Spin Diffusion Equation Research Articles

Spin Accumulation in Acoustically Excited Ni/GaAs/Ni Trilayers

In this paper, we report the first theoretical results on the acoustic generation of spin accumulation in ferromagnet-semiconductor-ferromagnet trilayers. As a representative material system, we consider a Ni/GaAs/Ni trilayer coupled to a piezoelectric transducer, which injects a planar acoustic wave into the adjoining Ni film. By combining an analytical solution of the spin diffusion equation in the GaAs spacer with results of numerical simulations of the coupled elastic and magnetic dynamics in the Ni films, we quantify an oscillating inhomogeneous spin accumulation in GaAs. It is found that both dc and ac parts of the mean spin accumulation vary nonmonotonically with the spacer thickness [Formula: see text], reaching maximal values at [Formula: see text] mostly close to 0.25 or 0.75 of the wavelength of the injected monochromatic acoustic wave. Remarkably, the transverse wave generates the spin accumulation much more efficiently than the longitudinal one. Our theoretical predictions provide guidelines for the development and optimization of energy-efficient acoustic spin injectors into semiconductors, which should have much lower power consumption than injectors driven by the microwave magnetic field.

Spin hydrodynamic generation in unsteady flows

We theoretically investigate a spin-mediated conversion from fluid dynamics to voltage, known as spin hydrodynamic generation (SHDG), in oscillatory and transient unsteady flows. We consider unsteady flows of liquid metal between two parallel infinite planes and then calculate its vorticity fields based on the Navier–Stokes equation for an incompressible viscous fluid. The spin accumulation and spin current generated by unsteady flows are derived using a spin-diffusion equation, including spin-vorticity coupling, which is a couple of angular momentum between electron spin and vorticity field in unsteady flows. The estimation of SHDG in liquid mercury flow suggests that an observable magnitude of voltage can be induced in unsteady flows. Our results are expected to enable the realization of high-speed spin devices with unsteady flows and broaden the range of fluid spintronics applicability.

Spin-accumulation and spin-transport effects induced by surface plasmon-polariton waves

A surface plasmon-polariton (SPP) wave forms highly inhomogeneous intensity distribution near the metal-dielectric interface, and this light field produces the inhomogeneous magnetization of the metal. Recently [Phys. Rev. B 101, 161404 (2020)PRBMDO0163-182910.1103/PhysRevB.101.161404; Phys. Rev. B 102, 125431 (2020)PRBMDO0163-182910.1103/PhysRevB.102.125431], the SPP-induced magnetization was considered theoretically as a source of purposeful excitation and control of the spin-transport phenomena. Here, this problem is revisited with the refined boundary conditions for the spin-diffusion equation. The improved theoretical description of the light-induced spin accumulation and spin current is formulated. The validity limits of the stationary spin-accumulation model are discussed and numerically estimated. Numerical simulations based on the Drude model for electron gas in metal confirm the general weakness of the SPP-induced spin-transport phenomena but also indicate possibilities of their enhancement and detectable manifestations via employment of high-power short-pulse excitation. The results can be useful for the studies and applications of the SPP-induced effects, in particular, for the development of optically driven spintronic devices.

Manipulation of spin Hall magnetoresistance and unidirectional spin Hall magnetoresistance in Ta/Pt/CoFeB multilayers

The interaction between spin current and magnetic moments in heavy metal (HM)/ferromagnetic (FM) structure results in the well-known spin orbit torque and the concomitant magnetoresistance effect: spin Hall magnetoresistance (SMR) and the unidirectional SMR (USMR). In this work, we simultaneously investigated the SMR and USMR in Ta/Pt/CoFeB/MgO multilayers with different Pt thickness (tPt) by the harmonic measurements. It is observed that at tPt≈0.9nm, the SMR is close to zero and the polarity of USMR reverses, which originate from the competition of the spin current from Ta and Pt layers. Under the theoretical framework of the spin diffusion equation, we derived the theoretical expressions of SMR and USMR in the tri-layer magnetic heterostructures. We found that the tPt dependences of the SMR and USMR can be well fitted by the expressions, from which the spin transport parameters of Pt were extracted. Our work provides a fundamental insight into spin transport in the system with two heavy-metal layers, and a possible way to control SMR and USMR of HM/FM systems.

Thermal spin–orbit torque in spintronics

Based on the spinor Boltzmann equation (SBE) formalism, we present a theory of temperature-dependent thermal spin–orbit torque for a system in the presence of Rashba spin–orbit interaction. Under the local equilibrium assumption, we can expand the distribution function of spinor Boltzmann equation around local equilibrium distribution; then, the spin diffusion equation can be derived from SBE, where the spin transfer torque, spin orbit torque as well as thermal spin–orbit torque can be naturally obtained. It is shown that this thermal spin–orbit torque originates from the temperature gradient of local equilibrium distribution function, which is explicit and straightforward than previous works. Finally, we illustrate them by an example of spin-polarized transport through a ferromagnet with Rashba spin–orbit coupling, in which those torques driven whatever by temperature gradient or bias are manifested quantitatively.Graphic abstractWe proposed a new expression for the thermal spin–orbit torque, which can be unified with the usual spin orbit torque as a generalized spin orbit torque.

Energy-efficient spin injector into semiconductors driven by elastic waves

Generation of spin imbalance in nonmagnetic semiconductors is crucial for the functioning of many spintronic devices. An attractive design of spin injectors into semiconductors is based on a spin pumping from a precessing ferromagnet, typically excited by a microwave magnetic field leading to a high power consumption of the device. Here we describe theoretically a spin injector with greatly reduced energy losses, in which the magnetic dynamics is excited by an elastic wave injected into a ferromagnet-semiconductor heterostructure. To demonstrate the efficient functioning of such an injector, we perform micromagnetoelastic simulations of the coupled elastic and magnetic dynamics in Ni films and Ni/GaAs bilayers. For thick Ni films, it is shown that a monochromatic acoustic wave generates a spin wave with the same frequency and wavelength, which propagates over distances of several micrometers at the excitation frequencies close to the frequency of ferromagnetic resonance. The simulations of Ni/GaAs bilayers with Ni thicknesses comparable to the wavelength of the injected acoustic wave demonstrate the development of a steady-state magnetization precession at the Ni/GaAs interface. The amplitude of such a precession has a maximum at Ni thickness amounting to three quarters of the wavelength of the elastic wave, which is explained by an analytical model. Using simulation data obtained for the magnetization precession at the Ni/GaAs interface, we evaluate the spin current pumped into GaAs and calculate the spin accumulation in it by solving the spin diffusion equation. Then the electrical signals resulting from the spin flow and the inverse spin Hall effect are determined via the numerical solution of the Laplace's equation. It is shown that amplitudes of these ac signals are large enough for experimental measurement, which indicates an efficient acoustically driven spin pumping into GaAs.

Spinor Boltzmann equation approach to domain-wall motion driven by spin-polarized current

Based on the nonequilibrium Green's function formalism, the spinor Boltzmann equation beyond gradient approximation is derived in a ferromagnetic metal with a single domain wall (DW). We further obtain the charge continuity equation and the spin diffusion equation by integrating over the momentum. By using the spin diffusion equation, we get a generalized spin transfer torque (STT), in which the usual STT is extended to the case beyond the gradient approximation and with inhomogeneous current. We also calculate numerically the physical observables such as charge density $n(x)$, spin accumulation $\mathbit{m}(x)$, current density $j(x)$, spin current density ${\mathbit{j}}_{s}(x)$, etc., by the use of the spinor distribution function. Along with the Landau-Lifshitz-Gilbert-Slonczewski equation that contains the above generalized STT, we can study the motion domain wall, and the critical electric field at an initial velocity of the DW is obtained in terms of the linear stability analysis method.

Spin drift-diffusion for two-subband quantum wells

Controlling the spin dynamics and spin lifetimes is one of the main challenges in spintronics. To this end, the study of the spin diffusion in two-dimensional electron gases (2DEGs) shows that when the Rashba and Dresselhaus spin-orbit couplings (SOC) are balanced, a persistent spin helix regime arises. There, a striped spin pattern shows a long lifetime, limited only by the cubic Dresselhaus SOC, and its dynamics can be controlled by in-plane drift fields. Here, we derive a spin diffusion equation for non-degenerate two-subbands 2DEGs. We show that the intersubband scattering rate, which is defined by the overlap of the subband densities, enters as a new nob to control the spin dynamics, and can be controlled by electric fields, being maximum for symmetric quantum wells. We find that for large intersubband couplings the dynamics follow an effective diffusion matrix given by approximately half of the subband-averaged matrices. This extra 1/2 factor arises from Matthiessen's rule summing over the intra- and intersubband scattering rates, and leads to a reduced diffusion constant and larger spin lifetimes. We illustrate our findings with numerical solutions of the diffusion equation with parameters extracted from realistic Schr\"odinger-Poisson calculations.

Ferrimagnetic resonance induced by the spin Hall effect

A bilayer consisting of a ferrimagnetic insulator (FiM) and a heavy metal (HM) can be used as a ``pump-probe'' system to analyze the magnetic properties of the FiM. An oscillating electric current in the HM induces spin current injection into the FiM via the spin Hall effect. The resulting magnetization dynamics of the FiM, in turn, causes spin pumping back into the HM and modifies its conductivity via the inverse spin Hall effect (ISHE). We present a phenomenological theory to model the ISHE output spectrum, in which the Landau-Lifshitz-Gilbert equations governing the FiM dynamics are coupled to the spin-diffusion equations governing the spin and charge transports in the HM. It is found that the ISHE signal is greatly enhanced at the magnetization compensation point of the FiM. Furthermore, the peak frequencies and amplitudes of the ISHE spectrum are strongly correlated to the magnetic properties of the FiM sublattices. Most interestingly, the ISHE output can be increased by several orders of magnitude by tuning the on-site-sublattice and cross-sublattice damping constants. Our analysis shows the possibility of using the FiM/HM bilayer system as a sensitive probe into the magnetic parameters of the FiM sublattices. Conversely, the drastically enhanced ISHE signal with appropriate tuning of the damping parameters suggests a means to substantially improve the spin torque efficiency in FiM-based heterostructures.

Optically induced electron spin currents in the Kretschmann configuration

We investigate electron spin currents induced optically via plasmonic modes in the Kretschmann configuration. By utilising the scattering matrix formalism, we take the plasmonic mode coupled to external laser drive into consideration and calculate induced magnetisation in the metal. The spatial distribution of the plasmonic mode is inherited by the induced magnetisation, which acts as inhomogeneous effective magnetic field and causes the Stern-Gerlach effect to drive electron spin currents in the metal. We solve spin diffusion equation with a source term to analyse the spin current as a function of the spin diffusion length of the metal, the frequency and the incident angle of the external drive.

Extrinsic spin Hall effect in inhomogeneous systems

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 spin Hall current. This spin source term reduces to the form of spin-vorticity coupling when the spin Hall angle is independent of impurity concentration. We also study the effects of long-range Coulomb interaction, which is indispensable when the process involves charge inhomogeneity and accumulation as in the present problem.

Discovery Principles and Materials for Symmetry-Protected Persistent Spin Textures with Long Spin Lifetimes

Summary Persistent spin textures (PSTs) in solid-state materials arise from a unidirectional spin-orbit field in momentum space and offer a route to deliver the necessary long carrier spin lifetimes utilized in future quantum microelectronic devices. Nonetheless, few bulk materials host PSTs owing to crystal symmetry and chemical requirements, with even fewer experimentally demonstrated examples. This scarcity makes both PST materials discovery and performance assessment challenging. Here, we demonstrate that bulk PSTs exist in the family of layered A3B2O7 oxides and that a persistent spin helix (PSH) occurs over a large region of the Brillouin zone—a feature essential to experimental realization. By solving the spin-diffusion equations in the strong coupling limit and using the PST performance criteria we formulate, we find that the spin lifetime of the PSH is ∼63 ns in Sr3Hf2O7—substantially larger than that predicted (∼10 ns) and demonstrated experimentally (∼600 ps) in GaAs/AlGaAs quantum wells.

Magnetization Reversal of Single-Molecular Magnets by a Spin-Polarized Current**Supported by the National Key R&D Program of China (Grant No. 2018YFA0305804), the National Natural Science Foundation of China (Grant No. 11834014), and the Strategetic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB28000000).

We study the magnetization reversal of single-molecular magnets by a spin-polarized current in the framework of the spinor Boltzmann equation. Because of the spin–orbit coupling, the spin-polarized current will impose a non-zero spin transfer torque on the single-molecular magnets, which will induce the magnetization switching of the latter. Via the s–d exchange interaction between the conducting electrons and single-molecular magnets, we can investigate the magnetization dynamics of single-molecular magnets. We demonstrate the dynamics of the magnetization based on the spin diffusion equation and the Heisenberg-like equation. The results show that when the current is large enough, the magnetization of the single-molecular magnets can be reversed. We also calculate the critical current density required for the magnetization reversal under different anisotropy and external magnetic fields, which is helpful for the corresponding experimental design.

Domain wall motion in a diffusive weak ferromagnet

We study the domain wall motion in a disordered weak ferromagnet, induced by injecting a spin current from a strong ferromagnet. Starting from the spin diffusion equation describing the spin accumulation in the weak ferromagnet, we calculate the force and torque acting on the domain wall. We also study the ensuing domain wall dynamics, and suggest a possible measurement method for detecting the domain wall motion via measuring the additional resistance.

Electron spin transport driven by surface plasmon polaritons

We propose a mechanism of angular momentum conversion from optical transverse spin in surface plasmon polaritons (SPPs) to conduction electron spin. Free electrons in the metal follow the transversally spinning electric field of SPP, and the resulting orbital motions create inhomogeneous static magnetisation in the metal. By solving the spin diffusion equation in the SPP, we find that the magnetisation field generates an electron spin current. We show that there exists a resonant condition where the spin current is resonantly enhanced, and the polarisation of the spin current is flipped. Our theory reveals a novel functionality of SPP as a spin current source.

Effects of surface plasmons on spin currents in a thin film system

We propose and analyze surface-plasmon-driven electron spin currents in a thin metallic film. The electron gas in the metal follows the transversely rotating electric fields of the surface plasmons (SPs), which leads to a static magnetization gradient. We consider herein SPs in a thin-film insulator–metal–insulator structure and solve the spin diffusion equation in the presence of a magnetization gradient. The results reveal that the SPs at the metal interfaces generate spin currents in the metallic film. For thinner film, the SPs become strongly hybridized, which increases the magnetization gradient and enhances the spin current. We also discuss how the spin current depends on SP wavelength and the spin-diffusion length of the metal. The polarization of the spin current can be controlled by tuning the wavelength of the SPs and/or the spin diffusion length.

Generalized spin-orbit torques in two-dimensional ferromagnets with spin-orbit coupling

Based on the spinor Boltzmann equation, we obtain the continuity equations of charge and spin accumulation in two-dimensional ferromagnets with Rashba or Dresselhaus spin-orbit coupling, in which the spin diffusion equation (the continuity equation of spin accumulation) gives the spin-orbit torques that have an extra term of charge density gradient that is paid little attention in previous studies. Numerical results indicate this term plays an important role in inhomogeneous systems. In order to study the magnetization dynamics of ferromagnets, we solve the Landau-Lifshitz-Gilbert equation together with the spin diffusion equation. Our results indicate that the magnetization in systems with Rashba SOI is easier to be switched by an external electric field than the system with Dresselhaus SOI, and for anisotropic ferromagnets, we study the critical external electric field that manipulate the reversal of magnetization.

On Guided Remagnetization in Layered Anoheterostructures

Field-guided magnetic dynamics in magnetic multilayer nanostructures involves interconnection of the control field with localized spin states, which can occur directly or indirectly depending on the nature of the field and spin polarization. At the control electromagnetic field, this interconnection can be directly induced by the photon-induced spin-flip processes and indirectly by a bias field during antiferromagnetic exchange relaxation. The control impact of electric field and electric current on the magnetic states occurs indirectly via the spin polarization and spin current in combination with the exchange interaction of these polarized spins with localized magnetic states. The corresponding description of the magnetic dynamics is based on the modified Landau-Lifshitz equation and spin diffusion equations, taking into account the spin Hall and the inverse spin Hall effects for systems with normal metal sublayers. In the case of the magnetic nanostructures with the Rashba spin-orbit interaction in interfaces, the electric field-controlled magnetization is realized via the Rashba field-induced spin polarization, and its exchange interaction with localized magnetic states. Corresponding description is based on a tight-binding model of spin-orbit-coupled electrons exchange coupled to the localized magnetic states.

Spin diffusion equation in superconductors in the vicinity of Tc

We microscopically derive the spin diffusion equation in an s-wave superconductor in the vicinity of the superconducting transition temperature Tc. Applying the general relation between the relaxation function and the response function to the present spin diffusion problem, we examine how the spin relaxation time and the spin diffusion coefficient are renormalized in the superconducting state. The analysis reveals that, below Tc, both the spin relaxation time and the spin diffusion coefficient are increased, resulting in an enhancement of the spin diffusion length. The present result may provide an explanation for the recent observation of an enhanced spin pumping signal below Tc in a Py/Nb/Pt system that is free from the coherence peak effect.

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.