Magnetoelectric responses are a fundamental characteristic of materials that break time-reversal and inversion symmetries (notably multiferroics) and, remarkably, of "topological insulators" in which those symmetries are unbroken. Previous work has shown how to compute spin and lattice contributions to the magnetoelectric tensor. Here we solve the problem of orbital contributions by computing the frozen-lattice electronic polarization induced by a magnetic field. One part of this response (the "Chern-Simons term") can appear even in time-reversal-symmetric materials and has been previously shown to be quantized in topological insulators. In general materials there are additional orbital contributions to all parts of the magnetoelectric tensor; these vanish in topological insulators by symmetry and also vanish in several simplified models without time-reversal and inversion those magnetoelectric couplings were studied before. We give two derivations of the response formula, one based on a uniform magnetic field and one based on extrapolation of a long-wavelength magnetic field, and discuss some of the consequences of this formula.
The kinetic magnetoelectric effect is an orbital analogue of the Edelstein effect and offers an additional degree of freedom to control magnetisation via the charge current. Here we theoretically propose a gigantic kinetic magnetoelectric effect in topological insulators and interpret the results in terms of topological surface currents. We construct a theory of the kinetic magnetoelectric effect for a surface Hamiltonian of a topological insulator, and show that it well describes the results by direct numerical calculation. This kinetic magnetoelectric effect depends on the details of the sur…
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…
We theoretically study magnetoelectric effects in a heterostructure of a generic band insulator and a ferromagnet. In contrast to the kinetic magnetoelectric effect in metals, referred to as the Edelstein effect or the inverse spin galvanic effect, our mechanism relies on virtual interband transitions between the valence and conduction bands and therefore immune to disorder or impurity scattering. By calculating electric field-induced magnetization by the linear response theory, we reveal that the magnetoelectric effect shows up without specific parameter choices. The magnetoelectric effect qu…
We report transport studies on (Bi,Sb)2Te3 topological insulator thin films with tunable electronic band structure. We find a doping and temperature regime in which the Hall coefficient is negative indicative of electron-type carriers, whereas the Seebeck coefficient is positive indicative of hole-type carriers. This sign anomaly is due to the distinct transport behaviors of the bulk and surface states: the surface Dirac fermions dominate magnetoelectric transport while the thermoelectric effect is mainly determined by the bulk states. These findings may inspire new ideas for designing topolog…
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 …
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…
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…
In this letter, we demonstrate switching of nanomagnets through RKKY interactions in topological insulator-based magnetoelectric devices. The switching speed is dependent on the size of the ferromagnets.
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…
Magnetoelectric effect in multiferroic metals via a direct spin-charge interaction
Key Lab of advanced optoelectronic quantum architecture and measurement (MOE), and School of Interdisciplinary Science, Beijing Institute of Technology（北京理工大学光电量子架构与测量教育部重点实验室，交叉科学研究院）
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…
We study the response of topological insulator films to strong magnetic and electric fields beyond the linear response theory. As a model, we use the three-dimensional lattice Wilson-Dirac Hamiltonian where we simultaneously introduce both magnetic field through the Peierls substitution and electric field as a potential energy depending on lattice coordinate. We compute the electron energy spectrum by numerically diagonalizing this Hamiltonian and obtain quantized magnetoelectric polarizability. In addition, we find that the magnetoelectric effect vanishes as the film width decreases due to th…
Topological magnetoelectric effect in a three-dimensional topological insulator is a novel phenomenon, where an electric field induces a magnetic field in the same direction, with a universal coefficient of proportionality quantized in units of $e^2/2h$. Here we propose that the topological magnetoelectric effect can be realized in the zero-plateau quantum anomalous Hall state of magnetic topological insulators or ferromagnet-topological insulator heterostructure. The finite-size effect is also studied numerically, where the magnetoelectric coefficient is shown to converge to a quantized value…
The interface between a topological insulator and a ferromagnetic insulator exhibits an interesting interplay of topological Dirac electrons and magnetism. As has been shown recently, the breaking of time-reversal invariance by magnetic order generates a Chern-Simons term in the action, that in turn leads to a Berry phase and a magnetoelectric effect of topological origin. Here, we consider the system in the presence of long-range Coulomb interaction between the Dirac electrons, and find that the magnetoelectric effect of the fluctuating electric field becomes non-local. We derive a Landau-Lif…
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…
We theoretically study the magnetoelectric coupling in a quantum anomalous Hall insulator state induced by interfacing a dynamic magnetization texture to a topological insulator. In particular, we propose that the quantum anomalous Hall insulator with a magnetic configuration of a domain wall, when contacted by electrical reservoirs, acts as a magnetoelectric piston. A moving domain wall pumps charge current between electrical leads in a closed circuit, while applying an electrical bias induces reciprocal domain-wall motion. This piston-like action is enabled by a finite reflection of charge c…
It has been proposed that topological insulators can be best characterized not as surface conductors, but as bulk magnetoelectrics that -- under the right conditions-- have a universal quantized magnetoelectric response coefficient $e^2/2h$. However, it is not clear to what extent these conditions are achievable in real materials that can have disorder, finite chemical potential, residual dissipation, and even inversion symmetry. This has led to some confusion and misconceptions. The primary goal of this work is to illustrate exactly under what real life scenarios and in what context topologic…
One of the hallmarks of time-reversal-symmetric topological insulators in three dimensions is the topological magnetoelectric effect (TME). So far, a time-reversal breaking variant of this effect has attracted much attention, in the sense that the induced electric charge changes sign when the direction of an externally applied magnetic field is reversed. Theoretically, this effect is described by the so-called axion term. Here, we discuss a time-reversal-symmetric TME, where the electric charge depends only on the magnitude of the magnetic field but is independent of its sign. We obtain this n…
Geometry-Driven Nonlinear Orbital Magnetoelectric Effect
University of Science and Technology of China（中国科学技术大学）；Hefei National Laboratory, University of Science and Technology of China（合肥国家实验室，中国科学技术大学）；Shenzhen University（深圳大学）；Quantum Science Center of Guangdong-Hongkong-Macao Greater Bay Area (Guangdong)（粤港澳大湾区量子科学中心（广东））；The University of Hong Kong（香港大学）
Profiles of the spin and orbital angular momentum accumulations induced by a longitudinally applied electric field are explored in nanoribbons of $p$-band systems with a honeycomb lattice. We show that nanoribbons with zigzag borders can exhibit orbital magnetoelectric effects. More specifically, we have found that purely orbital magnetization oriented perpendicularly to the ribbon may be induced in these systems by means of the external electric field, when sublattice symmetry is broken. The effect is rather general and may occur in other multi-orbital materials.
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 electrodynamics of topological insulators (TIs) is described by modified Maxwell's equations, which contain additional terms that couple an electric field to a magnetization and a magnetic field to a polarization of the medium, such that the coupling coefficient is quantized in odd multiples of $e^2 / 2 h c $ per surface. Here, we report on the observation of this so-called topological magnetoelectric (TME) effect. We use monochromatic terahertz (THz) spectroscopy of TI structures equipped with a semi-transparent gate to selectively address surface states. In high external magnetic fields,…
Magnetization can be induced by an electric field in systems without inversion symmetry $\mathcal{P}$ and time-reversal symmetry $\mathcal{T}$. This phenomenon is called the magnetoelectric (ME) effect. The spin ME effect has been actively studied in multiferroics. The orbital ME effect also exists and has been mainly discussed in topological insulators at zero temperature. In this paper, we study the intrinsic orbital ME response in metals at finite temperature using the Kubo formula. The intrinsic response originates from the Fermi sea and does not depend on the dissipation. Especially in sy…
While it is often assumed that the orbital response is suppressed and short-ranged due to strong crystal field potential and orbital quenching, we show that the orbital magnetoelectric response can be remarkably long-ranged in ferromagnets. In a bilayer consisting of a nonmagnet and a ferromagnet, spin injection from the interface results in spin accumulation and torque in the ferromagnet, which rapidly oscillate and decay by spin dephasing. In contrast, we find that even when an external electric field is applied only on the nonmagnet, we find substantially long-ranged orbital magnetoelectric…
Topological insulators (TIs) exhibit a quantized magnetoelectric response when time-reversal symmetry is broken on its surface. This unusual electromagnetic (EM) response is a unique macroscopic manifestation of the quantum Hall effect on the TI surface and it is described by a quantized Chern-Simons theory. In this paper, we construct the Green's function (GF) describing the EM response of two topological media separated by a cylindrical interface. This GF, in the appropriate limits, describes the magnetoelectric effect of both i) a cylindrical TI surrounded by a dielectric fluid, and ii) a T…
Magnetic skyrmions are topologically protected spin-whirl quasiparticles currently considered as promising components for ultra-dense memory devices. In the bulk they form lattices that are stable over just a few Kelvin below the ordering temperature. This narrow stability range presents a key challenge for applications, and finding ways to tune the SkL stability over a wider phase space is a pressing issue. Here we show experimentally that the skyrmion phase in the magnetoelectric insulator ${\text{Cu}_2 \text{O} \text{Se} \text{O}_3}$ can either expand or shrink substantially depending on th…
Superconducting magnetoelectric effect, which is the current-induced magnetization in a superconductor, mainly focused on the spin magnetization in previous studies, but ignore the effect of the orbital magnetic moments carried by the paired Bloch electrons. In this work, we show that orbital magnetic moments in superconductors can induce large orbital magnetization in the presence of a current. We constructed a unified description for the current-induced spin and orbital magnetization across the superconductivity normal metal phase transition. We find that in a superconductor with uniform pai…
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…
When time-reversal symmetry is broken on its surface, topological insulators exhibit a magnetoelectric response which is described by axion electrodynamics. A direct consequence of this theory is the appearance of a magnetic field that resembles the one produced by a magnetic image monopole when a point-like electric charge is located near the surface of the material. In this paper we investigate the more realistic problem when the point-like charge is replaced by a finite size sphere at constant potential. We calculate the electromagnetic fields using the potential formulation in a particular…
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…
We investigate the interaction between a topological insulator nanoparticle and a quantum dot in an impulse magnetic field. Since topological insulators are nonmagnetic, after the impulse has ended only the localised topological surface modes, which are quantised in terms of dipolar bosonic modes, thus coupling dipolarly to the quantum dot. Hence, the hybrid system can be treated as a single bosonic mode interacting with a two-level system, where the coupling strength is quantised in terms of the magnetoelectric polarizability. We implement the interaction of the hybrid with the environment th…
We theoretically study the coupling of electric charge and spin polarization in an equilibrium and nonequilibrium electric transport across a two dimensional Josephson configuration comprised of disordered surface channels of a three dimensional topological insulator. In the equilibriun state of the system we predict the Edelstein effect, which is much more pronounced than its counterpart in conventional spin orbit coupled materials. Employing a quasiclassical Keldysh technique, we demonstrate that the ground state of system can be experimentally shifted into arbitrary macroscopic superconduct…
Non-volatile Orbital magnetoelectric effect in a two-dimensional electron gas
CNR-SPIN（意大利国家研究委员会自旋物理研究所）；Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；CEA（法国原子能和替代能源委员会）；CNRS（法国国家科学研究中心）；Grenoble-INP（格勒诺布尔国立理工学院）；SPINTEC（自旋电子学与自旋器件实验室）；Laboratoire Albert Fert（阿尔伯特·费特实验室）；Thales（泰雷兹集团）；Université Paris-Saclay（巴黎萨克雷大学）；Photon Science Division, Paul Scherrer Institut（保罗谢尔研究所光子科学部）；Institut de Physique et Chimie des Matériaux de Strasbourg (IPCMS)（斯特拉斯堡材料物理与化学研究所）；Université de Strasbourg（斯特拉斯堡大学）；Department of Applied Physics, Indian Institute of Information Technology Vadodara（瓦多德拉印度信息技术学院应用物理系）；Dipartimento di Fisica, Università di Cagliari（卡利亚里大学物理系）；CNR-IOM（意大利国家研究委员会无机材料与光化学研究所）；Dipartimento di Fisica “E. R. Caianiello,” Universitá di Salerno（萨莱诺大学E.R.凯亚涅洛物理系）；Faculty of Environmental Life, Natural Science and Technology, Okayama University（冈山大学环境生命自然科学与技术学部）
The orbital magnetoelectric effect (OME) generically refers to the appearance of an orbital magnetization induced by an applied electric field. Here, we show that nanoribbons of transition metal dichalcogenides (TMDs) with zigzag (ZZ) edges may exhibit a sizeable OME activated by an electric field applied along the ribbons' axis. We examine nanoribbons extracted from a monolayer (1L) and a bilayer (2L) of MoS$_2$ in the trigonal (H) structural phase. Transverse profiles of the induced orbital angular momentum accumulations are calculated to first order in the longitudinally applied electric fi…
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 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…
We study the magnetization dynamics in ferromagnet$\mid$insulator$\mid$ferromagnet and ferromagnet$\mid$insulator$\mid$normal metal ultra-small tunnel junctions, and the associated voltage drop in the presence of an electromagnetic environment assisting the tunneling processes. We show that the environment strongly affects the resulting voltage, which becomes a highly non-linear function of the precession cone angle $θ$. We find that voltages comparable to the driving frequency $ω$ can be reached even for small precession cone angles $θ$, in stark contrast to the case where the environment is …
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…
We study the generalization of the three-dimensional two-band Hopf insulator to the case of many bands, where all the bands are separated from each other by band gaps. The obtained $\mathbb{Z}$ classification of such a $N$-band Hopf insulator is related to the quantized isotropic magnetoelectric coefficient of its bulk. The boundary of a $N$-band Hopf insulator can be fully gapped, and we find that there is no unique way of dividing a finite system into bulk and boundary. Despite this non-uniqueness, we find that the magnetoelectric coefficient of the bulk and the anomalous Hall conductivity o…
Topological insulators have been predicted to exhibit a variety of interesting phenomena including a quantized magnetoelectric response and novel spintronics effects due to spin textures on their surfaces. However, experimental observation of these phenomena has proved difficult due to the finite bulk carrier density which may overwhelm the intrinsic topological responses that are expressed at the surface. Here, we demonstrate a novel ionic gel gating technique to tune the chemical potential of Bi$_{2}$Se$_{3}$ thin films while simultaneously performing THz spectroscopy. We can tune the carrie…
