Extrinsic Spin-orbit Research Articles

Phase diagrams and topological mixed edge states in silicene with intrinsic and extrinsic Rashba effects

We investigate the topological phases and corresponding edge states in silicene with fixed intrinsic and extrinsic Rashba spin-orbit couplings. The results show that silicene can exhibit four distinct topological phases, such as two types of valley-polarized quantum anomalous Hall effect (QAHE) and two types of QAHE that can be effectively controlled by the exchange field and electric field. In addition, these topological phases only correspond to the valley edge states distributed at the boundary of silicene. In particular, when the off-resonant circularly polarized (ORCP) light is further considered, silicene can host richer topological phases that are characterized by the Chern numbers from C=−3 to C=3, where the corresponding mixed edge states—where the valley and nonvalley edge states could coexist—can remarkably appear. More interestingly, the special higher Chern numbers C=±3 determine two valley edge states and one nonvalley edge state, with the right-moving or left-moving direction. These peculiar edge states are attributed to the ORCP light destroying the collapse of the Dirac cones that the valley edge states bridge. Additionally, silicene can experience multiple edge-state transitions for identical Chern numbers C=±2 between the valley edge state, nonvalley edge state, and two types of mixed edge state, under the ORCP light. This work may provide a platform and point of view for investigating other forms of topological edge state and is beneficial for designing multifunctional topological devices. Published by the American Physical Society 2024

Spin-orbit torques due to extrinsic spin-orbit scattering of topological insulator surface states: out-of-plane magnetization

The origins of the spin-orbit torque (SOT) at ferromagnet/topological insulator interfaces are incompletely understood. The theory has overwhelmingly focussed on the Edelstein effect due to the surface states in the presence of a scalar scattering potential. We investigate here the contribution to the SOT due to extrinsic spin-orbit (SO) scattering of the surface states, focusing on the case of an out-of-plane magnetization. We show that SO scattering brings about a sizable renormalization of the field-like SOT, which exceeds 20 at larger strengths of the extrinsic SO parameter. The resulting SOT exhibits a maximum as a function of the Fermi energy, magnetization, and extrinsic SO strength. The field-like SOT decreases with increasing disorder strength, while the damping-like SOT is independent of the impurity density. With experimental observation in mind we also determine the role of extrinsic SO scattering on the anomalous Hall effect. Our results suggest extrinsic SO scattering is a significant contributor to the surface SOT stemming from the Edelstein effect when the magnetization is out of the plane.

Dirac imprints on the g -factor anisotropy in graphene

Dirac electrons in graphene are to lowest order spin 1/2 particles, owing to the orbital symmetries at the Fermi level. However, anisotropic corrections in the $g$-factor appear due to the intricate spin-valley-orbit coupling of chiral electrons. We resolve experimentally the $g$-factor along the three orthogonal directions in a large-scale graphene sample. We employ a Hall bar structure with an external magnetic field of arbitrary direction, and extract the effective $g$-tensor via resistively-detected electron spin resonance. We employ a theoretical perturbative approach to identify the intrinsic and extrinsic spin orbit coupling and obtain a fundamental parameter inherent to the atomic structure of $^{12}$C, commonly used in ab-initio models.

Quasiclassical theory of the spin–orbit magnetoresistance of three-dimensional Rashba metals

The magnetoresistance of a three-dimensional Rashba metal placed on top of a ferromagnetic insulator is theoretically investigated. In addition to the intrinsic Rashba spin–orbit interaction, we consider extrinsic spin–orbit coupling via side-jump and skew scattering, as well as Elliott–Yafet spin relaxation. The latter is anisotropic due to the mass anisotropy which reflects the noncentrosymmetric crystal structure. A quasiclassical approach is employed to derive coupled spin-diffusion equations, which are supplemented by boundary conditions that account for the spin-transfer torque at the interface of the bilayer. The magnetoresistance is fully determined by the current-induced spin polarization, i.e., it cannot in general be ascribed to a single (bulk) spin Hall angle. Our theory reproduces several features of the experiments, at least qualitatively, and contains established phenomenological results in the relevant limiting cases. In particular, the anisotropy of the Elliott–Yafet spin relaxation mechanism plays a major role for the interpretation of the observed magnetoresistance.

Dirac Cone Spin Polarization of Graphene by Magnetic Insulator Proximity Effect Probed with Outermost Surface Spin Spectroscopy

AbstractThe effects of the proximity contact with magnetic insulator on the spin‐dependent electronic structure of graphene are explored for the heterostructure of single‐layer graphene (SLG) and yttrium iron garnet Y3Fe5O12 (YIG) by means of outermost surface spin spectroscopy using a spin‐polarized metastable He atom beam. In the SLG/YIG heterostructure, the Dirac cone electrons of graphene are found to be negatively spin polarized in parallel to the minority spins of YIG with a large polarization degree, without giving rise to significant changes in the π band structure. Theoretical calculations reveal the electrostatic interfacial interactions providing a strong physical adhesion and the indirect exchange interaction causing the spin polarization of SLG at the interface with YIG. The Hall device of the SLG/YIG heterostructure exhibits a nonlinear Hall resistance attributable to the anomalous Hall effect, implying the extrinsic spin–orbit interactions as another manifestation of the proximity effect.

Spin properties of black phosphorus and phosphorene, and their prospects for spincalorics

Semiconducting black phosphorus attracts a lot of attention due to its extraordinary electronic properties. Its application to spincalorics requires the knowledge about the spin and thermal properties. Here, we describe first principles calculations of the spin–orbit coupling and spin scattering in phosphorene and bulk black phosphorus. We find that the intrinsic spin–orbit coupling is of the order of 20 meV for the valence and conduction band, both for phosphorene and bulk black phosphorus, and induces spin mixing with the probability –10−4. A strong anisotropy of b2 is observed. The calculated Elliott–Yafet spin relaxation times reach nanoseconds for realistic values of the momentum relaxation times. The extrinsic spin–orbit coupling, enabling the D’yakonov–Perel’ spin relaxation mechanism, is studied for phosphorene by application of a transverse electric field. We observe a strong anisotropy of the extrinsic effects for the valence band and much weaker for the conduction band. It is shown, that for small enough electric fields the spin relaxation is dominated by the Elliott–Yafet mechanism, while the D’yakonov–Perel’ matters for higher electric fields. Our theoretical results stay in a good agreement with the experimental findings, and indicates that long spin lifetimes in black phosphorus and phosphorene makes them prospective materials for spincalorics and spintronics.

Electromagnetic wave propagating along a space curve

Using the thin-layer approach, we derive the effective equation for the electromagnetic wave propagating along a space curve. We find intrinsic spin-orbit, extrinsic spin-orbit and extrinsic orbital angular momentum and intrinsic orbital angular momentum couplings induced by torsion, which can lead to geometric phase, spin and orbital Hall effects. And we show the helicity inversion induced by curvature that can convert the right-handed circularly polarized electromagnetic wave into left-handed polarized one, vice verse. Finally, we demonstrate that the gauge invariance of the effective dynamics is protected by the geometrically induced gauge potential.

Extrinsic Spin–Orbit Coupling-Induced Large Modulation of Gilbert Damping Coefficient in CoFeB Thin Film on the Graphene Stack with Different Defect Density

Control of Gilbert damping parameter is imperative for various spintronic and magnonic devices, and various schemes have been attempted to achieve that. We report a large tunability of Gilbert damping by varying the underlayer of CoFeB thin film from few-layer graphene (FLG) to graphite layer. We measured the ultrafast magnetization dynamics of CoFeB, FLG/CoFeB, and graphite/CoFeB by using time-resolved magneto-optical Kerr effect (TR-MOKE) magnetometry. While the magnetization precession frequency remained independent of the underlayer, a very large variation (∼200%) in the value of the Gilbert damping coefficient α is observed from FLG/CoFeB (α ≈ 0.035 ± 0.005) to graphite/CoFeB (α ≈ 0.008 ± 0.001). This large variation of the damping coefficient is understood in terms of the extrinsic spin–orbit interaction of FLG and graphite films, which is very large in FLG due to the presence of large amount of surface defects in it. A faster demagnetization time and fast relaxation time (τ1) were noted for graphit...

Weak Localization and Antilocalization in Topological Materials with Impurity Spin-Orbit Interactions.

Topological materials have attracted considerable experimental and theoretical attention. They exhibit strong spin-orbit coupling both in the band structure (intrinsic) and in the impurity potentials (extrinsic), although the latter is often neglected. In this work, we discuss weak localization and antilocalization of massless Dirac fermions in topological insulators and massive Dirac fermions in Weyl semimetal thin films, taking into account both intrinsic and extrinsic spin-orbit interactions. The physics is governed by the complex interplay of the chiral spin texture, quasiparticle mass, and scalar and spin-orbit scattering. We demonstrate that terms linear in the extrinsic spin-orbit scattering are generally present in the Bloch and momentum relaxation times in all topological materials, and the correction to the diffusion constant is linear in the strength of the extrinsic spin-orbit. In topological insulators, which have zero quasiparticle mass, the terms linear in the impurity spin-orbit coupling lead to an observable density dependence in the weak antilocalization correction. They produce substantial qualitative modifications to the magnetoconductivity, differing greatly from the conventional Hikami-Larkin-Nagaoka formula traditionally used in experimental fits, which predicts a crossover from weak localization to antilocalization as a function of the extrinsic spin-orbit strength. In contrast, our analysis reveals that topological insulators always exhibit weak antilocalization. In Weyl semimetal thin films having intermediate to large values of the quasiparticle mass, we show that extrinsic spin-orbit scattering strongly affects the boundary of the weak localization to antilocalization transition. We produce a complete phase diagram for this transition as a function of the mass and spin-orbit scattering strength. Throughout the paper, we discuss implications for experimental work, and, at the end, we provide a brief comparison with transition metal dichalcogenides.

Armchair-edged nanoribbon as a bottleneck to electronic total transmission through a topologically nontrivial graphene nanojunction

It is currently a promising approach to experimentally realize the topological insulator phase transition of graphene by introducing the extrinsic spin–orbit coupling (SOC). Then, electronic total transmission through various topological nontrivial graphene nanojunctions (GNJs) is obtainable, if the electronic transport is supported by the helical edge states. Though the bulk graphene is a gapless semiconductor, the inter-valley scattering could introduce a topological trivial gap in semiconducting armchair-edged graphene nanoribbon (GNR). The SOC should be strong enough to reopen a topological nontrivial gap before close such a trivial gap. Therefore, our theoretical study indicates that a semiconducting armchair-edged graphene nanoribbon (GNR) can not develop the helical edge states when the SOC strength is lower than a threshold, though the bulk phase is topological nontrivial. This implies a competition between the SOC and the inter-valley scattering. However, for a metallic armchair-edged GNR, a small SOC can always open a nontrivial gap. Nevertheless, the helical edge state is much less localized than that in a zigzag-edged GNR of the same width. As a result, and by numerically calculating the electronic transmission spectrum of step- and L-shaped GNJs, we conclude that when an armchair-edged GNR is a part of a GNJ, it is the weak point to realize the electronic total transmission even though the bulk phase of graphene is topologically insulating.

Strain-controlled spin and charge pumping in graphene devices via spin-orbit coupled barriers

We theoretically propose a graphene-based adiabatic quantum pump with intrinsic spin-orbit coupling (SOC) subject to strain where two time-dependent extrinsic spin-orbit coupled barriers drive spin and charge currents. We study three differing operation modes where i) location, ii) chemical potential, and iii) SOC of the two barriers oscillate periodically and out of phase around their equilibrium states. Our results demonstrate that the amplitude of adiabatically pumped currents highly depends on the considered operation mode. We find that such a device operates with highest efficiency and in a broader range of parameters where the barriers' chemical potential drives the quantum pump. Our results also reveal that by introducing strain to the system, one can suppress or enhance the charge and spin currents separately, depending on the strain direction.

Spin and pseudospin in monolayer graphene: Part II. Competition between exchange and spin–orbit interactions

Spin and pseudospin properties of monolayer graphene, when both the exchange and extrinsic spin–orbit (SO) interactions are taken into account, are analyzed within a framework of geometric algebra. The rotor equations for even and odd parts of electron bispinor are constructed in three-dimensional (3D) Euclidean space thus providing clear geometrical interpretation to the problem. It is shown that in the presence of combined action of exchange and SO interactions the spin and pseudospin fields in a monolayer graphene from two-dimensional become 3D, with spin and pseudospin components pointing out of the graphene plane. Also, the effect of both interactions on the Berry phase is considered analytically.

First-principles study of Bi and Sb intercalated graphene on SiC(0001) substrate

The atomic structures and electronic properties of antimony and bismuth intercalated graphene on SiC(0001) substrate were investigated using first-principles calculations. The results show that, at Bi and Sb coverages of 1ML, the Dirac cones were preserved. The π and π* bands of the graphene are barely split by extrinsic spin–orbit interaction from metal layers. At 2/3ML, the Bi and Sb metal layers form a honeycomb structure, and the Dirac cones remain intact. Furthermore, a notable band splitting at the M point contributed from metal layers was found at 1ML, whereas a similar splitting at the K point was found at 2/3ML. Finally, our results show that these two metal intercalations lead to n-type doping of graphene.

Onsager Relations in a Two-Dimensional Electron Gas with Spin-Orbit Coupling

Theory predicts for the two-dimensional electron gas with only a Rashba spin-orbit interaction a vanishing spin Hall conductivity and at the same time a finite inverse spin Hall effect. We show how these seemingly contradictory results are compatible with the Onsager relations: The latter do hold for spin and particle (charge) currents in the two-dimensional electron gas, although (i) their form depends on the experimental setup and (ii) a vanishing bulk spin Hall conductivity does not necessarily imply a vanishing spin Hall effect. We also discuss the situation in which extrinsic spin orbit from impurities is present and the bulk spin Hall conductivity can be different from zero.

Influence of the spin–orbit interaction in the impurity-band states of n-doped semiconductors

We study numerically the effects of an extrinsic spin–orbit interaction on the model of electrons in n-doped semiconductors of Matsubara and Toyozawa (MT). We focus on the analysis of the density of states (DOS) and the inverse participation ratio (IPR) of the spin–orbit perturbed states in the MT set of energy eigenstates in order to characterize the eigenstates with respect to their extended or localized nature. The finite sizes that we are able to consider necessitate an enhancement of the spin–orbit coupling strength in order to obtain a meaningful perturbation. The IPR and DOS are then studied as a function of the enhancement parameter.

Influence of impurity-induced spin–orbit scattering on persistent spin helix

The persistent spin helix discovered in intrinsic spin–orbit coupled systems previously is reexamined using the motion equations of Green's functions by considering the effect of extrinsic impurity-induced spin–orbit coupling. We find both the intrinsic and extrinsic spin–orbit couplings can increase the excitation energy of spin helix. They together can reduce drastically the lifetime of the spin helix, making it severely departure from the ideal infinite value. The effect of impurity density on spin helix is also analyzed. The results may be helpful to understand experimental measurements on spin helix.

Anomalous Hall effect in two-dimensional systems with magnetization: The roles of electron-impurity and electron-phonon scatterings

We present a kinetic equation approach to investigate the anomalous Hall effect in two-dimensional systems with magnetization considering both electron-impurity and electron-phonon scatterings. In our study, the spin-orbit interaction due to the external driving electric field as well as the extrinsic spin-orbit couplings induced by electron-impurity and electron-phonon scatterings are taken into account, while the intrinsic Rashba and Dresselhaus spin-orbit couplings are ignored. We derive the side-jump contributions to anomalous Hall current in terms of the distribution function and obtain the skew-scattering contribution by considering electron-impurity (and electron-phonon) scattering up to the second Born approximation. By performing a numerical calculation for InSb-based quantum wells, the temperature dependencies of the various components of anomalous Hall current are examined. We also discuss the roles of electron-impurity and electron-phonon scatterings in contributing to the total anomalous Hall current.

Anomalous Hall effect induced by a pure longitudinal spin injection

Recently, a nonvanishing anomalous Hall resistivity was observed accompanying a pure spin injection into aluminum through a ferromagnetic contact [Nature 442, (2006) 176]. In this paper, we present a quantitative explanation of this experimental observation considering intrinsic spin–orbit coupling, as well as an extrinsic spin–orbit interaction induced by electron-impurity scattering. In connection with this, we evaluate the band structure and wave function of aluminum by the orthogonalized-plane-wave method and determine the contributions to anomalous Hall resistivity from side-jump and skew scatterings after numerically solving a spatially dependent Boltzmann equation. The obtained Hall resistivity is in good agreement with the experimental data.

Anomalous Hall effect in Rashba two-dimensional electron systems based on narrow-band semiconductors: Side-jump and skew scattering mechanisms

We employ a helicity-basis kinetic equation approach to investigate the anomalous Hall effect in two-dimensional narrow-band semiconductors considering both Rashba and extrinsic spin-orbit (SO) couplings, as well as a SO coupling directly induced by an external driving electric field. Taking account of long-range electron-impurity scattering up to the second Born approximation, we find that the various components of the anomalous Hall current fit into two classes: (a) side-jump and (b) skew scattering anomalous Hall currents. The side-jump anomalous Hall current involves contributions not only from the extrinsic SO coupling but also from the SO coupling due to the driving electric field. It also contains a component which arises from the Rashba SO coupling and relates to the off-diagonal elements of the helicity-basis distribution function. The skew scattering anomalous Hall effect arises from the anisotropy of the diagonal elements of the distribution function and it is a result of both the Rashba and extrinsic SO interactions. Further, we perform a numerical calculation to study the anomalous Hall effect in a typical $\mathrm{InSb}∕\mathrm{AlInSb}$ quantum well. The dependencies of the side-jump and skew scattering anomalous Hall conductivities on magnetization and on the Rashba SO coupling constant are examined.