We present a unified theory of magnetic damping in itinerant electron ferromagnets at order $q^2$ including electron-electron interactions and disorder scattering. We show that the Gilbert damping coefficient can be expressed in terms of the spin conductivity, leading to a Matthiessen-type formula in which disorder and interaction contributions are additive. In a weak ferromagnet regime, electron-electron interactions lead to a strong enhancement of the Gilbert damping.
The Gilbert damping of ferromagnetic materials is arguably the most important but least understood phenomenological parameter that dictates real-time magnetization dynamics. Understanding the physical origin of the Gilbert damping is highly relevant to developing future fast switching spintronics devices such as magnetic sensors and magnetic random access memory. Here, we report an experimental study of temperature-dependent Gilbert damping in permalloy (Py) thin films of varying thicknesses by ferromagnetic resonance. From the thickness dependence, two independent contributions to the Gilbert…
The magnetization dynamics of a single domain ferromagnet in contact with a thermal bath is studied by scattering theory. We recover the Landau-Liftshitz-Gilbert equation and express the effective fields and Gilbert damping tensor in terms of the scattering matrix. Dissipation of magnetic energy equals energy current pumped out of the system by the time-dependent magnetization, with separable spin-relaxation induced bulk and spin-pumping generated interface contributions. In linear response, our scattering theory for the Gilbert damping tensor is equivalent with the Kubo formalism.
We present a systematic phenomenological description of Gilbert damping in two-sublattice magnets. Our theory covers the full range of materials from ferro- via ferri- to antiferromagnets. Following a Rayleigh dissipation functional approach within a Lagrangian classical field formulation, the theory captures intra- as well as cross-sublattice terms in the Gilbert damping, parameterized by a 2$\times$2 matrix. When spin-pumping into an adjacent conductor causes dissipation, we obtain the corresponding Gilbert damping matrix in terms of the interfacial spin-mixing conductances. Our model reprod…
We present a microscopic calculation of the Gilbert damping constant for the magnetization of a two-dimensional spin-polarized electron liquid in the presence of intrinsic spin-orbit interaction. First we show that the Gilbert constant can be expressed in terms of the auto-correlation function of the spin-orbit induced torque. Then we specialize to the case of the Rashba spin-orbit interaction and we show that the Gilbert constant in this model is related to the spin-channel conductivity. This allows us to study the Gilbert damping constant in different physical regimes, characterized by diffe…
The precession and damping of a collinear magnetization displaced from its equilibrium are described by the Landau-Lifshitz-Gilbert equation. For a noncollinear magnetization, it is not known how the damping should be described. We use first-principles scattering theory to investigate the damping in one-dimensional transverse domain walls (DWs) of the important ferromagnetic alloy Ni$_{80}$Fe$_{20}$ and interpret the results in terms of phenomenological models. The damping is found to depend not only on the magnetization texture but also on the specific dynamic modes of Bloch and Néel DWs. Eve…
In spintronics applications, ultrafast spin dynamics have to be controlled at femtosecond (fs) timescales via fs-laser radiation. At such ultrafast timescales, the effect of the Gilbert damping factor α on ultrafast demagnetization time should be considered. In previous explorations for the relationship between these two parameters, it was found that the theoretical calculations based on the local spin-flip scattering model do not agree with the experimental results. Here, we find that in Fe81Ga19(FeGa)/Ir20Mn80(IrMn) bilayers, the unconventional IrMn thickness dependence of α results from the…
We have measured the size effect in nonlocal Gilbert relaxation rate in FM(t$_{FM}$) / Cu (5nm) [/ Pt (2nm)] / Al(2nm) heterostructures, FM = \{ Ni$_{81}$Fe$_{19}$, Co$_{60}$Fe$_{20}$B$_{20}$, pure Co\}. Common behavior is observed for three FM layers, where the additional relaxation obeys both a strict inverse power law dependence $ΔG =K \:t^{n}$, $n=-\textrm{1.04}\pm\textrm{0.06}$ and a similar magnitude $K=\textrm{224}\pm\textrm{40 Mhz}\cdot\textrm{nm}$. As the tested FM layers span an order of magnitude in spin diffusion length $λ_{SDL}$, the results are in support of spin diffusion, rathe…
Gilbert damping is a key property governing magnetization dynamics in ordered magnets. We present a theoretical study of intrinsic Gilbert damping induced by magnon decay in antiferromagnetic metals through $s$-$d$ exchange interaction. Our theory delineates the qualitative features of damping in metallic antiferromagnets owing to their bipartite nature, in addition to providing analytic expressions for the damping parameters. Magnon-induced intraband electron scattering is found to predominantly cause magnetization damping, whereas the Néel field is found to be damped via disorder. Depending …
Using a formulation of first-principles scattering theory that includes disorder and spin-orbit coupling on an equal footing, we calculate the resistivity $ρ$, spin flip diffusion length $l_{sf}$ and the Gilbert damping parameter $α$ for Ni$_{1-x}$Fe$_x$ substitutional alloys as a function of $x$. For the technologically important Ni$_{80}$Fe$_{20}$ alloy, permalloy, we calculate values of $ρ= 3.5 \pm 0.15$ $μ$Ohm-cm, $l_{sf}=5.5 \pm 0.3$ nm, and $α= 0.0046 \pm 0.0001$ compared to experimental low-temperature values in the range $4.2-4.8$ $μ$Ohm-cm for $ρ$, $5.0-6.0$ nm for $l_{sf}$, and $0.00…
In-situ, device level measurement of thermal mag-noise spectral linewidths in 60nm diameter CPP-GMR spin-valve stacks of IrMn/ref/Cu/free, with reference and free layer of similar CoFe/CoFeGe alloy, are used to simultaneously determine the intrinsic Gilbert damping for both magnetic layers. It is shown that careful alignment at a "magic-angle" between free and reference layer static equilibrium magnetization can allow direct measurement of the broadband intrinsic thermal spectra in the virtual absence of spin-torque effects which otherwise grossly distort the spectral line shapes and require l…
We analyzed the enhancement of the Gilbert damping constant due to spin pumping in non-collinear ferromagnet / non-magnet / ferromagnet trilayer systems. We show that the Gilbert damping constant depends both on the precession angle of the magnetization of the free layer and on the direction of the magntization of the fixed layer. We find the condition to be satisfied to realize strong enhancement of the Gilbert damping constant.
Using a scattering matrix approach, the precession of the magnetization of a ferromagnet is shown to transfer spins into adjacent normal metal layers. This ``pumping'' of spins slows down the precession corresponding to an enhanced Gilbert damping factor in the Landau-Lifshitz equation. The damping is expressed in terms of the scattering matrix of the ferromagnet-normal metal interface, which is accessible to model and first-principles calculations. Our estimates for permalloy thin films explain the trends observed in recent experiments.
We study the spin dynamics of a ferromagnetic insulator on which graphene is placed. We show that the Gilbert damping is enhanced by the proximity exchange coupling at the interface. The modulation of the Gilbert damping constant is proportional to the product of the spin-up and spin-down densities of states of graphene. Consequently, the Gilbert damping constant in a strong magnetic field oscillates as a function of the external magnetic field that originates from the Landau level structure of graphene. We find that a measurement of the oscillation period enables the strength of the exchange …
Magnon chiral damping beyond the Dzyaloshinskii-Moriya interaction
Rheinland-Pfälzische Technische Universität Kaiserslautern-Landau（莱茵兰-普法尔茨技术大学凯泽斯劳滕-兰道分校）；Indian Institute of Technology (Indian School of Mines)（印度理工学院（印度矿业学院））；Uppsala University（乌普萨拉大学）；University of Münster（明斯特大学）
By means of first principles calculations, we have systematically investigated how the magnetodynamic properties Gilbert damping, magnetization and exchange stiffness are affected when permalloy (Py) (Fe$_{0.19}$Ni$_{0.81}$) is doped with 4d or 5d transition metal impurities. We find that the trends in the Gilbert damping can be understood from relatively few basic parameters such as the density of states at the Fermi level, the spin-orbit coupling and the impurity concentration. % The temperature dependence of the Gilbert damping is found to be very weak which we relate to the lack of intraba…
Spin valves form a key building block in a wide range of spintronic concepts and devices from magnetoresistive read heads to spin-transfer-torque oscillators. We elucidate the dependence of the magnetic damping in the free layer on the angle its equilibrium magnetization makes with that in the fixed layer. The spin pumping-mediated damping is anisotropic and tensorial, with Gilbert- and Bloch-like terms. Our investigation reveals a mechanism for tuning the free layer damping in-situ from negligible to a large value via the orientation of fixed layer magnetization, especially when the magnets a…
A keen interest towards technological implications of spin-orbit driven magnetization dynamics requests a proper theoretical description, especially in the context of a microscopic framework, to be developed. Indeed, magnetization dynamics is so far approached within Landau-Lifshitz-Gilbert equation which characterizes torques on magnetization on purely phenomenological grounds. Particularly, spin-orbit coupling does not respect spin conservation, leading thus to angular momentum transfer to lattice and damping as a result. This mechanism is accounted by the Gilbert damping torque which descri…
We present a microscopic theory for magnetization relaxation in metallic ferromagnets of nanoscopic dimensions that is based on the dynamic spin response matrix in the presence of spin-orbit coupling. Our approach allows the calculation of the spin excitation damping rate even for perfectly crystalline systems, where existing microscopic approaches fail. We demonstrate that the relaxation properties are not completely determined by the transverse susceptibility alone, and that the damping rate has a non-negligible frequency dependence in experimentally relevant situations. Our results indicate…
Starting from a model that consists of a semiclassical spin coupled to two leads we present a microscopic derivation of the Langevin equation for the direction of the spin. For slowly-changing direction it takes on the form of the stochastic Landau-Lifschitz-Gilbert equation. We give expressions for the Gilbert damping parameter and the strength of the fluctuations, including their bias-voltage dependence. At nonzero bias-voltage the fluctuations and damping are not related by the fluctuation-dissipation theorem. We find, however, that in the low-frequency limit it is possible to introduce a v…
Precessing magnetization in a thin film magnetic insulator pumps spins into adjacent metals; however, this phenomenon is not quantitatively understood. We present a theory for the dependence of spin-pumping on the transverse mode number and in-plane wave vector. For long-wavelength spin waves, the enhanced Gilbert damping for the transverse mode volume waves is twice that of the macrospin mode, and for surface modes, the enhancement can be ten or more times stronger. Spin-pumping is negligible for short-wavelength exchange spin waves. We corroborate our analytical theory with numerical calcula…
Exceptional point (EP) associated with the parity-time (PT) symmetry breaking is receiving considerable recent attention by the broad physics community. By introducing balanced gain and loss, it has been realized in photonic, acoustic, and electronic structures. However, the observation of magnonic EP remains elusive. The major challenge is to experimentally generate the negative Gilbert damping, which was thought to be highly unlikely but is demanded by the PT symmetry. In this work, we study the magneto-optical interaction of circularly-polarized lasers with a submicron magnet placed in an o…
We study how the shape of the spinwave resonance lines in rf-voltage induced FMR can be used to extract the spinwave density of states and the damping within the precessing layer in nanoscale tunnel junctions that possess perpendicular anisotropy. We work with a field applied along the easy axis to preserve the uniaxial symmetry of the system. We describe the set-up to study the susceptibility contributions of the spin waves in the field-frequency space. We then identify the maximum device size above which the spinwaves can no longer be studied in isolation as the linewidths of their responses…
The ability to controllably manipulate the laser-induced ultrafast magnetic dynamics is a prerequisite for future high speed spintronic devices. The optimization of devices requires the controllability of the ultrafast demagnetization time, , and intrinsic Gilbert damping, . In previous attempts to establish the relationship between and , the rare-earth doping of a permalloy film with two different demagnetization mechanism is not a suitable candidate. Here, we choose Co/Ni bilayers to investigate the relations between and by means of time-resolved magneto-optical Kerr effect (TRMOKE) via adju…
We present a theory of Landau-Lifshitz-Gilbert damping $α$ for a localized spin ${\vec S}$ in the junction coupled to the conduction electrons in both leads under an applied volatege $V$. We find the voltage dependence of the damping term reflecting the energy dependence of the density of states. We find the effect is linear in the voltage and cotrolled by particle-hole asymmetry of the leads.
We derive an approximate expression for the Gilbert damping coefficient α_G of itinerant electron ferromagnets which is based on their description in terms of spin-density-functional-theory (SDFT) and Kohn-Sham quasiparticle orbitals. We argue for an expression in which the coupling of magnetization fluctuations to particle-hole transitions is weighted by the spin-dependent part of the theory's exchange-correlation potential, a quantity which has large spatial variations on an atomic length scale. Our SDFT result for α_G is closely related to the previously proposed spin-torque correlation-fun…
We compute the Gilbert damping in (Ga,Mn)As based on the scattering theory of magnetization relaxation. The disorder scattering is included non-perturbatively. In the clean limit, the spin-pumping from the localized d-electrons to the itinerant holes dominates the relaxation processes. In the diffusive regime, the breathing Fermi-surface effect is balanced by the effects of interband scattering, which cause the Gilbert damping constant to saturate at around 0.005. In small samples, the system shape induces a large anisotropy in the Gilbert damping.
We theoretically study dynamic properties of thin ferromagnetic films in contact with normal metals. Moving magnetizations cause a flow of spins into adjacent conductors, which relax by spin flip, scatter back into the ferromagnet, or are absorbed by another ferromagnet. Relaxation of spins outside the moving magnetization enhances the overall damping of the magnetization dynamics in accordance with the Gilbert phenomenology. Transfer of spins between different ferromagnets by these nonequilibrium spin currents leads to a long-ranged dynamic exchange interaction and novel collective excitation…
We obtain the exact one-spin intrinsic localized excitation in an anisotropic Heisenberg ferromagnetic spin chain in a constant/variable external magnetic field with Gilbert damping included. We also point out how an appropriate magnitude spin current term in a spin transfer nano-oscillator (STNO) can stabilize the tendency towards damping. Further, we show how this excitation can be sustained in a recently suggested PT-symmetric magnetic nanostructure. We also briefly consider more general spin excitations.
Spin-transfer torques (STT), Gilbert damping (GD), and effective spin renormalization (ESR) are investigated microscopically in a 2D Rashba ferromagnet with spin-independent Gaussian white-noise disorder. Rashba spin-orbit coupling induced anisotropy of these phenomena is thoroughly analysed. For the case of two partly filled spin subbands, a remarkable relation between the anisotropic STT, GD, and ESR is established. In the absence of magnetic field and other torques on magnetization, this relation corresponds to a current-induced motion of a magnetic texture with the classical drift velocity…
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…
The domain wall depinning field represents the minimum magnetic field needed to move a domain wall, typically pinned by samples' disorder or patterned constrictions. Conventionally, such field is considered independent on the Gilbert damping since it is assumed to be the field at which the Zeeman energy equals the pinning energy barrier (both damping independent). Here, we analyse numerically the domain wall depinning field as function of the Gilbert damping in a system with perpendicular magnetic anisotropy and Dzyaloshinskii-Moriya interaction. Contrary to expectations, we find that the depi…
Gilbert damping in metallic ferromagnets from Schwinger-Keldysh field theory: Intrinsically nonlocal and nonuniform, and made anisotropic by spin-orbit coupling
The spin pumping mechanism and associated interfacial Gilbert damping are demonstrated in ion-beam sputtered Co2FeAl (CFA) /Mo bilayer thin films employing ferromagnetic resonance spectroscopy. The dependence of the net spin current transportation on Mo layer thickness, 0 to 10 nm, and the enhancement of the net effective Gilbert damping are reported. The experimental data has been analyzed using spin pumping theory in terms of spin current pumped through the ferromagnet /nonmagnetic metal interface to deduce the effective spin mixing conductance and the spin-diffusion length, which are estima…
We examine room-temperature magnetic relaxation in polycrystalline Fe films. Out-of-plane ferromagnetic resonance (FMR) measurements reveal Gilbert damping parameters of $\approx$ 0.0024 for Fe films with thicknesses of 4-25 nm, regardless of their microstructural properties. The remarkable invariance with film microstructure strongly suggests that intrinsic Gilbert damping in polycrystalline metals at room temperature is a local property of nanoscale crystal grains, with limited impact from grain boundaries and film roughness. By contrast, the in-plane FMR linewidths of the Fe films exhibit d…
We study time-dependent electronic and spin transport through an electronic level connected to two leads and coupled with a single-molecule magnet via exchange interaction. The molecular spin is treated as a classical variable and precesses around an external magnetic field. We derive expressions for charge and spin currents by means of the Keldysh non-equilibrium Green's functions technique in linear order with respect to the time-dependent magnetic field created by this precession. The coupling between the electronic spins and the magnetization dynamics of the molecule creates inelastic tunn…
The phenomenological Landau-Lifshitz-Gilbert (LLG) equation of motion remains as the cornerstone of contemporary magnetisation dynamics studies, wherein the Gilbert damping parameter has been attributed to first-order relativistic effects. To include magnetic inertial effects the LLG equation has previously been extended with a supplemental inertia term and the arising inertial dynamics has been related to second-order relativistic effects. Here we start from the relativistic Dirac equation and, performing a Foldy-Wouthuysen transformation, derive a generalised Pauli spin Hamiltonian that cont…
An analysis of a single-domain magnetic needle in the presence of an external magnetic field ${\bf B}$ is carried out with the aim of achieving a high precision magnetometer. We determine the uncertainty $ΔB$ of such a device due to Gilbert dissipation and the associated internal magnetic field fluctuations that gives rise to diffusion of the magnetic needle axis direction ${\bf n}$ and the needle orbital angular momentum. The levitation of the magnetic needle in a magnetic trap and its stability are also analyzed.
Giant Gilbert damping anisotropy is identified as a signature of strong Rashba spin-orbit coupling in a two-dimensional antiferromagnet on a honeycomb lattice. The phenomenon originates in spin-orbit induced splitting of conduction electron subbands that strongly suppresses certain spin-flip processes. As a result, the spin-orbit interaction is shown to support an undamped non-equilibrium dynamical mode that corresponds to an ultrafast in-plane Néel vector precession and a constant perpendicular-to-the-plane magnetization. The phenomenon is illustrated on the basis of a two dimensional $s$-$d$…
We report time-resolved measurements of current-induced reversal of a free magnetic layer in Py/Cu/Py elliptical nanopillars at temperatures T = 4.2 K to 160 K. Comparison of the data to Landau-Lifshitz-Gilbert macrospin simulations of the free layer switching yields numerical values for the spin torque and the Gilbert damping parameters as functions of T. The damping is strongly T-dependent, which we attribute to the antiferromagnetic pinning behavior of a thin permalloy oxide layer around the perimeter of the free layer. This adventitious antiferromagnetic pinning layer can have a major impa…
Ferromagnetic resonance in thin films is analyzed under the influence of spatiotemporal feedback effects. The equation of motion for the magnetization dynamics is nonlocal in both space and time and includes isotropic, anisotropic and dipolar energy contributions as well as the conserved Gilbert- and the non-conserved Bloch-damping. We derive an analytical expression for the peak-to-peak linewidth. It consists of four separate parts originated by Gilbert damping, Bloch-damping, a mixed Gilbert-Bloch component and a contribution arising from retardation. In an intermediate frequency regime the …
We present a novel process that allows the transfer of monocrystalline yttrium-iron-garnet microstructures onto virtually any kind of substrate. The process is based on a recently developed method that allows the fabrication of freestanding monocrystalline YIG bridges on gadolinium-gallium-garnet. Here the bridges' spans are detached from the substrate by a dry etching process and immersed in a watery solution. Using drop casting the immersed YIG platelets can be transferred onto the substrate of choice, where the structures finally can be reattached and thus be integrated into complex devices…
Relaxation effects are of primary importance in the description of magnetic excitations, leading to a myriad of methods addressing the phenomenological damping parameters. In this work, we consider several well-established forms of calculating the intrinsic Gilbert damping within a unified theoretical framework, mapping out their connections and the approximations required to derive each formula. This scheme enables a direct comparison of the different methods on the same footing and a consistent evaluation of their range of validity. Most methods lead to very similar results for the bulk ferr…
We investigate the spin dynamics including Gilbert damping in the ferromagnetic nanowire arrays. We have measured the ferromagnetic resonance of ferromagnetic nanowire arrays using vector-network analyzer ferromagnetic resonance (VNA-FMR) and analyzed the results with the micromagnetic simulations. We find excellent agreement between the experimental VNA-FMR spectra and micromagnetic simulations result for various applied magnetic fields. We find that the demagnetization factor for longitudinal conditions, Nz (Ny) increases (decreases) as decreasing the nanowire width in the micromagnetic simu…
Perpendicular magnetic materials with low damping constant and high thermal stability have great potential for realizing high-density, non-volatile, and low-power consumption spintronic devices, which can sustain operation reliability for high processing temperatures. In this work, we study the Gilbert damping constant (α) of perpendicularly magnetized W/CoFeB/MgO films with a high perpendicular magnetic anisotropy (PMA) and superb thermal stability. The α of these PMA films annealed at different temperatures is determined via an all-optical Time-Resolved Magneto-Optical Kerr Effect method. We…
In this paper, we investigate theoretically the spin-orbit torque as well as the Gilbert damping for a two band model of a 2D helical surface state with a Ferromagnetic (FM) exchange coupling. We decompose the density matrix into the Fermi sea and Fermi surface components and obtain their contributions to the electronic transport as well as the spin-orbit torque (SOT). Furthermore, we obtain the expression for the Gilbert damping due to the surface state of a 3D Topological Insulator (TI) and predicted its dependence on the direction of the magnetization precession axis.
A Landau-Lifshitz-Gilbert type equation has been derived by using s-d model in which the s-electron system is regarded as an environment coupled weakly with the localized spins. Based on the irreducible linear response theory, we show that the relaxation function of the s-electron spin leads to the Gilbert type damping term which corresponds to the retarded resistance function in the generalized Langevin equation. The Ohmic form of the Gilbert term stems from the fact that the imaginary part of the response function (spin susceptibility) of the itinerant electron system is proportional to the …
We analyze the Landau-Lifshitz-Gilbert equation when the precession motion of the magnetic moments is additionally subjected to an uniaxial anisotropy and is driven by a multiplicative coupled stochastic field with a finite correlation time $τ$. The mean value for the spin wave components offers that the spin-wave dispersion relation and its damping is strongly influenced by the deterministic Gilbert damping parameter $α$, the strength of the stochastic forces $D$ and its temporal range $τ$. The spin-spin-correlation function can be calculated in the low correlation time limit by deriving an e…
The dynamics of the domain wall (DW) in a biaxial ferromagnet interacting with a spin-polarized current are described by sine-gordon (SG) equation coupled with Gilbert damping term in this paper. Within our frame-work of this model, we obtain a threshold of the current in the motion of a single DW with the perturbation theory on kink soliton solution to the corresponding ferromagnetic system, and the threshold is shown to be dependent on the Gilbert damping term. Also, the motion properties of the DW are discussed for the zero- and nonzero-damping cases, which shows that our theory to describe…
The localized magnon modes of isolated $kπ$ skyrmions on a field-polarized background are analyzed based on the Landau-Lifshitz-Gilbert equation within the terms of an atomistic classical spin model, with system parameters based on the Pd/Fe biatomic layer on Ir(111). For increasing skyrmion order $k$ a higher number of excitation modes are found, including modes with nodes in the radial eigenfunctions. It is shown that at low fields $2π$ and $3π$ skyrmions are destroyed via a burst instability connected to a breathing mode, while $1π$ skyrmions undergo an elliptic instability. At high fields …
The conventional Landau-Lifshitz-Gilbert (LLG) equation is a widely used tool to describe dynamics of local magnetic moments, viewed as classical vectors of fixed length, with their change assumed to take place simultaneously with the cause. Here we demonstrate that recently developed [M. D. Petrović {\em et al.}, {\tt arXiv:1802.05682}] self-consistent coupling of the LLG equation to time-dependent quantum-mechanical description of electrons microscopically generates time-retarded damping in the LLG equation described by a memory kernel which is also spatially dependent. For sufficiently slow…
