Half-metallicity and low magnetic damping are perpetually sought for in spintronics materials and full Heusler alloys in this respect provide outstanding properties. However, it is challenging to obtain the well-ordered half-metallic phase in as-deposited full Heusler alloys thin films and theory has struggled to establish a fundamentals understanding of the temperature dependent Gilbert damping in these systems. Here we present a study of the temperature dependent Gilbert damping of differently ordered as-deposited Co2FeAl full Heusler alloy thin films. The sum of inter- and intraband electron scattering in conjunction with the finite electron lifetime in Bloch states govern the Gilbert damping for the well-ordered phase in contrast to the damping of partially-ordered and disordered phases which is governed by interband electronic scattering alone. These results, especially the ultralow room temperature intrinsic damping observed for the well-ordered phase provide new fundamental insights to the physical origin of the Gilbert damping in full Heusler alloy thin films.
Engineering Gilbert damping of ferromagnetic metal films is of great importance to exploit and design spintronic devices that are operated with an ultrahigh speed. Based on scattering theory of Gilbert damping, we extend the torque method originally used in studies of magnetocrystalline anisotropy to theoretically determine Gilbert dampings of ferromagnetic metals. This method is utilized to investigate Gilbert dampings of 3d transition metal ferromagnet iron, cobalt and nickel monolayers that are contacted by the prototypical topological insulator Bi2Se3. Amazingly, we find that their Gilbert…
An essential property of magnetic devices is the relaxation rate in magnetic switching which depends strongly on the damping in the magnetisation dynamics. It was recently measured that damping depends on the magnetic texture and, consequently, is a non-local quantity. The damping enters the Landau-Lifshitz-Gilbert equation as the phenomenological Gilbert damping parameter $α$, that does not, in a straight forward formulation, account for non-locality. Efforts were spent recently to obtain Gilbert damping from first principles for magnons of wave vector $\mathbf{q}$. However, to the best of ou…
We report an enhanced magnetoelastic contribution to the Gilbert damping in highly magnetostrictive Fe$_{0.7}$Ga$_{0.3}$ thin films. This effect is mitigated for perpendicular-to-plane fields, leading to a large anisotropy of the Gilbert damping in all of the films (up to a factor of 10 at room temperature). These claims are supported by broadband measurements of the ferromagnetic resonance linewidths over a range of temperatures (5 to 400 K), which serve to elucidate the effect of both the magnetostriction and phonon relaxation on the magnetoelastic Gilbert damping.
Confirming the origin of Gilbert damping by experiment has remained a challenge for many decades, even for simple ferromagnetic metals. In this Letter, we experimentally identify Gilbert damping that increases with decreasing electronic scattering in epitaxial thin films of pure Fe. This observation of conductivity-like damping, which cannot be accounted for by classical eddy current loss, is in excellent quantitative agreement with theoretical predictions of Gilbert damping due to intraband scattering. Our results resolve the longstanding question about a fundamental damping mechanism and off…
The Gilbert damping in ferromagnetic semiconductors is theoretically investigated based on the $s$-$d$ model. In contrast to the situation in metals, all the spin-conserving scattering in ferromagnetic semiconductors supplies an additional spin relaxation channel due to the momentum dependent effective magnetic field of the spin-orbit coupling, thereby modifies the Gilbert damping. In the presence of a pure spin current, we predict a new contribution due to the interplay of the anisotropic spin-orbit coupling and a pure spin current.
A method with which to calculate the Gilbert damping parameter from a real-space electronic structure method is reported here. The anisotropy of the Gilbert damping with respect to the magnetic moment direction and local chemical environment is calculated for bulk and surfaces of Fe$_{50}$Co$_{50}$ alloys from first principles electronic structure in a real space formulation. The size of the damping anisotropy for Fe$_{50}$Co$_{50}$ alloys is demonstrated to be significant. Depending on details of the simulations, it reaches a maximum-minimum damping ratio as high as 200%. Several microscopic …
Tailoring Gilbert damping of metallic ferromagnetic thin films is one of the central interests in spintronics applications. Here we report a giant Gilbert damping anisotropy in epitaxial Co$_{50}$Fe$_{50}$ thin film with a maximum-minimum damping ratio of 400 \%, determined by broadband spin-torque as well as inductive ferromagnetic resonance. We conclude that the origin of this damping anisotropy is the variation of the spin orbit coupling for different magnetization orientations in the cubic lattice, which is further corroborate from the magnitude of the anisotropic magnetoresistance in Co$_…
We report on a study of Gilbert damping due to particle-hole pair excitations in conducting ferromagnets. We focus on a toy two-band model and on a four-band spherical model which provides an approximate description of ferromagnetic (Ga,Mn)As. These models are sufficiently simple that disorder-ladder-sum vertex corrections to the long-wavelength spin-spin response function can be summed to all orders. An important objective of this study is to assess the reliability of practical approximate expressions which can be combined with electronic structure calculations to estimate Gilbert damping in …
We propose a systematic and sequential expansion of the Landau-Lifshitz-Gilbert equation utilizing the dependence of the Gilbert damping tensor on the angle between magnetic moments, which arises from multi-body scattering processes. The tensor consists of a damping-like term and a correction to the gyromagnetic ratio. Based on electronic structure theory, both terms are shown to depend on e.g. the scalar, anisotropic, vector-chiral and scalar-chiral products of magnetic moments: $\vec{e}_i\cdot\vec{e}_j$, $(\vec{n}_{ij}\cdot\vec{e}_i)(\vec{n}_{ij}\cdot\vec{e}_j)$, $\vec{n}_{ij}\cdot(\vec{e}_i…
The enhancement of Gilbert damping observed for Ni80Fe20 (Py) films in contact with the non-magnetic metals Cu, Pd, Ta and Pt, is quantitatively reproduced using first-principles scattering theory. The "spin-pumping" theory that qualitatively explains its dependence on the Py thickness is generalized to include a number of factors known to be important for spin transport through interfaces. Determining the parameters in this theory from first-principles shows that interface spin-flipping makes an essential contribution to the damping enhancement. Without it, a much shorter spin-flip diffusion …
Using broadband ferromagnetic resonance, we measure the damping parameter of [Co(5 Å)/Pt(3 Å)]${\times 6}$ multilayers whose growth was optimized to maximize the perpendicular anisotropy. Structural characterizations indicate abrupt interfaces essentially free of intermixing despite the miscible character of Co and Pt. Gilbert damping parameters as low as 0.021 can be obtained despite a magneto-crystalline anisotropy as large as $10^6~\textrm{J/m}^3$. The inhomogeneous broadening accounts for part of the ferromagnetic resonance linewidth, indicating some structural disorder leading to a equiva…
A recent theory by Chen and Zhang [Phys. Rev. Lett. 114, 126602 (2015)] predicts strongly anisotropic damping due to interfacial spin-orbit coupling in ultrathin magnetic films. Interfacial Gilbert-type relaxation, due to the spin pumping effect, is predicted to be significantly larger for magnetization oriented parallel to compared with perpendicular to the film plane. Here, we have measured the anisotropy in the Pt/Ni$_{81}$Fe$_{19}$/Pt system via variable-frequency, swept-field ferromagnetic resonance (FMR). We find a very small anisotropy of enhanced Gilbert damping with sign opposite to t…
The viscous Gilbert damping parameter governing magnetization dynamics is of primary importance for various spintronics applications. Although, the damping constant is believed to be anisotropic by theories. It is commonly treated as a scalar due to lack of experimental evidence. Here, we present an elaborate angle dependent broadband ferromagnetic resonance study of high quality epitaxial La$_{0.7}$Sr$_{0.3}$MnO$_{3}$ films. Extrinsic effects are suppressed and we show convincing evidence of anisotropic damping with twofold symmetry at room temperature. The observed anisotropic relaxation is …
The effects of rhenium doping in the range 0 to 10 atomic percent on the static and dynamic magnetic properties of Fe65Co35 thin films have been studied experimentally as well as with first principles electronic structure calculations focusing on the change of the saturation magnetization and the Gilbert damping parameter. Both experimental and theoretical results show that the saturation magnetization decreases with increasing Re doping level, while at the same time Gilbert damping parameter increases. The experimental low temperature saturation magnetic induction exhibits a 29 percent decrea…
We have experimentally and theoretically investigated the dependence of the intrinsic Gilbert damping parameter $α_0$ on the spin-orbital coupling strength $ξ$ by using L1$_{\mathrm{0}}$ ordered FePd$_{\mathrm{1-x}}$Pt$_{\mathrm{x}}$ ternary alloy films with perpendicular magnetic anisotropy. With the time-resolved magneto-optical Kerr effect, $α_0$ is found to increase by more than a factor of ten when $x$ varies from 0 to 1.0. Since changes of other leading parameters are found to be neglected, the $α_0$ has for the first time been proven to be proportional to $ξ^2$.
The magnetostatic mode (MSM) spectrum of a 300$μ$m diameter single crystalline sphere of yttrium iron garnet is investigated using broadband ferromagnetic resonance (FMR). The individual MSMs are identified via their characteristic dispersion relations and the corresponding mode number tuples $(nmr)$ are assigned. Taking FMR data over a broad frequency and magnetic field range allows to analyze both the Gilbert damping parameter~$α$ and the inhomogeneous line broadening contribution to the total linewidth of the MSMs separately. The linewidth analysis shows that all MSMs share the same Gilbert…
We report on broadband ferromagnetic resonance linewidth measurements performed on epitaxial Heusler thin films. A large and anisotropic two-magnon scattering linewidth broadening is observed for measurements with the magnetization lying in the film plane, while linewidth measurements with the magnetization saturated perpendicular to the sample plane reveal low Gilbert damping constants of $(1.5\pm0.1)\times 10^{-3}$, $(1.8\pm0.2)\times 10^{-3}$, and $<8\times 10^{-4}$ for Co$_2$MnSi/MgO, Co$_2$MnAl/MgO, and Co$_2$FeAl/MgO, respectively. The in-plane measurements are fit to a model combining G…
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…
We study the enhancement of the ferromagnetic relaxation rate in thin films due to the adjacent normal metal layers. Using linear response theory, we derive the dissipative torque produced by the s-d exchange interaction at the ferromagnet-normal metal interface. For a slow precession, the enhancement of Gilbert damping constant is proportional to the square of the s-d exchange constant times the zero-frequency limit of the frequency derivative of the local dynamic spin susceptibility of the normal metal at the interface. Electron-electron interactions increase the relaxation rate by the Stone…
The modification of the magnetization dissipation or Gilbert damping caused by an inhomogeneous magnetic structure and expressed in terms of a wave vector dependent tensor $\underlineα(\vec{q})$ is investigated by means of linear response theory. A corresponding expression for $\underlineα(\vec{q})$ in terms of the electronic Green function has been developed giving in particular the leading contributions to the Gilbert damping linear and quadratic in $q$. Numerical results for realistic systems are presented that have been obtained by implementing the scheme within the framework of the fully …
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.
Although both theoretical predications and experimental observations demonstrated that the damping factor is anisotropic at ferromagnet/semiconductor interface with robust interfacial spin-orbit coupling, it is not well understood whether non-local Gilbert damping driven by spin currents in heavy metal/ferromagnetic metal (HM/FM) bilayers is anisotropic or not. Here, we investigated the in-plane angular- and frequency- dependence of magnetic relaxation of epitaxial Fe/MgO(001) films with different capping layers of Pd and Cu. After disentangling the parasitic contributions, such as two-magnon …
We present an ab initio theory of the Gilbert damping in substitutionally disordered ferromagnetic alloys. The theory rests on introduced nonlocal torques which replace traditional local torque operators in the well-known torque-correlation formula and which can be formulated within the atomic-sphere approximation. The formalism is sketched in a simple tight-binding model and worked out in detail in the relativistic tight-binding linear muffin-tin orbital (TB-LMTO) method and the coherent potential approximation (CPA). The resulting nonlocal torques are represented by nonrandom, non-site-diago…
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…
The Landau-Lifshitz-Gilbert damping coefficient employed in the analysis of spin wave ferromagnetic resonance is related to the electrical conductivity of the sample. The changing magnetization (with time) radiates electromagnetic fields. The electromagnetic energy is then absorbed by the sample and the resulting heating effect describes magnetic dissipative damping. The ferromagnetic resonance relaxation rate theoretically depends on the geometry (shape and size) of the sample as well as temperature in agreement with experiment.
New terms to the dynamical equation of magnetization motion, associated with spin transport, have been reported over the past several years. Each newly identified term is thought to possess both a real and an imaginary effective field leading to fieldlike and dampinglike torques on magnetization. Here we show that three metallic ferromagnets possess an imaginary effective-field term which mirrors the well-known real effective-field term associated with exchange in spin waves. Using perpendicular standing spin wave resonance between 2-26 GHz, we evaluate the magnitude of the finite-wavenumber (…
We investigate the Gilbert damping parameter for rare earth (RE)-transition metal (TM) ferrimagnets over a wide temperature range. Extracted from the field-driven magnetic domain-wall mobility, the Gilbert damping parameter was as low as 0.0072 and was almost constant across the angular momentum compensation temperature, starkly contrasting previous predictions that the Gilbert damping parameter should diverge at the angular momentum compensation temperature due to vanishing total angular momentum. Thus, magnetic damping of RE-TM ferrimagnets is not related to the total angular momentum but is…
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…
Engineering of magnetic materials for developing better spintronic applications relies on the control of two key parameters: the spin polarization and the Gilbert damping responsible for the spin angular momentum dissipation. Both of them are expected to affect the ultrafast magnetization dynamics occurring on the femtosecond time scale. Here, we use engineered Co2MnAlxSi1-x Heusler compounds to adjust the degree of spin polarization P from 60 to 100% and investigate how it correlates with the damping. We demonstrate experimentally that the damping decreases when increasing the spin polarizati…
Gilbert Damping Parameters of Epitaxially-Stabilized Iron Gallium Thin Films from Ferromagnetic Resonance
The Ferroelectronics Laboratory, University of Michigan（密歇根大学铁电实验室）；Imec（imec研究所）；Department of Materials Science and Engineering, University of Wisconsin-Madison（威斯康星大学麦迪逊分校材料科学与工程系）
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…
We use time-dependent spin-density-functional theory to study dynamical magnetic phenomena. First, we recall that the local-spin-density approximation (LSDA) fails to account correctly for magnetic fluctuations in the paramagnetic state of iron and other itinerant ferromagnets. Next, we construct a gradient-dependent density functional that does not suffer from this problem of the LSDA. This functional is then used to derive, for the first time, the phenomenological Gilbert equation of micromagnetics directly from time-dependent density-functional theory. Limitations and extensions of Gilbert …
We investigate optically induced ultrafast magnetization dynamics in [Co(0.5 nm)/Pd(1 nm)]x5/NiFe(t) exchange-spring samples with tilted perpendicular magnetic anisotropy using a time-resolved magneto-optical Kerr effect magnetometer. The competition between the out-of-plane anisotropy of the hard layer, the in-plane anisotropy of the soft layer and the applied bias field reorganizes the spins in the soft layer, which are modified further with the variation in t. The spin-wave spectrum, the ultrafast demagnetization time, and the extracted damping coefficient all depend on the spin distributio…
The ability to tailor the damping factor is essential for spintronic and spin-torque applications. Here, we report an approach to manipulate the damping factor of FeGa/MgO(001) films by oblique deposition. Owing to the defects at the surface or interface in thin films, two-magnon scattering (TMS) acts as a non-Gilbert damping mechanism in magnetization relaxation. In this work, the contribution of TMS was characterized by in-plane angular dependent ferromagnetic resonance (FMR). It is demonstrated that the intrinsic Gilbert damping is isotropic and invariant, while the extrinsic mechanism rela…
In magnetic recording the signal-to-noise ratio (SNR) is a good indicator for the quality of written bits. However, a priori it is not clear which parameters have the strongest influence on the SNR. In this work, we investigate the role of the Gilbert damping on the SNR. Grains consisting of FePt like hard magnetic material with two different grain sizes $d_1=5\,$nm and $d_2=7\,$nm are considered and simulations of heat-assisted magnetic recording (HAMR) are performed with the atomistic simulation program VAMPIRE. The simulations display that the SNR saturates for damping constants larger or e…
Thin highly textured Fe$_{\mathrm{1+x}}$Co$_{\mathrm{2-x}}$Si ($0 \leq$ x $\leq 1$) films were prepared on MgO (001) substrates by magnetron co-sputtering. The magneto-optic Kerr effect (MOKE) and ferromagnetic resonance (FMR) measurements were used to investigate the composition dependence of the magnetization, the magnetic anisotropy, the gyromagnetic ratio and the relaxation of the films. The effective magnetization for the thin Fe$_{\mathrm{1+x}}$Co$_{\mathrm{2-x}}$Si films, determined by FMR measurements, are consistent with the Slater Pauling prediction. Both MOKE and FMR measurements re…
We study the magnetic excitations of itinerant helimagnets by applying time-resolved optical spectroscopy to Fe0.8Co0.2Si. Optically excited oscillations of the magnetization in the helical state are found to disperse to lower frequency as the applied magnetic field is increased; the fingerprint of collective modes unique to helimagnets, known as helimagnons. The use of time-resolved spectroscopy allows us to address the fundamental magnetic relaxation processes by directly measuring the Gilbert damping, revealing the versatility of spin dynamics in chiral magnets. (*These authors contributed …
Heusler alloys have been intensively studied due to the wide variety of properties that they exhibit. One of these properties is of particular interest for technological applications, i.e. the fact that some Heusler alloys are half-metallic. In the following, a systematic study of the magnetic properties of three different Heusler families $\textrm{Co}_2\textrm{Mn}\textrm{Z}$, $\text{Co}_2\text{Fe}\text{Z}$ and $\textrm{Mn}_2\textrm{V}\textrm{Z}$ with $\text{Z}=\left(\text{Al, Si, Ga, Ge}\right)$ is performed. A key aspect is the determination of the Gilbert damping from first principles calcu…
A Kubo-Greenwood-like equation for the Gilbert damping parameter $α$ is presented that is based on the linear response formalism. Its implementation using the fully relativistic Korringa-Kohn-Rostoker (KKR) band structure method in combination with Coherent Potential Approximation (CPA) alloy theory allows it to be applied to a wide range of situations. This is demonstrated with results obtained for the bcc alloy system Fe$_x$Co$_{1-x}$ as well as for a series of alloys of permalloy with 5d transition metals. To account for the thermal displacements of atoms as a scattering mechanism, an alloy…
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…
The pseudo-PT symmetric Dirac equation is proposed and analyzed by using a non-unitary Foldy-Wouthuysen transformations. A new spin operator PT symmetric expectation value (called the mean spin operator) for an electron interacting with a time-dependent electromagnetic field is obtained. We show that spin magnetization - which is the quantity usually measured experimentally - is not described by the standard spin operator but by this new mean spin operator to properly describe magnetization dynamics in ferromagnetic materials and the corresponding equation of motion is compatible with the phen…
Damping mechanisms in magnetic systems determine the lifetime, diffusion and transport properties of magnons, domain walls, magnetic vortices, and skyrmions. Based on the phenomenological Landau-Lifshitz-Gilbert equation, here the effective damping parameter in noncollinear magnetic systems is determined describing the linewidth in resonance experiments or the decay parameter in time-resolved measurements. It is shown how the effective damping can be calculated from the elliptic polarization of magnons, arising due to the noncollinear spin arrangement. It is concluded that the effective dampin…
We investigate the temperature dependence of the coefficients in the Landau-Lifshitz-Gilbert equation in ferromagnetic GaMnAs by employing the Zener model. We first calculate the hole spin relaxation time based on the microscopic kinetic equation. We find that the hole spin relaxation time is typically several tens femtoseconds and can present a nonmonotonic temperature dependence due to the variation of the interband spin mixing, influenced by the temperature related Zeeman splitting. With the hole spin relaxation time, we are able to calculate the coefficients in the Landau-Lifshitz-Gilbert …
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）
