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Observations of ferromagnetic resonance modes on FeCo-based nanocrystalline alloys
Journal of Magnetism and Magnetic Materials 323 (2011) 635–640
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Observations of ferromagnetic resonance modes on FeCo-based nanocrystalline alloys X. Wang n, L.J. Deng, J.L. Xie, D. Li State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu, China
a r t i c l e in f o
abstract
Article history: Received 20 May 2010 Received in revised form 19 October 2010 Available online 28 October 2010
Uniform and symmetric resonance modes (known as Aharoni’s exchange resonance modes) are derived from micromagnetic equilibrium condition in the linear approximation. To investigate the uniform and symmetric resonance modes in ferromagnetic nanoscale grains, the microwave permeability of FeCo-based nanocrystalline alloy particles/paraffin composites was measured and calculated in the range 0.5–18 GHz. The measured dynamic permeability curves exhibit a broad resonance band at 4–6 GHz; some curves also exhibit a narrow resonance band at 13 GHz. The former behavior is in qualitative agreement with the uniform mode, and the latter is attributed to the first eigenvalue mode of the symmetric resonance modes excited in nanocrystalline monodomain grains in FeCo-based alloys. The difference value (Do11) between the uniform resonance frequency and the first frequency eigenvalue of the symmetric resonance modes shows good agreement with experiment. & 2010 Elsevier B.V. All rights reserved.
Keywords: Nanocrystalline alloy Ferromagnetic resonance Uniform mode Symmetric modes
1. Introduction Current domain theory is based largely on the wall concept and on the 1935s paper of Landau and Lifshitz [1]. The domain theory works reasonably well with macroscopic scale magnetic material, but at a microscopic level, magnetic domains are difficult to be postulated and subsequently proved theoretically. The theory of micromagnetics should in principle give a self-consistent description of all magnetization processes, from which the domain and wall concepts when valid will emerge naturally, without having to be postulated [2]. However, micromagnetics rigorously deals with almost all energy terms, which make micromagnetic calculations extremely difficult. The hardest part in micromagnetics is the accuracy of magnetostatic energy calculation. To solve the problem, linearization of Brown’s equations has been proved possible in the study of the approach to saturation magnetization and ferromagnetic resonance estimation. Moreover, when the microscopic level is reached, especially the nanoscale size, the effects which could be neglected to simplify micromagnetic calculations in terms of magnetostatic energy happen to be negligible [3]. Over the past decades, mechanisms of ferromagnetic resonance have been studied with the theory of micromagnetics in ferromagnetic nanomaterials, For example [4–8], a multiresonance behavior, which could be qualitatively related to exchange resonance modes proposed from micromagnetic equilibrium conditions (Brown’s equations), has been detected with submicrometer
n
Corresponding author. E-mail address: [email protected] (X. Wang).
0304-8853/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2010.10.033
sized particles of various ferromagnetic compositions. However, those monodispersed, submicrometer sized particles have usually been obtained by precipitation in liquid polyols from cobalt hydroxide and polymetallic mixed hydroxides. The determination of the magnetic properties of metallic particles at high frequencies requires them to be insulated from each other so that the material is nonconducting. Consequently, the powders have usually been dispersed randomly in an epoxy resin or an elastomeric matrix with certain volume concentration. The irreversible aggregation of small particles to form clusters is a central problem in the preparation process, hence the determination of actual particle size becomes difficult. Furthermore, it is not feasible to apply a high saturation magnetic field to a sample when the microwave properties of the sample were measured with the open-ended coaxial transmission line technique. Hence the deviation of the size dependence in Ref. [5] could be explained by the unsaturated magnetization of spherical ferromagnetic particles, which makes the condition of linear approximation unsatisfied and equivalently reduces the size dependence. Nanocrystalline structures are refered to as polycrystalline materials with grain sizes in the range 1–50 nm. In this work, homogeneous nanocrystalline grains surrounded by magnetism intergranular phase have been proved equivalent to monodomain under optimal annealing condition. Therefore nanocrystalline ferromagnetic alloys can be considered as experimental observation of ferromagnetic resonance in micromagnetics. Considering structural properties and microwave characterizations, FeCo-based nanocrystalline flakes over a range of composition and grain size have been prepared and analyzed. Using micromagnetics, we have focused our attention on the mechanism of magnetic resonance of
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FeCo-based nanocrystalline ferromagnetic flake composites. By ignoring the magnetostatic potential gradient in the symmetric case, we have derived uniform and symmetric resonance modes from the micromagnetic equations, which describe the dynamic properties of the single-domain states to explain the mechanism of ferromagnetic resonance in FeCo-based nanocrystalline alloys.
The condition that these equations have a common, nonzero solution is that the determinant of the coefficients of mx0 and my0 vanishes. Expanding this determinant and substituting for Hz in Eq. (4), the resonance frequencies (o) are found as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi o 2K1 2K1 7 ¼ H0 þ ðNx Nz ÞMs þ H0 þðNy Nz ÞMs þ : ð6Þ Ms Ms g0
2. Micromagnetic solution of ferromagnetic resonance
Let the shape of the specimen is sphere, whereNx ¼ Ny ¼ Nz , then Eq. (6) becomes
For a ferromagnet without losses, the general theory of resonance was formulated by Brown as following (in cgs units) [2]: C 2 1 @ oa 1 dm m , ð1Þ þH ¼ r m Ms Ms @m g0 dt
o=g0 ¼ Hz þNx Ms ¼ Hz þ Ny Ms ¼ H0 þ 2K1 =Ms :
where H is the magnetic field composed of the applied external field and the field created by the volume and surface charges of the magnetization distribution; m is a unit vector in the direction of magnetization; g0 the gyromagnetic ratio; C the exchange constant; oa the anisotropy energy density; Ms the saturation magnetization and t the time. If a large dc field H0 is applied along the z-axis, the components of the magnetization along x and y axes, mx and my are small. To the first order approximation, these conditions lead to the linearized equations: C 2 1 dmy @Vin ¼ r Hz mx Ms g0 dt @x C 2 1 dmx @Vin ¼ , ð2Þ r Hz my þ Ms g0 dt @y where Vin is the potential only due to the transverse magnetization, and the potential due to the z component is included in Hz. For these linearized equations cubic or uniaxial anisotropies lead to the same expression, provided that z is the easy axis Hz ¼ H0 Nz Ms þ 2K1 =MS ,
2.1. Uniform mode
ð4Þ
o m ¼0 g0 y0
o m ðHz þ Ny Ms Þmy0 ¼ 0 g0 x0
In a recent article [8], Aharoni proposed a theoretical treatment of exchange resonance modes in fine ferromagnetic spheres, which gave an expression for their frequencies and predicted a size effect. Here it seems useful to recall the guidelines and perfect Aharoni’s theoretical treatment. Corresponding with uniform mode, we named it symmetric modes. Since strong but short-range exchange and weak but long-range magnetostatic interactions compete on nanostructural length scales, right-hand side of Eq. (3), the magnetostatic potential gradient, could be ignored if the diameter of sphere is small enough. Under the condition of this approximation, a good solution can be accomplished by means of simple mathematical methods for wave equation. The wave equation in terms of the spherical coordinates r, y and f becomes [9] 1 @ @U 1 @ @U r2 þ 2 r2 U þk2 U ¼ 2 sin y @r @y r @r r sin y @y þ
@2 U
1 2
r 2 sin y @f2
þ k2 U ¼ 0:
ð8Þ
Using the variable separation approach, the general solution U can be written as Uðr, y, fÞ ¼
1 X n X 1 X r r ½An cos rf þ Bn sin rf½Ci Jn ðki rÞ n¼0r¼0i¼0
r
þ Di Yn ðki rÞPn ðcos yÞ,
ð9Þ
where A, B, C, D and ki are real constants; Pn is the Legendre function; and Jn and Yn are the spherical Bessel functions; and r, n and i are integers. For the sphere, considering the simplest special solutions for Eq. (2) with right-hand side being neglected, the distribution of the dynamic magnetization along the coordinate x and y can be written as my ¼ B sinðotÞcosðfÞPn1 ðcos yÞJn ðmr=RÞ,
where mx0 and my0 are constant, o is the angular frequency. For this solution the boundary conditions are satisfied, and hx ¼ @Vin =@x and hy ¼ @Vin =@y are homogeneous linear functions of mx0 and my0, which involve the demagnetizing factors of the ellipsoid. Hence Eq. (2) is reduced to a pair of homogeneous linear simultaneous equations of mx0 and my0 as ðHz þNx Ms Þmx0
2.2. Symmetric modes
mx ¼ A cosðotÞcosðfÞPn1 ðcos yÞJn ðmr=RÞ
We can always find a solution mx ¼ mxo cos ot my ¼ myo sin ot,
Substituting Eq. (4) in Eq. (5), we obtain mx0 ¼ my0 .
ð3Þ
where K1 is the anisotropy constant, and Nz is the demagnetizing factor along z-axis. In the absence of losses, certain special forms of alternating magnetization with time-dependent factor can sustain themselves without the aid of an applied alternating field. The primary problem is to find these modes of natural oscillation; then we can find the response to an arbitrary applied alternating field by well known methods. Hence it is not impossible but particularly difficult to solve Eq. (2) and obtain analytical solution. However the particular solution can be found by the same method used for evaluating the eigenvalues of nucleation modes which are mixed by surface anisotropy in a sphere [4].
ð7Þ
ð5Þ
ð10Þ
where m is the real constant; R is the radius of sphere. Substituting Eq. (10) in Eq. (2), it is seen that the former is indeed a solution of the latter, provided C m2 o þ H A B ¼ 0 z g0 R2 Ms o C m2 A 2 þ Hz B ¼ 0, ð11Þ g0 R Ms The condition that these equations have a common, nonzero solution is that the determinant of the coefficients of A and B vanished. Expanding this determinant, the resonance frequencies
X. Wang et al. / Journal of Magnetism and Magnetic Materials 323 (2011) 635–640
637
o are
o C m2 ¼ þ Hz : g0 R2 Ms
ð12Þ
The differential equations are now satisfied, and it is only necessary to fulfill the boundary condition. In the absence of surface anisotropy, the parameter m must assume one of the roots of the equations djn ðuÞ=du ¼ 0,
for n Z 1,
ð13Þ
3. Experimental details Alloy ingots with different compositions [(Fe0.67Co0.33)78Nb6B15Cu1 and (Fe0.5Co0.5)78Nb6B15Cu1], were prepared by an induction melting technique. Amorphous alloys were produced in the form of ribbon by a single roller melt-spinning method. Subsequent isothermal heat treatments above the primary crystallization temperature, heat treatment at 500 1C for 0.5, 1, 1.5 and 2 h for (Fe0.67Co0.33)78Nb6B15Cu1, 550 1C for 1 h for (Fe0.5Co0.5)78Nb6B15Cu1 stabilized the nanocrystalline microstructure. Subsequently, these ribbons were milled for 24, 36 and 48 h with 250 rpm by the high-energy planetary ball milling. In the whole milling process, anhydrous acetone was added in the jar as medium. The ball-to-powder weight ratio is 20:1. The morphology and structure of the samples were analyzed by scanning electron microscope (SEM) (JSM-6490LV) and X-ray diffractometry (Philips X’Pert). The composite samples for high frequency measurement were prepared by mixing the alloy particles with paraffin with 20% volume concentration of the particles. The microwave properties of flakesparaffin composites were measured in the range 0.5–18 GHz with an APC7 coaxial line associated with an Agilent 8720ET vector network analyzer. This technique is conventionally referred to as the Nicolson– Ross method [10]. Measurements were conducted in the absence of an external magnetic field. In the following, the complex permeability of flake composites rather than intrinsic permeability of ferromagnetic particles are studied for analyzing microwave resonance, based on the prior observation of our laboratory study [7].
Fig. 1. Scanning electron microscopy (SEM) image of (Fe0.5Co0.5)78Nb6B15Cu1 flakes prepared by high-energy ball milling.
4. Results and discussion 4.1. Flake characterization Fig. 1 shows the morphologies of (Fe0.5Co0.5)78Nb6B15Cu1 annealed alloy after 24 h high-energy ball milling. Mechanical alloying is often used for fabricating alloys directly from elemental precursors by activating chemical reactions and structural changes. In this paper, it is used to deform as-quenched ribbons into thin flake particles so that the eddy currents can be reduced. The flakes have an ultrathin thickness of less than 0.2 mm, and the particle size is homogeneously distributed around 1 mm as shown in Fig. 1. Therefore the contributions of eddy currents on the permeability of our flakes in composite which is justified for mean diameter lower than 2 mm can be neglected [11]. Fig. 2 shows the typical X-ray diffraction spectra of the (Fe0.5Co0.5)78Nb6B15Cu1 alloy powders (all for 24 h ball milling) for different anneal time 0.5, 1, 1.5 and 2 h. The initial amorphous structure is virtually not affected by short-time annealing (0.5 h). With increasing the annealing time (1 h), the diffraction peaks of the nanocrystalline phases emerge from the amorphous phase. They have been assigned to a a0 -FeCo solid solution. For the longest annealing times (1.5 and 2 h), additional diffraction peaks related to a mixture of boride phases emerge from the background. The semiempirical Scherrer analysis can be used to estimate the grain size for the diffraction peaks using the equation [12] D ¼ 0:89l=ðB cos yÞ,
ð14Þ
Fig. 2. XRD data for alloyed flakes with different anneal times.
Table 1 Structural properties of nanocrystalline alloyed flakes. Anneal
0.5 h
1h
1.5 h
B (rad.) y (rad.) D (nm)
0.059 0.778 3.2
0.023 0.777 8.4
0.01 0.779 19.3
where the grain thickness D is inversely proportional to the X-ray half peak breadth B (in radians) and the cosine of the Bragg angle y. The Scherrer analysis for these alloys is summarized in Table 1.
4.2. Nanocrystalline grain Nanocrystalline alloys have particular ultrafine grain structures with a majority of grain diameters in a typical range of 1–50 nm. Amorphous alloys crystallizing at temperatures above their primary crystallization temperature but below the secondary crystallization temperature can yield nanocrystalline grains embedded in
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an amorphous matrix. Crystallization states can be optimized to yield 1–50 nm a-FeSi, a-Fe, a- or a0 -FeCo particles surrounded by a thin TL–TE–B–Cu intergranular phase, where TL denotes a late (ferromagnetic) transition metal element (Co, Ni, or Fe), TE is an early transition metal element (Zr, Nb, Hf, Ta, etc.). This is the principle of the processing routes used to produce the interesting FINEMET, NANOPERM and HITPERM alloys. These materials differ in the nanocrystalline phase which forms with positive exchange integral and a large atomic magnetic moment and in the relative amounts of non-magnetic addition to the original amorphous phase with negative exchange integral [13]. The theoretical single-domain critical size (radius R) of an isolated sphere is larger than [14] rffiffiffiffiffiffi q2 3C Rc0 ¼ ð15Þ , q2 ¼ 2:0816 Ms 4p and smaller than the smaller of the two entities pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2:2646 C=pMs2 ffi, Rc1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 11:4038ð9K1 9Þ=ðpMs2 Þ
ð16AÞ
and Rc2 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 Cð9K1 9 þ 8psMs2 Þ 8ð3s2ÞMs2
,
s ¼ 0:785398,
ð16BÞ
Fig. 3. Imaginary parts (m00 ) of the complex permeabilties of (Fe0.67Co0.33)78Nb6B15Cu1 flake composites prepared under varied conditions.
where C is the exchange constant. For a soft material, these bounds are close. For Fe the bounds imply a critical radius between 8.46 and 11.0 nm. Therefore, under optimal annealing condition, homogeneous nanocrystalline grains surrounded by magnetism intergranular phase can be considered as monodomain. 4.3. Experimental observation of resonance modes In Section 2 we have discussed two different resonance modes, uniform resonance mode and symmetric resonance mode, the former is an analytic solution on a homogeneously magnetized, single-domain, ellipsoidal shaped particle, and the latter has been demonstrated theoretically on submicrometer sized, homogeneously magnetized, ferromagnetic sphere shaped particles [5–7]. In the nucleation theory, only the one with the smallest eigenvalue has a physical meaning, but in case of symmetric resonance modes, several modes with different eigenvalues could be excited simultaneously. Therefore, multi-resonance peaks can be observed in a nanoscale sphere magnetized to saturation, which normally also exhibit a single resonance peak associated with uniform resonance mode. The dynamic permeability curves of (Fe0.67Co0.33)78Nb6B15Cu1 flake composites with different ball milling time 24, 36, 48 h after 1 h anneal at 500 1C are illustrated as the square symbols in Fig. 3, which exhibit a broad resonance band with several shoulders around 6 GHz, and present main resonance peaks with different frequencies. Sphere symbols in Fig. 3 show the variation of m00 (o) in terms of milling time for 1 h anneal at 550 1C. All three different m00 (o) spectra all exhibit two distinguishable resonance bands: a broad resonance peak at 4–6 GHz and a narrow resonance peak at 13 GHz. There is always a main broad resonance peak observed with flake composite of different milling times and different annealed temperatures. The resonance frequency of this main peak can be interpreted in terms of the influence of the magnetic anisotropy as for the uniform mode resonance. Consideration of frequency dependence of permittivity is useful in distinguishing actual dispersive behavior of permeability from measurement uncertainties. With false magnetic peaks, the permittivity shows an opposite loss peak at the same frequency so that the measured S-parameters are smoother functions of frequency [15]. As shown in Fig. 4, the imaginary parts (e00 ) of the complex permittivity of
Fig. 4. Frequency dependence of permittivity of (Fe0.67Co0.33)78Nb6B15Cu1 flake composites.
flake composites does not exhibit any resonance bands above 4 GHz. The magnetic anisotropy may arise from the shape of the particle, magnetocrystalline effects and stress. It is difficult to determine the value of all kinds of magnetic anisotropy of the flakes [16]. However, a0 -FeCo phase presents cubic magnetocrystalline, in which K1 and K2 are the first two anisotropy energy constants. Cubic magnetocrystalline may lead to double resonance peaks or one broad overlapping peak, which can be interpreted qualitatively as the difference values between K1 and K2. Let us consider that nanocrystalline grains in flake after 1 h annealing at 550 1C as independent monodomain particles. In Section 2, in a sphere-shape
X. Wang et al. / Journal of Magnetism and Magnetic Materials 323 (2011) 635–640
specimen, the uniform mode resonance frequency is given by
o=g0 ¼ H0 þ 2K1 =Ms ,
ð17Þ
where H0 is static magnetic field applied, hence the frequency width of main peaks can be written as
Do=g0 ¼ 2ðK1 K2 Þ=Ms :
ð18Þ
In terms of the static magnetic properties, these independent monodomain ferromagnetic particles equates to bulk alloy with the same composition as nanocrystalline grain. By approximately taking [17] K1 ¼48.1 103 J m 3 and K2 ¼12 103 J m 3 in bcc-Fe, and Ms ¼1.7 106 A m 1, the value of frequency width are calculated as Do ¼1.19 GHz. Nevertheless, the experimental resonance frequency widths are always wider than those calculated from uniform mode resonance for a sphere. This can be explained by complex magnetic anisotropy in flakes and heterogeneous internal demagnetizing field in non-ideal nanocrystalline two-phase alloy. Particularly, the narrow resonance peaks around 13 GHz are observed with different milling time after 1 h anneal at 550 1C. The shape, size and distribution of flakes in composites always vary with the milling time [18]. Therefore, narrow peaks can be related to phase structure rather than morphology of flakes because the only characteristic difference between samples with different annealed temperature is crystallization states. The relationship between this narrow resonance peaks and crystallization should be discovered by more precise experiments appropriately, we will study this topic later. The complex permittivities and permeabilities of (Fe0.5Co0.5)78 Nb6B15Cu1 flake composites with varied annealed holding time 0.5, 1, 1.5 and 2 h are presented in Fig. 5. After milling for 24 h, m00 (o) exhibits a broad resonance band with several shoulders near resonance peak around 5 GHz except for 1 h where there is a higher narrow distinct resonance peak around 12 GHz. Considering the effect of heat treatments on crystallization states, as shown in Fig. 5, the higher resonance peak might be interpreted as some kind of resonance mode,
639
symmetric resonance mode assumed, excited only in proper crystallization conditions. According to the symmetric modes in Section 2, the resonance frequencies in sphere and the saturation magnetization are related as
C m2 þ Hz R2 Ms
¼
o , g0
Hz ¼ H0 Nz Ms þ
2K1 , Ms
ð19Þ
ð20Þ
Again, considering nanocrystalline grains in flake particles with 1 h anneal are equivalent to independent monodomain particles. Hz is only attributed to magnetic anisotropy field because permeability measurements were performed without any static magnetic field applied. Furthermore, uniform magnetization for single monodomain particles always generates a demagnetizing field in the opposite direction of magnetization; however, the internal demagnetizing field induced by each grain could be vanished in randomly oriented nanocrystalline grains homogeneously distributed in a non-magnetic amorphous matrix. It is difficult to determine the value of the magnetic anisotropy field, therefore, as we have discussed above, the difference value of frequency between the broad resonance band and the narrow resonance peak is given by C m2 Do ¼ : g0 R2 Ms
ð21Þ
By approximately taking [17] Ms ¼1.7 106 A m 1, C ¼6.8 10 12 J/m, g0 ¼ g/2p ¼3.52 104 m A 1 s 1, the two first mkn roots which are m11 ¼2.08, m21 ¼3.34 and R¼8 10 9 m for 1 h annealed flakes combined with the flakes characterization shown in Table 1, the first two values of frequency difference between two kinds of modes are calculated to be Do11 ¼7.6 GHz, Do21 ¼19.5 GHz. The difference value of frequency between uniform mode and
Fig. 5. Complex permittivity and permeability of (Fe0.5Co0.5)78Nb6B15Cu1 flake composites with varied anneal holding times.
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X. Wang et al. / Journal of Magnetism and Magnetic Materials 323 (2011) 635–640
symmetric mode of the first eigenvalue Do11 shows a good agreement with experiment. The rest of permeability curves in Fig. 5 perform a broad resonance bands without the narrow resonance peak: according to the analysis of X-ray diffraction data, when annealed time is 0.5 h, a low crystallinity is observed and ferromagnetic elements are widely distributed among crystalline and amorphous phase, therefore the hypothesis in terms of equivalent sphere does not hold true; whereas when time is 1.5 or 2 h, high crystallinity with a mixture of boride phases formed brings about bigger grains and a non-uniform distribution of magnetic moment, which also make the hypothesis of monodomain invalid. However, the uniform mode is always qualitatively agreed with experimental observation. The magnetocrystalline anisotropy of most materials reflects the competition between electrostatic crystal-field interaction and spin–orbit coupling. The crystal field which reflects the local symmetry of the crystal or surface and acts on the orbits of the inner-shell d and f electrons is derived from short-range order [19]. There are several kinds of magnetic anisotropy exhibited in amorphous and nanocrystalline alloys, including magnetocrystalline anisotropy which is independent of the crystallinity, shape anisotropy and magnetoelastic anisotropy. Magnetocrystalline anisotropy contributes to the main peaks of uniform mode and the all other anisotropies broaden peaks.
By comparing the resonance modes deduced with micromagnetic solution approach, we found that the uniform mode is always qualitatively in agreement with experimental observation on FeCo-based nanocrystalline alloys, and contributed by anisotropy field; the symmetric resonance modes could be excited in nanocrystalline grains under optimal annealing condition. Finally, we proved that the difference value of frequency between uniform mode and the first eigenvalue Do11 of the symmetric mode shows good agreement with experiment. Accordingly, the micromagnetic solution in terms of ferromagnetic resonance mode in FeCo-based nanocrystalline alloys has been qualitatively related to the experimental observation.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
5. Conclusion In this paper, we have studied the solving approach of Brown’s differential equation in micromagnetics with the linear approximation. As a result, the uniform resonance mode and the symmetric resonance mode have been deduced. Experimentally, we have studied the structure characteristics and magnetic properties of FeCo-based nanocrystalline alloys. Homogeneous nanocrystalline grains surrounded by magnetism intergranular phase have been considered as monodomain under optimal annealing condition.
[12] [13] [14] [15] [16] [17] [18] [19]
L. Landau, E. Lifshits, Phys. Z. Sowjetunion 8 (1935) 153. W.F. Brown JR., J. Appl. Phys. 30 (4) (1959) 62S. A. Aharoni., Physica B. 306 (2001) 1. A. Aharoni, J. Appl. Phys. 81 (2) (1997) 830. G. Viau, F. Fievet-Vincent, F. Fievet, P. Toneguzzo, F. Ravel, O. Acher, J. Appl. Phys. 81 (6) (1997) 2749. Ph. Toneguzzo, O. Acher, G. Viau, F. Fievet-Vincent, F. Fievet, J. Appl. Phys. 81 (8) (1997) 5546. L.J. Deng, P.H. Zhou, J.L. Xie, L. Zhang, J. Appl. Phys. 101 (2007) 103916. A. Aharoni, J. Appl. Phys. 69 (11) (1991) 7762. N.O. Sadiku Matthew, Numerical Techniques in Electromagnetic, 2nd ed., CRC, 2001 (Chapter 2). A.M. Nicolson, G.F. Ross, IEEE Trans. Instrum. Meas. 19 (1970) 377. Fusheng Liang Qiao, Jianqiang Wen, Jianbo Wei, Wang, Fashen Li, J. Appl. Phys. 103 (2008) 063903. B.D. Cullity, Elements of X-ray Diffraction, Addison-Wesley, Reading, MA, 1978, p. 44. Michael E. Mchenry, Matthew A. Willard, David E. Laughlin, Prog. Mater. Sci. 44 (1999) 291. W.F. Brown Jr., J. Appl. Phys. 39 (1968) 993. A.N. Lagarkov, K.N. Rozanov, J. Magn. Magn. Mater. 321 (2009) 2082. E.C. Stoner, E.P. Wohlfarth, IEEE Trans. Magn. 27 (4) (1991) 3475. D.F. Wan, X.L. Ma, Magnetic Physics, Publishing House of Electronics Industry, Beijing, 1999. H.R. Madaah Hosseini, A. Bahrami, J. Mater. Sci. Eng. B 123 (2005) 78. R. Skomski, D.J. Sellmyer, Handbook of Advanced Magnetic Materials, vol. I, Tsinghua University Press, 2005 p.11.
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Observations of ferromagnetic resonance modes on FeCo-based nanocrystalline alloys X. Wang n, L.J. Deng, J.L. Xie, D. Li State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu, China
a r t i c l e in f o
abstract
Article history: Received 20 May 2010 Received in revised form 19 October 2010 Available online 28 October 2010
Uniform and symmetric resonance modes (known as Aharoni’s exchange resonance modes) are derived from micromagnetic equilibrium condition in the linear approximation. To investigate the uniform and symmetric resonance modes in ferromagnetic nanoscale grains, the microwave permeability of FeCo-based nanocrystalline alloy particles/paraffin composites was measured and calculated in the range 0.5–18 GHz. The measured dynamic permeability curves exhibit a broad resonance band at 4–6 GHz; some curves also exhibit a narrow resonance band at 13 GHz. The former behavior is in qualitative agreement with the uniform mode, and the latter is attributed to the first eigenvalue mode of the symmetric resonance modes excited in nanocrystalline monodomain grains in FeCo-based alloys. The difference value (Do11) between the uniform resonance frequency and the first frequency eigenvalue of the symmetric resonance modes shows good agreement with experiment. & 2010 Elsevier B.V. All rights reserved.
Keywords: Nanocrystalline alloy Ferromagnetic resonance Uniform mode Symmetric modes
1. Introduction Current domain theory is based largely on the wall concept and on the 1935s paper of Landau and Lifshitz [1]. The domain theory works reasonably well with macroscopic scale magnetic material, but at a microscopic level, magnetic domains are difficult to be postulated and subsequently proved theoretically. The theory of micromagnetics should in principle give a self-consistent description of all magnetization processes, from which the domain and wall concepts when valid will emerge naturally, without having to be postulated [2]. However, micromagnetics rigorously deals with almost all energy terms, which make micromagnetic calculations extremely difficult. The hardest part in micromagnetics is the accuracy of magnetostatic energy calculation. To solve the problem, linearization of Brown’s equations has been proved possible in the study of the approach to saturation magnetization and ferromagnetic resonance estimation. Moreover, when the microscopic level is reached, especially the nanoscale size, the effects which could be neglected to simplify micromagnetic calculations in terms of magnetostatic energy happen to be negligible [3]. Over the past decades, mechanisms of ferromagnetic resonance have been studied with the theory of micromagnetics in ferromagnetic nanomaterials, For example [4–8], a multiresonance behavior, which could be qualitatively related to exchange resonance modes proposed from micromagnetic equilibrium conditions (Brown’s equations), has been detected with submicrometer
n
Corresponding author. E-mail address: [email protected] (X. Wang).
0304-8853/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2010.10.033
sized particles of various ferromagnetic compositions. However, those monodispersed, submicrometer sized particles have usually been obtained by precipitation in liquid polyols from cobalt hydroxide and polymetallic mixed hydroxides. The determination of the magnetic properties of metallic particles at high frequencies requires them to be insulated from each other so that the material is nonconducting. Consequently, the powders have usually been dispersed randomly in an epoxy resin or an elastomeric matrix with certain volume concentration. The irreversible aggregation of small particles to form clusters is a central problem in the preparation process, hence the determination of actual particle size becomes difficult. Furthermore, it is not feasible to apply a high saturation magnetic field to a sample when the microwave properties of the sample were measured with the open-ended coaxial transmission line technique. Hence the deviation of the size dependence in Ref. [5] could be explained by the unsaturated magnetization of spherical ferromagnetic particles, which makes the condition of linear approximation unsatisfied and equivalently reduces the size dependence. Nanocrystalline structures are refered to as polycrystalline materials with grain sizes in the range 1–50 nm. In this work, homogeneous nanocrystalline grains surrounded by magnetism intergranular phase have been proved equivalent to monodomain under optimal annealing condition. Therefore nanocrystalline ferromagnetic alloys can be considered as experimental observation of ferromagnetic resonance in micromagnetics. Considering structural properties and microwave characterizations, FeCo-based nanocrystalline flakes over a range of composition and grain size have been prepared and analyzed. Using micromagnetics, we have focused our attention on the mechanism of magnetic resonance of
636
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FeCo-based nanocrystalline ferromagnetic flake composites. By ignoring the magnetostatic potential gradient in the symmetric case, we have derived uniform and symmetric resonance modes from the micromagnetic equations, which describe the dynamic properties of the single-domain states to explain the mechanism of ferromagnetic resonance in FeCo-based nanocrystalline alloys.
The condition that these equations have a common, nonzero solution is that the determinant of the coefficients of mx0 and my0 vanishes. Expanding this determinant and substituting for Hz in Eq. (4), the resonance frequencies (o) are found as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi o 2K1 2K1 7 ¼ H0 þ ðNx Nz ÞMs þ H0 þðNy Nz ÞMs þ : ð6Þ Ms Ms g0
2. Micromagnetic solution of ferromagnetic resonance
Let the shape of the specimen is sphere, whereNx ¼ Ny ¼ Nz , then Eq. (6) becomes
For a ferromagnet without losses, the general theory of resonance was formulated by Brown as following (in cgs units) [2]: C 2 1 @ oa 1 dm m , ð1Þ þH ¼ r m Ms Ms @m g0 dt
o=g0 ¼ Hz þNx Ms ¼ Hz þ Ny Ms ¼ H0 þ 2K1 =Ms :
where H is the magnetic field composed of the applied external field and the field created by the volume and surface charges of the magnetization distribution; m is a unit vector in the direction of magnetization; g0 the gyromagnetic ratio; C the exchange constant; oa the anisotropy energy density; Ms the saturation magnetization and t the time. If a large dc field H0 is applied along the z-axis, the components of the magnetization along x and y axes, mx and my are small. To the first order approximation, these conditions lead to the linearized equations: C 2 1 dmy @Vin ¼ r Hz mx Ms g0 dt @x C 2 1 dmx @Vin ¼ , ð2Þ r Hz my þ Ms g0 dt @y where Vin is the potential only due to the transverse magnetization, and the potential due to the z component is included in Hz. For these linearized equations cubic or uniaxial anisotropies lead to the same expression, provided that z is the easy axis Hz ¼ H0 Nz Ms þ 2K1 =MS ,
2.1. Uniform mode
ð4Þ
o m ¼0 g0 y0
o m ðHz þ Ny Ms Þmy0 ¼ 0 g0 x0
In a recent article [8], Aharoni proposed a theoretical treatment of exchange resonance modes in fine ferromagnetic spheres, which gave an expression for their frequencies and predicted a size effect. Here it seems useful to recall the guidelines and perfect Aharoni’s theoretical treatment. Corresponding with uniform mode, we named it symmetric modes. Since strong but short-range exchange and weak but long-range magnetostatic interactions compete on nanostructural length scales, right-hand side of Eq. (3), the magnetostatic potential gradient, could be ignored if the diameter of sphere is small enough. Under the condition of this approximation, a good solution can be accomplished by means of simple mathematical methods for wave equation. The wave equation in terms of the spherical coordinates r, y and f becomes [9] 1 @ @U 1 @ @U r2 þ 2 r2 U þk2 U ¼ 2 sin y @r @y r @r r sin y @y þ
@2 U
1 2
r 2 sin y @f2
þ k2 U ¼ 0:
ð8Þ
Using the variable separation approach, the general solution U can be written as Uðr, y, fÞ ¼
1 X n X 1 X r r ½An cos rf þ Bn sin rf½Ci Jn ðki rÞ n¼0r¼0i¼0
r
þ Di Yn ðki rÞPn ðcos yÞ,
ð9Þ
where A, B, C, D and ki are real constants; Pn is the Legendre function; and Jn and Yn are the spherical Bessel functions; and r, n and i are integers. For the sphere, considering the simplest special solutions for Eq. (2) with right-hand side being neglected, the distribution of the dynamic magnetization along the coordinate x and y can be written as my ¼ B sinðotÞcosðfÞPn1 ðcos yÞJn ðmr=RÞ,
where mx0 and my0 are constant, o is the angular frequency. For this solution the boundary conditions are satisfied, and hx ¼ @Vin =@x and hy ¼ @Vin =@y are homogeneous linear functions of mx0 and my0, which involve the demagnetizing factors of the ellipsoid. Hence Eq. (2) is reduced to a pair of homogeneous linear simultaneous equations of mx0 and my0 as ðHz þNx Ms Þmx0
2.2. Symmetric modes
mx ¼ A cosðotÞcosðfÞPn1 ðcos yÞJn ðmr=RÞ
We can always find a solution mx ¼ mxo cos ot my ¼ myo sin ot,
Substituting Eq. (4) in Eq. (5), we obtain mx0 ¼ my0 .
ð3Þ
where K1 is the anisotropy constant, and Nz is the demagnetizing factor along z-axis. In the absence of losses, certain special forms of alternating magnetization with time-dependent factor can sustain themselves without the aid of an applied alternating field. The primary problem is to find these modes of natural oscillation; then we can find the response to an arbitrary applied alternating field by well known methods. Hence it is not impossible but particularly difficult to solve Eq. (2) and obtain analytical solution. However the particular solution can be found by the same method used for evaluating the eigenvalues of nucleation modes which are mixed by surface anisotropy in a sphere [4].
ð7Þ
ð5Þ
ð10Þ
where m is the real constant; R is the radius of sphere. Substituting Eq. (10) in Eq. (2), it is seen that the former is indeed a solution of the latter, provided C m2 o þ H A B ¼ 0 z g0 R2 Ms o C m2 A 2 þ Hz B ¼ 0, ð11Þ g0 R Ms The condition that these equations have a common, nonzero solution is that the determinant of the coefficients of A and B vanished. Expanding this determinant, the resonance frequencies
X. Wang et al. / Journal of Magnetism and Magnetic Materials 323 (2011) 635–640
637
o are
o C m2 ¼ þ Hz : g0 R2 Ms
ð12Þ
The differential equations are now satisfied, and it is only necessary to fulfill the boundary condition. In the absence of surface anisotropy, the parameter m must assume one of the roots of the equations djn ðuÞ=du ¼ 0,
for n Z 1,
ð13Þ
3. Experimental details Alloy ingots with different compositions [(Fe0.67Co0.33)78Nb6B15Cu1 and (Fe0.5Co0.5)78Nb6B15Cu1], were prepared by an induction melting technique. Amorphous alloys were produced in the form of ribbon by a single roller melt-spinning method. Subsequent isothermal heat treatments above the primary crystallization temperature, heat treatment at 500 1C for 0.5, 1, 1.5 and 2 h for (Fe0.67Co0.33)78Nb6B15Cu1, 550 1C for 1 h for (Fe0.5Co0.5)78Nb6B15Cu1 stabilized the nanocrystalline microstructure. Subsequently, these ribbons were milled for 24, 36 and 48 h with 250 rpm by the high-energy planetary ball milling. In the whole milling process, anhydrous acetone was added in the jar as medium. The ball-to-powder weight ratio is 20:1. The morphology and structure of the samples were analyzed by scanning electron microscope (SEM) (JSM-6490LV) and X-ray diffractometry (Philips X’Pert). The composite samples for high frequency measurement were prepared by mixing the alloy particles with paraffin with 20% volume concentration of the particles. The microwave properties of flakesparaffin composites were measured in the range 0.5–18 GHz with an APC7 coaxial line associated with an Agilent 8720ET vector network analyzer. This technique is conventionally referred to as the Nicolson– Ross method [10]. Measurements were conducted in the absence of an external magnetic field. In the following, the complex permeability of flake composites rather than intrinsic permeability of ferromagnetic particles are studied for analyzing microwave resonance, based on the prior observation of our laboratory study [7].
Fig. 1. Scanning electron microscopy (SEM) image of (Fe0.5Co0.5)78Nb6B15Cu1 flakes prepared by high-energy ball milling.
4. Results and discussion 4.1. Flake characterization Fig. 1 shows the morphologies of (Fe0.5Co0.5)78Nb6B15Cu1 annealed alloy after 24 h high-energy ball milling. Mechanical alloying is often used for fabricating alloys directly from elemental precursors by activating chemical reactions and structural changes. In this paper, it is used to deform as-quenched ribbons into thin flake particles so that the eddy currents can be reduced. The flakes have an ultrathin thickness of less than 0.2 mm, and the particle size is homogeneously distributed around 1 mm as shown in Fig. 1. Therefore the contributions of eddy currents on the permeability of our flakes in composite which is justified for mean diameter lower than 2 mm can be neglected [11]. Fig. 2 shows the typical X-ray diffraction spectra of the (Fe0.5Co0.5)78Nb6B15Cu1 alloy powders (all for 24 h ball milling) for different anneal time 0.5, 1, 1.5 and 2 h. The initial amorphous structure is virtually not affected by short-time annealing (0.5 h). With increasing the annealing time (1 h), the diffraction peaks of the nanocrystalline phases emerge from the amorphous phase. They have been assigned to a a0 -FeCo solid solution. For the longest annealing times (1.5 and 2 h), additional diffraction peaks related to a mixture of boride phases emerge from the background. The semiempirical Scherrer analysis can be used to estimate the grain size for the diffraction peaks using the equation [12] D ¼ 0:89l=ðB cos yÞ,
ð14Þ
Fig. 2. XRD data for alloyed flakes with different anneal times.
Table 1 Structural properties of nanocrystalline alloyed flakes. Anneal
0.5 h
1h
1.5 h
B (rad.) y (rad.) D (nm)
0.059 0.778 3.2
0.023 0.777 8.4
0.01 0.779 19.3
where the grain thickness D is inversely proportional to the X-ray half peak breadth B (in radians) and the cosine of the Bragg angle y. The Scherrer analysis for these alloys is summarized in Table 1.
4.2. Nanocrystalline grain Nanocrystalline alloys have particular ultrafine grain structures with a majority of grain diameters in a typical range of 1–50 nm. Amorphous alloys crystallizing at temperatures above their primary crystallization temperature but below the secondary crystallization temperature can yield nanocrystalline grains embedded in
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an amorphous matrix. Crystallization states can be optimized to yield 1–50 nm a-FeSi, a-Fe, a- or a0 -FeCo particles surrounded by a thin TL–TE–B–Cu intergranular phase, where TL denotes a late (ferromagnetic) transition metal element (Co, Ni, or Fe), TE is an early transition metal element (Zr, Nb, Hf, Ta, etc.). This is the principle of the processing routes used to produce the interesting FINEMET, NANOPERM and HITPERM alloys. These materials differ in the nanocrystalline phase which forms with positive exchange integral and a large atomic magnetic moment and in the relative amounts of non-magnetic addition to the original amorphous phase with negative exchange integral [13]. The theoretical single-domain critical size (radius R) of an isolated sphere is larger than [14] rffiffiffiffiffiffi q2 3C Rc0 ¼ ð15Þ , q2 ¼ 2:0816 Ms 4p and smaller than the smaller of the two entities pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2:2646 C=pMs2 ffi, Rc1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 11:4038ð9K1 9Þ=ðpMs2 Þ
ð16AÞ
and Rc2 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 Cð9K1 9 þ 8psMs2 Þ 8ð3s2ÞMs2
,
s ¼ 0:785398,
ð16BÞ
Fig. 3. Imaginary parts (m00 ) of the complex permeabilties of (Fe0.67Co0.33)78Nb6B15Cu1 flake composites prepared under varied conditions.
where C is the exchange constant. For a soft material, these bounds are close. For Fe the bounds imply a critical radius between 8.46 and 11.0 nm. Therefore, under optimal annealing condition, homogeneous nanocrystalline grains surrounded by magnetism intergranular phase can be considered as monodomain. 4.3. Experimental observation of resonance modes In Section 2 we have discussed two different resonance modes, uniform resonance mode and symmetric resonance mode, the former is an analytic solution on a homogeneously magnetized, single-domain, ellipsoidal shaped particle, and the latter has been demonstrated theoretically on submicrometer sized, homogeneously magnetized, ferromagnetic sphere shaped particles [5–7]. In the nucleation theory, only the one with the smallest eigenvalue has a physical meaning, but in case of symmetric resonance modes, several modes with different eigenvalues could be excited simultaneously. Therefore, multi-resonance peaks can be observed in a nanoscale sphere magnetized to saturation, which normally also exhibit a single resonance peak associated with uniform resonance mode. The dynamic permeability curves of (Fe0.67Co0.33)78Nb6B15Cu1 flake composites with different ball milling time 24, 36, 48 h after 1 h anneal at 500 1C are illustrated as the square symbols in Fig. 3, which exhibit a broad resonance band with several shoulders around 6 GHz, and present main resonance peaks with different frequencies. Sphere symbols in Fig. 3 show the variation of m00 (o) in terms of milling time for 1 h anneal at 550 1C. All three different m00 (o) spectra all exhibit two distinguishable resonance bands: a broad resonance peak at 4–6 GHz and a narrow resonance peak at 13 GHz. There is always a main broad resonance peak observed with flake composite of different milling times and different annealed temperatures. The resonance frequency of this main peak can be interpreted in terms of the influence of the magnetic anisotropy as for the uniform mode resonance. Consideration of frequency dependence of permittivity is useful in distinguishing actual dispersive behavior of permeability from measurement uncertainties. With false magnetic peaks, the permittivity shows an opposite loss peak at the same frequency so that the measured S-parameters are smoother functions of frequency [15]. As shown in Fig. 4, the imaginary parts (e00 ) of the complex permittivity of
Fig. 4. Frequency dependence of permittivity of (Fe0.67Co0.33)78Nb6B15Cu1 flake composites.
flake composites does not exhibit any resonance bands above 4 GHz. The magnetic anisotropy may arise from the shape of the particle, magnetocrystalline effects and stress. It is difficult to determine the value of all kinds of magnetic anisotropy of the flakes [16]. However, a0 -FeCo phase presents cubic magnetocrystalline, in which K1 and K2 are the first two anisotropy energy constants. Cubic magnetocrystalline may lead to double resonance peaks or one broad overlapping peak, which can be interpreted qualitatively as the difference values between K1 and K2. Let us consider that nanocrystalline grains in flake after 1 h annealing at 550 1C as independent monodomain particles. In Section 2, in a sphere-shape
X. Wang et al. / Journal of Magnetism and Magnetic Materials 323 (2011) 635–640
specimen, the uniform mode resonance frequency is given by
o=g0 ¼ H0 þ 2K1 =Ms ,
ð17Þ
where H0 is static magnetic field applied, hence the frequency width of main peaks can be written as
Do=g0 ¼ 2ðK1 K2 Þ=Ms :
ð18Þ
In terms of the static magnetic properties, these independent monodomain ferromagnetic particles equates to bulk alloy with the same composition as nanocrystalline grain. By approximately taking [17] K1 ¼48.1 103 J m 3 and K2 ¼12 103 J m 3 in bcc-Fe, and Ms ¼1.7 106 A m 1, the value of frequency width are calculated as Do ¼1.19 GHz. Nevertheless, the experimental resonance frequency widths are always wider than those calculated from uniform mode resonance for a sphere. This can be explained by complex magnetic anisotropy in flakes and heterogeneous internal demagnetizing field in non-ideal nanocrystalline two-phase alloy. Particularly, the narrow resonance peaks around 13 GHz are observed with different milling time after 1 h anneal at 550 1C. The shape, size and distribution of flakes in composites always vary with the milling time [18]. Therefore, narrow peaks can be related to phase structure rather than morphology of flakes because the only characteristic difference between samples with different annealed temperature is crystallization states. The relationship between this narrow resonance peaks and crystallization should be discovered by more precise experiments appropriately, we will study this topic later. The complex permittivities and permeabilities of (Fe0.5Co0.5)78 Nb6B15Cu1 flake composites with varied annealed holding time 0.5, 1, 1.5 and 2 h are presented in Fig. 5. After milling for 24 h, m00 (o) exhibits a broad resonance band with several shoulders near resonance peak around 5 GHz except for 1 h where there is a higher narrow distinct resonance peak around 12 GHz. Considering the effect of heat treatments on crystallization states, as shown in Fig. 5, the higher resonance peak might be interpreted as some kind of resonance mode,
639
symmetric resonance mode assumed, excited only in proper crystallization conditions. According to the symmetric modes in Section 2, the resonance frequencies in sphere and the saturation magnetization are related as
C m2 þ Hz R2 Ms
¼
o , g0
Hz ¼ H0 Nz Ms þ
2K1 , Ms
ð19Þ
ð20Þ
Again, considering nanocrystalline grains in flake particles with 1 h anneal are equivalent to independent monodomain particles. Hz is only attributed to magnetic anisotropy field because permeability measurements were performed without any static magnetic field applied. Furthermore, uniform magnetization for single monodomain particles always generates a demagnetizing field in the opposite direction of magnetization; however, the internal demagnetizing field induced by each grain could be vanished in randomly oriented nanocrystalline grains homogeneously distributed in a non-magnetic amorphous matrix. It is difficult to determine the value of the magnetic anisotropy field, therefore, as we have discussed above, the difference value of frequency between the broad resonance band and the narrow resonance peak is given by C m2 Do ¼ : g0 R2 Ms
ð21Þ
By approximately taking [17] Ms ¼1.7 106 A m 1, C ¼6.8 10 12 J/m, g0 ¼ g/2p ¼3.52 104 m A 1 s 1, the two first mkn roots which are m11 ¼2.08, m21 ¼3.34 and R¼8 10 9 m for 1 h annealed flakes combined with the flakes characterization shown in Table 1, the first two values of frequency difference between two kinds of modes are calculated to be Do11 ¼7.6 GHz, Do21 ¼19.5 GHz. The difference value of frequency between uniform mode and
Fig. 5. Complex permittivity and permeability of (Fe0.5Co0.5)78Nb6B15Cu1 flake composites with varied anneal holding times.
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symmetric mode of the first eigenvalue Do11 shows a good agreement with experiment. The rest of permeability curves in Fig. 5 perform a broad resonance bands without the narrow resonance peak: according to the analysis of X-ray diffraction data, when annealed time is 0.5 h, a low crystallinity is observed and ferromagnetic elements are widely distributed among crystalline and amorphous phase, therefore the hypothesis in terms of equivalent sphere does not hold true; whereas when time is 1.5 or 2 h, high crystallinity with a mixture of boride phases formed brings about bigger grains and a non-uniform distribution of magnetic moment, which also make the hypothesis of monodomain invalid. However, the uniform mode is always qualitatively agreed with experimental observation. The magnetocrystalline anisotropy of most materials reflects the competition between electrostatic crystal-field interaction and spin–orbit coupling. The crystal field which reflects the local symmetry of the crystal or surface and acts on the orbits of the inner-shell d and f electrons is derived from short-range order [19]. There are several kinds of magnetic anisotropy exhibited in amorphous and nanocrystalline alloys, including magnetocrystalline anisotropy which is independent of the crystallinity, shape anisotropy and magnetoelastic anisotropy. Magnetocrystalline anisotropy contributes to the main peaks of uniform mode and the all other anisotropies broaden peaks.
By comparing the resonance modes deduced with micromagnetic solution approach, we found that the uniform mode is always qualitatively in agreement with experimental observation on FeCo-based nanocrystalline alloys, and contributed by anisotropy field; the symmetric resonance modes could be excited in nanocrystalline grains under optimal annealing condition. Finally, we proved that the difference value of frequency between uniform mode and the first eigenvalue Do11 of the symmetric mode shows good agreement with experiment. Accordingly, the micromagnetic solution in terms of ferromagnetic resonance mode in FeCo-based nanocrystalline alloys has been qualitatively related to the experimental observation.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
5. Conclusion In this paper, we have studied the solving approach of Brown’s differential equation in micromagnetics with the linear approximation. As a result, the uniform resonance mode and the symmetric resonance mode have been deduced. Experimentally, we have studied the structure characteristics and magnetic properties of FeCo-based nanocrystalline alloys. Homogeneous nanocrystalline grains surrounded by magnetism intergranular phase have been considered as monodomain under optimal annealing condition.
[12] [13] [14] [15] [16] [17] [18] [19]
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