## Abstract

The interplay of magnetism and topology is a key research subject in condensed matter physics, which offers great opportunities to explore emerging new physics, such as the quantum anomalous Hall (QAH) effect, axion electrodynamics, and Majorana fermions. However, these exotic physical effects have rarely been realized experimentally because of the lack of suitable working materials. Here, we predict a series of van der Waals layered MnBi_{2}Te_{4}-related materials that show intralayer ferromagnetic and interlayer antiferromagnetic exchange interactions. We find extremely rich topological quantum states with outstanding characteristics in MnBi_{2}Te_{4}, including an antiferromagnetic topological insulator with the long-sought topological axion states on the surface, a type II magnetic Weyl semimetal with one pair of Weyl points, as well as a collection of intrinsic axion insulators and QAH insulators in even- and odd-layer films, respectively. These notable predictions, if proven experimentally, could profoundly change future research and technology of topological quantum physics.

## INTRODUCTION

The correlation between topology and symmetry as a central fundamental problem of modern physics has attracted intensive research interests in condensed matter physics and materials science since the discovery of topological insulators (TIs) (*1*, *2*). One essential symmetry is the time-reversal symmetry (TRS), which is crucial to many kinds of topological quantum states of matter such as TIs and is broken in the presence of magnetism. Intriguingly, the interplay between magnetism and topology could generate a variety of exotic topological quantum states in materials (*1*–*23*), including the quantum anomalous Hall (QAH) effect, showing dissipationless chiral edge states (*3*–*6*), the topological axion states, displaying quantized magnetoelectric effects (*7*–*10*), and Majorana fermions, obeying non-Abelian statistics (*2*, *11*). In this context, great research effort has been devoted to exploring exotic topological quantum physics, which is of profound importance to fundamental science and future technologies, such as dissipationless topological electronics and topological quantum computation (*1*, *2*).

One key subject that is of crucial significance to the whole research community is the development of topological quantum materials (TQMs) showing the coexistence of topology and other quantum phases (e.g., magnetism, ferroelectricity, charge density wave, and superconductivity), coined “composite TQMs” (CTQMs), which include magnetic TQMs (MTQMs) as an important subset. Currently, a very limited number of MTQMs are experimentally available, including magnetically doped TIs and magnetic topological heterostructures (*5*, *6*, *24*–*31*), whose material fabrication, experimental measurement, and property optimization are quite challenging. Because the major topological properties of these existing MTQMs depend sensitively on delicate magnetic doping or proximity effects, only little preliminary experimental progress has been achieved until now, leaving many important physical effects not ready for practical use or still unproved. For instance, the QAH effect was only observed in magnetically doped (Bi* _{x}*Sb

_{1–x})

_{2}Te

_{3}thin films at very low working temperatures by fine-tuning chemical compositions (

*5*,

*6*). While a previous theory predicted the existence of topological axion states at the surface of a three-dimensional (3D) antiferromagnetic (AFM) TI (

*10*), the 3D AFM TI phase has not been realized experimentally, as far as we know, despite several attempts. For future research and applications, people should go beyond the existing strategy for building MTQMs and try to design intrinsic MTQMs without the need of introducing alloy/doping or heterostructures.

Noticeably, van der Waals (vdW) layered materials represent a large family of materials with greatly tunable properties by quantum size effects or vdW heterojunctions (*32*), in which a variety of quantum phases in different spatial dimensions have been discovered by state-of-the-art research (*21*, *33*, *34*). For instance, TI states were previously found in the tetradymite-type Bi_{2}Te_{3}-class materials (*33*), and 2D intrinsic magnetism was recently found in ultrathin films of CrI_{3} (*35*) and Cr_{2}Ge_{2}Te_{6} (*36*) . However, these topological materials are not magnetic, and those magnetic materials are not topological. It is highly desirable to incorporate magnetic and topological states together into the same vdW material to obtain layered intrinsic MTQMs, which are able to inherit advantages of vdW materials.

## RESULTS

In this work, on the basis of first-principles density functional theory (DFT) calculations, we predicted a series of layered intrinsic MTQMs from the tetradymite-type MnBi_{2}Te_{4}-related ternary chalcogenides (MB_{2}T_{4}: M = transition-metal or rare-earth element, B = Bi or Sb, T = Te, Se, or S) (Fig. 1A), in which the intralayer exchange coupling is ferromagnetic (FM), giving 2D ferromagnetism in the monolayer, while the interlayer exchange coupling is AFM, forming 3D *A*-type antiferromagnetism in their vdW layered bulk. Both 2D and 3D physics are easily accessible in this family of materials by varying film thickness. Moreover, magnetic states are tunable from AFM to FM under external magnetic fields or to paramagnetic (PM) at high temperatures, leading to different symmetry properties and thus distinct topological quantum states. Therefore, these materials provide a promising platform to investigate the interplay of dimension, symmetry, magnetism, and topology, as illustrated in Fig. 1B. They could host many unusual 2D and 3D topological quantum states, including QAH insulators (*4*–*6*), axion insulators (*7*–*9*), and quantum spin Hall insulators (*37*–*39*) in the thin films, as well as 3D QAH insulators (*12*), Weyl semimetals (WSMs) (*17*–*21*), Dirac semimetals (*22*, *23*), time-reversal invariant TIs (*33*), and AFM TIs (*10*) in the bulk. We demonstrated that almost all of those exotic physical effects could be realized in the representative material MnBi_{2}Te_{4}. As the bulk and thin films of this particular material have been successfully fabricated experimentally (*40*, *41*), our theoretical predictions are awaiting experimental proofs.

The tetradymite-type MnBi_{2}Te_{4} crystalizes in a rhombohedral layered structure with the space group *40*, *42*), and each layer has a triangular lattice with atoms ABC stacked along the out-of-plane direction, the same as Bi_{2}Te_{3}. Slightly differently, monolayer MnBi_{2}Te_{4} includes seven atoms in a unit cell, forming a Te-Bi-Te-Mn-Te-Bi-Te septuple layer, which can be viewed as intercalating a Mn-Te bilayer into the center of a Bi_{2}Te_{3} quintuple layer (Fig. 1A). Noticeably, the mixing between Mn and Bi in this compound would create unstable valence states Mn^{3+} and Bi^{2+}, which is energetically unfavorable. Thus, the formation of alloys could be avoided, leading to a stoichiometric compound. This physical mechanism also explains the stability of compounds such as SnBi_{2}Te_{4} and PbBi_{2}Te_{4} and is applicable to many other MnBi_{2}Te_{4}-family materials. Our DFT calculations predicted that a series of MnBi_{2}Te_{4} compounds are dynamically stable (at least metastable), including M = Ti, V, Mn, Ni, Eu, etc. (Fig. 1A), as confirmed by DFT phonon calculations (Supplementary Materials), implying that these materials could possibly be fabricated by experiment. These stable compounds are characterized by an insulating band gap in their single layers, offering a collection of layered intrinsic magnetic (topological) insulators. Comprehensive data on the atomic, magnetic, and electronic properties of MBi_{2}Te_{4}-family materials were presented in the Supplementary Materials.

In MnBi_{2}Te_{4}, Mn has a valence of +2 by losing its two 4*s* electrons. The remaining five 3*d* electrons fill up the spin-up Mn-*d* levels according to the Hund’s rule, introducing 5 μ* _{B}* magnetic moment (mostly from Mn) per unit cell. By comparing different magnetic structures, we found that the magnetic ground state of the monolayer is FM with an out-of-plane easy axis (depicted in Fig. 1A), in agreement with previous work (

*30*). Each Mn atom is bonded with six neighboring Te atoms, which form a slightly distorted edge-sharing octahedron. According to the Goodenough-Kanamori rule, the superexchange interactions between Mn-Te-Mn with a bonding angle of ∼86° are FM, similarly as in CrI

_{3}and Cr

_{2}Ge

_{2}Te

_{6}(

*35*,

*36*). We found that intralayer exchange couplings are FM in other stable MB

_{2}T

_{4}-family members, but the magnetic easy axis can be varied to in-plane (e.g., for M = V), displaying rich 2D magnetic features (Fig. 1A).

Intriguingly, magnetic and topological states are well combined in MnBi_{2}Te_{4}, where Mn introduces magnetism and the Bi-Te layers could generate topological states similarly as in Bi_{2}Te_{3} (*33*), as schematically depicted in Fig. 1C. The spin splitting between spin-up and spin-down Mn *d* bands is extremely large (>7 eV) because of the large magnetic moment of Mn. Thus, Mn *d*-bands are far away from the band gap, and only Bi/Te *p*-bands are close to the Fermi level. While the monolayer is a topologically trivial FM insulator with a direct band gap of 0.70 eV (Fig. 1D), extremely interesting topological quantum physics emerges in the bulk and thin films. Note that monolayer MnBi_{2}Te_{4} was introduced as an FM insulator to break the TRS of a TI surface to realize the QAH and topological magnetoelectric effects in the magnetic topological heterostructures (*30*, *31*). Topological properties of MnBi_{2}Te_{4} itself, however, were not discussed.

We found that the interlayer magnetic coupling is AFM, giving an *A*-type AFM ground state (depicted in Fig. 2A) (*41*), which is similar to that in CrI_{3} (*35*) and is explained by the interlayer super-superexchange coupling. In the AFM bulk, the spatial inversion symmetry *P* (i.e., *P*_{1} centered at *O*_{1}) is preserved, but the TRS Θ gets broken. Two new symmetries exist: *P*_{2}Θ and *S* = Θ*T*_{1/2}, where *P*_{2} is an inversion operation centered at O_{2} and *T*_{1/2} is a lattice translation (depicted in Fig. 2A). The band structures with and without spin-orbit coupling (SOC) are presented in Fig. 2 (B and C). Here, every band is (at least) doubly degenerate, which is ensured by *P*_{2}Θ (*22*). In the presence of *S*, a *Z*_{2} classification becomes feasible (*10*). However, in contrast to the time-reversal invariant case, there is a *Z*_{2} invariant for the *k _{z}* = 0 plane but not for the

*k*= π plane. Here,

_{z}*Z*

_{2}= 1 corresponds to a 3D AFM TI phase (

*10*), which has not been experimentally confirmed before.

Here, the parity criteria can be applied to determine *Z*_{2} due to the *P*_{1} symmetry (*43*). The parities of the valence band maximum and conduction band minimum at Γ are opposite, and both change signs by the SOC effects (Fig. 2, B and C), implying a band inversion. By varying the SOC strength, the band gap first closes and then reopens at Γ (Fig. 2D), showing one band inversion and thus implying a topological phase transition. The SOC-induced band reversal happens between Bi _{2}Te_{3} (*33*). Our Wannier charge center (WCC) calculations for the *k _{z}* = 0 plane revealed that

*Z*

_{2}= 1 when including SOC (Fig. 2E), confirming that MnBi

_{2}Te

_{4}is a 3D AFM TI. The global band gap is direct at

*Z*(∼0.16 eV), and the direct gap at Γ is ∼0.18 eV. One prominent feature of the AFM TI is the existence of 2D gapless surface states at side surfaces that have preserved

*S*symmetry, which is confirmed by the surface-state calculations (Fig. 2, F and G). The surface states are gapless on the (110) termination, showing Rashba-like spin textures (fig. S5B), but become gapped on the (111) termination due to the

*S*symmetry breaking, in agreement with previous theory (

*10*).

The intrinsically gapped (111) surfaces are promising for probing the long-sought topological axion states, which give the topological quantized magnetoelectric effects related to an axion field with θ = π (*7*–*10*). Previous works proposed to probe the axion states by adding opposite out-of-plane ferromagnetism onto the bottom and top surfaces of 3D TIs (*7*, *25*–*27*) or by applying magnetic fields on TI films (*44*). These TRS-broken surface states are naturally and intrinsically provided by even-layer MnBi_{2}Te_{4} films (when with negligible hybridizations between the top and bottom surface states), benefitting from their *A*-type AFM states. An additional requirement is to open a band gap at side surfaces, which is realized in relatively thin films [∼10 layers according to a simple estimation (Supplementary Materials)] or by applying magnetic fields that have nonzero components perpendicular to all side surfaces. The simplified proposal, together with the suitable material candidate MnBi_{2}Te_{4}, could greatly facilitate the research of axion electrodynamics.

As the interlayer exchange coupling is quite weak here, the AFM ground state of MnBi_{2}Te_{4} could be tuned, for instance, by applying an external magnetic field, to other magnetic configurations. Then, the material symmetry changes, leading to distinct topological phases. This concept is demonstrated by studying a simple FM ordering along the out-of-plane direction (Fig. 3A). The FM bulk has *P*_{1} but neither Θ nor *P*_{2}Θ, leading to spin-split bands, nonzero Berry curvatures, and possibly nonzero topological Chern numbers that correspond to nontrivial topological phases, such as 3D QAH insulators (*12*) and WSMs (*17*–*21*). The band structure of FM MnBi_{2}Te_{4} bulk (Fig. 3B) displays a pair of band crossings at W/W′ along the *C*_{3} rotational symmetry. Our WCC calculations found that the W point is a momentum-space monopole with a topological charge of +1 (i.e., a Berry phase of 2π) (Fig. 3D), and its time-reversal partner W′ has an opposite topological charge of −1, indicating that the system is a topological WSM. While in most of momentum space the electron pocket is located above the hole pocket (e.g., along the F-W-L line), the Weyl cones get tilted along the *21*). In contrast to the time-reversal invariant WSMs that must have even pairs of Weyl points, this magnetic WSM represents the simplest one, hosting only a pair of Weyl points. Moreover, our surface-state calculations demonstrated the existence of Fermi arcs on the (

VdW layered materials are featured by tunable quantum size effects. For AFM MnBi_{2}Te_{4} films, even layers do not possess *P* and Θ symmetries but have *P*_{2}Θ, ensuring double degeneracy in every band (*22*). Differently, odd layers have *P* but neither Θ nor *P*_{2}Θ, leading to spin-split bands. Different symmetries in even and odd layers lead to distinct topological properties. Specifically, the topological Chern number *C* = 0 is required by *P*_{2}Θ in even layers, but *C* ≠ 0 is allowed in odd layers. In contrast to the single- and three-layer films, where a trivial insulating gap is opened by quantum confinement effects, we found that the five-layer film is an intrinsic QAH insulator with *C* = 1, as confirmed by the appearance of a quantized Hall conductance and chiral edge states within the bulk band gap (Fig. 4, A to C). We also calculated a seven-layer film, which is also a QAH insulator with *C* = 1. For comparison, our calculations of a four-layer film show that there is no gapless edge states within the bulk gap, confirming that *C* = 0 for an even-layer film (Supplementary Materials).

Exchange splitting introduced by the magnetic Mn layers, together with the strong SOC effects in the Bi-Te bands, cooperatively induces the QAH effect, similarly as in the magnetically doped Bi_{2}Te_{3}-class TIs (*4*). However, the present material has an intrinsic magnetism and does not need uncommon mechanisms [e.g., the Van-Vleck mechanism (*4*)] to induce ferromagnetism in insulating states. Moreover, because the magnetic doping is not needed, disorder-induced magnetic domains and potential fluctuations, which deteriorate the QAH effect, are avoided. Furthermore, the QAH gap of the five-layer film is 38 meV, greater than the room temperature thermal energy of 26 meV, enabling a high working temperature.

The thickness dependence behaviors of films are illustrated in Fig. 4 (D and E). There are intrinsically gapped surface states on both sides of thick films, as obtained for the (111) semi-infinite surface. These surface gaps are opened by the TRS-breaking field near the surface, which have half-quantized Hall conductances σ_{xy} = *e*^{2}/2*h* or −*e*^{2}/2*h* when the magnetism in the surface layer is up- or down-oriented, respectively. Thus, the Hall conductances of the top and bottom surfaces are cancelled in even layers, giving axion insulators with *C* = 0 or topological axion states mentioned above (Fig. 4D), while they get added in odd layers, giving QAH insulators with *C* = 1 (Fig. 4E). This physical picture, consistent with our calculation results, suggests an oscillation of *C* in even and odd layers. Thus, chiral edge states always appear near step edges (Fig. 4F), which can be used for dissipationless conduction. On the basis of this unique feature, chiral edge states could be selectively patterned in the present material by controlling the film thickness or step edges, advantageous for building dissipationless circuits.

## DISCUSSION

Experimentally, the material is not intrinsically insulating due to the existence of dopants, but the Fermi level should be tuned to the bulk gap for observing the QAH and topological magnetoelectric effects. This could be realized by various techniques, such as alloying between MnBi_{2}Te_{4} and MnSb_{2}Te_{4} or ionic liquid gating. Moreover, defects in AFM materials could lead to uncompensated magnetic moment and localized spin canting. The effect, however, is usually localized and disordered, which hardly changes the global topological property. Another crucial issue is the influence of temperature on topological properties. The observation of magnetic topological effects depends crucially on the existence of surface gap induced by magnetization. When some spins are flipped due to thermal excitation, the net magnetization of surfaces becomes smaller, which decreases the topological surface gap. If the temperature is gradually increased, the magnetic configuration would change from AFM to PM at ∼20 K in the MnBi_{2}Te_{4} thin film (*41*), leading to a topological phase transition from AFM TI to TRS invariant TI. The latter phase has gapless surface states. Therefore, the emergence of magnetization-induced surface gap with decreasing temperature provides a smoking gun signature for the temperature-driven topological phase transition.

Looking back to the history of the TI research, the first- and second-generation TIs are the HgTe/CdTe quantum wells (*38*, *39*) and Bi-Sb alloys (*43*, *45*, *46*), respectively, which are very complex and difficult to study theoretically and experimentally. Research interests have increased exponentially since the discovery of the third-generation TIs in the intrinsic Bi_{2}Te_{3}-class materials (*33*). A very similar situation is faced by the research of magnetic topological physics. Currently, experimental works are mostly based on magnetically doped TIs (*5*, *6*) and magnetic topological heterostructures (*25*–*31*), which are quite challenging and have led to little preliminary progress. Looking forward, the research progress is expected to be greatly prompted by discovering intrinsic MTQMs that are simple and easy to control. The vdW layered MnBi_{2}Te_{4}-family materials satisfy all these material traits. This material class could host extremely rich topological quantum states in different spatial dimensions and is also promising for investigating other exotic emerging physics (such as Majorana fermions if interacting with superconductivity), which thus could serve as a next-generation material platform for future research.

Note that, after posting the preprint of this work (arXiv:1808.08608) online, we became aware of related works on MnBi_{2}Te_{4} bulk (*47*, *48*), MnBi_{2}Te_{4} thin films (*41*, *49*), and the MnBi_{2}Se_{4} counterparts (*50*), as well as the intrinsic anomalous Hall effect (*51*) and topological superconductivity (*52*) in the same material.

## METHODS

First-principles DFT calculations were mainly performed with the Vienna Ab initio Simulation Package (VASP) (*53*) using the projector-augmented wave method and the plane-wave basis with an energy cutoff of 350 eV. The Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional (*54*) was used in combination with the GGA+*U* method to treat the localized *d* and *f* orbitals. The *U* parameter was tested and selected to be *U* = 3, 3, 4, 4, and 5 eV for 3*d* or 4*f* orbitals of Ti, V, Mn, Ni, and Eu, respectively. In addition, the Heyd-Scuseria-Ernzerhof (HSE) hybrid functional (*55*) and the modified Becke-Johnson (mBJ) functional (*56*) were used. The structural optimization and phonon calculations were done by the GGA+*U* method. vdW corrections were included by the DFT-D3 method (*57*) when studying few-layer films or bulks. Band structures of AFM and FM MnBi_{2}Te_{4} bulks were computed by the GGA+*U* and mBJ methods, respectively, whose results were further verified by the more advanced HSE method. Electronic bands of monolayer structures were calculated by the HSE functional to correct the band gap problem by using the all-electron, full-potential code FHI-aims (*58*). The Monkhorst-Pack *k*-point meshes of 9 × 9 × 3, 9 × 9 × 5, and 12 × 12 × 1 were adopted for the calculations of AFM bulk, FM bulk, and monolayer/few-layer structures, respectively. The dense Monkhorst-Pack *k*-point meshes of 24 × 24 × 1 were used to calculate magnetocrystalline anisotropy energies of monolayer MB_{2}T_{4}. Structure optimizations were performed with a force criterion of 0.01 eV/A. Phonon dispersions were calculated by the frozen phonon method using 3 × 3 × 1 supercells. Surface- and edge-state calculations based on maximally localized Wannier functions were done using the WannierTools package (*59*).

## SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/6/eaaw5685/DC1

Supplementary Text

Section S1. Primary physical properties of MBi_{2}Te_{4}

Section S2. Quantum confinement effects on side surfaces

Fig. S1. Band structures of monolayer MBi_{2}Te_{4} calculated by DFT-HSE excluding SOC.

Fig. S2. Band structures of monolayer MBi_{2}Te_{4} calculated by DFT-HSE including SOC.

Fig. S3. Phonon dispersions of monolayer MBi_{2}Te_{4}.

Fig. S4. Band structures of AFM MBi_{2}Te_{4} bulks calculated by the GGA+*U* method with SOC.

Fig. S5. Topological surface states and spin textures for the side (110) surfaces of AFM MnBi_{2}Te_{4}.

Fig. S6. Band structure and edge states of the four-layer AFM MBi_{2}Te_{4}.

Table S1. Physical properties of monolayer MBi_{2}Te_{4} for varying elements M.

Table S2. Physical properties of bulk MBi_{2}Te_{4} for varying elements M.

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## REFERENCES AND NOTES

**Acknowledgments:**

**Funding:**This work was supported by the Basic Science Center Project of NSFC (grant no. 51788104), the Ministry of Science and Technology of China (grant nos. 2016YFA0301001, 2018YFA0307100, 2018YFA0305603, and 2017YFA0303303), the National Natural Science Foundation of China (grant nos. 11674188, 11334006, 11874035, and 51661135024), and the Beijing Advanced Innovation Center for Future Chip (ICFC). Y.X. acknowledges support from the National Thousand Young Talents Program and Tsinghua University Initiative Scientific Research Program.

**Author contributions:**Y.X., W.D., and K.H. conceived the project. J.L. performed the first-principles calculations with the help of Y.L., S.D., and Z.W. Y.X., W.D., K.H., B.-L.G., and S.-C.Z. analyzed the data. Y.X. wrote the manuscript with input from J.L., W.D., K.H., B.-L.G., and S.-C.Z. All authors discussed the results and commented on the manuscript.

**Competing interests:**The authors declare that they have no competing interests.

**Data and materials availability:**All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.

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