Anisotropic Dirac Fermions in BaMnBi2 and BaZnBi2

We investigate the electronic structure of BaMnBi2 and BaZnBi2 using angle-resolved photoemission spectroscopy and first-principles calculations. Although they share similar structural properties, we show that their electronic structure exhibit dramatic differences. A strong anisotropic Dirac dispersion is revealed in BaMnBi2 with a decreased asymmetry factor compared with other members of AMnBi2 (A = alkali earth or rare earth elements) family. In addition to the Dirac cones, multiple bands crossing the Fermi energy give rise to a complex Fermi surface topology for BaZnBi2. We further show that the strength of hybridization between Bi-p and Mn-d/Zn-s states is the main driver of the differences in electronic structure for these two related compounds.

Ba on the A-site leads to anisotropic Dirac bands, although with different asymmetry and anisotropy from (Ca,Sr) MnBi 2 compounds. In addition, substantial changes in the electronic structure are found by substituting Zn with Mn (3d 5 with Mn 2+ to 3d 10 with Zn 2+ , respectively), yielding more trivial bands at E F due to the increased hybridization between Zn and Bi states. Figure 1 shows the atomic structure of BaMnBi 2 and BaZnBi 2 . These materials consist of alternating layers of Bi-net and Mn (or Zn)-Bi tetrahedra separated by Ba layers located in mirror symmetric positions with respect to the Bi-net (upper right panel of Fig. 1a), satisfying I4/mmm space group symmetry. Table 1 compares the lattice constants and ionic radii of A-site ions for other AMnBi 2 compounds with BaZnBi 2 . With A-site cations of smaller ionic radius, the atomic structure takes up the P4/nmm space group, in which the A-site cations are located at staggered positions, breaking the mirror symmetry with respect to the Bi square net 13 . We find by X-ray diffraction that the space group of both BaMnBi 2 and BaZnBi 2 are I4/mmm, confirming the sensitivity of the space group to A-site. Moreover, for AMnBi 2 compounds with the I4/mmm space group, the c/a ratio is measured to increases with increasing A-site ionic radius, as expected. A unique feature of the crystal structure associated with this family of materials is that there are two types of Bi atoms in the unit cell: one consisting of the Bi-net (defined as Bi1) and the other forming the Bi-tetrahedra around the Mn (or Zn) cations (defined as Bi2). It has been pointed out that states near E F of AMnBi 2 compounds are dominated by Bi1-p orbitals whereas the Bi2-p states that hybridize with the Mn-d orbitals are away from the Fermi level 13 . We note that the c/a ratio of the BaZnBi 2 is dramatically smaller than those reported for the Mn family, a reduction that can be primarily attributed to the reduced Zn-Bi distance and that reflects differences in the nature of bonding between Zn-Bi and Mn-Bi. With the small binding energy of the Bi2-p states in AMnBi 2 13 , the expected change in the bonding character in BaZnBi 2 should induce a significant change in the states near the E F , which will be discussed later in our ARPES data and first-principles results.
Fermi surfaces (FSs) of BaMnBi 2 (Fig. 1b) and BaZnBi 2 (Fig. 1c) measured by ARPES show notably distinct features. The FS of BaMnBi 2 displays four crescent-shape hole pockets resulting from anisotropic Dirac bands along Γ-M directions, typical of the other anisotropic Dirac materials AMnBi 2 (A = Ca, Sr, Eu, and Yb) of this class 14,16 . On the other hand, the BaZnBi 2 FS shows new features, including double layers of large diamond-like pieces connecting the four X points in the Brillouin zone (BZ) and two concentric circle-like hole pockets around the Γ point. The four corners of the diamond-like FS features overlap with neighboring ones at the X points, generating additional small diamond-like electron pockets at these four X points in the BZ.
Further investigations of the electronic structures of BaMnBi 2 and BaZnBi 2 from our ARPES measurements are illustrated on Figs 2 and 3, respectively. Constant energy contours of BaMnBi 2 at different binding energies are shown in Fig. 2a. As the binding energy increases, the size of four crescent-shaped iso-energy surface also increases, indicating the hole-like nature of anisotropic Dirac bands in Γ-M directions. Meanwhile, the other four sets of new crescent-like pockets around the X point start to appear around binding energy E B = 100 meV. Our  To analyze these anisotropic Dirac bands further, we examine the measured electronic band dispersion along four cuts from A to D (Fig. 2b). The Dirac bands marked by yellow arrows exhibit a clear asymmetry with a steeper left branch of the band compared to the right one (Fig. 2b,c). The Fermi velocity of the left (v L ) and the right (v R ) branches are ~6 eV·Å and ~1.9 eV·Å, respectively, which yields an asymmetry value ((v L − v R )/v L ) ~0.68. This is smaller than the asymmetry values of CaMnBi 2 (0.80) and SrMnBi 2 (0.78) 16 . It is well known that this asymmetry originates from the strong spin-orbit coupling which also gaps out the Dirac band 13,16 . The doubly-degenerate Dirac cones associated with the low energy electronic structure along Γ-M and absence of states along Γ-X direction is consistent with our first-principles calculations, which will be discussed in more detail below.
The crossing points of Dirac bands are the highest in energy along Γ-M directions and shift downwards with momentum away from the Γ-M directions (from cut A to cut D). At the same time, on cut C and D, the Dirac band dissected along the perpendicular direction compared to the cut A appears around the Γ point. The slope of the Γ-point Dirac band is much less steep compared to the one in cut A, which clearly exhibits the strong anisotropic characteristics expected from a Dirac band. For a quantitative analysis of the anisotropy of Dirac bands, we extract the Fermi velocity along Γ-M (v ∥ = (v L+ v R )/2) and perpendicular to Γ-M (v ⊥ ), which yield ~4 eV·Å and ~0.3 eV·Å, respectively. The anisotropy value can be defined as v ∥ /v ⊥ ~ 13, which is smaller than CaMnBi 2 (64), YbMnBi 2 (209), but roughly equivalent to SrMnBi 2 (13) 14,16 . This illustrates a correlation of the strength of the anisotropy with space group, since CaMnBi 2 and YbMnBi 2 with P4/nmm display relatively stronger anisotropy than SrMnBi 2 and BaMnBi 2 that belong to I4/mmm.
We substitute Mn with Zn to investigate the effect of their different valence configurations on the band structure and anisotropy. The constant energy contours measured for BaZnBi 2 (Fig. 3a) display significant differences in the band features compared to those of BaMnBi 2 (Fig. 2a) with two concentric hole-like inner circular FS pockets surrounded by two concentric hole-like outer diamond pockets with electron-like FS at the X point that originate from band overlap. As the binding energy increases, circular and diamond bands expand and overlap with each other while the pockets at the X point shrink, as expected, from the binding energy-dependent nature of the hole-like and electron-like bands, respectively. The FS calculated from our first-principles DFT-GGA + SO calculations reproduce the main features, such as the increase in the number of bands crossing the E F and diamond-shaped bands; our calculations also show deviations compared with ARPES spectra, potentially due to missing correlation effects beyond DFT, indicated by a substantial band renormalization about 1.3 used for better agreement between the DFT and ARPES band structures (see the discussion in the calculation methods). In order to investigate the origin of the changes in the electronic structure associated with substituting Mn for Zn, we calculate and compare the projected density of the states (PDOS) of BaMnBi 2 and BaZnBi 2 (see Fig. 5a,b). In BaMnBi 2 , the Bi2-p states hybridize primarily with Mn-d shown from the overlap in the computed PDOS around −4.5 eV; whereas Bi2-p states in the Zn compound hybridize mainly with the Zn-s states over a broad energy range, from around −5 eV to 0-2 eV above E F . Since the bonding between the Zn-s and Bi2-p orbitals is stronger than Mn-d and Bi2-p orbitals due to the large spatial extent of the Zn-s orbitals and shorter Zn-Bi distance (see Table 1), the antibonding Bi2-p bands shifts upwards in BaZnBi 2 compared with BaMnBi 2 . The upward shifts in the Bi2-p bands result in additional band crossings at E F , shown by orbital-projected band structures in Fig. 5c,d. The Bi2-p states around the Γ point in BaZnBi 2 move up by about 0.5 eV compared with those in the Mn compound, resulting in Bi2-p derived bands crossing E F .
In order to distinguish between the effect of the change in structure (Mn/Zn to Bi distance) and the change in bonding character (d-p vs. s-p hybridization), we compare the band structures of BaMn(Zn)Bi 2 calculated with relaxed atomic structure of BaMn(Zn)Bi 2 and with deliberate changes of Mn(Zn) -Bi distance for that of BaZn(Mn)Bi 2 in Fig. 5e,f. The results demonstrate that there is little difference in the band dispersion due to the change in structure. Thus, we can conclude that the change in the band structure is mainly caused by the difference in bonding character. Our calculations also suggest that, to isolate the Dirac bands crossing E F , it would be ideal to choose transition metal ions with frontier orbitals that hybridize weakly with the Bi2-p orbitals and that are gapped at the Fermi level.
In conclusion, we have investigated the electronic structure of BaMnBi 2 and BaZnBi 2 using ARPES and first-principles calculations, focusing on the effect of substituting A-site cation with larger ionic radius and the role of the transition metal states on the band dispersion near E F . Compared with the isostructural compound SrMnBi 2 , substitution of the A-site cation with larger ionic radius results in a small decrease in the spin-orbit induced asymmetry and negligible change in the anisotropy of the Dirac cone, and maintains the same Fermi surface topology originating with the Bi-p states from the square net. However, we find that the substitution of the Mn with Zn gives rise to a drastic change in the dispersion near the Fermi level with Bi-p states derived from Bi square net and Zn-Bi complex due to the large hybridization between Zn-s and Bi-p states. Our results imply that transition metals with frontier orbitals weakly hybridizing with Bi-p states may be ideal for the isolation of the Dirac bands crossing at the Fermi level.

Method
Single crystal growth. Single crystals of BaMnBi 2 and BaZnBi 2 were grown from molten metallic fluxes as described previously 15,17 .   Electronic structure calculations. We perform first-principles density functional theory calculations with the generalized gradient approximation (GGA) method using the Vienna ab-initio simulation package 18,19 . The Perdew-Becke-Erzenhof (PBE) parametrization 20 are used for the GGA exchange correlation functional. Spin-orbit coupling is included self-consistently for all the calculations. We use the projector augmented wave method 21 with an energy cutoff of 500 eV and k-point sampling on a 6 × 6 × 2 grid. The atomic positions are fully relaxed until Hellmann-Feynman forces are less than 0.02 eV/Å. The lattice constants relaxed with the PBE functional are in good agreement with experiment for BaMnBi 2 with difference of 1.5% for volume and −0.04% for c/a ratio. For BaZnBi 2 the unit-cell volume calculated by the GGA shows reasonable agreement with 2.3% error with respect to the experimental value but the c/a ratio deviates more significantly, with an error about 8.5% compared with experimental value. The deviation in the c/a ratio does not change significantly with the inclusion of the long-range Coulomb interaction using hybrid functional 22 but decreases with van der Waals interaction (vdW-D3) 23 to 5%. Since there are only small changes in the band structures by including vdW, the band structures using GGA exchange correlation functional are presented both BaMnBi 2 and BaZnBi 2 . The band structures of BaMnBi 2 are calculated with checkerboard type of antiferromagnetic ordering for the Mn-d spins; the checkerboard order is calculated to have the lowest total energy which is 0.08 and 0.26 eV per formula unit lower than stripe-type antiferromagnetic ordering and ferromagnetic ordering, respectively, consistent with other AMnBi 2 compounds 13,24 . Fermi surfaces are calculated by interpolating energy dispersion using dense k-grid points (80 × 80) at k z = 0.