A unified description of collective magnetic excitations
Benjamin W. Zingsem, Michael Winklhofer, Ralf Meckenstock, Michael Farle
AA unified description of collective magnetic excitations
Benjamin W. Zingsem,
1, 2
Michael Winklhofer,
1, 3
Ralf Meckenstock, and Michael Farle
1, 4 Faculty of Physics and Center for Nanointegration (CENIDE),University Duisburg-Essen, 47057 Duisburg, Germany Ernst Ruska-Centre for Microscopy and Spectroscopywith Electrons and Peter Grünberg Institute,Forschungszentrum Jülich GmbH, 52425 Jülich, Germany IBU/School of Mathematics and Science, University of Oldenburg,Carl-von-Ossietzky-Strasse 9-11, D-26129 Oldenburg, Germany Center for Functionalized Magnetic Materials,Immanuel Kant Baltic Federal University,236041 Kaliningrad, Russian Federation
Abstract
In this work, we define a set of analytic tools to describe the dynamic response of the magneti-zation to small perturbations, which can be used on its own or in combination with micromagneticsimulations and does not require saturation. We present a general analytic description of the fer-romagnetic high frequency susceptibility tensor to describe angular as well as frequency dependentferromagnetic resonance spectra and account for asymmetries in the line shape. Furthermore, weexpand this model to reciprocal space and show how it describes the magnon dispersion. Finally wesuggest a trajectory dependent solving tool to describe the equilibrium states of the magnetization. a r X i v : . [ c ond - m a t . o t h e r] A p r . INTRODUCTION Many solutions of the Ferromagnetic high frequency susceptibility (Polder-) tensor havebeen formulated. Mostly these solutions represent simplified versions to suit particular prob-lems, such as a certain energy landscape, a certain kind of coupling or a specific symmetry.In this work we formulate a generalized linearization of the Landau-Lifshitz-Gilbert equa-tions (LLG), which does not require symmetry assumptions and is applicable regardless ofthe coupling as well as the types of damping present in the system. It allows one to startwith a general formulation of the free energy density of ferromagnets, including all magneticinderactions which might be present in the magnetic material, e.g. exchange, dipole-dipole,Dzyaloshinskii-Moriya interaction, anisotropies, etc. and can be expanded to antiferro-magnetis and multilayer artificial antiferromagnets in the usual way . The conventionalapproaches mostly involve solving large systems of equations, linearizing at different pointsin the calculation in order to formulate the high frequency susceptibility . This is avoidedhere, by applying a straight forward linearization through series expansion of the LLG.Furthermore this algorithm is formulated to cover the entire magnon dispersion, includingferromagnetic resonance modes as well as traveling waves with non-zero wave-vectors.In the second part we present a model that can be used to calculate the equilibriumorientations of the magnetization, using an algorithm that closely resembles the actual mea-surement procedures used in ferromagnetic resonance measurements. Following a gradientof the energy landscape imposed on the magnetization, this model can be used to describemeta-stable and stable equilibrium states of the magnetization even for fields that are notapplied along symmetry directions.Neglecting thermal fluctuations, a combination of those models can be used to makeaccurate predictions about the magnetodynamic properties of ferromagnetic systems. II. ANALYTIC MODELA. The ferromagnetic high frequency susceptibility tensor
In the derivation presented here we assume a system that is described by one macro-spin (cid:126)M which is subjected to one effective magnetic field (cid:126)B yielding one high frequency2usceptibility tensor χ hf . This model therefore as derived here is designed to describe a fullysaturated sample. It is not limited to a single magnetization though and can be applied toa set of macro-spins where the local effective magnetic field is known at each site. In thatcase the total high frequency susceptibility would be given as χ = (cid:80) n χ n where χ n is the highfrequency susceptibility of the n th macro-spin (cid:126)M n due to the field (cid:126)B n it is exposed to. Thiscan be used for non saturated systems and samples with inhomogeneous magnetization ormagnetic nanoparticle configurations.In order to derive the full tensor we start from the Landau-Lifshitz-Gilbert Equation 1using the Polder-Ansatz as shown in eq. 2 (cid:126)L := − γ (cid:126)M × (cid:126)B − αM (cid:126)M × ˙ (cid:126)M − ˙ (cid:126)M = 0 (1) (cid:126)M ( t ) := (cid:126)M ( M, θ M , ϕ M ) + (cid:126)m exp ( ıωt ) (cid:126)B ( t ) := (cid:126)B ( B, θ B , ϕ B ) + (cid:126)b exp ( ıωt ) (2)Considering the dynamic excitation and response quantities (cid:126)m and (cid:126)b to be sufficiently small,the ferromagnetic high-frequency susceptibility χ hf can be expressed as a linear tensor (cid:126)m = χ hf · (cid:126)b (3)where linear means, that χ hf does not depend on (cid:126)m and (cid:126)b . This is usually the case for mi-crowave fields (cid:126) (cid:107) b (cid:107) < . To obtain the magnetic flux that the magnetization is exposedto, we consider the magnetic contribution to the free energy per unit volume F (cid:16) (cid:126)B appl , (cid:126)M (cid:17) where (cid:126)B appl corresponds to the applied magnetic field and (cid:126)M is the magnetization vector asdiscussed in the literature (See for example ).The Helmholtz free energy density F usuallycontains an anisotropic contribution due to the crystal lattice, particularly spin orbit inter-action, as well as several other contributions that arise from surfaces/interfaces, the shapeof the sample and the Zeeman-Energy. In this generalized approach the nature of thesemagnetic energies almost does not matter. The only necessary requirement is that the first3nd second derivatives used in eq. 4 exist. The total magnetic flux is then given as (cid:126)B ( t ) = ∇ (cid:126)M F (cid:16) (cid:126)B appl , (cid:126)M (cid:17) + J (cid:126)M (cid:16) ∇ (cid:126)M F (cid:16) (cid:126)B appl , (cid:126)M (cid:17)(cid:17) · (cid:126)m exp ( ıωt )+ (cid:126)b exp ( ıωt ) (4)where ∇ (cid:126)M F (cid:16) (cid:126)B appl , (cid:126)M (cid:17) is the anisotropy-field and J (cid:126)M (cid:16) ∇ (cid:126)M F (cid:16) (cid:126)B appl , (cid:126)M (cid:17)(cid:17) the responsefunction that accounts for a field caused by a precessing (cid:126)m , where ∇ (cid:126)M is the gradient in (cid:126)M and J (cid:126)M the Jacobian matrix in (cid:126)M . Using this we can now go back to eq. 1 and obtain (cid:126)L → (cid:126)L (cid:16) (cid:126)b, (cid:126)m (cid:17) = − γ (cid:126)M ( t ) × (cid:126)B ( t ) − αM (cid:126)M ( t ) × ˙ (cid:126)M ( t ) − ˙ M ( t ) ! = 0 ∀ t (5)which defines the hyper-plane in which all dynamic motion of the magnetization takes place.Since (cid:126)m and (cid:126)b are small, as defined in 3 we can now approximate (cid:126)L (cid:16) (cid:126)b, (cid:126)m (cid:17) by using a Taylor-expansion around (cid:126)L (cid:16) (cid:126)b = (cid:126) , (cid:126)m = (cid:126) (cid:17) to obtain (cid:126)L (cid:16) (cid:126)b, (cid:126)m (cid:17) ≈ (cid:126)L (cid:16) (cid:126) , (cid:126) (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) (cid:126) + J (cid:126)b, (cid:126)m · ( b x , b y , b z , m x , m y , m z ) (cid:62) (6)This leads to the system of equations 7, (cid:126) ! = J (cid:126)b, (cid:126)m · ( b x , b y , b z , m x , m y , m z ) (cid:62) (7)where J (cid:126)b, (cid:126)m = J ( b x ,b y ,b z ,m x ,m y ,m z ) T (cid:16) L (cid:16) (cid:126)b, (cid:126)m (cid:17)(cid:17) is the Jacobian matrix of (cid:126)L in (cid:126)b and (cid:126)m . Eq. 7can then be further decomposed into (cid:126) J (cid:126)b · (cid:126)b + J (cid:126)m · (cid:126)m(cid:126)m = − (cid:18)(cid:16) J (cid:126)m (cid:17) − · J (cid:126)b (cid:19) · (cid:126)b (8)where J (cid:126)m and J (cid:126)b are the Jacobian matrices in (cid:126)m and (cid:126)b respectively. By comparison to eq.3 we find χ hf = − (cid:18)(cid:16) J (cid:126)m (cid:17) − · J (cid:126)b (cid:19) (9)which we refer to as the complete analytic solution of the ferromagnetic high-frequency4usceptibility. Note that this approach is independent of the form of the free energy func-tional. Since we obtain the full tensor without assumptions regarding its entries, we haveto project it on the unit vectors (cid:126)u b and (cid:126)u m that represent an excitation-measurement-pairof observables to obtain a representative spectrum. In a typical numerical evaluation onewould assume (cid:126)u b to be parallel to the unit-vector in φ direction of the applied field (cid:126)B and (cid:126)u m to be parallel to the unit-vector in φ direction of the magnetization vector in sphericalcoordinates. Nonparallel unit-vectors (cid:126)u b and (cid:126)u m can be used to account for nonuniform mi-crowave fields. The angle between (cid:126)u b and (cid:126)u m represents an effective phase shift ∆ betweenthe excitation and the response. This is illustrated in the inset in fig. 1. Such a phase shiftcan be created for instance by having the sample covered by a conductive layer in whichthe microwave creates an eddy current that in turn creates a phase shifted microwave signalthat superimposes with the original one as described in . The approach presented here wasused in to calculate asymmetric line shapes. B. Extension to reciprocal-( (cid:126)k -)space and description of the magnon dispersion
The model presented above can be extended to reciprocal space in order to obtain themagnon dispersion. Accordingly, the spatial contributions to the energy landscape are in-cluded in the energy density formulation. Also the Ansatz has to be changed such that thedynamic magnetization has a spatial dependence. We imagine that the spatial distributionof the magnetization can be described as a constant part and a dynamic part where thedynamic part is a Fourier series. In contrast to the description by Suhl, where this Ansatzappears we consider the amplitude for every (cid:126)k to be small, such that we can perturb thesystem with a single (cid:126)k at a time, yielding an Ansatz of the form (cid:126)M ( t, x ) := (cid:126)M ( M, θ M , ϕ M ) + (cid:126)m k exp (cid:16) ıωt − (cid:126)k · (cid:126)x (cid:17) (10)where (cid:126)k is the reciprocal vector for which the susceptibility is being calculated and (cid:126)x is thespatial coordinate at which the wave is observed.For example we can consider exchange energy contribution in a continuum model F ex = d B ex (cid:13)(cid:13)(cid:13) (cid:126)M (cid:13)(cid:13)(cid:13) (cid:16) (cid:126)M ( t, x ) · (cid:52) (cid:126)M ( t, x ) (cid:17) (11)5 - [ a. u. ] Δ = Δ = π
20 30 40 50 - [ mT ] A m p li t ude [ a . u . ] π π π π P ha s e s h i ft Δ Figure 1: Top: Amplitude of the susceptibility as a function of the applied magnetic field andthe phase-shift ∆ . At the angles π / and π / the line-shape is fully anti-symmetric similar to thederivative of a Lorenz line-shape. In the vicinity of those angles the signal is asymmetric. At theangles and π the signal is symmetric with positive amplitude and at π the signal is symmetricwith negative amplitude. Bottom: Selected lines at specific angles and π . The inset illustratesthe phase shift ∆ induced by choosing a nonparallel tuple of excitation measurement projectionvectors (cid:126)u b and (cid:126)u m shown in the precession cone of the time dependent magnetization. For simplicitythe the precession is indicated as a circular motion normalized to the length of the unit vectorsperpendicular to (cid:126)M . In general it would be approximated to be elliptical and the opening of thecone is much smaller compared to the Magnetization vector. and a (cid:126)k dependent dipolar coupling to include dynamic aspects of dipolar interactions F Demag = 12 µ (cid:13)(cid:13)(cid:13) (cid:126)M (cid:13)(cid:13)(cid:13) (cid:126)m k · (cid:126)k (cid:107) (cid:126)m (cid:107) (cid:13)(cid:13)(cid:13) (cid:126)k (cid:13)(cid:13)(cid:13) (cid:126)k · (cid:126)M ( t, x ) (12)where d is the distance between two neighboring spins, B ex is the exchange field they exerton each other and (cid:52) is the Laplace operator in real space. Adding this contribution to the6 . 0.5 1.log ( Amplitude ) [ a. u. ] l og ( f [ H z ]) Γ H P N Γ P N H10.411.512.613.7 l og ( f [ H z ]) Figure 2: The magnon dispersion calculated for a bcc structure including a cubic anisotropy ex-change and dipolar coupling (top) and for a similar system with an additional strong chiral coupling (bottom).
Helmholtz energy density we can proceed as before and calculate the susceptibility for every (cid:126)k in the Brillouin zone as shown exemplary in fig. 2. The results agree with the literature(see for instance ). For other spatial contributions such as anisotropic exchange andchiral coupling , the model can be applied in the same way.
C. The equilibrium position of the magnetization
In order to use the result in eq. 9 to obtain the susceptibility it is necessary for (cid:126)M ( θ M , φ M ) to locally minimize the free energy density. The orientation of the magnetization vector hasto be determined form the shape of the free energy landscape including an applied magneticfield. In the following we present our recursive method to efficiently find these minima. Interms of infinitesimals this method can be viewed as a trajectory depended analytic solution.Due to its infinite recursion along a chosen trajectory however it resembles a second ordernewton algorithm, which is a numerical tool, and we therefore tend to call it a semi-analytictrajectory dependent solution of the equilibrium states of the magnetization.7 M δ Magnetization Angle ф M F r ee E ne r g y D en s i t y [ a . u .] ф M δ ф M ф M δ ф M δ Figure 3: The environment of a minimum of the energy Landscape as a function of the Magneti-zation angle (blue) and the same energy landscape after changing the applied field angle φ B by asmall quantity δ (black) together with a Taylor expansion of the changed energy landscape aroundthe the position φ M (red dashed). Note that the series expansion fits perfectly to the black curvearound its minimum. For certain paths in the applied field space, where the trajectory passes sufficiently farbeyond a hard direction, the equilibrium angles are discontinuous if the Zeeman energydoes not overcome the anisotropic contributions of this hard direction. This can leadto a hysteretic behavior of the magnetization depending on the trajectory of (cid:126)B ( τ ) := (cid:126)B ( B ( τ ) , θ B ( τ ) , ϕ B ( τ )) . To account for this behavior a solution representing the equi-librium angles must depend on the trajectory (cid:126)B ( τ ) and not only on a momentary con-figuration of (cid:126)B . Without loss of generality we will only consider the equilibrium angles { θ M , ϕ M } of the magnetization in spherical coordinates to minimize the free energy, sincein many applications the norm of the magnetization may be considered constant. Oncea minimizer (cid:126) Ω (cid:16) (cid:126)B (0) (cid:17) = { θ M , ϕ M } of the free energy F (cid:16) (cid:126)B, (cid:126)M (cid:17) is known for a certainstarting configuration (cid:126)B (0) , a small change in (cid:126)B → (cid:126)B (0 + δ ) that yields a small changein the position of the minimum of F (cid:16) (cid:126)B, (cid:126)M (cid:17) can be accounted for by calculating a se-ries expansion of F (cid:16) (cid:126)B (0 + δ ) , (cid:126)M (cid:17) at the position { θ M , ϕ M } to the second order. Theposition of the minimum of this parabola will be close to the minimum { θ M , ϕ M } δ of F (cid:16) (cid:126)B (0 + δ ) , (cid:126)M (cid:17) . In fact as δ decreases the solution obtained this way will get closer tothe exact minimum. This procedure is illustrated in fig. 3, where the free energy was de-fined to be F = sin (2 φ M ) − φ B − φ M ) . Since the function obtained from the seriesexpansion is of quadratic order it can always be written in a form such that the vertex can8e directly extracted from the function. Therefore a recursive function of the form 13 (cid:126) Ω (cid:16) (cid:126)B (0 − δ ) (cid:17) = (cid:126) Ω (cid:16) (cid:126)B (0) (cid:17) − H − F (cid:12)(cid:12)(cid:12) (cid:126) Ω ( (cid:126)B (0 − δ ) ) · (cid:126) ∇ F (cid:12)(cid:12)(cid:12) (cid:126) Ω ( (cid:126)B (0 − δ ) ) (cid:126) Ω (cid:16) (cid:126)B (0 − δ ) (cid:17) = (cid:126) Ω (cid:16) (cid:126)B (0 − δ ) (cid:17) − (13) H − F (cid:12)(cid:12)(cid:12) (cid:126) Ω ( (cid:126)B (0 − δ ) ) · (cid:126) ∇ F (cid:12)(cid:12)(cid:12) (cid:126) Ω ( (cid:126)B (0 − δ ) ) ... can be derived to describe the position of a minimum for certain trajectories (cid:126)B ( τ ) , where H F is the Hessian Matrix of the free energy density that described the curvature and (cid:126) ∇ F the gradient that describes the slope of the free energy. Conceptually this can be considereda second order Newton algorithm with the exception that it starts from a known positionmaking the number of iterations required tend towards as δ gets small. To determine aminimizer that can be used as a starting point in eq. 13 the easiest approach in a numericalcalculation is to start at a field value sufficiently higher than the field at which the Zeemanenergy fully overcomes the anisotropy energy – in the sense that there is only one minimumand one maximum left in the energy landscape – and to assume that the magnetizationis parallel to the applied field in this configuration. This approach was implemented andfound to be very accurate in for fitting FMR spectra recorded at different microwavefrequencies. Figure 4 shows some calculated spectra using the solution presented above, withthe corresponding equilibrium angles calculated with this trajectory dependent algorithm.The overall calculation time was about five minutes for data points.Equation 9 in combination with eq. 13 describe a very fast algorithm to calculate thecomplete susceptibility for any given free energy density and any measurement trajectory.This algorithm however will not always align the magnetization in the absolute minimum ofthe free energy, in fact it will fall into meta-stable states if for instance a fourfold crystallineanisotropy is considered and the applied field is swept along the field angle rather than thefield amplitude, predicting the occurrence of ferromagnetic resonance in meta-stable states.9 - [ Arb. U. ] π π π π Applied Field [ mT ] I np l ane A ng l e ϕ B π π π π Applied Field [ mT ] I np l ane A ng l e ϕ B π π π π Applied Field [ mT ] I np l ane A ng l e ϕ B - π π M agne t i z a t i on A ng l e ϕ M Figure 4: Calculated spectra at typical X-Band frequencies: 10 GHz (top left), and 18GHz (topright) and the corresponding solutions for the magnetization angles (bottom). The model that hasbeen used for the free energy here is a cubic anisotropy K = 4 . · J / m where the 110-directionis perpendicular to the azimuthal plane ( θ = π ) together with an in-plane uni-axial anisotropy K u = − . · J / m and a 2-fold out of plane anisotropy K = 0 . · J / m according to withthe demagnetizing tensor of a thin film. The g-factor was set to . and the damping constant of α = 0 . was used. III. SUMMARY
We have devised a versatile analytic model, capable of accurately describing FMR experi-ments as well as modeling the full magnon dispersion. The model is simple in that it requiresonly derivatives. Condensed into a single operator χ hf , it is compact and thus easy to usein analytic and numeric applications. The formulation through an energy density allows foreasy modification of the model to adapt different types of interactions, such as dipole-dipole-interaction, spin-spin-interactions like the Dzyaloshinskiˇı-Moriya interaction and spin-orbitinteractions. It can also be applied directly to spatial dependent spin configurations ob-tained from micromagnetic simulations to retrieve information about the magnetodynamicproperties of spin textures. The model is not restricted to evaluating the magnon dispersionas a function ω ( k ) but instead yields the magnonic response amplitude χ ( ω, k ) as a Green’sfunction. In addition to this, the algorithm described in sec. II C makes it possible to applythe model on orientations of the magnetization which are non collinear with the symmetrydirections of the system or the applied magnetic field. This can be used to calculate angulardependent spectra, as well as identify meta-stable states and describe their magnetodynamic10ehavior. D. Polder, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science , 99 (1949), http://dx.doi.org/10.1080/14786444908561215, URL http://dx.doi.org/10.1080/14786444908561215 . L. D. Landau and J. M. Lifschitz, Phys. Zeitsch. der Sow. , 153 (1935). T. L. Gilbert, IEEE Transactions on Magnetics , 3443 (2004), ISSN 0018-9464. C. Kittel, Phys. Rev. , 565 (1951), URL http://link.aps.org/doi/10.1103/PhysRev.82.565 . F. Keffer and C. Kittel, Phys. Rev. , 329 (1952), URL http://link.aps.org/doi/10.1103/PhysRev.85.329 . S. V. Vonsovski˘ı,
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