Introduction

The investigations on topological materials (TMs) have been steadily gaining traction in the field of materials physics because they exhibit novel physical phenomena1,2. Intriguingly, the topological Dirac semimetal (TDSM) is an intermediate quantum state between trivial and non-trivial states3. Also, these new types of materials are a good playground to identify the new quantum phenomena in condensed matter physics from Dirac semimetals to Weyl semimetals4,5. TDSM is observed when the valence and conduction bands are degenerate at one critical point, defined as Dirac point in 3-dimension (3D) bulk structure near the Fermi level, and exhibits Lorentz symmetry breaking, which forms a Dirac cone6,40,41,\(\overline{3 }\)m1 (164), while the doped case belongs to P3m1 (156). For comparison, Fig. 3b also shows the band structure of CaCd2SbBi along the Γ to A under HSE06. Interestingly, the substituted Bi in CaCd2SbBi causes the breaking of the inversion symmetry while preserving the C3 rotational symmetry along the z-axis. From Fig. 3a–c, we observed the down shift of \({\overline{\Delta } }_{6}\) and upward shift of \({\overline{\Delta } }_{4}+{\overline{\Delta } }_{5}\). Two bands, \({\overline{\Delta } }_{4}+{\overline{\Delta } }_{5}\) and \({\overline{\Delta } }_{6}\), near the Fermi level closed and crossed over the Fermi level resulting in a pair of Dirac points (DP2 and DP3) in the first BZ, as shown by a pair of the blue points labeled in Fig. 2c. Surprisingly, the symmetry indicators at Γ and A are still the same as CaCd2Sb2, thus resulting in a trivial insulator phase. Totally, four emerging Dirac points are formed along -A to Γ to A. Eventually, one Dirac point remains as the material is changed to CaCd2Bi2.

To further elucidate the mechanism behind the TDSM in CaCd2Bi2, the band crossing between Γ and A points is expressed as an effective Hamiltonian. To prove that the fourfold point is a Dirac point, an analysis using the k · p model is discussed. The four-band effective Hamiltonian obeys the following three symmetries: threefold rotation symmetry \(({C}_{3}^{+})\), mirror symmetry (σ1v), and inversion and time reversal symmetry (IT)64. The irreducible representations of the four-band at D (Fig. 3c, d), \({\overline{\Delta } }_{4}+{\overline{\Delta } }_{5}\) and \({\overline{\Delta } }_{6}\), are chosen as the basis set. Using the Pauli matrices σ1, σ2, σ3, and 2 × 2 identity matrix σ0, the matrix representations of the three symmetries for the basis are − σ0, iσ3, − iσ2 and \(\frac{1}{2}\)(σ0 − i\(\sqrt{3}\)σ2), iσ3, − iσ2, respectively. Thus, the four-band effective Hamiltonian takes the form

$$\left[\begin{array}{cccc}{c}_{1}+{c}_{2}{k}_{z}+{c}_{3}{k}_{z}& 0& {c}_{4}{k}_{y}-i{c}_{5}{k}_{y}& {c}_{4}{k}_{x}-i{c}_{5}{k}_{x}\\ 0& {c}_{1}+{c}_{2}{k}_{z}+{c}_{3}{k}_{z}& {c}_{4}{k}_{x}+i{c}_{5}{k}_{x}& {-c}_{4}{k}_{y}-i{c}_{5}{k}_{y}\\ {c}_{4}{k}_{y}+i{c}_{5}{k}_{y}& {c}_{4}{k}_{x}-i{c}_{5}{k}_{x}& {c}_{1}+{c}_{2}{k}_{z}-{c}_{3}{k}_{z}& 0\\ {c}_{4}{k}_{x}+i{c}_{5}{k}_{x}& {-c}_{4}{k}_{y}+i{c}_{5}{k}_{y}& 0& {c}_{1}+{c}_{2}{k}_{z}-{c}_{3}{k}_{z}\end{array}\right]$$

The eigenvalues of the four-bands (c1 + c2kz)±\(\sqrt{({c}_{4}^{2}+{c}_{5}^{2}){k}_{x}^{2}+({c}_{4}^{2}+{c}_{5}^{2}){k}_{y}^{2}+{c}_{3}^{2}{k}_{z}^{2}}\) indicates that the four-fold point is indeed a Dirac point64.

Finally, the Green’s function-derived surface states projected on the (100) plane are calculated for CaCd2Bi2 (Fig. 4a) and CaCd2SbBi (Fig. 4c) to confirm the location of Dirac points. The 3D band structure is shown in Figure S12 highlighting two Dirac cones. The corresponding Fermi arcs highlighting the surface states of CaCd2Bi2 and CaCd2SbBi are shown in Fig. 4b, d, respectively. The connections to the Dirac points are projected onto different k locations in the surface BZ. The k points in the BZ projected on (100) surface are shown in Fig. 2c. The (100) surface of the Γ-centered BZ corresponds to the projection of the M-Γ line. In Fig. 4a, the calculated surface states connect two Dirac points (DP1) in CaCd2Bi2 and cross at Γ point. However, since the structure of CaCd2SbBi breaks the inversion symmetry, four emerging Dirac points are formed. In Fig. 4c, the surface states of CaCd2SbBi, in contrast with CaCd2Bi2, do not cross at a high symmetry point while connecting each pair of Dirac points (DP2 and DP3) in the BZ. The surface states connect the two Dirac points (DP2 and DP3) within Γ to A, while the other pair of surface states connect the two Dirac points within Γ to −A.

Figure 4
figure 4

Bulk (a) CaCd2Bi2 and (c) CaCd2SbBi surface states projected on (100) plane under HSE06 with SOC. 2D Fermi arcs for (b) CaCd2Bi2 and (d) CaCd2SbBi at the E−EF = −0.1 eV and 0.1 eV, respectively.

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Methods

We systematically explored 10 compounds of CaM2X2 where M = Zn or Cd; and X = N, P, As, Sb, or Bi, under the three different existing structures, totaling 30 compounds of Zintl materials: CaAl2Si2-, ThCr2Si2-, and BaCu2S2-type crystal phase44, through first-principles calculation as implemented in Vienna Ab initio Simulation Package (VASP) using projector-augmented wave (PAW) functions65,66 under the Perdew-Burke-Ernzerhof (PBE)67 functional with plane wave cut-off energy of 400 eV. The structures were allowed to relax until the residual Hellmann–Feynman forces acting on each atom were no greater than 10–2 eV/Å. The K-points sampling used is the Γ-centered grid Monkhorst–Pack with 24 × 24 × 12 kmesh within the first Brillouin zone (BZ) for all the self-consistent calculations. Phonon spectra calculations are performed using the Phonopy68 package. Spin-polarized and SOC were included in all the energetic calculations. In determining the magnetic properties, we treat magnetic moments (μ) greater than 0.1 as ferromagnetic (FM) states, while the rest are non-magnetic (NM) states. In NM state, we analyzed each case for the topological properties in 3D bulk structure using the SymTopo package55 which uses symmetry-based indicators that can categorize different topological properties, such as trivial insulator (I), conventional metal (M), topological insulator (TI), topological crystalline insulator (TCI), high-symmetry-point semi-metal (HSPSM), and high-symmetry-line semi-metal (HSLSM). To predict the topological properties of the investigate materials more accurately, Heyd-Scuseria-Ernzerhof (HSE06) hybrid functional69 is further included in the band structure calculations. Finally, the Maximally-Localised Wannier Functions (MLWF) of CaM2Bi2 are obtained from the Wannier9070 program, and the surface Green’s function is calculated using the WannierTools program71.

Conclusions

In this study, a systematic materials search coupled with first-principles calculations was performed to investigate 10 Zintl-phase compounds with a chemical formula of CaM2X2 (M = Zn or Cd; X = N, P, As, Sb, or Bi) and found all of these materials adapt CaAl2Si2-type structure. Symmetry-based indicators are used to evaluate their topological properties. Surprisingly, we found that CaM2Bi2 (M = Zn or Cd) are topological crystalline insulators (TCIs) under the GGA-PBE. Further calculations under the hybrid functional approach reveal that they exhibit topological Dirac semimetal (TDSM) states, where the four-fold degenerate Dirac points are located along the high symmetry line in-between Г to A points and is analyzed using the effective Hamiltonian derived by the k · p model. These findings are verified through Green's function surface state calculations under HSE06. Finally, phonon calculations were done to verify the thermodynamic stability of the CaCd2Bi2, implying possible synthesis. Our findings show that the Zintl phase of AM2X2 compounds can be a new playground in the search for new topological materials (TMs).