Inhomogeneous Gilbert damping from impurities and electron-electron interactions

Hankiewicz, Ewelina M.; Vignale, Giovanni; Tserkovnyak, Yaroslav

doi:10.1103/physrevb.78.020404

Cited by 54 publications

(53 citation statements)

References 16 publications

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“…Thus, also the Gilbert damping tensor is here a frequency and q-dependent quantity, in accordance with recent measurements [32]. Suitable expressions for χ have been considered previously in the context of Gilbert damping [13,16,45]. Linear-response formulations that express χ as a spin-spin correlation function include the Pauli and Van Vleck susceptibility contributions [61], and expressions for the orbital susceptibility have been derived as well [62].…”

Section: Analysis Of the Damping Expressionmentioning

confidence: 75%

Relativistic theory of spin relaxation mechanisms in the Landau-Lifshitz-Gilbert equation of spin dynamics

2016

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Starting from the Dirac-Kohn-Sham equation we derive the relativistic equation of motion of spin angular momentum in a magnetic solid under an external electromagnetic field. This equation of motion can be rewritten in the form of the well-known Landau-Lifshitz-Gilbert equation for a harmonic external magnetic field, and leads to a more general magnetization dynamics equation for a general time-dependent magnetic field. In both cases with an electronic spin-relaxation term which stems from the spin-orbit interaction. We thus rigorously derive, from fundamental principles, a general expression for the anisotropic damping tensor which is shown to contain an isotropic Gilbert contribution as well as an anisotropic Ising-like and a chiral, Dzyaloshinskii-Moriya-like contribution. The expression for the spin relaxation tensor comprises furthermore both electronic interband and intraband transitions. We also show that when the externally applied electromagnetic field possesses spin angular momentum, this will lead to an optical spin torque exerted on the spin moment.

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Section: Analysis Of the Damping Expressionmentioning

confidence: 75%

Relativistic theory of spin relaxation mechanisms in the Landau-Lifshitz-Gilbert equation of spin dynamics

2016

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“…While in this paper we focus on the longitudinal spin current noise, there is also a transverse contribution not captured by our analysis, which leads to spin-wave damping proportional to q 2 . 19 So far we have only considered thermal current noise; let us finally turn to shot noise. With the voltage U across the ferromagnet turned on, a nonzero current I flows in the y direction.…”

Section: ͑13͒mentioning

confidence: 99%

Current-induced noise and damping in nonuniform ferromagnets

et al. 2008

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In the presence of spatial variation in the magnetization direction, electric current noise causes a fluctuating spin-transfer torque that increases the fluctuations of the ferromagnetic order parameter. By the fluctuationdissipation theorem, the fluctuations at thermal equilibrium are related to the viscous magnetization damping, which in nonuniform ferromagnets acquires a nonlocal tensor structure. At the hand of spin spirals, we demonstrate that the current-induced noise and damping increase with the gradient of the magnetization texture and becomes significant for narrow domain walls. DOI: 10.1103/PhysRevB.78.140402 PACS number͑s͒: 72.70.ϩm, 72.25.Mk, 75.75.ϩa Three decades ago, Berger 1,2 showed that an electric current passing through a ferromagnetic domain wall exerts a torque on the wall. The spin of the electron that traverses the wall adiabatically adapts to the local exchange field, thereby transferring angular momentum to the magnetization. Subsequently, it was realized that the same effect also exists in magnetic multilayers.3 Sufficiently strong current-induced torques switch the magnetization direction in multilayers or move domain walls in bulk magnets. The early ideas have been confirmed both theoretically and experimentally. 4 Recently, the importance of noise for current-induced magnetization dynamics has drawn attention. Although often noise is undesired, it may in some cases be quite useful. Wetzels et al. 5 showed that current-induced magnetization reversal of spin valves is accelerated by an increased level of current noise. The noisy current exerts a fluctuating torque on the magnetization.6 Ravelosona et al. 7 reported observation of thermally assisted depinning of a narrow domain wall under an applied current. Thermally assisted current-driven domain-wall motion has also been studied theoretically. 8,9 The present paper addresses current-induced magnetization noise in nonuniformly magnetized ferromagnets. The spatial variation in the magnetization direction gives rise to increased magnetization noise; by a fluctuating spin-transfer torque, electric current noise causes fluctuations of the magnetic order parameter. The increased magnetization noise can be represented by introducing fictitious stochastic magnetic fields. By the fluctuation-dissipation theorem ͑FDT͒, the thermal stochastic field is related to the dissipation of energy, and thus the damping of the magnetization dynamics. Since the correlator of the stochastic field in general is inhomogenous and anisotropic, the damping is a nonlocal tensor. Ferromagnetic spin spirals are interesting model systems to study these effects since the field correlator and damping become spatially independent. It is shown that for spirals with relatively short wavelength ͑ϳ20 nm͒, the currentinduced noise and damping is substantial. We consider here disordered metallic ferromagnets in which the scattering mean-free path is smaller than the spatial scale of the ferromagnet. This implies that a spin spiral is a good model for a domain wall with...

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“…Here, we wish to elucidate the central role of such self-consistent dissipative spin currents, which govern the diffusionlike terms in the magnetic equation of motion in the limit of strong ferromagnetic exchange correlations. This paper is a follow up to our previous work, 9 providing additional technical details and offering a broader phenomenological base. Apart from assuming strong exchange correlation limit, our phenomenological approach and the main results of the paper should not be sensitive to the microscopic details and do not rely on the specific model of the ferromagnetic material ͑such as the Stoner or an s-d model, for example͒.…”

Section: Introductionmentioning

confidence: 99%

“…The main goals of this paper are as follows: ͑i͒ to put the results of Ref. 9 into a broader phenomenological perspective, ͑ii͒ to explicitly show that two quite different models-the spin-polarized itinerant-electron liquid ͑treated in Ref. 9͒ and the s-d model-lead to the same phenomenology and can be treated in parallel, and ͑iii͒ to make direct contact with the spin-pumping theory.…”

Section: Introductionmentioning

confidence: 99%

Transverse spin diffusion in ferromagnets

2009

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We discuss the dissipative diffusion-type term of the form m ϫ ٌ 2 ‫ץ‬ t m in the phenomenological LandauLifshitz equation of ferromagnetic precession, which describes enhanced Gilbert damping of finite-momentum spin waves. This term arises physically from itinerant-electron spin flows through a perturbed ferromagnetic configuration and can be understood to originate in the ferromagnetic spin pumping in the continuum limit. We develop a general phenomenology as well as provide microscopic theory for the Stoner and s-d models of ferromagnetism, taking into account disorder and electron-electron scattering. The latter is manifested in our problem through the Coulomb drag between the spin bands. The spin diffusion coefficient is identified with the transverse spin conductivity, in analogy with the Einstein relation in the kinetic theory.

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Inhomogeneous Gilbert damping from impurities and electron-electron interactions

Cited by 54 publications

References 16 publications

Relativistic theory of spin relaxation mechanisms in the Landau-Lifshitz-Gilbert equation of spin dynamics

Relativistic theory of spin relaxation mechanisms in the Landau-Lifshitz-Gilbert equation of spin dynamics

Current-induced noise and damping in nonuniform ferromagnets

Transverse spin diffusion in ferromagnets

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