Analysis of the Collective Behavior of a 10 by 10 Array of Fe3O4 Dots in a Large Micromagnetic Simulation
aa r X i v : . [ c ond - m a t . m e s - h a ll ] N ov Analysis of the Collective Behavior of a 10 by 10 Arrayof Fe O Dots in a Large Micromagnetic Simulation
Christine C. Dantas
Divis˜ao de Materiais (AMR), Instituto de Aeron´autica e Espa¸co (IAE), Departamentode Ciˆencia e Tecnologia Aeroespacial (DCTA), Brazil, FAX/Tel: 55 12 3941-2333
Abstract
We report a full (3D) micromagnetic simulation of a set of 100 ferrite (Fe O )cylindrical dots, arranged in a 10 by 10 square (planar) array of side 3 . µ m,excited by an external in-plane magnetic field. The resulting power spectrumof magnetic excitations and the dynamical magnetization field at the result-ing resonance modes were investigated. The absorption spectrum deviatesconsiderably from that of a single particle reference simulation, presenting amode-shifting and splitting effect. We found an inversion symmetry throughthe center of the array, in the sense that each particle and its inversioncounterpart share approximately the same magnetization mode behavior.Magnonic designs aiming at synchronous or coherent tunings of spin-waveexcitations at given spatially separated points within a regular square arraymay benefit from the new effects here described. Keywords: spin waves; micromagnetic simulations; thin films
Email address: [email protected] (Christine C. Dantas)
Preprint submitted to Physica E: Low-dimensional Systems and NanostructuresSeptember 5, 2018 . Introduction
For several decades, there has been a great interest in the study of thecollective spin excitations in magnetically ordered media, and recently the-oretical and experimental investigations have been thoroughly conducted[1, 2, 3, 4, 5]. These investigations established the prospect of controllingSWs in magnonic crystals (similarly to the control of light in photonic crystals[6]), motivating a whole new field, currently being referred to as magnonics(c.f. [7] and references therein).In magnonic crystals, dipolar (magnetostatic) interactions have an impor-tant physical role, since they couple excitation modes of individual, closely-spaced particles, affecting both the static and dynamic behavior of the mag-netization [8, 9, 10]. This effect results in the formation of collective modesin the form of Bloch waves [11, 12, 13, 14], leading to allowed and forbiddenmagnonic states at given frequencies, or band gaps [15, 16]. These and otherparticular characteristics stimulated new research directions in the study of“spin-waves” [17] (hereon SWs), given the possibility of designing filters andwaveguides for microwave nanotechnology applications [18, 19].However, only recently experimental and numerical investigations on themodification of normal modes of magnetic excitations in periodically ar-ranged nanomagnets have been carried out in some detail [20, 21, 22, 23,24, 25, 26, 27], given the need for advanced experimental and computationalcapabilities. The general theoretical formulation of magnetic phenomena atscales of ∼ − − − m is based on the Landau-Lifshitz-Gilbert (LLG)equation [28, 29, 30] for the magnetization dynamics. Note that the LLGequation only be solved analytically for special cases [31], hence computa-2ional micromagnetics with increasingly detailed simulations have been animportant aid at understanding micromagnetic phenomena [32, 33].In a series of papers by Kruglyak et al. [23, 24, 25, 26], particular attentionwas given to the investigation of the magnetization dynamics of square arraysof submicron elements of different sizes under a range of bias fields. Theseinvestigations involved the use of time-resolved scanning Kerr microscopyto probe the magnetic response of nanoelements, along with micromagneticsimulations to aid the analysis of the resulting spectra. These works havegenerally shown that the position of the mode frequencies as a function of theelement size as well as their relative absorption amplitudes present a com-plex dependence and follows a nonmonotonic behavior. It was also observedthat the position of mode frequencies did not follow the prediction of themacrospin model for an isolated element. It was inferred that a nonuniformdistribution of the demagnetizing field could be responsible for nonuniformprecession within the elements of the array, adding to the complex depen-dence the role of exchange interaction. It has also been noted that, as the sizeof the element in the array is decreased, the edge regions of a given elementpresent increasingly dominant modes confined by the demagnetizing field inrelation to uniform modes. In addition, in studies where the orientation ofthe external magnetic field was rotated in the element plane of the array,the effective magnetic field inside a given element also presented an “extrin-sic” anisotropic contribution due to the stray field from nearby elements,as contrasted to an “intrinsic” anisotropy occurring in an isolated element.Furthermore, a dynamical configurational anisotropy was necessary to quali-tatively explain the data. An additional important feature, specially observed3n micromagnetic simulations of arrays of nanoelements, is the splitting ofprecessional modes, a feature experimentally verified recently as detectablecollective spin wave modes extended throughout the array [27].Here we report a 100-particles micromagnetic simulation – to our knowl-edge, the largest detailed micromagnetic calculation ever performed at thegiven scale and number of particles, with a careful observation of the accuracyrequirements (see accuracy details in Section 2.3 of Ref. [34], which were alsoadopted here). This work is part of an ongoing project [35, 34] motivatedby the theoretical investigation and design of new nanostructured magneticconfigurations with interesting SWs or magnonic band gap behavior, suitablefor different applications in the microwave frequency range. We show thatthe collective magnetization behavior is constrained by an inversion sym-metry through the center of the array. In particular, this opens interestingpossibilities for applications that require spatially coordinated patterns.
2. Methods
The simulations were performed by using the freely available integrator
OOMMF (Object Oriented Micromagnetic Framework)[36], which was in-stalled and executed on a 3 GHz Intel Pentium Desktop PC, running Kuru-min Linux. The present 100-particles simulation turned out to be a computa-tionally demanding one (taking about 4 months for completion, consideringinterruptions), and no other variation of the parameters were attempted atthis time. We applied the same methodology described in our previous works[35, 34], based in the procedure given by Jung et al. [37].An incident in-plane magnetic field was applied uniformly to the ferrite4articles, composed by a static ( dc ) magnetic field ( B dc ≡ µ H dc ) of 100mT in the y direction, and a varying ( ac ) magnetic field ( B ac ≡ µ H ac )of small amplitude (1 mT) in the x direction, with the functional form: B ac = (1 − e − λt ) B ac, cos( ωt ). We varied the frequency ( f = ω/ (2 π )) from 2to 9 . . B ac field was discretized in intervalsof 0 .
005 ns.OOMMF performed the numerical integration of the LLG equation lead-ing to the evolution of the magnetization field of 100 ferrite particles regu-larly distributed in a square, 10 by 10 array of side 3 . µ m. The particleswere identical cylindrical dots (each with a diameter d = 0 . µ m and thick-ness δ = 85 nm). We adopted a small interparticle (edge-to-edge) spacing( s = 0 . µ m ≪ d ). The simulation was stopped at 5 ns, giving 1000 out-puts for each frequency run. A reference simulation of a single-particle witha diameter d = 0 . µ m was also conducted, with the same global parametersof the 100-particles simulation. Table 1 lists the parameters used to set upthe OOMMF integrator in both cases.The power spectra of magnetic excitations were obtained by excludingdata from the first 2 ns of the averaged magnetization vector in the x di-rection, h ~M i x ( t ≤ h ~M i x (2 < t ≤ . Results The power spectra of magnetic excitations for the 100-particles simulationand single-particle simulation are shown in Fig. 1. The resonance peak inthe power spectrum of the single-particle simulation is clearly split into fourdistinct peaks of lower amplitude in the 100-particles simulation, with twopeaks at a lower and the other two at a higher frequency with respect to thereference fundamental mode, which is located at the gap between resonances2 and 3 of the 100-particles simulation.We analysed the nature of the modes of interest by an inspection of thetime-dependent magnetization vector field configuration. Bitmaps or “snap-shots” of the corresponding simulations were generated, selected at two pointsof the ac field cycle, namely, at the highest ( τ ) and lowest ( τ + π ) representa-tive amplitudes of the equilibrium magnetization oscillation. We subsampledthe x -component of the magnetization field in order to show 9 vectors per cellelement, and different pixel tonalities correspond to different values of M x .In Fig. 1 (inset), the magnetization vector field of the one-particle simulationaround its resonance peak is shown. This should be contrasted with those ofFig. 2 (snapshots of the 100-particles simulation at the previously identifiedfour peaks of interest).Fig. 3(a) shows some zoom-out regions of the 100-particles simulation inorder to allow for the identification of several types of magnetization modebehavior, which will be discussed in more detail in the next section. On theother hand, from a visual inspection of the 10 by 10 snapshots, it is possibleto identify an inversion symmetry through the center of the array in sucha way that each particle and its inversion counterpart share approximately6he same mode behavior in the array. In other words, by setting a Cartesiangrid on the array, where the origin of the coordinates is the center of thearray, and where each element is centered at coordinates ( i, j ), an inversiontransformation ( i, j ) → ( − i, − j ) leaves the magnetization configuration ofthe array approximately invariant. Some examples are shown in Fig. 3(b).This global symmetry should arise from dipolar couplings, but the exact for-mulation is yet not clearly understood. This effect has already been pointedout in our previous work in the cases of 2 by 2 and 3 by 3 array simulations(c.f. Fig 8 of Ref. [35] and discussions in Ref. [34]), but it was not possibleat that time to extrapolate whether the effect would persist in a larger array.In order to address in a more quantitative way the magnetization dy-namics distribution in the array, we computed two simple estimators, whichnevertheless establish the relevance of the visually noted effect. The firstestimator is the modulus of the difference of average magnetization valuesat the points of the ac field cycle: m ( i,j ) ≡| h M x i ( i,j ) ( τ ) − h M x i ( i,j ) ( τ + π ) | ;where h M x i ( i,j ) is the average magnetization of a given particle at grid coor-dinates ( i, j ). The second estimator, σ ( i,j ) , is the modulus of the difference ofstandard deviation values of particle magnetizations within the array, thatis, the standard deviation is computed with reference to the average magne-tization of the whole array. Fig. 4 shows the resulting m ( i,j ) and σ ( i,j ) mapscomputed for the 4 modes of interest. Notice that each map pixel is labelledby the particle coordinate ( i, j ). We discuss these maps in more detail in thenext section. 7 . Discussion and Conclusion It is presently understood that particular features of the power spec-tra of magnetic excitations can be associated with the various types ofnodal behavior of the time-dependent magnetization field (see, e.g., Refs.[21, 9, 20, 23, 24, 25, 26]). A general “spin-wave” behavior (SWB) for theexcitations indicates that the magnetization vectors present small ampli-tude oscillations about a nonuniform static magnetization field. In addition,the following subclasses of excitations can be identified [20, 34]: (i) “Quasi-uniform” behavior (QUB) : the movement of each magnetization vector issimilar to that of its neighbors, except for the regions around the edge of theparticle; and (ii) “Edge-like” behavior (EDB) : the magnetization field in thecentral region of the particle is almost entirely static and aligned with thedirection of the external dc field; the nonuniformly distributed magnetizationvectors present small amplitude oscillations near the edges of the particle. Inparticular, these modes may be more affected by the dipolar couplings fromnearby particles.In the present work, the nature of the reference absorption peak is basi-cally due to QUB, whereas the 100-particles simulation presents all types ofbehavior (as can be seen from an inspection of the zoom-out regions exem-plified in Fig. 3(a)). In particular, peak 1 at 3 . . . . m ( i,j ) and σ ( i,j ) estimators shown in Fig. 4. It is clear that m ( i,j ) giveslarger values for larger amplitudes of the average particle magnetization,but also the original inversion symmetry through the center of the array ofthe magnetization configuration should be translated approximately into asymmetry around a central horizontal axis in a m ( i,j ) map. It is clear that ifthe standard deviation distribution is similar at the two opposite amplitudepoints, then a map of σ ( i,j ) should be uniform. This estimator also gives largervalues for particles whose behavior with respect to the whole array presentsa substantial difference at the two points of the cycle, thus furnishing anoverall measure of the degree of cycle “coherence”.The expected central horizontal axis symmetry is indeed seen in Fig. 4(left column of panels), which corroborates our visual analysis. In peak 1, m ( i,j ) is larger for particles localized in the top and bottom rows of the array.In peak 2, note the interesting regular, alternating magnetization excitationsalong specific rows of the array (also inferred from a visual inspection of thesnapshots). Peaks 3 and 4 show higher values of m ( i,j ) in the central regions,in contrast to peak 1. The σ ( i,j ) estimator results (Fig. 4, right column ofpanels) show that, excluding the top and bottom rows of the arrays in cases1, 3 and 4, the array behavior is reasonably similar at the two extremes of thecycle. However, case 2 shows again a pattern, where the second and ninthrows (related to particles with very low amplitudes of the average particle9agnetization, c.f. left column of panels) have standard deviations thatsignificantly differ at the two extremes of the cycle.Qualitatively, the results reported in the present work, specially the split-ting of the resonance mode and its decreased relative amplitude, as well asthe spatially nonuniform behavior of the elements in the array, are compati-ble with the behavior of the collective excitations reported in similar previousworks [23, 24, 25, 26, 27]. Particularly, in Ref. [27], for the first time the col-lective spin wave modes of the entire array has been experimentally detected.With the aid of a micromagnetic simulation of a 3 × ×
10 elements. Yet, as already mentioned, a symmetry pattern can beidentified. Indeed, the inversion symmetry here noted is compatible with theexperimentally observed “tilt” of the modes in regions of higher amplitude[27], relatively to the horizontal and vertical axes, reported in that work. Webelieve that such a “tilt” would be observed in a larger simulation for thatmaterial.In summary, our present results and previous indications allow us to inferthat, for interparticle (edge-to-edge) separations at least of the order of ∼
5. Acknowledgments
We thank the anonymous referee for useful comments and corrections.We acknowledge the support of Dr. Mirabel C. Rezende and FINEP/Brazil.
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Simulation Parameter/Option ParameterSaturation magnetization [A/m] 5 . × Exchange stiffness [J/m] 1 . × − Anisotropy constant [J/m ] − . × Anisotropy Type cubicFirst Anisotropy Direction (x,y,z) (1 1 1)Second Anisotropy Direction (x,y,z) (1 0 0)Damping constant 0 . . × Particle thickness [nm] 85 . . ig. 1 : Top panel:
Power spectrum of magnetic excitations of the 100-particles simulation.
Bottom panel:
Comparison of the former spectrum(diamond symbols) with that of the single-particle reference simulation (dotsymbols). Four distinct peaks in the spectrum of the 100-particles simulationare identified. “Snapshots” of the magnetization vector field of the one-particle simulation around the resonance peak are indicated by the arrow.Snapshot to the left refers to the highest amplitude of the oscillation and thesnapshot to the right, to the lowest one.
Fig. 2 : “Snapshots” of the magnetization vector field of the 100-particles(10 by 10 particle array) simulation around the four peaks identified in theprevious figure. For each peak, the upper snapshot shows the magnetizationstate of the array at the highest amplitude of the oscillation ( τ ) whereas thesnapshot immediately below, at the lowest amplitude ( τ + π ). Symmetryaxes are shown schematically at the top left of the figure. Fig. 3 : (a) Amplified particular selections (3 by 4 sub-arrays given bythe solid rectangular shown inside the 10 by 10 array snapshots), at the twopoints of the cycle. (b) Illustration of the symmetric mode behavior: oneexample of particle pairs is taken from each of the four peaks at τ . Gridcoordinates are indicated above the particles. Fig. 4 : Estimator maps ( m ( i,j ) and σ ( i,j ) distributions) for the arrays, asexplained in the main text. New symmetry axis is indicated.18 igure 1: igure 2: Figure 3: igure 4:igure 4: