Rashba Model Research Articles

Spin-orbit-coupled transport and spin torque in a ferromagnetic heterostructure

Ferromagnetic heterostructures provide an ideal platform to explore the nature of spin-orbit torques arising from the interplay mediated by itinerant electrons between a Rashba-type spin-orbit coupling and a ferromagnetic exchange interaction. For such a prototypic system, we develop a set of coupled diffusion equations to describe the diffusive spin dynamics and spin-orbit torques. We characterize the spin torque and its two prominent---out-of-plane and in-plane---components for a wide range of relative strength between the Rashba coupling and ferromagnetic exchange. The symmetry and angular dependence of the spin torque emerging from our simple Rashba model is in an agreement with experiments. The spin diffusion equation can be generalized to incorporate dynamic effects such as spin pumping and magnetic damping.

Tunable polarization in a beam splitter based on two-dimensional topological insulators

The typical bulk model describing 2D topological insulators (TI) consists of two types of spin-orbit terms, the so-called Dirac term, which induces out-of-plane spin polarization, and the Rashba term, which induces in-plane spin polarization. We show that for some parameters of the Fermi energy, the beam splitter device built on 2D TIs can achieve higher in-plane spin polarization than the one built on materials described by the Rashba model itself. Further, due to high tunability of the electron density and the asymmetry of the quantum well, spin polarization in different directions can be obtained. While in the normal (topologically trivial) regime the in-plane spin polarization would dominate, in the inverted regime, the out-of-plane polarization is more significant not only in the band gap but also for small Fermi energies above the gap. Further, we suggest a double beam splitter scheme to measure in-plane spin current all-electrically. Although we consider here as an example HgTe/CdTe quantum wells, this scheme could be also promising for InAs/GaSb QWs where the in- and out-of-plane polarization could be achieved in a single device.

Current-induced torques and interfacial spin-orbit coupling

In bilayer systems consisting of an ultrathin ferromagnetic layer adjacent to a metal with strong spin-orbit coupling, an applied in-plane current induces torques on the magnetization. The torques that arise from spin-orbit coupling are of particular interest. Here, we calculate the current-induced torque in a Pt-Co bilayer to help determine the underlying mechanism using first principles methods. We focus exclusively on the analogue to the Rashba torque, and do not consider the spin Hall effect. The details of the torque depend strongly on the layer thicknesses and the interface structure, providing an explanation for the wide variation in results found by different groups. The torque depends on the magnetization direction in a way similar to that found for a simple Rashba model. Artificially turning off the exchange spin splitting and separately the spin-orbit coupling potential in the Pt shows that the primary source of the "field-like" torque is a proximate spin-orbit effect on the Co layer induced by the strong spin-orbit coupling in the Pt.

Chirality from Interfacial Spin-Orbit Coupling Effects in Magnetic Bilayers

As nanomagnetic devices scale to smaller sizes, spin-orbit coupling due to the broken structural inversion symmetry at interfaces becomes increasingly important. Here, we study interfacial spin-orbit coupling effects in magnetic bilayers using a simple Rashba model. The spin-orbit coupling introduces chirality into the behavior of the electrons and through them into the energetics of the magnetization. In the derived form of the magnetization dynamics, all of the contributions that are linear in the spin-orbit coupling follow from this chirality, considerably simplifying the analysis. For these systems, an important consequence is a correlation between the Dzyaloshinskii-Moriya interaction and the spin-orbit torque. We use this correlation to analyze recent experiments.

Snaking states on a cylindrical surface in a perpendicular magnetic field

We calculate electronic states on a closed cylindrical surface as a model of a core-shell nanowire. The length of the cylinder can be infinite or finite. We define cardinal points on the circumference of the cylinder and consider a spatially uniform magnetic field perpendicular to the cylinder axis, in the direction South-North. The orbital motion of the electrons depends on the radial component of the field which is nonuniform around the circumference: it is equal to the total field at North and South, but vanishes at the West and East sides. For a strong field, when the magnetic length is comparable to the radius of the cylinder, the electronic states at North and South become localized cyclotron orbits, whereas at East and West the states become long and narrow snaking orbits propagating along the cylinder. The energy of the cyclotron states increases with the magnetic field whereas the energy of the snaking states is stable. Consequently, at high magnetic fields the electron density vanishes at North and South and concentrates at East and West. We include spin-orbit interaction with linear Rashba and Dresselhaus models. For a cylinder of finite length the Dresselhaus interaction produces an axial twist of the charge density relative to the center of the wire, which may be amplified in the presence of the Rashba interaction.

Publisher's Note: Spin-flip transitions and departure from the Rashba model in the Au(111) surface [Phys. Rev. B88, 125404 (2013)

We present a detailed analysis of the spin-flip excitations induced by a periodic time-dependent electric field in the Rashba prototype Au(111) noble metal surface. Our calculations incorporate the full spinor structure of the spin-polarized surface states and employ a Wannier-based scheme for the spin-flip matrix elements. We find that the spin-flip excitations associated with the surface states exhibit an important angular modulation which is completely absent in the standard Rashba model \cite{rashba}. Furthermore, we demonstrate that the maximum of the calculated spin-flip absorption rate is about twice the model prediction. These results show that although the Rashba model accurately describes the spectrum and spin polarization, it does not fully account for the dynamical properties of the surface states.

Spin-flip transitions and departure from the Rashba model in the Au(111) surface

The authors acknowledge financial support from UPV/EHU (Grant No. IT-366-07 and program UFI 11/55), the Spanish Mineco (Grants No. FIS2012-36673-C03-01 and No. FIS2009-12773-C02-01), and the Basque Government (Grant No. IT472-10).

Ultrafast electron dynamics in a metallic quantum well nanofilm with spin splitting

Using time- and angle-resolved two-photon photoemission spectroscopy, we investigate the energy- and momentum-dependent ultrafast electron dynamics in the Rashba spin-split quantum-well nanofilm Bi/Cu(111). We find an expected increase of electron lifetimes towards the band bottom due to a competition of intra- and interband scattering processes. In addition, we find an unexpected peculiar decrease of the lifetimes around the intersection of the split bands. We compare the experimental results with calculated lifetimes due to electron-electron scattering in a model system of a 2D electron gas including a Rashba interaction term and an effective statically screened Coulomb interaction. Although the Rashba model reproduces the increase of lifetimes towards the band bottom well, there is no indication of the experimentally observed decrease around the intersection point in this simple model system. To investigate spin-orbit coupling effects, beyond those contained in a pure Rashba model, we introduce a phenomenological $k$-dependent spin mixing that leads to a ``spin hot spot.'' It is shown that such a mixing would strongly increase the electron-electron scattering rate around the band intersection and thus improves the agreement with experiment.

Intrinsic spin Hall effect at asymmetric oxide interfaces: Role of transverse wave functions

An asymmetric triangular potential well provides the simplest model for the confinement of mobile electrons at the interface between two insulating oxides, such as LaAlO${}_{3}$ and SrTiO${}_{3}$ (LAO/STO). These electrons have been recently shown to exhibit a large spin-orbit coupling of the Rashba type, i.e., linear in the in-plane momentum. In this paper we study the intrinsic spin Hall effect due to Rashba coupling in an asymmetric triangular potential well. This is the minimal model that captures the asymmetry of the spin-orbit coupling on opposite sides of the interface. Besides splitting each subband into two branches of opposite chirality, the spin-orbit interaction causes the transverse wave function (i.e., the wave function in the $z$ direction, perpendicular to the plane of the quantum well) to depend on the in-plane wave vector $\mathbf{k}$. At variance with the standard Rashba model, the triangular well supports a nonvanishing intrinsic spin Hall conductivity, which is proportional to the square of the spin-orbit coupling constant and, in the limit of low carrier density, depends only on the effective mass renormalization associated with the $\mathbf{k}$ dependence of the transverse wave functions. The origin of the effects lies in the nonvanishing matrix elements of the spin current between subbands corresponding to different states of quantized motion perpendicular to the plane of the well.

Anomalous circular photogalvanic effect of the spin-polarized two-dimensional electron gas in Mg0.2Zn0.8O/ZnO heterostructures at room temperature

A spin-related photocurrent with swirly distribution and anomalous dependence of the total spin-related photocurrent on the incident angle were observed on spin-polarized two-dimensional gas in a Mg0.2Zn0.8O/ZnO heterostructure under illumination of circular polarized light at room temperature. The ferromagnetic two-dimensional Rashba model was adopted to interpret the results. It is demonstrated that a radial spin current induced by the gradient of the spin-polarized electron density is the origin of the anomalousness. This spin current only exists in spin polarized systems.

Current induced torques and interfacial spin-orbit coupling: Semiclassical modeling

In bilayer nanowires consisting of a ferromagnetic layer and a non-magnetic layer with strong spin-orbit coupling, currents create torques on the magnetization beyond those found in simple ferromagnetic nanowires. The resulting magnetic dynamics appear to require torques that can be separated into two terms, damping-like and field-like. The damping-like torque is typically derived from models describing the bulk spin Hall effect and the spin transfer torque, and the field-like torque is typically derived from a Rashba model describing interfacial spin-orbit coupling. We derive a model based on the Boltzmann equation that unifies these approaches. We also consider an approximation to the Boltzmann equation, the drift-diffusion model, that qualitatively reproduces the behavior, but quantitatively fails to reproduce the results. We show that the Boltzmann equation with physically reasonable parameters can match the torques for any particular sample, but in some cases, it fails to describe the experimentally observed thickness dependences.

Rashba Effect on the Structure of the Bi One-Bilayer Film: Fully Relativistic First-Principles Calculation

Using first-principles calculations, we study the spin–orbit interactions and spin textures of a Bi one-bilayer film, which attracts scientific interest because of the topological insulator and so on. The substrate effect is successfully mimicked by applying on electric field in the perpendicular direction of the film, which breaks the inversion symmetry. We study the highest occupied band around the Γ point. Although the vortex of the in-plane spin component is well explained on the basis of the conventional Rashba effect, we find a substantial out-of-plane component which cannot be explained by the conventional Rashba model. This spin texture is similar to that of a multi-bilayer Bi film, which has recently been observed using a spin-resolved angle-resolved photoemission spectroscopy experiment. We also find a spin vortex around the K point although this point has no time-reversal symmetry. We expect that a similar vortex appears in materials having the p3m1 symmetry, whose spin–orbit interactions have recently attracted scientific interest.

Direct Observation of Interband Spin-Orbit Coupling in a Two-Dimensional Electron System

We report the direct observation of interband spin-orbit (SO) coupling in a two-dimensional (2D) surface electron system, in addition to the anticipated Rashba spin splitting. Using angle-resolved photoemission experiments and first-principles calculations on Bi-Ag-Au heterostructures, we show that the effect strongly modifies the dispersion as well as the orbital and spin character of the 2D electronic states, thus giving rise to considerable deviations from the Rashba model. The strength of the interband SO coupling is tuned by the thickness of the thin film structures.

Orbital Rashba effect and its detection by circular dichroism angle-resolved photoemission spectroscopy

We show, by way of tight-binding and first-principles calculations, that a one-to-one correspondence between electron's crystal momentum k and non-zero orbital angular momentum (OAM) is a generic feature of surface bands. The OAM forms a chiral structure in momentum space much as its spin counterpart in Rashba model does, as a consequence of the inherent inversion symmetry breaking at the surface but not of spin-orbit interaction. Circular dichroism (CD) angle-resolved photoemission (ARPES) experiment is an efficient way to detect this new order, and we derive formulas explicitly relating the CD-ARPES signal to the existence of OAM in the band structure. The cases of degenerate p- and d-orbital bands are considered.

Three-Dimensional Spin Rotations at the Fermi Surface of a Strongly Spin-Orbit Coupled Surface System

The spin texture of the metallic two-dimensional electron system (sqrt[3]×sqrt[3])-Au/Ge(111) is revealed by fully three-dimensional spin-resolved photoemission, as well as by density functional calculations. The large hexagonal Fermi surface, generated by the Au atoms, shows a significant splitting due to spin-orbit interactions. The planar components of the spin exhibit a helical character, accompanied by a strong out-of-plane spin component with alternating signs along the six Fermi surface sections. Moreover, in-plane spin rotations toward a radial direction are observed close to the hexagon corners. Such a threefold-symmetric spin pattern is not described by the conventional Rashba model. Instead, it reveals an interplay with Dresselhaus-like spin-orbit effects as a result of the crystalline anisotropies.

Anomalous spin–orbit coupling in high-density two-dimensional electron gas confined in InGaAs/InAlAs quantum well

We study the magnetotransport property of a high-density two-dimensional electron gas confined in InGaAs/InAlAs quantum well. Both beating pattern in the Shubnikov–de Hass oscillation of resistivity and weak antilocalization effect are observed. From these two effects, Rashba spin-splitting energy is extracted. The extracted Rashba spin-splitting energy shows a nonmonotonic dependence on Fermi wave vector, contrary to the prevailing linear Rashba model. This anomalous behavior can be attributed to the nonlinear Rashba spin-splitting mechanism [Yang et al., Phys. Rev. B 74 (2006) 193314].

Theory of magnetoelectric photocurrent generated by direct interband transitions in a semiconductor quantum well

A linearly polarized light normally incident on a semiconductor quantum well with spin-orbit coupling may generate pure spin current via direct interband optical transition. An electric photocurrent can be extracted from the pure spin current when an in-plane magnetic field is applied, which has been recently observed in the InGaAs/InAlAs quantum well [Dai et al., Phys. Rev. Lett. 104, 246601 (2010)]. Here we present a theoretical study of this magnetoelectric photocurrent effect associated with the interband transition. By employing the density matrix formalism, we show that the photoexcited carrier density has an anisotropic distribution in $\mathbf{k}$ space, strongly dependent on the orientation of the electron wavevector and the polarization of the light. This anisotropy provides an intuitive picture of the observed dependence of the photocurrent on the magnetic field and the polarization of the light. We also show that the ratio of the pure spin photocurrent to the magnetoelectric photocurrent is approximately equal to the ratio of the kinetic energy to the Zeeman energy, which enables us to estimate the magnitude of the pure spin photocurrent. The photocurrent density calculated with the help of an anisotropic Rashba model and the Kohn-Luttinger model can produce all three terms in the fitting formula for measured current, with comparable order of magnitude, but discrepancies are still present and further investigation is needed.

Measurement of the Intrinsic Anomalous Hall Effect in a 2D Hole System with Rashba Spin-orbit Coupling

The anomalous Hall effect of two-dimensional holes in a GaAs/AlGaAs heterostructure was measured under a slightly tilted-from-parallel magnetic field (B). In addition to the dominant ordinary Hall resistivity, which is linear in B, there is a small anomalous Hall resistivity correlated with the perpendicular spin magnetization of the holes due to the subband depopulation. When the anomalous Hall conductivity (σ xy) is extracted from the experimental data, it exhibits a nonmonotonic dependence on B, a behavior expected for the intrinsic anomalous Hall effect in the 2D Rashba model. If σ xy is plotted as a function of the longitudinal conductivity (σxx), an intrinsic region in which σ xy is almost constant and is several tenths of e /h for σxx of several tens of e /h is identified.

Large spin-orbit splitting in light quantum films: Al/W(110)

We investigate the long-standing question whether or not a large spin-orbit splitting can be induced in a light quantum film by a substrate effect. It is shown that quantum-well states in Al films on W(110) display large Rashba-type spin-orbit splittings up to at least ten monatomic layers (MLs) in angle-resolved photoemission measurements with and without spin resolution. Moreover, at 10 ML thickness, a non-Rashba-type behavior of the splitting is observed attaining its maximum value $({\ensuremath{\Delta}}_{\text{so}}\ensuremath{\sim}0.3\text{ }\text{eV})$ for electron wave vectors around the border of the projected substrate band gap. This deviation from the Rashba model is explained as direct interaction of quantum-well states with spin-orbit split states of the substrate.

Spin splitting modulated by uniaxial stress in InAs nanowires

We theoretically study the electronic structure, spin splitting, effective mass,and spin orientation of InAs nanowires with cylindrical symmetry in thepresence of an external electric field and uniaxial stress. Using an eight-bandk·p theoretical model, we deduce a formula for the spin splitting in the system, indicating thatthe spin splitting under uniaxial stress is a nonlinear function of the momentum and theelectric field. The spin splitting can be described by a linear Rashba model when thewavevector and the electric field are sufficiently small. Our numeric results show that theuniaxial stress can modulate the spin splitting. With the increase of wavevector, theuniaxial tensile stress first restrains and then amplifies the spin splitting of the lowestelectron state compared to the no strain case. The reverse is true under a compression.Moreover, strong spin splitting can be induced by compression when the top ofthe valence band is close to the bottom of the conductance band, and the spinorientations of the electron stay almost unchanged before the overlap of the two bands.