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An approach to coercivity relating coercive field and activation volume
Physica B 319 (2002) 127–132
An approach to coercivity relating coercive field and activation volume V.T.M.S. Barthema, D. Givordb,*, M.F. Rossignolb, P. Tenaudc b
a Instituto de Fisica, Universidade Federal de Rio de Janeiro, Brazil Laboratoire Louis N!eel, C.N.R.S., B.P. 166 X 25, Avenue des Martyrs, 38042 Grenoble Cedex 9, France c UGIMAG S.A., BP 2, 38830-Saint-Pierre d’Allevard, France
Received 13 February 2002
Abstract In modern hard magnets, the reduction in the coercive field with respect to the main phase anisotropy field reveals the determining influence of structural defects. Yet, coercivity is usually discussed in reference with main phase anisotropy. An alternative approach is developed in this article in which the coercive field is related to another experimental coercivity parameter, the activation volume, va : Confrontation with experimental results on ferrite magnets and NdFeB magnets shows that this approach incorporates the essential features required to describe coercivity of hard materials. It emerges from the analysis that the anisotropy in the activation volume is closely related to the main phase anisotropy. An unexpected outcome of this is to explain the success of usual analyses of coercivity. r 2002 Published by Elsevier Science B.V. PACS: 75.60.G Keywords: Coercivity model; Magnetisation processes; Hard magnetic materials; Hard ferrites; NdFeB magnets
1. Introduction The concept of coercivity is intrinsically linked to that of magnetic anisotropy. Reversal by uniform rotation is expected for an ideal system, with the coercive field Hc being equal to the anisotropy field HA : Actually, in modern hard magnets, Hc is typically an order of magnitude smaller than HA : This reveals the determining influence of structural defects. Magnetisation reversal within each constitutive grain is a hetero*Corresponding author. Fax: +33-4-76-88-11-91. E-mail address: [email protected] (D. Givord).
geneous non-uniform process which develops from a small ‘‘critical nucleus’’ of volume vn : Coercivity in such systems is expected to depend on magnetic anisotropy within the critical nucleus. This anisotropy should depend at least in part on defect region properties and it has no a priori reason to be simply proportional to main phase magnetic anisotropy as it is usually assumed. We show in this paper than an expression may be established which ultimately relates the coercive field to the anisotropy within the critical nucleus. This relation is then used to analyse the temperature dependence of Hc in representative hard magnetic systems.
0921-4526/02/$ - see front matter r 2002 Published by Elsevier Science B.V. PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 1 1 1 5 - 8
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128
2. Model
tion volume va : The activation volume is related to Sv through [7–10]
Following usual analyses, the simplest expression for Hc ðTÞ is (see Refs. [1–4])
va ¼ kT=m0 Sv Ms :
m0 Hc ðTÞ ¼ am0 HA ðTÞ m0 Neff Ms ðTÞ:
ð1Þ
This can be expressed more conveniently as m0 Hcrit ðTÞ ¼ m0 Hc ðTÞ þ m0 Neff Ms ðTÞ ¼ am0 HA ðTÞ:
ð10 Þ
The field m0 Hcrit ðTÞ which determines reversal, is made of the experimental coercive field and of dipolar fields within matter represented by the term m0 Neff Ms ðTÞ; where Ms ðTÞ is the main phase magnetisation and Neff is a phenomenological parameter (experimentally Neff E1). Reversal occurs when m0 Hcrit ðTÞ amounts to am0 HA ðTÞ (experimentally aE0:1), i.e. it is assumed that the anisotropy field in the defect region is proportional to the main phase anisotropy field HA ðTÞ: Following Kools [1] and Hirosawa and Sagawa [2], a may be considered as a purely phenomenological parameter. Alternatively, within the micromagnetic model developed by Kronmuller . and co-workers, a is linked to the type of structural defects, which are critical for magnetisation reversal [3]. In the Global model developed by Givord et al. [5], it is explicitly considered that magnetisation reversal develops from an initial critical volume. The critical field is expressed as 0
m0 Hcrit ðTÞ ¼ a gn ðTÞ=ðvn ðTÞ
1=3
Ms ðTÞÞ:
ð2Þ
Relation (2) relates m0 Hcrit ðTÞ to the domain wall energy, gn, within the critical nucleus and to vn itself, with a0 being a phenomenological parameter. The so-called fluctuation field, m0 Hf ; which describes the reduction in coercivity due to thermal activation effects, is added to m0 Hcrit ðTÞ: m0 Hf E 25m0 Sv ; with Sv being the magnetic viscosity coefficient, deduced from magnetic after-effect measurements [6]. The fluctuation field is small in general with respect to other contributions to m0 Hcrit ðTÞ and the fact that it is neglected in relation (10 ) is of minor importance. At finite temperature, magnetisation reversal is thermally activated and vn in relation (2) can be identified with an experimental entity, the activa-
ð3Þ
The domain wall energy within the critical nucleus, gn, remains unknown. It is assumed to be proportional to g; the domain wall energy within the main phase. Relation (2) becomes [5] m0 Hcrit ðTÞ ¼ aG gðTÞ=ðva ðTÞ1=3 Ms ðTÞÞ;
ð20 Þ
where again aG is a phenomenological parameter. As long as uniform rotation does not occur, magnetisation reversal involves the formation of a non-uniform magnetisation configuration. In high anisotropy systems, a domain wall is the nonuniform configuration of lowest energy. Thus, the critical nucleus should be proportional to d3n ; where dn is the domain wall thickness within the critical nucleus [5,4]. Let us now assume that 2nd order anisotropy only is involved. p The domain ffiffiffiffiffiffiffiffiffiffiffiffiffiffi wall thickness dn is proportional to An =Kn (see Ref. [11] for instance) where Kn is the 2nd order anisotropy constant within the critical nucleus and An the corresponding exchange constant. Kn may thus be expressed as a function of va ( vn ): ð4Þ Kn pp2 An =v2=3 a pffiffiffiffiffiffiffiffiffiffiffiffi and gn ð¼ An Kn Þ may be related to dn and thus to va through gn ¼ 4pAn =dn p4pAn =v1=3 a :
ð5Þ
Combining Eqs. (2) and (5) leads to m0 Hcrit ðTÞ ¼ 4pa0 An ðTÞ=ðva ðTÞ2=3 Ms ðTÞÞ:
ð6Þ
In this relation, the exchange constant An ðTÞ is not accessible to experiment. Assuming that it is proportional to hard phase value, one obtains m0 Hcrit ðTÞ ¼ a0G AðTÞ=ðva ðTÞ2=3 Ms ðTÞÞ:
ð60 Þ
Expression (60 ) relates the critical field to the magnetocrystalline anisotropy within the critical volume, implicitly contained within the experimental parameter va : The main phase parameters introduced in Eq. (60 ) are the exchange constant, AðTÞ; and the magnetisation, Ms ðTÞ: These are not determining parameters for coercivity. Additionally (i) for temperatures up to typically 0:8TC ; the decrease with temperature of the exchange constant and that of the magnetisation are expected to
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be much less than that of the anisotropy and (ii) the ratio of exchange to magnetisation varies less with temperature than the exchange constant and magnetisation do separately.
The magnetic viscosity coefficient Sv ðTÞ was measured in the same ferrite magnets for which m0 Hc ðTÞ has been measured previously. This was done following the procedure described in Ref. [15] (Fig. 1-inset). In order to compare experimental data to the model, relation (60 ) was re-expressed as
3. Coercivity analyses in ferrite and NdFeB magnets
m0 H0 ðTÞ=m0 Ms ðTÞ ¼ a0G AðTÞ=ðva ðTÞ2=3 Ms ðTÞ2 Þ
Ferrite magnets, where 3d magnetism only is involved and the magnetocrystalline anisotropy is largely dominated by 2nd order terms, are appropriate systems to test the applicability of expression (60 ). In (Ba1xLax)(Fe12xCox)O19 or (Sr1xLax)(Fe12xCox)O19, the low temperature magnetocrystalline anisotropy increases with x; for sufficiently low x values. An increase in magnet coercivity follows the increase in anisotropy, with optimum room temperature properties being obtained for xE0:2 [12,13]. The temperature dependence of the coercive field, m0 Hc ðTÞ; was measured recently in such Ba-Ferrites for 0oxo0:4 (Fig. 1) [14]. It is much dependent on x value. This means that these magnets can constitute meaningful systems to test the relevance of the present approach.
with m0 H0 ðTÞ being the coercive field corrected for thermal activation effects:
Neff
m0 H0 ðTÞ ¼ m0 Hc ðTÞ þ 25m0 Sv ðTÞ:
ð8Þ
In relation (7), Ms ðTÞ was directly derived from magnetisation measurements and va ðTÞ was deduced from Sv ðTÞ (see expression (3)). The evaluation of the exchange constant AðTÞ in these ferrimagnetic compounds is not straightforward. From the temperature dependence of the hyperfine . field at each Fe site, measured by Mossbauer spectroscopy, van Loef et al. showed that two groups of Fe sites may be identified [16]. The first group (I) is made of five Fe atoms (two at tetrahedral sites, plus two at octahedral sites plus one with fivefold coordination) and the second group (II) is made of seven Fe atoms at octahedral sites. The temperature dependence of the hyperfine field, Hhyp;I and Hhyp;II at sites (I) and (II), respectively, are markedly different, the faster variation of Hhyp;II being characteristic of weaker exchange interactions. The exchange energy may thus be written as Eexch ¼ 12nI m2I þ nI2II mI mII þ 12nII m2II ;
Fig. 1. Temperature dependence of m0 Hc ðTÞ in (Ba1xLax)(Fe12xCox)O19 magnets.
ð7Þ
ð9Þ
where mI and mII are the finite temperature Fe magnetic moment within sublattice (I) and (II), respectively, nI and nII are the corresponding molecular field coefficients and nI2II is the molecular field coefficient representing the coupling between both sublattices. The molecular field coefficients were deduced by fitting the temperature dependences of the hyperfine fields. In this procedure, the zero Kelvin Fe moment, mFe 0 ; was taken as 5mB which corresponds to the expected value for Fe3+. Hhyp;I and Hhyp;II ; were assumed to be proportional to the Fe moments mI and mII and thus to follow the B5=2 ðxÞ variation, with xI ¼ Fe mFe 0 Hm;I= kT (Hm;I ¼ nI mI þ nI2II mII ) and xII ¼ m0
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Hm;II= kT (Hm;II ¼ nII mII þ nIII mI ). The value of the molecular field coefficients then derived are nI ¼ 26:5; nII ¼ 32:8 and nI2II¼ 54:8 K/m2B. Að0Þ ¼ 9:23 1012 J/m was deduced from TC and AðTÞ was assumed to be proportional to Eexch : Fig. 2 shows m0 H0 ðTÞ=m0 Ms ðTÞ versus AðTÞ=ðva ðTÞ2=3 m0 Ms ðTÞ2 Þ: Linear variations are obtained for the three different types of Ferrite magnets studied. For the three of them, it is deduced that aV E30 and Neff E1:5: The temperature dependence of 30AðTÞ=ðva ðTÞ2=3 m0 Ms ðTÞ2 Þ is plotted in Fig. 3. The corresponding curves are shifted from m0 H0 ðTÞ=m0 Ms ðTÞ curves by an approximate constant amount which is representative of Neff : It is striking that the main qualitative differences in the experimental temperature dependences of m0 H0 ðTÞ=m0 Ms ðTÞ from one sample to another are reproduced in the calculated temperature dependences of 30AðTÞ= ðva ðTÞ2=3 m0 Ms ðTÞ2 Þ: At low temperatures, m0 H0 ðTÞ= m0 Ms ðTÞ and 30AðTÞ=ðva ðTÞ2=3 m0 Ms ðTÞ2 Þ curves are shifted downward in the x ¼ 0 sample with respect to the two other samples. The increase of both m0 H0 ðTÞ=m0 Ms ðTÞ and 30AðTÞ= ðva ðTÞ2=3 m0 Ms ðTÞ2 Þ with temperature is maximum for x ¼ 0 and minimum for x ¼ 0:4: Thus different features of the experimental temperature depen-
Fig. 2. m0 H0 ðTÞ=m0 Ms ðTÞ versus AðTÞ=ðva ðTÞ2=3 m0 Ms ðTÞ2 Þ in (Ba1xLax)(Fe12xCox)O19 magnets.
Fig. 3. Temperature dependence of 30AðTÞ=ðva ðTÞ2=3 m0 Ms ðTÞ2 Þ and temperature dependence of m0 H0 ðTÞ=m0 Ms ðTÞ in (Ba1xLax)(Fe12xCox)O19 magnets.
dences of the coercive fields in different magnets are reproduced by the model. The RFeB magnets constitute the other main family of hard magnets. Measurements of the coercive field and magnetic viscosity were previously reported for these systems [17]. In NdFeB sintered magnets, m0 H0 ðTÞ=m0 Ms ðTÞ varies linearly with AðTÞ=ðva ðTÞ2=3 m0 Ms ðTÞ2 Þ from room temperature down to approximately 150 K (Fig. 4). At lower temperature, strong deviation from linearity develops which indicates that relation (60 ) is not valid anymore. As already stressed, this relation is expected to apply under the condition that anisotropy is dominated by 2nd order terms. The discrepancy between model and experimental data suggests that, at low temperature, higher-order anisotropy terms become large within the critical nucleus. This characteristic behaviour of NdFeB magnets may be linked to a well-known property of the Nd2Fe14B compound. The easy magnetisation direction in Nd2Fe14B is parallel to the c-axis of the tetragonal structure at room temperature but a magnetisation reorientation takes place at TSR ¼ 135 K [18]. When the moment direction leaves the c-axis, the 2nd order anisotropy terms vanish. At lower temperature, 4th order and 6th
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range within which 2nd order anisotropy terms are dominant.
4. Conclusion
Fig. 4. m0 H0 ðTÞ=m0 Ms ðTÞ versus AðTÞ=ðva ðTÞ2=3 m0 Ms ðTÞ2 Þ and versus gðTÞ=va ðTÞ1=3 m0 Ms ðTÞ2 in Nd15Fe77B8 magnets.
order anisotropy terms compete with 2nd order terms. The temperature TSR is determined by the subtle balance between various anisotropy terms (see Ref. [19]). The fact that the anomaly in m0 H0 ðTÞ=m0 Ms ðTÞ in magnets develops at the same temperature at which the spin reorientation in the Nd2Fe14B phase appears implies that the anisotropy properties within the critical nucleus do not differ strongly from main phase properties. The same conclusion was derived previously from measurements of the angular dependence of Hc [20] and from measurements of the temperature dependence of the activation volume [10,5]. The non-linearity observed at low temperature in m0 H0 ðTÞ=m0 Ms ðTÞ versus AðTÞ= ðva ðTÞ2=3 m0 Ms ðTÞ2 Þ is not observed in m0 H0 ðTÞ= m0 Ms ðTÞ versus gðTÞ=va ðTÞ1=3 m0 Ms ðTÞ2 which is actually linear in the whole temperature range (see Fig. 4) [17]. Expression (20 ) is more appropriate than expression (60 ) to describe the temperature dependence of the coercive field because (i) the present experimental results reveal that the anisotropy in the critical nucleus follows main phase anisotropy and thus it can be considered as proportional to it and (ii) the applicability of relation (20 ) is not restricted to the temperature
In the above discussion, the actual processes by which magnetisation reversal takes place were not considered. Possible processes for ferrite and NdFeB magnets were discussed in Ref. [4]. True nucleation is usually modelled as a uniform rotation within the critical nucleus. A significant reduction in anisotropy within the critical nucleus is then required to account for a coercive field, which is much weaker than the anisotropy field within the main phase. The present analysis led us to conclude that such a reduction does not occur. Passage and expansion mechanisms correspond to the growth of the initial nucleus into the main phase. In this case, a coercive field much weaker than the anisotropy field is consistent with the anisotropy in the critical volume being close to main phase anisotropy, in agreement with experimental results. The specificity of the analysis of coercivity presented in this article is to relate Hc and va ; two different experimental parameters characterising the coercive behaviours of hard materials. From the comparison of properties in a variety of different magnets, Barbier [21,7] noted that the room temperature coercive field Hc and the magnetic viscosity coefficient Sv were related through Hc pSv0:70 : We pointed out that this is approximately equivalent to Hc pv2=3 and we 1=3 related this to the fact that Hc pKn or g=va and 3 vn pdn [22,15]. Relation (60 ) of the present paper was derived from a discussion of the temperature dependence of the physical properties in a given system, using the same type of arguments. The adequation of the present model to describe the temperature dependence of Hc in ferrite magnets and also its inadequation to describe Hc ðTÞ in RFeB magnets establish that this model incorporates the essential features which are required to describe coercivity of hard magnetic materials. This model does not introduce an a priori relationship between critical volume
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anisotropy and main phase anisotropy. The results then obtained demonstrate that the anisotropy in the critical nucleus is closely related to hard phase anisotropy. An unexpected outcome of this analysis is to explain the success of usual models of coercivity.
References [1] F. Kools, J. Phys. 46 (1985) C6–349. [2] M. Sagawa, H. Hirosawa, J. Phys. (Paris) 49 (1988) C8–617. [3] H. Kronmuller, . Phys. Stat. Sol. B 144 (1987) 385. [4] Coercivity D.Givord, M.F. Rossignol, in: J.M.D. Coey (Ed.), Rare-Earth Iron Permanent Magnets, Clarendon Press, Oxford, 1996, pp. 219–284. [5] D. Givord, P. Tenaud, T. Viadieu, IEEE Trans. Magn. 24 (1988) 1921. [6] R. Street, J.C. Woolley, Proc. Phys. Soc. 62A (1949) 562. [7] E.P. Wohlfarth, J. Phys. F14 (1984) L155. [8] P. Gaunt, J. Appl. Phys. 59 (1986) 4129. [9] R. Street, P.G. McCormick, Y. Estrin, 10th International Workshop on Rare-earth Magnets and their applications, Kyoto, 1985, p. 83. [10] D. Givord, A. Li!enard, P. Tenaud, T. Viadieu, J. Magn. Magn. Mater. 67 (1987) L281.
[11] A.H. Morrish (Ed.), The Physical Principles of Magnetism, Wiley, New York, 1965. [12] H. Taguchi, T. Takeishi, K. Suwa, K. Mazuwa, Y. Minachi, J. Phys. IV 7 (1997) C1–C311. . [13] A. Morel, F. Kools, P. Tenaud, R. Grossinger, M.F. Rossignol, Proceedings of the ICF 8, IEEE Trans. Magn., 2001, in preparation. [14] V. Baltz, A. Morel, F. Kools, P. Tenaud, M.F. Rossignol (unpublished). [15] D. Givord, M.F. Rossignol, V. Villas-Boas, F. Cebollada, J.M. Gonzalez, in: F.P. Missell, V. Villas-Boas, H.R. Rechenberg, F.J.G. Landgraf (Ed.), Proceedings of the Ninth International Workshop on Magn. Anis. RE-TM alloys, Vol. 2, World Scientific, Singapore, p. 21. [16] J.J. van Loef, P.J.M. Franssen, Phys. Lett. 7 (1963) 225. [17] D.W. Taylor, V. Villas-Boas, Q. Lu, M.F. Rossignol, F.P. Missell, D. Givord, S. Hirosawa, J. Magn. Magn. Mater. 130 (1994) 225. [18] D. Givord, H.S. Li, R. Perrier de la B#athie, Solid State Commun. 51 (1984) 857. [19] J.J.M. Franse, R.J. Radwanski, in: J.M.D. Coey (Ed.), Rare-Earth Iron Permanent Magnets, Clarendon Press, Oxford, 1996, pp. 58–158. [20] F. Cebollada, M.F. Rossignol, D. Givord, V. Villas-Boas, J.M. Gonzalez, Phys. Rev. B 52 (1995) 13511. [21] J.C. Barbier, Ann. Phys. 9 (1950) 84. [22] D. Givord, Q. Lu, M.F. Rossignol, P. Tenaud, T. Viadieu, J. Magn. Magn. Mater. 83 (1990) 183.
An approach to coercivity relating coercive field and activation volume V.T.M.S. Barthema, D. Givordb,*, M.F. Rossignolb, P. Tenaudc b
a Instituto de Fisica, Universidade Federal de Rio de Janeiro, Brazil Laboratoire Louis N!eel, C.N.R.S., B.P. 166 X 25, Avenue des Martyrs, 38042 Grenoble Cedex 9, France c UGIMAG S.A., BP 2, 38830-Saint-Pierre d’Allevard, France
Received 13 February 2002
Abstract In modern hard magnets, the reduction in the coercive field with respect to the main phase anisotropy field reveals the determining influence of structural defects. Yet, coercivity is usually discussed in reference with main phase anisotropy. An alternative approach is developed in this article in which the coercive field is related to another experimental coercivity parameter, the activation volume, va : Confrontation with experimental results on ferrite magnets and NdFeB magnets shows that this approach incorporates the essential features required to describe coercivity of hard materials. It emerges from the analysis that the anisotropy in the activation volume is closely related to the main phase anisotropy. An unexpected outcome of this is to explain the success of usual analyses of coercivity. r 2002 Published by Elsevier Science B.V. PACS: 75.60.G Keywords: Coercivity model; Magnetisation processes; Hard magnetic materials; Hard ferrites; NdFeB magnets
1. Introduction The concept of coercivity is intrinsically linked to that of magnetic anisotropy. Reversal by uniform rotation is expected for an ideal system, with the coercive field Hc being equal to the anisotropy field HA : Actually, in modern hard magnets, Hc is typically an order of magnitude smaller than HA : This reveals the determining influence of structural defects. Magnetisation reversal within each constitutive grain is a hetero*Corresponding author. Fax: +33-4-76-88-11-91. E-mail address: [email protected] (D. Givord).
geneous non-uniform process which develops from a small ‘‘critical nucleus’’ of volume vn : Coercivity in such systems is expected to depend on magnetic anisotropy within the critical nucleus. This anisotropy should depend at least in part on defect region properties and it has no a priori reason to be simply proportional to main phase magnetic anisotropy as it is usually assumed. We show in this paper than an expression may be established which ultimately relates the coercive field to the anisotropy within the critical nucleus. This relation is then used to analyse the temperature dependence of Hc in representative hard magnetic systems.
0921-4526/02/$ - see front matter r 2002 Published by Elsevier Science B.V. PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 1 1 1 5 - 8
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2. Model
tion volume va : The activation volume is related to Sv through [7–10]
Following usual analyses, the simplest expression for Hc ðTÞ is (see Refs. [1–4])
va ¼ kT=m0 Sv Ms :
m0 Hc ðTÞ ¼ am0 HA ðTÞ m0 Neff Ms ðTÞ:
ð1Þ
This can be expressed more conveniently as m0 Hcrit ðTÞ ¼ m0 Hc ðTÞ þ m0 Neff Ms ðTÞ ¼ am0 HA ðTÞ:
ð10 Þ
The field m0 Hcrit ðTÞ which determines reversal, is made of the experimental coercive field and of dipolar fields within matter represented by the term m0 Neff Ms ðTÞ; where Ms ðTÞ is the main phase magnetisation and Neff is a phenomenological parameter (experimentally Neff E1). Reversal occurs when m0 Hcrit ðTÞ amounts to am0 HA ðTÞ (experimentally aE0:1), i.e. it is assumed that the anisotropy field in the defect region is proportional to the main phase anisotropy field HA ðTÞ: Following Kools [1] and Hirosawa and Sagawa [2], a may be considered as a purely phenomenological parameter. Alternatively, within the micromagnetic model developed by Kronmuller . and co-workers, a is linked to the type of structural defects, which are critical for magnetisation reversal [3]. In the Global model developed by Givord et al. [5], it is explicitly considered that magnetisation reversal develops from an initial critical volume. The critical field is expressed as 0
m0 Hcrit ðTÞ ¼ a gn ðTÞ=ðvn ðTÞ
1=3
Ms ðTÞÞ:
ð2Þ
Relation (2) relates m0 Hcrit ðTÞ to the domain wall energy, gn, within the critical nucleus and to vn itself, with a0 being a phenomenological parameter. The so-called fluctuation field, m0 Hf ; which describes the reduction in coercivity due to thermal activation effects, is added to m0 Hcrit ðTÞ: m0 Hf E 25m0 Sv ; with Sv being the magnetic viscosity coefficient, deduced from magnetic after-effect measurements [6]. The fluctuation field is small in general with respect to other contributions to m0 Hcrit ðTÞ and the fact that it is neglected in relation (10 ) is of minor importance. At finite temperature, magnetisation reversal is thermally activated and vn in relation (2) can be identified with an experimental entity, the activa-
ð3Þ
The domain wall energy within the critical nucleus, gn, remains unknown. It is assumed to be proportional to g; the domain wall energy within the main phase. Relation (2) becomes [5] m0 Hcrit ðTÞ ¼ aG gðTÞ=ðva ðTÞ1=3 Ms ðTÞÞ;
ð20 Þ
where again aG is a phenomenological parameter. As long as uniform rotation does not occur, magnetisation reversal involves the formation of a non-uniform magnetisation configuration. In high anisotropy systems, a domain wall is the nonuniform configuration of lowest energy. Thus, the critical nucleus should be proportional to d3n ; where dn is the domain wall thickness within the critical nucleus [5,4]. Let us now assume that 2nd order anisotropy only is involved. p The domain ffiffiffiffiffiffiffiffiffiffiffiffiffiffi wall thickness dn is proportional to An =Kn (see Ref. [11] for instance) where Kn is the 2nd order anisotropy constant within the critical nucleus and An the corresponding exchange constant. Kn may thus be expressed as a function of va ( vn ): ð4Þ Kn pp2 An =v2=3 a pffiffiffiffiffiffiffiffiffiffiffiffi and gn ð¼ An Kn Þ may be related to dn and thus to va through gn ¼ 4pAn =dn p4pAn =v1=3 a :
ð5Þ
Combining Eqs. (2) and (5) leads to m0 Hcrit ðTÞ ¼ 4pa0 An ðTÞ=ðva ðTÞ2=3 Ms ðTÞÞ:
ð6Þ
In this relation, the exchange constant An ðTÞ is not accessible to experiment. Assuming that it is proportional to hard phase value, one obtains m0 Hcrit ðTÞ ¼ a0G AðTÞ=ðva ðTÞ2=3 Ms ðTÞÞ:
ð60 Þ
Expression (60 ) relates the critical field to the magnetocrystalline anisotropy within the critical volume, implicitly contained within the experimental parameter va : The main phase parameters introduced in Eq. (60 ) are the exchange constant, AðTÞ; and the magnetisation, Ms ðTÞ: These are not determining parameters for coercivity. Additionally (i) for temperatures up to typically 0:8TC ; the decrease with temperature of the exchange constant and that of the magnetisation are expected to
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be much less than that of the anisotropy and (ii) the ratio of exchange to magnetisation varies less with temperature than the exchange constant and magnetisation do separately.
The magnetic viscosity coefficient Sv ðTÞ was measured in the same ferrite magnets for which m0 Hc ðTÞ has been measured previously. This was done following the procedure described in Ref. [15] (Fig. 1-inset). In order to compare experimental data to the model, relation (60 ) was re-expressed as
3. Coercivity analyses in ferrite and NdFeB magnets
m0 H0 ðTÞ=m0 Ms ðTÞ ¼ a0G AðTÞ=ðva ðTÞ2=3 Ms ðTÞ2 Þ
Ferrite magnets, where 3d magnetism only is involved and the magnetocrystalline anisotropy is largely dominated by 2nd order terms, are appropriate systems to test the applicability of expression (60 ). In (Ba1xLax)(Fe12xCox)O19 or (Sr1xLax)(Fe12xCox)O19, the low temperature magnetocrystalline anisotropy increases with x; for sufficiently low x values. An increase in magnet coercivity follows the increase in anisotropy, with optimum room temperature properties being obtained for xE0:2 [12,13]. The temperature dependence of the coercive field, m0 Hc ðTÞ; was measured recently in such Ba-Ferrites for 0oxo0:4 (Fig. 1) [14]. It is much dependent on x value. This means that these magnets can constitute meaningful systems to test the relevance of the present approach.
with m0 H0 ðTÞ being the coercive field corrected for thermal activation effects:
Neff
m0 H0 ðTÞ ¼ m0 Hc ðTÞ þ 25m0 Sv ðTÞ:
ð8Þ
In relation (7), Ms ðTÞ was directly derived from magnetisation measurements and va ðTÞ was deduced from Sv ðTÞ (see expression (3)). The evaluation of the exchange constant AðTÞ in these ferrimagnetic compounds is not straightforward. From the temperature dependence of the hyperfine . field at each Fe site, measured by Mossbauer spectroscopy, van Loef et al. showed that two groups of Fe sites may be identified [16]. The first group (I) is made of five Fe atoms (two at tetrahedral sites, plus two at octahedral sites plus one with fivefold coordination) and the second group (II) is made of seven Fe atoms at octahedral sites. The temperature dependence of the hyperfine field, Hhyp;I and Hhyp;II at sites (I) and (II), respectively, are markedly different, the faster variation of Hhyp;II being characteristic of weaker exchange interactions. The exchange energy may thus be written as Eexch ¼ 12nI m2I þ nI2II mI mII þ 12nII m2II ;
Fig. 1. Temperature dependence of m0 Hc ðTÞ in (Ba1xLax)(Fe12xCox)O19 magnets.
ð7Þ
ð9Þ
where mI and mII are the finite temperature Fe magnetic moment within sublattice (I) and (II), respectively, nI and nII are the corresponding molecular field coefficients and nI2II is the molecular field coefficient representing the coupling between both sublattices. The molecular field coefficients were deduced by fitting the temperature dependences of the hyperfine fields. In this procedure, the zero Kelvin Fe moment, mFe 0 ; was taken as 5mB which corresponds to the expected value for Fe3+. Hhyp;I and Hhyp;II ; were assumed to be proportional to the Fe moments mI and mII and thus to follow the B5=2 ðxÞ variation, with xI ¼ Fe mFe 0 Hm;I= kT (Hm;I ¼ nI mI þ nI2II mII ) and xII ¼ m0
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Hm;II= kT (Hm;II ¼ nII mII þ nIII mI ). The value of the molecular field coefficients then derived are nI ¼ 26:5; nII ¼ 32:8 and nI2II¼ 54:8 K/m2B. Að0Þ ¼ 9:23 1012 J/m was deduced from TC and AðTÞ was assumed to be proportional to Eexch : Fig. 2 shows m0 H0 ðTÞ=m0 Ms ðTÞ versus AðTÞ=ðva ðTÞ2=3 m0 Ms ðTÞ2 Þ: Linear variations are obtained for the three different types of Ferrite magnets studied. For the three of them, it is deduced that aV E30 and Neff E1:5: The temperature dependence of 30AðTÞ=ðva ðTÞ2=3 m0 Ms ðTÞ2 Þ is plotted in Fig. 3. The corresponding curves are shifted from m0 H0 ðTÞ=m0 Ms ðTÞ curves by an approximate constant amount which is representative of Neff : It is striking that the main qualitative differences in the experimental temperature dependences of m0 H0 ðTÞ=m0 Ms ðTÞ from one sample to another are reproduced in the calculated temperature dependences of 30AðTÞ= ðva ðTÞ2=3 m0 Ms ðTÞ2 Þ: At low temperatures, m0 H0 ðTÞ= m0 Ms ðTÞ and 30AðTÞ=ðva ðTÞ2=3 m0 Ms ðTÞ2 Þ curves are shifted downward in the x ¼ 0 sample with respect to the two other samples. The increase of both m0 H0 ðTÞ=m0 Ms ðTÞ and 30AðTÞ= ðva ðTÞ2=3 m0 Ms ðTÞ2 Þ with temperature is maximum for x ¼ 0 and minimum for x ¼ 0:4: Thus different features of the experimental temperature depen-
Fig. 2. m0 H0 ðTÞ=m0 Ms ðTÞ versus AðTÞ=ðva ðTÞ2=3 m0 Ms ðTÞ2 Þ in (Ba1xLax)(Fe12xCox)O19 magnets.
Fig. 3. Temperature dependence of 30AðTÞ=ðva ðTÞ2=3 m0 Ms ðTÞ2 Þ and temperature dependence of m0 H0 ðTÞ=m0 Ms ðTÞ in (Ba1xLax)(Fe12xCox)O19 magnets.
dences of the coercive fields in different magnets are reproduced by the model. The RFeB magnets constitute the other main family of hard magnets. Measurements of the coercive field and magnetic viscosity were previously reported for these systems [17]. In NdFeB sintered magnets, m0 H0 ðTÞ=m0 Ms ðTÞ varies linearly with AðTÞ=ðva ðTÞ2=3 m0 Ms ðTÞ2 Þ from room temperature down to approximately 150 K (Fig. 4). At lower temperature, strong deviation from linearity develops which indicates that relation (60 ) is not valid anymore. As already stressed, this relation is expected to apply under the condition that anisotropy is dominated by 2nd order terms. The discrepancy between model and experimental data suggests that, at low temperature, higher-order anisotropy terms become large within the critical nucleus. This characteristic behaviour of NdFeB magnets may be linked to a well-known property of the Nd2Fe14B compound. The easy magnetisation direction in Nd2Fe14B is parallel to the c-axis of the tetragonal structure at room temperature but a magnetisation reorientation takes place at TSR ¼ 135 K [18]. When the moment direction leaves the c-axis, the 2nd order anisotropy terms vanish. At lower temperature, 4th order and 6th
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range within which 2nd order anisotropy terms are dominant.
4. Conclusion
Fig. 4. m0 H0 ðTÞ=m0 Ms ðTÞ versus AðTÞ=ðva ðTÞ2=3 m0 Ms ðTÞ2 Þ and versus gðTÞ=va ðTÞ1=3 m0 Ms ðTÞ2 in Nd15Fe77B8 magnets.
order anisotropy terms compete with 2nd order terms. The temperature TSR is determined by the subtle balance between various anisotropy terms (see Ref. [19]). The fact that the anomaly in m0 H0 ðTÞ=m0 Ms ðTÞ in magnets develops at the same temperature at which the spin reorientation in the Nd2Fe14B phase appears implies that the anisotropy properties within the critical nucleus do not differ strongly from main phase properties. The same conclusion was derived previously from measurements of the angular dependence of Hc [20] and from measurements of the temperature dependence of the activation volume [10,5]. The non-linearity observed at low temperature in m0 H0 ðTÞ=m0 Ms ðTÞ versus AðTÞ= ðva ðTÞ2=3 m0 Ms ðTÞ2 Þ is not observed in m0 H0 ðTÞ= m0 Ms ðTÞ versus gðTÞ=va ðTÞ1=3 m0 Ms ðTÞ2 which is actually linear in the whole temperature range (see Fig. 4) [17]. Expression (20 ) is more appropriate than expression (60 ) to describe the temperature dependence of the coercive field because (i) the present experimental results reveal that the anisotropy in the critical nucleus follows main phase anisotropy and thus it can be considered as proportional to it and (ii) the applicability of relation (20 ) is not restricted to the temperature
In the above discussion, the actual processes by which magnetisation reversal takes place were not considered. Possible processes for ferrite and NdFeB magnets were discussed in Ref. [4]. True nucleation is usually modelled as a uniform rotation within the critical nucleus. A significant reduction in anisotropy within the critical nucleus is then required to account for a coercive field, which is much weaker than the anisotropy field within the main phase. The present analysis led us to conclude that such a reduction does not occur. Passage and expansion mechanisms correspond to the growth of the initial nucleus into the main phase. In this case, a coercive field much weaker than the anisotropy field is consistent with the anisotropy in the critical volume being close to main phase anisotropy, in agreement with experimental results. The specificity of the analysis of coercivity presented in this article is to relate Hc and va ; two different experimental parameters characterising the coercive behaviours of hard materials. From the comparison of properties in a variety of different magnets, Barbier [21,7] noted that the room temperature coercive field Hc and the magnetic viscosity coefficient Sv were related through Hc pSv0:70 : We pointed out that this is approximately equivalent to Hc pv2=3 and we 1=3 related this to the fact that Hc pKn or g=va and 3 vn pdn [22,15]. Relation (60 ) of the present paper was derived from a discussion of the temperature dependence of the physical properties in a given system, using the same type of arguments. The adequation of the present model to describe the temperature dependence of Hc in ferrite magnets and also its inadequation to describe Hc ðTÞ in RFeB magnets establish that this model incorporates the essential features which are required to describe coercivity of hard magnetic materials. This model does not introduce an a priori relationship between critical volume
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anisotropy and main phase anisotropy. The results then obtained demonstrate that the anisotropy in the critical nucleus is closely related to hard phase anisotropy. An unexpected outcome of this analysis is to explain the success of usual models of coercivity.
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