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The effect of grain-size distribution on coercivity in nanocrystalline soft magnetic alloys
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 272–276 (2004) 1445–1446
The effect of grain-size distribution on coercivity in nanocrystalline soft magnetic alloys Teruo Bitoha,*, Akihiro Makinoa, Akihisa Inoueb a
Department of Machine Intelligence, Faculty of Systems Science and Technology, Akita Prefectural University, 84–4 Ebinokuchi Tuchiya, Honjo 015-0055, Japan b Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
Abstract A simple model considering grain-size distribution has been fabricated based on the random anisotropy model. The model indicates that the effective magnetic anisotropy increases with increasing width of the grain-size distribution even if the mean grain size is constant. The coercivity (Hc) of some nanocrystalline Fe–Nb–B(–P–Cu) alloys with different grain-size distribution has been calculated. Our model explains well the dependence of Hc on the grain-size distribution. r 2003 Elsevier B.V. All rights reserved. PACS: 75.50.Tt; 75.50.Bb; 61.46.+w Keywords: Nanocrystalline soft magnetic alloy; Random anisotropy model; Grain-size distribution; Structural inhomogeneity
Since Herzer’s first application of the random anisotropy model (RAM) to nanocrystalline Fe–Si–B– Nb–Cu alloy [1], this model has been employed widely to explain the origin of magnetic softness in various nanocrystalline systems. In these analysis based on the RAM, the magnetic softness has been discussed using the mean grain size. However, the grain size has a distribution in actual nanocrystalline alloys. Recently, we reported that the Fe85Nb6B9 alloy has an asquenched structure composed of an amorphous phase and a-Fe grains with 20–45 nm in size, and has a crystallized nanostructure including the relatively coarse grains [2–4]. The effect of the structural inhomogeneities on soft magnetic properties is very interesting. In this paper, the effect of the grain-size distribution on the magnetic softness of nanocrystalline soft magnetic alloys is discussed based on the RAM. For simplicity, let us consider that the maximum grain size (Dm ) does not exceed the exchange correlation length (Lex ). Then the fluctuating part of the magnetocrystalline anisotropy (/K1 S) is given by using a *Corresponding author. Tel./fax: +81-184-27-2211. E-mail address: teruo [email protected] (T. Bitoh).
distribution function of the grain size (f ðDÞ), as [4] Z Dm 2 K1 D3 f ðDÞdD ; ð1Þ hK1 i ¼ v2c 6 L0 0 where vc is the volume fraction of the crystalline phase, K1 is the intrinsic magnetocrystalline anisotropy constant and L0 is the intrinsic exchange correlation length [5]. We assume further a log-normal distribution function as the grain-size distribution 1 ln2 Dr ; ð2Þ exp f ðDr Þ ¼ pffiffiffiffiffiffi 2s2D 2psD Dr where Dr ¼ D=D0 is the reduced grain size, D0 is the median, and sD is the geometric standard deviation. If f ðDr Þ is negligible small at Dr > Dm ; then Dm can be regarded as infinity and we obtain Lex ¼ L40 =fvc /DS3 expð3s2D Þg and h Di 6 hK1 i ¼ v2c K1 expð6s2D Þ; ð3Þ L0 where /DS ¼ D0 expðs2D =2Þ is the mean grain size. Eq. (3) indicates that /K1 S increases with increasing sD ; i.e., soft magnetic properties of nanocrystalline alloys deteriorates with increasing sD ; even if /DS is
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.12.368
ARTICLE IN PRESS 1446
T. Bitoh et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 1445–1446
constant. It should be noted that this result is essentially established in other distribution functions. Naturally, the effective anisotropy in the nanocrystalline alloys may have contributions from induced anisotropies such as magneto-elastic anisotropy other than the random magnetocrystalline anisotropy and hence the effective anisotropy constant in actual materials is more correctly /KS ¼ ðKu2 þ /K1 S2 Þ1=2 [6], where Ku is the induced uniaxial anisotropy conR 1=2 stant and /K1 S ¼ vc K1 =N 1=2 ¼ vc K1 fð D3 f ðDÞdDÞ= L3ex g1=2 (N is the number of grains in a magnetically coupled volume). Most of the nanocrystalline Fe–M–B (M=Zr, Hf, Nb) alloys with good soft magnetic properties exhibit remanence ratio (Jr =Js ) of around 0.5. This means that the magnetization process of the alloys is mostly governed by the induced anisotropies, i.e., Ku > /K1 S: In a limiting condition of Ku2 b/K1 S2 enables us to arrive at Lex ¼ L0 ðK1 =Ku Þ1=2 and pffiffiffiffiffiffiffiffiffiffiffi Ku K1 hDi 3 expð3s2D Þ: ð4Þ hK iEKu þ vc 2 L0 The D3 behavior of Hc is observed for the nanocrystalline Fe–Zr–B(–Cu) alloys with sufficiently small D [6]. The grain-size distribution evaluated by counting the a-Fe grains in TEM images are shown in Fig. 1 [4]. The log–normal distribution function reproduces well the observed grain-size distribution. In order to reproduce the grain-size distribution of the Fe85Nb6B9 alloy more correctly, we consider the bimodal distribution function (fb ðDÞ) expressed by superimposing the two log–normal distribution functions with the different medians (D0 and db D0 ; db > 1), the geometric standard deviations (sD and sb ) and the ratio of the distribution function for the large grains to that of the small grains (rb ) [4]. The coercivity (Hc ) is given as Hc ¼ 0:64ð/KS Ku Þ=Js [7] (Js is the saturation magnetization) when the induced anisotropies are dominant. The calculated results are also shown in Fig. 1 [4]. Here, vc valued are determined by X-ray diffraction profiles [4]. The calculated Hc with the unimodal log– normal distribution function for the Fe84Nb7B9 and Fe84.9Nb6B8P1Cu0.1 alloys are in good agreement with the measured ones. On the other hand, the calculated Hc for the Fe85Nb6B9 alloy is 23 times as large as the experimental one. This large difference originates in disregarding the existence of the coarse grains with about 40 nm in size. The calculated Hc is consistent with the experimental one by considering the existence of the coarse grains. Therefore, it can be said that our model explains well the dependence of Hc for the nanocrystalline Fe–Nb–B(–P–Cu) alloys on the grain-size distribution. These results also suggest that one should pay the attention on not only the mean grain size but also the grain-size distribution since the inhomogeneity of the grain size increases Hc :
Fig. 1. Grain-size distribution of Fe–Nb–B(–P–Cu) alloys. The % were obtained from TEM histograms and mean grain size ðDÞ images. The solid lines indicate fitting (a), (b) and (d) unimodal or (c) bimodal log–normal distribution functions. The inset in (c) is an enlarged view within 30–50 nm grain size. The fitting parameters and calculated results are also shown.
This work was entrusted with the ‘‘Nanohetero Metallic Materials’’ as a part of the Special Coordination Funds for Promoting Science and Technology from National Institute Materials Science.
References [1] G. Herzer, IEEE Trans. Magn. 25 (1989) 3327. [2] A. Makino, T. Bitoh, A. Inoue, T. Masumoto, Scr. Mater. 48 (2003) 869. [3] A. Makino, T. Bitoh, J. Appl. Phys. 93 (2003) 6522. [4] T. Bitoh, A. Makino, A. Inoue, Mater. Trans. 44 (2003) 2011. [5] G. Herzer, Scr. Metall. Mater. 33 (1995) 1741. [6] K. Suzuki, G. Herzer, J.M. Cadogan, J. Magn. Magn. Mater. 177–181 (1998) 949. [7] R.M. Bozorth, Ferromagnetism, Van Nostrand, New York, 1951, p. 811.
Journal of Magnetism and Magnetic Materials 272–276 (2004) 1445–1446
The effect of grain-size distribution on coercivity in nanocrystalline soft magnetic alloys Teruo Bitoha,*, Akihiro Makinoa, Akihisa Inoueb a
Department of Machine Intelligence, Faculty of Systems Science and Technology, Akita Prefectural University, 84–4 Ebinokuchi Tuchiya, Honjo 015-0055, Japan b Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
Abstract A simple model considering grain-size distribution has been fabricated based on the random anisotropy model. The model indicates that the effective magnetic anisotropy increases with increasing width of the grain-size distribution even if the mean grain size is constant. The coercivity (Hc) of some nanocrystalline Fe–Nb–B(–P–Cu) alloys with different grain-size distribution has been calculated. Our model explains well the dependence of Hc on the grain-size distribution. r 2003 Elsevier B.V. All rights reserved. PACS: 75.50.Tt; 75.50.Bb; 61.46.+w Keywords: Nanocrystalline soft magnetic alloy; Random anisotropy model; Grain-size distribution; Structural inhomogeneity
Since Herzer’s first application of the random anisotropy model (RAM) to nanocrystalline Fe–Si–B– Nb–Cu alloy [1], this model has been employed widely to explain the origin of magnetic softness in various nanocrystalline systems. In these analysis based on the RAM, the magnetic softness has been discussed using the mean grain size. However, the grain size has a distribution in actual nanocrystalline alloys. Recently, we reported that the Fe85Nb6B9 alloy has an asquenched structure composed of an amorphous phase and a-Fe grains with 20–45 nm in size, and has a crystallized nanostructure including the relatively coarse grains [2–4]. The effect of the structural inhomogeneities on soft magnetic properties is very interesting. In this paper, the effect of the grain-size distribution on the magnetic softness of nanocrystalline soft magnetic alloys is discussed based on the RAM. For simplicity, let us consider that the maximum grain size (Dm ) does not exceed the exchange correlation length (Lex ). Then the fluctuating part of the magnetocrystalline anisotropy (/K1 S) is given by using a *Corresponding author. Tel./fax: +81-184-27-2211. E-mail address: teruo [email protected] (T. Bitoh).
distribution function of the grain size (f ðDÞ), as [4] Z Dm 2 K1 D3 f ðDÞdD ; ð1Þ hK1 i ¼ v2c 6 L0 0 where vc is the volume fraction of the crystalline phase, K1 is the intrinsic magnetocrystalline anisotropy constant and L0 is the intrinsic exchange correlation length [5]. We assume further a log-normal distribution function as the grain-size distribution 1 ln2 Dr ; ð2Þ exp f ðDr Þ ¼ pffiffiffiffiffiffi 2s2D 2psD Dr where Dr ¼ D=D0 is the reduced grain size, D0 is the median, and sD is the geometric standard deviation. If f ðDr Þ is negligible small at Dr > Dm ; then Dm can be regarded as infinity and we obtain Lex ¼ L40 =fvc /DS3 expð3s2D Þg and h Di 6 hK1 i ¼ v2c K1 expð6s2D Þ; ð3Þ L0 where /DS ¼ D0 expðs2D =2Þ is the mean grain size. Eq. (3) indicates that /K1 S increases with increasing sD ; i.e., soft magnetic properties of nanocrystalline alloys deteriorates with increasing sD ; even if /DS is
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.12.368
ARTICLE IN PRESS 1446
T. Bitoh et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 1445–1446
constant. It should be noted that this result is essentially established in other distribution functions. Naturally, the effective anisotropy in the nanocrystalline alloys may have contributions from induced anisotropies such as magneto-elastic anisotropy other than the random magnetocrystalline anisotropy and hence the effective anisotropy constant in actual materials is more correctly /KS ¼ ðKu2 þ /K1 S2 Þ1=2 [6], where Ku is the induced uniaxial anisotropy conR 1=2 stant and /K1 S ¼ vc K1 =N 1=2 ¼ vc K1 fð D3 f ðDÞdDÞ= L3ex g1=2 (N is the number of grains in a magnetically coupled volume). Most of the nanocrystalline Fe–M–B (M=Zr, Hf, Nb) alloys with good soft magnetic properties exhibit remanence ratio (Jr =Js ) of around 0.5. This means that the magnetization process of the alloys is mostly governed by the induced anisotropies, i.e., Ku > /K1 S: In a limiting condition of Ku2 b/K1 S2 enables us to arrive at Lex ¼ L0 ðK1 =Ku Þ1=2 and pffiffiffiffiffiffiffiffiffiffiffi Ku K1 hDi 3 expð3s2D Þ: ð4Þ hK iEKu þ vc 2 L0 The D3 behavior of Hc is observed for the nanocrystalline Fe–Zr–B(–Cu) alloys with sufficiently small D [6]. The grain-size distribution evaluated by counting the a-Fe grains in TEM images are shown in Fig. 1 [4]. The log–normal distribution function reproduces well the observed grain-size distribution. In order to reproduce the grain-size distribution of the Fe85Nb6B9 alloy more correctly, we consider the bimodal distribution function (fb ðDÞ) expressed by superimposing the two log–normal distribution functions with the different medians (D0 and db D0 ; db > 1), the geometric standard deviations (sD and sb ) and the ratio of the distribution function for the large grains to that of the small grains (rb ) [4]. The coercivity (Hc ) is given as Hc ¼ 0:64ð/KS Ku Þ=Js [7] (Js is the saturation magnetization) when the induced anisotropies are dominant. The calculated results are also shown in Fig. 1 [4]. Here, vc valued are determined by X-ray diffraction profiles [4]. The calculated Hc with the unimodal log– normal distribution function for the Fe84Nb7B9 and Fe84.9Nb6B8P1Cu0.1 alloys are in good agreement with the measured ones. On the other hand, the calculated Hc for the Fe85Nb6B9 alloy is 23 times as large as the experimental one. This large difference originates in disregarding the existence of the coarse grains with about 40 nm in size. The calculated Hc is consistent with the experimental one by considering the existence of the coarse grains. Therefore, it can be said that our model explains well the dependence of Hc for the nanocrystalline Fe–Nb–B(–P–Cu) alloys on the grain-size distribution. These results also suggest that one should pay the attention on not only the mean grain size but also the grain-size distribution since the inhomogeneity of the grain size increases Hc :
Fig. 1. Grain-size distribution of Fe–Nb–B(–P–Cu) alloys. The % were obtained from TEM histograms and mean grain size ðDÞ images. The solid lines indicate fitting (a), (b) and (d) unimodal or (c) bimodal log–normal distribution functions. The inset in (c) is an enlarged view within 30–50 nm grain size. The fitting parameters and calculated results are also shown.
This work was entrusted with the ‘‘Nanohetero Metallic Materials’’ as a part of the Special Coordination Funds for Promoting Science and Technology from National Institute Materials Science.
References [1] G. Herzer, IEEE Trans. Magn. 25 (1989) 3327. [2] A. Makino, T. Bitoh, A. Inoue, T. Masumoto, Scr. Mater. 48 (2003) 869. [3] A. Makino, T. Bitoh, J. Appl. Phys. 93 (2003) 6522. [4] T. Bitoh, A. Makino, A. Inoue, Mater. Trans. 44 (2003) 2011. [5] G. Herzer, Scr. Metall. Mater. 33 (1995) 1741. [6] K. Suzuki, G. Herzer, J.M. Cadogan, J. Magn. Magn. Mater. 177–181 (1998) 949. [7] R.M. Bozorth, Ferromagnetism, Van Nostrand, New York, 1951, p. 811.