Electronic structure and topological properties of centrosymmetric MoAs2/WAs2 from first principles

We investigate the electronic structure of group VI-B transition metal di-arsenides (TAs2, T = Mo, W). By comparing the formation energies, the centrosymmetric di-arsenides compounds are energetically more stable, in contrast to the di-phosphorides (MoP2/WP2). Both compounds can be well described by a two-band model with a pair of well-separated electron/hole bands. The electron/hole carrier density is nearly compensated in MoAs2 (|n e − n h|/n h < 1%). The \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathscr{Z}}}_{2}$$\end{document}Z2 classification for all partially occupied bands are topologically strong (1;001), and therefore robust surface states are expected in these materials. Using the adaptive K-mesh method, no energy degenerate state could be found except the spin degeneracy in the whole Brillouin zone, excluding the possibility of intrinsic Dirac or Weyl points near the Fermi level in the system.

In this article, we report our latest first-principles calculation results of MoAs 2 and WAs 2 . The calculation shows that the electron/hole carrier densities are nearly compensated in MoAs 2 , which can serve as natural explanation for the large positive MR. We show that the Z 2 classification for these compounds are strongly nontrivial (1;001), and that no accidental degeneracy (Dirac or Weyl point) can be found in the whole Brillouin zone without external magnetic field.

Results and Discussion
MoP 2 /WP 2 were investigated previously by Autès et al. 17 . The phosphorus compounds are intrinsic WSMs hosting robust Weyl points due to their non-centrosymmetric crystal structure. In contrast, the arsenic compounds T As 2 (T = Mo, W) crystallize in the same structure as TaSb 2 ( Fig. 1(a)), which is base-centered monoclinic with centrosymmetric space group C2/m (#12) ( Table 1). Therefore, they cannot host Weyl points since both the time reversal symmetry and inversion symmetry are preserved in these compounds. We adopt the same bulk and surface Brillouine zone as well as the high symmetry points as defined in ref. 26. After full structure optimization, the lattice constants and internal atomic positions from calculation can be well compared with experimental values within 5% errorbar (Table 1). To further investigate the possible crystalline structures of these pnictides, we have also performed calculations for T P 2 with hypothetical centrosymmetric crystal structure (space group C2/m). The centrosymmetric MoP 2 (WP 2 ) is energetically ~80 meV/f.u. (30 meV/f.u.) higher than the actual non-centrosymmetric MoP 2 (WP 2 ). Similarly, the noncentrosymmetric MoAs 2 (WAs 2 ) is energetically ~120 meV/f.u. (205 meV/f.u.) higher than centrosymmetric MoAs 2 (WAs 2 ). Therefore, the noncentrosymmetric MoAs 2 /WAs 2 and centrosymmetric MoP 2 /WP 2 are energetically unstable at ambient pressure, consistent with the experimental observation.
Band Structure and Density of States. Before performing further analysis, it would be instructive to compare the band structure of T As 2 compounds calculated with/without spin-orbit coupling (SOC). Since Mo and W belong to the same family, the band structure of both compounds are very close to each other. Dirac-like dispersion features can be identified between X 1 and Y from both T As 2 band structure without SOC (red dashed box in Fig. 2a). The specific feature is ~0.1 eV above E F in MoAs 2 (~0.2 eV above E F in WAs 2 ). Please be noted that the feature (blue dashed box in Fig. 2b) at ~0.1 eV below E F in WAs 2 is not band-crossing feature (inset of Fig. 2(b)), although the two bands are extremely close to each other. More band-crossing features could be  identified at much lower energies (below E F -0.8 eV), between X 1 and Y as well as F 1 and L, but they are far from the Fermi level and therefore may not be relevant with transport properties. Close to the Fermi level, the electronic states of both compounds are dominated by Mo-4d or W-5d orbitals (red color in Fig. 2), with moderate As-4p contributions (blue color in Fig. 2). The band structure is then fitted to a tight-binding Hamiltonian using the maximally localized Wannier function (MLWF) method. Using the adaptive K-mesh searching method, it is then determined that the band-crossings close to E F form nodal rings in the BZ. All above mentioned band-crossing features are fully gapped out when SOC is considered in the calculation. For the MoAs 2 compound, since neither Mo nor As is heavy element, the SOC splitting is fairly small (~50 meV for the feature between X 1 and Y); while for the WAs 2 compound, the SOC splitting is much larger (~180 meV for the same feature). The SOC does not significantly alter the orbital component for either compound, thus the electronic states near E F are still dominated by Mo-4d/W-5d and As-4p orbitals. Using the adaptive K-mesh method, we searched the whole BZ and ruled out the possibility of accidental energy degeneracies in these systems. The electron/hole bands crossing the Fermi level are therefore well separated from each other, as well as from all other bands. Thus, the electron/hole type carrier densities can be evaluated by calculating the electron/ hole band contributions to the density of states (DOS). It is therefore estimated that the n e = 1.90 × 10 21 cm −3 and n h = 1.92 × 10 21 cm −3 in MoAs 2 (n e = 1.85 × 10 21 cm −3 and n h = 1.72 × 10 21 cm −3 in WAs 2 ), where n e and n h represents the carrier density of electrons and holes, respectively. Therefore, the deviation from electron-hole compensation is ~1% in MoAs 2 with slightly more holes, whereas the deviation is ~7.4% in WAs 2 with slightly more electrons. From the two-band model with both electron and hole contributions, the Hall coefficient at small field limit and the magnetoresistivity (MR) 27  where μ e and μ h are the mobilities of electron-type and hole-type carriers, respectively. Therefore, a nearly compensated metal with slightly more hole-type carrier can also exhibit negative R H if electron mobility is sufficiently larger than hole mobility (μ e /μ h > 1.005 for MoAs 2 ), as well as B 2 -dependent MR at small field. If the deviation from electron-hole compensation is very small, the magnetic field required to saturate MR can be very large. These arguments are in consistency with the experimental observations of MoAs 2 , and thus we believe that the system can be described by a two-band model with both electron and hole contributions.

Topological Invariants and Surface States.
Since the SOC gapped out all nodal points between the electron and hole bands, and separated them from all other bands, the topological invariant Z 2 can be well defined for the system. As the system has an inversion center, its Z 2 invariant can be determined by calculating the parities of band states at 8 time-reversal invariant momenta (TRIM) 28 (Table 2). The resulting Z 2 classification counting up to the partially filled hole (band 12)/electron band (band 13) is (0;000) and (1;001), respectively. Therefore the band structure of both MoAs 2 and WAs 2 are topologically strong, and robust surface states between band 13 and 14 immune to impurities are expected. However, since both MoAs 2 and WAs 2 are indeed very metallic systems, they are not topological insulators, and the appearance of these surface states are not guaranteed, similar to the case in TaSb 2 26 . Using the surface green's function method, we calculated the electronic states of MoAs 2 at [010] surfaces (Fig. 3). From Fig. 1b, it is clear that the bulk states from Γ, X, Y and X 1 will be projected to the 2D BZ center Γ; bulk states from Z, L, I and I 1 will be projected to Y; M to C; N to X. The surface states are completely overwhelmed by the bulk states along Γ to X, but they could be observed around Y, C and H, respectively. Unfortunately, all the surface states crossing the Fermi levels are not topological, since none of them connects between bulk band 13 and 14.
can be used to determine the Z 2 topological invariant, and the hole (electron) band crossing E F is n = 12 (13). Π 12 and Π 13 are exactly the same for both compounds. points can be induced by external magnetic field in TaAs 2 system 23 . Such phenomena is also possible in MoAs 2 or WAs 2 , leading to longitudinal negative MR observed in experiments. In addition, since the Z 2 invariant up to band 13 is strongly topological, topological insulator state is possible in these structures, if the system could be electron doped to band 13 while maintaining the direct gap between band 13 and 14.

Conclusion
Using the first-principles calculation method, we have studied the electronic structure and topological properties of MoAs 2 and WAs 2 . Despite of their similar chemical compositions to MoP 2 , these arsenides are centrosymmetric, and thus cannot hold intrinsic Weyl points. Both compounds could be regarded as two band system with both electron and hole carriers. The carriers are nearly compensated in MoAs 2 , which may lead to large positive MR proportional to B 2 . The Z 2 classification of these materials is (1;001), and robust surface states are expected. Additional degenerate states other than spin degeneracy are absent throughout the whole Brillouin zone, meaning the system do not hold intrinsic Dirac nor Weyl points.

Methods
The electronic structure of these compounds were calculated using density functional theory employing plane-wave basis projected augmented wave (PAW) method as implemented in Vienna Abinitio Simulation Package (VASP) 34,35 . After careful convergence tests, we found that an energy cut-off of 400 eV and a 16 × 16 × 7 Γ-centered K-mesh (over primitive cell) is sufficient to converge the total energy to 1 meV/atom. The electronic structure calculations were then performed using crystal structures with optimized lattice constants and internal atomic parameters. The density of states (DOS) were obtained with dense Γ-centered K-mesh of 32 × 32 × 14 and tetrahedron method. With the maximally localized wannier function (MLWF) method 36 , the DFT band structure is extrapolated to a 100 × 100 × 100 K-mesh to construct the Fermi surface. The resulting tight-binding Hamiltonian was also used to calculate the surface states using the surface Green's function 37 . Z 2 topological indices were calculated using the parity-check method 28 .
The band anti-crossing features, or energy degeneracy points (nodal points) in the current study, are identified using an adaptive K-mesh searching method. It begins with interpolating band energies on a dense 100 × 100 × 100 K grid using a well fitted tight-binding Hamiltonian (with MLWF method in the current study). Then the vicinity of the K-points yielding nearly degenerate eigen-energies (δε ε ε = − < . + 0 01eV k n k n 1 ) will be extrapolated to a 1000 times denser K grid (10 times in each direction) and the K-point yielding smallest δε will be kept as the initial guess of the next iteration. After 3 iterations, the K-points yielding δε < 1.0 × 10 −5 eV will be regarded as potential candidates for nodal points. Eventually, all candidates are cross-checked with DFT calculations to verify the energy differences.