MnBi2Te4-family intrinsic magnetic topological materials

MnBi2Te4 and its derivative compounds have received focused research interests recently for their inherent magnetic order and the rich, robust and tunable topological phases hosted in them. Here, I briefly introduce MnBi2Te4-family intrinsic magnetic topological materials—the electronic and magnetic properties, the topological phase diagrams and the research progress made on them in the past years. I try to present a simple picture to understand their rich electronic, magnetic and topological properties, and a concise guide to engineer them for intended topological phases and the quantum anomalous Hall effect at higher temperature.


INTRINSIC MAGNETIC TOPOLOGICAL INSULATOR MNBI 2 TE 4
In a first-principles calculation work by E. V. Chulkov's group from San Sebastián, Spain in 2017, it was found that if a MnTe bilayer is artificially inserted into the first quintuple layer of Bi 2 Te 3 , constructing a MnBi 2 Te 4 septuple layer (SL; Fig. 1a), a rather large magnetically induced energy gap (up to 77 meV, much larger than the <1 meV gap in usual magnetically doped TIs) is opened at the topological surface states (TSSs), which promises a robust QAH state 6 . The MnBi 2 Te 4 capping layer, then considered as a ferromagnetic insulator (FMI) by the authors, somehow allows the TSSs of Bi 2 Te 3 to pervade it and strongly interact with the Mn atoms (termed as "magnetic extension" by the authors), which leads to the huge magnetic gap.
Little was known on the material at that time because, as a metastable phase, its single-crystal samples are rather difficult to prepare 7 . At the end of 2017, our group first obtained thin films of MnBi 2 Te 4 with molecular beam epitaxy and checked their electronic structure with in situ angle-resolved photoemission spectroscopy (K.H., talk in Zhengzhou University International Forum for Young Scholars on December 17, 2017) 8 . Amazingly, the MnBi 2 Te 4 films exhibit archetypical Dirac-cone-shaped bands around Fermi level-a characteristic of a 3D TI. If MnBi 2 Te 4 is indeed a 3D TI (not considering its magnetism), the mysterious "magnetic extension" in MnBi 2 Te 4 -covered Bi 2 Te 3 becomes straightforward to understand. 3D TIs have a special property: put two 3D TIs together, you get one 3D TI, with the TSSs appearing only at the surface of the whole system 9 . Naturally, the TSSs will "extend" to the MnBi 2 Te 4 capping layer on Bi 2 Te 3 .
Indeed, according to the calculation results, the energy bands of MnBi 2 Te 4 near Fermi level are dominated by the p bands of Bi and Te, which contribute to the TI property [10][11][12] . The hybridization between the p bands and the d bands of Mn atoms, several electron volts away from Fermi level, is weak enough to retain the TI characteristic, but strong enough to induce a significant magnetic gap at the TSSs when the magnetic order is built (Fig. 1b, c). We call it an intrinsic magnetic TI because the magnetism comes from the compound itself, rather than from magnetic dopants or neighboring layers, which is crucial for the enough strong p-d hybridization inducing the large magnetic gap.
MnBi 2 Te 4 has a A-type antiferromagnetic (AFM) structure: every ferromagnetic (FM) SL, with out-of-plane easy magnetic axis, is weakly antiferromagnetically coupled with neighboring SLs. The interplay between the magnetic structure and the topological nontrivial bands endows the material with rich topological phases [10][11][12] . Actually, one can easily understand the topological phases of MnBi 2 Te 4 and other similar compounds MnBi 2n Te 3n+1 (n ≥ 1) based on the theoretical topological phase diagrams given by A. A. Burkov and L. Balents in 2011 by considering the materials as superlattices between MnTe (FMI) and Bi 2 Te 3 (TI) layers of different thicknesses 13 .
When the magnetic order vanishes at an enough high temperature, a FMI/TI superlattice degenerates into a normal insulator (NI)/TI superlattice. Its topological phase, as shown in Fig. 2a, depends on the competition between the hybridizations of the TSSs from the same TI layers (Δ S ) and from neighboring TI layers (Δ D ). Clearly, a superlattice with thinner NI layers (larger Δ D ) and thicker TI layers (smaller Δ S ) tends to be a TI, while that with thicker NI layers (smaller Δ D ) and thinner TI layers (larger Δ S ) tends to be a NI. When the FMI layers recover their ferromagnetism and are coupled antiferromagnetically (A-type AFM structure), the phase diagram is similar except that the TI phase is replaced by the AFM TI (Fig. 2b), which is exactly the ground state of MnBi 2 Te 4 . AFM TI is a TI protected by the combined time-reversal and translation symmetry, and possesses gapless TSSs only at surfaces parallel to the magnetization axis 14 .
In MnBi 2 Te 4 thin films, the A-type AFM structure makes the magnetization vectors of the top and bottom SLs parallel for the odd-number SL thickness, and antiparallel for the even-number SL thickness. In the former case, the top and bottom surfaces have topologically different surface states, and are thus bounded by a 1 State Key Laboratory of Low-Dimensional Quantum Physics, Department of Physics, Tsinghua University, 100084 Beijing, China. ✉ email: kehe@tsinghua.edu.cn gapless chiral edge channel at the film edge (Fig. 2e), which makes a QAH insulator with Chern number (C) of unity. In the latter case, the two surfaces are the same in their topological properties, and the film is an axion insulator characterized by the topological magnetoelectric effect (Fig. 2d) 11,15 .
If the FMI layers of a FMI/TI superlattice are magnetized in the same direction, its topological phase is determined by the competitions among Δ S , Δ D , and the magnetic gap of the TSSs (m), as shown in the phase diagram in Fig. 2c (ref. 13 ). When Δ S is large compared with m, while Δ D is small, the superlattice is a usual FMI. When Δ D is large compared with m, while Δ S is small, the superlattice is a FMI that becomes a TI when the FM order is lost above the Curie temperature. The thin films of such a phase will exhibit C = 1 QAH effect, similar to magnetically doped TI Crdoped (Bi,Sb) 2 Te 3 . The phase is denoted by "FMTI", since FM order and TI phase do not exist at the same time though in the same material 3,16 . When both Δ S and Δ D are large compared with m, the superlattice enters the magnetic Weyl semimetal (WSM) phase. Its thin films can show the QAH effect with Chern number (the number of chiral edge channels) depending on the film thickness (Fig. 2f, g) 17,18 . When Δ S and Δ D are both small compared with m, the TSSs are decoupled with each other, and the superlattice can be considered as many parallel and isolated QAH systems which is denoted by QAH multilayer (ML) 19 . The rich topological phases found in MnBi 2 Te 4 -family materials can be understood and engineered based on the phase diagrams.

RESEARCH PROGRESS ON MNBI 2 TE 4
Rapid and exciting experimental progress has been made in MnBi 2 Te 4 -family materials in the past years, constructing one of the hottest topics in condensed matter physics recently 8,20-36 . Single-crystal samples of MnBi 2 Te 4 -family materials have been successfully prepared by several groups [20][21][22][23][24][25][26][27] . The A-type AFM structure has been demonstrated with SQUID measurements and directly observed with magnetic force microscopy 28,29 . A magnetic field of several Tesla is enough to overcome the weak AFM interlayer coupling and make MnBi 2 Te 4 FM, as shown by the stepped hysteresis loops characterizing a van der Waals magnetic material 8 . The spectroscopic results of MnBi 2 Te 4 family materials, however, are controversial 8,[20][21][22][23][30][31][32][33] . There is still no consensus on whether the observed TSSs are gapped or gapless and why they are insensitive to temperature.
Above all, quantized transport properties have been observed in MnBi 2 Te 4 thin flakes exfoliated from MnBi 2 Te 4 single crystals by several groups above 1.5 K, higher than in usual magnetically doped TI samples [34][35][36] . In odd-SL thick flakes, Yuanbo Zhang and Xianhui Chen's group observed quantized plateaus of the Hall resistance, as well as diminishing longitudinal resistance, both in the AFM configuration around zero magnetic field and in the FM configuration at magnetic field of several Tesla 35 . Yayu Wang and Jinsong Zhang's group observed the topological phase transition between axion insulator and QAH insulator in even-SL thick MnBi 2 Te 4 flakes, as the magnetic configuration is changed between AFM to FM by magnetic field 35 . Jian Wang's group observed C = 2 plateau (at magnetic field of several Tesla) at relatively thick MnBi 2 Te 4 flakes, which provides a strong evidence for the magnetic WSM phase in FM MnBi 2 Te 4 . Furthermore, in some samples, nearly quantized Hall resistance (90.4% h/e 2 ) plateau is observed up to 45 K, even higher than the Néel temperature of MnBi 2 Te 4 (25 K) 36 . The seemingly counter-intuitive observation can be understood with the 2D magnetism of the material. In such a 2D magnetic system, according to the Mermin-Wagner theorem, the magnetic ordering temperature is determined by the magnetic anisotropy energy which suppresses magnetic fluctuations. An external magnetic field enhances the effective magnetic anisotropy energy, and can thus hold longrange FM order to higher temperature. The result implies that the QAH states in MnBi 2 Te 4 are indeed quite robust and may exist at higher temperature, with a more efficient way to suppress the magnetic fluctuations.
The key to control the topological phases in MnBi 2 Te 4 -family compounds is to engineer their magnetic structures, especially the interlayer magnetic coupling (IMC). A recent theoretical work has revealed that the IMC of MnBi 2 Te 4 -family materials is essentially the superexchange interaction and follows the Goodenough-Kanamori rules 37,38 . It enables us to obtain the magnetic WSM phase with FM ground state, as well as other interesting topological phases by constructing heterostructures between different MnBi 2 Te 4 -family materials 37 . Different from the short-ranged superexchange coupling in ferrites (usually mediated by only one oxygen ion), the IMC of MnBi 2 Te 4 -family materials is significant even with the neighboring FM layers spaced by six nonmagnetic atoms and a K. He van der Waals gap, reaching 16.5 meV (~190 K) in NiBi 2 Te 4 . The super-long-ranged superexchange coupling is possible because Bi and Te atoms have similar electronegativities, and both possess rather delocalized p orbitals, which greatly facilitates electron hopping between them 37 .
The result that IMC strength of van der Waals magnetic materials can far exceed the energy scale of liquid nitrogen temperature (77 K) indicates a way to achieve high temperature QAH effect in MnBi 2 Te 4 . Choosing an appropriate FMI neighboring layer that are strongly coupled with MnBi 2 Te 4 may significantly suppress magnetic fluctuations in it, without needing external magnetic field 37,39 . The Curie temperature should be mainly determined by the IMC strength, which can exceed 77 K. Sandwiching a MnBi 2 Te 4 film with two such FMI layers would thus lead to the QAH effect above the liquid nitrogen temperature.

CONCLUSION AND PROSPECTIVE
So, MnBi 2 Te 4 -family materials and heterostructures provide us a neat, simple, and versatile way to fuse topological energy bands with magnetic order, and unveil a feasible road map to push the occurring temperature of the QAH effect above 77 K. They may thus play a crucial role in achieving wide practical applications of the QAH effect, as well as other related quantum effects, opening up the gate toward a new concept electronics based on elegant topological quantum states.