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Hysteresis losses in nanocrystalline alloys with magnetic-field-induced anisotropy
Journal of Magnetism and Magnetic Materials 479 (2019) 19–26
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Research articles
Hysteresis losses in nanocrystalline alloys with magnetic-field-induced anisotropy
T
⁎
Yu.N. Starodubtseva,b, V.A. Kataeva, , K.O. Bessonovaa, V.S. Tsepeleva a b
Boris Yeltsin Ural Federal University, Mira str. 19, Yekaterinburg 620002, Russia Gammamet Research and Production Enterprise, Tatishchev str. 92, Yekaterinburg 620028, Russia
A R T I C LE I N FO
A B S T R A C T
Keywords: Hysteresis losses Induced anisotropy Soft magnet Nanocrystalline alloy
The hysteresis losses and their relation to the parameters of the minor static hysteresis loops were investigated in the nanocrystalline Fe67.5Co5Cu1Nb2Mo1.5Si14B9 alloy with magnetic-field-induced anisotropy. The analysis of experimental data was performed using a power function, which approximated the dependencies between the hysteresis parameters. It is shown that all the experimental dependences in cores with magnetic-field-induced anisotropy in a weak magnetic field are consistent with the formulas derived from the Rayleigh law. It is also shown, that in a nanocrystalline alloy with uniaxial induced anisotropy, certain empirical relationships obtained earlier for isotropic materials are not met, particularly, for the relation of the hysteresis losses to the remanence and the maximal magnetic induction. It has been found, that samples with longitudinal induced anisotropy demonstrate low initial permeability, and the magnetization process in the Rayleigh region is carried out by the domain walls’ displacement at distances comparable to the correlation length Lex. A magnetic hysteresis mechanism associated with the irreversible magnetization rotation in ferromagnetic clusters of nanocrystalline alloys is proposed. Formulas are proposed to calculate hysteresis losses in a soft magnetic nanocrystalline material with a different magnetic anisotropy.
1. Introduction
characterized by the local magnetic anisotropy constant 〈K1〉 and the randomly selected common easy magnetization axis [2]. For the exchange coupled grains of 10 nm size, the effective local magnetic anisotropy constant 〈K1〉 decreases by three orders compared to the magnetocrystalline anisotropy constant K1. Thus, we can assume that the nanocrystalline material consists of the exchange correlation spaces or ferromagnetic clusters. After annealing in a magnetic field, the local magnetic anisotropy of the nanocrystalline alloy is overlaid by the uniaxial magnetic-field-induced anisotropy. The resulting magnetic anisotropy constant is
Soft magnetic nanocrystalline alloys are widely used in devices that convert electrical energy [1]. The nanocrystalline alloy Finemet with the chemical composition Fe73.5Cu1Nb3Si13.5B9 consists of crystalline αFeSi grains of about 10 nm size, embedded into an amorphous matrix [2]. Such a structure is formed during crystallization of the amorphous precursor [3], which is obtained using the technology of ultrafast melt quenching. This nanocrystalline alloy has a high permeability and low magnetic losses if the crystal grain size is less than the magnetocrystalline exchange length [4]
L0 ≈
A/|K1| ,
(1)
where A – exchange stiffness, K1 – magnetocrystalline anisotropy constant. For α-FeSi grains with 20 at% Si we obtain L0 = 35 nm, taking into account K1 = 8·103 J/m3 and A = 10−11 J/m. The correlation length Lex
Lex ≈
A/ 〈K1〉
(2)
determines the exchange correlation space of grains, which can be ⁎
〈K 〉 ≈
Ku2 + β 2 〈K1 〉2 ,
(3)
where Ku – induced anisotropy constant, and the factor β depends on the crystals’ symmetry and is close to unity [5]. In the Finemet alloy, the predominant mechanism of the induced magnetic anisotropy is directional ordering of the solution atoms Si in nanocrystals [6]. In case of partial replacement of Fe by Co or Ni, the directional ordering of Co and Ni atoms makes an additional contribution to the induced anisotropy [7,8]. For the nanocrystalline alloys with induced anisotropy, mainly the nature of the anisotropy, as such, has been studied [5,9,10]. Magnetic
Corresponding author. E-mail address: [email protected] (V.A. Kataev).
https://doi.org/10.1016/j.jmmm.2019.02.018 Received 30 June 2018; Received in revised form 27 November 2018; Accepted 4 February 2019 Available online 05 February 2019 0304-8853/ © 2019 Elsevier B.V. All rights reserved.
Journal of Magnetism and Magnetic Materials 479 (2019) 19–26
Y.N. Starodubtsev et al.
properties of these alloys were investigated, for example, in [11–14], and the main attention in these papers was given to the dynamic magnetic properties in medium magnetic fields. In this paper, we investigated the hysteresis losses and their relation to the parameters of the static minor hysteresis loops in the soft magnetic nanocrystalline Fe67.5Co5Cu1Nb2Mo1.5Si14B9 alloy with magnetic-field-induced anisotropy. The measurement results beginning with the region of weak magnetic field, in which the Rayleigh law is met, are presented. 2. Experimental The experimental research of the minor magnetic hysteresis loops has been carried out on the soft magnetic nanocrystalline Fe67.5Co5Cu1Nb2Mo1.5Si14B9 alloy. The alloy was melted in a vacuum induction furnace. A ribbon of 25 μm thick and 10 mm wide with the amorphous structure was produced with the planar flow casting process. The ribbon was wound up onto toroidal cores with the 32 mm outer diameter and 20 mm inner diameter. To obtain the nanocrystalline structure the samples were annealed at a temperature of 823 К, 1 h in longitudinal and transverse magnetic field and also without any magnetic field. Accordingly, these samples were designated as L, T and O and their hysteresis loops are shown in the Fig. 1. The static hysteresis loops measurements have been carried out using the computer controlled measuring device “MMKS-05”. The magnetizing field was changed “in a step-like fashion” and the loops were obtained by a pointby-point procedure (ballistic method) [15]. Before measuring each loop, the sample was demagnetized with a decaying alternating magnetic field with 1 Hz frequency. The magnetic loops have been determined in the range of maximal magnetic induction of 0.003 T to magnetic saturation.
Fig. 2. Dependences of the hysteresis losses Wh on the maximal induction Bmax for cores annealed in a longitudinal (L) or transverse (T) magnetic field, and with no magnetic field (O). Numbers shown along the curves are the exponents s . of s for the approximating power function Wh = rBmax
axis directed along the length of the ribbon and circumferentially of the toroid. The core L domain structure consists of slab domains, separated by 180° Bloch walls [16]. If the external magnetic field H is directed along the anisotropy axis of the core L, then the magnetization process occurs by displacement of the domain walls. Slab domains may be also found in the core T, but the magnetic anisotropy axis in this core is directed across the length of the ribbon. If the external magnetic field H is directed along the toroid circumference, i. e. at the angle of 90° to the easy magnetization axis of the core T, then magnetization process occurs by means of the magnetization rotation to the direction of the magnetic field. After the heat treatment of Fe67.5Co5Cu1Nb2Mo1.5Si14B9 alloy without any magnetic field, the hysteresis loop has the remanence ratio of Br/Bmax close to 0.5, which corresponds to an almost isotropic magnetization distribution in the core O. Fig. 2 shows the dependences of the hysteresis losses Wh on the maximum induction Bmax, scaled logarithmically, for the L, T and O cores. The curves have linear sections corresponding to a constant value of the exponent s for the approximating power function
3. Results and discussion As can be seen from Fig. 1 after a heat treatment of Fe67.5Co5Cu1Nb2Mo1.5Si14B9 alloy in a magnetic field, the hysteresis loop becomes rectangular with the remanence ratio Br/Bmax close to value of 1 (core L) or linear with remanence ratio close to zero (core T). The core L has the longitudinal uniaxial magnetic anisotropy with the
s Wh = rBmax .
(4)
Numbers shown along the curves in Fig. 2 are the exponents of s for the power function (4) at the corresponding sections of the curves. The use of the power function is convenient, since it is scale invariant. This means that all power functions with the same exponent s are similar, although they may differ by scale coefficient r. In fact, a change of variable Bmax in (4) by a factor of k will result in the function‘s value change by a factor of ks, and the fundamental dependence, determined by the exponent s, will remain: Wh (kBmax ) = r (kBmax ) s = rk s (Bmax ) s . Therefore, a few dependences are well approximated by power functions with exponent of s, but different factors r, this proves a similarity of the processes they are based on. Contrarily, different exponent‘s s would mean different processes. Thus, using (4), we may assume the same magnetization process and its changes if a variable, e.g. Bmax, changes. As can be seen from Fig. 2 for Bmax less than 0.1 T, the core L with longitudinal uniaxial anisotropy has the largest hysteresis losses. For this core the boundary value Bmax = 0.0025 T is observed on the curve Wh = f(Bmax). At the boundary value the slope of the curve changes.
Fig. 1. Magnetic hysteresis loops for Fe67.5Co5Cu1Nb2Mo1.5Si14B9 nanocrystalline cores annealed in a longitudinal (L) or transverse (T) magnetic field, and with no magnetic field (O). Arrows indicate the external magnetic field direction during magnetization. 20
Journal of Magnetism and Magnetic Materials 479 (2019) 19–26
Y.N. Starodubtsev et al.
Fig. 3. Dependences of the coercivity Hc on the maximal induction Bmax for cores annealed in a longitudinal (L) or transverse (T) magnetic field, and also with no magnetic field (O). Numbers shown along the curves are the exponents s . of s for the approximating power function Hc = rBmax
When the induction is less than 0.0025 T, the exponent is 2.70 and close to the number 3, which follows from Rayleigh law [18],
Wh =
4η 3 . Bmax 3μ02 μi3
(5)
where μi is the initial permeability, η is the Rayleigh constant, and μ0 is the magnetic constant, which is equal to 4π·10−7 H/m. In the same region for Bmax less than 0.0025 T, the approximating s power function Hc = rBmax has the exponent s = 1.70 (see Fig. 3) that is close to the number 2. It also follows from the Rayleigh law,
Hc =
η 2 Bmax . 2μ02 μi3
(6)
At the same boundary value of Bmax = 0.0025 T, the shape of the hysteresis loop varies from rectangular to lens-like, typical for the Rayleigh region (see Fig. 4b). Thus, in a weak magnetic field, the magnetization process of the core L with a longitudinal induced anisotropy corresponds to the Rayleigh law as a whole. The change in the shape of the hysteresis loop from rectangular to lens-like may be interpreted as follows. If in the core L with the longitudinal induced anisotropy the external magnetic field H is directed along the circumference of the toroid, i. e., along the easy magnetization axis (see the inset in Fig. 1), then the magnetization process will be effected by the displacement of the 180° domain walls. The loop has a rectangular shape if most of the domain walls almost simultaneously overcome the maximum local potential barriers. The Rayleigh law was formulated as the first nonlinear approximation for a weak external effect. In the Rayleigh region, a reversible displacement of the domain walls predominates. The displacement should be small and not exceed the characteristic length of the local magnetic inhomogeneity of the material. In a nanocrystalline alloy, the magnetic inhomogeneity is associated with a random distribution of the easy magnetization axes in ferromagnetic clusters, i. e., the characteristic length is the correlation length Lex. Note that Lex is also proportional to the width of the 180° domain wall. In a model with flat 180° domain walls perpendicular to the surface of the ribbon, the increment of the magnetic induction ΔB is equal
Fig. 4. Minor magnetic hysteresis loops in a medium (a) and weak (b) magnetic field for the core L with longitudinal induced anisotropy.
ΔB =
2Bs Lex , D
(7)
if the domain walls are displaced by the distance Lex, where D – domain width, Bs – saturation magnetic induction. For the estimation we take the following values Bs = 1.2 T, D = 10−3 m [14] and correlation length Lex = 10−6 m according to the formula (2) for A = 10−11 J/m and 〈K1〉 = 10 J/m3. For these values, we obtain the increment of the magnetic induction ΔB = 0.0024 T. This value is close to the boundary value of 0.0025 T, obtained from Figs. 2 and 4 as a transition to the Rayleigh region in the core L. As follows from the formula (7), as the domain width D decreases the boundary induction value ΔB for transition to Rayleigh region increases. Note that, according to the model of the domain wall bulging in the no pinned places [19], the decrease in the domain width D should also lead to an increase in the initial permeability, since
μiw = 21
μ0 Ms2 h2 , 3γD
(8)
Journal of Magnetism and Magnetic Materials 479 (2019) 19–26
Y.N. Starodubtsev et al.
where Ms is the saturation magnetization, h is the distance between the fixing points of the domain wall, γ = 2 AK is the specific energy of the 180° Bloch domain wall. With a periodic change in the local density of magnetic energy ΔE in a ferromagnetic material, the coercive force associated with the irreversible displacement of the domain walls over these magnetic inhomogeneities can be expressed in the form [20]
Hc ≈
ΔE δ , μ0 Ms l
(9)
where δ – domain wall width, l – characteristic length of magnetic inhomogeneity. Kondorsky used the magnetoelastic energy density ΔE = λ sσ [20], and in a nanocrystalline alloy ΔE = 〈K1〉 and l ≈ Lex [21]. The greatest value of the coercive force is obtained under the condition l = δ. In a nanocrystalline alloy δ ≈ Lex and the limiting coercive force associated with the random distribution of the easy magnetization axes in ferromagnetic clusters is
〈K1 〉 Hc ≈ . μ0 Ms
Fig. 5. Schematic representation of the irreversible rotation of the magnetization in a nanocrystalline material with induced magnetic anisotropy (core T). 1 – direction of the magnetization vector in a ferromagnetic cluster for H = 0, 2 – direction of the magnetization vector in a ferromagnetic cluster for H > 0.
(10)
After annealing in a magnetic field, the easy magnetization axes in ferromagnetic clusters are concentrated within a certain solid angle, which narrows as the induced anisotropy constant Ku increases. At the same time, the differences between the orientations of the easy magnetization axes in neighboring clusters are smoothed out. Such smoothing is equivalent to the decrease in ΔE and the increase in the characteristic length of the magnetic inhomogeneity l. According to the relation (9), this should lead to the decrease in the coercive force. Perhaps this mechanism provides a low coercive force in the core L compared to the core O, which was annealed without a magnetic field. From Figs. 2 and 3 for the core L in region Bmax > 0.0025 T we 0.10 1.10 obtain Hc ∼ Bmax and Wh ∼ Bmax . From these two relations follows the proportionality of the hysteresis losses Wh to the product Bmax·Hc. The same relation Wh ∼ Bmax·Hc is obtained in a weak magnetic field, where 1.70 2.70 Hc ∼ Bmax and Wh ∼ Bmax . We note that in the medium magnetic field the hysteresis losses almost linearly depend on Bmax and have an exponent close to unity (s = 1.10). This value is significantly different from the value s = 1.6, obtained by Steinmetz for isotropic iron [17]. The core T has the transverse induced anisotropy. In an external magnetic field directed along the circumference of the toroid, i. e., at angle of 90° to the easy magnetization axis (see the inset in Fig. 1), the magnetization process will be carried out by magnetization rotation to the direction of the magnetic field. The experimental value of the initial permeability μi for the core T is 35000. If this value is associated only with the process of homogeneous rotation, then from the formula [20]
μir =
μ0 Ms2 2Ku
axes located randomly. In a sample with induced anisotropy Ku (cores L and T) the easy magnetization axes of ferromagnetic clusters (shown in Fig. 5 with a white bidirectional arrow) are concentrated within a certain solid angle. The axis of this solid angle coincides with the direction of the magnetic field during the heat treatment. In Fig. 5 Ms indicates the direction of the magnetization in the magnetic domain. Without an external magnetic field, the magnetization vectors in the neighboring clusters are close to the direction indicated by the number of 1. In an external magnetic field H directed at 90° to the easy magnetization axis Ku (core T), the magnetization vector of the clusters begins to turn to the direction referred to as H. In the left cluster in Fig. 5, a rotation from the direction 1 leads to an increase in the angle between the magnetization vector and the easy magnetization axis of the cluster, i.e., to an increase in the magnetic anisotropy energy. Therefore, beginning from a certain point the direction 1 becomes unstable and the magnetization vector irreversibly goes into the stable direction 2, which is the closest one to the direction of the external magnetic field. In the right cluster, the direction 1 is stable and the magnetization vector will rotate to the direction of the magnetic field without going to the direction 2. Thus, in about half of the ferromagnetic clusters, the irreversible magnetization rotation is possible. The mechanism of the irreversible magnetization rotation in a nanocrystalline material is analogous to the irreversible rotation arising in a particle with the uniaxial magnetic anisotropy [20]. The process of inhomogeneous magnetization rotation in nanocrystalline materials with a small magnetic anisotropy constant can be accompanied by a transformation of the domain structure. This transformation results from the displacement, nucleation and disappearance of the domain walls [9,14]. Thus, in the core T, the magnetic hysteresis can be associated with both, the irreversible magnetization rotation and the irreversible displacement of the domain walls. Since the samples T and L are made from the same alloy and annealed in the same regime except for the direction of the magnetic field, they probably have the same structural state and the same ferromagnetic clusters’ magnetic structure. In this case, the coincidence of the coercive force value in Fig. 3, for which the slope of the curves Hc = f (Bmax) in these samples varies, is not accidental. This coincidence indirectly confirms the correctness of the above arguments for the coercive force formation in the nanocrystalline samples with the uniaxial induced magnetic anisotropy. However, the same coercive force in the cores T and L is achieved at substantially different maximal induction of 0.08 and 0.0025 T respectively (Fig. 3). This difference takes place due to the significant contribution of the homogeneous (nonhysteretic) magnetization rotation to the increment of the magnetic induction in
(11) 3
we obtain the induced anisotropy constant Ku = 18 J/m at Ms = 106 A/m. For the core T, the dependence of the hysteresis losses on the magnetic induction in a weak magnetic field has the exponent s = 2.90 s for the approximating power function Wh = rBmax (Fig. 2). This value is close to the exponent of 3 in formula (5) for the Rayleigh region and it does not change until the magnetic induction reaches 0.08 T. Thus, the Rayleigh region in the core T is much wider than in the core L. At a higher induction, the exponent s in core T is reduced to the value of 1.87 and this value is higher than the one obtained by Steinmetz for the isotropic iron [17]. Thus for the core T, the change in the slope of the curve in Fig. 2 is observed. This change can be associated with the transition from the region in which the homogeneous reversible rotation of the magnetization prevails, to the region in which inhomogeneous rotation predominates. As is known from [2], the soft magnetic nanocrystalline material consists of ferromagnetic clusters with easy magnetization 22
Journal of Magnetism and Magnetic Materials 479 (2019) 19–26
Y.N. Starodubtsev et al.
Fig. 6. Magnetic hysteresis loops for the cores L, T and O in a weak magnetic field.
Fig. 7. Dependences of the hysteresis losses Wh on the corcivity Hc for cores annealed in a longitudinal (L) or transverse (T) magnetic field, and also with no magnetic field (O). Numbers shown along the curves are the exponents of s for the approximating power function Wh = rHcs .
the core T. The heat treatment of the core O was carried out with no magnetic field. But the cooling of the α-FeSi crystalline phase took place below the Curie point with the spontaneous magnetization and the establishment of a domain structure. Therefore, the directional ordering of the solution atoms Si and Co occurred in the ferromagnetic clusters’ local magnetic field. Such local directional ordering leads to an additional stabilization of the domain walls due to the enhancement of the local magnetic anisotropy. As a result of this additional stabilization, the core O coercive force is higher than in the L and T cores (Fig. 1). s s The induction dependences Wh = rBmax and Hc = rBmax for the cores O and T in Figs. 2 and 3 do not differ practically up to Bmax = 0.08 T. Further, the curves diverge due to the faster growth of the hysteresis losses and the coercive force in the core O. This proves that the magnetic hysteresis mechanisms in the O and T cores in a weak magnetic field are the same or close to each other. Fig. 6 shows the magnetic hysteresis loops in the cores O, T and L in a weak magnetic field. It is seen that the loops have a different inclination angle, which corresponds to the initial permeability of 58,000, 35,000 and 2900 for the cores O, T and L, respectively. Hence it may be assumed, that in a weak magnetic field in the core O, as well as in the core T, the role of the magnetization rotation is considerable. In this region in the core O, the magnetization rotation and the domain walls’ displacement are not isolated, but occur simultaneously, leading to the synergetic influence. Probably this synergetic influence can explain the higher initial permeability in the core O compared to the core T. Let's consider the relationship between the hysteresis losses and other parameters of the minor hysteresis loops. Fig. 7 shows the dependence of the hysteresis losses on the coercive force. In a weak magnetic field, all the dependencies have the exponent s close to 1.5. For the Rayleigh region it coincides with the exponent in formula
Wh =
8 2 μ0 μi1.5 3η0.5
Hc1.5,
It is worth mentioning, that the exponents s in Figs. 2, 3, and 7 are consistent. For example, in the region of medium magnetic fields, fol1.10 0.10 lowing from Figs. 2 and 3, Wh ∼ Bmax and Hc ∼ Bmax , hence we see 11 Wh ∼ Hc . The exponent s = 11 practically coincides with the experimentally determined value of 11.6 in Fig. 7. The dependence of the hysteresis losses Wh on the remanence Br for the approximating function Wh = rBrs is also of interest. In [22–26] it was found that the exponent s is close to 1.35 in a wide region of the magnetic field for various materials. It should be noted that all the magnetic materials in these works were isotropic. In Fig. 8 the
(12)
which can be obtained if we take into account formulas (5) and (6). Almost identical exponents s in the approximating power function Wh = rHcs indicate the similarity of the magnetization processes for all the cores T, O and L in a weak magnetic field. As shown in Fig. 7, in the cores L and T, the exponent s changes at Hc = 0,25 A/м. This value is smaller than in the core O, in which there is an additional stabilization of the domain walls.
Fig. 8. Dependences of the hysteresis losses Wh on the remanence Br for cores annealed in a longitudinal (L) or transverse (T) magnetic field, and also with no magnetic field (O). Numbers shown along the curves are the exponents of s for the approximating power function Wh = rBrs . 23
Journal of Magnetism and Magnetic Materials 479 (2019) 19–26
Y.N. Starodubtsev et al.
dependences of the hysteresis losses Wh on the remanence Br in the cores after heat treatment in a longitudinal (L) or transverse (T) magnetic field, and also with no magnetic field (O) is shown. In a weak magnetic field, the dependences have the exponent close to 1.5. In the Rayleigh region this number coincides with the exponent in formula
Wh =
8 2 Br1.5, 3(ημ0 )0.5
(13)
which can be obtained if we take into account formula (5) and relation [18]
Br =
1 2 ημ Hmax . 2 0
(14)
Outside the Rayleigh region in the core O, the exponent s = 1.33 and this value is close to 1.35 obtained in isotropic magnetic materials. In cores with induced magnetic anisotropy, the exponent s gets lower, s = 1.10 for the sample L, or gets greater, s = 2.82 for the sample T. Hence, one can not speak of the universal power dependence Wh = rBr1.35 obtained by Kobayashi [22–26]. The numerical value of 1.35 can be used as an average value for materials with isotropic magnetic properties. 4. Hysteresis losses calculation
Fig. 9. Hysteresis losses Wh measured and calculated by formulas (5), (18) and (20) versus the maximal induction Bmax in the core L.
Finally, let us consider the formulas for calculation of hysteresis losses that can be used for technical applications and estimate the accuracy of such calculations for L, T and O cores. The hysteresis loop of the core L has a rectangular shape, so the area of such a loop, i. e., hysteresis losses, can be estimated using
Wh = 4Bm Hc .
(15)
For the flat hysteresis loop of the core T, the loop area can be expressed as a combination of the triangles formed by connecting the characteristic points Bmax, Br and Hc
Wh = (Bmax + Br ) Hc + Br Hmax .
(16)
For the flat hysteresis loop Bmax > > Br, formula (16) can be simplified
Wh = Bmax Hc + Br Hmax .
(17)
For the calculations, we also use the relations obtained earlier using the method of dimensionless hysteresis quantities [27]
Wh =
8Bmax Hc , 3
(18)
Wh =
8Br Hmax , 3
(19)
Wh =
8 Br Bmax Hc Hmax , 3
(20)
Bmax Hc = 1. Br Hmax
Fig.10. Hysteresis losses Wh measured and calculated by formulas (16), (18), (19) and (20) versus the maximal induction Bmax in the core T.
(21)
Δ=
These relations were generally obtained without the assumption of anisotropy. Therefore, we apply relation (21) for further simplification (17) and obtain a formula for estimating the hysteresis loss in the core T
Wh = 2Bmax Hc .
Wcal − Wmeas ·100, Wmeas
(23)
averaged over the entire range of the magnetic induction. The table shows the average relative deviations of the measured and calculated hysteresis losses for different formulas (Table 1). According to the table, the best coincidence is obtained in the core O, having the magnetic hysteresis loop, typical for materials with isotropic magnetic properties. We note that in the core O, the average deviations calculated considering formulas (18) and (19) have similar values, but one of them is always negative. Based on the said above, we introduce the formula
(22)
In Figs. 9–11 the dependence of the hysteresis losses Wh on the maximal induction Bmax in the cores L, T and O measured and calculated using the above formulas is shown. The measured results are in line with the calculated curves Wh = f (Bmax) and allow us to consider the formulas as universal for a wide range of magnetic inductions. Furthermore, the accuracy of the formulas may be estimated relying on an integral characteristic, which is the relative deviation between the calculated and measured values of the hysteresis losses
Wh =
24
4(Bmax Hc + Br Hmax ) . 3
(24)
Journal of Magnetism and Magnetic Materials 479 (2019) 19–26
Y.N. Starodubtsev et al.
magnetic field. It is shown that all the experimental dependences in cores with magnetic-field-induced anisotropy in a weak magnetic field are consistent with the formulas derived from the Rayleigh law. It was found, that the core L with longitudinal induced anisotropy has a low permeability. The magnetization process in the core L Rayleigh region begins with the displacement of the domain walls at distances comparable to the correlation length Lex. In the core T with transverse induced anisotropy, the magnetic hysteresis can be associated with both irreversible magnetization rotation and with irreversible domain walls’ displacement. In the core O without induced anisotropy and the core T with transverse induced anisotropy, the dependence of the hysteresis losses Wh on the magnetic induction Bmax and the coercive force Hc is practically indistinguishable at Bmax < 0.08 T. This indicates a predominance of the magnetization rotation in this induction region both in the core T and in the core O. Cooling the core O without a magnetic field leads to an increase in the local magnetic anisotropy and a rise in the coercive force compared to the cores that were cooled in a uniform external magnetic field. In a soft magnetic nanocrystalline alloy with a uniaxial induced anisotropy, certain empirical relationships obtained earlier for isotropic magnetic materials are not met. So, in a medium s magnetic field, the exponent s in formula Wh = rBmax deviates noticeably from the standard value of 1.60 obtained by Steinmetz. In contrast to above, the values of s = 1.10 for the core L with longitudinal anisotropy and s = 1.87 for the core T with transverse anisotropy have been achieved. The exponent s for the dependence of hysteresis losses on the remanence in the formula Wh = rBrs deviates from the standard value 1.35 obtained by Kobayashi. Particularly, s = 1.10 for the core L with longitudinal anisotropy and s = 2.82 for the core T with transverse anisotropy may be shown. Formulas for hysteresis losses calculation are proposed. The formulas most accurately predict the measured hysteresis losses in the core O without uniaxial induced anisotropy. These formulas can be referred to as universal, since they can be used to calculate hysteresis losses in a wide region of the magnetic field as accurately as 0.5%. The calculations’ accuracy for the cores with uniaxial induced anisotropy is lower. For the core with transverse anisotropy, the accuracy of calculations may be significantly improved selecting appropriate scaling factors. For the cores with longitudinal anisotropy different formulas need to be used for calculations in the region of weak and medium magnetic fields.
Fig. 11. Hysteresis losses Wh measured and calculated by formulas (18), (19) and (20) versus the maximal induction Bmax in the core O. Table 1 Average relative deviations for measured and calculated hysteresis losses. Formulas for calculating hysteresis losses Wh
Wh = 8BmaxHc/3 Wh = 8BrHmax/3 Wh = 8[(BrBmaxHcHmax)0.5]/3 Wh = 4BmaxHc Wh = (Bmax + Br)Hc + BrHmax Wh = 2BmaxHc
(18) (19) (20) (15) (16) (22)
¯, % Average deviation, Δ Core L
Core T
Core O
−20.8 −10.5 −16.2 18.8 – –
7.9 17.9 12.5 – −13.5 −19.1
2.5 −2.4 0.5 – – –
The calculation by formula (24) results in a lower average deviation of 0.3%. Note that formulas (15), (18) and (22) differ only in the scale factor, which decreases from 4 to 2 when the loop shape changes from rectangular to linear. Thus, the relation Wh ∼ Bmax·Hc, which was derived earlier from Figs. 2 and 3, is sufficiently universal. Practical application of this relation requires a proper choice of the scale factor value. The average deviations for the cores L and T are much greater. Thus, for the core T, the calculation considering formula (22) results in ¯ = −19.1%. Replacing in this formula the scale factor 2 by 2.38 we Δ increase it by 19.1%. Such a replacement will reduce the average deviation down to −3.7%. Similarly, it is possible to reduce the average deviation by adjusting the scale factor in the other formulas in case of core T. Formula (16) presents an exception, which is the sum of two quantities. For the core L such a universal formula for Wh to obtain small deviations in a wide region of the magnetic field does not exist. Therefore formulas (18) and (20) result in an average deviation of about 3% for the magnetic induction less than 0.05 T, and formula (15) result in the same small average deviation for the magnetic induction > 0.05 T.
Acknowledgement This work was supported by the Ministry of Education and Science of the Russian Federation in the framework of state tasks No. 3.6121.2017/8.9 and No. 4.9541.2017/8.9. References [1] Y. Yoshizawa, Nanocrystalline soft magnetic materials and their applications, Springer, 2006, pp. 124–158. [2] G. Herzer, Nanocrystalline soft magnetic alloys, Elsevier B.V., Amsterdam, 1997, pp. 415–462. [3] Y. Yoshizawa, S. Oguma, K. Yamauchi, New Fe-based soft magnetic alloys composed of ultrafine grain structure, J. Appl. Phys. 64 (1988) 6044–6046. [4] G. Herzer, Grain structure and magnetism of nanocrystalline ferromagnets, IEEE Trans. Magn. 25 (1989) 3327–3329. [5] K. Suzuki, G. Herzer, Magnetic-field-induced anisotropies and exchange softening in Fe-rich nanocrystalline soft magnetic alloys, Scr. Mater. 67 (2012) 548–553. [6] G. Herzer, Magnetic field-induced anisotropy in nanocrystalline fFe-Cu-Nb-Si-B alloys, J. Magn. Magn. Mater. 133 (1994) 248–250. [7] Y. Yoshizawa, S. Fujii, D.H. Ping, M. Ohnuma, K. Hono, Magnetic properties of nanocrystalline FeMCuNbSiB alloys (M: Co, Ni), Scripta. Mater. 48 (2003) 863–868. [8] V.A. Kataev, Yu.N. Starodubtsev, E.A. Mikhalitsyna, V.Ya. Belozerov, R.V. Tsyngalov, Magnetic properties and induced anisotropy of nanocrystalline Fe72.5-xNixCu1.1Nb1.9Mo1.5Si14.3B8.7 alloys, Phys. Met. Metallogr. 118 (2017) 558–563. [9] S. Flohrer, R. Schäfer, C. Polak, G. Herzer, Interplay of uniform and random anisotropy in nanocrystalline soft magnetic alloys, Acta Mater. 53 (2005) 2937–2942. [10] N. Ito, A. Michels, J. Kohlbrecher, J.S. Garitaonandia, K. Suzuki, J.D. Cashion, Effect of magnetic field annealing on the soft magnetic properties of nanocrystalline materials, J. Magn. Magn. Mater. 316 (2007) 458–461.
5. Conclusion The hysteresis losses and their relation to the parameters of the minor static hysteresis loops were investigated in the soft magnetic nanocrystalline Fe67.5Co5Cu1Nb2Mo1.5Si14B9 alloy after heat treatment in a longitudinal or transverse magnetic field, and also with no 25
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[11] E. Ferrara, C. De Luigi, C. Beatrice, C. Appino, F. Fiorillo, Energy loss vs. magnetizing frequency in field-annealed nanocrystalline alloys, J. Magn. Magn. Mater. 215–216 (2000) 466–468. [12] L. Fratila, O. Geoffroy, J.L. Portesel, Multiplicatiom of domain walls by eddy currents in nanocrystalline soft magnetic materials, J. Magn. Magn. Mater. 242–245 (2002) 229–231. [13] S. Flohrer, R. Schäfer, J. McCord, S. Roth, L. Schultz, G. Herzer, Magnetization loss and domain structure in nanocrystalline tape wound cores, Acta Mater. 54 (2006) 3253–3259. [14] S. Flohrer, R. Schäfer, G. Herzer, Magnetic microstructure of nanocrystalline FeCuNbSiB soft magnets, J. Non-Cryst. Sol. 354 (2008) 5097–5100. [15] S. Fiorillo, Measurement and Characterization of Magnetic Materials, Elsevier Academic Press, 2004. [16] R. Schäfer, The magnetic microstructure of nanostructured materials, Springer, 2009, pp. 275–307. [17] C.P. Steinmetz. On the relationship between magnetic losses and domain structure law of hysteresis// Proc. IEEE vol. 72. pp.197–221, 1989, preprinted from the Amer. Inst. Electr. Eng. Trans. vol.9. pp.3–64, 1892. [18] E. Kneller, Ferromagnetismus, Springer-Verlag, 1962. [19] M. Kersten, Die Wölbung der Blochwand als Elementarvorgang reversibler
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[27]
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Magnetisierungsänderungen (Anfangspermeabilität und ΔE-Effekt), Z. angew. Phys. 8 (1956) 313–322. S. Chikazumi, Physics of Ferromagnetism, Oxford University Press, 1997. G. Herzer, Anisotropies in soft magnetic nanocrystalline alloys, J. Magn. Magn. Mater. 294 (2005) 99–106. S. Kobayashi, T. Fujiwara, S. Takahashi, H. Kikuchi, Y. Kamada, K. Ara, T. Shishido, The effect of temperature on laws of hysteresis loops in nickel single crystals with compressive deformation, Phil. Mag. 89 (2009) 651–664. S. Kobayashi, S. Takahashi, T. Shishido, Y. Kamada, H. Kikuchi, Low-field magnetic characterization of ferromagnets using a minor-loop scaling law, J. Appl. Phys. 107 (2010) #023908. S. Takahashi, S. Kobayashi, Y. Kamada, H. Kikuchi, L. Zhang, K. Ara, Analysis of minor hysteresis loops and dislocations in Fe, Phys. B 372 (2006) 190–193. S. Kobayashi, S. Takahashi, H. Kikuchi, Y. Kamada, Domain-wall pinning in Er and Dy studied by minor-loop scaling laws, J. Phys.: Conf. Ser. 266 (2011) #012015. V.S. Tsepelev, YuN. Starodubtsev, N.P. Tsepeleva, Hysteresis properties of the amorphous high permeability Co66Fe3Cr3Si15B13 alloy, AIP Advances 8 (2018) #047707. Yu.N. Starodubtsev, V.A. Kataev, V.S. Tsepelev, Dimensionless quantities of hysteresis loops”, J. Magn. Magn. Mater. 460 (2018) 146–152.
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Research articles
Hysteresis losses in nanocrystalline alloys with magnetic-field-induced anisotropy
T
⁎
Yu.N. Starodubtseva,b, V.A. Kataeva, , K.O. Bessonovaa, V.S. Tsepeleva a b
Boris Yeltsin Ural Federal University, Mira str. 19, Yekaterinburg 620002, Russia Gammamet Research and Production Enterprise, Tatishchev str. 92, Yekaterinburg 620028, Russia
A R T I C LE I N FO
A B S T R A C T
Keywords: Hysteresis losses Induced anisotropy Soft magnet Nanocrystalline alloy
The hysteresis losses and their relation to the parameters of the minor static hysteresis loops were investigated in the nanocrystalline Fe67.5Co5Cu1Nb2Mo1.5Si14B9 alloy with magnetic-field-induced anisotropy. The analysis of experimental data was performed using a power function, which approximated the dependencies between the hysteresis parameters. It is shown that all the experimental dependences in cores with magnetic-field-induced anisotropy in a weak magnetic field are consistent with the formulas derived from the Rayleigh law. It is also shown, that in a nanocrystalline alloy with uniaxial induced anisotropy, certain empirical relationships obtained earlier for isotropic materials are not met, particularly, for the relation of the hysteresis losses to the remanence and the maximal magnetic induction. It has been found, that samples with longitudinal induced anisotropy demonstrate low initial permeability, and the magnetization process in the Rayleigh region is carried out by the domain walls’ displacement at distances comparable to the correlation length Lex. A magnetic hysteresis mechanism associated with the irreversible magnetization rotation in ferromagnetic clusters of nanocrystalline alloys is proposed. Formulas are proposed to calculate hysteresis losses in a soft magnetic nanocrystalline material with a different magnetic anisotropy.
1. Introduction
characterized by the local magnetic anisotropy constant 〈K1〉 and the randomly selected common easy magnetization axis [2]. For the exchange coupled grains of 10 nm size, the effective local magnetic anisotropy constant 〈K1〉 decreases by three orders compared to the magnetocrystalline anisotropy constant K1. Thus, we can assume that the nanocrystalline material consists of the exchange correlation spaces or ferromagnetic clusters. After annealing in a magnetic field, the local magnetic anisotropy of the nanocrystalline alloy is overlaid by the uniaxial magnetic-field-induced anisotropy. The resulting magnetic anisotropy constant is
Soft magnetic nanocrystalline alloys are widely used in devices that convert electrical energy [1]. The nanocrystalline alloy Finemet with the chemical composition Fe73.5Cu1Nb3Si13.5B9 consists of crystalline αFeSi grains of about 10 nm size, embedded into an amorphous matrix [2]. Such a structure is formed during crystallization of the amorphous precursor [3], which is obtained using the technology of ultrafast melt quenching. This nanocrystalline alloy has a high permeability and low magnetic losses if the crystal grain size is less than the magnetocrystalline exchange length [4]
L0 ≈
A/|K1| ,
(1)
where A – exchange stiffness, K1 – magnetocrystalline anisotropy constant. For α-FeSi grains with 20 at% Si we obtain L0 = 35 nm, taking into account K1 = 8·103 J/m3 and A = 10−11 J/m. The correlation length Lex
Lex ≈
A/ 〈K1〉
(2)
determines the exchange correlation space of grains, which can be ⁎
〈K 〉 ≈
Ku2 + β 2 〈K1 〉2 ,
(3)
where Ku – induced anisotropy constant, and the factor β depends on the crystals’ symmetry and is close to unity [5]. In the Finemet alloy, the predominant mechanism of the induced magnetic anisotropy is directional ordering of the solution atoms Si in nanocrystals [6]. In case of partial replacement of Fe by Co or Ni, the directional ordering of Co and Ni atoms makes an additional contribution to the induced anisotropy [7,8]. For the nanocrystalline alloys with induced anisotropy, mainly the nature of the anisotropy, as such, has been studied [5,9,10]. Magnetic
Corresponding author. E-mail address: [email protected] (V.A. Kataev).
https://doi.org/10.1016/j.jmmm.2019.02.018 Received 30 June 2018; Received in revised form 27 November 2018; Accepted 4 February 2019 Available online 05 February 2019 0304-8853/ © 2019 Elsevier B.V. All rights reserved.
Journal of Magnetism and Magnetic Materials 479 (2019) 19–26
Y.N. Starodubtsev et al.
properties of these alloys were investigated, for example, in [11–14], and the main attention in these papers was given to the dynamic magnetic properties in medium magnetic fields. In this paper, we investigated the hysteresis losses and their relation to the parameters of the static minor hysteresis loops in the soft magnetic nanocrystalline Fe67.5Co5Cu1Nb2Mo1.5Si14B9 alloy with magnetic-field-induced anisotropy. The measurement results beginning with the region of weak magnetic field, in which the Rayleigh law is met, are presented. 2. Experimental The experimental research of the minor magnetic hysteresis loops has been carried out on the soft magnetic nanocrystalline Fe67.5Co5Cu1Nb2Mo1.5Si14B9 alloy. The alloy was melted in a vacuum induction furnace. A ribbon of 25 μm thick and 10 mm wide with the amorphous structure was produced with the planar flow casting process. The ribbon was wound up onto toroidal cores with the 32 mm outer diameter and 20 mm inner diameter. To obtain the nanocrystalline structure the samples were annealed at a temperature of 823 К, 1 h in longitudinal and transverse magnetic field and also without any magnetic field. Accordingly, these samples were designated as L, T and O and their hysteresis loops are shown in the Fig. 1. The static hysteresis loops measurements have been carried out using the computer controlled measuring device “MMKS-05”. The magnetizing field was changed “in a step-like fashion” and the loops were obtained by a pointby-point procedure (ballistic method) [15]. Before measuring each loop, the sample was demagnetized with a decaying alternating magnetic field with 1 Hz frequency. The magnetic loops have been determined in the range of maximal magnetic induction of 0.003 T to magnetic saturation.
Fig. 2. Dependences of the hysteresis losses Wh on the maximal induction Bmax for cores annealed in a longitudinal (L) or transverse (T) magnetic field, and with no magnetic field (O). Numbers shown along the curves are the exponents s . of s for the approximating power function Wh = rBmax
axis directed along the length of the ribbon and circumferentially of the toroid. The core L domain structure consists of slab domains, separated by 180° Bloch walls [16]. If the external magnetic field H is directed along the anisotropy axis of the core L, then the magnetization process occurs by displacement of the domain walls. Slab domains may be also found in the core T, but the magnetic anisotropy axis in this core is directed across the length of the ribbon. If the external magnetic field H is directed along the toroid circumference, i. e. at the angle of 90° to the easy magnetization axis of the core T, then magnetization process occurs by means of the magnetization rotation to the direction of the magnetic field. After the heat treatment of Fe67.5Co5Cu1Nb2Mo1.5Si14B9 alloy without any magnetic field, the hysteresis loop has the remanence ratio of Br/Bmax close to 0.5, which corresponds to an almost isotropic magnetization distribution in the core O. Fig. 2 shows the dependences of the hysteresis losses Wh on the maximum induction Bmax, scaled logarithmically, for the L, T and O cores. The curves have linear sections corresponding to a constant value of the exponent s for the approximating power function
3. Results and discussion As can be seen from Fig. 1 after a heat treatment of Fe67.5Co5Cu1Nb2Mo1.5Si14B9 alloy in a magnetic field, the hysteresis loop becomes rectangular with the remanence ratio Br/Bmax close to value of 1 (core L) or linear with remanence ratio close to zero (core T). The core L has the longitudinal uniaxial magnetic anisotropy with the
s Wh = rBmax .
(4)
Numbers shown along the curves in Fig. 2 are the exponents of s for the power function (4) at the corresponding sections of the curves. The use of the power function is convenient, since it is scale invariant. This means that all power functions with the same exponent s are similar, although they may differ by scale coefficient r. In fact, a change of variable Bmax in (4) by a factor of k will result in the function‘s value change by a factor of ks, and the fundamental dependence, determined by the exponent s, will remain: Wh (kBmax ) = r (kBmax ) s = rk s (Bmax ) s . Therefore, a few dependences are well approximated by power functions with exponent of s, but different factors r, this proves a similarity of the processes they are based on. Contrarily, different exponent‘s s would mean different processes. Thus, using (4), we may assume the same magnetization process and its changes if a variable, e.g. Bmax, changes. As can be seen from Fig. 2 for Bmax less than 0.1 T, the core L with longitudinal uniaxial anisotropy has the largest hysteresis losses. For this core the boundary value Bmax = 0.0025 T is observed on the curve Wh = f(Bmax). At the boundary value the slope of the curve changes.
Fig. 1. Magnetic hysteresis loops for Fe67.5Co5Cu1Nb2Mo1.5Si14B9 nanocrystalline cores annealed in a longitudinal (L) or transverse (T) magnetic field, and with no magnetic field (O). Arrows indicate the external magnetic field direction during magnetization. 20
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Fig. 3. Dependences of the coercivity Hc on the maximal induction Bmax for cores annealed in a longitudinal (L) or transverse (T) magnetic field, and also with no magnetic field (O). Numbers shown along the curves are the exponents s . of s for the approximating power function Hc = rBmax
When the induction is less than 0.0025 T, the exponent is 2.70 and close to the number 3, which follows from Rayleigh law [18],
Wh =
4η 3 . Bmax 3μ02 μi3
(5)
where μi is the initial permeability, η is the Rayleigh constant, and μ0 is the magnetic constant, which is equal to 4π·10−7 H/m. In the same region for Bmax less than 0.0025 T, the approximating s power function Hc = rBmax has the exponent s = 1.70 (see Fig. 3) that is close to the number 2. It also follows from the Rayleigh law,
Hc =
η 2 Bmax . 2μ02 μi3
(6)
At the same boundary value of Bmax = 0.0025 T, the shape of the hysteresis loop varies from rectangular to lens-like, typical for the Rayleigh region (see Fig. 4b). Thus, in a weak magnetic field, the magnetization process of the core L with a longitudinal induced anisotropy corresponds to the Rayleigh law as a whole. The change in the shape of the hysteresis loop from rectangular to lens-like may be interpreted as follows. If in the core L with the longitudinal induced anisotropy the external magnetic field H is directed along the circumference of the toroid, i. e., along the easy magnetization axis (see the inset in Fig. 1), then the magnetization process will be effected by the displacement of the 180° domain walls. The loop has a rectangular shape if most of the domain walls almost simultaneously overcome the maximum local potential barriers. The Rayleigh law was formulated as the first nonlinear approximation for a weak external effect. In the Rayleigh region, a reversible displacement of the domain walls predominates. The displacement should be small and not exceed the characteristic length of the local magnetic inhomogeneity of the material. In a nanocrystalline alloy, the magnetic inhomogeneity is associated with a random distribution of the easy magnetization axes in ferromagnetic clusters, i. e., the characteristic length is the correlation length Lex. Note that Lex is also proportional to the width of the 180° domain wall. In a model with flat 180° domain walls perpendicular to the surface of the ribbon, the increment of the magnetic induction ΔB is equal
Fig. 4. Minor magnetic hysteresis loops in a medium (a) and weak (b) magnetic field for the core L with longitudinal induced anisotropy.
ΔB =
2Bs Lex , D
(7)
if the domain walls are displaced by the distance Lex, where D – domain width, Bs – saturation magnetic induction. For the estimation we take the following values Bs = 1.2 T, D = 10−3 m [14] and correlation length Lex = 10−6 m according to the formula (2) for A = 10−11 J/m and 〈K1〉 = 10 J/m3. For these values, we obtain the increment of the magnetic induction ΔB = 0.0024 T. This value is close to the boundary value of 0.0025 T, obtained from Figs. 2 and 4 as a transition to the Rayleigh region in the core L. As follows from the formula (7), as the domain width D decreases the boundary induction value ΔB for transition to Rayleigh region increases. Note that, according to the model of the domain wall bulging in the no pinned places [19], the decrease in the domain width D should also lead to an increase in the initial permeability, since
μiw = 21
μ0 Ms2 h2 , 3γD
(8)
Journal of Magnetism and Magnetic Materials 479 (2019) 19–26
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where Ms is the saturation magnetization, h is the distance between the fixing points of the domain wall, γ = 2 AK is the specific energy of the 180° Bloch domain wall. With a periodic change in the local density of magnetic energy ΔE in a ferromagnetic material, the coercive force associated with the irreversible displacement of the domain walls over these magnetic inhomogeneities can be expressed in the form [20]
Hc ≈
ΔE δ , μ0 Ms l
(9)
where δ – domain wall width, l – characteristic length of magnetic inhomogeneity. Kondorsky used the magnetoelastic energy density ΔE = λ sσ [20], and in a nanocrystalline alloy ΔE = 〈K1〉 and l ≈ Lex [21]. The greatest value of the coercive force is obtained under the condition l = δ. In a nanocrystalline alloy δ ≈ Lex and the limiting coercive force associated with the random distribution of the easy magnetization axes in ferromagnetic clusters is
〈K1 〉 Hc ≈ . μ0 Ms
Fig. 5. Schematic representation of the irreversible rotation of the magnetization in a nanocrystalline material with induced magnetic anisotropy (core T). 1 – direction of the magnetization vector in a ferromagnetic cluster for H = 0, 2 – direction of the magnetization vector in a ferromagnetic cluster for H > 0.
(10)
After annealing in a magnetic field, the easy magnetization axes in ferromagnetic clusters are concentrated within a certain solid angle, which narrows as the induced anisotropy constant Ku increases. At the same time, the differences between the orientations of the easy magnetization axes in neighboring clusters are smoothed out. Such smoothing is equivalent to the decrease in ΔE and the increase in the characteristic length of the magnetic inhomogeneity l. According to the relation (9), this should lead to the decrease in the coercive force. Perhaps this mechanism provides a low coercive force in the core L compared to the core O, which was annealed without a magnetic field. From Figs. 2 and 3 for the core L in region Bmax > 0.0025 T we 0.10 1.10 obtain Hc ∼ Bmax and Wh ∼ Bmax . From these two relations follows the proportionality of the hysteresis losses Wh to the product Bmax·Hc. The same relation Wh ∼ Bmax·Hc is obtained in a weak magnetic field, where 1.70 2.70 Hc ∼ Bmax and Wh ∼ Bmax . We note that in the medium magnetic field the hysteresis losses almost linearly depend on Bmax and have an exponent close to unity (s = 1.10). This value is significantly different from the value s = 1.6, obtained by Steinmetz for isotropic iron [17]. The core T has the transverse induced anisotropy. In an external magnetic field directed along the circumference of the toroid, i. e., at angle of 90° to the easy magnetization axis (see the inset in Fig. 1), the magnetization process will be carried out by magnetization rotation to the direction of the magnetic field. The experimental value of the initial permeability μi for the core T is 35000. If this value is associated only with the process of homogeneous rotation, then from the formula [20]
μir =
μ0 Ms2 2Ku
axes located randomly. In a sample with induced anisotropy Ku (cores L and T) the easy magnetization axes of ferromagnetic clusters (shown in Fig. 5 with a white bidirectional arrow) are concentrated within a certain solid angle. The axis of this solid angle coincides with the direction of the magnetic field during the heat treatment. In Fig. 5 Ms indicates the direction of the magnetization in the magnetic domain. Without an external magnetic field, the magnetization vectors in the neighboring clusters are close to the direction indicated by the number of 1. In an external magnetic field H directed at 90° to the easy magnetization axis Ku (core T), the magnetization vector of the clusters begins to turn to the direction referred to as H. In the left cluster in Fig. 5, a rotation from the direction 1 leads to an increase in the angle between the magnetization vector and the easy magnetization axis of the cluster, i.e., to an increase in the magnetic anisotropy energy. Therefore, beginning from a certain point the direction 1 becomes unstable and the magnetization vector irreversibly goes into the stable direction 2, which is the closest one to the direction of the external magnetic field. In the right cluster, the direction 1 is stable and the magnetization vector will rotate to the direction of the magnetic field without going to the direction 2. Thus, in about half of the ferromagnetic clusters, the irreversible magnetization rotation is possible. The mechanism of the irreversible magnetization rotation in a nanocrystalline material is analogous to the irreversible rotation arising in a particle with the uniaxial magnetic anisotropy [20]. The process of inhomogeneous magnetization rotation in nanocrystalline materials with a small magnetic anisotropy constant can be accompanied by a transformation of the domain structure. This transformation results from the displacement, nucleation and disappearance of the domain walls [9,14]. Thus, in the core T, the magnetic hysteresis can be associated with both, the irreversible magnetization rotation and the irreversible displacement of the domain walls. Since the samples T and L are made from the same alloy and annealed in the same regime except for the direction of the magnetic field, they probably have the same structural state and the same ferromagnetic clusters’ magnetic structure. In this case, the coincidence of the coercive force value in Fig. 3, for which the slope of the curves Hc = f (Bmax) in these samples varies, is not accidental. This coincidence indirectly confirms the correctness of the above arguments for the coercive force formation in the nanocrystalline samples with the uniaxial induced magnetic anisotropy. However, the same coercive force in the cores T and L is achieved at substantially different maximal induction of 0.08 and 0.0025 T respectively (Fig. 3). This difference takes place due to the significant contribution of the homogeneous (nonhysteretic) magnetization rotation to the increment of the magnetic induction in
(11) 3
we obtain the induced anisotropy constant Ku = 18 J/m at Ms = 106 A/m. For the core T, the dependence of the hysteresis losses on the magnetic induction in a weak magnetic field has the exponent s = 2.90 s for the approximating power function Wh = rBmax (Fig. 2). This value is close to the exponent of 3 in formula (5) for the Rayleigh region and it does not change until the magnetic induction reaches 0.08 T. Thus, the Rayleigh region in the core T is much wider than in the core L. At a higher induction, the exponent s in core T is reduced to the value of 1.87 and this value is higher than the one obtained by Steinmetz for the isotropic iron [17]. Thus for the core T, the change in the slope of the curve in Fig. 2 is observed. This change can be associated with the transition from the region in which the homogeneous reversible rotation of the magnetization prevails, to the region in which inhomogeneous rotation predominates. As is known from [2], the soft magnetic nanocrystalline material consists of ferromagnetic clusters with easy magnetization 22
Journal of Magnetism and Magnetic Materials 479 (2019) 19–26
Y.N. Starodubtsev et al.
Fig. 6. Magnetic hysteresis loops for the cores L, T and O in a weak magnetic field.
Fig. 7. Dependences of the hysteresis losses Wh on the corcivity Hc for cores annealed in a longitudinal (L) or transverse (T) magnetic field, and also with no magnetic field (O). Numbers shown along the curves are the exponents of s for the approximating power function Wh = rHcs .
the core T. The heat treatment of the core O was carried out with no magnetic field. But the cooling of the α-FeSi crystalline phase took place below the Curie point with the spontaneous magnetization and the establishment of a domain structure. Therefore, the directional ordering of the solution atoms Si and Co occurred in the ferromagnetic clusters’ local magnetic field. Such local directional ordering leads to an additional stabilization of the domain walls due to the enhancement of the local magnetic anisotropy. As a result of this additional stabilization, the core O coercive force is higher than in the L and T cores (Fig. 1). s s The induction dependences Wh = rBmax and Hc = rBmax for the cores O and T in Figs. 2 and 3 do not differ practically up to Bmax = 0.08 T. Further, the curves diverge due to the faster growth of the hysteresis losses and the coercive force in the core O. This proves that the magnetic hysteresis mechanisms in the O and T cores in a weak magnetic field are the same or close to each other. Fig. 6 shows the magnetic hysteresis loops in the cores O, T and L in a weak magnetic field. It is seen that the loops have a different inclination angle, which corresponds to the initial permeability of 58,000, 35,000 and 2900 for the cores O, T and L, respectively. Hence it may be assumed, that in a weak magnetic field in the core O, as well as in the core T, the role of the magnetization rotation is considerable. In this region in the core O, the magnetization rotation and the domain walls’ displacement are not isolated, but occur simultaneously, leading to the synergetic influence. Probably this synergetic influence can explain the higher initial permeability in the core O compared to the core T. Let's consider the relationship between the hysteresis losses and other parameters of the minor hysteresis loops. Fig. 7 shows the dependence of the hysteresis losses on the coercive force. In a weak magnetic field, all the dependencies have the exponent s close to 1.5. For the Rayleigh region it coincides with the exponent in formula
Wh =
8 2 μ0 μi1.5 3η0.5
Hc1.5,
It is worth mentioning, that the exponents s in Figs. 2, 3, and 7 are consistent. For example, in the region of medium magnetic fields, fol1.10 0.10 lowing from Figs. 2 and 3, Wh ∼ Bmax and Hc ∼ Bmax , hence we see 11 Wh ∼ Hc . The exponent s = 11 practically coincides with the experimentally determined value of 11.6 in Fig. 7. The dependence of the hysteresis losses Wh on the remanence Br for the approximating function Wh = rBrs is also of interest. In [22–26] it was found that the exponent s is close to 1.35 in a wide region of the magnetic field for various materials. It should be noted that all the magnetic materials in these works were isotropic. In Fig. 8 the
(12)
which can be obtained if we take into account formulas (5) and (6). Almost identical exponents s in the approximating power function Wh = rHcs indicate the similarity of the magnetization processes for all the cores T, O and L in a weak magnetic field. As shown in Fig. 7, in the cores L and T, the exponent s changes at Hc = 0,25 A/м. This value is smaller than in the core O, in which there is an additional stabilization of the domain walls.
Fig. 8. Dependences of the hysteresis losses Wh on the remanence Br for cores annealed in a longitudinal (L) or transverse (T) magnetic field, and also with no magnetic field (O). Numbers shown along the curves are the exponents of s for the approximating power function Wh = rBrs . 23
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dependences of the hysteresis losses Wh on the remanence Br in the cores after heat treatment in a longitudinal (L) or transverse (T) magnetic field, and also with no magnetic field (O) is shown. In a weak magnetic field, the dependences have the exponent close to 1.5. In the Rayleigh region this number coincides with the exponent in formula
Wh =
8 2 Br1.5, 3(ημ0 )0.5
(13)
which can be obtained if we take into account formula (5) and relation [18]
Br =
1 2 ημ Hmax . 2 0
(14)
Outside the Rayleigh region in the core O, the exponent s = 1.33 and this value is close to 1.35 obtained in isotropic magnetic materials. In cores with induced magnetic anisotropy, the exponent s gets lower, s = 1.10 for the sample L, or gets greater, s = 2.82 for the sample T. Hence, one can not speak of the universal power dependence Wh = rBr1.35 obtained by Kobayashi [22–26]. The numerical value of 1.35 can be used as an average value for materials with isotropic magnetic properties. 4. Hysteresis losses calculation
Fig. 9. Hysteresis losses Wh measured and calculated by formulas (5), (18) and (20) versus the maximal induction Bmax in the core L.
Finally, let us consider the formulas for calculation of hysteresis losses that can be used for technical applications and estimate the accuracy of such calculations for L, T and O cores. The hysteresis loop of the core L has a rectangular shape, so the area of such a loop, i. e., hysteresis losses, can be estimated using
Wh = 4Bm Hc .
(15)
For the flat hysteresis loop of the core T, the loop area can be expressed as a combination of the triangles formed by connecting the characteristic points Bmax, Br and Hc
Wh = (Bmax + Br ) Hc + Br Hmax .
(16)
For the flat hysteresis loop Bmax > > Br, formula (16) can be simplified
Wh = Bmax Hc + Br Hmax .
(17)
For the calculations, we also use the relations obtained earlier using the method of dimensionless hysteresis quantities [27]
Wh =
8Bmax Hc , 3
(18)
Wh =
8Br Hmax , 3
(19)
Wh =
8 Br Bmax Hc Hmax , 3
(20)
Bmax Hc = 1. Br Hmax
Fig.10. Hysteresis losses Wh measured and calculated by formulas (16), (18), (19) and (20) versus the maximal induction Bmax in the core T.
(21)
Δ=
These relations were generally obtained without the assumption of anisotropy. Therefore, we apply relation (21) for further simplification (17) and obtain a formula for estimating the hysteresis loss in the core T
Wh = 2Bmax Hc .
Wcal − Wmeas ·100, Wmeas
(23)
averaged over the entire range of the magnetic induction. The table shows the average relative deviations of the measured and calculated hysteresis losses for different formulas (Table 1). According to the table, the best coincidence is obtained in the core O, having the magnetic hysteresis loop, typical for materials with isotropic magnetic properties. We note that in the core O, the average deviations calculated considering formulas (18) and (19) have similar values, but one of them is always negative. Based on the said above, we introduce the formula
(22)
In Figs. 9–11 the dependence of the hysteresis losses Wh on the maximal induction Bmax in the cores L, T and O measured and calculated using the above formulas is shown. The measured results are in line with the calculated curves Wh = f (Bmax) and allow us to consider the formulas as universal for a wide range of magnetic inductions. Furthermore, the accuracy of the formulas may be estimated relying on an integral characteristic, which is the relative deviation between the calculated and measured values of the hysteresis losses
Wh =
24
4(Bmax Hc + Br Hmax ) . 3
(24)
Journal of Magnetism and Magnetic Materials 479 (2019) 19–26
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magnetic field. It is shown that all the experimental dependences in cores with magnetic-field-induced anisotropy in a weak magnetic field are consistent with the formulas derived from the Rayleigh law. It was found, that the core L with longitudinal induced anisotropy has a low permeability. The magnetization process in the core L Rayleigh region begins with the displacement of the domain walls at distances comparable to the correlation length Lex. In the core T with transverse induced anisotropy, the magnetic hysteresis can be associated with both irreversible magnetization rotation and with irreversible domain walls’ displacement. In the core O without induced anisotropy and the core T with transverse induced anisotropy, the dependence of the hysteresis losses Wh on the magnetic induction Bmax and the coercive force Hc is practically indistinguishable at Bmax < 0.08 T. This indicates a predominance of the magnetization rotation in this induction region both in the core T and in the core O. Cooling the core O without a magnetic field leads to an increase in the local magnetic anisotropy and a rise in the coercive force compared to the cores that were cooled in a uniform external magnetic field. In a soft magnetic nanocrystalline alloy with a uniaxial induced anisotropy, certain empirical relationships obtained earlier for isotropic magnetic materials are not met. So, in a medium s magnetic field, the exponent s in formula Wh = rBmax deviates noticeably from the standard value of 1.60 obtained by Steinmetz. In contrast to above, the values of s = 1.10 for the core L with longitudinal anisotropy and s = 1.87 for the core T with transverse anisotropy have been achieved. The exponent s for the dependence of hysteresis losses on the remanence in the formula Wh = rBrs deviates from the standard value 1.35 obtained by Kobayashi. Particularly, s = 1.10 for the core L with longitudinal anisotropy and s = 2.82 for the core T with transverse anisotropy may be shown. Formulas for hysteresis losses calculation are proposed. The formulas most accurately predict the measured hysteresis losses in the core O without uniaxial induced anisotropy. These formulas can be referred to as universal, since they can be used to calculate hysteresis losses in a wide region of the magnetic field as accurately as 0.5%. The calculations’ accuracy for the cores with uniaxial induced anisotropy is lower. For the core with transverse anisotropy, the accuracy of calculations may be significantly improved selecting appropriate scaling factors. For the cores with longitudinal anisotropy different formulas need to be used for calculations in the region of weak and medium magnetic fields.
Fig. 11. Hysteresis losses Wh measured and calculated by formulas (18), (19) and (20) versus the maximal induction Bmax in the core O. Table 1 Average relative deviations for measured and calculated hysteresis losses. Formulas for calculating hysteresis losses Wh
Wh = 8BmaxHc/3 Wh = 8BrHmax/3 Wh = 8[(BrBmaxHcHmax)0.5]/3 Wh = 4BmaxHc Wh = (Bmax + Br)Hc + BrHmax Wh = 2BmaxHc
(18) (19) (20) (15) (16) (22)
¯, % Average deviation, Δ Core L
Core T
Core O
−20.8 −10.5 −16.2 18.8 – –
7.9 17.9 12.5 – −13.5 −19.1
2.5 −2.4 0.5 – – –
The calculation by formula (24) results in a lower average deviation of 0.3%. Note that formulas (15), (18) and (22) differ only in the scale factor, which decreases from 4 to 2 when the loop shape changes from rectangular to linear. Thus, the relation Wh ∼ Bmax·Hc, which was derived earlier from Figs. 2 and 3, is sufficiently universal. Practical application of this relation requires a proper choice of the scale factor value. The average deviations for the cores L and T are much greater. Thus, for the core T, the calculation considering formula (22) results in ¯ = −19.1%. Replacing in this formula the scale factor 2 by 2.38 we Δ increase it by 19.1%. Such a replacement will reduce the average deviation down to −3.7%. Similarly, it is possible to reduce the average deviation by adjusting the scale factor in the other formulas in case of core T. Formula (16) presents an exception, which is the sum of two quantities. For the core L such a universal formula for Wh to obtain small deviations in a wide region of the magnetic field does not exist. Therefore formulas (18) and (20) result in an average deviation of about 3% for the magnetic induction less than 0.05 T, and formula (15) result in the same small average deviation for the magnetic induction > 0.05 T.
Acknowledgement This work was supported by the Ministry of Education and Science of the Russian Federation in the framework of state tasks No. 3.6121.2017/8.9 and No. 4.9541.2017/8.9. References [1] Y. Yoshizawa, Nanocrystalline soft magnetic materials and their applications, Springer, 2006, pp. 124–158. [2] G. Herzer, Nanocrystalline soft magnetic alloys, Elsevier B.V., Amsterdam, 1997, pp. 415–462. [3] Y. Yoshizawa, S. Oguma, K. Yamauchi, New Fe-based soft magnetic alloys composed of ultrafine grain structure, J. Appl. Phys. 64 (1988) 6044–6046. [4] G. Herzer, Grain structure and magnetism of nanocrystalline ferromagnets, IEEE Trans. Magn. 25 (1989) 3327–3329. [5] K. Suzuki, G. Herzer, Magnetic-field-induced anisotropies and exchange softening in Fe-rich nanocrystalline soft magnetic alloys, Scr. Mater. 67 (2012) 548–553. [6] G. Herzer, Magnetic field-induced anisotropy in nanocrystalline fFe-Cu-Nb-Si-B alloys, J. Magn. Magn. Mater. 133 (1994) 248–250. [7] Y. Yoshizawa, S. Fujii, D.H. Ping, M. Ohnuma, K. Hono, Magnetic properties of nanocrystalline FeMCuNbSiB alloys (M: Co, Ni), Scripta. Mater. 48 (2003) 863–868. [8] V.A. Kataev, Yu.N. Starodubtsev, E.A. Mikhalitsyna, V.Ya. Belozerov, R.V. Tsyngalov, Magnetic properties and induced anisotropy of nanocrystalline Fe72.5-xNixCu1.1Nb1.9Mo1.5Si14.3B8.7 alloys, Phys. Met. Metallogr. 118 (2017) 558–563. [9] S. Flohrer, R. Schäfer, C. Polak, G. Herzer, Interplay of uniform and random anisotropy in nanocrystalline soft magnetic alloys, Acta Mater. 53 (2005) 2937–2942. [10] N. Ito, A. Michels, J. Kohlbrecher, J.S. Garitaonandia, K. Suzuki, J.D. Cashion, Effect of magnetic field annealing on the soft magnetic properties of nanocrystalline materials, J. Magn. Magn. Mater. 316 (2007) 458–461.
5. Conclusion The hysteresis losses and their relation to the parameters of the minor static hysteresis loops were investigated in the soft magnetic nanocrystalline Fe67.5Co5Cu1Nb2Mo1.5Si14B9 alloy after heat treatment in a longitudinal or transverse magnetic field, and also with no 25
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