Axion Insulator Research Articles

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

The intrinsic ferromagnetic (FM) axion insulators and Weyl semimetals (WSMs) with only single pair of Weyl points have drawn intensive attention but so far remain rare and elusive in real materials. Here, we propose a new class of Mn\emph{X}$_{2}$\emph{B}$_{2}$\emph{T}$_{6}$-B (\emph{X}=Ge, Sn, or Pb; \emph{B}=Sb or Bi; \emph{T}=Se or Te) family that is the stable structural form of this system. We find that the Mn\emph{X}$_{2}$\emph{B}$_{2}$\emph{T}$_{6}$-B family has not only the intrinsic FM axion insulators MnGe$_{2}$Bi$_{2}$Te$_{6}$-B, MnSn$_{2}$Bi$_{2}$Te$_{6}$-B, and MnPb$_{2}$Bi$_{2}$Te$_{6}$-B, but also the intrinsic WSM MnSn$_{2}$Sb$_{2}$Te$_{6}$-B with only a single pair of Weyl points. Thus, the Mn\emph{X}$_{2}$\emph{B}$_{2}$\emph{T}$_{6}$-B family can provide an ideal platform to explore the exotic topological magnetoelectric effect and the intrinsic properties related to Weyl points.

Progress on the antiferromagnetic topological insulator MnBi2Te4.

Topological materials, which feature robust surface and/or edge states, have now been a research focus in condensed matter physics. They represent a new class of materials exhibiting nontrivial topological phases, and provide a platform for exploring exotic transport phenomena, such as the quantum anomalous Hall effect and the quantum spin Hall effect. Recently, magnetic topological materials have attracted considerable interests due to the possibility to study the interplay between topological and magnetic orders. In particular, the quantum anomalous Hall and axion insulator phases can be realized in topological insulators with magnetic order. MnBi2Te4, as the first intrinsic antiferromagnetic topological insulator discovered, allows the examination of existing theoretical predictions; it has been extensively studied, and many new discoveries have been made. Here we review the progress made on MnBi2Te4 from both experimental and theoretical aspects. The bulk crystal and magnetic structures are surveyed first, followed by a review of theoretical calculations and experimental probes on the band structure and surface states, and a discussion of various exotic phases that can be realized in MnBi2Te4. The properties of MnBi2Te4 thin films and the corresponding transport studies are then reviewed, with an emphasis on the edge state transport. Possible future research directions in this field are also discussed.

Electronic Spectrum Features under the Transition from Axion Insulator Phase to Quantum Anomalous Hall Effect Phase in an Intrinsic Antiferromagnetic Topological Insulator Thin Film

In this paper, we investigate the electron topological states in a thin film of intrinsic antiferromagnetic topological insulator, focusing on their relationship with the magnetic texture. We consider a model for the film with an even number of septuple-layer blocks, which is subject to transition from the phase of an axion insulator to the phase of quantized Hall conductivity under an external magnetic field. In the continuum approach, we model an effective two-dimensional Hamiltonian of the thin film of a topological insulator with non-collinear magnetization, on the basis of which we obtain the energy spectrum and the Berry curvature. The analysis of topological indices makes it possible to construct a topological phase diagram depending on the parameters of the system and the degree of non-collinearity. For topologically different regions of the diagram, we describe the edge electronic states on the side face of the film. In addition, we investigate the spectrum of one-dimensional states on the domain wall separating domains with the opposite canting angle. We also discuss the results obtained and the experimental situation in thin films of the MnBi2Te4 compound.

Research progress of intrinsic magnetic topological insulator MnBi<sub>2</sub>Te<sub>4</sub>

The interaction between non-trivial topological states and the magnetic order of intrinsic magnetic topological insulators gives rise to various exotic physical properties, including the quantum anomalous Hall effect and axion insulator. These materials possess great potential applications in low-power topological spintronic devices and topological quantum computation. Since the first intrinsic magnetic topological insulator, MnBi<sub>2</sub>Te<sub>4</sub>, was discovered in 2019, this material system has received significant attention from researchers and sparked a research boom. This paper begins with discussing the fundamental properties of MnBi<sub>2</sub>Te<sub>4</sub> and then turns to important research findings related to this intrinsic magnetic topological insulator. Specifically, it focuses on the quantum anomalous Hall effect, axion insulating state, and Majorana zero energy mode exhibited by the MnBi<sub>2</sub>Te<sub>4</sub> series. Furthermore, this paper highlights other research directions and current challenges associated with this material system. Finally, this paper provides a summary and outlook for future research on MnBi<sub>2</sub>Te<sub>4</sub>, aiming to offer valuable references for researchers in related fields.

Quantum transport phenomena in magnetic topological insulators

In the past decade, magnetic topological insulators have been an important focus in condensed matter physics research. The intricate interplay between the nontrivial band topology and spin, orbit, charge, and dimensionality degrees of freedom can give rise to abundant exotic topological quantum states and topological phase transitions. Measuring the transport properties of magnetic topological insulators is a crucial approach to exploring their exotic properties, which is of significant scientific importance in deepening our understanding of topological quantum states. Simultaneously, it also holds substantial potential applications in the development of novel low-power electronic devices. In this work, experimental progress of transport researches of magnetic topological insulators is reviewed, including quantum anomalous Hall effect and topological quantum phase transitions in magnetically doped topological insulators, the quantum anomalous Hall phase, axion insulator phase and Chern insulator phase in intrinsic antiferromagnetic topological insulator MnBi<sub>2</sub>Te<sub>4</sub>, as well as the helical phase emerged from the Chern insulator in pulsed high magnetic fields. Finally, this work analyzes the future direction of development in magnetic topological insulators, and the transport phenomena that have not been understood in these systems, offering an insight into and perspectives on the potential breakthroughs to be achieved in this area of research.

Axion insulators protected by C2T symmetry, their K -theory invariants, and material realizations

Axion insulators are generally understood as magnetic topological insulators whose Chern-Simons axion coupling term is quantized and equal to $\pi$. Inversion and time reversal, or the composition of either one with a rotation or a translation, are symmetries which protect this invariant. In this work, we focus our attention on the composition of a 2-fold rotation with time reversal, and we show that insulators with this symmetry possess a Z2 invariant arising from Atiyah's real K-theory. We call this invariant the K-theory Kane-Mele invariant due to the similarities it has with the Kane-Mele invariant for systems with time-reversal symmetry. Whenever all Chern numbers vanish, we demonstrate that this invariant is equivalent to the Chern-Simons axion coupling, and in the presence of the inversion symmetry, we show how this invariant could be obtained from the eigenvalues of the inversion operator on its fixed points in momentum space. For the general case of non-trivial Chern numbers, the Chern-Simons axion coupling term incorporates information of the K-theory Kane-Mele invariant as well as information regarding bands with non-trival Chern numbers. An explicit formula in terms of K-theory generators is presented for the Chern-Simons axion coupling term, the relation with the K-theory Kane-Mele invariant is explained, and a formula in terms of eigenvalues of the inversion operator is obtained. Using an effective Hamiltonian model and first-principles calculations, we also show that the occurrence of bulk-band inversion and nontrivial K-theory Kane-Mele invariant index can be observed in axion insulators of the pnictides family. In particular, we demonstrate that NpBi can be classified as an axion insulator due to the detection of additional topological indicators such as the quantum spin Hall effect, gapped surface states, surface quantized anomalous Hall effect, and chiral hinge modes

Surface state evolution induced by magnetic order in axion insulator candidate EuIn2As2

Gapping of Dirac surface states through time reversal symmetry breaking may realize the axion insulator state in condensed matter. Despite tremendous efforts, only a few material systems fall in this category of intrinsic magnetic topological insulators (TI). Recent theoretical calculations proposed the antiferromagnetic EuIn$_2$As$_2$ to be a topologically non-trivial magnetic insulator with gapped surface states. Here we use scanning tunneling microscopy and spectroscopy (STM/STS) complemented with density-functional theory (DFT) calculations and modelling to probe the surface electronic states in EuIn$_2$As$_2$. We find a spin-orbit induced bulk gap of ~120 meV located only a few meV above the Fermi energy, within which topological surface states reside. Temperature dependent measurements provide evidence of the partial gapping (~40 meV) of the surface states at low temperatures below the AFM order, which decreases with increasing temperature but remains finite above $T_N$

Quantum interference in a superconductor- MnBi2Te4 -superconductor Josephson junction

We study the transport properties of a Josephson junction consisting of two identical $s$-wave superconductors separated by an even-layer ${\mathrm{MnBi}}_{2}{\mathrm{Te}}_{4}$ (MBT). Using recursive Green's function method, we calculate the supercurrent in the presence of a perpendicular magnetic field and find that its quantum interference exhibits distinct patterns when the MBT is in different magnetic states. In the antiferromagnetic state, the MBT is an axion insulator supporting an extended ``hinge'' supercurrent, which leads to a sinusoidal interference pattern decaying with the field strength. In the ferromagnetic state, the MBT is a Chern insulator and the unbalanced chiral supercurrents on opposite edges give rise to a highly asymmetric interference pattern. If the MBT turns into a metal as the Fermi level is tuned into the conduction band, the interference exhibits a Fraunhofer pattern due to the uniformly distributed bulk supercurrent. Our work unravels a strong indicator to identify different phases in the MBT and can be verified directly by experiments.

Direct Probe of the Ferromagnetic Proximity Effect at the Interface of SnTe/Fe Heterostructure by Polarized Neutron Reflectometry.

Introducing magnetic order into a topological insulator (TI) system has attracted much attention with an expectation of realizing exotic phenomena such as the quantum anomalous Hall effect (QAHE) and axion insulator states. The magnetic proximity effect (MPE) is one of the promising schemes to induce the magnetic order on the surface of a TI without introducing disorder accompanied by doping magnetic impurities in the TI. In this study, we investigate the MPE at the interface of a heterostructure consisting of the topological crystalline insulator (TCI) SnTe and Fe by employing polarized neutron reflectometry. The ferromagnetic order penetrates ∼2.2 nm deep into the SnTe layer from the interface with Fe, which persists up to room temperature. This is induced by the MPE on the surface of the TCI preserving the coherent topological states without introducing the disorder by doping magnetic impurities. This would open up a way for realizing next-generation spintronics and quantum computational devices.

Magnetization tunable Weyl states in EuB6

The interplay between magnetism and topological band structure offers extraordinary opportunities to realize rich exotic magnetic topological phases such as axion insulators, Weyl semimetals, and quantum anomalous Hall insulators, and therefore has attracted fast growing research interest. The rare-earth hexaboride $\mathrm{Eu}{\mathrm{B}}_{6}$ represents an interesting magnetic topological phase with tunable magnetizations along different crystallographic directions, while the correlation with the topological properties remains scarcely explored. In this work, combining magnetotransport measurements and first principles calculations, we demonstrate that $\mathrm{Eu}{\mathrm{B}}_{6}$ exhibits versatile magnetic topological phases along different crystallographic directions, which tightly correlate with the varied magnetizations. Moreover, by virtue of the weak magnetocrystalline anisotropy and the relatively strong coupling between the local magnetization and the conduction electrons, we show that the magnetic ground state of the system can be directly probed by the anisotropy in the magnetotransport properties. Our work thus introduces an excellent platform to study the rich topological phases that are tunable by magnetic orders.

Axionic surface wave in dynamical axion insulators

The electromagnetic response of three-dimensional topological insulators can be described by an effective axion action with a quantized axion field. Once both time-reversal and inversion symmetries are broken, for example, in antiferromagnetic topological insulators, the axion field becomes dynamical along with magnetic fluctuations. The dynamical axion field, when coupled to electromagnetic fields, can lead to rich physical phenomena. Here, based on the modified Maxwell's equations, we reveal the existence of an exotic type of polariton excitation, termed a surface axion polariton, which propagates in the interface between a dynamical axion insulator and a dielectric. When doping the dynamical axion insulator to be metallic, the coexistence of axion-photon coupling and plasmon-photon coupling will further generate a mixed surface axion plasmon polariton. We also propose a Kretschmann-Raether configuration to experimentally detect the surface axion polariton. Our result provides an alternative way to study the axion electrodynamics in condensed matter physics.

Evolution of the Electronic Structure of Ultrathin MnBi2Te4 Films.

Ultrathin films of intrinsic magnetic topological insulator MnBi2Te4 exhibit fascinating quantum properties such as the quantum anomalous Hall effect and the axion insulator state. In this work, we systematically investigate the evolution of the electronic structure of MnBi2Te4 thin films. With increasing film thickness, the electronic structure changes from an insulator type with a large energy gap to one with in-gap topological surface states, which is, however, still in drastic contrast to the bulk material. By surface doping of alkali-metal atoms, a Rashba split band gradually emerges and hybridizes with topological surface states, which not only reconciles the puzzling difference between the electronic structures of the bulk and thin-film MnBi2Te4 but also provides an interesting platform to establish Rashba ferromagnet that is attractive for (quantum) anomalous Hall effect. Our results provide important insights into the understanding and engineering of the intriguing quantum properties of MnBi2Te4 thin films.

Topological magnetoelectric response in ferromagnetic axion insulators.

The topological magnetoelectric effect (TME) is a hallmark response of the topological field theory, which provides a paradigm shift in the study of emergent topological phenomena. However, its direct observation is yet to be realized due to the demanding magnetic configuration required to gap all surface states. Here, we theoretically propose that axion insulators with a simple ferromagnetic configuration, such as the MnBi2Te4/(Bi2Te3)n family, provide an ideal playground to realize the TME. In the designed triangular prism geometry, all the surface states are magnetically gapped. Under a vertical electric field, the surface Hall currents give rise to a nearly half-quantized orbital moment, accompanied by a gapless chiral hinge mode circulating in parallel. Thus, the orbital magnetization from the two topological origins can be easily distinguished by reversing the electric field. Our work paves the way for direct observation of the TME in realistic axion-insulator materials.

Axion insulator states in a topological insulator proximitized to magnetic insulators: A tight-binding characterization

The recent discovery of axion states in materials such as antiferromagnetic topological insulators has boosted investigations of the magnetoelectric response in topological insulators and their promise towards realizing dissipationless topological electronics. In this paper, we develop a tight-binding methodology to explore the emergence of axion states in ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$ in proximity to magnetic insulators on the top and bottom surfaces. The topological protection of the surface states is lifted by a time-reversal-breaking perturbation due to the proximity of a magnetic insulator, and a gap is opened on the surfaces, giving rise to half-quantized Hall conductance and a zero Hall plateau---evidencing an axion insulator state. We developed a real-space tight-binding Hamiltonian for ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$ using first-principles data. Transport properties of the system were obtained within the Landauer-B\"uttiker formalism, and we discuss the creation of axion states through Hall conductance and a zero Hall plateau at the surfaces, as a function of proximitized magnetization and corresponding potentials at the surfaces, as well as the thickness of the topological insulator.

Evolution of surface states of antiferromagnetic topological insulator MnBi2Te4 with tuning the surface magnetization

The interplay between magnetism and topologically non-trivial electronic states is an important subject in condensed matter physics. Recently, the stoichiometric intrinsic magnetic material MnBi2Te4 provides an ideal platform to study the magnetic topological phenomena, such as quantum anomalous Hall effect, axion insulator state, topological magnetoelectric effect. However, it is still controversial whether the topological surface state in the (111) plane is gapped or not. Here, we develop an effective method to study different surface magnetizations based on first-principles calculations. Then we investigate the band dispersions, the Fermi surfaces (FSs), the quasiparticle interferences (QPIs) and the spin texture of topological surface states of MnBi2Te4 with tuning the surface magnetization. We find that the surface magnetization has significant effects on the surface states. Our results also indicate that the symmetry breaking of FSs and QPIs may be a useful way to determine the possible surface magnetization of MnBi2Te4.

Identifying axion insulator by quantized magnetoelectric effect in antiferromagnetic MnBi2Te4 tunnel junction

The authors propose a ${\mathrm{MnBi}}_{2}\phantom{\rule{0}{0ex}}{\mathrm{Te}}_{4}$ tunnel junction to unambiguously identify an axion insulator through the quantized topological magnetoelectric effect.

Dynamical magnetoelectric coupling in axion insulator thin films

Axion insulator is an exotic magnetic topological insulator with zero Chern number but a nonzero quantized Chern-Simons magnetoelectric coupling. A conclusive experimental evidence for axion insulators is still lacking due to the small signal of topological magnetoelectric effect (TME). Here we show that the dynamical magnetoelectric coupling can be induced by the \emph{out-of-plane} surface magnetization dynamics in axion insulator thin films, which further generates a polarization current in the presence of an external magnetic field in the same direction. Such a current is finite in the bulk and increases as the film thickness $d$ decreases, in opposite to TME current which decreases as $d$ decreases. Remarkably, the current in thin films at magnetic resonance is at least ten times larger than that of TME, and thus may serve as a smoking gun signature for axion insulators.

Observation of Magnetism-Induced Topological Edge State in Antiferromagnetic Topological Insulator MnBi4Te7.

Breaking time reversal symmetry in a topological insulator may lead to quantum anomalous Hall effect and axion insulator phase. MnBi4Te7 is a recently discovered antiferromagnetic topological insulator with TN ∼ 12.5 K, which is composed of an alternatively stacked magnetic layer (MnBi2Te4) and nonmagnetic layer (Bi2Te3). By means of scanning tunneling spectroscopy, we clearly observe the electronic state present at a step edge of a magnetic MnBi2Te4 layer but absent at nonmagnetic Bi2Te3 layers at 4.5 K. Furthermore, we find that as the temperature rises above TN the edge state vanishes, while the point defect induced state persists upon an increase in temperature. These results confirm the observation of magnetism-induced edge states. Our analysis based on an axion insulator theory reveals that the nontrivial topological nature of the observed edge state.

Ferromagnetic MnBi4Te7 obtained with low-concentration Sb doping: A promising platform for exploring topological quantum states

The tuning of the magnetic phase, chemical potential, and structure is crucial to observe diverse exotic topological quantum states in $\mathrm{Mn}{\mathrm{Bi}}_{2}{\mathrm{Te}}_{4}{({\mathrm{Bi}}_{2}{\mathrm{Te}}_{3})}_{m}$ $(m=0--3)$. Here we show a ferromagnetic (FM) phase with a chiral crystal structure in $\mathrm{Mn}{({\mathrm{Bi}}_{1\ensuremath{-}x}{\mathrm{Sb}}_{x})}_{4}{\mathrm{Te}}_{7}$, obtained via tuning the growth conditions and Sb concentration. Unlike previously reported $\mathrm{Mn}{({\mathrm{Bi}}_{1\ensuremath{-}x}{\mathrm{Sb}}_{x})}_{4}{\mathrm{Te}}_{7}$, which exhibits FM transitions only at high Sb doping levels, our samples show FM transitions $({T}_{\mathrm{C}}=13.5\phantom{\rule{0.16em}{0ex}}\mathrm{K})$ at 15%--27% doping levels. Furthermore, our single-crystal x-ray-diffraction structure refinements find Sb doping leads to a chiral structure with the space group of $P3$, contrasted with the centrosymmetric $P\overline{3}m1$ crystal structure of the parent compound ${\mathrm{MnBi}}_{4}{\mathrm{Te}}_{7}$. Through angle-resolved photoemission spectroscopy measurements, we also demonstrated that the nontrivial band topology is preserved in the Sb-doped FM samples. Given that the nontrivial band topology of this system remains robust for low Sb doping levels, our success in making FM $\mathrm{Mn}{({\mathrm{Bi}}_{1\ensuremath{-}x}{\mathrm{Sb}}_{x})}_{4}{\mathrm{Te}}_{7}$ with $x=0.15$, 0.175, 0.2, and 0.27 paves the way for realizing the predicted topological quantum states, such as the axion insulator and Weyl semimetals. Additionally, we also observed magnetic glassy behavior in both antiferromagnetic ${\mathrm{MnBi}}_{4}{\mathrm{Te}}_{7}$ and FM $\mathrm{Mn}{({\mathrm{Bi}}_{1\ensuremath{-}x}{\mathrm{Sb}}_{x})}_{4}{\mathrm{Te}}_{7}$ samples, which we believe originates from cluster spin-glass phases coexisting with long-range antiferromagnetic/FM orders. We have also discussed how the antisite Mn ions impact the interlayer magnetic coupling and how FM interlayer coupling is stabilized in this system.

Magnetization-tuned topological quantum phase transition in MnBi2Te4 devices

Recently, the intrinsic magnetic topological insulator MnBi2Te4 has attracted enormous research interest due to the great success in realizing exotic topological quantum states, such as the quantum anomalous Hall effect (QAHE), axion insulator state, high-Chern-number and high-temperature Chern insulator states. One key issue in this field is to effectively manipulate these states and control topological phase transitions. Here, by systematic angle-dependent transport measurements, we reveal a magnetization-tuned topological quantum phase transition from Chern insulator to magnetic insulator with gapped Dirac surface states in MnBi2Te4 devices. Specifically, as the magnetic field is tilted away from the out-of-plane direction by around 40-60 degrees, the Hall resistance deviates from the quantization value and a colossal, anisotropic magnetoresistance is detected. The theoretical analyses based on modified Landauer-Buttiker formalism show that the field-tilt-driven switching from ferromagnetic state to canted antiferromagnetic state induces a topological quantum phase transition from Chern insulator to magnetic insulator with gapped Dirac surface states in MnBi2Te4 devices. Our work provides an efficient means for modulating topological quantum states and topological quantum phase transitions.