Intrinsic magnetic topological insulators of the MnBi₂Te₄ family

Abstract

This short review appears on the occasion of the fifth anniversary of discovery of intrinsic magnetic topological insulators (MTIs) of the MnBi₂Te₄ family, which have attracted a great deal of attention recently. This family of materials has been discovered in attempts to increase the observation temperature of the quantum anomalous Hall effect as well as to facilitate the eventual realization of the topological magnetoelectric effect. Therefore, we first briefly introduce these effects, then describe the experimental state-of-the-art in the MTIs field just prior to MnBi₂Te₄ appearance, after which we discuss the basic properties of this material and its family. Finally, we overview the exciting progress made during 5 years of intense research in this field.

Introduction

Introducing magnetism in a nonmagnetic topological insulator (TI) is known to give rise to fascinating and technologically promising phenomena^1,2. One of them is the quantum anomalous Hall effect (QAHE)^3,4,5,6, which can be observed in the magnetic thin films featuring an inverted band gap due to spin-orbit coupling, i.e., two-dimensional magnetic TIs (MTIs). The bulk-edge correspondence⁷ dictates that at the physical border of such materials there must be a gapless topological edge state. Remarkably, an electron moving in such a state at any given edge of the sample can propagate only in one direction and cannot backscatter. Such a chirality of the edge state makes the QAHE attractive for the dissipationless transport applications. For example, it can be used to create interconnect devices that electrically connect the components of an integrated circuit⁸. The experimental hallmark of the QAHE is a vanishing longitudinal conductivity σ_xx along with a transversal conductivity σ_xy quantized to integer multiples of the conductance quantum, Ce²/h (for the resistivities, ρ_xx = 0 and ρ_yx = h/(Ce²)). Here, e is the electron charge, h is the Planck’s constant, and C is a dimensionless integer called the first Chern number, which is a topological invariant for these kind of systems^1,2,6.

Another fundamental phenomenon arising in MTIs is the topological magnetoelectric effect (TME)^2,9,10. When the TI surfaces are gapped due to magnetism, it should respond to the application of an external electric (magnetic) field by generation of a (an) magnetic (electric) polarization that appears to be quantized. An intriguing TME-related phenomenon is the solid-state embodiment of the axion electrodynamics^2,11,12, arising from a peculiar analogy with a field theory, where the axion field is said to generate electrodynamics with exactly the same Lagrangian that describes the TME in TIs. A material possessing these exotic properties is called axion insulator. Other interesting implications of the unique magnetolectric properties of MTIs are the quantized Kerr and Faraday effects^2,13.

The observation of the QAHE in the Cr-doped TI (Bi_xSb_1−x)₂Te₃^14,15 gave a strong impetus to a worldwide study of these systems. Nowadays, magnetically-doped TIs reproducibly show very robust high-precision QAHE at mK temperatures and they are currently being considered for metrological applications¹⁶. Besides, the QAHE is achieved not only in the MTIs, but also in the twisted bilayer graphene¹⁷ and transition metal dichalcogenides heterostructures¹⁸. Measuring TME in a magnetic TI proved more challenging, but it was eventually achieved, too¹⁹. However, a random distribution of the magnetic atoms in the magnetically-doped TIs leads to strongly inhomogeneous magnetic and electronic properties of these materials^20,21,22, restricting the observation of these effects to very low temperatures²³. More specifically, the atomic disorder leads to the fluctuation of both the size and energy position of the Dirac point gap across the surface of these materials, as imaged with scanning tunneling spectroscopy in ref. ²⁰. As a consequence, this gap has not been observed in angle-resolved photoemission spectroscopy (ARPES) for Cr- or V-doped (Bi_xSb_1−x)₂Te₃ with stoichiometries close to those showing QAHE^24,25.

As an alternative to the magnetically doped TIs, a new class of magnetic topological matter emerged: intrinsic MTI compounds, the first representative of this class being antiferromagnetic (AFM) van der Waals material MnBi₂Te₄^{26,27,28,29,30,31}. This discovery was preceded by both theoretical and experimental studies of the magnetic topological heterostructures such as MnBi₂Te₄/Bi₂Te₃(0001), MnBi₂Se₄/Bi₂Se₃(0001), and others that were based on the thin films of Mn(Bi/Sb)₂(Te/Se)₄^{32,33,34,35,36,37}. It is necessary to say that the AFM TI state of matter was for the first time theoretically proposed in 2010 by Mong, Essin, and Moore³⁸. They introduced a ${{\mathbb{Z}}}_{2}$ topological classification of AFM insulators, provided by the S = ΘT_1/2-symmetry, where Θ is time-reversal and T_1/2 is the primitive lattice translation. However, during almost a decade after the prediction no material satisfying these conditions was observed experimentally.

In this short review, we provide the current state-of-the-art in the research field that has emerged following the discovery of MnBi₂Te₄. Our aim is to offer a concise yet comprehensive overview that serves both as a quick update for researchers in the field and an accessible entry point for newcomers. We first introduce the basic properties of MnBi₂Te₄, followed by a discussion of the key topics, including the sample dependence of the gap in its topological surface state, native defects and their possible impact on the gap, and the surface electronic structure of MnBi₂Te₄ above the Néel temperature. This is followed by an exploration of the intrinsic MTIs of the MnBi₂Te₄ family and, eventually, the unique properties of MnBi₂Te₄ in the two-dimensional limit. Throughout the review, we examine the most significant and recent advancements in the field, while analyzing and highlighting its key challenges. Finally, in the outlook, we briefly discuss the exciting opportunities that the MnBi₂Te₄ family of compounds presents for future research and applications.

Basic properties of MnBi₂Te₄

The first reference to MnBi₂Te₄ dates back to 2013 when it was synthesized in the powder form and its structure as well as thermoelectric properties were studied³⁹. It crystallizes in the trigonal $R\bar{3}m$-group structure^39,40,41 comprising septuple layer (SL) blocks stacked in the ABCABC fashion, in which hexagonal atomic layers follow the Te-Bi-Te-Mn-Te-Bi-Te sequence (Fig. 1a). The bonds within the blocks are ionic covalent, whereas the neighboring SLs are connected by van der Waals forces. Density functional theory (DFT) calculations coupled to the Monte Carlo simulations predict^26,27,28 that below T_Néel = 25 K MnBi₂Te₄ develops the A-type interlayer AFM structure in which the ferromagnetically-ordered Mn layers are aligned antiparallel to each other (Fig. 1a). The magnetic anisotropy energy calculations revealed the easy axis with an out-of-plane orientation of the local magnetic moments of  ±4.6μ_B. For this magnetic ground state, the insulating spectrum with the fundamental bulk bandgap of about 0.2 eV was predicted for MnBi₂Te₄. Having gapped spectrum and being S-symmetric due to the fortunate combination of its crystal and magnetic structures (Fig. 1a), MnBi₂Te₄ falls within the ${{\mathbb{Z}}}_{2}$ topological classification of AFM insulators^38,42. As a result of an ab initio calculation, ${{\mathbb{Z}}}_{2}=1$ was found^26,27,28 thus classifying MnBi₂Te₄ as an AFM TI.

Fig. 1: Crystal and magnetic structures as well as the calculated low-energy surface spectra of MnBi₂Te₄.

a Crystal and magnetic structures of bulk MnBi₂Te₄. Red arrows denote Mn local moments, while the T_1/2 translation is shown in green. b, e Surface electronic band structures of MnBi₂Te₄ calculated for the (0001) (S-breaking; c) and for the $(10\overline{1}1)$ (S-preserving; d) surface terminations, respectively, using the ab-initio-based tight-binding approach. The regions with a continuous spectrum correspond to the three-dimensional bulk states projected onto a two-dimensional Brillouin zone. Observe the gapped (gapless) character of the S-breaking (S-preserving) surface. b, e are reproduced with permission from ref. ²⁶, Copyright Springer-Nature, 2019.

The combined S = ΘT_1/2 symmetry protects the degeneracy of the Dirac point of the topological surface state of an AFM TI³⁸. The surfaces that respect this symmetry are gapless, as shown in Fig. 1d, e for the ${{{\rm{MnBi}}}}_{2}{{{\rm{Te}}}}_{4}(10\overline{1}1)$ surface containing the T_1/2 translation. However, the S-breaking surfaces are gapped whenever there is a non-zero magnetization component perpendicular to them. Therefore the (0001) surface, which is MnBi₂Te₄’s natural cleavage plane, shows a band gap of several tens of meV according to the theoretical calculation (Fig. 1b, c). Large predicted magnetic gap at the Dirac point is an important characteristic making this material attractive for the QAHE and TME realization.

The magnetic and photoemission measurements performed on the bulk single crystals^26,43 and molecular-beam epitaxy grown thin films²⁹ confirm the AFM TI state in MnBi₂Te₄, which thus can be considered as the first intrinsic MTI. In agreement with the theoretical predictions, a long-range AFM ordering is observed in experiment^26,41,44,45, which can be identified by the λ-like shape of the magnetic susceptibility vs. temperature curve (χ(T); Fig. 2a) as well as by the presence of the characteristic spin-flop transitions in the M(H) curve at H_SF (Fig. 2b). The Néel temperature T_Néel ≈ 24–26 K is found, slightly varying from sample to sample. Subsequent neutron powder and single-crystal diffraction measurements confirmed the predicted A-type AFM ordering, as well as the out-of-plane easy axis^43,46,47. The ARPES experiments performed on the MnBi₂Te₄(0001) surface revealed the bulk band gap that hosts the linearly-dispersing topological surface state with an apparent splitting at the Dirac point²⁶ (see Fig. 2c).

Fig. 2: Experimentally-measured magnetic properties and surface electronic structure of MnBi₂Te₄.

a Magnetic susceptibility (left axis) of MnBi₂Te₄ as a function of temperature measured in an external magnetic field of μ₀H = 1 T in zero-field-cooled (ZFC) and field-cooled-warming (FCW) conditions, alongside the temperature-dependent reciprocal susceptibility (right axis) for H⊥(0001). b High-field magnetization data of MnBi₂Te₄ at 4 and 30 K with the field applied either along the c axis or in the ab plane. H_SF denotes the spin-flop magnetic field. Dispersion of the topological surface state of MnBi₂Te₄(0001) measured with (c) a conventional synchrotron ARPES (photon energy hν = 28 eV, T = 17 K) and (d, e) laser-ARPES (hν = 6.3 eV, T = 10 K). In (d, e) the data acquired on two samples synthesized by different groups are shown. In (e) the second derivative data, d²N(E)/dE², are presented. Note the reduced intensity at the expected Dirac point position in (c), indicative of the gap, which is also seen in (e), while, in contrast, the Dirac cone is gapless in (d). In the inset to (c), the second derivative (d²N(E)/dE²) of the data taken with a more bulk sensitive photons (hν = 9 eV, T = 17 K) is presented, where the bulk valence and conduction bands, separated by a gap of about 0.2 eV¹⁹⁰, can be seen. Magnetic structure of MnBi₂Te₄ below (f) and above (g) T^*, see text for the explanation. Note that the intermixing levels are intentionally enhanced for the visualization purposes. h Compilation of the literature data on the Dirac point gap size, as measured by different groups using laser-ARPES. Only the data reported for T < T_Néel are shown. Note that Shikin et al.⁵⁸ report data on 15 different MnBi₂Te₄ samples, as shown by blue circles. a, c are reproduced with permission from ref. ²⁶, Copyright Springer-Nature, 2019. b is reproduced with permission from ref. ⁴⁵, Copyright American Physical Society, 2021. d is reproduced from ref. ⁴⁹. e is reproduced with permission from ref. ⁵⁸, Copyright American Physical Society, 2021.

Sample-dependent Dirac point gap at MnBi₂Te₄(0001)

The above-described experimental evidences of the AFM TI state in MnBi₂Te₄ were further supported by the transport experiments performed on thin films that revealed phenomena of the topological origin (see below). Very surprisingly, however, the second wave of the ARPES studies (performed with laser radiation) uncovered an unexpected behavior of the MnBi₂Te₄’s Dirac cone, which contradicted the one expected based on the general theory of AFM TIs^38,42, DFT calculations made for MnBi₂Te₄^26,27,28, and the results of the magnetic measurements performed on it. Namely, while the early photoemission studies^26,44,48, performed with the conventional synchrotron ARPES, reported large gaps in agreement with the theoretical predictions, a number of the following laser-ARPES experiments found a dramatically reduced or even vanishingly small splitting of the MnBi₂Te₄ Dirac cone^49,50,51,52 (Fig. 2d). Figure 2h summarizes all the available data on the Dirac point gap measured by laser-ARPES^{49,50,51,53,54,55,56,57,58}. It can be seen that in the currently available MnBi₂Te₄ samples the Dirac point gap can acquire an arbitrary value between a few and several tens of meV. An example of the laser-ARPES spectrum showing a large Dirac point gap is presented in Fig. 2e.

Several scenarios have been proposed to account for this unexpected behavior^{49,52,59,60,61,62}. On the one hand, possible changes of magnetism near the MnBi₂Te₄ surface have been suggested, such as near-surface magnetic dead layer⁶⁰ or a deviation of the magnetic structure or easy magnetization axis from those in bulk^49,61. Recent scanning tunneling spectroscopy study reports asymmetric quasiparticle interference pattern of the topological surface state close to the Dirac point⁶³, which would be in line with the presence of the in-plane magnetization component at the MnBi₂Te₄(0001) surface. However, this is corroborated by neither the X-ray magnetic circular dichroism measurements^26,52 nor magnetic force microscopy experiments⁶⁴, that provide evidences of the persistence of uniaxial A-type AFM order up to the surface layers of the MnBi₂Te₄ single crystals.

On the other hand, possible structural factors have been considered, such as surface collapse during the mechanical exfoliation⁵⁹ or expansion of the first van der Waals gap⁵² (i.e., the increase of the spacing that separates the surface SL from the subsurface one). However, the former was only observed during the exfoliation in an inert gas environment and never in the ultra-high vacuum, while the latter needs to be quite large (about 15 %) to close the Dirac point gap, but such expansions have not been observed in experiment^65,66.

Mn-Bi intermixing and its possible impact on the Dirac point gap

Latest theoretical studies discuss Mn-Bi intermixing as the most likely cause of the Dirac point gap reduction in MnBi₂Te₄^66,67. To discuss the latter, let us first review the effect of the Mn-Bi intermixing on the MnBi₂Te₄ magnetic structure. High-field magnetization study⁴⁵ revealed that above the first magnetization plateau at ${\mu }_{0}{H}_{1}^{{{\rm{c}}}}\approx 8$ T (Fig. 2b) the magnetization smoothly increases further and eventually saturates to ${M}_{2}^{{{\rm{c}}}}\approx 4.6{\mu }_{{{\rm{B}}}}$/f.u. near 50 T. The reason for such evolution of M(H) from the first (${M}_{1}^{{{\rm{c}}}}$) to the second (${M}_{2}^{{{\rm{c}}}}$) plateau has been suggested a spin flip of Mn ions residing on the Bi site (Mn_Bi). In other words, in the ground state, each MnBi₂Te₄ SL block is “ferrimagnetic" (Fig. 2f), in which the local moments of the Mn_Bi atoms are coupled antiparallel to those of the central Mn layer, as has initially been found for MnSb₂Te₄ by neutron diffraction measurements⁶⁸.

Going back to the discussion of possible mechanisms of the Dirac point gap reduction, ref. ⁶⁶ argues that it is the opposite magnetization of the Mn_Bi antisites (with respect to the central Mn layer) that causes the Dirac point gap reduction. This happens because the magnetic antisites are introduced exactly in the regions of the topological surface state predominant localization in the real space, which is around the Bi layers. Thus the oppositely directed magnetic moments of the antisites can effectively counteract the magnetic effect of the central layer Mn atoms, around which the topological surface state weight is small. As illustrated by DFT calculations⁶⁶, even a moderate amount of the Mn_Bi antisites may result in a significant reduction of the Dirac point gap or even in its almost complete shrinking. In contrast, according to the DFT results in ref. ⁶⁷, the Mn-Bi intermixing “pushes" the topological surface state towards the subsurface SL, whereupon the magnetic gap is diminished heavily due to the AFM alignment between the surface and subsurface SL blocks. It should, however, be noted that both DFT studies^66,67 introduce the intermixing only in the topmost SL of the slabs used, while the other SLs are pristine. Thus, the calculations using more realistic structural models are needed to clarify which mechanism actually takes place in the experimental samples.

Recent scanning tunneling spectroscopy studies^69,70 of the Dirac point gap behavior in the molecular-beam epitaxy grown thin MnBi₂Te₄ films reveal the gap fluctuation across the surface. At that, the gap values ranging between 0 (gapless) and 70 meV have been measured. These studies differ, however, as far as the correlation of the gap size with the point defects distribution is concerned. While ref. ⁶⁹ reveals that the gap is absent (present) in the regions with a high (low) concentration of Mn_Bi defects, ref. ⁷⁰ concludes that the gap size is uncorrelated to individual Mn_Bi and Bi_Mn defects. Instead, the Dirac point gap fluctuation appears to take place on the nanometer scale. Remarkably, application of the external magnetic field of about 1 T significantly reduces the gap fluctuations and leads to the increase of the average gap from 26.3 meV to 44 meV⁷⁰. The sensitivity of the Dirac point gap to the moderate magnetic field could be consistent with a deviation of the near-surface magnetic structures from the bulk one, as mentioned above⁶³.

In this context, an important comment should be made. Out of all defects generated by the Mn-Bi intermixing, only the Mn_Bi antisites from the first Bi layer (i.e., the second atomic layer from the surface) are clearly seen in the scanning tunneling microscopy. Imaging of the Bi_Mn defects that lie in the fourth atomic layer from the surface is already quite difficult^66,70,71. As far as the the sixth atomic layer Mn_Bi defects are concerned, to the best of our knowledge, no feature that would be clearly attributed to them has been reported in the scanning tunneling microscopy studies. These features should undoubtedly be faint and, because their extension should be about several lattice parameters as a minimum, may laterally overlap with each other as well as with those from defects in other layers, making them hardly distinguishable. Nevertheless imaging of the sixth layer Mn_Bi antisites is highly desirable since, according to the DFT calculations⁶⁶, their negative effect on the Dirac point gap is expected to be much stronger than that of the second layer Mn_Bi. This is because the sixth atomic layer (again, counting from the surface) carries a much larger weight of the topological surface state in the real space⁶⁶.

Latest experimental data shed more light onto the magnetism of the MnBi₂Te₄ family, which might be relevant in the context of the Dirac point gap issue. The nuclear magnetic resonance and muon spin spectroscopy measurements not only directly confirm the ferrimagnetic structure of the MnBi₂Te₄ SLs (and those of MnBi₂Te₄ ⋅ nBi₂Te₃ discussed below), but also reveal that the static magnetic moment of the Mn_Bi antisite sublattice disappears at T^* < T_Néel, T^* being equal to 12.5 K in the MnBi₂Te₄ case⁷², i.e., below (above) T^* the Mn_Bi sublattice is ordered (paramagnetic), as shown in Fig. 2f, g. It would be very interesting to explore the Dirac point gap behavior across T^*, given that one of the proposed scenarios⁶⁶ attributes the gap reduction to the AFM coupling between the Mn_Bi and Mn_Mn sublattices when both of them are ordered (i.e., below T^*). The existing temperature-dependent photoemission data⁵⁸ do not elucidate this behavior since the lowest measurement temperature (~10 K) is too close to the expected T^*. Note that for the samples studied in ref. ⁵⁸ T^* are unknown. It is reasonable to assume that T^* can slightly vary from sample to sample (likely depending on the degree of intermixing), pretty much as the Néel temperature of MnBi₂Te₄ varies within the 24–26 K range^66,72. Taking also into account the instrument error in the temperature determination and the temperature step of about 1.7 K in ref. ⁵⁸, it is difficult to draw conclusions about the Dirac point gap behavior across T^* from that experiment. The compilation of the laser-ARPES results shown in Fig. 2h does not clarify this issue either, since the measurements have been performed by different groups on different samples. Meanwhile, using time- and angle-resolved photoemission spectroscopy, it has been found that opposite helicities of mid-infrared circularly polarized light result in substantially different Dirac mass gaps in the MnBi₂Te₄ topological surface state below T^* (at 8 K), despite the equilibrium Dirac cone being massless⁵⁷.

MnBi₂Te₄ surface electronic structure above T _Néel

At this point, the surface electronic structure of MnBi₂Te₄ above the Néel temperature should also be discussed. A number of ARPES studies, both synchrotron^26,44,48 and laser based^{26,50,52,58,73}, in which the Dirac point gap has been detected in the AFM state, reported its persistence in the paramagnetic (PM) phase, similarly to what was previously observed for the magnetically-doped Bi₂Se₃^74,75. At that, in majority of the cases, MnBi₂Te₄’s Dirac point gap size in the AFM and PM phases is practically the same, with only one laser-ARPES study reporting a  ~40% reduction of the gap (from 65 to 40 meV) upon heating from below T_Néel up to 35 K⁵⁸. Such a behavior challenges the idea of the magnetically induced gap, as the latter should disappear at or just above the critical point, pretty much as it happens with the exchange splitting of the first bulk conduction and valence bands observed for MnBi₂Te₄^51,73. However, in MnBi₂Te₄ the persistence of the Dirac point gap could be caused by strong short range order effects that exist up to about 60 K, as observed by electron spin resonance, as well as ferromagnetic and AFM resonance experiments^26,76,77. The measured magnetization data⁴⁵, revealing that MnBi₂Te₄ is not in the PM limit even at T ≈ 50 K, support the latter observations. Such a behavior is also consistent with the strong spin fluctuation-driven spin scattering above T_Néel found in a previous magneto-transport study of MnBi₂Te₄ in ref. ⁴⁴. Beyond 50–60 K, an unprecedentedly large anisotropy of the Mn spin relaxation rate in the PM state of MnBi₂Te₄^26,76 may give rise to an instantaneous (on the timescale of electron spin resonance) out-of-plane magnetic field at the surface, preventing the gap from closing on the much faster timescale of the ARPES experiment.

MTIs of the MnBi₂Te₄ family

In spite of the Dirac point gap puzzle, MnBi₂Te₄ has attracted significant interest of the research community due to its unusual and highly tunable properties. Multiple tuning knobs, not only extrinsic such as magnetic field, pressure, and temperature, but also intrinsic such as Mn-Mn interlayer distance, chemical composition, and defect engineering, can be used to realize various magnetic and topological states in MnBi₂Te₄ or systems on its basis. For example, pnictogen or chalcogen substitutions and Mn/Bi/Te stoichiometry alternations give rise to such materials as Mn(Bi_1−xSb_x)₂Te₄, MnSb₂Te₄, MnBi₂Se₄, or Mn₂Bi₂Te₅, whose magnetic and topological properties were studied both theoretically and experimentally^{78,79,80,81,82,83,84,85,86}. In particular, MnBi₂Se₄, while being AFM TI with the same A-type structure as MnBi₂Te₄, displays staggered magnetization within the Mn plane⁸², which may give rise to the one-dimensional topologically-protected flat bands pinned to the magnetic domain walls at its surface, as predicted by theory⁸⁷. Incidentally, recent experimental and theoretical X-ray magnetic circular dichroism study has revealed that the Mn_Bi antisites in the SLs of the MnBi₂Se₄/Bi₂Se₃ heterostructure are coupled ferromagnetically to the central Mn plane of the SL⁸⁸, in a stark contrast to the MnBi₂Te₄ case, where this coupling is AFM^45,72. This striking difference has been attributed to a stronger hybridization of the Mn_Bi-d states with the Se-p states in MnBi₂Se₄ as compared to that with Te-p in MnBi₂Te₄, due to the shorter bondlength in the former. Another material with the 1-2-4 stoichiometry, MnSb₂Te₄, seems to be on the verge of the topological phase transition, as the DFT calculations reveal a strong sensitivity of its topology to the crystal structure details^{27,68,89,90,91}, while the available experimental data are contradictory^80,92 and further studies are needed. Mixing Bi and Sb on the pnictogen sublattice to create Mn(Bi_1−xSb_x)₂Te₄, enables realization of the Weyl semimetal state in the fully polarized ferromagnetic (FM) state^27,28, which was experimentally achieved by applying a sufficiently strong external magnetic field that overcomes the AFM interlayer exchange coupling^81,93. Ge-, Sn-, and Pb-doped MnBi₂Te₄ single crystals were also synthesized^94,95,96, showing peculiar topological phase transition⁹⁷.

Furthermore, the van der Waals nature of MnBi₂Te₄ enables “intercalating" the adjacent SLs with Bi₂Te₃ quintuple layers, resulting in the MnBi₂Te₄ ⋅ nBi₂Te₃ family of compounds (n = 1 for MnBi₄Te₇, n = 2 for MnBi₆Te₁₀, and so on, up to n = 6)^40,98,99,100. The increasing distance between the SLs progressively weakens the interlayer exchange coupling with an increasing n¹⁰¹, which enables an effective tuning of the magnetic structure by moderate magnetic fields^{102,103,104,105,106,107}, or hydrostatic pressure¹⁰⁸, driving these compounds from the AFM to the FM state. Overall, apart from the 3D AFM TI state, under different conditions MnBi₂Te₄ ⋅ nBi₂Te₃ materials may show topological crystalline insulator state¹⁰² as well as higher order topologies such as Möbius¹⁰⁹ or ${{\mathbb{Z}}}_{4}$-axion^{109,110,111,112} insulators, 2D AFM second order TI¹¹³ or topological superconductor¹¹⁴ states.

Magnetic antisite defects in the materials of the MnBi₂Te₄ family are lately in the center of attention not only because of their possible negative impact on the Dirac point gap but also thanks to their strong influence on the magnetic and electronic structure: they are exploited as an effective tuning knob to purposely modify the latter^68,115,116. The cation intermixing between the manganese and pnictogen crystallographic sites is favored by closeness of their ionic radii. This is especially true for MnSb₂Te₄ that supports up to  ~40% of Sb atoms in the Mn layer, while, in turn, up to  ~15% of each Sb layer is occupied by Mn atoms⁶⁸. For MnBi₂Te₄ the intermixing levels are roughly three times lower⁴¹. Strong intermixing in MnSb₂Te₄ promotes the FM coupling between the adjacent SLs¹¹⁷. At that, the magnetic transition temperature jumps from T_Néel = 19 K in the AFM-like bulk MnSb₂Te₄ single crystals⁶⁸ to T_Curie= 73 K in the FM-like ones¹¹⁸, which is achieved by varying the growth conditions with a certain degree of control⁶⁸. According to the DFT calculations, while a moderate Mn-Sb intermixing promotes the Weyl semimetal state due to the change of the interblock coupling to FM, its increase eventually renders MnSb₂Te₄ gapped and topologically trivial^68,80. Apart from MnSb₂Te₄, the interlayer coupling can also become truly FM in the Sb-doped MnBi₂Te₄ ⋅ nBi₂Te₃^119,120 and even in MnBi₆Te₁₀, in the latter case via the Mn-Bi defects engineering under appropriate growth conditions^115,116, which may help to achieve an FM axion insulator state^109,110.

MnBi₂Te₄ in the two-dimensional limit

MnBi₂Te₄ properties in the two-dimensional limit are exciting, too. Magnetism and topology of thin MnBi₂Te₄ films are thickness-dependent, while different symmetries in even- and odd-SL films lead to distinct phenomena^{28,29,30,121,122}. The films with odd (even) number of SLs are uncompensated (fully compensated) interlayer antiferromagnets, which break (preserve) the PΘ symmetry²⁸, P being inversion. In the films made of even number of blocks, the PΘ symmetry leads to the Chern number C = 0, while a C ≠ 0 is allowed in the films with the odd number. According to DFT calculations^28,30, while the one-SL-thick film is topologically trivial, the 5- and 7-SL-thick ones are QAH insulators with C = 1 (the predictions on the 3-SL-thick film are contradictory^28,30). On the other hand, the 2-, 4-, and 6-SL-thick films were predicted to show the so-called zero plateau QAH state. The importance of the zero plateau QAH state is that it can be a suitable platform for realization of the TME, characteristic of the axion insulator phase¹²³.

Soon after the theoretical predictions^28,30, the zero plateau QAH state has been observed experimentally in the MnBi₂Te₄ thin flakes made of even number of SLs¹²⁴, see Fig. 3a, b. Previously, this state of matter was being sought for in the FM1/TI/FM2 QAH heterostructures, where a relatively thick TI spacer enables magnetization reversal of the individual FM layers that have different coercivities, leading to the overall AFM alignment and, consequently, to a zero plateau QAH state^125,126. In theory, the MnBi₂Te₄ thin films made of even number of SLs realize this state intrinsically, i.e., without the need of magnetic field application. In practice, however, a magnetic field of about 8 T is applied and swiped to 0 T to prepare a single-domain AFM state with the desired orientation of the topmost SL magnetization, up or down^127,128. It should also be noted that a thickness of 6 SLs¹²⁴ is likely insufficient for the TME to be quantized because of the finite-size effect^123,129,130, however, it should become fully quantized upon increasing the film thicknesses towards the 3D limit.

Fig. 3: Q(A)HE and its zero-plateau in MnBi₂Te₄ flakes.

Magnetic field driven transition from the zero-plateau QAH state (around the zero field, corresponding to ρ_yx = 0 (a) and very large ρ_xx (b)) to the Chern insulator with QHE (at ± 9 T, with the nearly quantized ρ_yx ≈ ∓h/e² (a) and vanishing ρ_xx (b)) measured in a 6-SL-thick MnBi₂Te₄ flake at T = 1.6 K. The measurement is performed using the back-gate voltage V_g = 30 V, which corresponds to the charge neutrality point when the Fermi level lies within the surface state gap. Quantum anomalous Hall effect in a 5-SL-thick MnBi₂Te₄ flake illustrated by the magnetic field dependence of ρ_yx (c) and ρ_xx (d), measured at T = 1.4 K and V_g = −200 V. At μ₀H = 0 T, ρ_yx reaches 0.97h/e², while ρ_xx is suppressed (0.061h/e²), evidencing the QAHE. The insets in (a, c) schematically show the magnetic structure of the flakes at zero and high fields. a, b are reproduced with permission from ref. ¹²⁴, Copyright Springer-Nature, 2020. c, d are reproduced with permission from ref. ¹⁴⁰, Copyright AAAS, 2020.

Further experimental work not only confirms the zero plateau QAH state in 2D even-layered MnBi₂Te₄,^127,131 but also reveals a series of other novel phenomena^{127,128,131,132,133}. One of them is the layer Hall effect, in which the electrons from the top and bottom layers of the film spontaneously deflect in the opposite directions, which is due to the opposite signs of the Berry curvature^127,134. Observation of the effect is enabled by applying external electric field that breaks PΘ, leading to the emergence of a finite anomalous Hall effect (AHE), which is dominated by either top or bottom layer, depending on the field direction. Moreover, the opposite sign of the Berry curvature of the top and bottom surfaces is responsible for the appearance of the axion (i.e., surface¹³⁵) contribution to the optical magneto-electric coupling, as recently predicted¹³⁶ and observed for the 2D even-layered MnBi₂Te₄¹³¹. This effect enables optical control of the AFM order, i.e., selective stabilization of one or another AFM domain (up-down or down-up) by the circularly polarized light with either opposite helicity or different frequencies that couples differently to the opposite AFM domains.

Another unusual phenomenon recently witnessed in the thin MnBi₂Te₄ flakes with even number of blocks is the second-order nonlinear AHE^132,133, in which an alternating current (a.c.) with frequency ω induces an a.c. Hall voltage with frequency 2ω. In this case, the Berry curvature is not the origin of the effect, since it is equal to zero as dictated by the PΘ symmetry. Instead, the effect is generated by the quantum metric of the gapped MnBi₂Te₄ Dirac cone^132,133. More recently, third-order nonlinear Hall effect in both longitudinal and transverse directions in thick MnBi₂Te₄ flakes has been reported¹³⁷. The longitudinal third-harmonic response ${V}_{xx}^{3\omega }$ has been found to appear due to the quantum metric quadrupole, while the transverse response ${V}_{xy}^{3\omega }$ due to the Berry curvature quadrupole. Thus, MnBi₂Te₄ enables probing both the real and the imaginary part of the quantum geometric tensor of the Bloch states in transport experiments.

As for the predicted QAHE (i.e., the quantum Hall effect (QHE) in zero magnetic field), its observation in the odd-layered 2D MnBi₂Te₄ has proven challenging. In fact, initially MnBi₂Te₄ has shown the quantized Hall effect only under external magnetic field^138,139, but not without it (note that refs. ^138,139 are preprint versions of^124,140). The field has been used to overcome the AFM interlayer exchange coupling thus forcing the FM state (Fig. 3a), while the gate bias is employed to find the charge neutrality point, which in MnBi₂Te₄ should correspond to the gap in the Dirac point. At the magnetic fields used (about 9 T), the magnetic moments of the Mn_Bi antisites are most likely still ordered oppositely to those of the central Mn layer⁴⁵. However, this does not suppress the quantized Hall effect. In fact, the effect can be observed already in the canted AFM state, just after the spin-flop transition¹⁴¹. Upon further increase of the magnetic field strength, the canted AFM state transforms into the forced FM state with the QHE observed continuously. The absence of the free carriers at the Fermi level translates into the absence of Landau levels (LLs) in spite of the field application (in contrast to the conventional QHE¹⁴²) and the QHE stems from the non-trivial topology of the MnBi₂Te₄’s band structure in the forced FM state. Envisioned by theory^28,30, this C = 1 QHE without LL formation in MnBi₂Te₄ has been repeatedly observed in experiment by different groups to this date^{124,127,140,141,143,144,145,146,147,148}, with a reasonable quantization persisting up to 30 K¹⁴³. Worth mentioning as well is the C = 2 state without LLs that onsets in the flakes thicker than 9 SLs of MnBi₂Te₄ under external magnetic field and persists up to about 10 K^91,143. Finally, the conventional QHE with LLs¹⁴² can be observed in the MnBi₂Te₄ flakes too¹⁴⁰, which, apart from a rather strong magnetic field, requires increasing the carrier concentration by applying bias voltage.

The zero-field QAHE has been observed in the 5-SL-thick film¹⁴⁰, as shown in Fig. 3c, d. However, further advances in this direction have been slow, likely due to the difficulties in obtaining high-quality samples with reduced levels of the Mn-Bi intermixing, although the device fabrication process seems to introduce further complications^59,149. A valuable insight into the quantized transport of the odd-layered MnBi₂Te₄ flakes has been provided by scanning superconducting quantum interference device microscopy used to image their current distribution¹⁵⁰. A chiral edge current in the 7-SL-thick flake at zero magnetic field has been observed, confirming its topological nature. However, the finite bulk conduction and edge-bulk scattering have been found to undermine the transport quantization. This is consistent with the scanning tunneling spectroscopy data that image both the edge state of the 5-SL-thick molecular-beam epitaxy grown MnBi₂Te₄ film and its coupling to the gapless two-dimensional bulk regions arising from band gap fluctuations⁷⁰. However, very recently, a new procedure in the device preparation, consisting in depositing an AlO_x layer on the MnBi₂Te₄ flake surface prior to nano-fabrication, has enabled a significant improvement of the AHE phase quality, with the true zero-field QAHE achieved in few samples^151,152. Applying a hydrostatic pressure also allowed approaching the QAHE regime in 2D MnBi₂Te₄¹⁵³.

Recently, an intriguing behavior has been reported for the molecular-beam epitaxy grown 5 SL MnBi₂Te₄ film showing the QHE without LLs in the forced FM state. Tuning the gate voltage through the QHE region up until the quantization breakdown, when ρ_yx is no longer equal to h/e², and then switching the field off (for the material to adopt the AFM state) leads to reappearance of a nearly quantized ρ_yx plateau of  ≈h/e²¹⁵⁴. This happens for the Fermi level located in the valence band and not in the global gap. Despite the latter, as well as nonzero ρ_xx ≈0.85h/e², this state shows tan(θ_H) vs. T behavior (θ_H = ρ_yx/ρ_xx is the Hall angle), which is more similar to that of the zero-field QAHE in MnBi₂Te₄^140,153, rather than the insulating zero field state of the very same epitaxial film at the charge neutrality point. The appearance of this phase, dubbed “reentrant QAHE", has been attributed to the exchange-induced Berry curvature splitting and disorder-induced Anderson localization^154,155.

Outlook

Let us now briefly outline other exciting phenomena that compounds of the MnBi₂Te₄ family have been reported to (or may potentially) host. While we do not delve into details—an undoubtedly worthwhile endeavor that lies beyond the scope of this short review—we nevertheless highlight these intriguing studies as a roadmap for future research. Apart from the above-described variants of the Hall effect, the compounds of the MnBi₂Te₄-family were reported to show the QAHE in the 3D limit¹⁵⁶, as well as the topological^157,158 and planar Hall effects¹⁵⁹. Pending the observation are the predicted half-integer QHE³⁸, Θ-breaking quantum spin Hall effect¹⁶⁰, and its “hinged" version¹⁶¹, interfacial crystal Hall effect¹⁶², orbital Hall effect¹⁶³, as well the quantized version of the layer Hall effect¹⁶⁴. Besides, MnBi₂Te₄-based systems could be a platform for novel topological heterostructures^{32,34,160,165,166,167,168,169}, anomalous Nernst effect¹⁷⁰, high-Chern number phases¹⁷¹, skyrmions¹⁷² (in particular, coexisting with QAHE, which may generate novel topological phases¹⁷³), and Majorana fermions¹⁷⁴. Recent studies are seeking to induce superconductivity in MnBi₂Te₄ either via proximitizing it to a conventional superconductor NbSe₂¹⁷⁵ or via interfacing MnBi₂Te₄ film to that of a non-superconducting AFM material FeTe¹⁷⁶.

Novel device proposals based on the systems of the MnBi₂Te₄-family have already been put forward^177,178, such as rectifiers, spin filters, negative differential resistive devices, photoelectric sensors, photovoltaic, magneto-optoelectronic devices. The layer Hall effect makes MnBi₂Te₄ a potential platform to explore the ‘layertronics’ to encode, process, and store information¹⁷⁹, while the recently observed AFM diode effect^128,133 may enable a field-effect transistor and harvesting of wireless electromagnetic energy¹²⁸. Quantum computing and sensing applications have also been envisioned^180,181.

It should be said, however, that this research field is only in its infancy, and many results are still to be confirmed and understood, while many new exciting discoveries are likely to come soon. The most important task appears to be achieving an exhaustive explanation of the unexpected behavior of the MnBi₂Te₄ topological surface state gap and further detailed studies are needed. However, since the Mn-Bi intermixing seems to be detrimental in this sense, suppressing it could hopefully allow getting rid of the Dirac point gap issue in MnBi₂Te₄. Steps in this direction are being taken currently^{69,145,148,182,183}. Achieving this objective might translate into the improved quality of the experimental samples and, hence, to an observation of a plethora of phenomena and states of matter the MnBi₂Te₄ family is capable of hosting. In particular, it could enable realization of the robust intrinsic QAHE in MnBi₂Te₄, whose observation temperature may be expected to jump significantly above the currently reported 1.5 K¹⁴⁰.

For further detailed reading on the intrinsic MTIs of the MnBi₂Te₄-family we recommend the following review articles^{183,184,185,186,187}.

We have recently learned of the experimental observation of the topological Anderson Chern insulator phase in the [MnBi₂Te₄]_1SL/[Bi₂Te₃]_1QL heterostructure (where QL stands for quintuple layer), as reported in ref. ¹⁸⁸. More recently, the observation of a dynamical axion quasiparticle, i.e., a solid-state incarnation of the high-energy axion particle, has been reported in the 2D MnBi₂Te₄. This discovery may ultimately pave the way for the experimental detection of the elusive axion particle¹⁸⁹.