Intrinsic Hall effects, such as the anomalous Hall effect, originate from the orbital quantum geometry of Bloch states. However, in layered materials, the combined action of out-of-plane electric and magnetic fields couples to layer polarization and orbital moment, generating a mixed layer-orbital quantum geometry in field-dressed Bloch states. We show that this geometry produces an intrinsic magnetoelectric Hall effect that is bilinear in the electric and magnetic fields. The response is scattering-time independent and can arise in nonmagnetic systems without spin-orbit coupling. Its origin lies in interband coherence involving layer polarization and orbital moment, leading to a finite, non-quantized Hall response that persists in the band gap. The Hall coefficient is odd under gate reversal and tracks layer polarization. A symmetry analysis identifies the classes of layered materials that host this effect. As a representative realization, we demonstrate the effect in rhombohedral pentalayer graphene, where the conductivity reaches values of order $0.05\,e^2/h$. These results establish mixed layer-orbital quantum geometry as a mechanism for intrinsic magnetoelectric Hall transport and a direct probe of layer-resolved quantum geometry in Bloch bands.
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…
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…
A system with a topological insulator slab sandwiched by two one-dimensional photonic crystals as a defect is investigated. A giant topological magnetoelectric effect is proposed which is characterized by two resonant peaks with half-unit transmissions and rich polarization behaviors in the photonic band gap. We also predict a giant optical Hall effect in our system, in which the transverse shift of the transmitted light could be thousand times larger than the wavelength. These effects are purely from the $Θ$-domain walls at the topological insulator interfaces and could be observed easily. Th…
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…
Topological insulators (TIs) in three space dimensions can be characterized by a quantized magnetoelectric coefficient. However, this coupling does not have experimentally observable consequences in the presence of time-reversal symmetry, because the contributions of both bulk and surface states cancel each other. Instead, the characteristic response of a TI is a nonlinear magnetoelectric effect. Field theoretic aspects of the nonlinear magnetoelectric response and numerical calculations for experimentally relevant geometries are discussed. Distinct from this effect, the magnetoelectric coupli…
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…
Electrodynamic responses from three-dimensional (3D) topological insulators (TIs) are characterized by the universal magnetoelectric $E\cdot B$ term constituent of the Lagrangian formalism. The quantized magnetoelectric coupling, which is generally referred to as topological magnetoelectric (TME) effect, has been predicted to induce exotic phenomena including the universal low-energy magneto-optical effects. Here we report the experimental demonstration of the long-sought TME effect, which is exemplified by magneto-optical Faraday and Kerr rotations in the quantum anomalous Hall (QAH) states o…
The magnitude of the inverse Faraday effect (IFE), a static magnetization due to an ac electric field, can be strongly increased in a mesoscopic sample, sensitive to time-reversal symmetry (TRS) breaking. Random rectification of ac voltages leads to a magnetization flux, which can be detected by an asymmetry of Hall resistances in a multi-terminal setup. In the absence of applied magnetic field through a chaotic quantum dot the IFE scale, quadratic in voltage, is found as an analytic function of the ac frequency, screening, and coupling to the contacts and floating probes, and numerically it d…
We present a comprehensive quasiclassical approach for studying transport properties of superconducting diffusive hybrid structures in the presence of extrinsic spin-orbit coupling. We derive a generalized Usadel equation and boundary conditions that in the normal state reduce to the drift-diffusion theory governing the spin-Hall effect in inversion symmetric materials. These equations predict the non-dissipative spin-galvanic effect, that is the generation of supercurrents by a spin-splitting field, and its inverse -- the creation of magnetic moment by a supercurrent. These effects can be see…
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…
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…
Breaking the time-reversal symmetry of a topological insulator (TI) by ferromagnetism can induce exotic magnetoelectric phenomena such as quantized anomalous Hall (QAH) effect. Experimental observation of QAH effect in a magnetically doped TI requires ferromagnetism not relying on the charge carriers. We have realized the ferromagnetism independent of both polarity and density of carriers in Cr-doped BixSb2-xTe3 thin films grown by molecular beam epitaxy. Meanwhile, the anomalous Hall effect is found significantly enhanced with decreasing carrier density, with the anomalous Hall angle reaching…
We present spin transport studies on a low-field, room-temperature magnetoelectric multiferroic polycrystalline $\text{Sr}_{3}\text{Co}_{2}\text{Fe}_{24}\text{O}_{41}$ (SCFO)|Pt heterostructure wherein a highly tunable transverse conical magnetic phase is responsible for static and dynamic magnetoelectric coupling. We measured angular dependence of spin Hall magnetoresistance (SMR) at constant magnetic fields ($H$) in the range of 50 to 100 kOe. Application of field below the critical value (2.5 kOe), yielded negative SMR and the $H$-evolution of normalized SMR exhibited a negative gradient. F…
We study the kinetic magnetoelectric effect (current-induced magnetization including both the orbital and spin contributions) in three-dimensional conductors, specializing to the case of p-doped trigonal tellurium. We include both intrinsic and extrinsic contributions to the effect, which stem from the band structure of the crystal, and from disorder scattering, respectively. Specifically, we determine the dependence of the kinetic magnetoelectric response on the hole doping in tellurium, and show that the intrinsic and extrinsic effects dominate for low and high levels of doping, respectively…
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…
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…
Electrostatic charges placed near the interface between ordinary and topological insulators induce magnetic fields, through the so-called topological magnetoelectric effect. Here, we present a numerical implementation of the associated Maxwell equations. The resulting model is simple, fast and quantitatively as accurate as the image charge method, but with the advantage of providing easy access to elaborate geometries when pursuing specific effects. The model is used to study how magnetoelectric fields are influenced by the dimensions and the shape of the most common semiconductor nanostructur…
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）
In a thin strip of a two-dimensional semiconductor electronic system, spin-orbit coupling may be induced near both edges of the strip due to the substantial spatial variation of the confining potential in the boundary regions. In this paper we show that, in the presence of boundary-confinement induced spin-orbit coupling, a longitudinal charge current circulating through a 2D semiconductor strip may cause \textit{strong} non-equilibrium spin accumulation near both edges of the strip. The spins will be polarized along the normal of the 2DEG plane but in opposite directions at both edges of the …
Recent experiments in semiconductor nanowires with a spin-orbit coupling and proximity-induced superconductivity exhibit signatures of Majorana bound states predicted to exist in the topological phase. In this work we predict that these nanowire systems exhibit unconventional magnetoelectric effects showing a sharp crossover behavior at the topological phase transition. We find that magnetic fields with a component parallel to the spin-orbit field can give rise to currents in equilibrium. Surprisingly, also fields perpendicular to the spin-orbit field may induce currents and can be employed in…
The anomalous Hall effect is investigated experimentally and theoretically for ferromagnetic thin films of Mn$_5$Ge$_3$. We have separated the intrinsic and extrinsic contributions to the experimental anomalous Hall effect, and calculated the intrinsic anomalous Hall conductivity from the Berry curvature of the Bloch states using first-principles methods. The intrinsic anomalous Hall conductivity depends linearly on the magnetization, which can be understood from the long wavelength fluctuations of the spin orientation at finite temperatures. The quantitative agreement between theory and exper…
The dynamic interplay of transport, electrostatic, and magnetic effects in the resonant tunneling through ferromagnetic quantum wells is theoretically investigated. It is shown that the carrier-mediated magnetic order in the ferromagnetic region not only induces, but also takes part in intrinsic, robust, and sustainable high-frequency current oscillations over a large window of nominally steady bias voltages. This phenomenon could spawn a new class of quantum electronic devices based on ferromagnetic semiconductors.
Relativistic band theoretical calculations reveal that intrinsic spin Hall conductivity in hole-doped archetypical semiconductors Ge, GaAs and AlAs is large $[\sim 100 (\hbar/e)(Ωcm)^{-1}]$, showing the possibility of spin Hall effect beyond the four band Luttinger Hamiltonian. The calculated orbital-angular-momentum (orbital) Hall conductivity is one order of magnitude smaller, indicating no cancellation between the spin and orbital Hall effects in bulk semiconductors. Furthermore, it is found that the spin Hall effect can be strongly manipulated by strains, and that the $ac$ spin Hall conduc…
The spin Hall angle in Pt is evaluated in Pt/NiFe bilayers by spin torque ferromagnetic resonance (ST-FMR) measurements, and is found to increase with increasing the NiFe thickness. To extract the intrinsic spin Hall angle in Pt by estimating the total spin current injected into NiFe from Pt, the NiFe thickness dependent measurements are performed and the spin diffusion in the NiFe layer is taken into account. The intrinsic spin Hall angle of Pt is determined to be 0.068 at room temperature, and is found to be almost constant in the temperature range 13 - 300 K.
We propose a mechanism of intrinsic spin Hall effect (SHE). In this mechanism, local orbital angular momentum (OAM) induces electron position shift and couples with the bias electric field to generate orbital Hall effect (OHE). SHE then emerges as a concomitant effect of OHE through the atomic spin-orbit coupling. Spin Hall conductivity due to this mechanism is estimated to be comparable to experimental values for heavy metals. This mechanism predicts the sign change of the spin Hall conductivity as the spin-orbit polarization changes its sign, and also correlation between the spin Hall conduc…
We study the dependence of the intrinsic spin Hall effect on the crystal symmetry and geometry of experiment. The spin current is obtained and the Hall voltage caused by the polarization of the electron spins is computed. The unique dependence of the effect on the crystal symmetry permits the choice of geometry in which the spin Hall effect can be unambiguously distinguished from the effects due to the orbital motion of charge carriers and due to the magnetic field generated by the transport current.
Detecting the orientation of the Néel vector is a major research topic in antiferromagnetic spintronics. Here we recognize the intrinsic nonlinear Hall effect, which is independent of the relaxation time, as a prominent contribution to the time-reversal-odd second order conductivity and can be used to detect the flipping of the Néel vector. In contrast, the Berry-curvature-dipole-induced nonlinear Hall effect depends linear on relaxation time and is time-reversal-even. We study the intrinsic nonlinear Hall effect in an antiferromagnetic metal: tetragonal CuMnAs, and show that its nonlinear Hal…
The intrinsic anomalous Hall effect is one of the most exciting manifestations of the geometric properties of the electronic wave-function. Here, we predict that the electronic wave-function's geometric nature also gives rise to a purely quantum mechanical, {\it intrinsic} (scattering time-independent) component of the Hall conductivities in the presence of a magnetic field. We show that the orbital magnetic moment and the anomalous Hall velocity combine to generate a scattering time-independent contribution to the Hall effect, in addition to the Lorentz force induced scattering time-dependent…
We consider spin Hall effect in a system of massless Dirac fermions in a graphene lattice. Two types of spin-orbit interaction, pertinent to the graphene lattice, are taken into account - the intrinsic and Rashba terms. Assuming perfect crystal lattice, we calculate the topological contribution to spin Hall conductivity. When both interactions are present, their interplay is shown to lead to some peculiarities in the dependence of spin Hall conductivity on the Fermi level.
Multiferroic materials with coupled ferroelectric and ferromagnetic properties are important for multifunctional devices due to their potential ability of controlling magnetism via electric field, and vice versa. The recent discoveries of two-dimensional ferromagnetic and ferroelectric materials have ignited tremendous research interest and aroused hope to search for two-dimensional multiferroics. However, intrinsic two-dimensional multiferroic materials and, particularly, those with strong magnetoelectric couplings are still rare to date. In this paper, using first-principles simulations, we …
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…
In a kagome lattice, the time reversal symmetry can be broken by a staggered magnetic flux emerging from ferromagnetic ordering and intrinsic spin-orbit coupling, leading to several wellseparated nontrivial Chern bands and intrinsic quantum anomalous Hall effect. Based on this idea and ab initio calculations, we propose the realization of the intrinsic quantum anomalous Hall effect in the single layer Cs2Mn3F12 kagome lattice and on the (001) surface of a Cs2LiMn3F12 single crystal by modifying the carrier coverage on it, where the band gap is around 20 meV. Moreover, a simplified tight bindin…
We report observation of intrinsic inverse spin Hall effect in un-doped GaAs multiple quantum wells with a sample temperature of 10 K. A transient ballistic pure spin current is injected by a pair of laser pulses through quantum interference. By time-resolving the dynamics of the pure spin current, the momentum relaxation time is deduced, which sets the lower limit of the scattering time between electrons and holes. The transverse charge current generated by the pure spin current via the inverse spin Hall effect is simultaneously resolved. We find that the charge current is generated well befo…
First-principles calculations are applied to study spin Hall effect in semiconductors and simple metals. We found that intrinsic spin Hall conductivity (ISHC) in realistic materials shows rich sign changes, which may be used to distinguish the effect from the extrinsic one. The calculated ISHC in n-doped GaAs can be well compared with experiment, and it differs from the sign obtained from the extrinsic effect. On the other hand, the ISHC in W and Au, which shows opposite sign respectively, is robust and not sensitive to the disorder.
We describe a new effect in semiconductor spintronics that leads to dissipationless spin-currents in paramagnetic spin-orbit coupled systems. We argue that in a high mobility two-dimensional electron system with substantial Rashba spin-orbit coupling, a spin-current that flows perpendicular to the charge current is intrinsic. In the usual case where both spin-orbit split bands are occupied, the spin-Hall conductivity has a universal value.
Valley-related multiple Hall effect in 2D lattice is a fundamental transport phenomenon in the fields of condensed-matter physics and material science. So far, most proposals for its realization are limited to toy models or extrinsic effects. Here, based on tight-binding model and first-principles calculations, we report the discovery of intrinsic valley-related multiple Hall effect in 2D organometallic lattice of NbTa-benzene. Protected by the breaking of both time-reversal and inversion symmetry, NbTa-benzene exhibits large valley polarization spontaneously in both the conduction and valence…
Spin Hall effect in metallic Pt is studied with first-principles relativistic band calculations. It is found that intrinsic spin Hall conductivity (SHC) is as large as $\sim 2000 (\hbar/e)(Ω{\rm cm})^{-1}$ at low temperature, and decreases down to $\sim 200 (\hbar/e)(Ω{\rm cm})^{-1}$ at room temperature. It is due to the resonant contribution from the spin-orbit splitting of the doubly degenerated $d$-bands at high-symmetry $L$ and $X$ points near the Fermi level. By modeling these near degeneracies by effective Hamiltonian, we show that SHC has a peak near the Fermi energy and that the vertex…
In monolayer group-VI transition metal dichalcogenides (TMDC), charge carriers have spin and valley degrees of freedom, both associated with magnetic moments. On the other hand, the layer degree of freedom in multilayers is associated with electrical polarization. Here, we show that TMDC bilayers offer an unprecedented platform to realize a strong coupling between the spin, layer pseudospin, and valley degrees of freedom of holes. Such coupling not only gives rise to the spin Hall effect and spin circular dichroism in inversion symmetric bilayer, but also leads to a variety of magnetoelectric …
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…
Topological antiferromagnetic (AFM) spintronics is an emerging field of research, which involves the topological electronic states coupled to the AFM order parameter known as the N$\acute{\rm e}$el vector. The control of these states is envisioned through manipulation of the N$\acute{\rm e}$el vector by spin-orbit torques driven by electric currents. Here we propose a different approach favorable for low-power AFM spintronics, where the control of the topological states in a two-dimensional material, such as graphene, is performed via the proximity effect by the voltage induced switching of th…
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…
A single two-dimensional Dirac cone with a mass gap produces a quantized (spin-) Hall step in the absence of magnetic field. What happens in strong electric fields? This question is investigated by analyzing time evolution and dynamics of the (spin-) Hall effect. After switching on a longitudinal electric field, a stationary Hall current is reached through damped oscillations. The Hall conductivity remains quantized as long as the electric field (E) is too weak to induce Landau-Zener transitions, but quantization breaks down for strong fields and the conductivity decreases as 1/sqrt{E}. These …
We study theoretically the role of quenched magnetic disorder at the surface of a topological insulator by numerical simulation and scaling analysis. It is found that all the surface states are localized while the transverse conductivity is quantized to be $\pm {{e^2} \over {2 h}}$ as long as the Fermi energy is within the bulk gap. This greatly facilitates the realization of the topological magnetoelectric effect proposed by Qi {\it et al.} (Phys. Rev. B\textbf{78}, 195424 (2008)) with the surface magnetization direction being controlled by the simultaneous application of magnetic and electri…
Dynamic dipolar toroidal response is demonstrated by an optical plasmonic metamaterial composed of double disks. This response with a hotspot of localized E-field concentration is a well-behaved toroidal cavity mode that exhibits a large Purcell factor due to its deep-subwavelength mode volume. All-optical Hall effect (photovoltaic) due to this optical toroidal moment is demonstrated numerically, in mimicking the magnetoelectric effect in multiferroic systems. The result shows a promising avenue to explore various optical phenomena associated with this intriguing dynamic toroidal moment.
In time-reversal invariant systems, all charge Hall effects predicted so far are extrinsic effects due to the dependence on the relaxation time. We explore intrinsic Hall signatures by studying quantum noise spectrum of the Hall current in time-reversal invariant systems, and discover intrinsic thermal Hall noises in both linear and nonlinear regimes. As the band geometric characteristics, quantum geometric tensor and Berry curvature play critical roles in various Hall effects, so are their quantum fluctuations. It is found that the thermal Hall noise in linear order of the electric field is p…
University of Science and Technology of China（中国科学技术大学）；Shenzhen University（深圳大学）；Hefei National Laboratory, University of Science and Technology of China（中国科学技术大学合肥国家实验室）；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）
In a two dimensional electron gas with Rashba spin-orbit coupling, the external electric field may cause a spin Hall current in the direction perpendicular to the electric field. This effect was called the intrinsic spin Hall effect. In this paper, we investigate the influences of spin accumulation on this intrinsic spin Hall effect. We show that due to the existence of boundaries in a real sample, the spin Hall current generated by the intrinsic spin Hall effect will cause spin accumulation near the edges of the sample, and in the presence of spin accumulation, the spin Hall conductivity will…
