Gilbert Damping in Single and Multilayer Ultrathin Films: Role of Interfaces in Nonlocal Spin Dynamics (original) (raw)
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Gilbert damping in magnetic multilayers
Physical Review B, 2003
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 Stoner factor squared. We attribute the large anisotropic enhancements of the relaxation rate observed recently in multilayers containing palladium to this mechanism. For free electrons, the present theory compares favorably with recent spin-pumping result of Tserkovnyak et al. [Phys. Rev. Lett. 88,117601 (2002)].
Role of spin current in magnetic relaxations of metallic multilayer films
Journal of Magnetism and Magnetic Materials, 2003
In our recent studies the relaxation processes were investigated by ferromagnetic resonance (FMR) using magnetic single, Au/Fe/GaAs(0 0 1), and double layer, Au/Fe/Au/Fe/GaAs(0 0 1), structures prepared by molecular beam epitaxy (MBE). Single and double layer structures were grown on (4 Â 6) reconstructed GaAs(0 0 1) templates. The samples have high crystalline quality with the magnetic properties corresponding to bulk Fe, only modified by interface anisotropies. The magnetic relaxation in these structures is well described by Gilbert damping with no extrinsic contributions to the FMR line width. These structures provide an excellent opportunity to investigate non-local damping which is caused by spin transport across a non-magnetic spacer. In the double layer structures thin Fe layers were separated from a second thick Fe layer of 40 monolayer (ML) thickness by a 40 ML thick Au(0 0 1) spacer. The main result of these studies is that the ultrathin Fe films in magnetic double layers acquire an additional Gilbert damping. The additional FMR line width followed an inverse dependence on the Fe film thickness which is expected for interface damping. The spin pump model will be discussed and it will be shown that it is closely related to the dynamic part of interlayer exchange coupling. r
Intrinsic Spin Relaxation Processes in Metallic Magnetic Multilayers
Journal of Superconductivity and Novel Magnetism, 2007
Spin relaxation processes in metallic magnetic nanostructures are reviewed. First a brief review of the phenomenology of magnetic damping is presented using the Landau Lifshitz Gilbert (LLG) equations of motion. It is shown that the Gilbert damping in bulk metallic layers is caused by the spin orbit interaction and itinerant character of 3d and 4s-p electrons. Spin dynamics in magnetic nanostructures acquires an additional nonlocal damping. This means that a part of the magnetic damping is not given by the local Gilbert damping but arises from the proximity to other layers. Spin pumping and spin sink concepts will be introduced and used to describe the interface nonlocal Gilbert damping in magnetic multilayers. The modified LLG equation of motion in magnetic multilayers will be introduced and tested against the ferromagnetic resonance (FMR) data around the accidental crossover of FMR fields. The spin pumping theory will be compared to the early theories introduced in the 1970s for the interpretation of transmission electron spin resonance (TESR) measurements across ferromagnet/normal metal sandwiches.
Gilbert damping in binary magnetic multilayers
Physical Review B, 2017
We present quantum mechanical calculations of the Gilbert damping constant α in ultrathin L1 0 [Co/NM] N superlattices and (001) fcc [Co/NM] N magnetic multilayers built of cobalt and nonmagnetic metals NM = Cu, Ag, Pd, Pt, and Au. The calculations are performed within a realistic nine-orbital tight-binding model of the band structure including spin-orbit interaction. The dependence of α on the stacking number N , ferromagnetic and nonmagnetic layer thicknesses as well as the electron scattering rate is investigated. The damping constant is shown to be the sum of a constant term (bulklike) and a 1/N term (due to external surfaces) which arise from interand intraband electronic transitions, respectively. The calculated α is found to be enhanced in the considered multilayers in comparison with its values for bulk Co and their bilayer counterparts with the same total Co thickness. The origin of this enhancement and the variation of α with the geometric structure of the multilayers are further investigated by analyzing the damping contributions from individual atomic layers. The obtained theoretical results for the damping constant are shown to be in good agreement with previous experimental observations in magnetic multilayers. In particular, the experimentally observed linear dependence on the ratio of NM (Pd or Pt) and Co layer thicknesses is reproduced in the present calculations.
Quantum mechanism of nonlocal Gilbert damping in magnetic trilayers
Physical Review B, 2015
A fully quantum-mechanical calculation of the Gilbert damping constant α in magnetic trilayers is done by employing the torque-correlation formula within a realistic tight-binding model. A remarkable enhancement of α in Co/NM 1 /NM 2 trilayers is obtained due to adding the caps NM 2 =Pd, Pt, and it decays with the thickness of the spacers NM 1 =Cu, Ag, Au in agreement with experiment. Nonlocal origin of the Gilbert damping is visualized with its atomic layer contributions. It is shown that magnetization in Co is damped remotely by strong spin-orbit coupling in NM 2 via quantum states with large amplitude in both Co and NM 2 .
Physica B-condensed Matter, 2006
The Landau-Lifschitz-Gilbert (LLG) equation is used widely in device design to describe spin motions in magnetic nanoscale structures. The damping term in this equation plays an essential role in the description of the magnetization dynamics. The form of this term is simple and appealing, but it is derived through use of elementary phenomenological considerations. An important question is whether or not it provides a proper description of the damping of the magnetization in real materials. Recently, it was predicted that a mechanism called two magnon damping should contribute importantly to linewidths and consequently spin damping in ultrathin ferromagnetic films. This process yields ferromagnetic resonance (FMR) linewidths whose frequency dependence is incompatible with the linear variation expected from the Landau-Lifschitz equation. This prediction has now been confirmed experimentally. Furthermore, subsequent experimental and theoretical studies have demonstrated that the damping rate depends strongly on wave vector as well. It is thus clear that for many samples, the LLG equation fails to account for the systematics of the damping of the magnetization in ultrathin ferromagnets, at the linear response level. The paper will review the recent literature on this topic relevant to this issue. One must then inquire into the nature of a proper phenomenology to describe these materials. At the linear response level, the theory of the two magnon mechanism is sufficiently complete that one can describe the response of these systems without resort to LLG phenomenology. However, currently there is very great interest in the large amplitude response of the magnetization in magnetic nanostructures. In the view of the authors, it is difficult to envision a generally applicable extension of linear response theory into the large amplitude regime. r 2006 Published by Elsevier B.V.
Gilbert damping in magnetic layered systems
Physical Review B, 2014
The Gilbert damping constant present in the phenomenological Landau-Lifshitz-Gilbert equation describing the dynamics of magnetization is calculated for ferromagnetic metallic films as well as Co/nonmagnet (NM) bilayers. The calculations are done within a realistic nine-orbital tight-binding model including spin-orbit coupling. The convergence of the damping constant expressed as a sum over the Brillouin zone is remarkably improved by introducing finite temperature into the electronic occupation factors and subsequent summation over the Matsubara frequencies. We investigate how the Gilbert damping constant depends on the ferromagnetic film thickness as well as on the thickness of the nonmagnetic cap in Co/NM bilayers (NM = Cu, Pd, Ag, Pt, and Au). The obtained theoretical dependence of the damping constant on the electron-scattering rate, describing the average lifetime of electronic states, varies substantially with the ferromagnetic film thickness and it differs significantly from the dependence for bulk ferromagnetic metals. The presence of nonmagnetic caps is found to largely enhance the magnetic damping in Co/NM bilayers in accordance with experimental data. Unlike Cu, Ag, and Au a particularly strong enhancement is obtained for Pd and Pt caps. This is attributed to the combined effect of the large spin-orbit couplings of Pd and Pt and the simultaneous presence of d states at the Fermi level in these two metals. The calculated Gilbert damping constant also shows an oscillatory dependence on the thicknesses of both ferromagnetic and nonmagnetic parts of the investigated systems which is attributed to quantum-well states. Finally, the expression for contributions to the damping constant from individual atomic layers is derived. The obtained distribution of layer contributions in Co/Pt and Co/Pd bilayers proves that the enhanced damping which affects the dynamics of the magnetization in the Co film originates mainly from a region within the nonmagnetic part of the bilayer. Such a nonlocal damping mechanism, related to spin pumping, is almost absent in other investigated bilayers: Co/Cu, Co/Ag, and Co/Au.
Magnetic relaxation in metallic films: Single and multilayer structures
Journal of Applied Physics, 2002
The intrinsic magnetic relaxations in metallic films will be discussed. It will be shown that the intrinsic damping mechanism in metals is caused by incoherent scattering of itinerant electron-hole pair excitations by phonons and magnons. Berger ͓L. Berger, Phys. Rev. B 54, 9353 ͑1996͔͒ showed that the interaction between spin waves and itinerant electrons in multilayers can lead to interface Gilbert damping. Ferromagnetic resonance ͑FMR͒ studies were carried out using magnetic single and double layer films. The FMR linewidth of the Fe films in the double layer structures was found to always be larger than the FMR linewidth measured for the single Fe films having the same thickness. The increase in the FMR linewidth scaled inversely with the film thickness, and was found to be linearly dependent on the microwave frequency. These results are in agreement with Berger's predictions.
Calculation of Gilbert damping in ferromagnetic films
EPJ Web of Conferences, 2013
The Gilbert damping constant present in the phenomenological Landau-Lifshitz-Gilbert equation describing the dynamics of magnetization is calculated for ferromagnetic metallic films as well as Co/nonmagnet (NM) bilayers. The calculations are done within a realistic nine-orbital tight-binding model including spin-orbit coupling. The convergence of the damping constant expressed as a sum over the Brillouin zone is remarkably improved by introducing finite temperature into the electronic occupation factors and subsequent summation over the Matsubara frequencies. We investigate how the Gilbert damping constant depends on the ferromagnetic film thickness as well as on the thickness of the nonmagnetic cap in Co/NM bilayers (NM = Cu, Pd, Ag, Pt, and Au). The obtained theoretical dependence of the damping constant on the electron-scattering rate, describing the average lifetime of electronic states, varies substantially with the ferromagnetic film thickness and it differs significantly from the dependence for bulk ferromagnetic metals. The presence of nonmagnetic caps is found to largely enhance the magnetic damping in Co/NM bilayers in accordance with experimental data. Unlike Cu, Ag, and Au a particularly strong enhancement is obtained for Pd and Pt caps. This is attributed to the combined effect of the large spin-orbit couplings of Pd and Pt and the simultaneous presence of d states at the Fermi level in these two metals. The calculated Gilbert damping constant also shows an oscillatory dependence on the thicknesses of both ferromagnetic and nonmagnetic parts of the investigated systems which is attributed to quantum-well states. Finally, the expression for contributions to the damping constant from individual atomic layers is derived. The obtained distribution of layer contributions in Co/Pt and Co/Pd bilayers proves that the enhanced damping which affects the dynamics of the magnetization in the Co film originates mainly from a region within the nonmagnetic part of the bilayer. Such a nonlocal damping mechanism, related to spin pumping, is almost absent in other investigated bilayers: Co/Cu, Co/Ag, and Co/Au.
Micromagnetic theory of spin relaxation and ferromagnetic resonance in multilayered metallic films
Spin relaxation in the ultrathin metallic films of stacked microelectronic devices is investigated on the basis of a modified Landau-Lifshitz equation of micromagnetic dynamics in which the damping torque is treated as originating from the coupling between precessing magnetization-vector and the introduced stress-tensors of intrinsic and extrinsic magnetic anisotropy. Particular attention is given to the time of exponential relaxation and ferromagnetic resonance linewidth which are derived in analytic form from the equation of magnetization energy loss and Gabor uncertainty relation between the full-width-at-halfmaximum in resonance-shaped line and lifetime of resonance excitation. The potential of developed theory is briefly discussed in the context of recent measurements.