Axion insulators are magnetic topological insulators in which the non-trivial $\mathbb{Z}_2$ index is protected by inversion symmetry instead of time-reversal symmetry. The naturally gapped surfaces of axion insulators give rise to a half-quantized surface anomalous Hall conductivity (AHC), but the sign of the surface AHC cannot be determined from topological arguments. In this paper, we consider topological phenomena at the surface of an axion insulator. To be explicit, we construct a minimal tight-binding model on the pyrochlore lattice and investigate the all-in-all-out (AIAO) and ferromagnetic (FM) spin configurations. We also implement a recently proposed approach for calculating the surface AHC directly, which allows us to explore how the interplay between surface termination and magnetic ordering determines the sign of the half-quantized surface AHC. In the case of AIAO ordering, we show that it is possible to construct a topological state with no protected metallic states on boundaries of any dimension (surfaces, hinges, or corners), although chiral hinge modes do occur for many surface configurations. In the FM case, rotation of the magnetization by an external field offers a promising means of control of chiral hinge modes, which can also appear on surface steps or where bulk domain walls emerge at the surface.
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…
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…
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…
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…
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）
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…
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…
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…
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 …
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…
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…
A universal mechanism to generate chiral hinge states in the ferromagnetic axion insulator phase is proposed, which leads to an exotic transport phenomena, the quantum anomalous Hall effect (QAHE) on some particular surfaces determined by both the crystalline symmetry and the magnetization direction. A realistic material system Sm doped $\text{Bi}_2\text{Se}_3$ is then proposed to realize such exotic hinge states by combing the first principle calculations and the Green's function techniques. A physically accessible way to manipulate the surface QAHE is also proposed, which makes it very diffe…
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…
Realizing axion insulator state with a uniform magnetization considerably facilitates experimental explorations of the intriguing topological magnetoelectric effect, a hallmark of three-dimensional (3D) topological insulators (TIs). Through density functional theory calculations and four-band model studies, we find that magnetic ions Cr3+ in monolayer CrI3 and Mn2+ in septuple-layer MnBi2Se4 have opposite exchange couplings to the topological surface states of 3D TI Bi2Se3. As an exciting result of such opposite exchange couplings, axion insulator state is realized by a uniform magnetization i…
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…
Detecting axion dynamics on the surface of magnetic topological insulators
University of California, Berkeley（加州大学伯克利分校）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；Kavli Institute for Theoretical Physics, University of California, Santa Barbara（加州大学圣塔芭芭拉分校卡弗里理论物理研究所）
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…
Based on density functional calculation with LDA+U method, we propose that hypothetical Osmium compounds such as CaOs2O4 and SrOs2O4 can be stabilized in the geometrically frustrated spinel crystal structure. They also show some exotic electronic and magnetic properties in a reasonable range of on-site Coulomb correlation U such as ferromagnetism and orbital magnetoelectric effect characteristic to Axion electrodynamics. Other electronic phases including 3D Dirac metal and Mott insulator exist and would make perspective 5d spinels ideal for applications.
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 tet…
Beijing National Laboratory for Condensed Matter Physics, and Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所）；University of Chinese Academy of Sciences（中国科学院大学）；Condensed Matter Physics Data Center, Chinese Academy of Sciences（中国科学院凝聚态物理数据中心）；Department of Physics, Hong Kong University of Science and Technology（香港科技大学物理系）
Topological insulators have been proposed to be best characterized as bulk magnetoelectric materials that show response functions quantized in terms of fundamental physical constants. Here we lower the chemical potential of three-dimensional (3D) Bi$_2$Se$_3$ films to $\sim$ 30 meV above the Dirac point, and probe their low-energy electrodynamic response in the presence of magnetic fields with high-precision time-domain terahertz polarimetry. For fields higher than 5 T, we observed quantized Faraday and Kerr rotations, whereas the DC transport is still semi-classical. A non-trivial Berry phase…
We study theoretically the interplay between magnetism and topology in three-dimensional HgTe/MnTe superlattices stacked along the (001) axis. Our results show the evolution of the magnetic topological phases with respect to the magnetic configurations. An axion insulator phase is observed for the antiferromagnetic order with the out-of-plane Néel vector direction below a critical thickness of MnTe, which is the ground state amongst all magnetic configurations. Defining $T$ as the time-reversal symmetry, this axion insulator phase is protected by a magnetic two-fold rotational symmetry $C_2{\c…
Topological insulator with antiferromagnetic order can serve as an ideal platform for the realization of axion electrodynamics. In this paper, we report a systematic study of the axion topological insulator candidate EuIn$_2$As$_2$. A linear energy dispersion across the Fermi level confirms the existence of the proposed hole-type Fermi pocket. Spin-flop transitions occur with magnetic fields applied within the $ab$-plane while are absent for fields parallel to the $c$-axis. Anisotropic magnetic phase diagrams are observed and the orientation of the ground magnetic moment is found to be within …
EuIn2Sb2: epitaxially stabilized axion insulator candidate with strong spin-orbit coupling
Institute of Science Tokyo（东京科学大学）；RIKEN Center for Emergent Matter Science (CEMS)（理化学研究所 新兴物质科学中心）；Toyota Physical and Chemical Research Institute（丰田物理化学研究所）
Based on first-principles calculations and symmetry analysis, we propose that EuIn$_2$As$_2$ is a long awaited axion insulator with antiferromagnetic (AFM) long range order. Characterized by the parity-based invariant $\mathbb Z_4=2$, the topological magneto-electric effect is quantized with $θ=π$ in the bulk, with a band gap as large as 0.1 eV. When the staggered magnetic moment of the AFM phase is along $a/b$ axis, it's also a TCI phase. Gapless surface states emerge on (100), (010) and (001) surfaces, protected by mirror symmetries (nonzero mirror Chern numbers). When the magnetic moment is…
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 MnBi$_{2}$Te$_{4}$ family has attracted significant attention due to its rich topological states such as the quantum anomalous Hall (QAH) insulator state, the axion insulator state, and the magnetic Weyl semimetal state. Nevertheless, the intrinsic antiferromagnetic (AFM) interlayer coupling in MnBi$_{2}$Te$_{4}$ partly hinders the realization of "high-temperature" QAH effect. Here, by using first-principles electronic structure calculations, we design a new class of materials Mn\emph{X}$_{2}$\emph{B}$_{2}$\emph{T}$_{6}$ (\emph{X}=Ge, Sn, or Pb; \emph{B}=Sb or Bi; \emph{T}=Se or Te) based …
The higher order topological insulator (HOTI) has enticed enormous research interests owing to its novelty in supporting gapless states along the hinges of the crystal. Despite several theoretical predictions, enough experimental confirmation of HOTI state in crystalline solids is still lacking. It has been well known that interplay between topology and magnetism can give rise to various magnetic topological states including HOTI and Axion insulator states. Here using the high-resolution angle-resolved photoemission spectroscopy (ARPES) combined with the first-principles calculations, we repor…
The magnetic topological materials have attracted significant attention due to their potential realization of variety of novel quantum phenomena. EuIn$_2$As$_2$ has recently been theoretically recognized as a long awaited intrinsic antiferromagnetic bulk axion insulator. However, the experimental study on transport properties arising from the topological states in this material is scarce. In this paper, we perform the detailed magnetoresistance (MR) and Hall measurements to study the magnetotransport properties of this material. We find that the transport is strongly influenced by the spin con…
Modification of the gap at the Dirac point (DP) in antiferromagnetic (AFM) axion topological insulator MnBi$_2$Te$_4$ and its electronic and spin structure has been studied by angle- and spin-resolved photoemission spectroscopy (ARPES) under laser excitation with variation of temperature (9-35~K), light polarization and photon energy. We have distinguished both a large (62-67~meV) and a reduced (15-18~meV) gap at the DP in the ARPES dispersions, which remains open above the Néel temperature ($T_\mathrm{N}=24.5$~K). We propose that the gap above $T_\mathrm{N}$ remains open due to short-range ma…
The second-order nonlinear optical (NLO) processes, such as the photogalvanic effect and second-order harmonic generation (SHG), play crucial roles in probing and controlling light-matter interactions for energy and device applications. To date, most studies of second-order NLO processes focus on materials with broken spatial inversion symmetry, such as proper ferroelectrics and noncentrosymmetric Weyl semimetals. Nevertheless, inversion symmetry of Shubnikov groups can be broken via spin-ordering in centrosymmetric crystals. Unfortunately, these materials are less common, and their NLO respon…
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 …
Layer-dependent antiferromagnetic Chern and axion insulating states in UOTe
Howard University（霍华德大学）；S. N. Bose National Centre for Basic Sciences（S.N. Bose基础科学国家中心）；Harvard University（哈佛大学）；Washington University in St. Louis（圣路易斯华盛顿大学）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）；University of Michigan（密歇根大学）；Northeastern University（东北大学）
The non-trivial third homotopy class of three-dimensional topological insulators leads to quantized, magneto-electric coefficient or axion angle $θ= n π$, with $n \in \mathbb{Z}$. In Part I, we developed tools for computing $n$ from a staggered symmetry-indicator $κ_{AF,j}$ and Wilson loops of non-Abelian, Berry connection in momentum-space, which clearly distinguished between magneto-electrically trivial ($n=0$), and non-trivial ($n=2s$) topological crystalline insulators. In this work, we perform $\mathbb{Z}$-classification of real-space, topological response or $θ$ by carrying out thought e…
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…
Intrinsic magnetic topological insulators from MnBi2Te4 family promise realization of axion insulator, a theoretically predicted state of matter capable to mutually convert electric and magnetic signals, and whose properties are reminiscent of cosmological axion dark matter. Here we report the first experimental realization of room-temperature axionic single-spin switch in topological metamaterial MnBi2Te3. The metamaterial was in situ created in the scanning tunneling microscopy (STM) experiment by removing Te atomic layer from MnBi2Te4 surface using STM tip. Room temperature STM study of MnB…
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 constituted of alternatively stacked magnetic layer (MnBi2Te4) and non-magnetic 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 non-magnetic Bi2Te3 layers at 4.5 K. Furthermore, we find that as the temperature rises above TN, the edge state vanishes, wh…
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…
Realization of the quantum anomalous Hall effect and axion electrodynamics in topological materials are among the paradigmatic phenomena in condensed matter physics. Recently, signatures of both phases are observed to exist in thin films of MnBi$_2$Te$_4$, a stoichiometric antiferromagnetic topological insulator. Direct evidence of the bulk topological magnetoelectric response in an axion insulator requires an energy gap at its topological surface state (TSS). However, independent spectroscopic experiments revealed that such a surface gap is absent, or much smaller than previously thought, in …
Gapless criteria that can efficiently determine whether a crystal is gapless or not are particularly useful for identifying topological semimetals. In this work, we propose a sufficient gapless criterion for three-dimensional non-interacting crystals, based on the simplified expressions for the bulk average value of the static axion field. The brief logic is that two different simplified expressions give the same value in an insulator, and thus the gapless phase can be detected by the mismatch of them. We demonstrate the effectiveness of the gapless criterion in the magnetic systems with space…
A striking feature of time reversal symmetry (TRS) protected topological insulators (TIs) is that they are characterized by a half integer quantum Hall effect on the boundary when the surface states are gapped by time reversal breaking perturbations. While time reversal symmetry (TRS) protected TIs have become increasingly under control, magnetic analogs are still largely unexplored territories with novel rich structures. In particular, topological magnetic insulators can also host a quantized axion term in the presence of lattice symmetries. Since these symmetries are naturally broken on the …
We search for dynamical magnetoelectric phenomena in three-dimensional correlated systems with spin-orbit coupling. We focus on the antiferromagnetic insulator phases where the dynamical axion field is realized by the fluctuation of the antiferromagnetic order parameter. It is shown that the dynamical chiral magnetic effect, an alternating current generation by magnetic fields, emerges due to the time dependences of the order parameter such as antiferromagnetic resonance. It is also shown that the anomalous Hall effect arises due to the spatial variations of the order parameter such as antifer…
Introducing magnetism into topological insulators breaks time-reversal symmetry, and the magnetic exchange interaction can open a gap in the otherwise gapless topological surface states. This allows various novel topological quantum states to be generated, including the quantum anomalous Hall effect (QAHE) and axion insulator states. Magnetic doping and magnetic proximity are viewed as being useful means of exploring the interaction between topology and magnetism. However, the inhomogeneity of magnetic doping leads to complicated magnetic ordering and small exchange gaps, and consequently the …
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$_{2}$T…
Topological insulators are new states of matter with a bulk gap and robust gapless surface states protected by time-reversal symmetry. When time-reversal symmetry is broken, the surface states are gapped, which induces a topological response of the system to electromagnetic field--the topological magnetoelectric effect. In this paper we study the behavior of topological surface states in heterostructures formed by a topological insulator and a magnetic insulator. Several magnetic insulators with compatible magnetic structure and relatively good lattice matching with topological insulators ${\r…
We present a magneto-optical study of the three-dimensional topological insulator, strained HgTe using a technique which capitalizes on advantages of time-domain spectroscopy to amplify the signal from the surface states. This measurement delivers valuable and precise information regarding the surface state dispersion within <1 meV of the Fermi level. The technique is highly suitable for the pursuit of the topological magnetoelectric effect and axion electrodynamics.
Topological magnets are a new family of quantum materials providing great potential to realize emergent phenomena, such as quantum anomalous Hall effect and axion-insulator state. Here we present our discovery that stoichiometric ferromagnet MnBi8Te13 with natural heterostructure MnBi2Te4-(Bi2Te3)3 is an unprecedented half-magnetic topological insulator, with the magnetization existing at the MnBi2Te4 surface but not at the opposite surface terminated by triple Bi2Te3 layers. Our angle-resolved photoemission spectroscopy measurements unveil a massive Dirac gap at the MnBi2Te4 surface, and gapl…
