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Effects of Ga precipitates in nanocrystalline Nd–Fe–Ga–B magnets studied by micromagnetics
Journal of Magnetism and Magnetic Materials 247 (2002) 15–18
Effects of Ga precipitates in nanocrystalline Nd–Fe–Ga–B magnets studied by micromagnetics Zhao Sufen*, Jin Hanmin, Wang Xuefeng, Yan Yu Department of Physics, Jilin University, Jiefang Road 119, 130021 Changchun, China Received 6 July 2001; received in revised form 22 October 2001
Abstract The effects of the Ga precipitates in nanocrystalline Nd–Fe–Ga–B magnet on the demagnetization curve and spin distributions during the demagnetization process are studied using the finite element technique of micromagnetics. It is concluded that the drastic increase in coercivity by the Ga doping is due to the stress fields originating from the coherent precipitation of pure Ga. The void effect of the precipitates on coercivity is negligibly small. r 2002 Elsevier Science B.V. All rights reserved. PACS: 75.50.Kj; 75.50.Bb; 75.50.Ww; 75.60.Ej Keywords: Nanocrystalline; Fe and its alloys; Permanent magnets; Magnetization curves; Rare-earth alloys
1. Introduction The substitution of 1% of Fe by Ga in the meltspun Nd18Fe76B6 magnet is found to lead to a drastic increase in coercivity m0i Hc to as large as >4.6 T [1]. The effect is only weakly dependent on the grain size L. It has been observed that the pure Ga precipitates epitaxially as particles of B5 nm mean diameter only within the Nd2Fe14B crystallites with no evidence of its presence in the Nd-rich grain boundary phase [1]. The similar but less marked increase in m0i Hc has also been reported for Nb-contained Nd–Fe–B magnets, in which the dispersed Nb-rich coherent precipitates within the Nd2Fe14B crystallites have dimensions of 5–10 nm [2]. It has been thought that the precipitates having *Corresponding author. Tel.: +431-8922331-3533. E-mail address: [email protected] (Z. Sufen).
dimensions comparable with the domain wall width d ¼ 4:2 nm along with associated strain fields would act as pinning agents for domain walls [2]. The demagnetization curve as a function of L [3,4] and of temperature [5] for the nanocrystalline Nd2Fe14B magnet have been simulated quantitatively well by use of the finite element technique of micromagnetics on the basis of the model magnets composed of cubic grains. This work will study the effects of the precipitates on the demagnetization process basing on the similar model also by use of the finite element technique of micromagnetics.
2. Model and calculations The cubic magnet consists of N ¼ n n n (n ¼ 5; N ¼ 125) cubic Nd2Fe14B grains of L ¼ 20 nm which is the same with the experimental
0304-8853/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 2 ) 0 0 0 2 6 - 4
Z. Sufen et al. / Journal of Magnetism and Magnetic Materials 247 (2002) 15–18
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value in Ref. [1]. For the magnet of stoichiometric composition, the grains contact with each other and the inter-grain exchange interaction is the same with the inner-grain exchange interaction (denote by b ¼ 1). The another extreme case is for the Nd-rich magnet in which the grains are separated by a thick enough Nd-rich paramagnetic grain boundary phase so that the inter-grain exchange interaction is zero (denote by b ¼ 0). The c-axes of the grains are distributed uniformly in the angle space and are assigned to the grains randomly to simulate the isotropy of the real magnets. Each grain is divided into m m m (m ¼ 14) cubic elements of dimension L=m (E1:43 nmEd=3) and has a non-magnetic cubic precipitate of dimension d ¼ kL=m (k ¼ 0; 2, 4 and 6) at the center. Each magnetic single domain element is exchange coupled with the six adjacent elements. The magnet is placed at the center of cubic space B of dimension Lðnþ2 4=mÞ symmetrically, and the lattice grid consisted of the edges of the elements in the magnet are extended to the whole space B (Fig. 1). The applied field m0 Hap is decreased from 5 T by steps and the magnetic polarization for the magnet J at each m0 Hap is obtained from minimization of the Gibb’s free energy G: G is the sum of the exchange interaction energy, the magnetocrystalline anisotropy energy, the Zeeman energy and the stray field energy FM : Z " X G¼ A ðrai Þ þ K1 sin2 y/1 0 0S i
#
~ dv þ FM ; þK2 sin y/1 0 0S J~s H 4
where ai ði ¼ x; y; zÞ are the direction cosines of the magnetic polarization J~s and y/1 0 0S is the angle between J~s and the c-axis. Replacing FM by Z , 1 1 L 3 2 ~ ½r A ð~ r Þ J s ð~ r Þ dvE W¼ 2m0 B 2m0 m X , 2 ~ðiÞ J s ðiÞ ; ½r A i
the minimization is performed with respect to the polar angles of J~s ðiÞ; (yi ; ji ), and the three ~ðiÞ; Ax ðiÞ; Ay ðiÞ and Az ðiÞ (i is components of A through all elements within space B) under the ~ðjÞ at the grid points j boundary conditions that A on the boundary faces of B space are the magnetic potential vector produced by the magnetic moments of the elements, i.e. 3 X ~ J s ðiÞ ½~ r ðjÞ ~ r ðiÞ
~ðjÞ ¼ L A : 3 m ~ 4pj~ r ðjÞ r ðiÞj i The partial differentials of qAa =qg ða; g ¼ x; t; zÞ are approximated by the ratio of the finite differences DAa to Dg between the nearest neighboring grid points [6]. Under the minimum condition of G; the relation W ¼ FM holds and ~ within the space B becomes the magnetic vector A potential [7]. The exchange interaction between the neighboring elements is calculated by assuming that J~s varies linearly between the centers of the elements [3,4]. By considering the magnet as a uniformly magnetized sphere, the demagnetization ~M field is approximated by J~=3m0 ð¼ ðr A ~ J~Þ=m0 Þ; A M is the potential within the magnet). The values of the magnetic constants at room temperature are Js ¼ 1:63 T, A ¼ 7:7 1012 J/m, K1 ¼ 4:3 106 J/m3 and K2 ¼ 0:65 106 J/m3 [8]. The minimization of energy was obtained by using the conjugate gradient method [9].
3. Results and discussions
Fig. 1. A corner of the magnet and space B.
Fig. 2 shows the demagnetization curves calculated for d ¼ 0; 2:9 nm (k ¼ 2), 5.7 nm (k ¼ 4) and 8.6 nm (k ¼ 6). It can be seen that the value of m0i Hc is not affected by the precipitates while JðHÞ decreases with increase of the volume of the precipitates in proportion. Fig. 3a and b
Z. Sufen et al. / Journal of Magnetism and Magnetic Materials 247 (2002) 15–18
demonstrate the spin distributions at m0 Hap ¼ 1:87 T (J ¼ 0:017 T) and m0 Hap ¼ 1:88 T (J ¼ 0:009 T), respectively, on some area of the section cut parallel to the surface of the magnet and the applied field direction through the centers
Fig. 2. Demagnetization curve calculated for b ¼ 1 and d ¼ 0; 2.9, 5.7 and 8.6 nm.
17
of the grains. No sign of domain wall pinning is observed at the precipitates, and it would be concluded that the increase of m0i Hc is caused not by the geometrical effect of the non-magnetic precipitates but by the stress fields caused by the coherent precipitates. Quantitative analysis of the stress effects is impossible because among other things the magnetostriction constants are not constants but are dependent on the applied field strikingly. For instance the magnetostriction l measured perpendicular to the c-axis varies by as large as B110 106 without any sign of saturation when the field applied along the easy c-axis increases from 0 to 14 T [10] in contrast to that l should be essentially unchanged if the constants are field independent. A qualitative analysis is given as follows. Consider a spherical grain containing a spherical Ga precipitate of radius r0 at the center. By neglecting the constant term, the magnetostriction, nearly isotropic in the c-plane
Fig. 3. Spin distribution at Hap ¼ 1:87 T and J ¼ 0:017 T (figure a) and at Hap ¼ 1:88 T and J ¼ 0:009 T (figure b). b ¼ 1 and d ¼ 5:7 nm (k ¼ 4). The bold and thin arrows denote the opposite directions of the vector components vertical to the section. The twospearhead arrow at the upper-left corner of each grain denotes the direction of the c-axis.
Z. Sufen et al. / Journal of Magnetism and Magnetic Materials 247 (2002) 15–18
18
demagnetization curve calculated for K1eff ¼ K1 þ Ks with Ks ¼ 1:58 106 J/m3 (E0:36K1 ) simulates the experimental curve of m0i Hc ¼ 4:6 T [1] well as is showed in Fig. 4. The curves calculated for Ks ¼ 0 and b ¼ 0 and 1 are also presented in the figure for reference. In conclusion, the drastic increase in coercivity by the Ga doping would be due to the stress fields originating from the coherent precipitation of Ga. The void effect of the non-magnetic precipitates on coercivity is negligibly small. Fig. 4. Demagnetization curves. Full curve: experiment [1]; symbols: calculations for d ¼ 5:7 nm (k ¼ 4).
Acknowledgements [11], as a function of the directions of J~s and measurement is formulated as
This work is supported by National Science Foundation of China.
lða; bÞ ¼ A cos2 y/1 0 0S þ B cos2 ymeas þ C cos2 ymeas cos2 y/1 0 0S ; where ymeas is the angle between the direction of the measurement and the c-axis, and A, B and C are independent of y/1 0 0S and ymeas : The Young’s modulus being nearly isotropic [12], the stress s at position (r; yr ; jr ) in the coordinate system with the origin at the center and the z-axis parallel to the c-axis is radial and is sðrÞ ¼ sr0 ðr0 =rÞ2 ; where sr0 is the stress on the precipitate surface. The magnetoelastic energy then is r 2 0 Fs ðrÞ ¼ sr0 ½B cos2 yr r ðA þ C cos2 yr Þ sin2 y/1 0 0S
and effective first magnetic anisotropy constant at the position is r 2 r0 K1eff ¼ K1 þ sr0 ðA þ C cos2 yr Þ: r It can be seen from the relation that a striking increase in m0i Hc occurs if the average value of sr0 ðr0 =rÞ2 ðA þ C cos2 yr Þ is positive and large. The
References [1] M.A. Al-Khafaji Ahmad, H.A. Davies, W.M. Rainforth, R.A. Buckley, J. Magn. Magn. Mater. 145 (1995) L19. [2] S.F.H. Parker, R.J. Pollard, D.G. Lord, P.J. Grundy, IEEE Trans. Magn. 23 (1987) 210. [3] M.K. Griffiths, J.E.L. Bishop, J.W. Tucker, H.A. Davies, J. Magn. Magn. Mater. 183 (1998) 49. [4] Jin Hanmin, Wang Xuefeng, Zhao Sufen, Yan Yu, Chin. Phys. 10 (2001) 862. [5] Zhao Sufen, Jin Hanmin, Wang Xuefeng, Yan Yu, J. Phys.: Condens. Matter 13 (2001) 3865. [6] Jin Hanmin, Y.B. Kim, Wang Xuefeng, J. Phys.: Condens. Matter 10 (1998) 7243. [7] P. Asselin, A.A. Thiele, IEEE Trans. Magn. MAG-22 (1986) 1876. [8] R. Fischer, T. Schrefl, H. Kronmuller, . J. Fidler, J. Magn. Magn. Mater. 153 (1996) 35. [9] P. Gill, W. Murray, M.H. Wright, Practical optimization, Academic Press, London, 1981. [10] P.A. Algarabel, M.R. Iberra, C. Marquina, A. Del Moral, S. Zemirli, J. Magn. Magn. Mater. 84 (1990) 109. [11] A.V. Andreev, A.V. Deryagin, N.V. Kudrevatikh, N.V. Mushinikov, V.A. Reimer, S.V. Terentev, Zh. Elesp. Teor. Fiz. 90 (1986) 1042. [12] Y. Luo, Y. Hou, Proceedings of the Second International Symposium on Physics and Magnetic Materials, 1992, Beijing, China, p. 671.
Effects of Ga precipitates in nanocrystalline Nd–Fe–Ga–B magnets studied by micromagnetics Zhao Sufen*, Jin Hanmin, Wang Xuefeng, Yan Yu Department of Physics, Jilin University, Jiefang Road 119, 130021 Changchun, China Received 6 July 2001; received in revised form 22 October 2001
Abstract The effects of the Ga precipitates in nanocrystalline Nd–Fe–Ga–B magnet on the demagnetization curve and spin distributions during the demagnetization process are studied using the finite element technique of micromagnetics. It is concluded that the drastic increase in coercivity by the Ga doping is due to the stress fields originating from the coherent precipitation of pure Ga. The void effect of the precipitates on coercivity is negligibly small. r 2002 Elsevier Science B.V. All rights reserved. PACS: 75.50.Kj; 75.50.Bb; 75.50.Ww; 75.60.Ej Keywords: Nanocrystalline; Fe and its alloys; Permanent magnets; Magnetization curves; Rare-earth alloys
1. Introduction The substitution of 1% of Fe by Ga in the meltspun Nd18Fe76B6 magnet is found to lead to a drastic increase in coercivity m0i Hc to as large as >4.6 T [1]. The effect is only weakly dependent on the grain size L. It has been observed that the pure Ga precipitates epitaxially as particles of B5 nm mean diameter only within the Nd2Fe14B crystallites with no evidence of its presence in the Nd-rich grain boundary phase [1]. The similar but less marked increase in m0i Hc has also been reported for Nb-contained Nd–Fe–B magnets, in which the dispersed Nb-rich coherent precipitates within the Nd2Fe14B crystallites have dimensions of 5–10 nm [2]. It has been thought that the precipitates having *Corresponding author. Tel.: +431-8922331-3533. E-mail address: [email protected] (Z. Sufen).
dimensions comparable with the domain wall width d ¼ 4:2 nm along with associated strain fields would act as pinning agents for domain walls [2]. The demagnetization curve as a function of L [3,4] and of temperature [5] for the nanocrystalline Nd2Fe14B magnet have been simulated quantitatively well by use of the finite element technique of micromagnetics on the basis of the model magnets composed of cubic grains. This work will study the effects of the precipitates on the demagnetization process basing on the similar model also by use of the finite element technique of micromagnetics.
2. Model and calculations The cubic magnet consists of N ¼ n n n (n ¼ 5; N ¼ 125) cubic Nd2Fe14B grains of L ¼ 20 nm which is the same with the experimental
0304-8853/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 2 ) 0 0 0 2 6 - 4
Z. Sufen et al. / Journal of Magnetism and Magnetic Materials 247 (2002) 15–18
16
value in Ref. [1]. For the magnet of stoichiometric composition, the grains contact with each other and the inter-grain exchange interaction is the same with the inner-grain exchange interaction (denote by b ¼ 1). The another extreme case is for the Nd-rich magnet in which the grains are separated by a thick enough Nd-rich paramagnetic grain boundary phase so that the inter-grain exchange interaction is zero (denote by b ¼ 0). The c-axes of the grains are distributed uniformly in the angle space and are assigned to the grains randomly to simulate the isotropy of the real magnets. Each grain is divided into m m m (m ¼ 14) cubic elements of dimension L=m (E1:43 nmEd=3) and has a non-magnetic cubic precipitate of dimension d ¼ kL=m (k ¼ 0; 2, 4 and 6) at the center. Each magnetic single domain element is exchange coupled with the six adjacent elements. The magnet is placed at the center of cubic space B of dimension Lðnþ2 4=mÞ symmetrically, and the lattice grid consisted of the edges of the elements in the magnet are extended to the whole space B (Fig. 1). The applied field m0 Hap is decreased from 5 T by steps and the magnetic polarization for the magnet J at each m0 Hap is obtained from minimization of the Gibb’s free energy G: G is the sum of the exchange interaction energy, the magnetocrystalline anisotropy energy, the Zeeman energy and the stray field energy FM : Z " X G¼ A ðrai Þ þ K1 sin2 y/1 0 0S i
#
~ dv þ FM ; þK2 sin y/1 0 0S J~s H 4
where ai ði ¼ x; y; zÞ are the direction cosines of the magnetic polarization J~s and y/1 0 0S is the angle between J~s and the c-axis. Replacing FM by Z , 1 1 L 3 2 ~ ½r A ð~ r Þ J s ð~ r Þ dvE W¼ 2m0 B 2m0 m X , 2 ~ðiÞ J s ðiÞ ; ½r A i
the minimization is performed with respect to the polar angles of J~s ðiÞ; (yi ; ji ), and the three ~ðiÞ; Ax ðiÞ; Ay ðiÞ and Az ðiÞ (i is components of A through all elements within space B) under the ~ðjÞ at the grid points j boundary conditions that A on the boundary faces of B space are the magnetic potential vector produced by the magnetic moments of the elements, i.e. 3 X ~ J s ðiÞ ½~ r ðjÞ ~ r ðiÞ
~ðjÞ ¼ L A : 3 m ~ 4pj~ r ðjÞ r ðiÞj i The partial differentials of qAa =qg ða; g ¼ x; t; zÞ are approximated by the ratio of the finite differences DAa to Dg between the nearest neighboring grid points [6]. Under the minimum condition of G; the relation W ¼ FM holds and ~ within the space B becomes the magnetic vector A potential [7]. The exchange interaction between the neighboring elements is calculated by assuming that J~s varies linearly between the centers of the elements [3,4]. By considering the magnet as a uniformly magnetized sphere, the demagnetization ~M field is approximated by J~=3m0 ð¼ ðr A ~ J~Þ=m0 Þ; A M is the potential within the magnet). The values of the magnetic constants at room temperature are Js ¼ 1:63 T, A ¼ 7:7 1012 J/m, K1 ¼ 4:3 106 J/m3 and K2 ¼ 0:65 106 J/m3 [8]. The minimization of energy was obtained by using the conjugate gradient method [9].
3. Results and discussions
Fig. 1. A corner of the magnet and space B.
Fig. 2 shows the demagnetization curves calculated for d ¼ 0; 2:9 nm (k ¼ 2), 5.7 nm (k ¼ 4) and 8.6 nm (k ¼ 6). It can be seen that the value of m0i Hc is not affected by the precipitates while JðHÞ decreases with increase of the volume of the precipitates in proportion. Fig. 3a and b
Z. Sufen et al. / Journal of Magnetism and Magnetic Materials 247 (2002) 15–18
demonstrate the spin distributions at m0 Hap ¼ 1:87 T (J ¼ 0:017 T) and m0 Hap ¼ 1:88 T (J ¼ 0:009 T), respectively, on some area of the section cut parallel to the surface of the magnet and the applied field direction through the centers
Fig. 2. Demagnetization curve calculated for b ¼ 1 and d ¼ 0; 2.9, 5.7 and 8.6 nm.
17
of the grains. No sign of domain wall pinning is observed at the precipitates, and it would be concluded that the increase of m0i Hc is caused not by the geometrical effect of the non-magnetic precipitates but by the stress fields caused by the coherent precipitates. Quantitative analysis of the stress effects is impossible because among other things the magnetostriction constants are not constants but are dependent on the applied field strikingly. For instance the magnetostriction l measured perpendicular to the c-axis varies by as large as B110 106 without any sign of saturation when the field applied along the easy c-axis increases from 0 to 14 T [10] in contrast to that l should be essentially unchanged if the constants are field independent. A qualitative analysis is given as follows. Consider a spherical grain containing a spherical Ga precipitate of radius r0 at the center. By neglecting the constant term, the magnetostriction, nearly isotropic in the c-plane
Fig. 3. Spin distribution at Hap ¼ 1:87 T and J ¼ 0:017 T (figure a) and at Hap ¼ 1:88 T and J ¼ 0:009 T (figure b). b ¼ 1 and d ¼ 5:7 nm (k ¼ 4). The bold and thin arrows denote the opposite directions of the vector components vertical to the section. The twospearhead arrow at the upper-left corner of each grain denotes the direction of the c-axis.
Z. Sufen et al. / Journal of Magnetism and Magnetic Materials 247 (2002) 15–18
18
demagnetization curve calculated for K1eff ¼ K1 þ Ks with Ks ¼ 1:58 106 J/m3 (E0:36K1 ) simulates the experimental curve of m0i Hc ¼ 4:6 T [1] well as is showed in Fig. 4. The curves calculated for Ks ¼ 0 and b ¼ 0 and 1 are also presented in the figure for reference. In conclusion, the drastic increase in coercivity by the Ga doping would be due to the stress fields originating from the coherent precipitation of Ga. The void effect of the non-magnetic precipitates on coercivity is negligibly small. Fig. 4. Demagnetization curves. Full curve: experiment [1]; symbols: calculations for d ¼ 5:7 nm (k ¼ 4).
Acknowledgements [11], as a function of the directions of J~s and measurement is formulated as
This work is supported by National Science Foundation of China.
lða; bÞ ¼ A cos2 y/1 0 0S þ B cos2 ymeas þ C cos2 ymeas cos2 y/1 0 0S ; where ymeas is the angle between the direction of the measurement and the c-axis, and A, B and C are independent of y/1 0 0S and ymeas : The Young’s modulus being nearly isotropic [12], the stress s at position (r; yr ; jr ) in the coordinate system with the origin at the center and the z-axis parallel to the c-axis is radial and is sðrÞ ¼ sr0 ðr0 =rÞ2 ; where sr0 is the stress on the precipitate surface. The magnetoelastic energy then is r 2 0 Fs ðrÞ ¼ sr0 ½B cos2 yr r ðA þ C cos2 yr Þ sin2 y/1 0 0S
and effective first magnetic anisotropy constant at the position is r 2 r0 K1eff ¼ K1 þ sr0 ðA þ C cos2 yr Þ: r It can be seen from the relation that a striking increase in m0i Hc occurs if the average value of sr0 ðr0 =rÞ2 ðA þ C cos2 yr Þ is positive and large. The
References [1] M.A. Al-Khafaji Ahmad, H.A. Davies, W.M. Rainforth, R.A. Buckley, J. Magn. Magn. Mater. 145 (1995) L19. [2] S.F.H. Parker, R.J. Pollard, D.G. Lord, P.J. Grundy, IEEE Trans. Magn. 23 (1987) 210. [3] M.K. Griffiths, J.E.L. Bishop, J.W. Tucker, H.A. Davies, J. Magn. Magn. Mater. 183 (1998) 49. [4] Jin Hanmin, Wang Xuefeng, Zhao Sufen, Yan Yu, Chin. Phys. 10 (2001) 862. [5] Zhao Sufen, Jin Hanmin, Wang Xuefeng, Yan Yu, J. Phys.: Condens. Matter 13 (2001) 3865. [6] Jin Hanmin, Y.B. Kim, Wang Xuefeng, J. Phys.: Condens. Matter 10 (1998) 7243. [7] P. Asselin, A.A. Thiele, IEEE Trans. Magn. MAG-22 (1986) 1876. [8] R. Fischer, T. Schrefl, H. Kronmuller, . J. Fidler, J. Magn. Magn. Mater. 153 (1996) 35. [9] P. Gill, W. Murray, M.H. Wright, Practical optimization, Academic Press, London, 1981. [10] P.A. Algarabel, M.R. Iberra, C. Marquina, A. Del Moral, S. Zemirli, J. Magn. Magn. Mater. 84 (1990) 109. [11] A.V. Andreev, A.V. Deryagin, N.V. Kudrevatikh, N.V. Mushinikov, V.A. Reimer, S.V. Terentev, Zh. Elesp. Teor. Fiz. 90 (1986) 1042. [12] Y. Luo, Y. Hou, Proceedings of the Second International Symposium on Physics and Magnetic Materials, 1992, Beijing, China, p. 671.