Spin-orbit torques induced by spin Hall and spin swapping currents of a separate ferromagnet in a magnetic trilayer

Spin-orbit torques (SOTs) have been investigated most widely in normal metal/ferromagnet bilayers where the spin Hall effect of normal metal is a main source of spin currents. Recently, ferromagnets are found to also serve as spin-current sources through spin-orbit coupling. In this work, we theoretically investigate SOT acting on ferromagnet₂ in ferromagnet₁/normal metal/ferromagnet₂ trilayers, which is caused by the spin Hall and spin swapping effects of ferromagnet₁. Our result provides an analytical expression of SOT in the trilayers, which may be useful for quantifying the spin Hall and spin swapping effects of ferromagnets and also for designing and interpreting SOT experiments where a ferromagnet is used as a spin-current source instead of a normal metal.     

Interfacial spin Hall current in a Josephson junction with Rashba spin—orbit coupling

We theoretically investigate the spin transport properties of the Cooper pairs in a conventional Josephson junction with Rashba spin-orbit coupling considered in one of the superconducting leads.It is found that an angle-resolved spin supercurrent flows through the junction and a nonzero interfacial spin Hall current driven by the superconducting phase difference also appears at the interface.The physical origin of this is that the Rashba spin-orbit coupling can induce a triplet order parameter in the s-wave superconductor.The interfacial spin Hall current dependences on the system parameters are also discussed.     

Dynamical spin polarization in single-layer graphene

In the present work the dynamical behavior of $\mathrm{\pi }\mathrm{-}\mathrm{electronic}$ spin in graphene is investigated. The $\mathrm{\pi }\mathrm{-}\mathrm{electron}$ is under the influence of a normal uniform magnetic field and the Rashba spin–orbit coupling. Introducing a Casimir operator, we show that the governing Hamiltonian and, consequently, the time-evolution matrix is block-diagonal. We then proceed to calculate the temporal behavior of different spin components, when it is initially in-plane polarized. Our calculations show that the spin is dynamically polarized in a plane normal to the graphene sheet and follows the patterns of collapse-revivals. The dependence of amplitudes as well as the collapse-revivals’ periods on the external field and the Rashba spin–orbit coupling is also reported.     

Manipulating edge current spin polarization in zigzag MoS2 nanoribbons

The electronic structure and quantum transport of a zigzag monolayer molybdenum disulfide (MoS₂) nanoribbon are investigated using a six-band tight-binding model. For metallic edge modes, considering both an intrinsic spin–orbit coupling and local exchange field effects, spin degeneracy and spin inversion symmetry are broken and spin selective transport is possible. Our model is a three-terminal field effect transistor with a circular-shaped gate voltage in the middle of scattering region. One terminal measures the top edge current and the other measures the bottom edge current separately. By controlling the circular gate voltage, each terminal can detect a totally spin-polarized edge current. The radius of the circular gate and the strength of the exchange field are important, because the former determines the size of the channel in both S-terminated (top) and Mo-terminated (bottom) edges and the latter is strongly related to unbalancing of the density of spin states. The results presented here suggest that it should be possible to construct spin filters using implanted MoS₂ nanoribbons.     

Spin current pumping with two orthogonal electric fields in quantum wire

We study the problem of spin current pumping in a one-dimensional quantum wire when there exist two orthogonal Rashba spin-orbit couplings (SOCs) in different regions which evolve with time and can be induced by the perpendicular electric fields. On one hand, we demonstrate that the time-evolving Rashba SOC is equivalent to the spin-dependent electric field and the scheme may lead to the pure spin current associated with well suppressed charge current. On the other hand, we adopt the non-equilibrium Green's function method and numerically find that the parameter loop must satisfy certain condition for the successful pumping. We also study the effect of the Fermi energy and the inevitable disorder on the spin current. The implications of these results are discussed.     

True spin and pseudo spin entanglement around Dirac Points in graphene with Rashba spin–orbit interaction

We analytically obtained the Schmidt decomposition of the entangled state between the pseudo spin and the true spin in graphene with Rashba spin–orbit coupling. The entangled state has the standard form of the Bell state, where the SU(2) spin symmetry is broken. These states can be explicitly expressed as the superposition of two nonorthogonal, but mirror symmetrical spin states entangled with the pseudo spin states. Because of the closely locking between the pseudo spin and the true spin, it is found that the orbit curve in the spin-polarization parameter space for the fixed equi-energy contour around Dirac points has the same shape as the $\delta \overrightarrow{k}$-contour. Due to the spin–orbit coupling that cause the topological transition in the local geometry of the dispersion relation, the new equi-energy contours around the new emergent Dirac Points can be obtained by squeezing the one around the original Dirac point. The spin texture in the momentum space around the Dirac points is analyzed under the Rashba spin–orbit interaction and it is found that the orientation of the spin polarization at each crystal momentum $\overrightarrow{k}$ is independent of the Rashba coupling strength.     

Rashba coupling induced spin susceptibility and magnetic phase transition of conduction electrons in monolayer graphene

Isolated graphene cannot be obtained by the known synthesis processes and it should be placed on a substrate. This substrate introduces a new type of spin–orbit interaction known as Rashba coupling. Using the Kubo formalism, the magnetic properties of the system in the linear regime have been investigated. Mainly the effect of non-magnetic substrate on the spin susceptibility is calculated. Results show that the Rashba coupling has a central role in the magnetic response function of the system and it is really remarkable since this type of spin orbit coupling can be effectively controlled by an external gate voltage. Most importantly, it was shown that, in the presence of the Rashba interaction a magnetic phase transition could be observed. This magnetic phase corresponds to a magnetic order of conduction electrons that takes place at some special frequencies of external magnetic field.     

Spin current and its heat effect in a multichannel quantum wire with Rashba spin-orbit coupling

Using the perturbation method,we theoretically study the spin current and its heat effect in a multichannel quantum wire with Rashba spin-orbit coupling.The heat generated by the spin current is calculated.With the increase of the width of the quantum wire,the spin current and the heat generated both exhibit period oscillations with equal amplitudes.When the quantum-channel number is doubled,the oscillation periods of the spin current and of the heat generated both decrease by a factor of 2.For the spin current j s,xy,the amplitude increases with the decrease of the quantum channel;while the amplitude of the spin current j s,yx remains the same.Therefore we conclude that the effect of the quantum-channel number on the spin current j s,xy is greater than that on the spin current j s,yx.The strength of the Rashba spin-orbit coupling is tunable by the gate voltage,and the gate voltage can be varied experimentally,which implies a new method of detecting the spin current.In addition,we can control the amplitude and the oscillation period of the spin current by controlling the number of the quantum channels.All these characteristics of the spin current will be very important for detecting and controlling the spin current,and especially for designing new spintronic devices in the future.     

Spin current and its heat effect in a multichannel quantum wire with Rashba spinben orbit coupling

Using the perturbation method, we theoretically study the spin current and its heat effect in a multichannel quantum wire with Rashba spin—orbit coupling. The heat generated by the spin current is calculated. With the increase of the width of the quantum wire, the spin current and the heat generated both exhibit period oscillations with equal amplitudes. When the quantum-channel number is doubled, the oscillation periods of the spin current and of the heat generated both decrease by a factor of 2. For the spin current j_s,xy, the amplitude increases with the decrease of the quantum channel; while the amplitude of the spin current j_s,yx remains the same. Therefore we conclude that the effect of the quantum-channel number on the spin current j_s,xy is greater than that on the spin current j_s,yx. The strength of the Rashba spin—orbit coupling is tunable by the gate voltage, and the gate voltage can be varied experimentally, which implies a new method of detecting the spin current. In addition, we can control the amplitude and the oscillation period of the spin current by controlling the number of the quantum channels. All these characteristics of the spin current will be very important for detecting and controlling the spin current, and especially for designing new spintronic devices in the future.     

Magnetization of two-dimensional heavy holes with boundaries in a perpendicular magnetic field

The magnetisation of heavy holes in III--V semiconductor quantum wells with Rashba spin-orbit coupling (SOC) in an external perpendicular magnetic field is studied theoretically. We concentrate on the effects on the magnetisation induced by the system boundary, the Rashba SOC and the temperature. It is found that the sawtooth-like de Haas--van Alphen (dHvA) oscillations of the magnetisation will change dramatically in the presence of such three factors. Especially, the effects of the edge states and Rashba SOC on the magnetisation are more evident when the magnetic field is smaller. The oscillation center will shift when the boundary effect is considered and the Rashba SOC will bring beating patterns to the dHvA oscillations. These effects on the dHvA oscillations are preferably observed at low temperatures. With increasing temperature, the dHvA oscillations turn to be blurred and eventually disappear.     

δ势垒对多臂量子环中持续电流的影响

提出了中臂弯曲的多臂量子环模型,且是上臂最短和下臂最长的不等臂量子环.研究发现：总磁通为零时,持续电流随半导体环增大发生非周期性振荡,并与电极的磁矩方向及隧穿电子的自旋方向相关,下臂因为最长而获得最小的平均持续电流.AB磁通增强时,持续电流会发生周期性振荡,各臂之间明显出现相互制约的现象.各臂持续电流之间的差异与臂长和磁通分布相关,Rashba自旋轨道耦合具有改变持续电流相位和相位差的效应.在一定条件下,两种波函数所对应的持续电流是可分离的.     

Interplay of Rashba effect and spin Hall effect in perpendicular Pt/Co/MgO magnetic multilayers

The interplay of the Rashba effect and the spin Hall effect originating from current induced spin–orbit coupling was investigated in the as-deposited and annealed Pt/Co/MgO stacks with perpendicular magnetic anisotropy. The above two effects were analyzed based on Hall measurements under external magnetic fields longitudinal and vertical to dc current,respectively. The coercive field as a function of dc current in vertical mode with only the Rashba effect involved decreases due to thermal annealing. Meanwhile, spin orbit torques calculated from Hall resistance with only the spin Hall effect involved in the longitudinal mode decrease in the annealed sample. The experimental results prove that the bottom Pt/Co interface rather than the Co/MgO top one plays a more critical role in both Rashba effect and spin Hall effect.     

特殊结构的多臂量子环的持续电流

提出了中臂弯曲的多臂量子环模型,且是上臂最短和下臂最长的不等臂量子环.研究发现：总磁通为零时,持续电流随半导体环增大发生非周期性振荡,并与电极的磁矩方向及隧穿电子的自旋方向相关,下臂因为最长而获得最小的平均持续电流.AB磁通增强时,持续电流会发生周期性振荡,各臂之间明显出现相互制约的现象.各臂持续电流之间的差异与臂长和磁通分布相关,Rashba自旋轨道耦合具有改变持续电流相位和相位差的效应.在一定条件下,两种波函数所对应的持续电流是可分离的. 关键词： 多臂量子环 持续电流 Rashba自旋轨道耦合     

Effect of conduction band nonprabolicity on the Rashba spin splitting of AlGaN/GaN QWs

By using the perturbation expansion method and self-consistent iterative method, we evaluate the effect of the conduction band nonprabolicity on the wave vector (k_t) dependent Rashba coefficient (α) and nonlinear Rashba spin splitting (ΔE) in the Al_0.5Ga_0.5N/GaN quantum well (QW). The effective mass (energy) under the first order approximation m_t1 (E_k1) is in proximity to the iterative result m_tp (E_kp) and m_t1>m_tp, E_k1<E_kp, showing the higher order contributions to m_t (E_k) are small. The sign of the nonparabolic correction to E_k is just opposite to that of the correction to m_t. The increase of α and ΔE due to the conduction band nonparabolicity reaches about 3% at k_t=1 nm⁻¹. Around the left heterointerface, the probability density is high and E_k0>E_kp>E_k1, so α₀<α_p<α₁, ΔE₀<ΔE_p<ΔE₁. With increasing k_t, α decreases, and ΔE increases slowly. For small k_t, α₀ (ΔE₀), α₁ (ΔE₁) and α_p (ΔE_p) are nearly the same. While for large k_t, the difference between α₀ and α₁ (α_p) increases rapidly, but the difference between ΔE₀ and ΔE₁ (ΔE_p) increases slowly.     

Effect of eccentricity in the spin accumulation effect induced in semiconductors rings with Rashba spin orbit interaction

A spin accumulation effect (SAE) is induced in a semiconductor nanoring with Rashba spin orbit interaction and pierced by a magnetic flux. We show that when the sample is not perfectly symmetric, the profile of the SAE can be highly inhomogeneous along specific orientations. In particular, we analyze the anisotropy generated in the angular profile by a finite eccentricity. We discuss the feasibility of detecting the effect with usual magneto optical techniques for a number of electrons and values of magnetic fluxes experimentally accessible.     

Current-induced spin polarization in a spin-polarized two-dimensional electron gas with spin-orbit coupling

The current-induced spin polarization (CISP) is investigated in a combined Rashba-Dresselhaus spin-orbit-coupled two-dimensional electron gas, subjected to a homogeneous out-of-plane magnetization. It is found that, in addition to the usual collision-related in-plane parts of CISP, there are two impurity-density-free contributions, arising from intrinsic and disorder-mediated mechanisms. The intrinsic parts of spin polarization are related to the Berry curvature, analogous with the anomalous and spin Hall effects. For short-range collision, the disorder-mediated spin polarizations completely cancel the intrinsic ones and the total in-plane components of CISP equal those for systems without magnetization. However, for remote disorders, this cancellation does not occur and the total in-plane components of CISP strongly depend on the spin-orbit interaction coefficients and magnetization for both pure Rashba and combined Rashba-Dresselhaus models.     

Origin of large spin Hall magnetoresistance in Fe/CuOx bilayers

We have compared the spin Hall magnetoresistance (SMR) in Fe/Pt and Fe/CuO_x (with natural oxidation) bilayers with varying the thickness of Fe layer. A larger SMR in Fe/CuO_x bilayers has been found when the thickness of Fe layer is 3 nm. Moreover, the SMR of the two bilayers decrease with increasing the thickness of Fe from 3 nm to 10 nm, but that of Fe/CuO_x drops more sharply due to shunting current effect. Through harmonic measurements, the emergent spin current is proved to be generated in the Fe/CuO_x bilayers. The mixed phase of CuO_x has been confirmed including CuO, Cu₂O and Cu, which performs strong spin-orbit coupling and produce large spin current. On the other hand, the interface-generated spin current should be ruled out. All the results have been compared with those in Fe/Al₂O₃ bilayers with negligible spin current.     

Polarized current changes the exchange bias in a current-in-plane spin valve

A spin-polarized current changes the strength and direction of the exchange bias in spin valves with a current-in-plane geometry. The exchange bias can be manipulated and systematically changed by applying current pulses. The changes are nonmonotonic and asymmetric with respect to the directions of the applied field and current pulses. For different current pulses, different exchange-bias fields can be achieved in the same sample. Furthermore, for samples with different exchange bias, the bias field exhibits a dependence on the applied pulse. Since the strength of exchange bias is highly correlated to the micromagnetic state distribution of the antiferromagnet, we explain our observations by the spin torque exerted on the interfacial antiferromagnetic moments, excluding Joule heating and training effects.     

Persistent spin current in a quantum wire with weak Dresselhaus spin——orbit coupling

The spin current in a parabolically confined semiconductor heterojunction quantum wire with Dresselhaus spin--orbit coupling is theoretically studied by using the perturbation method. The formulae of the elements for linear and angular spin current densities are derived by using the recent definition for spin current based on spin continuity equation. It is found that the spin current in this Dresselhaus spin--orbit coupling quantum wire is antisymmetrical, which is different from that in Rashba model due to the difference in symmetry between these two models. Some numerical examples for the result are also demonstrated and discussed.     

Spin Hall effect and spin current generation in two-dimensional systems with random Rashba spin–orbit coupling

We consider a new effect induced by spin–orbit coupling in a two-dimensional electron gas confined in a semiconductor quantum well, i.e. the possibility of spin current generation by fluctuating random Rashba spin–orbit interaction, with the corresponding mean value of the interaction being equal to zero. Our main results suggest that – in contrast to the spatially uniform Rashba spin–orbit interaction – the spin Hall effect does not vanish for typical disorder strengths. We also point out some other possibilities of using such a random Rashba coupling for the generation of spin density and spin current in two-dimensional nonmagnetic structures.