Observation of the universal magnetoelectric effect in a 3D topological insulator

Layer Control of Magneto-Optical Effects and Their Quantization in Spin-Valley Splitting Antiferromagnets

Magneto-optical effects (MOE), interfacing the fundamental interplay between magnetism and light, have served as a powerful probe for magnetic order, band topology, and valley index. Here, based on multiferroic and topological bilayer antiferromagnets (AFMs), we propose a layer control of MOE (L-MOE), which is created and annihilated by layer-stacking or an electric field effect. The key character of L-MOE is the sign-reversible response controlled by ferroelectric polarization, the Néel vector, or the electric field direction. Moreover, the sign-reversible L-MOE can be quantized in topologically insulating AFMs. We reveal that the switchable L-MOE originates from the combined contributions of spin-conserving and spin-flip interband transitions in spin-valley splitting AFMs, a phenomenon not observed in conventional AFMs. Our findings bridge the ancient MOE to the emergent realms of layertronics, valleytronics, and multiferroics and may hold immense potential in these fields.

Optical Tellegen metamaterial with spontaneous magnetization

The nonreciprocal magnetoelectric effect, also known as the Tellegen effect, promises a number of groundbreaking phenomena connected to fundamental (e.g., electrodynamics of axion and relativistic matter) and applied physics (e.g., magnetless isolators). We propose a three-dimensional metamaterial with an isotropic and resonant Tellegen response in the visible frequency range. The metamaterial is formed by randomly oriented bi-material nanocylinders in a host medium. Each nanocylinder consists of a ferromagnet in a single-domain magnetic state and a high-permittivity dielectric operating near the magnetic Mie-type resonance. The proposed metamaterial requires no external magnetic bias and operates on the spontaneous magnetization of the nanocylinders. By leveraging the emerging magnetic Weyl semimetals, we further show how a giant bulk effective magnetoelectric effect can be achieved in a proposed metamaterial, exceeding that of natural materials by almost four orders of magnitude.

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.

Surface properties of 1T-TaS2 and contrasting its electron-phonon coupling with TlBiTe2 from helium atom scattering

We present a detailed helium atom scattering study of the charge-density wave (CDW) system and transition metal dichalcogenide 1T-TaS2. In terms of energy dissipation, we determine the electron-phonon (e-ph) coupling, a quantity that is at the heart of conventional superconductivity and may even "drive" phase transitions such as CDWs. The e-ph coupling of TaS2 in the commensurate CDW phase (λ = 0.59 ± 0.12) is compared with measurements of the topo-logical insulator TlBiTe2 (λ = 0.09 ± 0.01). Furthermore, by means of elastic He diffraction and resonance/interference effects in He scattering, the thermal expansion of the surface lattice, the surface step height, and the three-dimensional atom-surface interaction potential are determined including the electronic corrugation of 1T-TaS2. The linear thermal expansion coefficient is similar to that of other transition-metal dichalcogenides. The He-TaS2 interaction is best described by a corrugated Morse potential with a relatively large well depth and supports a large number of bound states, comparable to the surface of Bi2Se3, and the surface electronic corrugation of 1T-TaS2 is similar to the ones found for semimetal surfaces.

Theoretical studies of magneto-optical Kerr and Faraday effects in two-dimensional second-order topological insulators

Optical approaches are useful for studying the electronic and spin structure of materials. Here, based on the tight-binding model and linear response theory, we investigate the magneto-optical Kerr and Faraday effects in two-dimensional second-order topological insulators (SOTI) with external magnetization. We find that orbital-dependent Zeeman term induces band crossings for SOTI phase, which are absent for trivial phase. In the weak-magnetization regime, these crossings give rise to giant jumps (peaks) of Kerr and Faraday angles (ellipticity) for SOTI phase. In the strong-magnetization regime, we find that two nearly flat bands are formed at the high-symmetry point of Brillouin zone of SOTI phase. These flat bands give rise to two successive giant jumps (peaks) of Kerr and Faraday angles (ellipticity). These phenomena provide new possibilities to characterize and detect the two-dimensional SOTI phase.

Loss-gain compensated anti-Hermitian magnetodielectric medium to realize Tellegen nihility effects

Generalized duality transformations significantly modify the constitutive relations of electromagnetic media, preserving principal electromagnetic properties. Here, we contemplate transformation of Tellegen nihility as a specific type of extreme-property nonreciprocal bi-isotropic media and show that some intriguing electromagnetic properties of that medium can be realized in a particular class of isotropic magnetodielectric media without magnetoelectric coupling. We show that the permittivity and permeability of the corresponding transformed medium have equal absolute values and opposite signs. Depending on the value of the Tellegen parameter of the original medium, the transformed magnetodielectric medium can be Hermitian, non-Hermitian, or anti-Hermitian, which simultaneously exhibits loss and gain. Focusing on the latter class of anti-Hermitian media, we theoretically and numerically demonstrate that this extraordinary medium allows propagation of electromagnetic plane waves having zero time-averaged Poynting vector, similarly to the original Tellegen nihility media. Hopefully, this work can open novel opportunities for manipulating electromagnetic fields.

Topological signatures in the entanglement of a topological insulator-quantum dot hybrid

In the present work, we consider a hybrid plexciton composed of a semiconductor quantum dot interacting with a topological insulator nanoparticle subject to an external magnetic field. Due to the topological magnetoelectricity of the nanoparticle, long-living plasmonic surface modes are induced, which are quantized and coupled with the quantum dot through its polarization operator. We consider the hybrid as an open quantum system, such that environment effects are accounted by the master equation in the Born-Markov approximation. Then, we apply the Peres' positive partial transpose criterion to quantify the entanglement of the hybrid. We show that this entanglement is a direct signature of the [Formula: see text] invariant of topological insulators.

Topological Insulator Films for Terahertz Photonics

We discuss experimental and theoretical studies of the generation of the third terahertz (THz) frequency harmonic in thin films of Bi₂Se₃ and Bi_2-xSb_xTe_3-ySe_y (BSTS) topological insulators (TIs) and the generation of THz radiation in photoconductive antennas based on the TI films. The experimental results, supported by the developed kinetic theory of third harmonic generation, show that the frequency conversion in TIs is highly efficient because of the linear energy spectrum of the surface carriers and fast energy dissipation. In particular, the dependence of the third harmonic field on the pump field remains cubic up to the pump fields of 100 kV/cm. The generation of THz radiation in TI-based antennas is obtained and described for the pump, with the energy of photons corresponding to the electron transitions to higher conduction bands. Our findings open up possibilities for advancing TI-based films into THz photonics as efficient THz wave generators and frequency converters.

Quantum Imaging of Magnetic Phase Transitions and Spin Fluctuations in Intrinsic Magnetic Topological Nanoflakes

Topological materials featuring exotic band structures, unconventional current flow patterns, and emergent organizing principles offer attractive platforms for the development of next-generation transformative quantum electronic technologies. The family of MnBi₂Te₄ (Bi₂Te₃)_n materials is naturally relevant in this context due to their nontrivial band topology, tunable magnetism, and recently discovered extraordinary quantum transport behaviors. Despite numerous pioneering studies to date, the local magnetic properties of MnBi₂Te₄ (Bi₂Te₃)_n remain an open question, hindering a comprehensive understanding of their fundamental material properties. Exploiting nitrogen-vacancy (NV) centers in diamond, we report nanoscale quantum imaging of the magnetic phase transitions and spin fluctuations in exfoliated MnBi₄Te₇ flakes, revealing the underlying spin transport physics and magnetic domains at the nanoscale. Our results highlight the unique advantage of NV centers in exploring the magnetic properties of emergent quantum materials, opening new opportunities for investigating the interplay between topology and magnetism.

Topology-tuned light scattering around Fano resonances by a core-shell cylinder

The topological magnetoelectric (TME) effect is a novel optical response from topological insulators. This effect shows that magnetic (electric) polarization can be induced by an applied electric (magnetic) field, and it is characterized by the fine structure constant. However, the TME effect is generally very weak and still a challenge to be observed in the experiment. In this paper, we showed that the far-field scattering of a core-shell topological cylinder can be tuned by the TME effect which was enhanced at the surface of plasmonic core around Fano resonance. The interference of broad dipolar mode and narrow quadrupole mode is changed with the topological magnetoelectric polarizability. We demonstrated the reversal of optical responses associated with the TME effect in both far-field and near field. Our results may offer an alternative way to observe the TME effect in topological insulators.

Topological spintronics and magnetoelectronics

Topological electronic materials, such as topological insulators, are distinct from trivial materials in the topology of their electronic band structures that lead to robust, unconventional topological states, which could bring revolutionary developments in electronics. This Perspective summarizes developments of topological insulators in various electronic applications including spintronics and magnetoelectronics. We group and analyse several important phenomena in spintronics using topological insulators, including spin-orbit torque, the magnetic proximity effect, interplay between antiferromagnetism and topology, and the formation of topological spin textures. We also outline recent developments in magnetoelectronics such as the axion insulator and the topological magnetoelectric effect observed using different topological insulators.

The Magnetic Susceptibility Bifurcation in the Ni-Doped Sb2Te3 Topological Insulator with Antiferromagnetic Order Accompanied by Weak Ferromagnetic Alignment

The magnetic susceptibility reveals a discontinuity at Néel temperature and a hysteresis loop with low coercive field was observed below Néel temperature. The magnetic susceptibility of zero field cool and field cool processes coincide at a temperature above the discontinuity, and they split at temperature blow the discontinuity. The magnetic susceptibility splitting is larger at lower external magnetic fields. No more magnetic susceptibility splitting was observed at a magnetic field above 7000 Oe which is consistent with the magnetic anisotropy energy. Our study supports that these magnetic susceptibility characteristics originate from an antiferromagnetic order accompanied by weak ferromagnetism.

First Principles Calculation of the Topological Phases of the Photonic Haldane Model

Photonic topological materials with a broken time-reversal symmetry are characterized by nontrivial topological phases, such that they do not support propagation in the bulk region but forcibly support a nontrivial net number of unidirectional edge-states when enclosed by an opaque-type boundary, e.g., an electric wall. The Haldane model played a central role in the development of topological methods in condensed-matter systems, as it unveiled that a broken time-reversal symmetry is the essential ingredient to have a quantized electronic Hall phase. Recently, it was proved that the magnetic field of the Haldane model can be imitated in photonics with a spatially varying pseudo-Tellegen coupling. Here, we use Green’s function method to determine from “first principles” the band diagram and the topological invariants of the photonic Haldane model, implemented as a Tellegen photonic crystal. Furthermore, the topological phase diagram of the system is found, and it is shown with first principles calculations that the granular structure of the photonic crystal can create nontrivial phase transitions controlled by the amplitude of the pseudo-Tellegen parameter.

Magnetic Topological Insulator Heterostructures: A Review

Topological insulators (TIs) provide intriguing prospects for the future of spintronics due to their large spin-orbit coupling and dissipationless, counter-propagating conduction channels in the surface state. The combination of topological properties and magnetic order can lead to new quantum states including the quantum anomalous Hall effect that was first experimentally realized in Cr-doped (Bi,Sb)₂ Te₃ films. Since magnetic doping can introduce detrimental effects, requiring very low operational temperatures, alternative approaches are explored. Proximity coupling to magnetically ordered systems is an obvious option, with the prospect to raise the temperature for observing the various quantum effects. Here, an overview of proximity coupling and interfacial effects in TI heterostructures is presented, which provides a versatile materials platform for tuning the magnetic and topological properties of these exciting materials. An introduction is first given to the heterostructure growth by molecular beam epitaxy and suitable structural, electronic, and magnetic characterization techniques. Going beyond transition-metal-doped and undoped TI heterostructures, examples of heterostructures are discussed, including rare-earth-doped TIs, magnetic insulators, and antiferromagnets, which lead to exotic phenomena such as skyrmions and exchange bias. Finally, an outlook on novel heterostructures such as intrinsic magnetic TIs and systems including 2D materials is given.

Super-Resonant Transport of Topological Surface States Subjected to In-Plane Magnetic Fields

Magnetic oscillations of Dirac surface states of topological insulators are typically expected to be associated with the formation of Landau levels or the Aharonov-Bohm effect. We instead study the conductance of Dirac surface states subjected to an in-plane magnetic field in the presence of a barrier potential. Strikingly, we find that, in the case of large barrier potentials, the surface states exhibit pronounced oscillations in the conductance when varying the magnetic field, in the absence of Landau levels or the Aharonov-Bohm effect. These novel magnetic oscillations are attributed to the emergence of super-resonant transport by tuning the magnetic field, in which many propagating modes cross the barrier with perfect transmission. In the case of small and moderate barrier potentials, we identify a positive magnetoconductance due to the increase of the Fermi surface by tilting the surface Dirac cone. Moreover, we show that for weak magnetic fields, the conductance displays a shifted sinusoidal dependence on the field direction with period π and phase shift determined by the tilting direction with respect to the field direction. Our predictions can be applied to various topological insulators, such as HgTe and Bi_{2}Se_{3}, and provide important insights into exploring and understanding exotic magnetotransport properties of topological surface states.

The Casimir Effect in Topological Matter

We give an overview of the work done during the past ten years on the Casimir interaction in electronic topological materials, our focus being solids, which possess surface or bulk electronic band structures with nontrivial topologies, which can be evinced through optical properties that are characterizable in terms of nonzero topological invariants. The examples we review are three-dimensional magnetic topological insulators, two-dimensional Chern insulators, graphene monolayers exhibiting the relativistic quantum Hall effect, and time reversal symmetry-broken Weyl semimetals, which are fascinating systems in the context of Casimir physics. Firstly, this is for the reason that they possess electromagnetic properties characterizable by axial vectors (because of time reversal symmetry breaking), and, depending on the mutual orientation of a pair of such axial vectors, two systems can experience a repulsive Casimir–Lifshitz force, even though they may be dielectrically identical. Secondly, the repulsion thus generated is potentially robust against weak disorder, as such repulsion is associated with the Hall conductivity that is topologically protected in the zero-frequency limit. Finally, the far-field low-temperature behavior of the Casimir force of such systems can provide signatures of topological quantization.

Time-domain terahertz spectroscopy in high magnetic fields

There are a variety of elementary and collective terahertz-frequency excitations in condensed matter whose magnetic field dependence contains significant insight into the states and dynamics of the electrons involved. Often, determining the frequency, temperature, and magnetic field dependence of the optical conductivity tensor, especially in high magnetic fields, can clarify the microscopic physics behind complex many-body behaviors of solids. While there are advanced terahertz spectroscopy techniques as well as high magnetic field generation techniques available, a combination of the two has only been realized relatively recently. Here, we review the current state of terahertz time-domain spectroscopy (THz-TDS) experiments in high magnetic fields. We start with an overview of time-domain terahertz detection schemes with a special focus on how they have been incorporated into optically accessible high-field magnets. Advantages and disadvantages of different types of magnets in performing THz-TDS experiments are also discussed. Finally, we highlight some of the new fascinating physical phenomena that have been revealed by THz-TDS in high magnetic fields.

Gapless Criterion for Crystals from Effective Axion Field

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 noninteracting 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 groups 26 and 13, where mirror, glide, and inversion symmetries provide the simplified expressions. In particular, the gapless criterion can identify gapless phases that are missed by the symmetry-representation approach, as illustrated by space group 26. Our proposal serves as a guiding principle for future discovery of topological semimetals.

Topological magneto-optical effects and their quantization in noncoplanar antiferromagnets

Reflecting the fundamental interactions of polarized light with magnetic matter, magneto-optical effects are well known since more than a century. The emergence of these phenomena is commonly attributed to the interplay between exchange splitting and spin-orbit coupling in the electronic structure of magnets. Using theoretical arguments, we demonstrate that topological magneto-optical effects can arise in noncoplanar antiferromagnets due to the finite scalar spin chirality, without any reference to exchange splitting or spin-orbit coupling. We propose spectral integrals of certain magneto-optical quantities that uncover the unique topological nature of the discovered effect. We also find that the Kerr and Faraday rotation angles can be quantized in insulating topological antiferromagnets in the low-frequency limit, owing to nontrivial global properties that manifest in quantum topological magneto-optical effects. Although the predicted topological and quantum topological magneto-optical effects are fundamentally distinct from conventional light-matter interactions, they can be measured by readily available experimental techniques.

Quantization in Chiral Higher Order Topological Insulators: Circular Dichroism and Local Chern Marker

The robust quantization of observables in units of universal constants is a hallmark of topological phases. We show that chiral higher order topological insulators (HOTIs), bulk insulators with chiral hinge states, present two unusual features related to quantization. First, we show that circular dichroism is quantized to an integer or zero depending on the orientation of the sample. This probe locates the hinge states, and can be used to distinguish different types of chiral HOTIs. Second, we find that the average of the local Chern marker over a single surface, an observable related to the surface Hall conductivity known to be quantized in the infinite slab geometry, is nonuniversal for a finite surface. This is due to a nonuniversal contribution of the hinge states, previously unaccounted for, that distinguishes surfaces of chiral HOTIs from Chern insulators. Our findings are relevant to establish higher order topology in systems such as the axion insulator candidate EuIn_{2}As_{2}, and cold atomic realizations.

Proposal to Detect Dark Matter using Axionic Topological Antiferromagnets

Antiferromagnetically doped topological insulators (ATI) are among the candidates to host dynamical axion fields and axion polaritons, weakly interacting quasiparticles that are analogous to the dark axion, a long sought after candidate dark matter particle. Here we demonstrate that using the axion quasiparticle antiferromagnetic resonance in ATIs in conjunction with low-noise methods of detecting THz photons presents a viable route to detect axion dark matter with a mass of 0.7 to 3.5 meV, a range currently inaccessible to other dark matter detection experiments and proposals. The benefits of this method at high frequency are the tunability of the resonance with applied magnetic field, and the use of ATI samples with volumes much larger than 1  mm^{3}.

Fully Spin-Polarized Nodal Loop Semimetals in Alkaline Metal Monochalcogenide Monolayers

Topological semimetals in ferromagnetic materials have attracted an enormous amount of attention due to potential applications in spintronics. Using first-principles density functional theory together with an effective lattice model, here we present a new family of topological semimetals with a fully spin-polarized nodal loop in alkaline metal monochalcogenide MX (M = Li, Na, K, Rb, or Cs; X = S, Se, or Te) monolayers. The half-metallic ferromagnetism can be established in MX monolayers, in which one nodal loop formed by two crossing bands with the same spin components is found at the Fermi energy. This nodal loop half-metal survives even when considering the spin-orbit coupling owing to the symmetry protection provided by the Mz mirror plane. The quantum anomalous Hall state and Weyl-like semimetal in this system can be also achieved by rotating the spin from the out-of-plane to the in-plane direction. The MX monolayers hosting rich topological phases thus offer an excellent platform for realizing advanced spintronic concepts.

Axion Insulator State in a Ferromagnet/Topological Insulator/Antiferromagnet Heterostructure

We propose the use of 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 fields. Using density functional calculations and model Hamiltonian simulations, we find that the top and bottom surfaces have opposite half-quantum Hall conductances, [Formula: see text] and [Formula: see text], with a sizable global spin gap of 5.1 meV opened for the topological surface states of Bi₂Se₃. Our work provides a new strategy for the search of axion insulators by using van der Waals antiferromagnetic insulators along with three-dimensional topological insulators.

Landau level spectroscopy of valence bands in HgTe quantum wells: effects of symmetry lowering

The Landau level spectroscopy technique has been used to explore the electronic structure of the valence band in a series of p-type HgTe/HgCdTe quantum wells with both normal and inverted ordering of bands. We find that the standard axial-symmetric 4-band Kane model, which is nowadays widely applied in physics of HgTe-based topological materials, does not fully account for the complex magneto-optical response observed in our experiments-notably, for the unexpected avoided crossings of excitations and for the appearance of transitions that are electric-dipole forbidden within this model. Nevertheless, reasonable agreement with experiments is achieved when the standard model is expanded to include effects of bulk and interface inversion asymmetries. These remove the axial symmetry, and among other, profoundly modify the shape of valence bands.

Ultrafast manipulation of topologically enhanced surface transport driven by mid-infrared and terahertz pulses in Bi2Se3

Topology-protected surface transport of ultimate thinness in three-dimensional topological insulators (TIs) is breaking new ground in quantum science and technology. Yet a challenge remains on how to disentangle and selectively control surface helical spin transport from the bulk contribution. Here we use the mid-infrared and terahertz (THz) photoexcitation of exclusive intraband transitions to enable ultrafast manipulation of surface THz conductivity in Bi₂Se₃. The unique, transient electronic state is characterized by frequency-dependent carrier relaxations that directly distinguish the faster surface channel than the bulk with no complication from interband excitations or need for reduced bulk doping. We determine the topological enhancement ratio between bulk and surface scattering rates, i.e., γ_BS/γ_SS ~3.80 in equilibrium. The ultra-broadband, wavelength-selective pumping may be applied to emerging topological semimetals for separation and control of the protected transport connected with the Weyl nodes from other bulk bands.

It remains challenging on how to selectively control terahertz conductivity at surface from the bulk contribution in topological insulators. Here, Luo et al. discover and manipulate topologically enhanced surface transport due to helical spin structure using mid-infrared and terahertz ultrafast photoexcitations.

Systematic Study of Ferromagnetism in CrxSb2-xTe3 Topological Insulator Thin Films using Electrical and Optical Techniques

Ferromagnetic ordering in a topological insulator can break time-reversal symmetry, realizing dissipationless electronic states in the absence of a magnetic field. The control of the magnetic state is of great importance for future device applications. We provide a detailed systematic study of the magnetic state in highly doped Cr_xSb_2−xTe₃ thin films using electrical transport, magneto-optic Kerr effect measurements and terahertz time domain spectroscopy, and also report an efficient electric gating of ferromagnetic order using the electrolyte ionic liquid [DEME][TFSI]. Upon increasing the Cr concentration from x = 0.15 to 0.76, the Curie temperature (T_c) was observed to increase by ~5 times to 176 K. In addition, it was possible to modify the magnetic moment by up to 50% with a gate bias variation of just ±3 V, which corresponds to an increase in carrier density by 50%. Further analysis on a sample with x = 0.76 exhibits a clear insulator-metal transition at T_c, indicating the consistency between the electrical and optical measurements. The direct correlation obtained between the carrier density and ferromagnetism - in both electrostatic and chemical doping - using optical and electrical means strongly suggests a carrier-mediated Ruderman-Kittel-Kasuya-Yoshida (RKKY) coupling scenario. Our low-voltage means of manipulating ferromagnetism, and consistency in optical and electrical measurements provides a way to realize exotic quantum states for spintronic and low energy magneto-electronic device applications.

Realization of the Axion Insulator State in Quantum Anomalous Hall Sandwich Heterostructures

The "magnetoelectric effect" arises from the coupling between magnetic and electric properties in materials. The Z_{2} 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 magnetic TI layers are separated by an undoped TI layer. Magnetic force microscopy images of the same sample reveal sequential magnetization reversals of the top and bottom layers at different coercive fields, a consequence of the weak interlayer exchange coupling due to the spacer. When the magnetization is antiparallel, both the Hall resistance and Hall conductance show zero plateaus, accompanied by a large longitudinal resistance and vanishing longitudinal conductance, indicating the realization of an axion insulator state. Our findings thus show evidence for a phase of matter distinct from the established QAH state and provide a promising platform for the realization of the TME effect.

Topological states of condensed matter

Topological states of quantum matter have been investigated intensively in recent years in materials science and condensed matter physics. The field developed explosively largely because of the precise theoretical predictions, well-controlled materials processing, and novel characterization techniques. In this Perspective, we review recent progress in topological insulators, the quantum anomalous Hall effect, chiral topological superconductors, helical topological superconductors and Weyl semimetals.

Tailoring tricolor structure of magnetic topological insulator for robust axion insulator

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 the 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 exotic magnetoelectric coupling and can be expected to realize the TME effect as an axion insulator. We designed a magnetic TI with a tricolor structure where a nonmagnetic layer of (Bi, Sb)₂Te₃ is sandwiched by a soft ferromagnetic Cr-doped (Bi, Sb)₂Te₃ and a hard ferromagnetic V-doped (Bi, Sb)₂Te₃. Accompanied by the quantum anomalous Hall (QAH) effect, we observe zero Hall conductivity plateaus, which are a hallmark of the axion insulator state, in a wide range of magnetic fields between the coercive fields of Cr- and V-doped layers. The resistance of the axion insulator state reaches as high as 10⁹ ohms, leading to a gigantic magnetoresistance ratio exceeding 10,000,000% upon the transition from the QAH state. The tricolor structure of the TI may not only be an ideal arena for the topologically distinct phenomena but can also provide magnetoresistive applications for advancing dissipation-less topological electronics.

Scaling of the Quantum Anomalous Hall Effect as an Indicator of Axion Electrodynamics

We report on the scaling behavior of V-doped (Bi,Sb)_{2}Te_{3} samples in the quantum anomalous Hall regime for samples of various thickness. While previous quantum anomalous Hall measurements showed the same scaling as expected from a two-dimensional integer quantum Hall state, we observe a dimensional crossover to three spatial dimensions as a function of layer thickness. In the limit of a sufficiently thick layer, we find scaling behavior matching the flow diagram of two parallel conducting topological surface states of a three-dimensional topological insulator each featuring a fractional shift of 1/2e^{2}/h in the flow diagram Hall conductivity, while we recover the expected integer quantum Hall behavior for thinner layers. This constitutes the observation of a distinct type of quantum anomalous Hall effect, resulting from 1/2e^{2}/h Hall conductance quantization of three-dimensional topological insulator surface states, in an experiment which does not require decomposition of the signal to separate the contribution of two surfaces. This provides a possible experimental link between quantum Hall physics and axion electrodynamics.

Theory of topological insulator waveguides: polarization control and the enhancement of the magneto-electric effect

Topological insulators subject to a time-reversal-symmetry-breaking perturbation are predicted to display a magneto-electric effect that causes the electric and magnetic induction fields to mix at the material's surface. This effect induces polarization rotations of between ≈1-10 mrad per interface in an incident plane-polarized electromagnetic wave normal to a multilayered structure. Here we show, theoretically and numerically, that by using a waveguide geometry with a topological insulator guide layer and magneto-dielectric cladding it is possible to achieve rotations of ≈100 mrad and generate an elliptical polarization with only a three-layered structure. This geometry is beneficial, not only as a way to enhance the magneto-electric effect, rendering it easier to observe, but also as a method for controlling the polarization of electromagnetic radiation.

Terahertz spectroscopy on Faraday and Kerr rotations in a quantum anomalous Hall state

Electrodynamic responses from three-dimensional topological insulators are characterized by the universal magnetoelectric term constituent of the Lagrangian formalism. The quantized magnetoelectric coupling, which is generally referred to as topological magnetoelectric effect, has been predicted to induce exotic phenomena including the universal low-energy magneto-optical effects. Here we report the experimental indication of the topological magnetoelectric effect, which is exemplified by magneto-optical Faraday and Kerr rotations in the quantum anomalous Hall states of magnetic topological insulator surfaces by terahertz magneto-optics. The universal relation composed of the observed Faraday and Kerr rotation angles but not of any material parameters (for example, dielectric constant and magnetic susceptibility) well exhibits the trajectory towards the fine structure constant in the quantized limit.