Dirac and Weyl Semimetal in XYBi (X = Ba, Eu; Y = Cu, Ag and Au)

Weyl and Dirac semimetals recently stimulate intense research activities due to their novel properties. Combining first-principles calculations and effective model analysis, we predict that nonmagnetic compounds BaYBi (Y = Au, Ag and Cu) are Dirac semimetals. As for the magnetic compound EuYBi, although the time reversal symmetry is broken, their long-range magnetic ordering cannot split the Dirac point into pairs of Weyl points. However, we propose that partially substitute Eu ions by Ba ions will realize the Weyl semimetal.

Following the discovery of topological insulator (TI) 1,2 , there has been considerable research interest in studying the Weyl semimetal (WSM), the first metallic topologically nontrivial matter [3][4][5][6] . In WSM, non-degenerate valence and conduction bands touch at an accidental degeneracy point in a three-dimensional (3D) Brillouin zone, and its low energy physics is approximated by the Weyl equation 3,4 . Weyl points, the nondegenerate linear touchings of the bulk bands, always come in pair, and they are robust due to the protection by the topology of the band structure. The most remarkable feature of WSM is the Fermi arc surface states 3 . Several compounds, including pyrochlore iridates 3 , TI based heterostructures 7 , HgCr 2 Se 4 8 and many other systems [9][10][11][12][13] had been theoretically predicted as promising WSMs. The indication about realization of WSM have been reported [14][15][16] . Very recently, the noncentrosymmetric and nonmagnetic transition-metal monophosphide are predicted as WSM 17,18 and the Fermi arc, as the smoking-gun feature, has been confirmed experimentally [19][20][21][22][23] .
Same as the WSM, the Dirac semimetal (DSM) is also a 3D analog of graphene [24][25][26][27] . But in contrast with Weyl point, the Dirac point has four-fold degeneracy, and does not possess a topological number, consequently the Dirac point is not robust against the external perturbations and usually hard to be realized. Thus the 3D DSM receive much less attention until the discovery of Na 3 Bi 28 and Cd 3 As 2 29 . Wang et al. find that there is a paired 3D bulk Dirac points exist on the k z axis of Na 3 Bi 28 and Cd 3 As 2 29 , and these Dirac points are protected by the crystal symmetry thus are stable 28,29 . The theoretical prediction of DSM in Na 3 Bi and Cd 3 As 2 28,29 had been quickly confirmed by the subsequent photoemission measurement [30][31][32][33] . This immmediately triggers a new wave of research to explore the unique properties associated with the 3D Dirac points in the DSM [30][31][32][33][34][35][36] . Unfortunately, Na 3 Bi is not stable in air while arsenic limits the application of Cd 3 As 2 . Therefore searching new 3D DSM that is stable in nature and less toxic is of both fundamental and technological importance.
In this paper, based on the density functional theory (DFT) calculations and effective low energy models, we predict that BaYBi (Y = Au, Ag and Cu) are promising 3D Dirac materials. For BaAuBi, the nontrivial topology is due to the band inversion of the Bi-p bonding and antibonding states, while for the BaAgBi and BaCuBi, the band inversion happens between the Ag/Cu s and Bi p orbital. Protected by the C 3 rotation symmetry, the Dirac points locate along the Γ − A line. The magnetic configuration in EuYBi indeed break the time reversal symmetry, however cannot split the Dirac point into two Weyl points. We propose that partially substituting Eu by Ba, i.e. alloy compound Eu x Ba (1−x) Ag(Au)Bi, which could be grown using molecular beam epitaxy (MBE) technique, is a promising way to realize the WSM.

Results and Discussion
BaYBi (Y = Au, Ag, Cu) crystallize in the same hexagonal ZrBeSi type structure with space group / ( ) P mmc D 6 sites respectively 37 . There is no free internal coordinates, and the lattice constants are the only structural parameter for BaYBi lattice. We optimize the lattice parameter and for all of the three compounds, our numerical lattice constants are in good agreements with experiments, and the small discrepancy between the numerical and experimental structure has negligible effect on the electronic structure. Hence, the following results are obtained based on the experimental structure, unless stated specifically.
We first calculate the electronic structure of BaAuBi, and show the results in Fig. 2(a). The Ba in BaAuBi is highly ionic, has negligible contribution to the states around Fermi level. Au-6s and 5d bands  mainly located at − 4 to − 1 eV, and − 6 to − 4 eV, respectively. The Bi-6s is basically located about − 11 eV below the Fermi level. The valence and conduction bands are dominated by the Bi-6p bonding and antibonding states. Checking the wave function, we find that at the Γ point the Bi-6p antibonding state is higher than the Bi-6p bonding state, however at the A point, the odd-parity state is about 0.545 eV lower in energy than even-parity state.
In order to understand the mechanism of the band inversion, we illustrate the band evolution at the Γ point of BaAuBi in Fig. 2(b). As discussed above, the states near Fermi level are primarily contributed by the Bi-6p orbital, with also the Au-6s state. Since the two Bi atoms (Bi and Bi′ ) in the unit cell are related to each other by the inversion symmetry, similar with ref. 38,39, we combine the Bi-6p orbitals to form the hybridized states and label the bonding and antibonding states as , , respectively, where the superscripts + /− denote the parity of the corresponding states. According to the point group symmetry, the p z orbital split from the p x and p y orbitals while the latter two are still degenerate as shown in the Fig. 2 Along Γ − A line the C 3 symmetry is preserved, by the symmetry analysis the two relevant bands along this line belong to different representations (Δ 7 and Δ 8 as shown in Fig. 2(a). See Additional Data for the detail). Thus the hybridization between these bands is strictly forbidden, which results in the protected band crossing as shown in Fig. 2(a). The linear band dispersions associated with Dirac points near the Fermi surface will contribute a high-field unsaturated linear magnetoresistance 40,41 , we thus believe such novel properties may also be observed in BaAuBi, although the Dirac point is located slightly below the Fermi level as shown in Fig. 2(a).
Since the topological nature is determined by the Δ 7 and Δ 8 bands, based on the projection-operator method (see Additional Data), we build the effective Hamiltonian by using the four relevant states as basis vectors (in the order of We neglect all of other states, since they are far from the Fermi level and do not involve into the band inversion, and the Hamiltonian can be written as: 2 and k ± = k x ± ik y . The parameters in the above formula are material dependent, and by fitting the DFT calculated band dispersion, we obtain C 0 = − 0.06978 eV, Solving the above eigenvalue problem, we obtain ε ( ) = ( ) ± ( ) + ( ) + , we get the gapless solutions. In the vicinity of k c and neglect the high-order terms, E(k′) would be equal to 2 (δk x,y,z are small displacement from k c ), which is a linear dispersion and suggests in neighbourhood of k c , our effective Hamiltonian is nothing but 3D anisotropic massless Dirac fermions. A 3D Dirac semimetal state can also be realized at the critical point of the topological phase transition between a band insulator and a 3D topological insulator 42,43 . Different from this case, the Dirac points in BaAuBi are protected by the C 3 rotation symmetry, thus very robust. We also investigate the BaAgBi and BaCuBi. The electronic properties of BaCuBi are very similar with that of BaAgBi, we thus only report results of BaAgBi. As shown in Fig. 3(b), significantly different with the Au-6s state in BaAuBi, the Ag-5s orbit in BaAgBi is higher in energy than Bi-6p state, consequently the states closed to the Fermi level become | 〉, | 〉 The Dirac points in BaAgBi are doubly degenerate due to inversion and time-reversal symmetries, and upon breaking the time reversal symmetry 3 or inversion symmetry 42,43 a Dirac cone will split into two Weyl nodes separated in momentum space. This family of intermetallic compound with hexagonal structure indeed has several members with magnetic ion Eu: EuXY (X = Cu, Au, Ag; Y = As, Sb, Bi) 37,44,45 . Experiments confirm that some of them indeed possess long-range magnetic configuration 44 . Unfortunately, the Eu 2+ spins align ferromagnetically with the ab plane, but antiferromagnetically along the c-axis 45 , therefore the exchange field is exactly cancelled at the XY-plane of EuXY. Thus breaking the time reversal symmetry by this type of antiferromagnetic configuration cannot split the Dirac points, and the compounds of EuXY have no chance to become WSM.
We, however, expect that substituting part of Eu ions by Ba ions, the two antiferromagnetically coupled Eu plane in Eu x Ba 1−x AgBi may not exactly cancel each other, and then there is a chance the compound becomes WSM. To confirm this expectation, we then performed another calculation on Eu 0.5 Ba 0.5 AgBi, in which we replace one of the two antiferromagnetically coupled Eu plane in the unit cell by Ba. According to our calculation, the c axis is the magnetic easy axis, this magnetization does not break the C 3 rotation symmetry, consequently as shown in Fig. 4(a), the Dirac point indeed splits into two Weyl points as marked by red circle. There is also two other Weyl points slightly above/below the Weyl nodes marked by red circle. The Weyl nodes are very close to the Fermi energy as shown in Fig. 4(a), thus, the phenomena associated with the chiral anomaly 46-48 also exist in Eu 0.5 Ba 0.5 AgBi. Fig. 4(b) shows the Fermi arcs which connect projected bulk Weyl points of opposite chirality. Thus we believe to grow it by the cutting edge film growth technique like MBE and to explore the possible WSM are a very interesting topic.
In summary, based on density-functional calculation and effective model analysis, we propose that the BaYBi (Y = Au, Ag and Cu) are 3D Dirac semimetals. The nontrivial topological feature is due to p-p inversion for BaAuBi and s-p band inversion for BaAgBi and BaCuBi, and their Dirac points are protected by the C 3 rotation symmetry and thus are very robust. Their magnetic cousins, i.e EuYBi (Y = Au, Ag and Cu) are not Weyl semimetals. However, partial substitution of Eu with Ba ions in EuYBi could result in the Weyl semimetal. Furthermore, our numerical calculation also confirm that a uniaxial strain along a-axis, which breaks the C 3 rotation symmetry, will drive BaAgBi into topological insulator.
Note. When finalizing our work, we became aware of a recent study by Borisenko et al. 49 , in which the authors also predict BaAgBi is a possible 3D DSM, agreeing with our conclusion.

Methods
The electronic band structure calculations have been carried out using the full potential linearized augmented plane wave method as implemented in WIEN2K package 50 . The modified Becke-Johnson exchange potential together with local-density approximation for the correlation potential (MBJLDA) has been used here to obtain accurate band inversion strength and band order 51 . A 16 × 16 × 7 mesh is used for the Brillouin zone integral. Using the second-order variational procedure, we include the spin-orbital coupling (SOC) interaction.
The tight-binding model has been established by using Slater-Koster method 52 . 5 s orbit of Ag atom and three 6p orbits of Bi atom are taken as basis. The tight-binding model has been used to simulate the bulk band structure with Weyl points as found by our LSDA + SO + U calculation. To calculate the surface state and Fermi arc, we build the (010) slab of a thickness of 85 unit-cells.

Additional Data
Effective Hamiltonian for BaAuBi. The conduction and valence bands of BaAuBi are mainly con-  Γ matrices representation T  The Dirac Γ matrices can be written as 39 . The projection operator is defined as where g is the group order, l i is the dimension of the ith representation, R denotes the group element i.e. the symmetry operation, χ i (R) represent the character of group element R in ith representation, P R is the operator of group element R.
The double group of D h 6 4 has 18 classes, and their irreducible representations are denoted as R 1 to R 18 53 , and its character table can be found in ref. 45. Based on the basis mentioned above, one can easily work out the transformation matrix D R for symmetry operator P R , which allow us to apply the projection operator p i on , consequently determine the irreducible representation of Γ a . Using the same process, one can also determine the irreducible representation for the polynomials of k up to O(k 3 ). We present the irreducible representation of Dirac Γ matrices and polynomials of k, and their transformation under time reversal in Table 1.
With the Table 1   belong to representation Γ − 8 , which had been marked in Fig. 2(a). Different from Γ point, the symmetry of Γ − A line is C v  Table 3. It is clear that the representation Γ + 7 and Γ − 8 , evolute to Δ 7 and Δ 8 , respectively. For BaAgBi, the valence/conduction states at the Γ point of BZ belong Γ /Γ + − 8 7 , and will change to Δ 8 and Δ 7 along Γ − A line according to Table 3.