Gilbert damping in magnetic multilayers

Šimánek, E.; Heinrich, B.

doi:10.1103/physrevb.67.144418

Cited by 148 publications

(114 citation statements)

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“…A possible explanation is being offered by the Stoner enhancement factor which enhances the strength of spin pumping, see Simanek and Heinrich. 17 Recently we studied the increase of the Gilbert damping in GaAs/16Fe/10Pd/20Au͑001͒, where integers represent the number of atomic layers. This sample was prepared by molecular beam epitaxy ͑MBE͒ where atomic intermixing between Fe and Pd is kept at its minimum.…”

Section: ͑9͒mentioning

confidence: 99%

Semiclassical theory of spin transport in magnetic multilayers

Urban

Heinrich

Woltersdorf

2003

Journal of Applied Physics

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A semiclassical model of the spin momentum transfer in ferromagnetic film ͑FM͒/normal metal ͑NM͒ structures is presented. It is based on the Landau-Lifshitz equation of motion and the exchange interaction in FM, and on the spin diffusion equation in NM. The internal magnetic field is treated by employing Maxwell's equations. A precessing magnetization in FM creates a spin current which is described by spin pumping proposed by Tserkovnyak et al. In our recent ferromagnetic resonance ͑FMR͒ studies [1][2][3][4] it was shown that the transfer of the spin momentum across ferromagnetic ͑FM͒/normal metal ͑NM͒ interfaces can result in nonlocal interface Gilbert damping GЈ. The generation of spin momentum in magnetic ultrathin films was theoretically described by Tserkovnyak et al. 5 and the effect was called ''spin pumping.'' The presence of a second magnetic layer creates a spin sink. 3,4,6,7 The combination of spin pump and spin sink in the ballistic limit leads to an additional interface Gilbert damping. In this article we extend the spin pump and spin sink mechanisms to the nonballistic electron transport which includes a full treatment of the Landau-Lifshitz ͑LL͒ equation of motion in FM and diffusion equation in NM and Maxwell's equations accounting for a finite penetration of the rf fields.The coordinate system was chosen in such a way that the sample normal is parallel to the z axis. The external dc field, H, lies in the sample plane and is parallel to the y axis, and the internal electromagnetic rf fields are hϭ(h,0,0), e ϭ(0,e,0). The LL equations of motion in FM and NM layers can be written aswhere ␥ is the absolute value of the electron gyromagnetic ratio, M s is the saturation magnetization of FM, G 0 is the intrinsic Gilbert damping, D is the diffusion constant in NM (Dϭv F 2 el /6, v F is the Fermi velocity and el is the electron momentum relaxation time͒, s f is the spin-flip relaxation time, and ␦M N ϭM N Ϫ P h is the excess magnetization in NM, where P is the Pauli susceptibility. The effective field H eff F is derived from the total Gibbs free energy which contains the external fields, magnetocrystaline anisotropies, and exchange interaction. 8 The effective field H eff N in NM contains only the external dc, internal field H, and the demagnetizing field perpendicular to the sample plane. Equations ͑1͒ and ͑2͒ were solved in a small angle approximation using

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Section: ͑9͒mentioning

confidence: 99%

Semiclassical theory of spin transport in magnetic multilayers

Urban

Heinrich

Woltersdorf

2003

Journal of Applied Physics

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“…A large number of theoretical approaches to the Gilbert damping has been worked out during the last two decades; here we mention only schemes within the oneelectron theory of itinerant magnets, [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] where the most important effects of electron-electron interaction are captured by means of a local spin-dependent exchangecorrelation (XC) potential. These techniques can be naturally combined with existing first-principles techniques based on the density-functional theory, which leads to parameter-free calculations of the Gilbert damping tensor of pure ferromagnetic metals, their ordered and disordered alloys, diluted magnetic semiconductors, etc.…”

Section: Introductionmentioning

confidence: 99%

“…One part of these approaches is based on a static limit of the frequency-dependent spin-spin correlation function of a ferromagnet. [5][6][7][8]15,16 Other routes to the Gilbert damping employ relaxations of occupation numbers of individual Bloch electron states during quasistatic nonequilibrium processes or transition rates between different states induced by the spin-orbit (SO) interaction. [9][10][11][12]14,20 The dissipation of magnetic energy accompanying the slow magnetization dynamics, evaluated within a scattering theory or the Kubo linear response formalism, leads also to explicit expressions for the Gilbert damping tensor.…”

Section: Introductionmentioning

confidence: 99%

Nonlocal torque operators inab initiotheory of the Gilbert damping in random ferromagnetic alloys

2015

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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-diagonal and spin-independent matrices, which simplifies the configuration averaging. The CPA-vertex corrections play a crucial role for the internal consistency of the theory and for its exact equivalence to other first-principles approaches based on the random local torques. This equivalence is also illustrated by the calculated Gilbert damping parameters for binary NiFe and FeCo random alloys, for pure iron with a model atomic-level disorder, and for stoichiometric FePt alloys with a varying degree of L10 atomic long-range order.

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“…In order to arrive at the adiabatic (Gilbert) damping the magnetization dynamics has to be sufficiently slow such that u j u j t ÿ t _ u j t . Since m 2 1 and hence _ m m 0 [10] PRL 101, 037207 (2008) P…”

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Scattering Theory of Gilbert Damping

2008

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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. DOI: 10.1103/PhysRevLett.101.037207 PACS numbers: 75.40.Gb, 72.25.Mk, 76.60.Es Magnetization relaxation is a collective many-body phenomenon that remains intriguing despite decades of theoretical and experimental investigations. It is important in topics of current interest since it determines the magnetization dynamics and noise in magnetic memory devices and state-of-the-art magnetoelectronic experiments on current-induced magnetization dynamics [1]. Magnetization relaxation is often described in terms of a damping torque in the phenomenological Landau-Lifshitz-Gilbert (LLG) equationwhere M is the magnetization vector, g B =@ is the gyromagnetic ratio in terms of the g factor and the Bohr magneton B , and M s jMj is the saturation magnetization. Usually, the Gilbert dampingG M is assumed to be a scalar and isotropic parameter, but in general it is a symmetric 3 3 tensor. The LLG equation has been derived microscopically [2] and successfully describes the measured response of ferromagnetic bulk materials and thin films in terms of a few material-specific parameters that are accessible to ferromagnetic-resonance (FMR) experiments [3]. We focus in the following on small ferromagnets in which the spatial degrees of freedom are frozen out (macrospin model). Gilbert damping predicts a strictly linear dependence of FMR linewidths on frequency. This distinguishes it from inhomogenous broadening associated with dephasing of the global precession, which typically induces a weaker frequency dependence as well as a zerofrequency contribution.The effective magnetic field H eff ÿ@F=@M is the derivative of the free energy F of the magnetic system in an external magnetic field H ext , including the classical magnetic dipolar field H d . When the ferromagnet is part of an open system as in Fig. 1, ÿ@F=@M can be expressed in terms of a scattering S matrix, quite analogous to the interlayer exchange coupling between ferromagnetic layers [4]. The scattering matrix is defined in the space of the transport channels that connect a scattering region (the sample) to thermodynamic (left and right) reservoirs by electric contacts that are modeled by ideal leads. Scattering matrices also contain information to describe giant magnetoresistance, spin pumping and spin battery, and current-induced magnetization dynamics in layered normal-metal N j ferromagnet (F) systems [4 -6].In the following we demonstrate that scattering theory can be also used to compute ...

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Gilbert damping in magnetic multilayers

Cited by 148 publications

References 32 publications

Semiclassical theory of spin transport in magnetic multilayers

Semiclassical theory of spin transport in magnetic multilayers

Nonlocal torque operators inab initiotheory of the Gilbert damping in random ferromagnetic alloys

Scattering Theory of Gilbert Damping

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