We propose a new quantum-mechanical formalism to calculate spin torques based on the gradient expansion, which naturally involves spacetime gradients of the magnetization and electromagnetic fields. We have no assumption in the small-amplitude formalism or no difficulty in the SU($2$) gauge transformation formalism. As a representative, we calculate the spin renormalization, Gilbert damping, spin-transfer torque, and $β$-term in a three-dimensional ferromagnetic metal with nonmagnetic and magnetic impurities being taken into account within the self-consistent Born approximation. Our results serve as a first-principles formalism for spin torques.
To study the effect of spin-orbit coupling (SOC) on spin-transfer torque in magnetic materials, we have implemented two theoretical formalisms that can accommodate SOC. Using the "charge-pumping" formalism, we find two contributions to the out-of-plane spin-transfer torque parameter $β$ in ballistic Ni domain walls (DWs). For short DWs, the nonadiabatic reflection of conduction electrons caused by the rapid spatial variation of the exchange potential results in an out-of-plane torque that increases rapidly with decreasing DW length. For long DWs, the Fermi level conduction channel anisotropy t…
Spin-Transfer Torque on Curved Surfaces: A Generalized Thiele Formalism
Universidade Federal de Viçosa（维索萨联邦大学）；Universidad de Chile（智利大学）；Leibniz-Institut für Festkörper- und Werkstoffforschung（莱布尼茨固体与材料研究所）；Bogolyubov Institute for Theoretical Physics of the National Academy of Sciences of Ukraine（乌克兰国家科学院博戈留博夫理论物理研究所）；Universidad Diego Portales（迭戈·波特莱斯大学）
In metallic ferromagnets, the spin-transfer torque and spin-motive force are known to exhibit a reciprocal relationship. Recent experiments on ferromagnets with strong spin-orbit coupling have revealed a rich complexity in the interaction between itinerant charge carriers and magnetization, but a full understanding of this coupled dynamics is lacking. Here, we develop a general phenomenology of the two reciprocal processes of charge pumping by spin-motive forces and current-driven magnetization dynamics. The formalism is valid for spin-orbit coupling of any strength and presents a systematic s…
We generalize magnetoelectronic circuit theory to account for spin transfer to and from the atomic lattice via interfacial spin-orbit coupling. This enables a proper treatment of spin transport at interfaces between a ferromagnet and a heavy-metal non-magnet. This generalized approach describes spin transport in terms of drops in spin and charge accumulations across the interface (as in the standard approach), but additionally includes the responses from in-plane electric fields and offsets in spin accumulations. A key finding is that in-plane electric fields give rise to spin accumulations an…
We address the nature of spin-orbit torques at the magnetic surfaces of topological insulators, using both Kubo formula and Keldysh formalism. Through the analysis of the derived spin-charge equations, we find that the so-called Dirac torques possess a different symmetry compared to their Rashba counterpart, as well as a high anisotropy as a function of the magnetization direction. In particular, the damping torque vanishes when the magnetization lies in the plane of the topological insulator surface. This peculiarity has important consequences in terms of magnetization dynamics and switching,…
Using the Green function formalism we calculate a current-induced spin polarization of weakly magnetized graphene with Rashba spin-orbit interaction. In a general case, all components of the current-induced spin polarization are nonzero, contrary to the nonmagnetic limit, where the only nonvanishing component of spin polarization is that in the graphene plane and normal to electric field. When the induced spin polarization is exchange-coupled to the magnetization, it exerts a spin-orbit torque on the latter. Using the Green function method we have derived some analytical formulas for the spin …
We introduce the general formalism to describe spin torques induced by the supercurrents injected from the adjacent superconducting electrodes into the spin-textured ferromagnets. By considering the adiabatic limit for the equal-spin superconducting correlations in the ferromagnet we show that the supercurrent can generate both the field-like spin transfer torque and the spin-orbital torque. These dissipationless spin torques are expressed through the current-induced corrections to the effective field derived from the system energy. The general formalism is applied to show that the supercurren…
We investigate the influence of magnons on the temperature-dependence and the anisotropy of the spin-orbit torque (SOT). For this purpose we use 3rd order perturbation theory in the framework of the Keldysh formalism in order to derive suitable equations to compute the magnonic SOT. We find several contributions to the magnonic SOT, which depend differently on the spin-wave stiffness $\mathcal{A}$ and on the temperature $T$, with the dominating contribution scaling like $T^{2}/\mathcal{A}^{2}$. Based on this formalism we compute the magnonic SOT in the ferromagnetic Rashba model. For large Ras…
We use a perturbative approach to study the effects of interfacial spin-orbit coupling in magnetic multilayers by treating the two-dimensional Rashba model in a fully three-dimensional description of electron transport near an interface. This formalism provides a compact analytic expression for current-induced spin-orbit torques in terms of unperturbed scattering coefficients, allowing computation of spin-orbit torques for various contexts, by simply substituting scattering coefficients into the formulas. It applies to calculations of spin-orbit torques for magnetic bilayers with bulk magnetis…
We propose the use of resonant tunneling as a route to enhance the spin-transfer torque switching characteristics of magnetic tunnel junctions. The proposed device structure is a resonant tunneling magnetic tunnel junction based on a MgO-semiconductor heterostructure sandwiched between a fixed magnet and a free magnet. Using the non-equilibrium Green's function formalism coupled self consistently with the Landau-Lifshitz-Gilbert-Slonczewski equation, we demonstrate enhanced tunnel magneto-resistance characteristics as well as lower switching voltages in comparison with traditional trilayer dev…
Current-driven spin-orbit torques are investigated in a heterostructure composed of a ferromagnet deposited on top of a three dimensional topological insulator using the linear response formalism. We develop a tight-binding model of the heterostructure adopting a minimal interfacial hybridization scheme that promotes induced magnetic exchange on the topological surface states, as well as induced Rashba-like spin-orbit coupling in the ferromagnet. Therefore, our model accounts for spin Hall effect from bulk states together with inverse spin galvanic and magnetoelectric effects at the interface …
The phenomenon of spin transfer torque (STT) has attracted a great deal of interests due to its promising prospects in practical spintronic devices. In this paper, we report a theoretical investigation of STT in a noncollinear magnetic tunnel junction under ac modulation based on the nonequilibrium Green's function formalism, and derive a closed-formulation for predicting the time-averaged STT. Using this formulation, the ac STT of a carbon-nanotube-based magnetic tunnel junction is analyzed. Under ac modulation, the low-bias linear (quadratic) dependence of the in-plane (out-of-plane) torque …
We develop a general phenomenology describing the interplay between coherent and incoherent dynamics in ferromagnetic insulators. Using the Onsager reciprocity and Neumann's principle, we derive expressions for the local thermomagnonic torques exerted by thermal magnons on the order-parameter dynamics and the reciprocal pumping processes, which are in close analogy to the spin-transfer torque and spin pumping at metallic interfaces. Our formalism is applicable to general long-wavelength dynamics and, although here we explicitly focus on ferromagnetic insulators possessing U(1) symmetry, our ap…
Within the spinor Boltzmann equation (SBE) formalism, we derived a temperature dependent thermal spin-orbit torque based on local equilibrium assumption in a system with Rashba spin-orbit interaction. If we expand the distribution function of spinor Boltzmann equation around local equilibrium distribution, we can obtain the spin diffusion equation from SBE, then the spin transfer torque, spin orbit torque as well as thermal spin-orbit torque we seek to can be read out from this equation. It exhibits that this thermal spin-orbit torque originates from the temperature gradient of local equilibri…
We study deterministic magnetic reversal of a perpendicularly magnetized Co layer in a Co/MgO/Ta nano-square driven by spin Hall torque from an in-plane current flowing in an underlying Pt layer. The rate-limiting step of the switching process is domain-wall (DW) depinning by spin Hall torque via a thermally-assisted mechanism that eventually produces full reversal by domain expansion. An in-plane applied magnetic field collinear with the current is required, with the necessary field scale set by the need to overcome DW chirality imposed by the Dzyaloshinskii-Moriya interaction. Once Joule hea…
A new physical mechanism for generating spin-transfer torque is proposed. It is due to interference of bias driven nonequilibrium electrons incident on a switching junction with the electrons reflected from an insulating barrier inserted in the junction after the switching magnet. It is shown using the rigorous Keldysh formalism that this new out-of-plane torque $T_{\perp}$ is proportional to an applied bias and is as large as the torque in a conventional junction generated by a strong charge current. However, the charge current and the in-plane torque $T_{\parallel}$ are almost completely sup…
We consider theoretically domain wall motion driven by spin-orbit and spin Hall torques. We find that it is possible to achieve universal absence of Walker breakdown for all spin-orbit torques using experimentally relevant spin-orbit coupling strengths. For spin-orbit torques other than the pure Rashba spin-orbit torque, this gives a linear current-velocity relation instead of a saturation of the velocity at high current densities. The effect is very robust and is found in both soft and hard magnetic materials, as well as in the presence of the Dzyaloshinskii-Moriya interaction and in coupled …
Benchmarking the generalized Kadanoff-Baym ansatz and second-order adiabatic expansion using time-dependent spintronic effects: Spin pumping, torque, and inertia
University of Delaware（特拉华大学）；Universidad Nacional de Colombia（哥伦比亚国立大学）
The angular dependence of spin-orbit torque in a disordered Co/Pt bilayer is calculated using a first-principles non-equilibrium Green's function formalism with an explicit supercell averaging over Anderson disorder. In addition to the usual dampinglike and fieldlike terms, the odd torque contains a sizeable planar Hall-like term $(\mathbf{m\cdot E})\mathbf{m}\times(\mathbf{z}\times\mathbf{m})$ whose contribution to current-induced damping is consistent with experimental observations. The dampinglike and planar Hall-like torquances depend weakly on disorder strength, while the fieldlike torqua…
In this work we demonstrate explicit analytical expressions for both charge and spin currents which constitute the 2x2 spinor in magnetic tunnel junctions with noncollinear magnetizations under applied voltage. The calculations have been performed within the free electron model in the framework of the Keldysh formalism and WKB approximation. We demonstrate that spin/charge currents and spin transfer torques are all explicitly expressed through only three irreducible quantities, without further approximations. The conditions and mechanisms of deviation from the conventional sine angular depende…
We propose to use oscillating spin currents with slowly varying frequency (chirp) to manipulate and control the magnetization dynamics in a nanomagnet. By recasting the Landau-Lifshitz-Slonczewski equation in a quantum-like two-level formalism, we show that a chirped spin current polarized in the direction normal to the anisotropy axis can induce a stable precession of the magnetic moment at any angle (up to $90^\circ$) with respect to the anisotropy axis. The drive current can be modest ($10^6\,\rm A/cm^2$ or lower) provided the chirp rate is sufficiently slow. The induced precession is stabl…
We demonstrate how magnetic droplet soliton pairs, nucleated by two separated nano-contact (NC) spin torque oscillators, can merge into a single droplet soliton. A detailed description of the magnetization dynamics of this merger process is obtained by micromagnetic simulations: A droplet pair with a steady-state in-phase spin precession is generated through the spin-transfer torque effect underneath two separate NCs, followed by a gradual expansion of the droplets volume and the out phase of magnetization on the inner side of the two droplets, resulting in the droplets merging into a larger d…
Using the Kubo linear-response formalism we derive expressions to calculate the heat current generated by magnetization dynamics in magnets with broken inversion symmetry and spin-orbit interaction (SOI). The effect of producing heat currents by magnetization dynamics constitutes the Onsager reciprocal of the thermal spin-orbit torque (TSOT), i.e., the generation of torques on the magnetization due to temperature gradients. We find that the energy current driven by magnetization dynamics contains a contribution from the Dzyaloshinskii-Moriya interaction (DMI), which needs to be subtracted from…
We investigate scaling of technologically relevant magnetic tunnel junction devices in the trilayer and pentalayer configurations by varying the cross-sectional area along the transverse direction using the non-equilibrium Green's function spin transport formalism. We study the geometry dependence by considering square and circular cross-sections. As the transverse dimension in each case reduces, we demonstrate that the transverse mode energy profile plays a major role in the resistance-area product. Both types of devices show constant tunnel magnetoresistance at larger cross-sectional areas b…
We examine a modified drift-diffusion formalism to describe spin transport near an ultrathin magnet whose thickness is similar to or less than the spin dephasing length. Most of the previous theories on spin torque assume the transverse component of a injected spin current dephases perfectly thus are fully absorbed into the ferromagnet. However, in the state-of-art multilayer systems under consideration of recent studies, the thicknesses of ferromagnets are on the order of or less than a nanometer, thus one cannot safely assume the spin dephasing to be perfect. To describe the effects of a fin…
Spin transfer torque nano-oscillators are potential candidates for replacing the traditional inductor based voltage controlled oscillators in modern communication devices. Typical oscillator designs are based on trilayer magnetic tunnel junctions which are disadvantaged by low power outputs and poor conversion efficiencies. In this letter, we theoretically propose to use resonant spin filtering in pentalayer magnetic tunnel junctions as a possible route to alleviate these issues and present device designs geared toward a high microwave output power and an efficient conversion of the d.c. input…
We have developed a drift-diffusion equation of spin transport in collinear bipartite metallic antiferromagnets. Starting from a model tight-binding Hamiltonian, we obtain the quantum kinetic equation within Keldysh formalism and expand it to the lowest order in spatial gradient using Wigner expansion method. In the diffusive limit, these equations track the spatio-temporal evolution of the spin accumulations and spin currents on each sublattice of the antiferromagnet. We use these equations to address the nature of spin transfer torque in (i) a spin-valve composed of a ferromagnet and an anti…
The thickness dependence of spin-orbit torque and magnetoresistance in ferromagnet/heavy-metal bilayers is studied using the first-principles non-equilibrium Green's function formalism combined with the Anderson disorder model. A systematic expansion in orthogonal vector spherical harmonics is used for the angular dependence of the torque. The damping-like torque in Co/Pt and Co/Au bilayers can be described as a sum of the spin-Hall contribution, which increases with thickness in agreement with the spin-diffusion model, and a comparable interfacial contribution. The magnetoconductance in the p…
Motivated by the importance of understanding competing mechanisms to current-induced spin-orbit torque in complex magnets, we develop a unified theory of current-induced spin-orbital coupled dynamics. The theory describes angular momentum transfer between different degrees of freedom in solids, e.g., the electron orbital and spin, the crystal lattice, and the magnetic order parameter. Based on the continuity equations for the spin and orbital angular momenta, we derive equations of motion that relate spin and orbital current fluxes and torques describing the transfer of angular momentum betwee…
We report that synthetic antiferromagnets (SAFs) can be efficiently switched by spin-orbit torques (SOTs) and the switching scheme does not obey the usual SOT switching rule. We show that both the positive and negative spin Hall angle (SHA)-like switching can be observed in Pt/SAF structures with only positive SHA, depending on the strength of applied in-plane fields. A new switching mechanism directly arising from the asymmetric domain expansion is proposed to explain the anomalous switching behaviors. Contrary to the macrospin-based switching model that the SOT switching direction is determi…
A consistent theory to describe the correlated dynamics of quantum mechanical itinerant spins and semiclassical local magnetization is given. We consider the itinerant spins as quantum mechanical operators, whereas local moments are considered within classical Lagrangian formalism. By appropriately treating fluctuation space spanned by basis functions, including a zero-mode wave function, we construct coupled equations of motion for the collective coordinate of the center-of-mass motion and the localized zero-mode coordinate perpendicular to the domain wall plane. By solving them, we demonstra…
The time-dependent spin current mediated spin transfer torque behaviour has been investigated via scattering formalism within density functional theory framework supported by Green's function. Quantum magnetotransport characteristics have been revealed in a model semiconducting MoS$_2$/phosphorene van der Waals heterostructure. The dynamics of spin current channelized heterolayer transport has been studied with rotational variation in magnetization angle. It is observed that the time-dependent spin transport torque remains invariant irrespective of magnetization angle direction. The polarized …
In collinear magnets lacking inversion symmetry application of electric currents induces torques on the magnetization and conversely magnetization dynamics induces electric currents. The two effects, which both rely on spin-orbit interaction (SOI), are reciprocal to each other and denoted direct spin-orbit torque (SOT) and inverse spin-orbit torque (ISOT), respectively. We derive expressions for SOT and ISOT within the Kubo linear response formalism. We show that expressions suitable for density-functional theory calculations can be derived either starting from a Kohn-Sham Hamiltonian with tim…
A fully quantum mechanical description of the precessional damping of Pt/Co bilayer is presented in the framework of the Keldysh Green function approach using {\it ab initio} electronic structure calculations. In contrast to previous calculations of classical Gilbert damping ($α_{GD}$), we demonstrate that $α_{GD}$ in the quantum case does not diverge in the ballistic regime due to the finite size of the total spin, $S$. In the limit of $S\rightarrow\infty$ we show that the formalism recovers the torque correlation expression for $α_{GD}$ which we decompose into spin-pumping and spin-orbital t…
An applied electric current through a space-inversion asymmetric magnet induces spin-orbit torques (SOTs) on the magnetic moments, which holds much promise for future memory devices. We discuss general Green's function expressions suitable to compute the linear-response SOT in disordered ferromagnets. The SOT can be decomposed into an even and an odd component with respect to magnetization reversal, where in the limit of vanishing disorder the even SOT is given by the constant Berry curvature of the occupied states, while the odd part exhibits a divergence with respect to disorder strength. Wi…
Deterministic magnetization switching using spin-orbit torque (SOT) has recently emerged as an efficient means to electrically control the magnetic state of ultrathin magnets. The SOT switching still lacks in oscillatory switching characteristics over time, therefore, it is limited to bipolar operation where a change in polarity of the applied current or field is required for bistable switching. The coherent rotation based oscillatory switching schemes cannot be applied to SOT because the SOT switching occurs through expansion of magnetic domains. Here, we experimentally achieve oscillatory sw…
We study the spin relaxation torque in nonmagnetic or ferromagnetic metals with nonuniform spin-orbit coupling within the Keldysh Green's function formalism. In non-magnet, the relaxation torque is shown to arise when the spin-orbit coupling is not uniform. In the absence of an external field, the spin current induced by the relaxation torque is proportional to the vector chirality of Rashba-type spin-orbit field (RSOF). In the presence of an external field, on the other hand, spin relaxation torque arises from the coupling of the external field and vector chirality of RSOF. Our result indicat…
We present theoretical studies of the intrinsic spin orbit torque (SOT) in a single domain ferromagnetic layer with Rashba spin-orbit coupling (SOC) using the non-equilibrium Green's function formalism for a model Hamiltonian. We find that, to the first order in SOC, the intrinsic SOT has only the field-like torque symmetry and can be interpreted as the longitudinal spin current induced by the charge current and Rashba field. We analyze the results in terms of the material related parameters of the electronic structure, such as band filling, band width, exchange splitting, as well as the Rashb…
Using the Boltzmann formalism based on the first principles electronic structure and scattering rates, we investigate the current-induced spin accumulation and spin-orbit torques in FePt/Pt and Co/Cu bilayers in the presence of substitutional impurities. In FePt/Pt bilayers we consider the effect of intermixing of Fe and Pt atoms in the FePt layer, and find a crucial dependence of spin accumulation and spin-orbit torques on the details of the distribution of these defects. In Co/Cu bilayers we predict that the magnitude and sign of the spin-orbit torque and spin accumulation depend very sensit…
Motivated by recent experiments observing spin-orbit torque (SOT) acting on the magnetization $\vec{m}$ of a ferromagnetic (F) overlayer on the surface of a three-dimensional topological insulator (TI), we investigate the origin of the SOT and the magnetization dynamics in such systems. We predict that lateral F/TI bilayers of finite length, sandwiched between two normal metal leads, will generate a large antidamping-like SOT per very low charge current injected parallel to the interface. The large values of antidamping-like SOT are {\it spatially localized} around the transverse edges of the …
Using the linear response formalism for the spin-orbit torque (SOT) we compute from first principles the SOT in a system of two layers of L1$_0$-FePt(001) deposited on an fcc Pt(001) substrate of varying thickness. We find that at room temperature the values of the SOTs that are even and odd with respect to magnetization generally lie in the range of values measured and computed for Co/Pt bilayers. We also observe that the even SOT is much more robust with respect to changing the number of layers in the substrate, and as a function of energy it follows the general trend of the even SOT exerted…
We have measured the relaxation time of a thermally unstable ferromagnetic nanoparticle incorporated into a magnetic tunnel junction (MTJ) as a function of applied magnetic field, voltage V (-0.38 V < V < +0.26 V), and temperatures (283 K< T< 363 K) . By analyzing the results within the framework of a modified Néel-Brown formalism we determine the effective attempt time of the nanoparticle and also the bias dependences of the in-plane and out-of-plane spin torques. There is a significant linear modification of the effective temperature with voltage due to the in-plane torque and a significant …
We report that by measuring current-induced hysteresis loop shift versus in-plane bias magnetic field, the spin Hall effect (SHE) contribution of the current-induced effective field per current density, $χ_{SHE}$, can be estimated for Pt and Ta-based magnetic heterostructures with perpendicular magnetic anisotropy (PMA). We apply this technique to a Pt-based sample with its ferromagnetic (FM) layer being wedged-deposited and discover an extra effective field contribution, $χ_{Wedged}$, due to the asymmetric nature of the deposited FM layer. We confirm the correlation between $χ_{Wedged}$ and t…
Symmetry and magnitude of spin-orbit torques (SOT), i.e., current-induced torques on the magnetization of systems lacking inversion symmetry, are investigated in a fully relativistic linear response framework based on the Kubo formalism. By applying all space-time symmetry operations contained in the magnetic point group of a solid to the relevant response coefficient, the torkance expressed as torque-current correlation function, restrictions to the shape of the direct and inverse response tensors are obtained. These are shown to apply to the corresponding thermal analogues as well, namely th…
We formulate a generalized scattering field theory a la Buttiker describing particles transport in magnetic/superconducting heterostructures. The proposed formalism, characterized by a four- component spinorial wavefunction of the Bogoliubov de Gennes theory, allows to describe the spin flipping phenomena induced by noncollinear magnetizations in the scattering region. As a specific application of the theory, we analyze the conductance, the magnetoresistance and the generation of spin-torque produced by an applied voltage in a spin-valve system. Quantum size effects and quantum beating pattern…
A strategy to drive skyrmion motion by a combination of an anisotropy gradient and spin Hall effect has recently been demonstrated. Here, we study the fundamental properties of this type of motion by combining micromagnetic simulations and a generalized Thiele equation. We find that the anisotropy gradient drives the skyrmion mainly along the direction perpendicular to the gradient, due to the conservative part of the torque. There is some slower motion along the direction parallel to the anisotropy gradient due to damping torque. When an appropriate spin Hall torque is added, the skyrmion vel…
In this contribution, we start off from a fully relativistic description of a single electron non-minimally coupled to an external electromagnetic field. Making direct use of the field equation, instead of canonically deriving from the Lagrangian density, the relativistic total angular momentum is attained, where the effect of the relativistic torque due to the external sources is also taken into account. Both the spin density and the orbital angular momentum tensors are identified in the relativistic-covariant approach. Furthermore, the symmetric and gauge invariant energy-momentum tensor is …
