Recent reports of spin-orbit coupling enhancement in chemically modified graphene have opened doors to studies of the spin Hall effect with massless chiral fermions. Here, we theoretically investigate the interaction and impurity density dependence of the extrinsic spin Hall effect in spin-orbit coupled graphene. We present a nonperturbative quantum diagrammatic calculation of the spin Hall response function in the strong-coupling regime that incorporates skew scattering and anomalous impurity density-independent contributions on equal footing. The spin Hall conductivity dependence on Fermi energy and electron-impurity interaction strength reveals the existence of experimentally accessible regions where anomalous quantum processes dominate. Our findings suggest that spin-orbit-coupled graphene is an ideal model system for probing the competition between semiclassical and bona fide quantum scattering mechanisms underlying the spin Hall effect.
We report the observation of the crossover between the extrinsic and intrinsic spin Hall effect induced by alloying. We found that the spin Hall angle, the ratio of the spin Hall conductivity to the electric conductivity, changes drastically by tuning the composition of Au-Cu alloy. The spin Hall angle changes the sign only in a limited range of the Cu concentration due to the extrinsic skew scattering, while the intrinsic contribution becomes dominant with increasing the Cu concentration. This observation provides essential information for fundamental understanding of spin-orbit physics.
We determine the composition of intrinsic as well as extrinsic contributions to the anomalous Hall effect (AHE) in the isoelectronic L1o FePd and FePt alloys. We show that the AHE signal in our 30 nm thick epitaxially deposited films of FePd is mainly due to extrinsic side-jump, while in the epitaxial FePt films of the same thickness and degree of order the intrinsic contribution is dominating over the extrinsic mechanisms of the AHE. We relate this crossover to the difference in spin-orbit strength of Pt and Pd atoms and suggest that this phenomenon can be used for tuning the origins of the A…
We study the extrinsic spin Hall effect induced by Ir impurities in Cu by injecting a pure spin current into a CuIr wire from a lateral spin valve structure. While no spin Hall effect is observed without Ir impurity, the spin Hall resistivity of CuIr increases linearly with the impurity concentration. The spin Hall angle of CuIr, $(2.1 \pm 0.6)$% throughout the concentration range between 1% and 12%, is practically independent of temperature. These results represent a clear example of predominant skew scattering extrinsic contribution to the spin Hall effect in a nonmagnetic alloy.
The drift-diffusion formalism for spin-polarized carrier transport in semiconductors is generalized to include spin-orbit coupling. The theory is applied to treat the extrinsic spin Hall effect using realistic boundary conditions. It is shown that carrier and spin diffusion lengths are modified by the presence of spin-orbit coupling and that spin accumulation due to the extrinsic spin Hall effect is strongly and qualitatively influenced by boundary conditions. Analytical formulas for the spin-dependent carrier recombination rates and inhomogeneous spin densities and currents are presented.
Recent experimental observations of unexpectedly large thermal Hall conductivities, $κ_H$, in insulating materials, including the parent compounds of the high temperature superconducting cuprates, likely reflect an extrinsic contribution from a yet to be identified extrinsic source of skew scattering of acoustic phonons. We show that resonant scattering of phonons from a certain class of three-level systems produces strong skew scattering in the presence of a modest magnetic field. We interpret this as a first step towards understanding the experiments.
Intrinsic and extrinsic spin Hall effects are considered together on an equal theoretical footing for the Rashba spin-orbit coupling in two-dimensional (2D) electron and hole systems, using the diagrammatic method for calculating the spin Hall conductivity. Our analytic theory for the 2D holes shows the expected lowest-order additive result for the spin Hall conductivity. But, the 2D electrons manifest a very surprising result, exhibiting a non-analyticity in the Rashba coupling strength $α$ where the strictly extrinsic spin Hall conductivity (for $α= 0$) cannot be recovered from the $α\to 0$ …
A key step towards dissipationless transport devices is the quantum anomalous Hall effect, which is characterized by an integer quantized Hall conductance in a ferromagnetic insulator with strong spin-orbit coupling. In this work, the anomalous Hall effect due to the impurity scattering, namely the extrinsic anomalous Hall effect, is studied when the Fermi energy crosses with the topologically nontrivial conduction band of a quantum anomalous Hall system. Two major extrinsic contributions, the side-jump and skew-scattering Hall conductivities, are calculated using the diagrammatic techniques i…
In order to describe correctly the interplay of extrinsic and intrinsic spin-orbit mechanisms to the spin Hall effect, it is necessary to consider different sources of spin relaxation. We take into account the spin relaxation time $τ_{DP}$ due to the Dyakonov-Perel mechanism as well as the Elliot-Yafet spin-relaxation time $τ_s$ due to the spin-orbit scattering from impurities. The total spin Hall conductivity depends crucially on the ratio $τ_s /τ_{DP}$.
We show that the extrinsic spin Hall effect can be engineered in monolayer graphene by decoration with small doses of adatoms, molecules, or nanoparticles originating local spin-orbit perturbations. The analysis of the single impurity scattering problem shows that intrinsic and Rashba spin-orbit local couplings enhance the spin Hall effect via skew scattering of charge carriers in the resonant regime. The solution of the transport equations for a random ensemble of spin-orbit impurities reveals that giant spin Hall currents are within the reach of the current state of the art in device fabrica…
We study the effect of anisotropy of the Rashba coupling on the extrinsic spin Hall effect due to spin-orbit active adatoms on graphene. In addition to the intrinsic spin-orbit coupling, a generalized anisotropic Rashba coupling arising from the breakdown of both mirror and hexagonal symmetries of pristine graphene is considered. We find that Rashba anisotropy can strongly modify the dependence of the spin Hall angle on carrier concentration. Our model provides a simple and general description of the skew scattering mechanism due to the spin-orbit coupling that is induced by proximity to large…
We investigate the spin Hall effect (SHE) of electrons described by the Dirac equation, which is used as an effective model near the $L$-points in bismuth. By considering short-range nonmagnetic impurities, we calculate the extrinsic as well as intrinsic contributions on an equal footing. The vertex corrections are taken into account within the ladder type and the so-called skew-scattering type. The intrinsic SHE which we obtain is consistent with that of Fuseya et al. [J. Phys. Soc. Jpn. 81, 93704 (2012)]. It is found that the extrinsic contribution dominates the intrinsic one when the system…
We report an experimental study of a gold-tungsten alloy (7% at. W concentration in Au host) displaying remarkable properties for spintronics applications using both magneto-transport in lateral spin valve devices and spin-pumping with inverse spin Hall effect experiments. A very large spin Hall angle of about 10% is consistently found using both techniques with the reliable spin diffusion length of 2 nm estimated by the spin sink experiments in the lateral spin valves. With its chemical stability, high resistivity and small induced damping, this AuW alloy may find applications in the nearest …
The distribution of the Hall voltage induced by low-frequency AC current is studied theoretically in the incoherent linear transport of quantum Hall systems. It is shown that the Hall-voltage distribution makes a crossover from the uniform distribution to a concentrated-near-edges distribution as the frequency is increased or the diagonal conductivity is decreased. This crossover is also reflected in the frequency dependence of AC magnetoresistance.
We study the intrinsic and extrinsic Hall effects in disordered magnetic Weyl semimetals numerically. We show that in Weyl metals, where the Fermi energy deviates from the Weyl point, the Hall and longitudinal conductances exhibit a specific relation, which is distinguished from the well-known relation in integer quantum Hall systems. Around the Weyl point, the Hall conductance increases with increasing longitudinal conductance. This increasing behavior indicates the existence of additional contributions to the Hall conductance from the subbands of Weyl cones besides that from the bulk Berry c…
We present a rigorous microscopic theory of the extrinsic spin Hall effect in disordered graphene based on a nonperturbative quantum diagrammatic treatment incorporating skew scattering and anomalous---impurity concentration-independent---quantum corrections on equal footing. The leading skew scattering contribution to the spin Hall conductivity is shown to quantitatively agree with Boltzmann transport theory over a wide range of parameters. Our self-consistent approach---where all topologically equivalent noncrossing diagrams are resummed---unveils that the skewness generated by spin--orbit-a…
Charge-to-spin conversion in inhomogeneous systems is studied theoretically. We consider free electrons subject to impurities with spin-orbit interaction and with spatially modulated distribution, and calculate spin accumulation and spin current induced by an external electric field. It is found that the spin accumulation is induced by the vorticity of electron flow through the side-jump and skew-scattering processes, and the differences of the two processes are discussed. The results can be put in a form of generalized spin diffusion equation with a spin source term given by the divergence of…
The kinetics of the extrinsic spin Hall conductivity induced by the skew scattering is performed from the fully microscopic kinetic spin Bloch equation approach in $(001)$ GaAs symmetric quantum well. In the steady state, the extrinsic spin Hall current/conductivity vanishes for the linear-$\mathbf k$ dependent spin-orbit coupling and is very small for the cubic-$\mathbf k$ dependent spin-orbit coupling. The spin precession induced by the Dresselhaus/Rashba spin-orbit coupling plays a very important role in the vanishment of the extrinsic spin Hall conductivity in the steady state. An in-plane…
A unified theory of the anomalous Hall effect (AHE) is presented for multi-band ferromagnetic metallic systems with dilute impurities. In the clean limit, the AHE is mostly due to the extrinsic skew-scattering. When the Fermi level is located around anti-crossing of band dispersions split by spin-orbit interaction, the intrinsic AHE to be calculated ab initio is resonantly enhanced by its non-perturbative nature, revealing the extrinsic-to-intrinsic crossover which occurs when the relaxation rate is comparable to the spin-orbit interaction energy.
We investigate the extrinsic spin Hall effect in the electron gas model due to magnetic impurities, by focusing on Ce- and Yb-impurities. In the dilute limit, the skew scattering term dominates the side jump term. For Ce-impurities, the spin Hall angle $α_{\rm SH}$ due to skew scattering is given by $-8π\sinδ_2/7$, where $δ_2 (\ll 1)$ is the phase shift for$d (l=2)$ partial wave. Since $α_{\rm SH}$ reaches $O(10^{-1})$ if $δ_2 \simge 0.03$, the spin Hall effect is anticipated to be considerable in metals with rare-earth impurities. The giant extrinsic SHE originates from the large orbital angu…
The 2-- to 1--dimensional crossover of the localisation length of electrons confined to a disordered quantum wire of finite width $L_y$ is studied in a model of electrons moving in the potential of uncorrelated impurities. An analytical formula for the localisation length is derived, describing the dimensional crossover as function of width $L_y$, conductance $g$ and perpendicular magnetic field $B$ . On the basis of these results, the scaling analysis of the quantum Hall effect in high Landau levels, and the delocalisation transition in a quantum Hall wire are reconsidered.
We show that pure spin Hall current, flowing out of a four-terminal phase-coherent two-dimensional electron gas (2DEG) within inversion asymmetric semiconductor heterostructure, contains contributions from both the extrinsic mechanisms (spin-orbit dependent scattering off impurities) and the intrinsic ones (due to the Rashba coupling). While the extrinsic contribution vanishes in the weakly and strongly disordered limits, and the intrinsic one dominates in the quasiballistic limit, in the crossover transport regime the spin Hall conductance, exhibiting sample-to-sample large fluctuations and s…
In this paper, we study topological properties and Hall conductivities in PbC/MnSe heterostructure under the illumination of a circularly polarized light. At high frequency regime, energy gap, Chern numbers, and Hall conductivities are studied based on the Floquet theory and Green's function formalism, respectively. The interplay between spin orbit coupling and light leads to topological phase transition between anomalous Hall states and spin Hall states, which is related to the emission and absorption of two virtual photons. The anomalous Hall conductivities are dependent on polarization of l…
We investigate the extrinsic spin Hall effect in the electron gas model due to transition-metal impurities based on the single-impurity Anderson model with orbital degrees of freedom. Both the skew scattering and side jump mechanisms are analyzed in a unified way, and the significant role of orbital degrees of freedom are clarified. The obtained spin Hall conductivities are in proportion to the spin-orbit polarization at the Fermi level ($R = <l \cdot s>$) as is the case with the intrinsic spin Hall effect: skew scattering term $σ^{ss} ~ R δ_1/ρ$, and side jump term $σ^{sj} ~ R$, where $δ_1$ i…
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…
We present a new theoretical approach for the integer quantum Hall effect, which is able to describe the inter-plateau transitions as well as the transition to the Hall insulator. We find two regimes (metallic and insulator like) of the top Landau level, in which the dissipative bulk current appears in different directions. The regimes are separated by a temperature invariant point.
We theoretically study the crossover between spin Hall effect and spin swapping, a recently predicted phenomenon that consists in the interchange between the current flow and its spin polarization directions [Lifshits and Dyakonov, Phys. Rev. Lett. 103, 186601 (2009)]. Using a tight-binding model with spin-orbit coupled disorder, spin Hall effect, spin relaxation and spin swapping are treated on equal footing. We demonstrate that spin Hall effect and spin swapping present very different dependences as a function of the spin-orbit coupling and disorder strengths. As a consequence, we show that …
Motivated by recent experiments on the phonon contribution to the thermal Hall effect in the cuprates, we present an analysis of chiral phonon transport. We assume the chiral behavior arises from a non-zero phonon Hall vicosity, which is likely induced by the coupling to electrons. Phonons with a non-zero phonon Hall viscosity have an intrinsic thermal Hall conductivity, but Chen et al. (Phys. Rev. Lett. 124, 167601 (2020)) have argued that a significantly larger thermal Hall conductivity can arise from an extrinsic contribution which is inversely proportional to the density of impurities. We …
Extrinsic Orbital Hall Effect and Orbital Relaxation in Mesoscopic Devices
Universidade Federal Rural de Pernambuco（巴西联邦农村大学）；Pohang University of Science and Technology (POSTECH)（韩国科学技术院）；Centro Brasileiro de Pesquisas Físicas（巴西物理研究中心）；Physics Center of Minho and Porto Universities (CF-UM-UP)（米尼奥与波尔图大学物理中心）；International Iberian Nanotechnology Laboratory (INL)（伊比利亚纳米技术国际实验室）
We incorporate quantum size effect to investigate the extrinsic spin-Hall effect in ultrathin metal films. A Lippmann-Schwinger formalism based theoretical method, accounting for quantum confinement and surface roughness scattering, is developed to calculate both spin-Hall and longitudinal resistivities and spin-Hall angle. The presence of quantum confinement gives rise to a linear relation $ρ_{sH}=αρ+β$ between the extrinsic spin-Hall resistivity $ρ_{sH}$ and longitudinal charge resistivity $ρ$. The linear term $αρ$ originates from side jump, and the constant $β$ is due to skew scattering. Th…
We develop the microscopic theory of the extrinsic spin Hall conductivity of a two-dimensional electron gas, including skew-scattering, side-jump, and Coulomb interaction effects. We find that while the spin-Hall conductivity connected with the side-jump is independent of the strength of electron-electron interactions, the skew-scattering term is reduced by the spin-Coulomb drag, so the total spin current and the total spin-Hall conductivity are reduced for typical experimental mobilities. Further, we predict that in paramagnetic systems the spin-Coulomb drag reduces the spin accumulations in …
We propose a new theory of the topological Hall effect (THE) in systems with chiral magnetization vortices such as magnetic skyrmions. We solve the problem of electron scattering on a magnetic skyrmion exactly, for an arbitrary strength of exchange interaction and the skyrmion size. We report the existence of different regimes of THE and resolve the apparent contradiction between the adiabatic Berry phase theoretical approach and the perturbation theory for THE. We traced how the topological charge Hall effect transforms into the spin Hall effect upon varying the exchange interaction strength …
We present a theory of the anomalous Hall effect (AHE) in a doped Weyl semimetal, or Weyl metal, including both intrinsic and extrinsic (impurity scattering) contributions. We demonstrate that a Weyl metal is distinguished from an ordinary ferromagnetic metal by the absence of the extrinsic and the Fermi surface part of the intrinsic contributions to the AHE, as long as the Fermi energy is sufficiently close to the Weyl nodes. The AHE in a Weyl metal is thus shown to be a purely intrinsic, universal property, fully determined by the location of the Weyl nodes in the first Brillouin zone.
We present a comparative study of the (magneto)transport properties, including Hall effect, of bulk, thin film and nanostructured MnSi. In order to set our results in relation to published data we extensively characterize our materials, this way establishing a comparatively good sample quality. Our analysis reveals that in particular for thin film and nanostructured material, there are extrinsic and intrinsic contributions to the electronic transport properties, which by modeling the data we separate out. Finally, we discuss our Hall effect data of nanostructured MnSi under consideration of th…
Based on a newly derived microscopic fractional quantum Hall edge model, we study its thermodynamics at finite temperature. For the dressed energy spectrum a critical energy scale determined by the temperature exists, below which the refractive dispersion which is essential to the model's bosonization to a Luttinger liquid is smeared. According to this observation, a temperature-dependent crossover picture is proposed, and applied to the analysis of a recent tunneling experiment in comparison with the standard Luttinger liquid theory, and a better fit to the features of the measured conductanc…
We extend the electrodynamics of two dimensional electron gases to account for the extrinsic spin Hall effect (SHE). The theory is applied to doped graphene decorated with a random distribution of absorbates that induce spin-orbit coupling (SOC) by proximity. The formalism extends previous semiclassical treatments of the SHE to the non-local dynamical regime. Within a particle-number conserving approximation, we compute the conductivity, dielectric function, and spin Hall angle in the small frequency and wave vector limit. The spin Hall angle is found to decrease with frequency and wave number…
Suspended graphene nano-ribbons formed during current annealing of suspended graphene flakes have been investigated experimentally. Transport measurements show the opening of a transport gap around charge neutrality due to the formation of "Coulomb islands", coexisting with quantum Hall conductance plateaus appearing at moderate values of magnetic field $B$. Upon increasing $B$, the transport gap is rapidly suppressed, and is taken over by a much larger energy gap due to electronic correlations. Our observations show that suspended nano-ribbons allow the investigation of phenomena that could n…
We theoretically study the contribution of superconducting fluctuation to extrinsic spin Hall effects in two- and three-dimensional electron gas and intrinsic spin Hall effects in two-dimensional electron gas with Rashba-type spin-orbit interaction. The Aslamazov-Larkin, Density-of-States, Maki-Thompson terms have logarithmic divergence $\lnε$ in the limit $ε=(T- T_{\mathrm{c}})/T_{\mathrm{c}} \rightarrow +0$ in two-dimensional systems for both extrinsic and intrinsic spin Hall effects except the Maki-Thompson terms in extrinsic effect, which are proportional to $(ε-γ_φ)^{-1}\lnε$ with a cutof…
The spin Hall effect (SHE), induced by spin-orbit interaction in nonmagnetic materials, is one of the promising phenomena for conversion between charge and spin currents in spintronic devices. The spin Hall (SH) angle is the characteristic parameter of this conversion. We have performed experiments of the conversion from spin into charge currents by the SHE in lateral spin valve structures. We present experimental results on the extrinsic SHEs induced by doping nonmagnetic metals, Cu or Ag, with impurities having a large spin-orbit coupling, Bi or Pb, as well as results on the intrinsic SHE of…
Injection of unpolarized charge current through the longitudinal leads of a four-terminal two-dimensional electron gas with the Rashba spin-orbit (SO) coupling and/or SO scattering off extrinsic impurities is responsible not only for the pure spin Hall current in the transverse leads, but also for random time-dependent current fluctuations. We employ the scattering approach to current-current correlations in multiterminal nanoscale conductors to analyze the shot noise of transverse pure spin Hall and zero charge current, or transverse spin current and non-zero charge Hall current, driven by un…
New low-lying excitations are observed by inelastic light scattering at filling factors $ν=p/(ϕp \pm 1)$ of the fractional quantum Hall regime with $ϕ=4$. Coexisting with these modes throughout the range $ν\leq 1/3$ are $ϕ=2$ excitations seen at 1/3. Both $ϕ=2$ and $ϕ=4$ excitations have distinct behaviors with temperature and filling factor. The abrupt first appearance of the new modes in the low energy excitation spectrum at $ν\lesssim 1/3$ suggests a marked change in the quantum ground state on crossing the $ϕ=2 \to ϕ=4$ boundary at $ν= 1/3$.
Auto-oscillations of magnetization driven by direct spin current have been previously observed in multiple quasi-zero-dimensional (0D) ferromagnetic systems such as nanomagnets and nanocontacts. Recently, it was shown that pure spin Hall current can excite coherent auto-oscillatory dynamics in quasi-one-dimensional (1D) ferromagnetic nanowires but not in quasi-two-dimensional (2D) ferromagnetic films. Here we study the 1D to 2D dimensional crossover of current-driven magnetization dynamics in wire-based Pt/$\mathrm{Ni}_{80}\mathrm{Fe}_{20}$ bilayer spin Hall oscillators via varying the wire wi…
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…
Electrons hopping in two-dimensional honeycomb lattices possess a valley degree of freedom in addition to charge and spin. In the absence of inversion symmetry, these systems were predicted to exhibit opposite Hall effects for electrons from different valleys. Such valley Hall effects have been achieved only by extrinsic means, such as substrate coupling, dual gating, and light illuminating. Here, we report the first observation of intrinsic valley Hall transport without any extrinsic symmetry breaking in the non-centrosymmetric monolayer and trilayer MoS2, evidenced by considerable nonlocal r…
We propose a new scaling law for anomalous Hall effect in ferromagnetic thin films by distinguishing three scattering sources, namely, bulk impurity, phonon, and more importantly a rough surface. This new scaling law fits the recent experimental data excellently with constant coefficients that are independent of temperature and film thickness. This is in stark constrast with previous scaling laws that use temperature/thickness dependent fitting coefficients, and is a strong indicator that this law captures the essential physics. By intepretating the experiments for Fe, Co, and Ni with this new…
An analytic form for the conductivity tensor in crossover between two quantum Hall plateaux is derived, which appears to be in good agreement with existing experimental data. The derivation relies on an assumed symmetry between quantum Hall states, a generalisation of the law of corresponding states from rational filling factors to complex conductivity, which has a mathematical expression in terms of an action of the modular group on the upper-half complex conductivity plane. This symmetry implies universality in quantum Hall crossovers. The assumption that the $β$-function for the complex con…
