# Fileset [PhysRevApplied.21.024027.pdf](https://mdr.nims.go.jp/filesets/21522b74-33ce-47cf-a5a5-795b2327ecb7/download) ## Creator [Donghyeon Lee](https://orcid.org/0000-0002-3796-1919), Jungmin Park, Donghyeon Han, Suzuki Ippei, [Takahashi Yukiko](https://orcid.org/0000-0001-9197-7236), Sujung Noh, Jisung Lee, JoonHyun Kwon, Hansaem Lee, [Sanghoon Kim](https://orcid.org/0000-0002-4577-8866) ## Rights [In Copyright](http://rightsstatements.org/vocab/InC/1.0/) ## Other metadata [Statistical approach based on the Stoner-Wohlfarth model for the switching field in a misoriented magnet array](https://mdr.nims.go.jp/datasets/845cf9a1-492b-4301-af5f-3f44cce3521a) ## Fulltext Statistical approach based on the Stoner-Wohlfarth model for the switching field in a misoriented magnet arrayPHYSICAL REVIEW APPLIED 21, 024027 (2024)Statistical approach based on the Stoner-Wohlfarth model for the switching fieldin a misoriented magnet arrayDonghyeon Lee ,1 Jungmin Park,2,* Donghyeon Han,3 Suzuki Ippei,4 Takahashi Yukiko,4Sujung Noh,5 Jisung Lee,5 JoonHyun Kwon,5 Hansaem Lee,5 and Sanghoon Kim 1,†1Department of Physics, University of Ulsan, Ulsan 44610, Korea2Department of Physics, KAIST, Daejeon 34141, Korea3Materials Science and Engineering, KAIST, Daejeon 34141, Korea4National Institute for Materials Science (NIMS), Tsukuba 305-0017, Japan5Research & Development Division, Hyundai Motor Company, Hwaseong 18280, Korea (Received 27 September 2023; revised 18 December 2023; accepted 24 January 2024; published 13 February 2024)The Stoner-Wohlfarth model (SW model) is used to explain the relationship between the external mag-netic field and the magnetization reversal of a single magnetic domain. However, it is inappropriate forinhomogeneous magnetic films or bulk materials, such as granular and defective films, because the mag-netization reversal of the films is affected by magnetic particles or grains, which could have a differentanisotropy for uniaxial magnetic anisotropy (UMA) of the films. In this paper, we propose a model basedon the SW model to understand the magnetization switching behavior of magnetic films with misorientedgrains. Our model considers the distribution effect of different anisotropies originating from magneticdefects or domains against the UMA of a magnetic thin film, and a Gaussian function is employed forthe probability distribution of tilted anisotropy. Finally, we experimentally demonstrate that our modeldescribes the magnetic switching behavior of a granular magnetic film (FePt-C) with perpendicular mag-netic anisotropy. This study provides an alternative analytical method for the magnetization reversal inmagnetic devices.DOI: 10.1103/PhysRevApplied.21.024027I. INTRODUCTIONOver the last decade, the scaling down of ferromagnetshas been an essential issue for high-density memory appli-cations such as magnetoresistive random access memory(MRAM) and hard disk drives (HDD) [1–4]. By reducingthe scale of a magnetic device with perpendicular magneticanisotropy (PMA) up to tens of nanometers smaller thana single magnetic domain, the Stoner-Wohlfarth model(SW model) explains the switching behaviors of a devicein a single domain state [5–7]. However, considering thewafer-scale fabrication for commercializing MRAMs, orthe existence of nanometer-sized magnetic particles in themagnetic media of HDDs, the SW model cannot per-fectly describe the total magnetization behaviors of billionsof nanometer-sized ferromagnets for the following rea-sons [8–19]. In the cases of MRAM, magnetic propertiesinevitably degrade during device fabrication, such as etch-ing damage at the edge of each device, resulting in differentmagnetic characteristics among magnetic tunnel junction(MTJ) devices on a 300-nm wafer. On the other hand,*jungmin0123@kaist.ac.kr†sanghoon.kim@ulsan.ac.krthe magnetic media in HDDs consists of magnetic gran-ular films such as FePt-SiO2, FePt-C, and CoCrPt-SiO2, inwhich the magnetic easy axes between particles are misori-ented. The magnetic properties of granular films depend ongrain size, grain interaction, and space between grains [20].Therefore, the conventional SW model cannot providerepresentative information on nanosized pattern arrays orgranular magnetic thin films. In addition to these systems,the magnetization behaviors of bulk hard magnets consist-ing of nanocomposites do not follow the SW model or thesimple domain-wall propagation model [8,9] owing to therandomly oriented magnetic anisotropy of the compositein the magnets.In this study, we develop a statistical model based onthe SW model considering a situation in which the uniax-ial magnetic anisotropy (UMA) direction of each particlein a system randomly deviates from the standard axisperpendicular to the plane. The statistical SW model repro-duces the magnetization switching of a magnetic granularfilm, which has uncountable magnetic particles with amisorientation exhibiting strong PMA. The best fitting ofthe angle-dependent coercivity of the granular magneticfilm (FePt-C) using the model quantitatively provided thedegree of deviation of PMA. Our model can be practically2331-7019/24/21(2)/024027(8) 024027-1 © 2024 American Physical Societyhttps://orcid.org/0000-0002-3796-1919https://orcid.org/0000-0002-4577-8866https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevApplied.21.024027&domain=pdf&date_stamp=2024-02-13http://dx.doi.org/10.1103/PhysRevApplied.21.024027DONGHYEON LEE et al. PHYS. REV. APPLIED 21, 024027 (2024)utilized to analyze the magnetic properties of patterned orself-assembled magnetic arrays.II. EXPERIMENTAL AND SIMULATION METHODA 4-nm-thick FePt-C nanogranular layer was epitax-ially grown on a MgO (100) substrate by co-sputteringFePt and C targets. Pt (5 nm) was deposited as a cap-ping layer. Here, the chemical composition of FePt wasestimated to be approximately 50:50. A Hall bar geometryof the FePt-C thin film was defined by using photolithog-raphy and ion milling. The channel width and length ofthe device were 10 and 90 μm, respectively. Ru (100 nm)was employed as a contact pad for electrical measurement.All electrical measurements were performed in a cryogen-free measurement system (cryogenic LTD) with a sourcemeter (Keithley 6221) and nanovoltmeter (Keithley 2182)at room temperature. An Al wire was used for electricalcontact with the Ru pads of the device. A dc current of1 mA was applied for electrical transport.The micromagnetic simulation, MuMax3, was employedfor comparison with our statistical model. In this simula-tion, the magnetic properties of FePt-C are referenced fromRef. [11]. The granular structure is developed by modi-fied Voronoi tessellation with the perpendicular bisectorsand asymmetric overlapping of the two layers as shown inFig. 4(a). Here, unit cell and plane size are 1 × 1 × 1 nm3,and 200 × 200 nm2, respectively. The thickness of thefilm is 4 nm. The average distance between grains (whiteshape) is 7 nm. We assume that magnetization satura-tion is 1.0 × 106 A/m, and the anisotropy constant is5.0 × 106 J/m3. Also, we consider the demagnetizationeffect in addition to a misorientation of the magneticgrains.III. RESULTS AND DISCUSSIONA. Statistical model based on the SW modelThe SW model considers two energies, the magneticanisotropy and Zeeman energies in a single-domain fer-romagnet. When an external magnetic field is applied, thetotal energy is given by [5,6],Etot = Ku sin2(θ − ϕ) − μ0MHex cos ϕ, (1)where the Ku is the uniaxial anisotropy, M is the magne-tization, and μ0 is the vacuum permeability. The first andsecond terms indicate the contributions of the anisotropyenergy and the Zeeman energy, respectively.According to Eq. (1), the magnetization of a singledomain rotates under an external magnetic field (Hex) com-peting with Ku. The switching field (hs) depends on theangle θ between the magnetic axis and Hex as shown inFig. 2(b). Then, the angle-dependent hs for a single domainobtained from the minima of the Zeeman energy is givenas [5]hs(θ) = (1 − tan2/3θ + tan4/3θ)1/21 + tan2/3θ. (2)This is a solution from the conventional SW model tounderstand the magnetic switching behavior of a singledomain.Now, we consider a magnetic particle with another uni-axial anisotropy (K ′u), which is tilted from the Ku axis,as shown in Fig. 1. Although factors such as dipole-dipole interaction and domain wall can affect magnetiza-tion reversal by external magnetic field in the system withmany magnetic domains or grains, we ignore these factorsto focus on the statistic effect of tilted anisotropy of thegrains. Then, a normalized magnetization vector with K ′ucan be defined as follows:nx = ωx√ωx2 + ωy2 + 1, ny = ωy√ωx2 + ωy2 + 1, nz= 1√ωx2 + ωy2 + 1. (3)Here, ω is introduced to consider the tilted weight of K ′u.We set the z axis as the standard axis. Then, we can definethe angle (δ) between K ′u and Hex as in Eq. (4) because Hexis applied in the x-z plane (Hex = (H sin θ , 0, H cos θ)) asshown in Fig. 1(b).Hex × n̂ = H cos δ = Hωx × sin θ + cos θ√ωx2 + ωy2 + 1. (4)The value of δ is determined by θ and ωx,y as follows:cos δ = ωx × sin θ + cos θ√ωx2 + ωy2 + 1. (5)If there is no difference between δ and θ , ω = 0 becausethe directions of Ku and K ′u are the same as shown inFig. 1(b). Therefore, hs considered with the K ′u of themagnetic particles can be expressed as follows:hs(δ(θ , ωx, ωy)) = (1 − tan2/3(δ(θ , ωx, ωy)) + tan4/3(δ(θ , ωx, ωy)))1/21 + tan2/3(δ(θ , ωx, ωy)). (6)024027-2STATISTICAL APPROACH BASED ON . . . PHYS. REV. APPLIED 21, 024027 (2024)(b)(a)FIG. 1. Illustration of our model and the SW model. (a) In a magnetic device, magnetic grains can have different magneticanisotropies. The red arrows indicate the anisotropy of a single domain. The black arrow indicates UMA. (b) Conceptualization ofour model along with the SW model illustrating the magnetization of a single domain. ϕ indicates the angle of magnetization (M )indicating Zeeman energy. θ is the angle between the Ku of the UMA and Hex. δ is an angle related to the K ′u of magnetic particlesagainst the UMA of the magnetic device.Here, we set the values of ωx and ωy in the range of −5to 5. In this range, the tilting angle (δ − θ) of K ′u variesfrom −78.6◦ to 78.6◦ toward the x-y plane.Figure 2 shows ω-dependent hs determined using Eq.(6) in terms of θ when −5 < ωx,y < +5. To clarify ouranalytical calculation results, we first focus on hs valuesas a function of ωx and ωy when Hex is parallel to thenormal direction to the x-y plane (θ = 0◦). Here, δ is nolonger a function of θ . Then, δ is only determined byωx,y = 0 according to Eq. (5). The estimated hs plot in(a) (b) (c)(d) (e) (f)FIG. 2. Plot of ω-dependent hs and P. hs as a function of ω at θ = 0◦ (a), 45◦ (b), and 90◦ (c) according to Eq. (7). The maximumvalue of hs indicates the perpendicular between Hex and the K ′u of magnetic particles. The Gaussian distribution as a function of t ωat σ = 0.10 (d), 0.20 (e), and 0.50 (f) from Eq. (7). The magnitudes of hs and the probability distribution are expressed as a spectrum(the black color indicates the minimum values, and the white color indicates the maximum values).024027-3DONGHYEON LEE et al. PHYS. REV. APPLIED 21, 024027 (2024)Fig. 2(a) shows centrosymmetric behavior in terms of ωx,y .In the case of θ = 45◦ in the x-z plane, the estimated hs interms of ω is noncentrosymmetric as shown in Fig. 2(b).When the K ′u axis is tilted by −45◦ from the z axis underHex, hs becomes maximum because K ′u is perpendicularto Hex(δ = 90◦). This behavior is natural according to theSW model. Therefore, the magnetization with ωx = −1should have a maximum hs independent of ωy (Hex isalways orthogonal to the y axis) as shown in Fig. 2(b).If K ′u is parallel to Hex indicating δ = 0, the switchingfield can also have the maximum value. Hence, there isa peak when ωx = 1 and ωy = 0. Figure 2(c) shows the ω-dependent hs values under Hex with θ = 90◦. This systemalso demonstrates the maximum value of hs as the perpen-dicular (ωx = 0) and parallel (ωx > 5, ωy = 0) betweenHex and K ′u, corresponding to the parabolic characteristicof hs in the SW model well.In the case of a system with several magnetic particleswith various tilting angles (δ − θ) to the z axis, we intro-duce the probability distribution function in addition to theconcept of ω. Here, we consider that the magnetic easyaxis of each magnetic particle in the system has a ran-dom variable that depends on ω. Therefore, the Gaussiandistribution, which is typically a continuous probabilitydistribution function (P), was employed in our model asfollows:P(ωx, ωy) = 1σ√2πexp(−12(ωxσ)2)× 1σ√2πexp(−12(ωyσ)2). (7)Here, σ is the standard deviation of P.Figures 2(d)–2(f) indicate the ω-dependent P(ωx, ωy)in terms of σ . Considering both P and ω, the expectedvalue of hs in the magnetic system for a large number ofmagnetic objects with various K ′u axes can be determinedas〈hs(θ)〉 =∑ωx∑ωyhs(δ(θ , ωx, ωy))P(ωx, ωy). (8)To describe our model intuitively, the overlapping graphof hs and P of magnetic particles is shown in Fig. 3(a)when θ is 45◦ and σ is 0.5 in Eq. (8). The dotted black cir-cle indicates P with σ = 0.5 in Fig. 3(a). Then, the valueof hs in the magnetic system is expected to be within theprobability distribution range because the ω of magneticanisotropy, considering the range of standard deviationof the probability distribution, is directly related to thehs of magnetic particles with different easy axis. In otherwords, the perspective of our model is how many parti-cles, which have different hs for each K ′u, are distributed inthe magnetic system.Based on Eq. (8), the expected value of hs as a functionof θ with various values of σ is shown in Fig. 3(b). Forthe analytical calculation, the constraint range of ω is −5to 5, which corresponds to δ ≤ |81.95◦| at θ = 0◦. In thecase of the conventional SW model, the maximum valueof hs is obtained when θ = 0◦ and 90◦, and the minimumvalue is obtained when θ = 45◦. On the other hand, the θdependence of hs weakens with increasing σ in our sta-tistical SW model. As an increase in σ indicates that therange of the distribution of misoriented grains (the rangeof tilted weight) is widened, the expected value of hs is lessaffected by θ . Therefore, hs at θ = 45◦ gradually increaseswith increasing σ as shown in Fig. 3(c). From Eq. (8), theθ for the minimum value of hs is shifted from 45◦ to 20◦with increasing σ as shown in Fig. 3(d).B. Micromagnetic simulationWe also conduct the micromagnetic simulation to sup-port our statistical SW model. Figure 4(a) shows thegranular structure of a magnetic thin film with perpen-dicular magnetic anisotropy as mentioned in the experi-mental section. In addition to the concept of our statisticalmodel, we incorporate the demagnetization effect of mag-netic grains into the switching field. The θ dependenceof the demagnetization field in each direction is shownin Fig. 4(b). Here, θ is defined as the angle between themagnetic axis and the external magnetic field in Fig. 1(b).Figures 5(a)–5(e) indicate the magnetization as a func-tion of the external magnetic field obtained from themicromagnetic simulation depending on σ and θ . Here,the magnetization is normalized with saturation magneti-zation. The key view in Figs. 5(a)–5(e) is that a perpendic-ular magnetic anisotropy becomes weaker as σ increases,meaning that each anisotropy of magnetic grains more tiltsagainst uniaxial magnetic anisotropy of the magnetic gran-ular films. From these results, we can obtain the hs of themagnetic granular structure depending on various σ andθ . Figure 5(f) shows the angle dependence of hs with ourstatistical SW model and the micromagnetic simulation.The squares indicate the results of the micromagnetic sim-ulation, and the color dash lines show the statistical SWmodel. The micromagnetic simulations also show a ten-dency to increase the minimum hs with increasing σ as inthe statistical SW model [Fig. 3(b)].C. Experimental resultsTo verify our analytical model, we applied it for theexperimental observation of an FePt-C granular film [11],as shown in Figs. 6(a). FePt-C is a good template toevaluate our model because it contains large amounts ofnanometer-sized magnetic particles with randomly dis-tributed K ′u [21]. Figure 6(b) shows the normalized anoma-lous Hall effect (AHE) loops of the FePt-C granular filmdepending on the angle θ . The AHE loops exhibited a024027-4STATISTICAL APPROACH BASED ON . . . PHYS. REV. APPLIED 21, 024027 (2024)(a) (b)(c) (d)FIG. 3. Plot of expected values of hs depending on σ . (a) Overlapping graph showing hs and a Gaussian distribution with 95%uncertainty at θ = 45◦ and σ = 0.5 based on Eq. (8). The total value of hs within the dotted black circle indicates the expected valueof hs under those conditions. (b) hs as a function of θ with various values of σ . Increase in hs with increasing standard deviation atθ = 45◦. (c) hs at θ = 45◦ as a function of σ . Increase in the value of hs with increasing σ . (d) Angle of minimum hs versus thestandard deviation. Decrease in θ with increasing σ .typical PMA behavior. Figure 6(c) shows the normalizedhs of the granular film as a function of θ . Here, hs wasobtained from the coercive field of the AHE data. Unlikein the conventional SW model, the maximum value of hsis observed at θ = 85°, and the minimum value appearsat θ = 30°. The maximum and minimum hs values are1.2 and 0.8, respectively. That is, the hs values of theFePt-C film do not exactly follow the SW model, as(a) (b)FIG. 4. Micromagnetic simulation for hsof the granular magnetic film. (a) The image of a granular structure with perpendicularmagnetic anisotropy. The white and gray colors in the structure indicate a magnetic material FePt and insulator C, respectively. Thisstructure is created using modified Voronoi tessellation. (b) The angle dependence of the demagnetization field (Bdemag) with eachdirection (x, y, z, s) in the granular structure. Here, s means the component parallel to saturation magnetization of the magneticgranular film (Bdemag,s = Bdemag,x sin(θ) + Bdemag,zcos(θ)) and the Bdemag,s decrease as increasing with θ .024027-5DONGHYEON LEE et al. PHYS. REV. APPLIED 21, 024027 (2024)(a) (b) (c)(d) (e) (f)FIG. 5. The simulation results of the M -H curve with various angle θ . Magnetization as a function of a perpendicular magnetic fieldwith 2KU/MS = 10, where KU is the anisotropy energy and MS indicates the saturation magnetization, by (a) σ = 0.00 (meaning theSW model), (b) σ = 0.10, (c) σ = 0.15, (d) σ = 0.20, and (e) σ = 0.50. (f) The angle dependence of hs obtained from our modeland micromagnetic simulation. The dashed lines and squares indicate our statistical model and the micromagnetic simulation results,respectively. The black dashed line means the SW model. Simulation results of the granular structure are similar to our statisticalmodel even though there is a demagnetization effect.shown by the black dotted line in Fig. 6(c). This behaviorhas been reported for several magnetic granular systems[11–13]. Unlike the conventional SW model, our statisti-cal SW model can fit the observed angle dependence ofhs when σ = 0.15 [see the blue dotted line in Fig. 6(c)].The square indicates the data of the simulation in Fig. 6(c).The results of the simulation well describe our model andexperiment although the demagnetization effect is includedin the simulation condition. It means that the anisotropy ofgrains is dominant to the switching field in the magneticgranular film and our statistical SW model becomesparticularly applicable when the magnetic anisotropy sig-nificantly surpasses the energy associated with interparti-cle interactions. Finally, we consider the dispersion weightof ωx and the z axis to determine PMA properties to cal-culate a tilted angle (δ − θ) of anisotropy of magneticgrains as shown in Fig. 6. Average tilted angle can beobtained from angle δ̄ = ∑ωx ,ωyωx ,ωyδ(ωx, ωy) · P(ωx, ωy) andthen, we can obtain the average value of the tilted angledepend on σ . In our statistical SW model, σ = 0.15 means(a) (b) (c)FIG. 6. Anomalous Hall effect of FePt-C granular film with PMA. (a) Illustration of the FePt-C Hall bar device. The AHE ismeasured at room temperature. (b) AHE as a function of the magnetic field B at various values of θ . The AHE loops are normalized.(c) The angle dependence of the coercive field of the FePt-C granular film. It is not fitted with the SW model (black dotted line).024027-6STATISTICAL APPROACH BASED ON . . . PHYS. REV. APPLIED 21, 024027 (2024)ω(∣∣∣√ω2x + ω2y∣∣∣)= 0.187966. This value indicates that themagnetic anisotropy in this FePt-C granular film is tilted by10.8358° (on average) from the z axis.D. DiscussionThe conventional SW model explains the magnetiza-tion behavior of a particle with a single-domain state underHex. Here, we added two factors to apply the SW model toreal situations as follows. First, we consider a system withmultiple particles. Second, we add an additional frame K ′uto the frame of the conventional SW model, as shown inFig. 1(b). Tilted magnetic anisotropy against UMA hasbeen an issue because it is related to the magnetic char-acteristics in various magnetic systems. Fischbacher et al.[22] theoretically showed that the misorientation of theanisotropy axis, which is caused by the demagnetizingfield at the grain boundary, affects the reduction in thecoercive field by nucleation in permanent magnets. Thismisorientation effect is more dominant than the thermalfluctuation. In addition, the misoriented grains in a hardmagnet influence local magnetization switching [23,24].In the case of magnetic thin films, misoriented grains areformed primarily at the interface between the magneticthin film and the substrate [21,25,26], and at grain bound-aries [27]. For example, strain relaxation occurs in an FePtfilm deposited on a TiN substrate, leading to the formationof misoriented FePt grains [28]. This induces a change inthe coercive field of the FePt film. As mentioned in theIntroduction section, the tilted anisotropy of the magneticgrains decreases the thermal stability of an MTJ, which isa unit device in MRAM. This is because tilted magneticgrains weaken the UMA of magnetic thin films [29,30].As this thermal stability is related to the write and readerror rate, our statistical SW model can be used to eval-uate the magnetic properties of magnetic pattern arraysand those of magnetic granular films for ultra-high-densitymagnetic recording media toward achieving tens of terabitsper square inch [31–34].Meanwhile, previous results based on the SW modelcannot explain the angle dependence of the switching fieldaround 90° [35,36]. To overcome these limitations, thedistribution of magnetization was considered on satura-tion magnetization and hysteresis loop [37,38]. However,the aforementioned problem with high angle has not beenfully solved yet because a polar angle is solely considered.However, we statistically deal with Gaussian-type distri-bution in the magnetic easy axis considering an azimuthalangle as well as the polar angle as shown in Figs. 2and 3. Our model and micromagnetic simulation repro-duce the switching field within a full range of the exter-nal field angle in Fig. 6(c). Therefore, we attribute ourresults of switching field around 90° to the Gaussian-typedistribution with an azimuthal and polar angle.IV. SUMMARYIn this study, we introduce a statistical method for differ-ent anisotropy effects to analyze the magnetization reversalof magnetically inhomogeneous arrays extending the SWmodel. The expected value of hs calculated by our modelwell reproduces the experimentally observed angle depen-dence of the coercivity of the granular film. The statisticalSW model can be widely applied for the quantitative anal-yses of the magnetic characteristics of various magneticarray systems or bulk-hard magnets.ACKNOWLEDGMENTSWe acknowledge Grant No. NRF-2019R1C1C1010345from the Samsung Research Funding Center of SamsungElectronics under Project No. SRFC-IT1901-11. J. 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Mater. 345, 147 (2013).024027-8https://doi.org/10.1063/1.4817274https://doi.org/10.1103/PhysRevB.61.1311https://doi.org/10.1063/1.3393960https://doi.org/10.1063/1.5113321https://doi.org/10.1063/1.364985https://doi.org/10.1063/1.3623752https://doi.org/10.1103/PhysRevB.84.214427https://doi.org/10.1109/TMAG.2018.2863225https://doi.org/10.1063/1.361531https://doi.org/10.1016/j.actamat.2015.03.007https://doi.org/10.1016/j.scriptamat.2017.11.020https://doi.org/10.1038/s41524-020-00361-zhttps://doi.org/10.1088/2515-7639/aaf26dhttps://doi.org/10.1016/j.actamat.2019.07.017https://doi.org/10.1016/j.physb.2011.06.028https://doi.org/10.1088/0022-3727/46/1/015002https://doi.org/10.1109/TED.2020.3025749https://doi.org/10.1063/1.3075986https://doi.org/10.1088/1361-6463/ab680dhttps://doi.org/10.1109/TMAG.2003.816282https://doi.org/10.3390/cryst10040263https://doi.org/10.1016/j.jmmm.2015.07.035https://doi.org/10.1016/j.jmmm.2013.06.028 I. INTRODUCTION II. EXPERIMENTAL AND SIMULATION METHOD III. RESULTS AND DISCUSSION A. Statistical model based on the SW model B. Micromagnetic simulation C. Experimental results D. Discussion IV. SUMMARY ACKNOWLEDGMENTS . 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