Intrinsic ferromagnetic axion states and a single pair of Weyl fermions in the stable-state MnX₂B₂T₆-family materials

$Introduction$: The discovery of the intrinsic antiferromagnetic (AFM) topological insulator (TI) MnBi₂Te₄ has attracted intensive attention in recent years 1Otrokov ; 2Deng ; 3Gong ; 4Li ; 5Zhang ; 6Otrokov ; 7Liu ; 8Hao ; 9Zeugner ; 10Li ; 11Chen ; 12LiYan ; 13Zhang . On one hand, it has tunability in layer thickness and external magnetic field as well as various novel magnetic topological phases such as the quantum anomalous Hall insulators 2Deng ; 4Li , the axion insulators (AXIs) 5Zhang ; 6Otrokov ; 7Liu , and the magnetic Weyl semimetals (WSMs) 4Li ; 5Zhang . On the other hand, its unique way of introducing magnetism can provide new insights and vitality in the search for materials with coexisting long-range magnetic order and nontrivial band topology. The MnBi₂Te₄ is formed by inserting an electrically neutral [MnTe] atomic layer into its parent TI Bi₂Te₃, which avoids the introduction of additional charges of magnetic impurity atoms and only introduces the magnetism of Mn²⁺ ions. But, the intrinsic A-type AFM MnBi₂Te₄ can only realize the AFM axion insulator state and the Weyl semimetal state with only a pair of Weyl points (WPs) under external magnetic field 5Zhang ; 14Li . It is worth noting that the ferromagnetic (FM) axion insulators have recently been proposed as an ideal platform to achieve desirable topological magnetoelectric responses 15Wan , while the ideal Weyl semimetals 16Bernevig with the minimum number of WPs can serve as the simplest template to study the intrinsic physical properties of the WPs and the application of related devices based on the WPs. However, the practical materials with the intrinsic FM axion insulators 17Hu ; 18Gao and intrinsic WSM states with only a pair of Weyl fermions 19Nie have so far been very rare.

Inspired by MnBi₂Te₄, in our previous work we designed a class of magnetic axion insulators MnX₂B₂T₆-A (X=Ge, Sn, or Pb; B=Sb or Bi; T=Se or Te) family 18Gao , which is composed of the parent TI X₂B₂T₅-A family and [MnTe] atomic intercalation. However, we found that the parent material X₂B₂T₅ family possesses two distinct stacking phases. Taking Pb₂Bi₂Te₅ as an example, both experiments 20Petrov ; 21Chatterjee and theory 22Ma ; 23Silkin confirm that it has two different stacking sequences A (-Te-Pb-Te-Bi-Te-Bi-Te-Pb-Te-) and B (-Te-Bi-Te-Pb-Te-Pb-Te-Bi-Te-), labeled as Pb₂Bi₂Te₅-A and Pb₂Bi₂Te₅-B, respectively, and the X₂B₂T₅ and MnX₂B₂T₆ families also adopt similar labels to distinguish these two different stacking sequences. We can see that the difference between X₂B₂T₅-A and X₂B₂T₅-B is that the X (Ge, Sn, or Pb) and B (Sb or Bi) atoms have exchanged positions with each other [see Figs. 1(a) and 1(b)], so the [XT] atomic layer of X₂B₂T₅-A is in the outermost layer, while the [XT] atomic layer of X₂B₂T₅-B in the innermost layer. Previous studies 22Ma have shown that Pb₂Bi₂Te₅-B is the stable structural form of Pb₂Bi₂Te₅-A in the range of temperatures 1000 K, and the two stacking phases show significant differences in electronic properties. Then one may naturally ask whether the MnX₂B₂T₆-B family obtained by inserting the [MnTe] atomic layer into the stable parent TI X₂B₂T₅-B material is the stable structure with lower energy than the MnX₂B₂T₆-A family? Could there be a new topological phase in MnX₂B₂T₆-B due to the structural difference between the MnX₂B₂T₆-B and MnX₂B₂T₆-A families?

In this work, we systematically investigate the stability, magnetic, electronic and topological properties of the MnX₂B₂T₆-B family by first-principles electronic structure calculations. Our calculations indicate that the MnX₂B₂T₆-B family has indeed a stable structure with lower energy than the MnX₂B₂T₆-A family. Surprisingly, we find not only a series of intrinsic ferromagnetic (FM) axion insulators MnGe₂Bi₂Te₆-B, MnSn₂Bi₂Te₆-B, and MnPb₂Bi₂Te₆-B, but also an intrinsic Weyl semimetal (WSM) MnSn₂Sb₂Te₆-B with only a pair of Weyl fermions, which has not been reported in the MnX₂B₂T₆-A family. We further investigate the effect of lattice strain on the topological properties of the MnSn₂Sb₂Te₆-B bulk materials. Compared with the MnBi₂Te₄ and the MnX₂B₂T₆-A family, the MnX₂B₂T₆-B family can provide a more ideal platform for future experiments to explore the unique topological quantum physics of the FM axion insulators and WSMs.

The electronic structures of the MnX₂B₂T₆-B (X=Ge, Sn, or Pb; B=Sb or Bi; T=Se or Te) family were investigated with the projector augmented wave method 24PBE as implemented in the VASP package 25Kresse in the framework of density functional theory (DFT). The generalized gradient approximation (GGA) of the Perdew-Burke-Ernzerhof (PBE) type 26Perdew was adopted for the exchange-correlation functional. The zero damping DFT-D3 method 27Grimme was chosen for the interlayer vdW interaction. The PBE+$U$ method was utilized to treat the localized 3d orbitals of Mn by selecting $U$= 4.0 eV. The $6\times 6\times 6$, $8\times 8\times 8$ and $12\times 12\times 1$ k-point meshes 28Monkhorst were adopted for the primitive cell calculations of the AFM bulks, FM bulks, and the monolayers, respectively. An 18 Å vacuum layer was used to avoid the residual interactions between neighboring image layers. The energy and force convergence criteria were set to $10^{-6}$ eV and 0.001 eV/Å, respectively. The phonon spectrum calculations were performed by using the DS-PAW software integrated in the Device Studio program 29Hongzhiwei , in which the $4\times 4\times 1$ supercell was used for the monolayer structures. The topological properties of the MnX₂B₂T₆-B family were calculated by using the Wannier90 30Mostofi and WannierTools 31Wu packages. An ab initio evolutionary algorithm, as implemented in the USPEX code 32Oganov ; 33Lyakhov , was employed to search for the stable bulk compounds of the MnPb₂Bi₂Te₆ system.

$Results$: To investigate the structural stability of the MnX₂B₂T₆-B family, we have calculated the phonon spectra of the MnX₂B₂T₆-B monolayers and bulks. As shown in Figs. S1 and S2 of Supplemental Material (SM), there are no soft phonon modes in the whole Brillouin zone (BZ), indicating that MnGe₂Sb₂Se₆-B, MnGe₂Sb₂Te₆-B, MnGe₂Bi₂Se₆-B, MnGe₂Bi₂Te₆-B, MnSn₂Sb₂Te₆-B, MnSn₂Bi₂Te₆-B, and MnPb₂Bi₂Te₆-B are dynamically stable. Next, we mainly focus on these seven MnX₂B₂T₆-B bulk compounds.

Considering that the interlayer Mn-Mn distance in the MnX₂B₂T₆-B bulk is very large ($\sim$20 Å) and interrupted by the vdW gap, we first consider the various magnetic structures in their monolayers (see Fig. 1 in Ref. 18Gao ) and then check the interlayer magnetic coupling to determine the magnetic ground states of the MnX₂B₂T₆-B compounds. In our previous work on MnX₂B₂T₆-A, various Hubbard $U$ values (such as $U$=3, 4, and 5 eV) on the Mn 3d orbitals were tested and we found that these selected $U$ values had little effect on the structural, magnetic, and electronic properties. Thus, in this work, we adopt $U$=4 eV as in Refs. 4Li ; 34Li . From Table 1, we can see that the energy of nonmagnetic (NM) state is much higher than those of other magnetic configurations, indicating that these structures all have magnetic interactions. Further, our calculations show that the FM order has the lowest energy among various magnetic configurations, suggesting that the magnetic ground state of MnX₂B₂T₆-B monolayers is the FM state with an out-of-plane easy magnetization axis. Moreover, the seven MnX₂B₂T₆-B monolayers in the FM ground state all exhibit good FM semiconductor properties in the presence of spin-orbit coupling (SOC) [see Fig. S4 in the SM], which may have potential applications in spintronic devices 35MacDonald ; 36Sarma .

The crystal structures of MnX₂B₂T₆-B [Fig. 1(d)] and MnX₂B₂T₆-A [Fig. 1(c)] bulks are very similar and have the same space group D${}_{3d}^{5}$ (No. 166), both of which are formed by ABC stacking of 11 atomic-layers building blocks along the c-axis via the vdW interaction. However, their difference in the structure is that the X (Ge, Sn, or Pb) and B (Sb or Bi) atoms exchange positions with each other. Obviously, this is derived from the difference between their parent X₂B₂T₅-B [Fig. 1(b)] and X₂B₂T₅-A [Fig. 1(a)] structures. For instance, Pb₂Bi₂Te₅-A and Pb₂Bi₂Te₅-B, which have been prepared experimentally 20Petrov ; 21Chatterjee , exchange the positions of Pb and Bi atoms. It is worth noting that the Pb₂Bi₂Te₅-B is a stable structural form of Pb₂Bi₂Te₅-A in the range of temperatures 1000 K 22Ma . Therefore, we focus on the stable parent X₂B₂T₅-B structures. A natural question is whether the MnX₂B₂T₆-B family derived from this stable parent X₂B₂T₅-B can have a stable structural form with lower energy than that of the MnX₂B₂T₆-A family? To this end, we firstly searched the MnBi₂Te₄ system based on the USPEX evolutionary algorithm and quickly obtained the real synthesized MnBi₂Te₄ crystals 37Lee with the space group R$\bar{3}$m [see Fig. S3 in the SM], which confirmed the effectiveness of the method. After that, we carried out a systematic search for the MnPb₂Bi₂Te₆ system, and found that the energy of MnPb₂Bi₂Te₆-B phase is indeed lower than that of the MnPb₂Bi₂Te₆-A phase 18Gao by nearly 526.9 meV/Mn (see Table 2). The MnPb₂Bi₂Te₆-B phase is located at the lowest energy (Fig. 2), showing that the MnPb₂Bi₂Te₆-B phase is the ground state of the MnPb₂Bi₂Te₆ system.

After confirming the intralayer FM ground state of MnX₂B₂T₆-B monolayer, we only need to consider the interlayer magnetic interaction to determine the magnetic ground state of MnX₂B₂T₆-B bulks. From Table 2, we can see that MnGe₂Sb₂Se₆-B and MnGe₂Bi₂Se₆-B bulk exhibit the A-type AFM magnetic ground state similar to MnBi₂Te₄ 3Gong ; 4Li ; 5Zhang , while the remaining five MnX₂B₂T₆-B bulks all have the FM ground state. From the magnetic anisotropy energy (MAE), the seven bulk compounds all host the same out-of-plane easy axis of magnetization as their monolayers. Meanwhile, we note that the energies of the seven MnX₂B₂T₆-B bulks are 42.9-526.9 meV/Mn lower than those of the MnX₂B₂T₆-A bulks in their respective ground states, suggesting that MnX₂B₂T₆-B is the stable structural form of the MnX₂B₂T₆ system.

The most interesting property of the MnX₂B₂T₆-B bulks is its electronic band structure with the abundant topological phases. Figure 3 shows the spin-polarized band structures and density of states (DOS) of MnSn₂Sb₂Te₆-B and MnPb₂Bi₂Te₆-B bulks in the FM ground state. We can find that they both are FM semiconductors with narrow band gaps (0.23 and 0.61 eV), and their energy bands near the top of valence bands mainly come from the contributions of the p orbitals of Te, Sb/Bi, and Sn/Pb atoms. The spin-polarized band structures of the other five MnX₂B₂T₆-B bulks in their respective ground states are shown in Fig. S5 of the SM.

When the SOC effect is taken into account, the band structure of MnSn₂Sb₂Te₆-B bulk in the FM ground state with an out-of-plane easy magnetization axis is shown in Fig. 4(a). One can see that it has only a minimal number of two type-I Weyl points along the Z-$\Gamma$-Z high-symmetry path near the Fermi level without other interfering bands [inset of Fig. 4(a)]. The band crossings are mainly caused by the hybridization between the p orbitals of the Te and Sb atoms and protected by the C₃ rotational symmetry. Our Wannier charge center (WCC) calculations show that the two Weyl points (WP₁ and WP₂) exhibit opposite chirality carrying the topological charges of +1 and -1 [Fig. 4(b)], respectively, indicating that MnSn₂Sb₂Te₆-B is an ideal topological WSM. In addition, the calculated Chern numbers $C=1$ at $k_{z}=0$ plane and $C=0$ at $k_{z}=\pi$ plane are also consistent with the feature of an ideal WSM. On the other hand, we also investigated the effect of strain on the topological properties of MnSn₂Sb₂Te₆-B bulk by applying uniaxial stress from -5$\%$ (compression) to 5$\%$ (tensile) along the c-axis. As shown in Fig. 4(c), for MnSn₂Sb₂Te₆-B bulk in the FM ground state, the energy gap between the valence band maximum (VBM) and the conduction band minimum (CBM) at the $\Gamma$ point increases monotonically with the increase of compressive stress and becomes an FM axion insulator (AXI). However, under tensile stress within 3$\%$ it is still a WSM with a single pair of Weyl points. Once the tensile stress exceeds 3$\%$, the system becomes a trivial FM insulator (FMI). Figure 4(d) shows the projected band structure with orbital weights for MnPb₂Bi₂Te₆-B bulk in the FM ground state with an out-of-plane easy magnetization axis, we can see that there is a band inversion between the p orbitals of Te and Bi atoms at the $\Gamma$ point near the Fermi level, indicating that the MnPb₂Bi₂Te₆-B bulk may have nontrivial topological properties. For the other five MnX₂B₂T₆-B bulks in their respective ground states, their band structures calculated with the SOC are shown in Fig. S6 of the SM.

Since the MnX₂B₂T₆-B family has the inversion symmetry under different ground state magnetic configurations, a parity-based Z₄ symmetry indicator can be used to characterize the topological properties of MnX₂B₂T₆-B family. The Z₄ invariant is given by 38Turner ; 39Ono ; 40Watanabe

where $\Lambda_{\alpha}$ are the eight inversion-symmetric crystal momenta, $\xi_{n}(\Lambda_{\alpha})$ is the parity eigenvalue (+1 or -1) of the nth occupied band at the $\Lambda_{\alpha}$, and $n_{occ}$ is the total number of occupied bands. Z₄=1 or 3 corresponds to a Weyl semimetal (WSM) phase 38Turner , while Z₄=2 indicates an axion insulator (AXI) [the axion angle $\theta=\pi$] in the case of the Chern numbers on all the 2D planes in the BZ are zeros 41Huan ; 42Xu . From Table 2, we can find that the Z₄ invariant of MnSn₂Sb₂Te₆-B equals to 1, indicating that it is a WSM, which is consist with our calculated band structures [Fig. 4(a)] and WCCs [Fig. 4(b)]. While the MnGe₂Bi₂Te₆-B, MnSn₂Bi₂Te₆-B, and MnPb₂Bi₂Te₆-B bulks correspond to Z₄=2, the calculated Chern numbers at $k_{z}=0$ and $k_{z}=\pi$ planes both equal to 0, implying that they are a class of FM axion insulators. Furthermore, the Z₄ invariants of MnGe₂Sb₂Se₆-B, MnGe₂Sb₂Te₆-B and MnGe₂Bi₂Se₆-B equal to 0, suggesting that they are a class of trivial insulators.

Both WSMs and FM axion insulators exhibit unique topological surface states. Considering that MnX₂B₂T₆-B bulks are formed by ABC stacking of 11 atomic-layers building blocks along the c-axis via the vdW interaction, the (111) surface of their primitive cells located in the vdW gap is their natural cleavage plane. So, we calculated the surface states for the (111) surfaces of MnSn₂Sb₂Te₆-B and MnPb₂Bi₂Te₆-B. We can find that the two Weyl points (WP₁ and WP₂) of MnSn₂Sb₂Te₆-B bulk are exactly projected to the same point $\bar{\Gamma}$ on the (111) surface, thus the two surface Fermi arcs are displayed at the $\bar{\Gamma}$ point [Fig. 5(a)]. However, for axion insulator, in addition to the above axion angle $\theta=\pi$, a gapped surface state is also required. We take MnPb₂Bi₂Te₆-B as an example to calculate the surface bands in the projected (111) surface [see Fig. 5(b)]. One can see that the surface states at the $\bar{\Gamma}$ point near the Fermi level do open an energy gap ($\sim$20 meV) and are accompanied by a triangular Fermi surface [see Fig. 5(c)], which is similar to the surface state of MnGe₂Sb₂Te₆-A that we previously reported 18Gao . Notably, MnPb₂Bi₂Te₆-B has only the gapped surface state at the Fermi level without other interfering bulk states.

$Discussion$: It is well known that the search for the ideal Weyl semimetals with a single pair of Weyl points and the intrinsic FM axion insulators in magnetic materials with highly desirable topological states has been a challenging task. Despite a long search, their candidates are still rare so far 17Hu ; 18Gao ; 19Nie . Fortunately, both of the two unique topological states can be found in the MnX₂B₂T₆-B family. Compared with our previous work on MnX₂B₂T₆-A, the MnX₂B₂T₆-B family has the following advantages. First, not only the energy of the MnX₂B₂T₆-B family is lower than that of the MnX₂B₂T₆-A family, but also the former is the stable structural form of MnX₂B₂T₆ system. Second, a new topological phase emerges from the MnX₂B₂T₆-B family, that is, the ideal magnetic Weyl state with only a single pair of Weyl points. Third, MnPb₂Bi₂Te₆-B has only the gapped surface state at the Fermi level without other interfering bulk states. Thus, it can provide a more ideal platform for future experiments to explore the long-sought quantized magnetoelectric effect based on the FM axion insulators. Finally, we emphasize that our design scheme can provide new ideas for realizing more vdW-type magnetic topological materials.

$Summary$: To summarize, we propose a new class of MnX₂B₂T₆-B (X=Ge, Sn, or Pb; B=Sb or Bi; T=Se or Te) family that is the stable structural form of the MnX₂B₂T₆-A materials we reported before. We systematically investigate the stability, magnetic, electronic and topological properties of the MnX₂B₂T₆-B family and find that MnX₂B₂T₆-B monolayers are narrow-bandgap ferromagnetic (FM) semiconductors in their FM ground state with an out-of-plane easy magnetization axis, while MnX₂B₂T₆-B bulks not only have the intrinsic FM axion insulators MnGe₂Bi₂Te₆-B, MnSn₂Bi₂Te₆-B, and MnPb₂Bi₂Te₆-B, but also the intrinsic WSM MnSn₂Sb₂Te₆-B with only a single pair of Weyl points that has not appeared in the MnX₂B₂T₆-A family. Thus, the new MnX₂B₂T₆-B family can provide an ideal platform for future experiments to explore the long-sought quantized magnetoelectric effect and the intrinsic properties related to Weyl points.