Quantum spin Hall insulating phase and van Hove singularities in Zintl single-quintuple-layer AM₂X₂ (A = Ca, Sr, or Ba; M = Zn or Cd; X = Sb or Bi) family

The discovery of the quantum Hall effect (QHE)¹ in 1980 provided the first observation of a topologically nontrivial state of matter and spurred rigorous development of its theoretical underpinnings.^2–6 Another milestone in this connection is the discovery of graphene,⁷ which led to Kane and Mele's model⁸ that exhibits the quantum spin Hall effect (QSHE)⁹ driven by intrinsic spin–orbit coupling (SOC). Exfoliating graphene⁷ from graphite has inspired further studies of materials with honeycomb-like structures that exhibit the QSHE^10–16 and also opened up the field of 2D materials more generally because these materials offer unique opportunities for hosting properties that set them apart from their 3D counterparts.¹⁷ 

Three-dimensional topological insulators (3D TIs) are unique in that they support conducting surface states, while 2D TIs similarly host symmetry-protected conducting edge states with an insulating interior.^18,19 In the 2D TIs, the SOC preserves the time-reversal symmetry (TRS), and, as a result, the conducting edge states are protected from backscattering (BS).¹⁹ Notably, 2D systems can support van Hove singularities (vHss),²⁰ which are well-known to play an important role in enhancing superconductivity,²¹ ferromagnetism,²² and antiferromagnetism.²³ The coexistence of vHss and TI phases has been observed in the 2D-regime, implying the possibility of 2D topological superconductors (TSCs),^24,25 which are relevant for low-power consuming electronic devices, quantum computation, Majorana physics, and spintronics. In this connection, 2D materials such as the group IIIA−VA compounds, pristine IVA and VA-based honeycombs,^14,26–29 nitrides,³⁰ Janus materials,^31–33 metal oxides,³⁴ MXenes,³⁵ phosphorenes,³⁶ ternary transition metal chalcogenides (TTMCs),^37,38 and transition metal dichalcogenides (TMDs)^39–41 have gained interest because their 2D cousins have been identified for hosting nontrivial phases.^8,29,42–44

Another family of materials, the Zintl compounds, is gaining attention because of their interesting applications in thermoelectrics.⁴⁵ Zintl compounds with AM₂X₂ stoichiometry can exist in orthorhombic BaCu₂S₂ (Pnma), tetragonal ThCr₂Si₂ (I4/mmm), and trigonal CaAl₂Si₂ (P$3¯$m1) structures.^45–47 Among these, the CaAl₂Si₂-type compounds have the advantage of substantial tunability.⁴⁵ The Zintl CaAl₂Si₂ structure consists of 2D rafts of Ca²⁺ ions separated by tightly bound Al₂Si₂²⁻ layers.⁴⁸ The AM₂X₂ structure can, thus, be viewed as consisting of 2D A²⁺ ions separated by M₂X₂²⁻ layers or two stacked MX-layers. As early as the 1980s, several bulk AM₂X₂ compounds including CaZn₂Sb₂, CaCd₂Sb₂, and BaCd₂Sb₂ were synthesized in the CaAl₂Si₂-type structure.^47,49–51 Theoretical studies have reported a rich tapestry of electronic properties of bulk CaZn₂X₂,⁵² CaCd₂X₂,⁵³ and SrZn₂X₂ (X = N, P, As, Sb, or Bi)⁵⁴ for optoelectronics applications, while ACd₂Sb₂ (A = Ca, Ba, or Sr)⁵⁵ and XCr₂Bi₂ (X = Ca or Sr)⁵⁶ have the potential for thermoelectric applications. Intriguingly, there have been recent experimental confirmations that bulk CaAl₂Si₂ hosts topological Dirac semimetal properties⁵⁷ and nontrivial topological states,⁵⁸ using angle-resolved photoemission spectroscopy (ARPES)⁵⁷ and magnetotransport measurements,⁵⁸ respectively, although the topological properties were only observed below the Fermi level. These results suggest the possibility that other Zintl CaAl₂Si₂-type compounds could host nontrivial topological states.

A recent study demonstrated the successful synthesis of a bilayer MX honeycomb via Li-intercalation in 3D ZnSb.⁵⁹ Then, using the concept of bidimensional polymorphism, 2D-layered Zintl compounds were fabricated from their 3D counterparts via electron transfer.^60,61 Another plausible route to synthesize 2D AM₂X₂ is through atomic substitution, which leads to interesting Janus-type materials (AM₂XY).^62–66 These novel synthesis techniques for manipulating structural dimensionality should make the synthesis of 2D-layered Zintl AM₂X₂ thin films possible.⁶⁷ 

While the focus to date has been on bulk Zintl AM₂X₂ materials and their stability and electronic properties for various applications,^{47,49–51,58} a comprehensive investigation of the stability and electronic and topological properties of single-quintuple-layer (1-QL) Zintl CaAl₂Si₂-type compounds is still lacking. So motivated, here, we discuss systematic first-principles calculations on 12 1-QL Zintl AM₂X₂ CaAl₂Si₂-type compounds composed of alkaline earth (A = Ca, Sr, or Ba), transition metal (M = Zn or Cd), and pnictogen atoms (X = Sb or Bi).

The stability of the investigated compounds was ascertained via formation energy calculations, which reveal that the most stable stacking configuration occurs when the alkaline earth (A) is sandwiched by two honeycomb-like MX layers as M-X-A-X-M. Our phonon dispersion computations further confirm the thermodynamic stability of all investigated compounds. Six 1-QL films were found to host nontrivial topological states, with five of these (CaZn₂Bi₂, SrZn₂Bi₂, CaCd₂Bi₂, SrCd₂Bi₂, and BaCd₂Bi₂) harboring 2D TIs with large bandgaps at the Γ-point, ranging from 0.500 to 0.774 eV, while BaZn₂Bi₂ displays a bandgap of 0.025 eV. Finally, our Z₂ number and edge-state calculations using the hybrid functional confirm our findings related to the nontrivial phases. Interestingly, we identified the presence of vHss in 1-QL films of CaCd₂Bi₂ and BaCd₂Bi₂, suggesting the possible coexistence of topological insulator and superconducting phases. Surprisingly, the thermodynamically stable 1-QL Janus BaCd₂SbBi is found to exhibit both TI and Rashba-splitting effects. Our study, thus, opens up new pathways for designing exotic materials supporting superconducting and topological states.

Density-functional-theory (DFT) based first-principles calculations implemented in the Vienna ab initio simulation package (VASP)^68,69 were performed with projected augmented wave (PAW) potentials.⁷⁰ The Perdew–Burke–Ernzerhof (PBE) generalized-gradient approximation (GGA) was used to treat exchange-correlation effects.^71–75 A 400 eV cutoff energy was used throughout the calculations. The residual forces were set to be no greater than 10⁻³eV/Å in optimizing the atomic positions. SOC effects were included in our band structure calculations. The energy convergence criterion was set to 10⁻⁵eV. A vacuum layer of 15.0 Å was inserted along the z-direction to prevent interactions among the repeated monolayers in the supercell. For all the atomic structure relaxations, a Γ-centered Monkhorst and Pack⁷⁶ grid of 12 × 12 × 1 in the Brillouin zone (BZ) was used. To accurately describe the bandgap and band topology, a denser grid of 24 × 24 × 1 was used for self-consistency cycles and band structure calculations. Band topologies based on the GGA were verified in all cases following the methods used in previous studies^77,78 for calculating the Z₂ topological invariants. Phonon dispersion calculations were performed using the VASP-Phonopy software.⁷⁹ Since the GGA normally underestimates bandgaps, the electronic and topological properties of all systems were recalculated under the Heyd–Scuseria–Ernzerhof (HSE06) hybrid functional⁸⁰ with a vacuum layer of 8.0 Å and a k-point grid of 6 × 6 × 1. Under HSE06, the maximally localized Wannier functions (MLWF) for the 1-QL AM₂X₂ films were obtained using the Wannier90 package,⁸¹ and the Z₂ invariants and surface Green's function (SGF) were calculated using the WannierTools program.⁸² Finally, to confirm the presence of vHss, we generated constant energy contour plots using a very dense k-mesh of 300 × 300 × 1.⁷⁶ 

The structure of Zintl AM₂X₂ CaAl₂Si₂-type compounds shows alkaline earth atoms (A) sandwiched by two honeycomb-like MX layers in Figs. 1(a) and 1(b) with perspective side and top view, respectively. We considered three different atomic stackings for the 1-QL AM₂X₂ materials with various exposed atoms on the surface as follows: (i) M-X-A-X-M (with the transition metal M exposed), (iii) X-M-A-M-X (with the pnictide X exposed), and (iii) X-M-M-X-A (with the pnictide X and the alkali A exposed on the top and bottom, respectively) as shown in Figs. 1(c)–1(e). The 3D and 2D first Brillouin zones (BZs) are shown in Figs. 1(f) and 1(g), respectively. The M-X-A-X-M stacking was found to be the most stable configuration with the lowest formation energy per formula unit (eV/f.u.), see Table I, and for this reason, this stacking was used in all further calculations. The corresponding lattice parameters for the 12 1-QL Zintl compounds are shown in Table II. Phonon dispersions in all films exhibit positive phonon modes and, thus, indicate that these films are thermodynamically stable, see supplementary material, Fig. S1. A possible fabrication method to synthesize 2D AM₂X₂ films is through intercalation or alloying of A atoms into bulk MX compounds along the lines of Ref. 59, where Li atoms were intercalated into a 3D ZnSb compound to synthesize 2D LiZnSb. It should be possible to synthesize our proposed 1-QL AM₂X₂ films by using the alkaline earth metals (Ca, Sr, and Ba) as the intercalation atoms. Another possible synthesis route would be to use strong acids such as the hydrofluoric aqueous solution to break the atomic bonds, which is an approach commonly used to fabricate MXenes.^83,84

After establishing the stability of the 1-QL AM₂X₂ compounds, we investigated their electronic properties. The electronic band structures of the 12 1-QL AM₂X₂ films calculated using HSE06 and PBE are shown in the supplementary material, Figs. S2–S5. Focusing on the Sb-containing compounds (AM₂Sb₂), a decreasing trend in the bandgap is seen with increasing A-atom size from Ca to Ba, see the supplementary material, Figs. S2(PBE) and S3(HSE06) with SOC. Referring to Table II, which is based on HSE06, the 1-QL AZn₂Sb₂ films are indirect gap insulators, while 1-QL ACd₂Sb₂ films are direct gap insulators with bandgaps ranging from 0.904 to 1.345 eV and 0.785 to 0.978 eV, respectively. The availability of a range of bandgaps in 1-QL AM₂Sb₂ films will make them suitable for low-power-consuming applications and optoelectronic devices such as infrared detectors and emitters.^46,52–55

Interestingly, the band structures of Bi-containing 1-QL AM₂X₂ films (AM₂Bi₂) undergo significant changes due to SOC effects. As seen in the supplementary material, Fig. S4, the valence band maximum (VBM) and the conduction band minimum (CBM) under PBE (without SOC) are touching. But, upon inclusion of the SOC, a gap opens between the VBM and CBM, suggesting the emergence of a nontrivial topological phase. To investigate this possibility, the Z₂ number for the stable structural phases of 1-QL AM₂X₂ with M-X-A-X-M stacking was calculated, see Table III. All 1-QL AM₂Sb₂ films have Z₂ = 0 and are, thus, trivial insulators. On the other hand, the 1-QL AM₂Bi₂ films (except for BaZn₂Bi₂) have Z₂ = 1 and are nontrivial (under PBE). The nontrivial topology of AM₂Bi₂ films was further confirmed using the hybrid functional (HSE06), see supplementary material Fig. S5 and Table III. Under HSE06, all 1-QL AM₂Bi₂ films exhibit Z₂ = 1 and are 2D TIs. The band structures under HSE06 (with SOC) in supplementary material Fig. S5 show band-opening centered at the Γ-point for 1-QL CaZn₂Bi₂, SrZn₂Bi₂, BaZn₂Bi₂, SrCd₂Bi₂, and BaCd₂Bi₂ with direct bandgaps (eV) of 0.571, 0.500, 0.025, 0.650, and 0.655, respectively, while 1-QL CaCd₂Bi₂ has a system-wide bandgap of 0.819 eV.

To understand the mechanism underlying the emergence of the nontrivial phases in 1-QL AM₂Bi₂ films, we have carried out a detailed orbital analysis. Taking BaCd₂Bi₂ film as an example, the band structures without [Fig. 2(a)] and with SOC [Fig. 2(b)] as well as the partial band projections [Figs. 2(c) and 2(d)] show that the topological phases in all cases investigated are driven by the band inversion at the Γ-point near the Fermi level involving Bi-(p_x+p_y) orbitals, see supplementary material Figs. S6 and S7.

To gain further insight into the topological properties of 1-QL AM₂Bi₂ films, we have also calculated the edge states using WannierTools⁸² under HSE06. In particular, we considered the 1-QL CaZn₂Bi₂, SrZn₂Bi₂, BaZn₂Bi₂, CaCd₂Bi₂, SrCd₂Bi₂, and BaCd₂Bi₂ films at their equilibrium lattice constants and performed SGF calculations to obtain their surface electronic spectra. Again, using BaCd₂Bi₂ as a representative film, we confirmed the presence of a topologically protected edge and the 2D topological insulating phase, see Fig. 3. supplementary material Figs. S8 and S9 present the topologically protected edge states for the other nontrivial 1-QL films.

Another unique feature that we observe in the band structures of 1-QL CaCd₂Bi₂ and BaCd₂Bi₂ under PBE [Figs. 4(a) and 4(b)] is the presence of saddle points near the Fermi level, which would generate vHs with diverging density of states (DOSs) and the possibility of inducing superconductivity in these films.⁸⁵ In this connection, we show band structures for the 1-QL CaCd₂Bi₂ and BaCd₂Bi₂ films along with the corresponding DOSs [Figs. 4(a) and 4(b)] and the associated constant-energy contour plots in Figs. 4(c) and 4(d)]. Diverging DOS in Fig. 4 (red arrows) and the saddle points in the contour plots (green dashed circles) near the Fermi level indicate the possible emergence of superconductivity in these films.^26,86,87 The preceding theoretical results related to the vHss are robust in that these features also occur when we compute the band structures and DOSs for the 1-QL CaCd₂Bi₂ and BaCd₂Bi₂ films using the HSE06 functional.

With our predicted thermodynamic stability and interesting topological properties of the 1-QL AM₂Bi₂ films in mind, we comment on how these films could be realized experimentally. Figure 5 presents a possible synthesis route, where we consider a film of BaCd₂Sb₂ because bulk BaCd₂Sb₂ has been synthesized.⁸⁸ First, it should be possible to synthesize along the lines of Ref. 59 a 1-QL BaCd₂Sb₂ film [Fig. 5(a)], which, however, is predicted to be a trivial insulator with Z₂ = 0 (Table III). Next, we can dope with Bi to substitute one Sb atom to create the 1-QL Janus structure BaCd₂SbBi [Fig. 5(b)], which is still predicted to be thermodynamically stable [Fig. 5(f)] and host a topological insulating phase [Fig. 5(e)] and conducting edge states (Fig. 6) as well as the Rashba splitting [Figs. 5(g)–5(i)]. The band structures and Z₂ numbers under PBE for other possible Janus 1-QL AM₂XY materials are presented in supplementary material Figs. S10 and S11 and Tables S3 and S4. Finally, via full Bi substitution, the 1-QL BaCd₂Bi₂ [Fig. 5(c)] could be realized, which is thermodynamically stable [Fig. S1(1)], possesses vHss [Figs. 4(b) and 4(d)], and hosts a large, gapped 2D TI (Table II and Fig. 2) state.

Finally, we illustrate our results in the family of 1-QL Zintl family as demonstrated in Fig. 7(a), which considers a schematic sheet of 1-QL AM₂Sb₂ with a region (leftmost portion) of normal insulator (1-QL BaCd₂Sb₂) that changes into a Janus AM₂SbBi film (middle portion) hosting a TI phase with Rashba spin-splittings via substitution of Sb by Bi (Janus 1-QL BaCd₂SbBi). Upon full Bi substitution (rightmost region), a 1-QL AM₂Bi₂ film with a large, gapped TI phase and vHs (1-QL BaCd₂Bi₂) is then realized. Figure 7(b) schematically shows the evolution of the band structures corresponding to the three regions of the film in Fig. 7(a).

We have studied the crystal and electronic structures of 12 different Zintl 1-QL AM₂X₂ films in the CaAl₂Si₂-structure composed of alkaline earth (A = Ca, Sr, or Ba), transition metal (M = Zn or Cd), and pnictogen atoms (X = Sb or Bi) with three different stackings (M-X-A-X-M, X-M-A-M-X, and X-M-M-X-A). Our formation energy computations show that the most energetically favorable stacking is M-X-A-X-M. The phonon dispersion results show that all the investigated films are thermodynamically stable. Nontrivial topological phases are hosted by 1-QL films of CaZn₂Bi₂, CaCd₂Bi₂, SrZn₂Bi₂, BaZn₂Bi₂, SrCd₂Bi₂, and BaCd₂Bi₂ with bandgaps at the Γ-point (eV) of 0.571, 0.500, 0.025, 0.774, 0.650, and 0.655, respectively. Our in-depth analysis reveals that the formation of topological phases is driven primarily by a band inversion at Γ involving the Bi-(p_x+p_y) orbitals. We confirm the nontrivial nature of the electronic states through edge-state calculations using the hybrid functional. Interestingly, we find the presence of van Hove singularities in 1-QL films of CaCd₂Bi₂ and BaCd₂Bi₂, suggesting the possibility of realizing coexisting insulating and superconducting topological phases in these films. Finally, we comment on the strategies for the synthesis of 2D films of the investigated Zintl compounds. Our study identifies a potentially new pathway for designing exotic materials platforms that support superconducting and topological states through the use of films of the Zintl family of compounds.

See the supplementary material for the formation energy formula, calculated band structures and bandgap values obtained using PBE and HSE06 functionals, phonon dispersion spectra, partial band projections concerning orbital contributions, the surface electronic spectra of the calculated materials, and the crystal and band structures of the proposed Janus 1-QL AM₂XY.

F.-C.C. acknowledges support from the National Center for Theoretical Sciences and the Ministry of Science and Technology of Taiwan under Grant Nos. MOST-107-2628-M-110-001-MY3 and MOST-110-2112-M-110-013-MY3. He is also grateful to the National Center for High-performance Computing for computer time and facilities. The work at Northeastern University was supported by the Air Force Office of Scientific Research under Award No. FA9550-20-1-0322, and it benefited from the computational resources of Northeastern University's Advanced Scientific Computation Center (ASCC) and the Discovery Cluster.

The authors have no conflicts to disclose.

F.C.C. conceived and initiated the study. M.N.R.P., R.A.B.V., L.Y.F., A.B.M., C.P.C., C.H.C., and Z.Q.H. conducted calculations. All the authors performed the detailed analysis and contributed to discussions, and wrote and reviewed the manuscript. M.N.R.P. and R.A.B.V. contributed equally to this work.

The data that support the findings of this study are available from the corresponding author upon reasonable request.