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2019-09-10
Spin Pumping from Permalloy into Uncompensated Antiferromagnetic Co doped Zinc Oxide
Heterostructures of Co-doped ZnO and Permalloy were investigated for their static and dynamic magnetic interaction. The highly Co-doped ZnO is paramagentic at room temperature and becomes an uncompensated antiferromagnet at low temperatures, showing a narrowly opened hysteresis and a vertical exchange bias shift even in the absence of any ferromagnetic layer. At low temperatures in combination with Permalloy an exchange bias is found causing a horizontal as well as vertical shift of the hysteresis of the heterostructure together with an increase in coercive field. Furthermore, an increase in the Gilbert damping parameter at room temperature was found by multifrequency FMR evidencing spin pumping. Temperature dependent FMR shows a maximum in magnetic damping close to the magnetic phase transition. These measurements also evidence the exchange bias interaction of Permalloy and long-range ordered Co-O-Co structures in ZnO, that are barely detectable by SQUID due to the shorter probing times in FMR.
1909.04362v3
2020-02-27
Ultrafast magnetization dynamics in half-metallic Co$_2$FeAl Heusler alloy
We report on optically induced, ultrafast magnetization dynamics in the Heusler alloy $\mathrm{Co_{2}FeAl}$, probed by time-resolved magneto-optical Kerr effect. Experimental results are compared to results from electronic structure theory and atomistic spin-dynamics simulations. Experimentally, we find that the demagnetization time ($\tau_{M}$) in films of $\mathrm{Co_{2}FeAl}$ is almost independent of varying structural order, and that it is similar to that in elemental 3d ferromagnets. In contrast, the slower process of magnetization recovery, specified by $\tau_{R}$, is found to occur on picosecond time scales, and is demonstrated to correlate strongly with the Gilbert damping parameter ($\alpha$). Our results show that $\mathrm{Co_{2}FeAl}$ is unique, in that it is the first material that clearly demonstrates the importance of the damping parameter in the remagnetization process. Based on these results we argue that for $\mathrm{Co_{2}FeAl}$ the remagnetization process is dominated by magnon dynamics, something which might have general applicability.
2002.12255v1
2020-06-05
Controlling the nonlinear relaxation of quantized propagating magnons in nanodevices
Relaxation of linear magnetization dynamics is well described by the viscous Gilbert damping processes. However, for strong excitations, nonlinear damping processes such as the decay via magnon-magnon interactions emerge and trigger additional relaxation channels. Here, we use space- and time-resolved microfocused Brillouin light scattering spectroscopy and micromagnetic simulations to investigate the nonlinear relaxation of strongly driven propagating spin waves in yttrium iron garnet nanoconduits. We show that the nonlinear magnon relaxation in this highly quantized system possesses intermodal features, i.e., magnons scatter to higher-order quantized modes through a cascade of scattering events. We further show how to control such intermodal dissipation processes by quantization of the magnon band in single-mode devices, where this phenomenon approaches its fundamental limit. Our study extends the knowledge about nonlinear propagating spin waves in nanostructures which is essential for the construction of advanced spin-wave elements as well as the realization of Bose-Einstein condensates in scaled systems.
2006.03400v2
2021-05-16
Anatomy of inertial magnons in ferromagnets
We analyze dispersion relations of magnons in ferromagnetic nanostructures with uniaxial anisotropy taking into account inertial terms, i.e. magnetic nutation. Inertial effects are parametrized by damping-independent parameter $\beta$, which allows for an unambiguous discrimination of inertial effects from Gilbert damping parameter $\alpha$. The analysis of magnon dispersion relation shows its two branches are modified by the inertial effect, albeit in different ways. The upper nutation branch starts at $\omega=1/ \beta$, the lower branch coincides with FMR in the long-wavelength limit and deviates from the zero-inertia parabolic dependence $\simeq\omega_{FMR}+Dk^2$ of the exchange magnon. Taking a realistic experimental geometry of magnetic thin films, nanowires and nanodiscs, magnon eigenfrequencies, eigenvectors and $Q$-factors are found to depend on the shape anisotropy. The possibility of phase-matched magneto-elastic excitation of nutation magnons is discussed and the condition was found to depend on $\beta$, exchange stiffness $D$ and the acoustic velocity.
2105.07376v1
2021-11-16
Ultrathin ferrimagnetic GdFeCo films with very low damping
Ferromagnetic materials dominate as the magnetically active element in spintronic devices, but come with drawbacks such as large stray fields, and low operational frequencies. Compensated ferrimagnets provide an alternative as they combine the ultrafast magnetization dynamics of antiferromagnets with a ferromagnet-like spin-orbit-torque (SOT) behavior. However to use ferrimagnets in spintronic devices their advantageous properties must be retained also in ultrathin films (t < 10 nm). In this study, ferrimagnetic Gdx(Fe87.5Co12.5)1-x thin films in the thickness range t = 2-20 nm were grown on high resistance Si(100) substrates and studied using broadband ferromagnetic resonance measurements at room temperature. By tuning their stoichiometry, a nearly compensated behavior is observed in 2 nm Gdx(Fe87.5Co12.5)1-x ultrathin films for the first time, with an effective magnetization of Meff = 0.02 T and a low effective Gilbert damping constant of {\alpha} = 0.0078, comparable to the lowest values reported so far in 30 nm films. These results show great promise for the development of ultrafast and energy efficient ferrimagnetic spintronic devices.
2111.08768v1
2021-11-30
First and second order magnetic anisotropy and damping of europium iron garnet under high strain
Understanding and tailoring static and dynamic properties of magnetic insulator thin films is important for spintronic device applications. Here, we grow atomically flat epitaxial europium iron garnet (EuIG) thin films by pulsed laser deposition on (111)-oriented garnet substrates with a range of lattice parameters. By controlling the lattice mismatch between EuIG and the substrates, we tune the strain in EuIG films from compressive to tensile regime, which is characterized by X-ray diffraction. Using ferromagnetic resonance, we find that in addition to the first-order perpendicular magnetic anisotropy which depends linearly on the strain, there is a significant second-order one that has a quadratic strain dependence. Inhomogeneous linewidth of the ferromagnetic resonance increases notably with increasing strain, while the Gilbert damping parameter remains nearly constant (~ 2x10^-2). These results provide valuable insight into the spin dynamics in ferrimagnetic insulators and useful guidance for material synthesis and engineering of next-generation spintronics applications.
2111.15142v1
2022-10-01
Nonlinear features of the superconductor--ferromagnet--superconductor $\varphi_0$ Josephson junction in ferromagnetic resonance region
We demonstrate the manifestations of the nonlinear features in magnetic dynamics and IV-characteristics of the $\varphi_0$ Josephson junction in the ferromagnetic resonance region. We show that at small values of system parameters, namely, damping, spin-orbit interaction, and Josephson to magnetic energy ratio, the magnetic dynamics is reduced to the dynamics of the scalar Duffing oscillator, driven by the Josephson oscillations. The role of increasing superconducting current in the resonance region is clarified. Shifting of the ferromagnetic resonant frequency and the reversal of its damping dependence due to nonlinearity are demonstrated by the full Landau-Lifshitz-Gilbert-Josephson system of equations, and in its different approximations. Finally, we demonstrate the negative differential resistance in the IV--characteristics, and its correlation with the foldover effect.
2210.00366v1
2023-12-16
Spin-torque nano-oscillator based on two in-plane magnetized synthetic ferrimagnets
We report the dynamic characterization of the spin-torque-driven in-plane precession modes of a spin-torque nano-oscillator based on two different synthetic ferrimagnets: a pinned one characterized by a strong RKKY interaction which is exchange coupled to an antiferromagnetic layer; and a second one, non-pinned characterized by weak RKKY coupling. The microwave properties associated with the steady-state precession of both SyFs are characterized by high spectral purity and power spectral density. However, frequency dispersion diagrams of the damped and spin transfer torque modes reveal drastically different dynamical behavior and microwave emission properties in both SyFs. In particular, the weak coupling between the magnetic layers of the non-pinned SyF raises discontinuous dispersion diagrams suggesting a strong influence of mode crossing. An interpretation of the different dynamical features observed in the damped and spin torque modes of both SyF systems was obtained by solving simultaneously, in a macrospin approach, a linearized version of the Landau-Lifshitz-Gilbert equation including the spin transfer torque term.
2312.10451v2
2013-08-17
Thickness and power dependence of the spin-pumping effect in Y3Fe5O12/Pt heterostructures measured by the inverse spin Hall effect
The dependence of the spin-pumping effect on the yttrium iron garnet (Y3Fe5O12, YIG) thickness detected by the inverse spin Hall effect (ISHE) has been investigated quantitatively. Due to the spin-pumping effect driven by the magnetization precession in the ferrimagnetic insulator YIG film a spin-polarized electron current is injected into the Pt layer. This spin current is transformed into electrical charge current by means of the ISHE. An increase of the ISHE-voltage with increasing film thickness is observed and compared to the theoretically expected behavior. The effective damping parameter of the YIG/Pt samples is found to be enhanced with decreasing YIG film thickness. The investigated samples exhibit a spin mixing conductance of g=(7.43 \pm 0.36) \times 10^{18} m^{-2} and a spin Hall angle of theta_{ISHE} = 0.009 \pm 0.0008. Furthermore, the influence of nonlinear effects on the generated voltage and on the Gilbert damping parameter at high excitation powers are revealed. It is shown that for small YIG film thicknesses a broadening of the linewidth due to nonlinear effects at high excitation powers is suppressed because of a lack of nonlinear multi-magnon scattering channels. We have found that the variation of the spin-pumping efficiency for thick YIG samples exhibiting pronounced nonlinear effects is much smaller than the nonlinear enhancement of the damping.
1308.3787v1
2020-05-28
Spintronics meets nonadiabatic molecular dynamics: Geometric spin torque and damping on noncollinear classical magnetism due to electronic open quantum system
We analyze a quantum-classical hybrid system of steadily precessing slow classical localized magnetic moments, forming a head-to-head domain wall, embedded into an open quantum system of fast nonequilibrium electrons. The electrons reside within a metallic wire connected to macroscopic reservoirs. The model captures the essence of dynamical noncollinear and noncoplanar magnetic textures in spintronics, while making it possible to obtain the exact time-dependent nonequilibrium density matrix of electronic system and split it into four contributions. The Fermi surface contribution generates dissipative (or damping-like in spintronics terminology) spin torque on the moments, and one of the two Fermi sea contributions generates geometric torque dominating in the adiabatic regime. When the coupling to the reservoirs is reduced, the geometric torque is the only nonzero contribution. Locally it has both nondissipative (or field-like in spintronics terminology) and damping-like components, but with the sum of latter being zero, which act as the counterparts of geometric magnetism force and electronic friction in nonadiabatic molecular dynamics. Such current-independent geometric torque is absent from widely used micromagnetics or atomistic spin dynamics modeling of magnetization dynamics based on the Landau-Lifshitz-Gilbert equation, where previous analysis of Fermi surface-type torque has severely underestimated its magnitude.
2005.14153v2
2020-09-29
Structural Phase Dependent Giant Interfacial Spin Transparency in W/CoFeB Thin Film Heterostructure
Pure spin current has transfigured the energy-efficient spintronic devices and it has the salient characteristic of transport of the spin angular momentum. Spin pumping is a potent method to generate pure spin current and for its increased efficiency high effective spin-mixing conductance (Geff) and interfacial spin transparency (T) are essential. Here, a giant T is reported in Sub/W(t)/Co20Fe60B20(d)/SiO2(2 nm) heterostructures in \beta-tungsten (\beta-W) phase by employing all-optical time-resolved magneto-optical Kerr effect technique. From the variation of Gilbert damping with W and CoFeB thicknesses, the spin diffusion length of W and spin-mixing conductances are extracted. Subsequently, T is derived as 0.81 \pm 0.03 for the \beta-W/CoFeB interface. A sharp variation of Geff and T with W thickness is observed in consonance with the thickness-dependent structural phase transition and resistivity of W. The spin memory loss and two-magnon scattering effects are found to have negligible contributions to damping modulation as opposed to spin pumping effect which is reconfirmed from the invariance of damping with Cu spacer layer thickness inserted between W and CoFeB. The observation of giant interfacial spin transparency and its strong dependence on crystal structures of W will be important for pure spin current based spin-orbitronic devices.
2009.14143v1
2023-12-31
Molecular Hybridization Induced Antidamping and Sizable Enhanced Spin-to-Charge Conversion in Co20Fe60B20/$β$-W/C60 Heterostructures
Development of power efficient spintronics devices has been the compelling need in the post-CMOS technology era. The effective tunability of spin-orbit-coupling (SOC) in bulk and at the interfaces of hybrid materials stacking is a prerequisite for scaling down the dimension and power consumption of these devices. In this work, we demonstrate the strong chemisorption of C60 molecules when grown on the high SOC $\beta$-W layer. The parent CFB/$\beta$-W bilayer exhibits large spin-to-charge interconversion efficiency, which can be ascribed to the interfacial SOC observed at the Ferromagnet/Heavy metal interface. Further, the adsorption of C60 molecules on $\beta$-W reduces the effective Gilbert damping by $\sim$15% in the CFB/$\beta$-W/C60 heterostructures. The anti-damping is accompanied by a gigantic $\sim$115% enhancement in the spin-pumping induced output voltage owing to the molecular hybridization. The non-collinear Density Functional Theory calculations confirm the long-range enhancement of SOC of $\beta$-W upon the chemisorption of C60 molecules, which in turn can also enhance the SOC at the CFB/$\beta$-W interface in CFB/$\beta$-W/C60 heterostructures. The combined amplification of bulk as well interfacial SOC upon molecular hybridization stabilizes the anti-damping and enhanced spin-to-charge conversion, which can pave the way for the fabrication of power efficient spintronics devices.
2401.00486v1
2002-04-25
Statics and Fast Dynamics of Nanomagnets with Vortex Structure
Within the framework of the Landau-Lifshitz-Gilbert equation, using permalloy parameters, we study the statics and dynamics of flat circular magnetic nano-structures with an in-plane magnetic vortex configuration, putting particular emphasis on the (planar) vorticity of the magnetic state and on the (perpendicular) polarisation of the vortex center (which may be shifted with respect to the center of the circle). These binary degrees of freedom can in principle be used to manipulate two independent bits of information. Studying switching processes induced by in-plane and out-of plane field pulses we find that it is possible to switch the vorticity of the magnetic dot on a time scale of 40 ps in strong enough and short enough perpendicular external field pulses (B_z^ext \approx 0.5 T, duration \approx 40 ps). But for realistically small values of the Gilbert damping, only the vorticity can be switched this fast, and it turns out that it is better to dismiss the center of the circle totally, concentrating on flat 'nano-rings' with an inner radius R_1 and an outer radius R_2. On these 'nano-rings' the vortex state is more stable, and with respect to the switching of the vorticity these structures have similar properties as circular dots.
0204541v3
2007-03-15
Functional Keldysh Theory of Spin Torques
We present a microscopic treatment of current-induced torques and thermal fluctuations in itinerant ferromagnets based on a functional formulation of the Keldysh formalism. We find that the nonequilibrium magnetization dynamics is governed by a stochastic Landau-Lifschitz-Gilbert equation with spin transfer torques. We calculate the Gilbert damping parameter $\alpha$ and the non-adiabatic spin transfer torque parameter $\beta$ for a model ferromagnet. We find that $\beta \neq \alpha$, in agreement with the results obtained using imaginary-time methods of Kohno, Tatara and Shibata [J. Phys. Soc. Japan 75, 113706 (2006)]. We comment on the relationship between $s-d$ and isotropic-Stoner toy models of ferromagnetism and more realistic density-functional-theory models, and on the implications of these relationships for predictions of the $\beta/\alpha$ ratio which plays a central role in domain wall motion. Only for a single-parabolic-band isotropic-Stoner model with an exchange splitting that is small compared to the Fermi energy does $\beta/\alpha$ approach one. In addition, our microscopic formalism incorporates naturally the fluctuations needed in a nonzero-temperature description of the magnetization. We find that to first order in the applied electric field, the usual form of thermal fluctuations via a phenomenological stochastic magnetic field holds.
0703414v2
2010-10-04
Thermal fluctuation field for current-induced domain wall motion
Current-induced domain wall motion in magnetic nanowires is affected by thermal fluctuation. In order to account for this effect, the Landau-Lifshitz-Gilbert equation includes a thermal fluctuation field and literature often utilizes the fluctuation-dissipation theorem to characterize statistical properties of the thermal fluctuation field. However, the theorem is not applicable to the system under finite current since it is not in equilibrium. To examine the effect of finite current on the thermal fluctuation, we adopt the influence functional formalism developed by Feynman and Vernon, which is known to be a useful tool to analyze effects of dissipation and thermal fluctuation. For this purpose, we construct a quantum mechanical effective Hamiltonian describing current-induced domain wall motion by generalizing the Caldeira-Leggett description of quantum dissipation. We find that even for the current-induced domain wall motion, the statistical properties of the thermal noise is still described by the fluctuation-dissipation theorem if the current density is sufficiently lower than the intrinsic critical current density and thus the domain wall tilting angle is sufficiently lower than pi/4. The relation between our result and a recent result, which also addresses the thermal fluctuation, is discussed. We also find interesting physical meanings of the Gilbert damping alpha and the nonadiabaticy parameter beta; while alpha characterizes the coupling strength between the magnetization dynamics (the domain wall motion in this paper) and the thermal reservoir (or environment), beta characterizes the coupling strength between the spin current and the thermal reservoir.
1010.0478v2
2015-06-03
Antidamping spin-orbit torque driven by spin-flip reflection mechanism on the surface of a topological insulator: A time-dependent nonequilibrium Green function approach
Motivated by recent experiments observing spin-orbit torque (SOT) acting on the magnetization $\vec{m}$ of a ferromagnetic (F) overlayer on the surface of a three-dimensional topological insulator (TI), we investigate the origin of the SOT and the magnetization dynamics in such systems. We predict that lateral F/TI bilayers of finite length, sandwiched between two normal metal leads, will generate a large antidamping-like SOT per very low charge current injected parallel to the interface. The large values of antidamping-like SOT are {\it spatially localized} around the transverse edges of the F overlayer. Our analysis is based on adiabatic expansion (to first order in $\partial \vec{m}/\partial t$) of time-dependent nonequilibrium Green functions (NEGFs), describing electrons pushed out of equilibrium both by the applied bias voltage and by the slow variation of a classical degree of freedom [such as $\vec{m}(t)$]. From it we extract formulas for spin torque and charge pumping, which show that they are reciprocal effects to each other, as well as Gilbert damping in the presence of SO coupling. The NEGF-based formula for SOT naturally splits into four components, determined by their behavior (even or odd) under the time and bias voltage reversal. Their complex angular dependence is delineated and employed within Landau-Lifshitz-Gilbert simulations of magnetization dynamics in order to demonstrate capability of the predicted SOT to efficiently switch $\vec{m}$ of a perpendicularly magnetized F overlayer.
1506.01303v3
2015-07-11
Realization of the thermal equilibrium in inhomogeneous magnetic systems by the Landau-Lifshitz-Gilbert equation with stochastic noise, and its dynamical aspects
It is crucially important to investigate effects of temperature on magnetic properties such as critical phenomena, nucleation, pinning, domain wall motion, coercivity, etc. The Landau-Lifshitz-Gilbert (LLG) equation has been applied extensively to study dynamics of magnetic properties. Approaches of Langevin noises have been developed to introduce the temperature effect into the LLG equation. To have the thermal equilibrium state (canonical distribution) as the steady state, the system parameters must satisfy some condition known as the fluctuation-dissipation relation. In inhomogeneous magnetic systems in which spin magnitudes are different at sites, the condition requires that the ratio between the amplitude of the random noise and the damping parameter depends on the magnitude of the magnetic moment at each site. Focused on inhomogeneous magnetic systems, we systematically showed agreement between the stationary state of the stochastic LLG equation and the corresponding equilibrium state obtained by Monte Carlo simulations in various magnetic systems including dipole-dipole interactions. We demonstrated how violations of the condition result in deviations from the true equilibrium state. We also studied the characteristic features of the dynamics depending on the choice of the parameter set. All the parameter sets satisfying the condition realize the same stationary state (equilibrium state). In contrast, different choices of parameter set cause seriously different relaxation processes. We show two relaxation types, i.e., magnetization reversals with uniform rotation and with nucleation.
1507.03075v1
2017-01-12
Dynamic coupling of ferromagnets via spin Hall magnetoresistance
The synchronized magnetization dynamics in ferromagnets on a nonmagnetic heavy metal caused by the spin Hall effect is investigated theoretically. The direct and inverse spin Hall effects near the ferromagnetic/nonmagnetic interface generate longitudinal and transverse electric currents. The phenomenon is known as the spin Hall magnetoresistance effect, whose magnitude depends on the magnetization direction in the ferromagnet due to the spin transfer effect. When another ferromagnet is placed onto the same nonmagnet, these currents are again converted to the spin current by the spin Hall effect and excite the spin torque to this additional ferromagnet, resulting in the excitation of the coupled motions of the magnetizations. The in-phase or antiphase synchronization of the magnetization oscillations, depending on the value of the Gilbert damping constant and the field-like torque strength, is found in the transverse geometry by solving the Landau-Lifshitz-Gilbert equation numerically. On the other hand, in addition to these synchronizations, the synchronization having a phase difference of a quarter of a period is also found in the longitudinal geometry. The analytical theory clarifying the relation among the current, frequency, and phase difference is also developed, where it is shown that the phase differences observed in the numerical simulations correspond to that giving the fixed points of the energy supplied by the coupling torque.
1701.03201v2
2018-10-16
Superfluid spin transport in ferro- and antiferromagnets
This paper focuses on spin superfluid transport, observation of which was recently reported in antiferromagnet Cr$_2$O$_3$ [Yuan et al., Sci. Adv. 4, eaat1098 (2018)]. This paper analyzes the role of dissipation in transformation of spin current injected with incoherent magnons to a superfluid spin current near the interface where spin is injected. The Gilbert damping parameter in the Landau-Lifshitz-Gilbert theory does not describe dissipation properly, and the dissipation parameters are calculated from the Boltzmann equation for magnons scattered by defects. The two-fluid theory is developed similar to the two-fluid theory for superfluids. This theory shows that the influence of temperature variation in bulk on the superfluid spin transport (bulk Seebeck effect) is weak at low temperatures. The scenario that the results of Yuan et al. are connected with the Seebeck effect at the interface between the spin detector and the sample is also discussed. The Landau criterion for an antiferromagnet put in a magnetic field is derived from the spectrum of collective spin modes. The Landau instability starts in the gapped mode earlier than in the Goldstone gapless mode, in contrast to easy-plane ferromagnets where the Goldstone mode becomes unstable. The structure of the magnetic vortex in the geometry of the experiment is determined. The vortex core has the skyrmion structure with finite magnetization component normal to the magnetic field. This magnetization creates stray magnetic fields around the exit point of the vortex line from the sample, which can be used for experimental detection of vortices.
1810.07020v4
2020-02-20
Stoner-Wohlfarth switching of the condensate magnetization in a dipolar spinor gas and the metrology of excitation damping
We consider quasi-one-dimensional dipolar spinor Bose-Einstein condensates in the homogeneous-local-spin-orientation approximation, that is with unidirectional local magnetization. By analytically calculating the exact effective dipole-dipole interaction, we derive a Landau-Lifshitz-Gilbert equation for the dissipative condensate magnetization dynamics, and show how it leads to the Stoner-Wohlfarth model of a uni-axial ferro-magnetic particle, where the latter model determines the stable magnetization patterns and hysteresis curves for switching between them. For an external magnetic field pointing along the axial, long direction, we analytically solve the Landau-Lifshitz-Gilbert equation. The solution explicitly demonstrates that the magnetic dipole-dipole interaction {\it accelerates} the dissipative dynamics of the magnetic moment distribution and the associated dephasing of the magnetic moment direction. Under suitable conditions, dephasing of the magnetization direction due to dipole-dipole interactions occurs within time scales up to two orders of magnitude smaller than the lifetime of currently experimentally realized dipolar spinor condensates, e.g., produced with the large magnetic-dipole-moment atoms ${}^{166} \textrm{Er}$. This enables experimental access to the dissipation parameter $\Gamma$ in the Gross-Pitaevski\v\i~mean-field equation, for a system currently lacking a complete quantum kinetic treatment of dissipative processes and, in particular, an experimental check of the commonly used assumption that $\Gamma$ is a single scalar independent of spin indices.
2002.08723v2
2022-06-20
First-principles calculation of the parameters used by atomistic magnetic simulations
While the ground state of magnetic materials is in general well described on the basis of spin density functional theory (SDFT), the theoretical description of finite-temperature and non-equilibrium properties require an extension beyond the standard SDFT. Time-dependent SDFT (TD-SDFT), which give for example access to dynamical properties are computationally very demanding and can currently be hardly applied to complex solids. Here we focus on the alternative approach based on the combination of a parameterized phenomenological spin Hamiltonian and SDFT-based electronic structure calculations, giving access to the dynamical and finite-temperature properties for example via spin-dynamics simulations using the Landau-Lifshitz-Gilbert (LLG) equation or Monte Carlo simulations. We present an overview on the various methods to calculate the parameters of the various phenomenological Hamiltonians with an emphasis on the KKR Green function method as one of the most flexible band structure methods giving access to practically all relevant parameters. Concerning these, it is crucial to account for the spin-orbit coupling (SOC) by performing relativistic SDFT-based calculations as it plays a key role for magnetic anisotropy and chiral exchange interactions represented by the DMI parameters in the spin Hamiltonian. This concerns also the Gilbert damping parameters characterizing magnetization dissipation in the LLG equation, chiral multispin interaction parameters of the extended Heisenberg Hamiltonian, as well as spin-lattice interaction parameters describing the interplay of spin and lattice dynamics processes, for which an efficient computational scheme has been developed recently by the present authors.
2206.09969v1
2023-09-25
Ultrafast Demagnetization through Femtosecond Generation of Non-thermal Magnons
Ultrafast laser excitation of ferromagnetic metals gives rise to correlated, highly non-equilibrium dynamics of electrons, spins and lattice, which are, however, poorly described by the widely-used three-temperature model (3TM). Here, we develop a fully ab-initio parameterized out-of-equilibrium theory based on a quantum kinetic approach--termed (N+2) temperature model--that describes magnon occupation dynamics due to electron-magnon scattering. We apply this model to perform quantitative simulations on the ultrafast, laser-induced generation of magnons in iron and demonstrate that on these timescales the magnon distribution is non-thermal: predominantly high-energy magnons are created, while the magnon occupation close to the center of the Brillouin zone even decreases, due to a repopulation towards higher energy states via a so-far-overlooked scattering term. We demonstrate that the simple relation between magnetization and temperature computed at equilibrium does not hold in the ultrafast regime and that the 3TM greatly overestimates the demagnetization. The ensuing Gilbert damping becomes strongly magnon wavevector dependent and requires a description beyond the conventional Landau-Lifshitz-Gilbert spin dynamics. Our ab-initio-parameterized calculations show that ultrafast generation of non-thermal magnons provides a sizable demagnetization within 200fs in excellent comparison with experimentally observed laser-induced demagnetizations. Our investigation emphasizes the importance of non-thermal magnon excitations for the ultrafast demagnetization process.
2309.14167v3
2023-12-12
Sliding Dynamics of Current-Driven Skyrmion Crystal and Helix in Chiral Magnets
The skyrmion crystal (SkX) and helix (HL) phases, present in typical chiral magnets, can each be considered as forms of density waves but with distinct topologies. The SkX exhibits gyrodynamics analogous to electrons under a magnetic field, while the HL state resembles topological trivial spin density waves. However, unlike the charge density waves, the theoretical analysis of the sliding motion of SkX and HL remains unclear, especially regarding the similarities and differences in sliding dynamics between these two spin density waves. In this work, we systematically explore the sliding dynamics of SkX and HL in chiral magnets in the limit of large current density. We demonstrate that the sliding dynamics of both SkX and HL can be unified within the same theoretical framework as density waves, despite their distinct microscopic orders. Furthermore, we highlight the significant role of gyrotropic sliding induced by impurity effects in the SkX state, underscoring the impact of nontrivial topology on the sliding motion of density waves. Our theoretical analysis shows that the effect of impurity pinning is much stronger in HL compared with SkX, i.e., $\chi^{SkX}/\chi^{HL}\sim \alpha^2$ ($\chi^{SkX}$, $\chi^{HL}$: susceptibility to the impurity potential, $\alpha$ ($\ll 1$) is the Gilbert damping). Moreover, the velocity correction is mostly in the transverse direction to the current in SkX. These results are further substantiated by realistic Landau-Lifshitz-Gilbert simulations.
2312.07116v2
2023-08-14
Temperature Evolution of Magnon Propagation Length in Tm$_3$Fe$_5$O$_{12}$ Thin Films: Roles of Magnetic Anisotropy and Gilbert Damping
The magnon propagation length ($\langle\xi\rangle$) of a ferro/ferrimagnet (FM) is one of the key factors that controls the generation and propagation of thermally-driven spin current in FM/heavy metal (HM) bilayer based spincaloritronic devices. Theory predicts that for the FM layer, $\langle\xi\rangle$ is inversely proportional to the Gilbert damping ($\alpha$) and the square root of the effective magnetic anisotropy constant ($K_{\rm eff}$). However, direct experimental evidence of this relationship is lacking. To experimentally confirm this prediction, we employ a combination of longitudinal spin Seebeck effect (LSSE), transverse susceptibility, and ferromagnetic resonance experiments to investigate the temperature evolution of $\langle\xi\rangle$ and establish its correlation with the effective magnetic anisotropy field, $H_K^{\rm eff}$ ($\propto K_{\rm eff}$) and $\alpha$ in Tm$_3$Fe$_5$O$_{12}$ (TmIG)/Pt bilayers. We observe concurrent drops in the LSSE voltage and $\langle\xi\rangle$ below 200$^\circ$K in TmIG/Pt bilayers regardless of TmIG film thickness and substrate choice and attribute it to the noticeable increases in $H_K^{\rm eff}$ and $\alpha$ that occur within the same temperature range. From the TmIG thickness dependence of the LSSE voltage, we determined the temperature dependence of $\langle\xi\rangle$ and highlighted its correlation with the temperature-dependent $H_K^{\rm eff}$ and $\alpha$ in TmIG/Pt bilayers, which will be beneficial for the development of rare-earth iron garnet-based efficient spincaloritronic nanodevices.
2308.07236v3
2014-01-08
Dynamic exchange via spin currents in acoustic and optical modes of ferromagnetic resonance in spin-valve structures
Two ferromagnetic layers magnetically decoupled by a thick normal metal spacer layer can be, nevertheless, dynamically coupled via spin currents emitted by the spin-pump and absorbed through the spin-torque effects at the neighboring interfaces. A decrease of damping in both layers due to a partial compensation of the angular momentum leakage in each layer was previously observed at the coincidence of the two ferromagnetic resonances. In case of non-zero magnetic coupling, such a dynamic exchange will depend on the mutual precession of the magnetic moments in the layers. A difference in the linewidth of the resonance peaks is expected for the acoustic and optical regimes of precession. However, the interlayer coupling hybridizes the resonance responses of the layers and therefore can also change their linewidths. The interplay between the two mechanisms has never been considered before. In the present work, the joint influence of the hybridization and non-local damping on the linewidth has been studied in weakly coupled NiFe/CoFe/Cu/CoFe/MnIr spin-valve multilayers. It has been found that the dynamic exchange by spin currents is different in the optical and acoustic modes, and this difference is dependent on the interlayer coupling strength. In contrast to the acoustic precession mode, the dynamic exchange in the optical mode works as an additional damping source. A simulation in the framework of the Landau-Lifshitz-Gilbert formalism for two ferromagnetic layers coupled magnetically and by spin currents has been done to separate the effects of the non-local damping from the resonance modes hybridization. In our samples both mechanisms bring about linewidth changes of the same order of magnitude, but lead to a distinctly different angular behavior. The obtained results are relevant for a broad class of coupled magnetic multilayers with ballistic regime of the spin transport.
1401.1672v1
2020-08-29
Exploring a quantum-information-relevant magnonic material: Ultralow damping at low temperature in the organic ferrimagnet V[TCNE]x
Quantum information science and engineering requires novel low-loss magnetic materials for magnon-based quantum-coherent operations. The search for low-loss magnetic materials, traditionally driven by applications in microwave electronics near room-temperature, has gained additional constraints from the need to operate at cryogenic temperatures for many applications in quantum information science and technology. Whereas yttrium iron garnet (YIG) has been the material of choice for decades, the emergence of molecule-based materials with robust magnetism and ultra-low damping has opened new avenues for exploration. Specifically, thin-films of vanadium tetracyanoethylene (V[TCNE]x) can be patterned into the multiple, connected structures needed for hybrid quantum elements and have shown room-temperature Gilbert damping ({\alpha} = 4 \times 10^-5) that rivals the intrinsic (bulk) damping otherwise seen only in highly-polished YIG spheres (far more challenging to integrate into arrays). Here, we present a comprehensive and systematic study of the low-temperature magnetization dynamics for V[TCNE]x thin films, with implications for their application in quantum systems. These studies reveal a temperature-driven, strain-dependent magnetic anisotropy that compensates the thin-film shape anisotropy, and the recovery of a magnetic resonance linewidth at 5 K that is comparable to room-temperature values (roughly 2 G at 9.4 GHz). We can account for these variations of the V[TCNE]x linewidth within the context of scattering from very dilute paramagnetic impurities, and anticipate additional linewidth narrowing as the temperature is further reduced.
2008.13061v3
2003-04-18
Elementary Excitations of Ferromagnetic Metal Nanoparticles
We present a theory of the elementary spin excitations in transition metal ferromagnet nanoparticles which achieves a unified and consistent quantum description of both collective and quasiparticle physics. The theory starts by recognizing the essential role played by spin-orbit interactions in determining the energies of ferromagnetic resonances in the collective excitation spectrum and the strength of their coupling to low-energy particle-hole excitations. We argue that a crossover between Landau-damped ferromagnetic resonance and pure-state collective magnetic excitations occurs as the number of atoms in typical transition metal ferromagnet nanoparticles drops below approximately $10^4$, approximately where the single-particle level spacing, $\delta$, becomes larger than, $\sqrt{\alpha} E_{\rm res}$, where $E_{\rm res}$ is the ferromagnetic resonance frequency and $\alpha$ is the Gilbert damping parameter. We illustrate our ideas by studying the properties of semi-realistic model Hamiltonians, which we solve numerically for nanoparticles containing several hundred atoms. For small nanoparticles, we find one isolated ferromagnetic resonance collective mode below the lowest particle-hole excitation energy, at $E_{\rm res} \approx 0.1$ meV. The spectral weight of this pure excitation nearly exhausts the transverse dynamical susceptibility spectral weight. As $\delta$ approaches $\sqrt{\alpha} E_{\rm res}$, the ferromagnetic collective excitation is more likely to couple strongly with discrete particle-hole excitations. In this regime the distinction between the two types of excitations blurs. We discuss the significance of this picture for the interpretation of recent single-electron tunneling experiments.
0304427v1
2014-08-02
Tunnel magnetoresistance and spin-transfer-torque switching in polycrystalline Co2FeAl full-Heusler alloy magnetic tunnel junctions on Si/SiO2 amorphous substrates
We studied polycrystalline B2-type Co2FeAl (CFA) full-Heusler alloy based magnetic tunnel junctions (MTJs) fabricated on a Si/SiO2 amorphous substrate. Polycrystalline CFA films with a (001) orientation, a high B2 ordering, and a flat surface were achieved using a MgO buffer layer. A tunnel magnetoresistance (TMR) ratio up to 175% was obtained for an MTJ with a CFA/MgO/CoFe structure on a 7.5-nm-thick MgO buffer. Spin-transfer torque induced magnetization switching was achieved in the MTJs with a 2-nm-thick polycrystalline CFA film as a switching layer. Using a thermal activation model, the intrinsic critical current density (Jc0) was determined to be 8.2 x 10^6 A/cm^2, which is lower than 2.9 x 10^7 A/cm^2, the value for epitaxial CFA-MTJs [Appl. Phys. Lett. 100, 182403 (2012)]. We found that the Gilbert damping constant evaluated using ferromagnetic resonance measurements for the polycrystalline CFA film was ~0.015 and was almost independent of the CFA thickness (2~18 nm). The low Jc0 for the polycrystalline MTJ was mainly attributed to the low damping of the CFA layer compared with the value in the epitaxial one (~0.04).
1408.0341v1
2018-02-20
Ultrafast magnetization dynamics in pure and doped Heusler and inverse Heusler alloys
By using a multiscale approach based on first-principles density functional theory combined with atomistic spin dynamics, we investigate the electronic structure and magnetization dynamics of an inverse Heusler and a Heusler compound and their alloys, i. e. Mn$_{2-x}Z_x$CoAl and Mn$_{2-x}Z_x$VAl, where $Z$ = Mo, W, Os and Ru, respectively. A signature of the ferrimagnetic ordering of Mn$_{2}$CoAl and Mn$_{2}$VAl Heusler alloys is reflected in the calculated Heisenberg exchange constants. They decay very rapidly with the interatomic distance and have short range, which is a consequence of the existence of the finite gap in the minority spin band. The calculated Gilbert damping parameter of both Mn$_2$CoAl and Mn$_2$VAl is high compared to other half-metals, but interestingly in the particular case of the inverse Mn$_{2}$CoAl alloys and due to the spin-gapless semiconducting property, the damping parameters decrease with the doping concentration in clear contradiction to the general trend. Atomistic spin dynamics simulations predict ultrafast magnetisation switching in Mn$_{2}$CoAl and Mn$_{2}$VAl under the influence of an external magnetic field, starting from a threshold field of $2\text{T}$. Our overall finding extends with Heusler and inverse Heusler alloys, the class of materials that exhibits laser induced magnetic switching.
1802.07195v1
2018-04-10
GONG Catalog of Solar Filament Oscillations Near Solar Maximum
We have catalogued 196 filament oscillations from the GONG $H{\alpha}$ network data during several months near the maximum of solar cycle 24 (January - June 2014). Selected examples from the catalog are described in detail, along with our statistical analyses of all events. Oscillations were classified according to their velocity amplitude: 106 small-amplitude oscillations (SAOs), with velocities $<10\mathrm{\, km \; s^{-1}}$, and 90 large-amplitude oscillations (LAOs), with velocities $>10\mathrm{\, km \; s^{-1}}$. Both SAOs and LAOs are common, with one event of each class every two days on the visible side of the Sun. For nearly half of the events we identified their apparent trigger. The period distribution has a mean value of 58$\pm$15 min for both types of oscillations. The distribution of the damping time per period peaks at $\tau/P=1.75$ and $1.25$ for SAOs and LAOs respectively. We confirmed that LAO damping rates depend nonlinearly on the oscillation velocity. The angle between the direction of motion and the filament spine has a distribution centered at $27^\circ$ for all filament types. This angle agrees with the observed direction of filament-channel magnetic fields, indicating that most of the catalogued events are longitudinal (i.e., undergo field-aligned motions). We applied seismology to determine the average radius of curvature in the magnetic dips, $R\approx89$ Mm, and the average minimum magnetic-field strength, $B\approx16$ G. The catalog is available to the community online, and is intended to be expanded to cover at least 1 solar cycle.
1804.03743v1
2018-10-16
Spin-wave-induced lateral temperature gradient in a YIG thin film/GGG system excited in an ESR cavity
Lateral thermal gradient of an yttrium iron garnet (YIG) film under the microwave application in the cavity of the electron spin resonance system (ESR) was measured at room temperature by fabricating a Cu/Sb thermocouple onto it. To date, thermal transport in YIG films caused by the Damon-Eshbach mode (DEM) - the unidirectional spin-wave heat conveyer effect - was demonstrated only by the excitation using coplanar waveguides. Here we show that effect exists even under YIG excitation using the ESR cavity - tool often employed to realize spin pumping. The temperature difference observed around the ferromagnetic resonance (FMR) field under the 4 mW microwave power peaked at 13 mK. The observed thermoelectric signal indicates the imbalance of the population between the DEMs that propagate near the top and bottom surfaces of the YIG film. We attribute the DEM population imbalance to the different magnetic damping near the top and bottom YIG surfaces. Additionally, the spin wave dynamics of the system were investigated using the micromagnetic simulations. The micromagnetic simulations confirmed the existence of the DEM imbalance in the system with the increased Gilbert damping at one of the YIG interfaces. The reported results are indispensable for the quantitative estimation of the electromotive force in the spin-charge conversion experiments using ESR cavities.
1810.06875v1
2019-11-21
Low damping and microstructural perfection of sub-40nm-thin yttrium iron garnet films grown by liquid phase epitaxy
The field of magnon spintronics is experiencing an increasing interest in the development of solutions for spin-wave-based data transport and processing technologies that are complementary or alternative to modern CMOS architectures. Nanometer-thin yttrium iron garnet (YIG) films have been the gold standard for insulator-based spintronics to date, but a potential process technology that can deliver perfect, homogeneous large-diameter films is still lacking. We report that liquid phase epitaxy (LPE) enables the deposition of nanometer-thin YIG films with low ferromagnetic resonance losses and consistently high magnetic quality down to a thickness of 20 nm. The obtained epitaxial films are characterized by an ideal stoichiometry and perfect film lattices, which show neither significant compositional strain nor geometric mosaicity, but sharp interfaces. Their magneto-static and dynamic behavior is similar to that of single crystalline bulk YIG. We found, that the Gilbert damping coefficient alpha is independent of the film thickness and close to 1 x 10-4, and that together with an inhomogeneous peak-to-peak linewidth broadening of delta H0|| = 0.4 G, these values are among the lowest ever reported for YIG films with a thickness smaller than 40 nm. These results suggest, that nanometer-thin LPE films can be used to fabricate nano- and micro-scaled circuits with the required quality for magnonic devices. The LPE technique is easily scalable to YIG sample diameters of several inches.
1911.09400v1
2021-08-24
Shape anisotropy effect on magnetization reversal induced by linear down chirp pulse
We investigate the influence of shape anisotropy on the magnetization reversal of a single-domain magnetic nanoparticle driven by a circularly polarized linear down-chirp microwave field pulse (DCMP). Based on the Landau-Lifshitz-Gilbert equation, numerical results show that the three controlling parameters of DCMP, namely, microwave amplitude, initial frequency and chirp rate, decrease with the increase of shape anisotropy. For certain shape anisotropy, the reversal time significantly reduces. These findings are related to the competition of shape anisotropy and uniaxial magnetocrystalline anisotropy and thus to the height of energy barrier which separates the two stable states. The result of damping dependence of magnetization reversal indicates that for a certain sample shape, there exists an optimal damping situation at which magnetization is fastest. Moreover, it is also shown that the required microwave field amplitude can be lowered by applying the spin-polarized current simultaneously. The usage of an optimum combination of both microwave field pulse and current is suggested to achieve cost efficiency and faster switching. So these findings may provide the knowledge to fabricate the shape of a single domain nanoparticle for the fast and power-efficient magnetic data storage device.
2108.10965v2
2021-11-23
Resonant dynamics of skyrmion lattices in thin film multilayers: Localised modes and spin wave emission
The spectral signatures of magnetic skyrmions under microwave field excitation are of fundamental interest and can be an asset for high frequency applications. These topological solitons can be tailored in multilayered thin films, but the experimental observation of their spin wave dynamics remains elusive, in particular due to large damping. Here, we study Pt/FeCoB/AlO$_x$ multilayers hosting dense and robust skyrmion lattices at room temperature with Gilbert damping of $\sim 0.02$. We use magnetic force microscopy to characterise their static magnetic phases and broadband ferromagnetic resonance to probe their high frequency response. Micromagnetic simulations reproduce the experiments with accuracy and allow us to identify distinct resonant modes detected in the skyrmion lattice phase. Low ($<$ 2 GHz) and intermediate frequency ($2-8$ GHz) modes involve excitations localised to skyrmion edges in conjunction with precession of the uniform background magnetisation, while a high frequency ($>$ 12 GHz) mode corresponds to in-phase skyrmion core precession emitting spin waves into uniform background with wavelengths in the 50--80 nm range commensurate with the lattice structure. These findings could be instrumental in the investigation of room temperature wave scattering and the implementation of novel microwave processing schemes in reconfigurable arrays of solitons.
2111.11797v2
2022-05-20
Effects of Crystalline Disorder on Interfacial and Magnetic Properties of Sputtered Topological Insulator/Ferromagnet Heterostructures
Thin films of Topological insulators (TIs) coupled with ferromagnets (FMs) are excellent candidates for energy-efficient spintronics devices. Here, the effect of crystalline structural disorder of TI on interfacial and magnetic properties of sputter-deposited TI/FM, Bi2Te3/Ni80Fe20, heterostructures is reported. Ni and a smaller amount of Fe from Py was found to diffuse across the interface and react with Bi2Te3. For highly crystalline c-axis oriented Bi2Te3 films, a giant enhancement in Gilbert damping is observed, accompanied by an effective out-of-plane magnetic anisotropy and enhanced damping-like spin-orbit torque (DL-SOT), possibly due to the topological surface states (TSS) of Bi2Te3. Furthermore, a spontaneous exchange bias is observed in hysteresis loop measurements at low temperatures. This is because of an antiferromagnetic topological interfacial layer formed by reaction of the diffused Ni with Bi2Te3 which couples with the FM, Ni80Fe20. For increasing disorder of Bi2Te3, a significant weakening of exchange interaction in the AFM interfacial layer is found. These experimental results Abstract length is one paragraph.
2205.09913v1
2022-12-24
Anatomy of ultrafast quantitative magneto-acoustics in freestanding nickel thin films
We revisit the quantitative analysis of the ultrafast magneto-acoustic experiment in a freestanding nickel thin film by Kim and Bigot [1] by applying our recently proposed approach of magnetic and acoustic eigenmodes decomposition by Vernik et al. [2]. We show that the application of our modeling to the analysis of time-resolved reflectivity measurements allows for the determination of amplitudes and lifetimes of standing perpendicular acoustic phonon resonances with unprecedented accuracy. The acoustic damping is found to scale as $\propto\omega^2$ for frequencies up to 80~GHz and the peak amplitudes reach $10^{-3}$. The experimentally measured magnetization dynamics for different orientations of an external magnetic field agrees well with numerical solutions of magneto-elastically driven magnon harmonic oscillators. Symmetry-based selection rules for magnon-phonon interactions predicted by our modeling approach allow for the unambiguous discrimination between spatially uniform and non-uniform modes, as confirmed by comparing the resonantly enhanced magneto-elastic dynamics simultaneously measured on opposite sides of the film. Moreover, the separation of time scales for (early) rising and (late) decreasing precession amplitudes provide access to magnetic (Gilbert) and acoustic damping parameters in a single measurement.
2212.12673v1
2023-05-16
Non-Hermitian Casimir Effect of Magnons
There has been a growing interest in non-Hermitian quantum mechanics. The key concepts of quantum mechanics are quantum fluctuations. Quantum fluctuations of quantum fields confined in a finite-size system induce the zero-point energy shift. This quantum phenomenon, the Casimir effect, is one of the most striking phenomena of quantum mechanics in the sense that there are no classical analogs and has been attracting much attention beyond the hierarchy of energy scales, ranging from elementary particle physics to condensed matter physics, together with photonics. However, the non-Hermitian extension of the Casimir effect and the application to spintronics have not yet been investigated enough, although exploring energy sources and developing energy-efficient nanodevices are its central issues. Here we fill this gap. By developing a magnonic analog of the Casimir effect into non-Hermitian systems, we show that this non-Hermitian Casimir effect of magnons is enhanced as the Gilbert damping constant (i.e., the energy dissipation rate) increases. When the damping constant exceeds a critical value, the non-Hermitian Casimir effect of magnons exhibits an oscillating behavior, including a beating one, as a function of the film thickness and is characterized by the exceptional point. Our result suggests that energy dissipation serves as a key ingredient of Casimir engineering.
2305.09231v1
2002-09-07
Neural network analysis of the magnetization reversal in magnetic dot arrays
We simulated the remagnetization dynamics of the ultra-dense and ultra-thin magnetic dot array system with dipole-dipole and exchange coupling interactions. Within the proposed 2D XY superlattice model, the square dots are modeled by the spatially modulated exchange-couplings. The dipole-dipole interactions were approximated by the hierarchical sums and dynamics was reduced to damping term of the Landau-Lifshitz-Gilbert equation. The simulation of 40 000 spin system leads to nonequilibrium nonuniform configurations with soliton-antisoliton pairs detected at intra-dot and inter-dot scales. The classification of intra-dot magnetic configurations was performed using the self-adaptive neural networks with varying number of neurons.
0209186v1
2005-04-06
Macrospin Models of Spin Transfer Dynamics
The current-induced magnetization dynamics of a spin valve are studied using a macrospin (single domain) approximation and numerical solutions of a generalized Landau-Lifshitz-Gilbert equation. For the purpose of quantitative comparison with experiment [Kiselev {\it et al.} Nature {\bf 425}, 380 (2003)], we calculate the resistance and microwave power as a function of current and external field including the effects of anisotropies, damping, spin-transfer torque, thermal fluctuations, spin-pumping, and incomplete absorption of transverse spin current. While many features of experiment appear in the simulations, there are two significant discrepancies: the current dependence of the precession frequency and the presence/absence of a microwave quiet magnetic phase with a distinct magnetoresistance signature. Comparison is made with micromagnetic simulations designed to model the same experiment.
0504142v1
2006-02-01
Mapping Monte Carlo to Langevin dynamics: A Fokker-Planck approach
We propose a general method of using the Fokker-Planck equation (FPE) to link the Monte-Carlo (MC) and the Langevin micromagnetic schemes. We derive the drift and disusion FPE terms corresponding to the MC method and show that it is analytically equivalent to the stochastic Landau-Lifshitz-Gilbert (LLG) equation of Langevin-based micromagnetics. Subsequent results such as the time quantification factor for the Metropolis MC method can be rigorously derived from this mapping equivalence. The validity of the mapping is shown by the close numerical convergence between the MC method and the LLG equation for the case of a single magnetic particle as well as interacting arrays of particles. We also found that our Metropolis MC is accurate for a large range of damping factors $\alpha$, unlike previous time-quantified MC methods which break down at low $\alpha$, where precessional motion dominates.
0602011v2
2006-04-21
Dynamic approach for micromagnetics close to the Curie temperature
In conventional micromagnetism magnetic domain configurations are calculated based on a continuum theory for the magnetization which is assumed to be of constant length in time and space. Dynamics is usually described with the Landau-Lifshitz-Gilbert (LLG) equation the stochastic variant of which includes finite temperatures. Using simulation techniques with atomistic resolution we show that this conventional micromagnetic approach fails for higher temperatures since we find two effects which cannot be described in terms of the LLG equation: i) an enhanced damping when approaching the Curie temperature and, ii) a magnetization magnitude that is not constant in time. We show, however, that both of these effects are naturally described by the Landau-Lifshitz-Bloch equation which links the LLG equation with the theory of critical phenomena and turns out to be a more realistic equation for magnetization dynamics at elevated temperatures.
0604508v1
2007-02-20
Spin dynamics in a superconductor / ferromagnet proximity system
The ferromagnetic resonance of thin sputtered Ni80Fe20 films grown on Nb is measured. By varying the temperature and thickness of the Nb the role of the superconductivity on the whole ferromagnetic layer in these heterostructures is explored. The change in the spin transport properties below the superconducting transition of the Nb is found to manifest itself in the Ni80Fe20 layer by a sharpening in the resonance of the ferromagnet, or a decrease in the effective Gilbert damping co-efficient. This dynamic proximity effect is in contrast to low frequency studies in these systems, where the effect of the superconductor is confined to a small region in the ferromagnet. We interpret this in terms of the spin pumping model.
0702461v1
2007-02-21
Domain wall mobility, stability and Walker breakdown in magnetic nanowires
We present an analytical calculation of the velocity of a single 180 degree domain wall in a magnetic structure with reduced thickness and/or lateral dimension under the combined action of an external applied magnetic field and an electrical current. As for the case of field-induced domain wall propagation in thick films, two motion regimes with different mobilities are obtained, below and far above the so-called Walker field. Additionally, for the case of current induced motion, a Walker-like current density threshold can be defined. When the dimensions of the system become comparable to the domain wall width, the threshold field and current density, stating the wall's internal structure stability, are reduced by the same geometrical demagnetising factor which accounts for the confinement. This points out the fact that the velocity dependence over an extended field/current range and the knowledge of the Walker breakdown are mandatory to draw conclusions about the phenomenological Gilbert damping parameter tuning the magnetisation dynamics.
0702492v1
2001-01-09
Hysteresis in layered spring magnets
This article addresses a problem of micromagnetics: the reversal of magnetic moments in layered spring magnets. A one-dimensional model is used of a film consisting of several atomic layers of a soft material on top of several atomic layers of a hard material. Each atomic layer is taken to be uniformly magnetized, and spatial inhomogeneities within an atomic layer are neglected. The state of such a system is described by a chain of magnetic spin vectors. Each spin vector behaves like a spinning top driven locally by the effective magnetic field and subject to damping (Landau-Lifshitz-Gilbert equation). A numerical integration scheme for the LLG equation is presented that is unconditionally stable and preserves the magnitude of the magnetization vector at all times. The results of numerical investigations for a bilayer in a rotating in-plane magnetic field show hysteresis with a basic period of $2\pi$ at moderate fields and hysteresis with a basic period of $\pi$ at strong fields.
0101077v1
2005-01-01
Equatorial and related non-equilibrium states in magnetization dynamics of ferromagnets: Generalization of Suhl's spin-wave instabilities
We investigate the nonlinear dynamics underlying the evolution of a 2-D nanoscale ferromagnetic film with uniaxial anisotropy in the presence of perpendicular pumping. Considering the associated Landau-Lifshitz spin evolution equation with Gilbert damping together with Maxwell equation for the demagnetization field, we study the dynamics in terms of the stereographic variable. We identify several new fixed points for suitable choice of external field in a rotating frame of reference. In particular, we identify explicit equatorial and related fixed points of the spin vector in the plane transverse to the anisotropy axis when the pumping frequency coincides with the amplitude of the static parallel field. We then study the linear stability of these novel fixed points under homogeneous and spin wave perturbations and obtain a generalized Suhl's instability criterion, giving the condition for exponential growth of P-modes under spin wave perturbations. Two parameter phase diagrams (in terms of amplitudes of static parallel and oscillatory perpendicular magnetic fields) for stability are obtained, which differ qualitatively from those for the conventional ferromagnetic resonance near thermal equilibrium and are amenable to experimental tests.
0501002v2
2002-12-30
Stochastic resonance in periodic potentials: realization in a dissipative optical lattice
We have observed the phenomenon of stochastic resonance on the Brillouin propagation modes of a dissipative optical lattice. Such a mode has been excited by applying a moving potential modulation with phase velocity equal to the velocity of the mode. Its amplitude has been characterized by the center-of-mass (CM) velocity of the atomic cloud. At Brillouin resonance, we studied the CM-velocity as a function of the optical pumping rate at a given depth of the potential wells. We have observed a resonant dependence of the CM velocity on the optical pumping rate, corresponding to the noise strength. This corresponds to the experimental observation of stochastic resonance in a periodic potential in the low-damping regime.
0212156v1
2007-05-03
Planar spin-transfer device with a dynamic polarizer
In planar nano-magnetic devices magnetization direction is kept close to a given plane by the large easy-plane magnetic anisotropy, for example by the shape anisotropy in a thin film. In this case magnetization shows effectively in-plane dynamics with only one angle required for its description. Moreover, the motion can become overdamped even for small values of Gilbert damping. We derive the equations of effective in-plane dynamics in the presence of spin-transfer torques. The simplifications achieved in the overdamped regime allow to study systems with several dynamic magnetic pieces (``free layers''). A transition from a spin-transfer device with a static polarizer to a device with two equivalent magnets is observed. When the size difference between the magnets is less than critical, the device does not exhibit switching, but goes directly into the ``windmill'' precession state.
0705.0406v1
2007-05-03
Effective attraction induced by repulsive interaction in a spin-transfer system
In magnetic systems with dominating easy-plane anisotropy the magnetization can be described by an effective one dimensional equation for the in-plane angle. Re-deriving this equation in the presence of spin-transfer torques, we obtain a description that allows for a more intuitive understanding of spintronic devices' operation and can serve as a tool for finding new dynamic regimes. A surprising prediction is obtained for a planar ``spin-flip transistor'': an unstable equilibrium point can be stabilized by a current induced torque that further repels the system from that point. Stabilization by repulsion happens due to the presence of dissipative environment and requires a Gilbert damping constant that is large enough to ensure overdamped dynamics at zero current.
0705.0508v1
2007-06-21
Spin pumping by a field-driven domain wall
We calculate the charge current in a metallic ferromagnet to first order in the time derivative of the magnetization direction. Irrespective of the microscopic details, the result can be expressed in terms of the conductivities of the majority and minority electrons and the non-adiabatic spin transfer torque parameter $\beta$. The general expression is evaluated for the specific case of a field-driven domain wall and for that case depends strongly on the ratio of $\beta$ and the Gilbert damping constant. These results may provide an experimental method to determine this ratio, which plays a crucial role for current-driven domain-wall motion.
0706.3160v3
2007-09-18
Theory of current-driven magnetization dynamics in inhomogeneous ferromagnets
We give a brief account of recent developments in the theoretical understanding of the interaction between electric currents and inhomogeneous ferromagnetic order parameters. We start by discussing the physical origin of the spin torques responsible for this interaction and construct a phenomenological description. We then consider the electric current-induced ferromagnetic instability and domain-wall motion. Finally, we present a microscopic justification of the phenomenological description of current-driven magnetization dynamics, with particular emphasis on the dissipative terms, the so-called Gilbert damping $\alpha$ and the $\beta$ component of the adiabatic current-driven torque.
0709.2937v2
2008-02-12
Temperature dependent magnetization dynamics of magnetic nanoparticles
Recent experimental and theoretical studies show that the switching behavior of magnetic nanoparticles can be well controlled by external time-dependent magnetic fields. In this work, we inspect theoretically the influence of the temperature and the magnetic anisotropy on the spin-dynamics and the switching properties of single domain magnetic nanoparticles (Stoner-particles). Our theoretical tools are the Landau-Lifshitz-Gilbert equation extended as to deal with finite temperatures within a Langevine framework. Physical quantities of interest are the minimum field amplitudes required for switching and the corresponding reversal times of the nanoparticle's magnetic moment. In particular, we contrast the cases of static and time-dependent external fields and analyze the influence of damping for a uniaxial and a cubic anisotropy.
0802.1740v1
2008-05-21
Non-equilibrium thermodynamic study of magnetization dynamics in the presence of spin-transfer torque
The dynamics of magnetization in the presence of spin-transfer torque was studied. We derived the equation for the motion of magnetization in the presence of a spin current by using the local equilibrium assumption in non-equilibrium thermodynamics. We show that, in the resultant equation, the ratio of the Gilbert damping constant, $\alpha$, and the coefficient, $\beta$, of the current-induced torque, called non-adiabatic torque, depends on the relaxation time of the fluctuating field $\tau_{c}$. The equality $\alpha=\beta$ holds when $\tau_c$ is very short compared to the time scale of magnetization dynamics. We apply our theory to current-induced magnetization reversal in magnetic multilayers and show that the switching time is a decreasing function of $\tau_{c}$.
0805.3306v1
2008-06-28
Theory of spin magnetohydrodynamics
We develop a phenomenological hydrodynamic theory of coherent magnetic precession coupled to electric currents. Exchange interaction between electron spin and collective magnetic texture produces two reciprocal effects: spin-transfer torque on the magnetic order parameter and the Berry-phase gauge field experienced by the itinerant electrons. The dissipative processes are governed by three coefficients: the ohmic resistance, Gilbert damping of the magnetization, and the "beta coefficient" describing viscous coupling between magnetic dynamics and electric current, which stems from spin mistracking of the magnetic order. We develop general magnetohydrodynamic equations and discuss the net dissipation produced by the coupled dynamics. The latter in particular allows us to determine a lower bound on the magnetic-texture resistivity.
0806.4656v2
2008-09-25
The theory of magnetic field induced domain-wall propagation in magnetic nanowires
A global picture of magnetic domain wall (DW) propagation in a nanowire driven by a magnetic field is obtained: A static DW cannot exist in a homogeneous magnetic nanowire when an external magnetic field is applied. Thus, a DW must vary with time under a static magnetic field. A moving DW must dissipate energy due to the Gilbert damping. As a result, the wire has to release its Zeeman energy through the DW propagation along the field direction. The DW propagation speed is proportional to the energy dissipation rate that is determined by the DW structure. An oscillatory DW motion, either the precession around the wire axis or the breath of DW width, should lead to the speed oscillation.
0809.4311v1
2008-10-08
Transverse spin diffusion in ferromagnets
We discuss the dissipative diffusion-type term of the form $\mathbf{m}\times\nabla^2\partial_t\mathbf{m}$ in the phenomenological Landau-Lifshitz 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 is identified in terms of the transverse spin conductivity, in analogy with the Einstein relation in the kinetic theory.
0810.1340v2
2008-10-16
Interaction of reed and acoustic resonator in clarinetlike systems
Sound emergence in clarinetlike instruments is investigated in terms of instability of the static regime. Various models of reed-bore coupling are considered, from the pioneering work of Wilson and Beavers ["Operating modes of the clarinet", J. Acoust. Soc. Am. 56, 653--658 (1974)] to more recent modeling including viscothermal bore losses and vena contracta at the reed inlet. The pressure threshold above which these models may oscillate as well as the frequency of oscillation at threshold are calculated. In addition to Wilson and Beavers' previous conclusions concerning the role of the reed damping in the selection of the register the instrument will play on, the influence of the reed motion induced flow is also emphasized, particularly its effect on playing frequencies, contributing to reduce discrepancies between Wilson and Beavers' experimental results and theory, despite discrepancies still remain concerning the pressure threshold. Finally, analytical approximations of the oscillating solution based on Fourier series expansion are obtained in the vicinity of the threshold of oscillation. This allows to emphasize the conditions which determine the nature of the bifurcation (direct or inverse) through which the note may emerge, with therefore important consequences on the musical playing performances.
0810.2870v1
2008-11-13
Intrinsic Coupling between Current and Domain Wall Motion in (Ga,Mn)As
We consider current-induced domain wall motion and, the reciprocal process, moving domain wall-induced current. The associated Onsager coefficients are expressed in terms of scattering matrices. Uncommonly, in (Ga,Mn)As, the effective Gilbert damping coefficient $\alpha_w$ and the effective out-of-plane spin transfer torque parameter $\beta_w$ are dominated by spin-orbit interaction in combination with scattering off the domain wall, and not scattering off extrinsic impurities. Numerical calculations give $\alpha_w \sim 0.01$ and $\beta_w \sim 1$ in dirty (Ga,Mn)As. The extraordinary large $\beta_w$ parameter allows experimental detection of current or voltage induced by domain wall motion in (Ga,Mn)As.
0811.2235v2
2008-11-21
Spin Transfer Torque as a Non-Conservative Pseudo-Field
In this paper we show that the spin transfer torque can be described by a pseudo magnetic field, proportional to the magnetic moment of the itinerant electrons that enters the Landau-Lifshitz-Gilbert equation in the same way as other external or internal magnetic fields. However, unlike an ordinary magnetic field, which is always conservative in nature, the spin torque induced pseudo field may have both conservative and non-conservative components. We further show that the magnetic moment of itinerant electrons develops an out-of-plane component only at non-equilibrium and this component is responsible for the Slonczewski type switching that acts against the damping and is always non-conservative. On the other hand, the in-plane components of the pseudo field exist both at equilibrium and out-of-equilibrium, and are responsible for the field like term. For tunnel based devices, this term results in lower switching current for anti-parallel (AP) to parallel (P) switching compared to P to AP, even when the torque magnitudes are completely symmetric with voltage.
0811.3472v1
2008-12-13
Non-Adiabatic Spin Transfer Torque in Real Materials
The motion of simple domain walls and of more complex magnetic textures in the presence of a transport current is described by the Landau-Lifshitz-Slonczewski (LLS) equations. Predictions of the LLS equations depend sensitively on the ratio between the dimensionless material parameter $\beta$ which characterizes non-adiabatic spin-transfer torques and the Gilbert damping parameter $\alpha$. This ratio has been variously estimated to be close to 0, close to 1, and large compared to 1. By identifying $\beta$ as the influence of a transport current on $\alpha$, we derive a concise, explicit and relatively simple expression which relates $\beta$ to the band structure and Bloch state lifetimes of a magnetic metal. Using this expression we demonstrate that intrinsic spin-orbit interactions lead to intra-band contributions to $\beta$ which are often dominant and can be (i) estimated with some confidence and (ii) interpreted using the "breathing Fermi surface" model.
0812.2570v1
2009-05-01
Spin excitations in a monolayer scanned by a magnetic tip
Energy dissipation via spin excitations is investigated for a hard ferromagnetic tip scanning a soft magnetic monolayer. We use the classical Heisenberg model with Landau-Lifshitz-Gilbert (LLG)-dynamics including a stochastic field representing finite temperatures. The friction force depends linearly on the velocity (provided it is small enough) for all temperatures. For low temperatures, the corresponding friction coefficient is proportional to the phenomenological damping constant of the LLG equation. This dependence is lost at high temperatures, where the friction coefficient decreases exponentially. These findings can be explained by properties of the spin polarization cloud dragged along with the tip.
0905.0112v2
2009-05-29
Ferromagnetic resonance linewidth in ultrathin films with perpendicular magnetic anisotropy
Transition metal ferromagnetic films with perpendicular magnetic anisotropy (PMA) have ferromagnetic resonance (FMR) linewidths that are one order of magnitude larger than soft magnetic materials, such as pure iron (Fe) and permalloy (NiFe) thin films. A broadband FMR setup has been used to investigate the origin of the enhanced linewidth in Ni$|$Co multilayer films with PMA. The FMR linewidth depends linearly on frequency for perpendicular applied fields and increases significantly when the magnetization is rotated into the film plane. Irradiation of the film with Helium ions decreases the PMA and the distribution of PMA parameters. This leads to a great reduction of the FMR linewidth for in-plane magnetization. These results suggest that fluctuations in PMA lead to a large two magnon scattering contribution to the linewidth for in-plane magnetization and establish that the Gilbert damping is enhanced in such materials ($\alpha \approx 0.04$, compared to $\alpha \approx 0.002$ for pure Fe).
0905.4779v2
2009-10-01
Spin motive forces and current fluctuations due to Brownian motion of domain walls
We compute the power spectrum of the noise in the current due to spin motive forces by a fluctuating domain wall. We find that the power spectrum of the noise in the current is colored, and depends on the Gilbert damping, the spin transfer torque parameter $\beta$, and the domain-wall pinning potential and magnetic anisotropy. We also determine the average current induced by the thermally-assisted motion of a domain wall that is driven by an external magnetic field. Our results suggest that measuring the power spectrum of the noise in the current in the presence of a domain wall may provide a new method for characterizing the current-to-domain-wall coupling in the system.
0910.0163v1
2009-10-08
Fast domain wall propagation under an optimal field pulse in magnetic nanowires
We investigate field-driven domain wall (DW) propagation in magnetic nanowires in the framework of the Landau-Lifshitz-Gilbert equation. We propose a new strategy to speed up the DW motion in a uniaxial magnetic nanowire by using an optimal space-dependent field pulse synchronized with the DW propagation. Depending on the damping parameter, the DW velocity can be increased by about two orders of magnitude compared the standard case of a static uniform field. Moreover, under the optimal field pulse, the change in total magnetic energy in the nanowire is proportional to the DW velocity, implying that rapid energy release is essential for fast DW propagation.
0910.1477v2
2009-11-24
Origin of adiabatic and non-adiabatic spin transfer torques in current-driven magnetic domain wall motion
A consistent theory to describe the correlated dynamics of quantum mechanical itinerant spins and semiclassical local magnetization is given. We consider the itinerant spins as quantum mechanical operators, whereas local moments are considered within classical Lagrangian formalism. By appropriately treating fluctuation space spanned by basis functions, including a zero-mode wave function, we construct coupled equations of motion for the collective coordinate of the center-of-mass motion and the localized zero-mode coordinate perpendicular to the domain wall plane. By solving them, we demonstrate that the correlated dynamics is understood through a hierarchy of two time scales: Boltzmann relaxation time when a non-adiabatic part of the spin-transfer torque appears, and Gilbert damping time when adiabatic part comes up.
0911.4628v1
2010-01-26
Strategies and tolerances of spin transfer torque switching
Schemes of switching nanomagnetic memories via the effect of spin torque with various polarizations of injected electrons are studied. Simulations based on macrospin and micromagnetic theories are performed and compared. We demonstrate that switching with perpendicularly polarized current by short pulses and free precession requires smaller time and energy than spin torque switching with collinear in plane spin polarization; it is also found to be superior to other kinds of memories. We study the tolerances of switching to the magnitude of current and pulse duration. An increased Gilbert damping is found to improve tolerances of perpendicular switching without increasing the threshold current, unlike in plane switching.
1001.4578v1
2010-03-31
Magnonic Crystal with Two-Dimensional Periodicity as a Waveguide for Spin Waves
We describe a simple method of including dissipation in the spin wave band structure of a periodic ferromagnetic composite, by solving the Landau-Lifshitz equation for the magnetization with the Gilbert damping term. We use this approach to calculate the band structure of square and triangular arrays of Ni nanocylinders embedded in an Fe host. The results show that there are certain bands and special directions in the Brillouin zone where the spin wave lifetime is increased by more than an order of magnitude above its average value. Thus, it may be possible to generate spin waves in such composites decay especially slowly, and propagate especially large distances, for certain frequencies and directions in ${\bf k}$-space.
1003.6092v1
2010-07-20
Precessing vortices and antivortices in ferromagnetic elements
A micromagnetic numerical study of the precessional motion of the vortex and antivortex states in soft ferromagnetic circular nanodots is presented using Landau-Lifshitz-Gilbert dynamics. For sufficiently small dot thickness and diameter, the vortex state is metastable and spirals toward the center of the dot when its initial displacement is smaller than a critical value. Otherwise, the vortex spirals away from the center and eventually exits the dot which remains in a state of in-plane magnetization (ground state). In contrast, the antivortex is always unstable and performs damped precession resulting in annihilation at the dot circumference. The vortex and antivortex frequencies of precession are compared with the response expected on the basis of Thiele's theory of collective coordinates. We also calculate the vortex restoring force with an explicit account of the magnetostatic and exchange interaction on the basis of the 'rigid' vortex and 'two-vortices side charges free' models and show that neither model explains the vortex translation mode eigenfrequency for nanodots of sufficiently small size.
1007.3508v1
2010-08-03
Determination of the spin-flip time in ferromagnetic SrRuO3 from time-resolved Kerr measurements
We report time-resolved Kerr effect measurements of magnetization dynamics in ferromagnetic SrRuO3. We observe that the demagnetization time slows substantially at temperatures within 15K of the Curie temperature, which is ~ 150K. We analyze the data with a phenomenological model that relates the demagnetization time to the spin flip time. In agreement with our observations the model yields a demagnetization time that is inversely proportional to T-Tc. We also make a direct comparison of the spin flip rate and the Gilbert damping coefficient showing that their ratio very close to kBTc, indicating a common origin for these phenomena.
1008.0674v1
2010-10-07
Power optimization for domain wall motion in ferromagnetic nanowires
The current mediated domain-wall dynamics in a thin ferromagnetic wire is investigated. We derive the effective equations of motion of the domain wall. They are used to study the possibility to optimize the power supplied by electric current for the motion of domain walls in a nanowire. We show that a certain resonant time-dependent current moving a domain wall can significantly reduce the Joule heating in the wire, and thus it can lead to a novel proposal for the most energy efficient memory devices. We discuss how Gilbert damping, non-adiabatic spin transfer torque, and the presence of Dzyaloshinskii-Moriya interaction can effect this power optimization.
1010.1537v1
2011-03-30
Spin motive forces due to magnetic vortices and domain walls
We study spin motive forces, i.e, spin-dependent forces, and voltages induced by time-dependent magnetization textures, for moving magnetic vortices and domain walls. First, we consider the voltage generated by a one-dimensional field-driven domain wall. Next, we perform detailed calculations on field-driven vortex domain walls. We find that the results for the voltage as a function of magnetic field differ between the one-dimensional and vortex domain wall. For the experimentally relevant case of a vortex domain wall, the dependence of voltage on field around Walker breakdown depends qualitatively on the ratio of the so-called $\beta$-parameter to the Gilbert damping constant, and thus provides a way to determine this ratio experimentally. We also consider vortices on a magnetic disk in the presence of an AC magnetic field. In this case, the phase difference between field and voltage on the edge is determined by the $\beta$ parameter, providing another experimental method to determine this quantity.
1103.5858v3
2011-07-04
Influence of randomness and retardation on the FMR-linewidth
The theory predicts that the spin-wave lifetime $\tau_L$ and the linewidth of ferromagnetic resonance $\Delta B$ can be governed by random fields and spatial memory. To that aim the effective field around which the magnetic moments perform a precession is superimposed by a stochastic time dependent magnetic field with finite correlation time. The magnetization dynamics is altered by inclusion of a spatial memory effect monitoring a non-local interaction of size $\xi$. The underlying Landau-Lifshitz-Gilbert equation (LLG) is modified accordingly. The stochastic LLG is equivalent to a Fokker-Planck equation which enables to calculate the mean values of the magnetization vector. Within the spin-wave approximation we present an analytical solution for the excitation energy and its damping. The lifetime and the linewidth are analyzed depending on the strength of the random field $D$ and its correlation time $\tau_c$ as well as the retardation strength $\Gamma_0$ and the size $\xi$. Whereas $\tau_L$ decreases with increasing $D$, retardation strength $\Gamma_0$ and $\tau_c$, the lifetime is enhanced for growing width $\xi$ of the spatial retardation kernel. In the same manner we calculate the experimentally measurable linewidth $\Delta B$ is increased strongly when the correlation time $\tau_c$ ranges in the nanosecond interval.
1107.0638v1
2011-11-18
Charge and Spin Transport in Magnetic Tunnel Junctions: Microscopic Theory
We study the charge and spin currents passing through a magnetic tunnel junction (MTJ) on the basis of a tight-binding model. The currents are evaluated perturbatively with respect to the tunnel Hamiltonian. The charge current has the form $A[\bm M_1(t)\times\dot{\bm M}_1(t)]\cdot\bm M_2+B\dot{\bm M}_1(t)\cdot\bm M_2$, where $\bm M_1(t)$ and $\bm M_2$ denote the directions of the magnetization in the free layer and fixed layer, respectively. The constant $A$ vanishes when one or both layers are insulators, {while the constant $B$ disappears when both layers are insulators or the same ferromagnets.} The first term in the expression for charge current represents dissipation driven by the effective electric field induced by the dynamic magnetization. In addition, from an investigation of the spin current, we obtain the microscopic expression for the enhanced Gilbert damping constant $\varDelta \alpha$. We show that $\varDelta\alpha$ is proportional to the tunnel conductance and depends on the bias voltage.
1111.4295v2
2012-01-17
Magnetic vortex echoes: application to the study of arrays of magnetic nanostructures
We propose the use of the gyrotropic motion of vortex cores in nanomagnets to produce a magnetic echo, analogous to the spin echo in NMR. This echo occurs when an array of nanomagnets, e.g., nanodisks, is magnetized with an in-plane (xy) field, and after a time \tau a field pulse inverts the core magnetization; the echo is a peak in M_{xy} at t=2\tau. Its relaxation times depend on the inhomogeneity, on the interaction between the nanodots and on the Gilbert damping constant \alpha. Its feasibility is demonstrated using micromagnetic simulation. To illustrate an application of the echoes, we have determined the inhomogeneity and measured the magnetic interaction in an array of nanodisks separated by a distance d, finding a d^{-n} dependence, with n\approx 4.
1201.3553v1
2012-02-15
Current-induced motion of a transverse magnetic domain wall in the presence of spin Hall effect
We theoretically study the current-induced dynamics of a transverse magnetic domain wall in bi-layer nanowires consisting of a ferromagnet on top of a nonmagnet having strong spin-orbit coupling. Domain wall dynamics is characterized by two threshold current densities, $J_{th}^{WB}$ and $J_{th}^{REV}$, where $J_{th}^{WB}$ is a threshold for the chirality switching of the domain wall and $J_{th}^{REV}$ is another threshold for the reversed domain wall motion caused by spin Hall effect. Domain walls with a certain chirality may move opposite to the electron-flow direction with high speed in the current range $J_{th}^{REV} < J < J_{th}^{WB}$ for the system designed to satisfy the conditions $J_{th}^{WB} > J_{th}^{REV}$ and \alpha > \beta, where \alpha is the Gilbert damping constant and \beta is the nonadiabaticity of spin torque. Micromagnetic simulations confirm the validity of analytical results.
1202.3450v1
2012-04-23
Rotating skyrmion lattices by spin torques and field or temperature gradients
Chiral magnets like MnSi form lattices of skyrmions, i.e. magnetic whirls, which react sensitively to small electric currents j above a critical current density jc. The interplay of these currents with tiny gradients of either the magnetic field or the temperature can induce a rotation of the magnetic pattern for j>jc. Either a rotation by a finite angle of up to 15 degree or -- for larger gradients -- a continuous rotation with a finite angular velocity is induced. We use Landau-Lifshitz-Gilbert equations extended by extra damping terms in combination with a phenomenological treatment of pinning forces to develop a theory of the relevant rotational torques. Experimental neutron scattering data on the angular distribution of skyrmion lattices suggests that continuously rotating domains are easy to obtain in the presence of remarkably small currents and temperature gradients.
1204.5051v1
2012-07-09
Thermal vortex dynamics in thin circular ferromagnetic nanodisks
The dynamics of gyrotropic vortex motion in a thin circular nanodisk of soft ferromagnetic material is considered. The demagnetization field is calculated using two-dimensional Green's functions for the thin film problem and fast Fourier transforms. At zero temperature, the dynamics of the Landau-Lifshitz-Gilbert equation is simulated using fourth order Runge-Kutta integration. Pure vortex initial conditions at a desired position are obtained with a Lagrange multipliers constraint. These methods give accurate estimates of the vortex restoring force constant $k_F$ and gyrotropic frequency, showing that the vortex core motion is described by the Thiele equation to very high precision. At finite temperature, the second order Heun algorithm is applied to the Langevin dynamical equation with thermal noise and damping. A spontaneous gyrotropic motion takes place without the application of an external magnetic field, driven only by thermal fluctuations. The statistics of the vortex radial position and rotational velocity are described with Boltzmann distributions determined by $k_F$ and by a vortex gyrotropic mass $m_G=G^2/k_F$, respectively, where $G$ is the vortex gyrovector.
1207.2192v2
2013-02-19
Chirality Sensitive Domain Wall Motion in Spin-Orbit Coupled Ferromagnets
Using the Lagrangian formalism, we solve analytically the equations of motion for current-induced domain-wall dynamics in a ferromagnet with Rashba spin-orbit coupling. An exact solution for the domain wall velocity is provided, including the effect of non-equilibrium conduction electron spin-density, Gilbert damping, and the Rashba interaction parameter. We demonstrate explicitly that the influence of spin-orbit interaction can be qualitatively different from the role of non-adiabatic spin-torque in the sense that the former is sensitive to the chirality of the domain wall whereas the latter is not: the domain wall velocity shows a reentrant behavior upon changing the chirality of the domain wall. This could be used to experimentally distinguish between the spin-orbit and non-adiabatic contribution to the wall speed. A quantitative estimate for the attainable domain wall velocity is given, based on an experimentally relevant set of parameters for the system.
1302.4744v1
2013-12-17
Control of the in-plane anisotropy in off-stoichiometric NiMnSb
NiMnSb is a ferromagnetic half-metal which, because of its rich anisotropy and very low Gilbert damping, is a promising candidate for applications in information technologies. We have investigated the in-plane anisotropy properties of thin, MBE-grown NiMnSb films as a function of their Mn concentration. Using ferromagnetic resonance (FMR) to determine the uniaxial and four-fold anisotropy fields, 2KU/Ms and 2K1/Ms, we find that a small variation in composition is sufficient to change the film from primarily four-fold to primarily uniaxial behavior, allowing for continuous tuning of the anisotropy. This provides valuable flexibility in designing new device geometries.
1312.4781v2
2014-05-09
Current-induced magnetization dynamics in two magnetic insulators separated by a normal metal
We study the dynamics of spin valves consisting of two layers of magnetic insulators separated by a normal metal in the macrospin model. A current through the spacer generates a spin Hall current that can actuate the magnetization via the spin-transfer torque. We derive expressions for the effective Gilbert damping and the critical currents for the onset of magnetization dynamics including the effects of spin pumping that can be tested by ferromagnetic resonance experiments. The current generates an amplitude asymmetry between the in-phase and out-of-phase modes. We briefly discuss superlattices of metals and magnetic insulators.
1405.2267v1
2014-05-25
Spin Hall phenomenology of magnetic dynamics
We study the role of spin-orbit interactions in the coupled magnetoelectric dynamics of a ferromagnetic film coated with an electrical conductor. While the main thrust of this work is phenomenological, several popular simple models are considered microscopically in some detail, including Rashba and Dirac two-dimensional electron gases coupled to a magnetic insulator, as well as a diffusive spin Hall system. We focus on the long-wavelength magnetic dynamics that experiences current-induced torques and produces fictitious electromotive forces. Our phenomenology provides a suitable framework for analyzing experiments on current-induced magnetic dynamics and reciprocal charge pumping, including the effects of magnetoresistance and Gilbert-damping anisotropies, without a need to resort to any microscopic considerations or modeling. Finally, some remarks are made regarding the interplay of spin-orbit interactions and magnetic textures.
1405.6354v2
2014-08-21
Brownian motion of massive skyrmions forced by spin polarized currents
We report on the thermal effects on the motion of current-driven massive magnetic skyrmions. The reduced equation for the motion of skyrmion has the form of a stochastic generalized Thiele's equation. We propose an ansatz for the magnetization texture of a non-rigid single skyrmion that depends linearly with the velocity. By utilizing this ansatz it is is found that the mass of skyrmion is closely related to intrinsic skyrmion parameters, such as Gilbert damping, skyrmion-charge and dissipative force. We have found an exact expression for the average drift velocity as well as the mean-square velocity of the skyrmion. The longitudinal and transverse mobility of skyrmions for small spin-velocity of electrons is also determined and found to be independent of the skyrmion mass.
1408.4861v2
2014-11-11
Capturing of a Magnetic Skyrmion with a Hole
Magnetic whirls in chiral magnets, so-called skyrmions, can be manipulated by ultrasmall current densities. Here we study both analytically and numerically the interactions of a single skyrmion in two dimensions with a small hole in the magnetic layer. Results from micromagnetic simulations are in good agreement with effective equations of motion obtained from a generalization of the Thiele approach. Skyrmion-defect interactions are described by an effective potential with both repulsive and attractive components. For small current densities a previously pinned skyrmion stays pinned whereas an unpinned skyrmion moves around the impurities and never gets captured. For higher current densities, j_c1 < j < j_c2, however, single holes are able to capture moving skyrmions. The maximal cross section is proportional to the skyrmion radius and to Sqrt(alpha), where alpha is the Gilbert damping. For j > j_c2 all skyrmions are depinned. Small changes of the magnetic field strongly change the pinning properties, one can even reach a regime without pinning, j_c2=0. We also show that a small density of holes can effectively accelerate the motion of the skyrmion and introduce a Hall effect for the skyrmion.
1411.2857v1
2014-12-01
Dissipation due to pure spin-current generated by spin pumping
Based on spin-dependent transport theory and thermodynamics, we develop a generalized theory of the Joule heating in the presence of a spin current. Along with the conventional Joule heating consisting of an electric current and electrochemical potential, it is found that the spin current and spin accumulation give an additional dissipation because the spin-dependent scatterings inside bulk and ferromagnetic/nonmagnetic interface lead to a change of entropy. The theory is applied to investigate the dissipation due to pure spin-current generated by spin pumping across a ferromagnetic/nonmagnetic/ferromagnetic multilayer. The dissipation arises from an interface because the spin pumping is a transfer of both the spin angular momentum and the energy from the ferromagnet to conduction electrons near the interface. It is found that the dissipation is proportional to the enhancement of the Gilbert damping constant by spin pumping.
1412.0688v1
2015-01-30
Head-to-Head Domain Wall Structures in Wide Permalloy Strips
We analyze the equilibrium micromagnetic domain wall structures encountered in Permalloy strips of a wide range of thicknesses and widths, with strip widths up to several micrometers. By performing an extensive set of micromagnetic simulations, we show that the equilibrium phase diagram of the domain wall structures exhibits in addition to the previously found structures (symmetric and asymmetric transverse walls, vortex wall) also double vortex and triple vortex domain walls for large enough strip widths and thicknesses. Also several metastable domain wall structures are found for wide and/or thick strips. We discuss the details of the relaxation process from random magnetization initial states towards the stable domain wall structure, and show that our results are robust with respect to changes of e.g. the magnitude of the Gilbert damping constant and details of the initial conditions.
1501.07731v1
2015-02-19
Characterization of spin relaxation anisotropy in Co using spin pumping
Ferromagnets are believed to exhibit strongly anisotropic spin relaxation, with relaxation lengths for spin longitudinal to magnetization significantly longer than those for spin transverse to magnetization. Here we characterize the anisotropy of spin relaxation in Co using the spin pumping contribution to Gilbert damping in noncollinearly magnetized Py$_{1-x}$Cu$_{x}$/Cu/Co trilayer structures. The static magnetization angle between Py$_{1-x}$Cu$_{x}$ and Co, adjusted under field bias perpendicular to film planes, controls the projections of longitudinal and transverse spin current pumped from Py$_{1-x}$Cu$_{x}$ into Co. We find nearly isotropic absorption of pure spin current in Co using this technique; fits to a diffusive transport model yield the longitudinal spin relaxation length $< 2$ nm in Co. The longitudinal spin relaxation lengths found are an order of magnitude smaller than those determined by current-perpendicular-to-planes giant magnetoresistance measurements, but comparable with transverse spin relaxation lengths in Co determined by spin pumping.
1502.05687v3
2015-03-26
Thermophoresis of an Antiferromagnetic Soliton
We study dynamics of an antiferromagnetic soliton under a temperature gradient. To this end, we start by phenomenologically constructing the stochastic Landau-Lifshitz-Gilbert equation for an antiferromagnet with the aid of the fluctuation-dissipation theorem. We then derive the Langevin equation for the soliton's center of mass by the collective coordinate approach. An antiferromagentic soliton behaves as a classical massive particle immersed in a viscous medium. By considering a thermodynamic ensemble of solitons, we obtain the Fokker-Planck equation, from which we extract the average drift velocity of a soliton. The diffusion coefficient is inversely proportional to a small damping constant $\alpha$, which can yield a drift velocity of tens of m/s under a temperature gradient of $1$ K/mm for a domain wall in an easy-axis antiferromagnetic wire with $\alpha \sim 10^{-4}$.
1503.07854v2
2015-04-01
Multiscale modeling of ultrafast element-specific magnetization dynamics of ferromagnetic alloys
A hierarchical multiscale approach to model the magnetization dynamics of ferromagnetic ran- dom alloys is presented. First-principles calculations of the Heisenberg exchange integrals are linked to atomistic spin models based upon the stochastic Landau-Lifshitz-Gilbert (LLG) equation to calculate temperature-dependent parameters (e.g., effective exchange interactions, damping param- eters). These parameters are subsequently used in the Landau-Lifshitz-Bloch (LLB) model for multi-sublattice magnets to calculate numerically and analytically the ultrafast demagnetization times. The developed multiscale method is applied here to FeNi (permalloy) as well as to copper- doped FeNi alloys. We find that after an ultrafast heat pulse the Ni sublattice demagnetizes faster than the Fe sublattice for the here-studied FeNi-based alloys.
1504.00199v1
2015-05-04
High-topological-number magnetic skyrmions and topologically protected dissipative structure
The magnetic skyrmion with the topological number of unity ($Q=1$) is a well-known nanometric swirling spin structure in the nonlinear $\sigma$ model with the Dzyaloshinskii-Moriya interaction. Here, we show that magnetic skyrmion with the topological number of two ($Q=2$) can be created and stabilized by applying vertical spin-polarized current though it cannot exist as a static stable excitation. Magnetic skyrmion with $Q=2$ is a nonequilibrium dynamic object, subsisting on a balance between the energy injection from the current and the energy dissipation by the Gilbert damping. Once it is created, it becomes a topologically protected object against fluctuations of various variables including the injected current itself. Hence, we may call it a topologically protected dissipative structure. We also elucidate the nucleation and destruction mechanisms of the magnetic skyrmion with $Q=2$ by studying the evolutions of the magnetization distribution, the topological charge density as well as the energy density. Our results will be useful for the study of the nontrivial topology of magnetic skyrmions with higher topological numbers.
1505.00522v2
2015-08-06
Large spin-wave bullet in a ferrimagnetic insulator driven by spin Hall effect
Due to its transverse nature, spin Hall effects (SHE) provide the possibility to excite and detect spin currents and magnetization dynamics even in magnetic insulators. Magnetic insulators are outstanding materials for the investigation of nonlinear phenomena and for novel low power spintronics applications because of their extremely low Gilbert damping. Here, we report on the direct imaging of electrically driven spin-torque ferromagnetic resonance (ST-FMR) in the ferrimagnetic insulator Y$_3$Fe$_5$O$_{12}$ based on the excitation and detection by SHEs. The driven spin dynamics in Y$_3$Fe$_5$O$_{12}$ is directly imaged by spatially-resolved microfocused Brillouin light scattering (BLS) spectroscopy. Previously, ST-FMR experiments assumed a uniform precession across the sample, which is not valid in our measurements. A strong spin-wave localization in the center of the sample is observed indicating the formation of a nonlinear, self-localized spin-wave `bullet'.
1508.01427v1
2015-12-02
Bose-Einstein Condensation of Magnons Pumped by the Bulk Spin Seebeck Effect
We propose inducing Bose-Einstein condensation of magnons in a magnetic insulator by a heat flow oriented toward its boundary. At a critical heat flux, the oversaturated thermal gas of magnons accumulated at the boundary precipitates the condensate, which then grows gradually as the thermal bias is dialed up further. The thermal magnons thus pumped by the magnonic bulk (spin) Seebeck effect must generally overcome both the local Gilbert damping associated with the coherent magnetic dynamics as well as the radiative spin-wave losses toward the magnetic bulk, in order to achieve the threshold of condensation. We quantitatively estimate the requisite bias in the case of the ferrimagnetic yttrium iron garnet, discuss different physical regimes of condensation, and contrast it with the competing (so-called Doppler-shift) bulk instability.
1512.00557v1
2016-01-10
Interfacial Dzyaloshinskii-Moriya interaction, surface anisotropy energy,and spin pumping at spin orbit coupled Ir/Co interface
The interfacial Dzyaloshinskii-Moriya interaction (iDMI), surface anisotropy energy, and spin pumping at the Ir/Co interface are experimentally investigated by performing Brillouin light scattering. Contrary to previous reports, we suggest that the sign of the iDMI at the Ir/Co interface is the same as in the case of the Pt/Co interface. We also find that the magnitude of the iDMI energy density is relatively smaller than in the case of the Pt/Co interface, despite the large strong spin-orbit coupling (SOC) of Ir. The saturation magnetization and the perpendicular magnetic anisotropy (PMA) energy are significantly improved due to a strong SOC. Our findings suggest that an SOC in an Ir/Co system behaves in different ways for iDMI and PMA. Finally, we determine the spin pumping effect at the Ir/Co interface, and it increases the Gilbert damping constant from 0.012 to 0.024 for 1.5 nmthick Co.
1601.02210v3
2016-02-23
Relaxation of a classical spin coupled to a strongly correlated electron system
A classical spin which is antiferromagnetically coupled to a system of strongly correlated conduction electrons is shown to exhibit unconventional real-time dynamics which cannot be described by Gilbert damping. Depending on the strength of the local Coulomb interaction, the two main electronic dissipation channels, transport of excitations via correlated hopping and via excitations of correlation-induced magnetic moments, become active on largely different time scales. We demonstrate that this can lead to a prethermalization scenario which so far has been observed in purely electronic systems only and which is governed here by proximity to the divergent magnetic time scale in the infinite-U limit.
1602.07317v2
2016-04-24
Coupled Spin-Light dynamics in Cavity Optomagnonics
Experiments during the past two years have shown strong resonant photon-magnon coupling in microwave cavities, while coupling in the optical regime was demonstrated very recently for the first time. Unlike with microwaves, the coupling in optical cavities is parametric, akin to optomechanical systems. This line of research promises to evolve into a new field of optomagnonics, aimed at the coherent manipulation of elementary magnetic excitations by optical means. In this work we derive the microscopic optomagnonic Hamiltonian. In the linear regime the system reduces to the well-known optomechanical case, with remarkably large coupling. Going beyond that, we study the optically induced nonlinear classical dynamics of a macrospin. In the fast cavity regime we obtain an effective equation of motion for the spin and show that the light field induces a dissipative term reminiscent of Gilbert damping. The induced dissipation coefficient however can change sign on the Bloch sphere, giving rise to self-sustained oscillations. When the full dynamics of the system is considered, the system can enter a chaotic regime by successive period doubling of the oscillations.
1604.07053v3
2016-05-12
Classical limit of Rabi nutations in spins of ferromagnets
Rabi oscillations describe the interaction of a two-level system with a rotating electromagnetic field. As such, they serve as the principle method for manipulating quantum bits. By using a combination of femtosecond laser pulses and microwave excitations, we have observed the classical form of Rabi nutations in a ferromagnetic system whose equations of motion mirror the case of a precessing quantum two-level system. Key to our experiments is the selection of a subset of spins that is in resonance with the microwave excitation and whose coherence time is thereby extended. Taking advantage of Gilbert damping, the relaxation times are further increased such that mode-locking takes place. The observation of such Rabi nutations is the first step towards potential applications based on phase-coherent spin manipulation in ferromagnets.
1605.03996v1
2016-05-21
Landau-Lifshitz theory of the magnon-drag thermopower
Metallic ferromagnets subjected to a temperature gradient exhibit a magnonic drag of the electric current. We address this problem by solving a stochastic Landau-Lifshitz equation to calculate the magnon-drag thermopower. The long-wavelength magnetic dynamics result in two contributions to the electromotive force acting on electrons: (1) An adiabatic Berry-phase force related to the solid angle subtended by the magnetic precession and (2) a dissipative correction thereof, which is rooted microscopically in the spin-dephasing scattering. The first contribution results in a net force pushing the electrons towards the hot side, while the second contribution drags electrons towards the cold side, i.e., in the direction of the magnonic drift. The ratio between the two forces is proportional to the ratio between the Gilbert damping coefficient $\alpha$ and the coefficient $\beta$ parametrizing the dissipative contribution to the electromotive force.
1605.06578v1
2016-06-07
The temperature dependence of FeRh's transport properties
The finite-temperature transport properties of FeRh compounds are investigated by first-principles Density Functional Theory-based calculations. The focus is on the behavior of the longitudinal resistivity with rising temperature, which exhibits an abrupt decrease at the metamagnetic transition point, $T = T_m$ between ferro- and antiferromagnetic phases. A detailed electronic structure investigation for $T \geq 0$ K explains this feature and demonstrates the important role of (i) the difference of the electronic structure at the Fermi level between the two magnetically ordered states and (ii) the different degree of thermally induced magnetic disorder in the vicinity of $T_m$, giving different contributions to the resistivity. To support these conclusions, we also describe the temperature dependence of the spin-orbit induced anomalous Hall resistivity and Gilbert damping parameter. For the various response quantities considered the impact of thermal lattice vibrations and spin fluctuations on their temperature dependence is investigated in detail. Comparison with corresponding experimental data finds in general a very good agreement.
1606.02072v1
2016-09-05
Coarsening dynamics of topological defects in thin Permalloy films
We study the dynamics of topological defects in the magnetic texture of rectangular Permalloy thin film elements during relaxation from random magnetization initial states. Our full micromagnetic simulations reveal complex defect dynamics during relaxation towards the stable Landau closure domain pattern, manifested as temporal power-law decay, with a system-size dependent cut-off time, of various quantities. These include the energy density of the system, and the number densities of the different kinds of topological defects present in the system. The related power-law exponents assume non-trivial values, and are found to be different for the different defect types. The exponents are robust against a moderate increase in the Gilbert damping constant and introduction of quenched structural disorder. We discuss details of the processes allowed by conservation of the winding number of the defects, underlying their complex coarsening dynamics.
1609.01094v1
2016-09-27
Anomalous Feedback and Negative Domain Wall Resistance
Magnetic induction can be regarded as a negative feedback effect, where the motive-force opposes the change of magnetic flux that generates the motive-force. In artificial electromagnetics emerging from spintronics, however, this is not necessarily the case. By studying the current-induced domain wall dynamics in a cylindrical nanowire, we show that the spin motive-force exerting on electrons can either oppose or support the applied current that drives the domain wall. The switching into the anomalous feedback regime occurs when the strength of the dissipative torque {\beta} is about twice the value of the Gilbert damping constant {\alpha}. The anomalous feedback manifests as a negative domain wall resistance, which has an analogy with the water turbine.
1609.08250v1
2016-10-04
Magnetomechanical coupling and ferromagnetic resonance in magnetic nanoparticles
We address the theory of the coupled lattice and magnetization dynamics of freely suspended single-domain nanoparticles. Magnetic anisotropy generates low-frequency satellite peaks in the microwave absorption spectrum and a blueshift of the ferromagnetic resonance (FMR) frequency. The low-frequency resonances are very sharp with maxima exceeding that of the FMR, because their magnetic and mechanical precessions are locked, thereby suppressing Gilbert damping. Magnetic nanoparticles can operate as nearly ideal motors that convert electromagnetic into mechanical energy. The Barnett/Einstein-de Haas effect is significant even in the absence of a net rotation.
1610.01072v2
2016-10-05
Finite-dimensional colored fluctuation-dissipation theorem for spin systems
When nano-magnets are coupled to random external sources, their magnetization becomes a random variable, whose properties are defined by an induced probability density, that can be reconstructed from its moments, using the Langevin equation, for mapping the noise to the dynamical degrees of freedom. When the spin dynamics is discretized in time, a general fluctuation-dissipation theorem, valid for non-Markovian noise, can be established, even when zero modes are present. We discuss the subtleties that arise, when Gilbert damping is present and the mapping between noise and spin degrees of freedom is non--linear.
1610.01622v1