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Predicting Dirac semimetals based on sodium ternary compounds

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 well-known Dirac semimetal, Na3Bi, three compounds in one family, namely Na2MgSn, Na2MgPb, and Na2CdSn, are located. Furthermore, hybrid-functional calculations with improved accuracy for estimation of band inversion show that Na2MgPb and Na2CdSn 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 three-dimensional (3D) analogs of graphene6 with and only with Dirac nodes on the Fermi level. These Dirac nodes are formed by band crossing, and the low-energy 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 high-symmetric 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 triple-point 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 DSMs11 is achieved in the series of studies on Na3Bi2,4 and Cd3As2,3,18,19,20,21 both of which were first proposed through first-principles 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 Na3Bi and Cd3As2 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 Na3Bi as a model system for tuning the chemical degree of freedom. Three sodium ternary compounds, Na2MgSn, Na2MgPb, and Na2CdSn, 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 Na3Bi2,7 can be viewed as the AB stacking of honeycomb layers along the c-axis, as shown in Fig. 1(a). For each honeycomb layer, one Na(1) atom and one Bi atom take the A and B sub-lattice 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 well-understood DSM, its low-energy 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 Γ-A direction forming Dirac nodes are dominated by Na(1)-s orbitals and Bi 6px,y orbitals.2 At Γ point the Na(1)-s bands are lower than those of Bi 6px,y mainly due to two things. One is that the heavy Bi has a relatively high on-site energy for 6p orbitals. The other is the interlayer coupling leads to splittings between the bonding and anti-bonding states for both s and p bands along Γ-A. These two crossing bands with different orbital characters have different irreducible representations along the Γ-A direction and the Dirac nodes are protected.

Fig. 1
figure1

a Crystal structure of Na3Bi with Na(1), Na(2), and Bi sites indicated. b Top view of the Na2MgSn unit-cell with Mg and Sn replacing Na(1) and Bi atoms in a, respectively. c The bulk Brillouin zone and the projected surface Brillouin zone for (100), (010), and (001) surfaces

Inspired by the above understanding, we notice that Na3Bi can be regarded as Na2Na1Bi. 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 band-filling, Na(1) should be substituted with atoms having two-valence electrons, such as alkaline-earth metal and II-B elements like Mg, Ca, Sr, Zn, Cd, and Hg. Thus, three sodium-containing ternary compounds reported experimentally, namely Na2MgSn, Na2MgPb, and Na2CdSn, are naturally and immediately located. Na2MgSn and Na2MgPb have been successfully synthesized recently,22,23 while Na2CdSn has been synthesized and investigated in 1980.24

Similar to Na3Bi, all these compounds crystallize in hexagonal lattice with the space group P63/mmc (#194, \(D_{6h}^4\)). We take Na2MgSn as an example, as demonstrated in Fig. 1(b). There are four Na atoms, two Mg atoms, and two Sn atoms in each unit-cell. The shortest bonds are those in the Mg-Sn layer. Na and Sn atoms align along the c-axis 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

Table 1 Optimized lattice constants, and lengths of the two shortest bonds (in-plane Mg/Cd-Sn/Pb bonds and vertical Na-Sn/Pb bonds) for Na2MgSn, Na2MgPb, and Na2CdSn

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 Na3Bi is structural dynamic stability. For Na3Bi, the P63/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 Na3Bi is still under debate.29,30

Fig. 2
figure2

Phonon dispersion for a Na2MgSn, b Na2MgPb, and c Na2CdSn

Electronic structures

The calculated electronic structures of all three materials using the Perdew-Burke-Ernzerhof (PBE) functional and the Heyd-Scuseria-Ernzerhof (HSE) hybrid-functional 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 Γ-A, where the band inversion and Dirac nodes happen in Na3Bi.

Fig. 3
figure3

Calculated electronic structures for a Na2MgSn, b Na2MgPb, and c Na2CdSn using the PBE functional without spin-orbit coupling (top panel), and hybrid-functional without (middle panel) and with (bottom panel) spin-orbit coupling. The fatted bands with the weight of atomic orbital projection near the Fermi level are present in the middle panel. The two arrows point out the two Dirac cones formed by band crossings from s-band and bonding, anti-bonding px,y − bands along Γ-A

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 unit-cell within both PBE and HSE calculations. The overestimation of band inversion in PBE is improved by HSE calculation. One can find that the lightest Na2MgSn has no band inversion and it is a normal semiconductor in HSE case. Na2MgPb has the same total mass as Na3Bi and is slightly lighter than the heaviest Na2CdSn, but all of them have the similar band inversion along Γ-A.

The spin-orbit coupling (SOC) is further included and the band structures of them are shown in the bottom panel in Fig. 3. Both Na2MgPb and Na2CdSn are DSMs with Dirac nodes on the path Γ-A, while Na2MgSn is an indirect band gap of 0.13 eV. For Na2MgPb and Na2CdSn, one notable difference from Na3Bi is that there are two pairs of Dirac nodes since the one s-orbital band inverts with both the bonding and anti-bonding px,y-orbital bands. The s-band belongs to Γ7 representation while the two px,y bands belong to Γ9 representation. The splitting in the bonding and anti-bonding px,y (in-plane orbitals) bands along Γ-A (z-direction) seems quite small, indicating the weak interlayer coupling among these in-plane orbitals along the stacking direction.

Surface states

Similar to Na3Bi, there will be surface states for DSMs Na2MgPb and Na2CdSn. To simulate surface states to be observed by the angle-resolved 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 Na3Bi,2 WC-type ZrTe,33 and KHgAs.34 The projected surface density of states for the (100), (010), and (001) surfaces of Na2MgPb 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 Na3Bi.2,4

Fig. 4
figure4

Surface band structure for a (100), b (010), and c (001) surfaces of Na2MgPb. The arrow points out the bulk Dirac cone, and the circle labels the topological surface states due to Z2 = 1 in kz = 0 plane. The corresponding Fermi surface with Fermi level at bulk Dirac point (61 meV) is shown in df

Fig. 5
figure5

Surface band structure for a (100), b (010), and c (001) surfaces of Na2CdSn. The arrow points out the bulk Dirac cone, and the circle labels the topological surface states. The corresponding Fermi surface with Fermi level at bulk Dirac point (40 meV) is shown in df

The projected surface density of states for the (100), (010), and (001) surfaces of Na2CdSn 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 anti-bonding px,y − orbital bands, the nontrivial surface states of Na2CdSn are not as clear as those in Na2MgPb. 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 Na2MgPb.

In this paper, we demonstrate an approach for searching new DSM materials by tuning the chemical degree of freedom based on material design of well-known DSM Na3Bi. By keeping both the crystal and electronic structures essentially identical to Na3Bi, three compounds Na2MgSn, Na2MgPb, and Na2CdSn 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 Na3Bi, paving the way for experimental verification. The hybrid-functional calculations with spin-orbit coupling show that Na2MgSn 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 Na2MgPb. For Na2CdSn, the band inversion is induced by replacing Mg with heavier Cd in Na2MgSn. 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 Na2CdSn 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 anti-bonding s bands are lower than the px,y bands along the whole path Γ-A. And we have confirmed that in this case it is a TCI of Z12 = 836 with mirror Chern number 2 in m001 plane.

Methods

First-principles calculations are performed using the Vienna ab-initio simulation package (VASP)37 based on density functional theory (DFT). The generalized gradient approximation (GGA) in the PBE parameterization for the exchange-correlation functional is used for structural relaxation. A plane-wave basis set is employed with kinetic energy cutoff of 500 eV. We use the projector-augmented-wave method and the related pseudo-potential for each element. A 11 × 11 × 5 q-mesh is used during structural relaxation for the unit-cell until the energy difference is converged within 10−6 eV, with a Hellman-Feynman force convergence threshold of 10−4 eV/Å. To improve the underestimation of band gap in the PBE functional, hybrid-functional method based on the HSE method are adopted.3840 The harmonic interatomic force constants (IFCs) are obtained by density functional perturbation theory using a 3 × 3 × 2 supercell with a 3 × 3 × 3 q-mesh. The phonon dispersion is calculated from the harmonic IFCs using the PHONOPY code.41,42 The Wannier functions43 for Cd/Mg s-orbital and Sn/Pb s-and p-orbitals 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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Correspondence to Hao Zhang or Hongming Weng.

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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/s41524-018-0124-5

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