Abstract
Predicting new Dirac semimetals, as well as other topological materials, is challenging since the relationship between crystal structure, atoms and band topology is complex and elusive. Here, we demonstrate an approach to design Dirac semimetals via exploring chemical degree of freedom. Based on understanding of the wellknown Dirac semimetal, Na_{3}Bi, three compounds in one family, namely Na_{2}MgSn, Na_{2}MgPb, and Na_{2}CdSn, are located. Furthermore, hybridfunctional calculations with improved accuracy for estimation of band inversion show that Na_{2}MgPb and Na_{2}CdSn have the band topology of Dirac semimetals. The nontrivial surface states with Fermi arcs on the (100) and (010) surfaces are shown to connect the projection of bulk Dirac nodes. Most importantly, the candidate compounds are dynamically stable and have been experimentally synthesized. The ideas in this work could stimulate further predictions of topological materials based on understanding of existing ones.
Introduction
Dirac semimetals (DSMs)^{1,2,3,4,5} are the threedimensional (3D) analogs of graphene^{6} with and only with Dirac nodes on the Fermi level. These Dirac nodes are formed by band crossing, and the lowenergy excitation around them leads to quasiparticles described by Dirac equation as emergent massless Dirac fermions.^{5,7,8,9,10,11} Up to now, there have been three classes of DSM proposed. One is the Dirac nodes with fourfold essential degeneracy, which is enforced by the nonsymmorphic symmetry at the highsymmetric momenta on the boundary of the Brillouin zone.^{1} The second is the accidental degenerate Dirac nodes, which appears as the topological phase transition critical point between different topological insulating states.^{12} The third one is also an accidental DSM, but the band crossing points are caused by band inversion and protected by proper crystal symmetry.^{2,11} DSMs serve as a singular point of various topological states, such as topological insulators, Weyl semimetals, nodal line semimetals, and triplepoint semimetals.^{13} DSMs exhibit many novel properties, such as high carrier mobility,^{14} unique surface states with Fermi arcs,^{2,15} and negative longitudinal magnetoresistivity due to the chiral anomaly.^{16,17}
The breakthrough in the search for stable DSMs^{11} is achieved in the series of studies on Na_{3}Bi^{2,4} and Cd_{3}As_{2},^{3,18,19,20,21} both of which were first proposed through firstprinciples calculations. They present good examples of the realization of the DSM in the above third class. The Dirac nodes are induced by band inversion and protected by proper axial rotational symmetry.^{2,11} Such protection makes the Dirac nodes quite robust within a finite range of Hamiltonian parameters, which is exactly the reason why this class of DSM is experimentally available while the other two remain to be found.
Despite the success in identifying Na_{3}Bi and Cd_{3}As_{2} and the intensive studies on them, to identify more DSMs remains a big challenge. How to locate a specific material among thousands of known compounds is not clear. Here, we demonstrate a chemically intuitive approach for searching new DSMs to show the underlying physics and ideas. We choose the first DSM Na_{3}Bi as a model system for tuning the chemical degree of freedom. Three sodium ternary compounds, Na_{2}MgSn, Na_{2}MgPb, and Na_{2}CdSn, are naturally selected. Further theoretical calculations reveal that the chemical trend in the elements of the same column in periodic table plays an important role in band inversion. The proposed general design principle can be used for finding new DSMs, as well as other topological materials.
Results and Discussion
Material design
The crystal structure of Na_{3}Bi^{2,7} can be viewed as the AB stacking of honeycomb layers along the caxis, as shown in Fig. 1(a). For each honeycomb layer, one Na(1) atom and one Bi atom take the A and B sublattice site, respectively. There are two additional Na(2) atoms above and below the Na(1)Bi honeycomb layer to connect the Bi atoms in the neighboring layers. As a wellunderstood DSM, its lowenergy electronic band structure has been found to be mostly determined by the Na(1) and Bi atoms in the honeycomb layer. The two crossing bands along the ΓA direction forming Dirac nodes are dominated by Na(1)s orbitals and Bi 6p_{x,y} orbitals.^{2} At Γ point the Na(1)s bands are lower than those of Bi 6p_{x,y} mainly due to two things. One is that the heavy Bi has a relatively high onsite energy for 6p orbitals. The other is the interlayer coupling leads to splittings between the bonding and antibonding states for both s and p bands along ΓA. These two crossing bands with different orbital characters have different irreducible representations along the ΓA direction and the Dirac nodes are protected.
Inspired by the above understanding, we notice that Na_{3}Bi can be regarded as Na_{2}Na_{1}Bi. The first two Na are on Na(2) site, which support the 3D lattice structure and also supply two electrons to the Na(1)Bi honeycomb layer. If the crystal structure and the electronic structure are similar to those of Na3Bi, one can get a new DSM material. Thus, this leads to the idea to find other potential DSMs by simply changing the atoms in the Na(1)Bi layer. To induce band inversion, Bi should be substituted with other similar heavy metal atoms such as Pb and Sn. Since Pb and Sn have one fewer valence electron than Bi, to maintain the same bandfilling, Na(1) should be substituted with atoms having twovalence electrons, such as alkalineearth metal and IIB elements like Mg, Ca, Sr, Zn, Cd, and Hg. Thus, three sodiumcontaining ternary compounds reported experimentally, namely Na_{2}MgSn, Na_{2}MgPb, and Na_{2}CdSn, are naturally and immediately located. Na_{2}MgSn and Na_{2}MgPb have been successfully synthesized recently,^{22,23} while Na_{2}CdSn has been synthesized and investigated in 1980.^{24}
Similar to Na_{3}Bi, all these compounds crystallize in hexagonal lattice with the space group P6_{3}/mmc (#194, \(D_{6h}^4\)). We take Na_{2}MgSn as an example, as demonstrated in Fig. 1(b). There are four Na atoms, two Mg atoms, and two Sn atoms in each unitcell. The shortest bonds are those in the MgSn layer. Na and Sn atoms align along the caxis connected by the second shortest bonds. The optimized lattice constants and bond lengths are listed in Table 1, which are in good agreements with previous experimental results.^{22,23,24}
For future experimental explorations, the stability of these three structures is an important aspect.^{25,26,27} A material is dynamically stable when there is no imaginary phonon frequency existing in its phonon spectrum. As shown in Fig. 2, no imaginary phonon frequency is found in all three materials, indicating their dynamical stability at 0 K. This is consistent with the existence of them reported by experiments. As possible candidates for DSMs, one main advantage of these sodium ternary compounds compared to Na_{3}Bi is structural dynamic stability. For Na_{3}Bi, the P6_{3}/mmc phase has been found dynamically unstable at the ground state due to large imaginary phonon frequencies.^{28} In fact, even now the ground state of Na_{3}Bi is still under debate.^{29,30}
Electronic structures
The calculated electronic structures of all three materials using the PerdewBurkeErnzerhof (PBE) functional and the HeydScuseriaErnzerhof (HSE) hybridfunctional are shown in the top and middle panels of Fig. 3, respectively. The fatted bands with the weight of projected atomic orbitals are also shown in the middle panel for each of them. We focus on the band structures along ΓA, where the band inversion and Dirac nodes happen in Na_{3}Bi.
In general, the strength of band inversion between the bands composed of s orbitals (of Mg or Cd on Na(1) site) and p orbitals (of Sn or Pb on Bi site) follows the order of total atomic number (mass) of the atoms in the unitcell within both PBE and HSE calculations. The overestimation of band inversion in PBE is improved by HSE calculation. One can find that the lightest Na_{2}MgSn has no band inversion and it is a normal semiconductor in HSE case. Na_{2}MgPb has the same total mass as Na_{3}Bi and is slightly lighter than the heaviest Na_{2}CdSn, but all of them have the similar band inversion along ΓA.
The spinorbit coupling (SOC) is further included and the band structures of them are shown in the bottom panel in Fig. 3. Both Na_{2}MgPb and Na_{2}CdSn are DSMs with Dirac nodes on the path ΓA, while Na_{2}MgSn is an indirect band gap of 0.13 eV. For Na_{2}MgPb and Na_{2}CdSn, one notable difference from Na_{3}Bi is that there are two pairs of Dirac nodes since the one sorbital band inverts with both the bonding and antibonding p_{x,y}orbital bands. The sband belongs to Γ_{7} representation while the two p_{x,y} bands belong to Γ_{9} representation. The splitting in the bonding and antibonding p_{x,y} (inplane orbitals) bands along ΓA (zdirection) seems quite small, indicating the weak interlayer coupling among these inplane orbitals along the stacking direction.
Surface states
Similar to Na_{3}Bi, there will be surface states for DSMs Na_{2}MgPb and Na_{2}CdSn. To simulate surface states to be observed by the angleresolved photoemission spectroscopy (ARPES), we use an iterative surface Green’s function method,^{31,32} where the HSE + SOC band structures are used in generating the maximally localized Wannier functions. The Brillouin zone of bulk and the projected surface Brillouin zones of (100), (010), and (001) planes are exactly the same as those of Na_{3}Bi,^{2} WCtype ZrTe,^{33} and KHgAs.^{34} The projected surface density of states for the (100), (010), and (001) surfaces of Na_{2}MgPb are shown in Fig. 4(a)–(c). On both (100) and (010) side surfaces, the projection of bulk Dirac cone (pointed by the arrow) is well separated from the topological surface Dirac cone (labeled by the circle). The surface Dirac cone has its branches merging into the bulk states at the projection of 3D Dirac point, which leads to the arc like Fermi surface when the Fermi level is set at the bulk Dirac nodal point. There are two Fermi arcs touch each other at the surface projection of bulk Dirac point at 61 meV, as shown in Fig. 4(d, e). For the (001) surface, the projection of bulk Dirac nodes overlaps with the surface Dirac cone as shown in Fig. 4(c), which is similar to the case in Na_{3}Bi.^{2,4}
The projected surface density of states for the (100), (010), and (001) surfaces of Na_{2}CdSn are shown in Fig. 5. For both the (100) and (010) surfaces, the bulk Dirac cone is closer to the Γ point. Due to the smaller band splitting between the bonding and antibonding p_{x,y} − orbital bands, the nontrivial surface states of Na_{2}CdSn are not as clear as those in Na_{2}MgPb. For the (100) surface, the Fermi arcs are hidden within the projection of the bulk states on the surface. They can be well revealed in the Fermi surface plot on the (010) surface with Fermi level at bulk Dirac point of 40 meV, as shown in Fig. 5(e). For the (001) surface, the surface projection of the bulk states is superposed with the nontrivial surface states, which is similar to the case in Na_{2}MgPb.
In this paper, we demonstrate an approach for searching new DSM materials by tuning the chemical degree of freedom based on material design of wellknown DSM Na_{3}Bi. By keeping both the crystal and electronic structures essentially identical to Na_{3}Bi, three compounds Na_{2}MgSn, Na_{2}MgPb, and Na_{2}CdSn are naturally located and two of them are identified as DSM candidates based on our theoretical calculations. The phonon calculations confirm that these compounds are stable than Na_{3}Bi, paving the way for experimental verification. The hybridfunctional calculations with spinorbit coupling show that Na_{2}MgSn is an indirect band gap normal semiconductor. By substituting Sn by heavier Pb, the band inversion occurs, and the Dirac nodes due to band crossing are protected by crystal symmetry in Na_{2}MgPb. For Na_{2}CdSn, the band inversion is induced by replacing Mg with heavier Cd in Na_{2}MgSn. Moreover, the coexistence of both a bulk 3D Dirac cone and topological surface states can be observed in the projected surface density of states for side surfaces (100) and (010), which can be used as a reference for further experimental validation in ARPES or scanned tunneling microscopy measurements. We hope the idea in this example would lead to more material design efforts based on known topological materials for more successful and efficient predictions.
During the preparation of this manuscript, ref. ^{35} proposed that Na_{2}CdSn is a topological crystalline insulator (TCI) candidate, which is consistent with our PBE + SOC calculation. From Fig. 3(c), it is seen that both bonding and antibonding s bands are lower than the p_{x,y} bands along the whole path ΓA. And we have confirmed that in this case it is a TCI of Z_{12} = 8^{36} with mirror Chern number 2 in m_{001} plane.
Methods
Firstprinciples calculations are performed using the Vienna abinitio simulation package (VASP)^{37} based on density functional theory (DFT). The generalized gradient approximation (GGA) in the PBE parameterization for the exchangecorrelation functional is used for structural relaxation. A planewave basis set is employed with kinetic energy cutoff of 500 eV. We use the projectoraugmentedwave method and the related pseudopotential for each element. A 11 × 11 × 5 qmesh is used during structural relaxation for the unitcell until the energy difference is converged within 10^{−6} eV, with a HellmanFeynman force convergence threshold of 10^{−4} eV/Å. To improve the underestimation of band gap in the PBE functional, hybridfunctional method based on the HSE method are adopted.^{38–40} The harmonic interatomic force constants (IFCs) are obtained by density functional perturbation theory using a 3 × 3 × 2 supercell with a 3 × 3 × 3 qmesh. The phonon dispersion is calculated from the harmonic IFCs using the PHONOPY code.^{41,42} The Wannier functions^{43} for Cd/Mg sorbital and Sn/Pb sand porbitals are generated, which are used in the surface state calculations.
Data availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
We thank the valuable discussions with X. Wan, C. Fang, Z. Song, and T. Zhang. This work is supported by the the National Key Research and Development Program of China (No. 2016YFA0300600 and 2018YFA0305700), the National Natural Science Foundation of China (Grant Nos. 11374063 and 11674369) and the “Strategic Priority Research Program (B)” of the Chinese Academy of Sciences (Grant No. XDB07020100).
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H.M.W. and H.Z. designed the research. B.P., C.M.Y., and H.Z. performed the calculations. B.P., H.M.W., Z.F., and H.Z. analyzed and discussed the results. B.P. and W.H.M. wrote the text of the manuscript. B.P. and C.M.Y. contributed equally to this work. All authors commented on the manuscript.
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Peng, B., Yue, C., Zhang, H. et al. Predicting Dirac semimetals based on sodium ternary compounds. npj Comput Mater 4, 68 (2018). https://doi.org/10.1038/s4152401801245
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