Topological states of quantum matter have attracted great attention in condensed matter physics and materials science. The study of time-reversal-invariant (TRI) topological states in quantum materials has made tremendous progress in both theories and experiments. As a great success, thousands of TRI topological materials are predicted through sweeping search. Richer exotic phenomena are expected to appear in magnetic topological materials because of varied magnetic configurations, but this study falls much behind due to the complex magnetic structures and transitions. Here, we predict the tetradymite-type compound MnBi$_2$Te$_4$ and its related materials host interesting magnetic topological states. The magnetic ground state of MnBi$_2$Te$_4$ is an antiferromagnetic phase which leads to an antiferromagetic topological insulator state with a large topologically non-trivial energy gap ($\sim$0.2~eV). It is the parent state for the axion state, which has gapped bulk and surface states, and quantized topological magnetoelectric effect. The ferromagnetic phase of MnBi$_2$Te$_4$ leads to an ideal minimal type-II Weyl semimetal with two Weyl points accompanied by one hole-type and one electron-type Fermi pocket at the Fermi level, which has never been discovered elsewhere. We further present a simple and unified continuum model to capture the salient topological features of this kind of materials.
Intrinsic magnetic topological insulator ${\mathrm{MnBi}}_{2}{\mathrm{Te}}_{4}$ is believed to be an axion insulator in its antiferromagnetic ground state. However, direct identification of axion insulators remains experimentally elusive because the observed vanishing Hall resistance, while indicating the onset of the axion field, is inadequate to distinguish the system from a trivial normal insulator. Using numerical Green's functions, we theoretically demonstrate the quantized magnetoelectric current in a tunnel junction of atomically thin ${\mathrm{MnBi}}_{2}{\mathrm{Te}}_{4}$ sandwiched be…
Axion, first postulated as a hypothetical particle in high-energy physics, is now extended to describe a novel topological magnetoelectric effect derived from the Chern-Simons theory in condensed matter systems. The recent discovered intrinsic magnetic topological insulators MnBi2Te4 and its derivatives have attracted great attention because of their potential as a material platform to realize such a quantized axion field. Since the magnetic exchange gap can bring the half-quantized anomalous Hall effect at the surface, an axion insulator manifests as quantum anomalous Hall and zero Hall plate…
More than forty years ago, axion was postulated as an elementary particle with a low mass and weak interaction in particle physics to solve the strong $\mathcal{CP}$ (charge conjugation and parity) puzzle. Axions are also considered as a possible component of dark matter of the universe. However, the existence of axions in nature has not been confirmed. Interestingly, axions arise as pseudoscalar fields derived from the Chern-Simons theory in condensed matter physics. In antiferromagnetic insulators, the axion field can become dynamical induced by spin-wave excitations and exhibits rich exotic…
Chromia is a magnetoelectric insulator whose bulk magnetoelectric susceptibility contains a non-zero pseudoscalar component, like that present in magnetized topological insulators. We reveal the dynamic axion field of chromia by measuring the temperature dependence of its non-linear magnetoelectric response using a lock-in technique with an AC electric field stimulus. The electric field dependence, magnetic field dependence, and temperature behavior of the second harmonic lock-in signal are all in agreement with theoretical expectations. Our results demonstrate that chromia magnetoelectric res…
Axion is a hypothetical elementary particle which was initially postulated to solve the charge conjugation-parity problem in particle physics. Interestingly, the axion state has emerged in effective theory of topological insulators and has attracted extensive attention in condensed matter physics. Time-reversal or inversion symmetry constrains the axion field $θ$ to be quantized. When both the time-reversal and inversion symmetries are broken by, say, an antiferromagnetic order, the axion field $θ$ could become unquantized and dynamical along with magnetic fluctuations, which is termed the dyn…
We study the critical properties of the in-plane magnetic field-induced quantum anomalous Hall (QAH) plateau transition in axion insulators. Take even septuple layer film MnBi$_2$Te$_4$ as a concrete example, we find the mirror symmetry breaking from in-plane field could induce axion insulator to QAH insulator transition, which belongs to generic integer quantum Hall plateau transition. The chiral Majorana fermion does not necessarily emerge at the QAH plateau transition in MnBi$_2$Te$_4$ due to strong exchange field, but may be quite feasible in its descendent materials MnBi$_4$Te$_7$ and MnB…
The orbital motion of electrons in a three-dimensional solid can generate a pseudoscalar magnetoelectric coupling $θ$, a fact we derive for the single-particle case using a recent theory of polarization in weakly inhomogeneous materials. This polarizability $θ$ is the same parameter that appears in the "axion electrodynamics" Lagrangian $Δ{\cal L}_{EM} = (θe^2 / 2 πh) {\bf E} \cdot {\bf B}$, which is known to describe the unusual magnetoelectric properties of the three-dimensional topological insulator ($θ=π$). We compute $θ$ for a simple model that accesses the topological insulator and discu…
Being the first intrinsic antiferromagnetic(AFM) topological insulator(TI), MnBi2Te4 is argued to be a topological axion state in its even-layer form due to the antiparallel magnetization between the top and bottom layers. Here we combine both transport and scanning microwave impedance microscopy(sMIM) to investigate such axion state in atomically thin MnBi2Te4 with even-layer thickness at zero magnetic field. While transport measurements show a zero Hall plateau signaturing the axion state, sMIM uncovers an unexpected edge state raising questions regarding the nature of the 'axion state'. Bas…
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 the surface states. Here, we theoretically propose that the axion insulators with a simple ferromagnetic configuration, such as MnBi2Te4/(Bi2Te3)n family, provide an ideal playground to realize TME. In a designed triangular prism geometry, all the surface states are magnetically gapped. Under a vertical …
We propose an intrinsic nonlinear spin magnetoelectric effect in magnetic materials, offering the potential of all-electric control of spin degree of freedom in centrosymmetric magnets, which reside outside of the current paradigm based on linear spin response. We reveal the band geometric origin of this effect in the momentum and magnetization space Berry connection polarizabilities, and clarify its symmetry characters. As an intrinsic effect, it is determined solely by the material's band structure and represents a material characteristic. Combining our theory with first-principles calculati…
The axion is a hypothetical but experimentally undetected particle. Recently, the antiferromagnetic topological insulator MnBi$_2$Te$_4$ has been predicted to host the axion insulator, but the experimental evidence remains elusive. Specifically, the axion insulator is believed to carry "half-quantized" chiral currents running antiparallel on its top and bottom surfaces. However, it is challenging to measure precisely the half-quantization. Here, we propose a nonlocal surface transport device, in which the axion insulator can be distinguished from normal insulators without a precise measurement…
The interplay of magnetism and topology is a key research subject in condensed matter physics and material science, which offers great opportunities to explore emerging new physics, like the quantum anomalous Hall (QAH) effect, axion electrodynamics and Majorana fermions. However, these exotic physical effects have rarely been realized in experiment, due to the lacking of suitable working materials. Here we predict that van der Waals layered MnBi$_2$Te$_4$-family materials show two-dimensional (2D) ferromagnetism in the single layer and three-dimensional (3D) $A$-type antiferromagnetism in the…
The interplay of non-trivial band topology and magnetism gives rise to a series of exotic quantum phenomena, such as the emergent quantum anomalous Hall (QAH) effect and topological magnetoelectric effect. Many of these quantum phenomena have local manifestations when the global symmetry is broken. Here, we report local signatures of the thickness dependent topology in intrinsic magnetic topological insulator MnBi$_2$Te$_4$(MBT), using scanning tunneling microscopy and spectroscopy on molecular beam epitaxy grown MBT thin films. A thickness-dependent band gap with an oscillatory feature is rev…
The intricate interplay between nontrivial topology and magnetism in two-dimensional (2D) materials has led to the emergence of many novel phenomena and functionalities. An outstanding example is the quantum anomalous Hall (QAH) effect, which was realized in magnetically doped topological insulators (TIs) in the absence of magnetic field. Recently, the layered van der Waals compound MnBi2Te4 has been theoretically predicted and experimentally verified to be a TI with interlayer antiferromagnetic (AFM) order. It is a rare stoichiometric material with coexisting topology and magnetism, thus repr…
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 bul…
Three-dimensional topological insulators or axion insulators exhibit the topological magnetoelectric effect, which is isotropic with a universal coefficient of proportionality quantized in units of $e^2/2h$. Here we study the finite-size effect of topological magnetoelectric effect, and find the magnetoelectric coefficients are anisotropic, namely $α_{xx}\neqα_{zz}$. Both of them are shown to converge to a quantized value when the thickness of topological insulator film $d$ increases reaching the three-dimensional bulk limit. The nonzero value of $(α_{xx}-α_{zz})\propto1/d$ could be measured b…
The recent discovery of antiferromagnetic (AFM) topological insulator (TI) MnBi$_2$Te$_4$ has triggered great research efforts on exploring novel magnetic topological physics. Based on first-principles calculations, we find that the manipulation of magnetic orientation and order not only significantly affects material symmetries and orbital hybridizations, but also results in variant new magnetic topological phases in MnBi$_2$Te$_4$. We thus predict a series of unusual topological quantum phase transitions that are magnetically controllable in the material, including phase transitions from AFM…
Floquet engineering is an important way to manipulate the electronic states of condensed matter physics. Recently, the discovery of the magnetic topological insulator MnBi$_2$Te$_4$ and its family provided a valuable platform to study magnetic topological phenomena, such as, the quantum anomalous Hall effect, the axion insulator state and the topological magnetoelectric effect. In this work, based on the effective model and first-principles calculations in combination with the Floquet theory, we reveal that the circularly polarized light (CPL) induces the sign reversal of the Chern number of o…
The intrinsic antiferromagnetic topological insulator MnBi2Te4 provides an ideal platform for exploring exotic topological quantum phenomena. Recently, the Chern insulator and axion insulator phases have been realized in few-layer MnBi2Te4 devices at low magnetic field regime. However, the fate of MnBi2Te4 in high magnetic field has never been explored in experiment. In this work, we report transport studies of exfoliated MnBi2Te4 flakes in pulsed magnetic fields up to 61.5 T. In the high-field limit, the Chern insulator phase with Chern number C = -1 evolves into a robust zero Hall resistance…
The recently discovered antiferromagnetic topological insulator MnBi$_2$Te$_4$ hosts a variety of exotic topological quantum phases such as the axion insulator and Chern insulator states. Here we report systematic gate voltage dependent magneto transport studies in six septuple-layer MnBi$_2$Te$_4$. In p-type carrier regime, we observe positive linear magnetoresistance when MnBi$_2$Te$_4$ is polarized in the ferromagnetic state by an out-of-plane magnetic field. Whereas in n-type regime, distinct negative magnetoresistance behaviors are observed. The magnetoresistance in both regimes is highly…
A quantized version of the magnetoelectric effect, known as the topological magnetoelectric effect, can exist in a time-reversal invariant topological insulator with all its surface states gapped out by magnetism. This topological phase, called the axion insulator phase, has been theoretically proposed but is still lack of conclusive experimental evidence due to the small signal of topological magnetoelectric effect. In this work, we propose that the dynamical in-plane magnetization in an axion insulator can generate a "pseudo-electric field", which acts on the surface state of topological ins…
The combination of topology and magnetism is attractive to produce exotic quantum matters, such as the quantum anomalous Hall state, axion insulators and the magnetic Weyl semimetals. MnBi2Te4, as an intrinsic magnetic topological insulator, provides a platform for the realization of various topological phases. Here we report the intermediate Hall steps in the magnetic hysteresis of MnBi2Te4, where four distinguishable magnetic memory states at zero magnetic field are revealed. The gate and temperature dependence of the magnetic intermediate states indicates the noncollinear spin structure in …
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 w…
Even-Odd Layer-Dependent Exchange Bias Effect in MnBi2Te4 Chern Insulator Devices
The Pennsylvania State University（宾夕法尼亚州立大学）；Nanjing University（南京大学）；Nankai University（南开大学）；National Institute for Materials Science（国立材料研究所）；Oak Ridge National Laboratory（橡树岭国家实验室）；University of California, Riverside（加州大学河滨分校）
The dynamical axion field is a new state of quantum matter where the magnetoelectric response couples strongly to its low-energy magnetic fluctuations. It is fundamentally different from an axion insulator with a static quantized magnetoelectric response. The dynamical axion field exhibits many exotic phenomena such as axionic polariton and axion instability. However, these effects have not been experimentally confirmed due to the lack of proper topological magnetic materials. Here by combining analytic models and first-principles calculations, we predict a series of van der Waal layered Mn$_2…
Recently, the search for an axion insulator state in the ferromagnet-3D topological insulator (TI) heterostructure and $\mathrm{MnBi_2Te_4}$ has attracted intensive interest. However, its detection remains difficult in experiments. We systematically investigate the disorder-induced phase transition of the axion insulator state in a 3D TI with antiparallel magnetization alignment surfaces. It is found that there exists a 2D disorder-induced phase transition which shares the same universality class with the quantum Hall plateau to plateau transition. Then, we provide a phenomenological theory wh…
Topological insulator (TI) thin films with surface magnetism are expected to exhibit a quantized anomalous Hall effect (QAHE) when the magnetizations on the top and bottom surfaces are parallel, and a quantized topological magnetoelectric (QTME) response when the magnetizations have opposing orientations (axion insulator phase) and the films are sufficiently thick. We present a unified picture of both effects that associates deviations from exact quantization of the QTME caused by finite thickness with non-locality in the side-wall current response function. Using realistic tight-binding model…
The quantum anomalous Hall (QAH) effect has recently been realized in thin films of intrinsic magnetic topological insulators (IMTIs) like MnBi$_2$Te$_4$. Here we point out that that the QAH gaps of these IMTIs can be optimized, and that both axion insulator/semimetal and Chern insulator/semimetal transitions can be driven by electrical gate fields on the $\sim 10$ meV/nm scale. This effect is described by combining a simplified coupled-Dirac-cone model of multilayer thin films with Schr{ö}dinger-Poisson self-consistent-field equations.
Topological surface states with intrinsic magnetic ordering in the MnBi$_2$Te$_4$(Bi$_2$Te$_3$)$_n$ compounds have been predicted to host rich topological phenomena including quantized anomalous Hall effect and axion insulator state. Here we use scanning tunneling microscopy to image the surface Dirac fermions in MnBi$_2$Te$_4$ and MnBi$_4$Te$_7$. We have determined the energy dispersion and helical spin texture of the surface states through quasiparticle interference patterns far above Dirac energy, which confirms its topological nature. Approaching the Dirac point, the native defects in the …
We propose to use ferromagnetic insulator MnBi2Se4/Bi2Se3/antiferromagnetic insulator Mn2Bi2Se5 heterostructures for the realization of the axion insulator state. Importantly, the axion insulator state in such heterostructures only depends on the magnetization of the ferromagnetic insulator and hence can be observed in a wide range of external magnetic field. Using density functional calculations and model Hamiltonian simulations, we find that the top and bottom surfaces have opposite half-quantum Hall conductance, with a sizable global spin gap of 5.1 meV opened for the topological surface st…
The interplay between magnetism and non-trivial topology in magnetic topological insulators (MTI) is expected to give rise to a variety of exotic topological quantum phenomena, such as the quantum anomalous Hall (QAH) effect and the topological axion states. A key to assessing these novel properties is to tune the Fermi level in the exchange gap of the Dirac surface band. MnBi$_2$Te$_4$ possesses non-trivial band topology with intrinsic antiferromagnetic (AFM) state that can enable all of these quantum states, however, highly electron-doped nature of the MnBi$_2$Te$_4$ crystals obstructs the e…
We propose MnBi$_{2n}$Te$_{3n+1}$ as a magnetically tunable platform for realizing various symmetry-protected higher-order topology. Its canted antiferromagnetic phase can host exotic topological surface states with a Möbius twist that are protected by nonsymmorphic symmetry. Moreover, opposite surfaces hosting Möbius fermions are connected by one-dimensional chiral hinge modes, which offers the first material candidate of a higher-order topological Möbius insulator. We uncover a general mechanism to feasibly induce this exotic physics by applying a small in-plane magnetic field to the antifer…
One of the intriguing properties characteristic to three-dimensional topological materials is the topological magnetoelectric phenomena arising from a topological term called the $θ$ term. Such magnetoelectric phenomena are often termed the axion electrodynamics, since the $θ$ term has exactly the same form as the action describing the coupling between a hypothetical elementary particle, axion, and a photon. The axion was proposed about forty years ago to solve the so-called strong CP problem in quantum chromodynamics, and is now considered as a candidate for dark matter. In this tutorial, we …
Unveiling the impact of anti-site defects in magnetic transitions of few-layer MnBi2Te4 by operando heating
State Key Laboratory of Surface Physics and Department of Physics, Fudan University（复旦大学表面物理国家重点实验室和物理系）；Institute for Nanoelectronic Devices and Quantum Computing, Fudan University（复旦大学纳米电子器件与量子计算研究所）；Key Laboratory of Micro and Nano Photonic Structures (MOE), Fudan University（复旦大学微纳光子结构教育部重点实验室）；Shanghai Branch, Hefei National Laboratory（合肥国家实验室上海分部）；Shanghai Research Center for Quantum Sciences（上海量子科学研究中心）；New Cornerstone Science Laboratory（新基石科学实验室）
Exploration of novel electromagnetic phenomena is a subject of great interest in topological quantum materials. One of the unprecedented effects to be experimentally verified is topological magnetoelectric (TME) effect originating from an unusual coupling of electric and magnetic fields in materials. A magnetic heterostructure of topological insulator (TI) hosts such an exotic magnetoelectric coupling and can be expected to realize the TME effect as an axion insulator. Here we designed a magnetic TI with tricolor structure where a non-magnetic layer of (Bi, Sb)2Te3 is sandwiched by a soft ferr…
Axion Insulator State in Hundred-Nanometer-Thick Magnetic Topological Insulator Sandwich Heterostructures
Department of Physics, The Pennsylvania State University（宾夕法尼亚州立大学物理系）；Department of Physics, Hong Kong University of Science and Technology（香港科技大学物理系）；Materials Research Institute, The Pennsylvania State University（宾夕法尼亚州立大学材料研究所）
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 str…
We study theoretically a three-dimensional correlated and spin-orbit coupled system, the half-filled extended Fu-Kane-Mele-Hubbard model on a diamond lattice, focusing on the topological magnetoelectric response of the antiferromagnetic insulator phase. In the antiferromagnetic insulator phase, the Dirac-like low-energy effective Hamiltonian is obtained. Then the theta term, which results in the magnetoelectric response, is derived as a consequence of the chiral anomaly. The realization of the dynamical axion field in our model is discussed. The relation with a symmetry broken phase induced by…
The 'magnetoelectric effect' arises from the coupling between magnetic and electric properties in materials. The Z2 invariant of topological insulators (TIs) leads to a quantized version of this phenomenon, known as the topological magnetoelectric (TME) effect. This effect can be realized in a new topological phase called an 'axion insulator' whose surface states are all gapped but the interior still obeys time reversal symmetry. We demonstrate such a phase using electrical transport measurements in a quantum anomalous Hall (QAH) sandwich heterostructure, in which two compositionally different…
The topological surface states of magnetic topological systems, such as Weyl semimetals and axion insulators, are associated with unconventional transport properties such as nonzero or half-quantized surface anomalous Hall effect. Here we study the surface anomalous Hall effect and its spectral signatures in different magnetic topological phases using both model Hamiltonian and first-principles calculations. We demonstrate that by tailoring the magnetization and interlayer electron hopping, a rich three-dimensional topological phase diagram can be established, including three types of topologi…
