Magnetism, originating from the moving charges and spin of elementary particles, has revolutionized important technologies such as data storage and biomedical imaging, and continues to bring forth new phenomena in emergent materials and reduced dimensions. The recently discovered two-dimensional (2D) magnetic van der Waals crystals provide ideal platforms for understanding 2D magnetism, the control of which has been fueling opportunities for atomically thin, flexible magneto-optic and magnetoelectric devices (such as magnetoresistive memories and spin field-effect transistors). The seamless integration of 2D magnets with dissimilar electronic and photonic materials opens up exciting possibilities for unprecedented properties and functionalities. We review the progress in this area and identify the possible directions for device applications, which may lead to advances in spintronics, sensors, and computing.

Two-dimensional magnetic crystals and emergent heterostructure devices

Magnetic topological materials represent a class of compounds whose properties are strongly influenced by the topology of the electronic wavefunctions coupled with the magnetic spin configuration. Such materials can support chiral electronic channels of perfect conduction, and can be used for an array of applications from information storage and control to dissipationless spin and charge transport. Here, we review the theoretical and experimental progress achieved in the field of magnetic topological materials beginning with the theoretical prediction of the Quantum Anomalous Hall Effect without Landau levels, and leading to the recent discoveries of magnetic Weyl semimetals and antiferromagnetic topological insulators. We outline the recent theoretical progress that resulted in the tabulation, for the first time, of all magnetic symmetry group representations and topology. We describe several experiments realizing Chern insulators, Weyl and Dirac magnetic semimetals, and an array of axionic and higher-order topological phases of matter as well as survey future perspectives. 

Progress and prospects in magnetic topological materials

The search for new two-dimensional monolayers with diverse electronic properties has attracted growing interest in recent years. Here, we present an approach to construct MA2Z4 monolayers with a septuple-atomic-layer structure, that is, intercalating a MoS2-type monolayer MZ2 into an InSe-type monolayer A2Z2. We illustrate this unique strategy by means of first-principles calculations, which not only reproduce the structures of MoSi2N4 and MnBi2Te4 that were already experimentally synthesized, but also predict 72 compounds that are thermodynamically and dynamically stable. Such an intercalated architecture significantly reconstructs the band structures of the constituents MZ2 and A2Z2, leading to diverse electronic properties for MA2Z4, which can be classified according to the total number of valence electrons. The systems with 32 and 34 valence electrons are mostly semiconductors. Whereas, those with 33 valence electrons can be nonmagnetic metals or ferromagnetic semiconductors. In particular, we find that, among the predicted compounds, (Ca,Sr)Ga2Te4 are topologically nontrivial by both the standard density functional theory and hybrid functional calculations. While VSi2P4 is a ferromagnetic semiconductor and TaSi2N4 is a type-I Ising superconductor. Moreover, WSi2P4 is a direct gap semiconductor with peculiar spin-valley properties, which are robust against interlayer interactions. Our study thus provides an effective way of designing septuple-atomic-layer MA2Z4 with unusual electronic properties to draw immediate experimental interest.

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Intercalated architecture of MA2Z4 family layered van der Waals materials with emerging topological, magnetic and superconducting properties.

The quantum Hall effect, discovered by von Klitzing more than 40 years ago, requires strong magnetic fields for its realization. More recently it was found that the effect can also be realized in zero magnetic field as a result of spontaneous time-reversal symmetry breaking. This Colloquium discusses the physics underlying this quantum anomalous Hall effect, the materials it is observed in, and potential applications. 

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Colloquium : Quantum anomalous Hall effect

This work is supported by Saint Petersburg State University project for scientific investigations (ID No. 51126254, https://spin.lab.spbu.ru) and Russian Science Foundation (Grant no. 18-12-00062 in part of the photoemission measurements and 18-12-00169 in part of calculations of topological invariants, investigation of dependence of the electronic spectra on SOC strength, and tight-binding band structure calculations). Russian Foundation for Basic Research (Grant nos. 20-32-70179 and 18-52-06009) and Science Development Foundation under the President of the Republic of Azerbaijan (Grant no. EIF-BGM-4-RFTF-1/2017-21/04/1-M-02) are acknowledged. We also acknowledge the support by the Basque Departamento de Educacion, UPV/EHU (Grant no. IT-756-13), Spanish Ministerio de Ciencia e Innovacion (Grant no. PID2019-103910GB-I00), the Fundamental Research Program of the State Academies of Sciences (line of research III.23.2.9) and Tomsk State University competitiveness improvement program (project no. 8.1.01.2018). I.P.R. acknowledge support from Ministry of Education and Science of the Russian Federation (State Task No. 0721-2020-0033) (tight-binding calculations). The calculations were performed in Donostia International Physics Center and in the Research park of St. Petersburg State University Computing Center (http://cc.spbu.ru).

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Tunable 3D/2D magnetism in the (MnBi2Te4)(Bi2Te3)m topological insulators family

Magnetically doped topological insulators enable the quantum anomalous Hall effect (QAHE), which provides quantized edge states for lossless charge-transport applications1–8. The edge states are hosted by a magnetic energy gap at the Dirac point2, but hitherto all attempts to observe this gap directly have been unsuccessful. Observing the gap is considered to be essential to overcoming the limitations of the QAHE, which so far occurs only at temperatures that are one to two orders of magnitude below the ferromagnetic Curie temperature, TC (ref. 8). Here we use low-temperature photoelectron spectroscopy to unambiguously reveal the magnetic gap of Mn-doped Bi2Te3, which displays ferromagnetic out-of-plane spin texture and opens up only below TC. Surprisingly, our analysis reveals large gap sizes at 1 kelvin of up to 90 millielectronvolts, which is five times larger than theoretically predicted9. Using multiscale analysis we show that this enhancement is due to a remarkable structure modification induced by Mn doping: instead of a disordered impurity system, a self-organized alternating sequence of MnBi2Te4 septuple and Bi2Te3 quintuple layers is formed. This enhances the wavefunction overlap and size of the magnetic gap10. Mn-doped Bi2Se3 (ref. 11) and Mn-doped Sb2Te3 form similar heterostructures, but for Bi2Se3 only a nonmagnetic gap is formed and the magnetization is in the surface plane. This is explained by the smaller spin–orbit interaction by comparison with Mn-doped Bi2Te3. Our findings provide insights that will be crucial in pushing lossless transport in topological insulators towards room-temperature applications. In theory, the anomalous quantum Hall effect is observed in edge channels of topological insulators when there is a magnetic energy gap at the Dirac point; this gap has now been observed by low-temperature photoelectron spectroscopy in Mn-doped Bi2Te3.

Large magnetic gap at the Dirac point in Bi2Te3/MnBi2Te4 heterostructures.

Magnetically doped topological insulators enable the quantum anomalous Hall effect (QAHE) which provides quantized edge states for lossless charge transport applications. The edge states are hosted by a magnetic energy gap at the Dirac point but all attempts to observe it directly have been unsuccessful. The gap size is considered crucial to overcoming the present limitations of the QAHE, which so far occurs only at temperatures one to two orders of magnitude below its principle limit set by the ferromagnetic Curie temperature $T_C$. Here, we use low temperature photoelectron spectroscopy to unambiguously reveal the magnetic gap of Mn-doped Bi$_2$Te$_3$ films, which is present only below $T_C$. Surprisingly, the gap turns out to be $\sim$90 meV wide, which not only exceeds $k_BT$ at room temperature but is also 5 times larger than predicted by density functional theory. By an exhaustive multiscale structure characterization we show that this enhancement is due to a remarkable structure modification induced by Mn doping. Instead of a disordered impurity system, it forms an alternating sequence of septuple and quintuple layer blocks, where Mn is predominantly incorporated in the septuple layers. This self-organized heterostructure substantially enhances the wave-function overlap and the size of the magnetic gap at the Dirac point, as recently predicted. Mn-doped Bi$_2$Se$_3$ forms a similar heterostructure, however, only a large, nonmagnetic gap is formed. We explain both differences based on the higher spin-orbit interaction in Bi$_2$Te$_3$ with the most important consequence of a magnetic anisotropy perpendicular to the films, whereas for Bi$_2$Se$_3$ the spin-orbit interaction it is too weak to overcome the dipole-dipole interaction. Our findings provide crucial insights for pushing the lossless transport properties of topological insulators towards room-temperature applications.

Large magnetic gap at the Dirac point in a Mn-induced Bi$_2$Te$_3$ heterostructure

The structure and morphology of BimSen epitaxial layers with compositions ranging from Bi2Se3 to the Bi1Se1 grown by molecular beam epitaxy with different flux compositions are investigated by transmission electron microscopy, high-resolution x-ray diffraction, and atomic force microscopy. It is shown that the lattice structure changes significantly as a function of the beam flux composition, i.e., Se/BiSe flux ratio that determines the stoichiometry of the layers. A perfect Bi2Se3 phase is formed only with a sufficiently high additional Se flux, whereas Bi1Se1 is obtained when only a BiSe compound source without additional Se is used. For intermediate values of the excess Se flux during growth, Bi2Se3-delta layers are obtained with the Se deficit delta varying between 0 and 1. This Se deficit is accommodated by incorporation of additional Bi-Bi double layers into the Bi2Se3 structure that otherwise exclusively consists of Se-Bi-Se-Bi-Se quintuple layers. While a periodic insertion of such Bi double layers would result in the formation of natural BimSen superlattices, we find that this Bi double-layer insertion is rather stochastic with a high degree of disorder depending on the film composition. Therefore, the structure of such epilayers is better described by a one-dimensional paracrystal model, consisting of disordered sequences of quintuple and double layers rather than by strictly periodic natural superlattices. From detailed analysis of the x-ray diffraction data, we determine the dependence of the lattice parameters a and c and distances of the individual (0001) planes d(j) as a function of composition, evidencing that only the in-plane lattice parameter a shows a linear dependence on composition. The simulation of the diffraction curves with the random stacking paracrystal model yields an excellent agreement with the experimental data and it brings quantitative information on the randomness of the stacking sequence, which is compared to growth modeling using Monte Carlo simulations. The analysis of transmission electron microscopy data furthermore confirms that the Bi-Bi bilayers contain a large amount of vacancies of up to 25%. Conductivity and Hall data confirm that BimSen phases containing Bi-Bi double layers exhibit a rather semimetallic behavior.

Structural disorder of natural BimSen superlattices grown by molecular beam epitaxy

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Large magnetic gap at the Dirac point in Bi2Te3/MnBi2Te4 heterostructures.

Large magnetic gap at the Dirac point in a Mn-induced Bi$_2$Te$_3$ heterostructure

Structural disorder of natural BimSen superlattices grown by molecular beam epitaxy