contours near $L^{\prime}$ show no sizeable BC spots.

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We now try to explain the spots of large BC, for short "hot spots", with the physical concepts developed for Weyl semimetals [15]. It is not a priori clear if these concepts also apply to non-collinear antiferromagnetic metals, but we assume they do. There is first the fact that the Chern number C of a band is non-zero between two Weyl points of opposite chirality. Thus, the band No. 50 for Mn₃Sn (see fig. 3) and band No. 70 for Mn₃Ge (see figs. 7 and 8) at the Fermi energy have $C\ne 0$. That is the hot spots are a property of the non-zero Chern number. The next higher band, No. 51 for Mn₃Sn and No. 71 for Mn₃Ge, do produce uniform BC but no spots. A simple formula valid for a model Weyl semimetal relates the AHC to the difference vector between two Weyl points of opposite chirality [15]. Using the Weyl points obtained here, we obtain about $\sigma_{yz}= 230\mathrm{ S/cm}$ for Mn₃Sn and $\sigma_{zx}=390\mathrm{ S/cm}$ for Mn₃Ge. It must be remarked that these values are fortuitously close to the observed ones, since none of the assumptions underlying their derivation are valid here, except for the Chern number of the bands. But the range of validity may be larger than assumed. This opens another question: Could it be that the chiral anomaly exists for non-collinear antiferromagnets? Presumably this question can only be answered by experiments. Concerning the thermoelectric effects it is to be seen whether they are as large in Mn₃Ge as in Mn₃Sn.