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Calculations of the magnetic properties of R2M14B intermetallic compounds (R=rare earth, M=Fe, Co)
Journal of Magnetism and Magnetic Materials 400 (2016) 379–383
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Calculations of the magnetic properties of R2M14B intermetallic compounds (R ¼ rare earth, M¼ Fe, Co) Masaaki Ito a,b,c,n, Masao Yano c, Nora M. Dempsey a,b, Dominique Givord a,b,d a
CNRS, Institut Néel, 25 rue des Martyrs, BP166, 38042 Grenoble, France University Grenoble Alpes, Institut Néel, 38042 Grenoble, France c Advanced Material Engineering Division, Toyota Motor Corporation, Susono 410-1193, Japan d Instituto de Fisica, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil b
art ic l e i nf o
a b s t r a c t
Article history: Received 22 June 2015 Received in revised form 11 August 2015 Accepted 14 August 2015 Available online 19 August 2015
The hard magnetic properties of “R–M–B” (R ¼rare earth, M ¼mainly Fe) magnets derive from the specific intrinsic magnetic properties encountered in Fe-rich R2M14B compounds. Exchange interactions are dominated by the 3d elements, Fe and Co, and may be modeled at the macroscopic scale with good accuracy. Based on classical formulae that relate the anisotropy coefficients to the crystalline electric field parameters and exchange interactions, a simple numerical approach is used to derive the temperature dependence of anisotropy in various R2Fe14B compounds (R¼ Pr, Nd, Dy). Remarkably, a unique set of crystal field parameters give fair agreement with the experimentally measured properties of all compounds. This implies reciprocally that the properties of compounds that incorporate a mixture of different rare-earth elements may be predicted accurately. This is of special interest for material optimization that often involves the partial replacement of Nd with another R element and also the substitution of Co for Fe. & 2015 Elsevier B.V. All rights reserved.
Keywords: Molecular field calculations Crystalline-electric field interactions R2M14B intermetallic compounds
1. Introduction The hard magnetic properties of “R–M–B” (R ¼rare earth, M ¼mainly Fe) magnets derive from the specific intrinsic magnetic properties encountered in Fe-rich R2M14B compounds [1,2]. These are large magnetization and high magnetocrystalline uniaxial anisotropy occurring at room temperature and above. The reference ternary phase used in these magnets is Nd2Fe14B. Material optimization very often involves the partial substitution of another R element for Nd and that of Co for Fe. It is of special interest in such cases to be able to predict what will be the impact of a given atom substitution on the values of magnetization and anisotropy and on their temperature dependence. The magnetic properties of most R2Fe14B compounds across the rare-earth series (lanthanides þY) [3–7], as well as those of a number of R2Co14B compounds [8–10], have been measured, in most cases using single crystal samples. The analysis developed involved a molecular field approach for the exchange interactions and a single-ion model for the crystalline-electric field (CEF) interactions [3,11,12]. From the individual temperature dependence n Corresponding author at: CNRS, Institut Néel, 25 rue des Martyrs, BP166, 38042 Grenoble, France. E-mail address: [email protected] (M. Ito).
http://dx.doi.org/10.1016/j.jmmm.2015.08.065 0304-8853/& 2015 Elsevier B.V. All rights reserved.
of the rare-earth and transition metal magnetic moments, the temperature dependence of the compound’s spontaneous magnetization can be easily derived. The connection between temperature independent crystalline-electric field parameters and temperature dependent anisotropy coefficients and anisotropy constants is more subtle. The latter are intrinsically related to coercivity and, for this reason, knowledge of them is very important from the point of view of applications. In the present study, the model used to derive the molecular field and CEF parameters in R2M14B compounds is recalled. A classical approach is used to macroscopically derive the anisotropy coefficients and anisotropy constants describing the material’s magnetocrystalline anisotropy from the CEF parameters. The calculated temperature dependence of the magnetization and anisotropies are compared to experimental values from literature.
2. Molecular field and CEF analysis of R2M14B compounds Due to the itinerant character of 3d electrons, a realistic description of the 3d magnetism in R–M compounds is much more difficult than that of the rare-earth ions, in which the magnetic 4f electrons are localized. In previous studies, two complementary expressions were used to analyse the magnetic properties of these
380
M. Ito et al. / Journal of Magnetism and Magnetic Materials 400 (2016) 379–383
Table 1 Curie temperature, R-M molecular field coefficients (nRM and n0 RM) and CEF parameters for various R2Fe14B compounds. nRM and n0 RM are related by nRM ¼ (2(gJ–1)/gJ) n0 RM where gJ is the Landé factor. R2Fe14B
nRM×μ0 (SI)
TC (K)
Ce Pr Nd Gd Dy
427 565 592 664 589
0 138.9 205.0 149.1 67.1
n0 RM×μ0 (SI)
0 277.7 273.3 149.1 134.2
Bn0(K /R‐ion)
B20
B40
B60
A20
A 40
A60
0 7.7 2.2 ― 1.5
0 3.4×10 2 1.1×10 2 ― 1.2×10 3
0 2.5×10 3 1.3×10 3 ― 8.2×10 6
0 305 305 ― 293
0 13.5 13.5 ― 13.1
0 2.2 2.2 ― 1.3
materials [4]. The state of the transition metal moment was expressed phenomenologically as:
EM = KM sin2 ϑ − ( nRM < MR >T + μ 0 Happ ) < MM >T
(1)
In this expression, KM is the second order anisotropy constant, usually assumed equal to the transition metal anisotropy ( KFe or KCo ) in La2M14B or Y2M14B (neither La nor Y are magnetic), ϑ is the angle between the M moment and the c-axis of the tetragonal structure, nRM is the molecular field coefficient between the R and M moments (discussed below), Happ is the applied field,T and T are the M and R magnetization values at the concerned temperature. In general, T is assumed to vary with τ ( τ = T /TC ), as found experimentally in compounds with nonmagnetic R. Alternatively, the phenomenological formula proposed by Kuz’min [13] may be used: 3
5
m (τ ) = [1 − sτ 2 − ( 1 − s ) τ 2 ]1/3
(2)
where m (τ )=Ms (T ) /Ms (0) is the normalized magnetization ( Ms (T ) is the spontaneous magnetization at temperature T and Ms (0) is the magnetization at absolute saturation). Expression (2) reproduces closely the real temperature dependence of the compounds, and in particular describes the low temperature dependence of the magnetization governed by the 3/2 spin-wave coefficient as well as the critical exponent 1/3 close to the Curie temperature. Expression (2) was used in the present study, with s = 0.6 in all cases [14]. The only free parameter in Expression (2) is the Curie temperature, which was taken from [15]. The state of the R ion is described with the Hamiltonian representing the combined action of the exchange interactions and CEF interactions on the rare-earth J multiplet:
/R =
∑ n, m
Bnm Onm
− nRM ⟨MM ⟩T MR
An0 (Ka0−n)
(3)
in which Onm represent the Stevens coefficients and Bnm are CEF parameters. The parameter nRM represents indirect coupling between the 3d and the 4f electrons. It should be written more rigorously as nRM = n3d − 5d ·n5d − 4f , where the first term represents the coupling between the M 3d electrons and the R 5d ones and the second term represents the on-site R coupling between the 5d and 4f electrons. The coefficient n5d − 4f depends on the nature of the R element considered: (i) the coupling is not between the R moments but between the spins, thus leading to the so-called de Gennes law, (ii) the distance between the 4f and 5d shells varies strongly across the R series and thus also the associated exchange interactions between the electrons involved [15]. The Onm Stevens operators in expression (3) are function of J and Jz, where J is the total angular momentum of the considered R ion and Jz the projection of J along the quantization axis, which corresponds in the present study to the c-axis of the tetragonal structure. The coefficient n (¼2, 4, or 6) is related to the order of the anisotropy term considered and the integer parameter m obeys the rules m ≤ n and m ≤ 4 in tetragonal symmetry.
From expression (3), the energy splitting of the rare-earth J multiplet may be obtained, with each state corresponding to a given Jz value. The population of each state is then simply given by Boltzmann statistics. The thermal average of the R magnetic moments,T , directly proportional to the rare-earth moments of order n, are readily derived. This determines the magnetic properties of the R ion at each temperature. The free parameters in the analysis are the molecular field coefficient nRM and the crystal field parameters Bnm , the most significant of which are B20 B40 and B60, representing the uniaxial anisotropy terms of 2nd, 4th and 6th orders. The Bnm parameters may be expressed as Bnm=θn Anm < rn > where θn is the Stevens factor of order n for the considered R ion, is the 4f shell radius of order n and Anm represents the environment. The TC and nRM, Bn0 and An0 parameters listed in Table 1 (R¼ Ce, Pr, Nd, Gd and Dy) correspond to average values taken from two different studies of R2Fe14B compounds [5,7]. The An0 parameters vary little between compounds, as expected given the identical crystal structure across the series. In the forthcoming analysis, the parameters characterizing Nd2Fe14B are used to calculate the properties of the other compounds (Pr2Fe14B and Gd2Fe14B).
3. From CEF parameters to anisotropy constants and anisotropy coefficients For practical purposes, the anisotropy energy of hard magnetic materials is often expressed as:
EA = K1 sin2 ϑ + K2 sin4 ϑ + K3 sin6 ϑ
(4)
The Ki’s are the anisotropy constants of different orders and ϑ is the angle between the magnetization direction and the c-axis of the tetragonal structure. Each anisotropy constant in Eq. (4) may be the sum of a contribution due to the M element and another contribution due to the R element. Only axial terms in the development of the anisotropy energy are considered here. Non-axial terms that represent in-plane anisotropy are smaller than the axial terms. Besides this, the easy magnetization axis of hard materials involved in magnet applications must be aligned along the c-axis, thus in-plane anisotropy effects are not relevant. The functions involved in the expansion of the anisotropy energy in terms of the powers of sin ϑ are not orthogonal. It results that the strength of the constants derived in the analysis depends on the value of the order n, at which the expansion is limited. An expansion in terms of the Legendre polynomials, Pn0 , that form a set of orthogonal functions, is more appropriate. The anisotropy energy is written:
Ea = k20 P20 (cos ϑ) + k 40 P40 (cos ϑ) + k60 P60 (cos ϑ)
(5)
where the k n0 are the so-called anisotropy coefficients. The anisotropy constants, Ki (T ) and coefficients are related through [16]
M. Ito et al. / Journal of Magnetism and Magnetic Materials 400 (2016) 379–383
K1= −
1 (3k20+10k 40+21k60 ) 2
1 K2= (35k 40+189k60 ) 8 K3= − (
231 0 k6 ) 16
381
(6a)
(6b)
(6c)
Expression (6) is valid for both the M and R anisotropy constants. However, the contribution of the M elements to anisotropy is reasonably well described within a development to first order in Ki (or 2nd order in k i0 ). Thus, the distinction between anisotropy constants and anisotropy coefficients become irrelevant. Concerning the R contribution to anisotropy, the anisotropy coefficients may be related to the thermal average values of the Stevens operators [16]
k n0 (T ) = Bn0T
(7)
Expression (7) illustrates the fact that the anisotropy is not directly linked to temperature but indirectly via the temperature dependence of the R magnetic moment [17]. This suggests a procedure to derive the temperature dependence of anisotropy. At each temperature, the Boltzmann population of the split states of the J multiplet can be calculated, and the thermal average of the R magnetic moment and Stevens operators of various orders are directly deduced. Assuming that Bn0 is temperature independent, which is certainly valid to first order, this gives access to k n0 (T ) and through relations (6) to the temperature dependent anisotropy constants that can be compared to experimental values. The present approach does not offer the inherent elegance of analytical expressions, that have been proposed to describe the temperature dependence of anisotropy [11,17] but it has the advantage that it involves simple numerical calculations.
4. Temperature dependence of the anisotropy in the R2M14B compounds Using values given in Table 1, the calculated temperature dependences of the spontaneous magnetization in all compounds are compared in Fig. 1 to the experimental temperature dependences [10]. The calculated temperature dependences of the anisotropy coefficients of order 2, 4 and 6 are shown in Fig. 2 for Nd2Fe14B. n (n + 1)
They follow approximately the power law k n0 (T )∝m (T ) 2 as predicted by Akulov [18]. From these coefficients, the anisotropy constants Ki (T ) in Nd2Fe14B were derived using expression (6). They are compared in Fig. 3a to experimental constants taken from
Fig. 2. Continuous line: temperature dependence of the anisotropy coefficients, kn (T ) , of order n¼ 2, 4 and 6 in Nd2Fe14B, derived from the CEF parameters known for this compound. Dots: temperature dependence of mn (n + 1) /2 , where m is the reduced magnetization. Each curve is normalized to the zero temperature value of the anisotropy coefficient to which it should be compared.
[4,10]. The calculated temperature dependences reproduce qualitatively experimental values but not quantitatively, in particular at low temperatures. The Fe contribution to the 2nd order anisotropy (inset to Fig. 3b) [10] was not included in the calculated K1 (T ). It represents a small but sizeable contribution to anisotropy at room temperature but it is negligible at low temperature and in no case can explain the disagreement between experiment and calculation. However, the calculated temperature dependence of the total anisotropy energy ET (T )=K1 (T )+K2 (T )+K3 (T ) is in excellent agreement with experimental anisotropy (Fig. 3b). This suggests that part of the disagreement comes from the difficulty in separating experimental anisotropy constants of various orders from each other. The field dependence of the magnetization M (Happ ) under Happ perpendicular to the easy axis, was calculated in a classical approach, in which the total magnetic energy was written as:
ET = KFe sin2 ϑ Fe + K1 sin2 ϑ Nd − nRFe ⟨MNd ⟩T ⟨MFe ⟩T cos ( ϑ Fe − ϑ Nd ) −μ 0 Happ ⟨MNd ⟩T sin ( ϑ Nd ) − μ 0 Happ ⟨MFe ⟩T sin ( ϑ Fe)
At each field value, the energy minimum was determined, considering the angles ϑ Fe and ϑNd as independent variables. Remarkably, the calculated magnetization curve M (Happ ) closely follow the experimental curve (Fig. 4 at 300 K) [19]. During reversal the angles ϑ Fe and ϑNd may differ by at most 4° (Fig. 4 inset). The calculated magnetization curve obtained when forcing collinearity ( ϑ Fe ¼ ϑNd ) is shown also in Fig. 4 for comparison. The anisotropy coefficients and constants for Pr2Fe14B and Dy2Fe14B were derived using the exact same An0 values as in Nd2Fe14B. The calculated temperature dependence K1 (T ) are shown in Fig. 5. No experimental temperature dependence exists for either of these compounds in the entire temperature range studied. The calculated K1 (300) in Pr2Fe14B is to be compared to the experimental value K1 = 5. 5 MJ/m3 at the same temperature, derived from μ0 HA = 8. 7 T assuming collinearity between the moments and that only second order anisotropy is present [10]. The anisotropy constant K1 (T ) derived from the slope, α, of the field dependence of magnetization in Dy2Fe14B using the classical expression α =
Fig. 1. Temperature dependence of the spontaneous magnetization in a series of R2Fe14B compounds. Dots: experimental data, continuous line: calculation.
(8)
μ0 Ms2 2K1
is systematically much smaller than the cal-
culated anisotropy constant. This discrepancy illustrates the fact that in ferrimagnetic compounds where the R and M moments are antiparallel, non-collinearity effects under a field applied perpendicular to the easy direction, drastically modify the magnetization variation, thus invalidating the usual procedure to derive
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M. Ito et al. / Journal of Magnetism and Magnetic Materials 400 (2016) 379–383
Fig. 3. (a) Temperature dependence of the anisotropy constants in Nd2Fe14B. Dots: experimental data. Continuous lines: variations derived from the values of anisotropy coefficients. (b) Temperature dependence of the total anisotropy energy in Nd2Fe14B. Dots: experimental data. Continuous line: derived from calculated anisotropy constants Ki’s. Inset: Experimental temperature dependence of anisotropy in Y2Fe14B.
Fig. 4. Field dependence of magnetization under an applied field perpendicular to the easy c-axis in Nd2Fe14B at 300 K. Blue dots: experiment, red curve: field dependence of the magnetization in a classical model, allowing for non-collinearity between the Nd and Fe moments. Green curve: same as red curve except that collinearity is imposed ( ϑFe ¼ ϑNd ). Inset: scheme illustrating a non-collinear magnetization process. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 6. Field dependence of the magnetization under an applied field perpendicular to the easy c-axis in Dy2Fe14B at 4.2 K. Blue dots: experiment, red curve: field dependence of the magnetization in a classical model, allowing for non-collinearity. Green curve: same as red curve except that collinearity is imposed ( ϑFe ¼ ϑDy ). Inset: scheme illustrating a non-collinear magnetization process. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
agreement is excellent thus confirming the relevance of the analysis here proposed.
5. Conclusions
Fig. 5. Temperature dependence of the anisotropy constants in Pr2Fe14B and Dy2Fe14B compared to Nd2Fe14B. These constants were derived by assuming that their CEF parameters are identical to those of Nd2Fe14B.
Using the known exchange parameters in R2Fe14B and the crystal-field parameters determined previously in Nd2Fe14B, we have derived the temperature dependence of the magnetization and of the anisotropy constants in a series of R2Fe14B compounds. The results are in good agreement with experiment. A classical model then permits calculations of the field dependence of the magnetization, which reproduce experimental data rather reasonably. This approach may now be extended to the calculation of the properties of R–M–B compounds including a partial substitution of another R element for Nd and of Co for Fe. This will contribute to material optimization, often required during magnet fabrication for specified applications.
Acknowledgments anisotropy constants experimentally. The calculated field dependence of the magnetization at 4.2 K in Dy2Fe14B using expression (8) is compared in Fig. 6 to experimental data [5,20]. The
This work was carried out in the framework of the Magnetic Materials for High-Efficient Motors (MagHEM) project.
M. Ito et al. / Journal of Magnetism and Magnetic Materials 400 (2016) 379–383
References [1] M. Sagawa, S. Fujimura, N. Togawa, H. Yamamoto, Y. Matsuura, J. Appl. Phys. 55 (1984) 2083. [2] J.F. Herbst, Rev. Mod. Phys. 63 (1991) 819. [3] J.J.M. Franse, R.J. Radwanski, Rare-Earth Fe High Performance Magnets, in: J.M. D. Coey (ed.). [4] J.M. Cadogan, J.P. Gavigan, D. Givord, H.S. Li, J. Phys. F 18 (1988) 779. [5] D. Givord, H.S. Li, J.M. Cadogan, J.M.D. Coey, J.P. Gavigan, O. Yamada, H. Maruyama, M. Sagawa, S. Hirosawa, J. Appl. Phys. 63 (1988) 3713. [6] M. Yamada, H. Kato, H. Yamamoto, Y. Nakagawa, Phys. Rev. B 38 (1988) 620. [7] M. Loewenhaupt, I. Sosnowska, J. Appl. Phys. 70 (1991) 5967. [8] H. Hiroyoshi, M. Yamada, N. Saito, H. Kato, Y. Nakagawa, M. Sagawa, J. Magn. Magn. Mater. 70 (1987) 337. [9] K.H.J. Buschow, D.B. de Mooij, S. Sinnema, R.J. Radwanski, J.J.M. Franse, J. Magn. Magn. Mater. 51 (1985) 211.
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[10] S. Hirosawa, K. Tokuhara, H. Yamamoto, S. Fujimura, M. Sagawa, J. Appl. Phys. 61 (1987) 3571. [11] M.D. Kuz’min, A.M. Tishin, Handbook of Magnetic Materials, in: K.H.J. Buschow (ed.), vol. 17, 2008, p. 149. [12] J.J.M. Franse, R.J. Radwanski, Handbook of Magnetic Materials, in: K.H.J. Buschow (ed.), vol. 7, 1993, p. 307. [13] M.D. Kuz’min, Phys. Rev. Lett. 94 (2005) 107204. [14] M.D. Kuz’min, D. Givord, V. Skumryev, J. Appl. Phys. 107 (2010) 1. [15] E. Belorizky, M.A. Frémy, J.P. Gavigan, D. Givord, H.S. Li, J. Appl. Phys. 61 (1987) 3971. [16] R.J. Radwanski, J.J.M. Franse, J. Magn. Magn. Mater. 80 (1989) 14. [17] H.B. Callen, E. Callen, J. Phys. Chem. Solids 27 (1966) 1271. [18] Z. Akulov, Physik 100 (1936) 197. [19] S. Chikazumi, Physics of Ferromagnetism, Oxford Science Publications, Oxford, 1997. [20] Y.B. Kim, Han-min Jin, J. Magn. Magn. Mater. 189 (1998) 251.
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Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Calculations of the magnetic properties of R2M14B intermetallic compounds (R ¼ rare earth, M¼ Fe, Co) Masaaki Ito a,b,c,n, Masao Yano c, Nora M. Dempsey a,b, Dominique Givord a,b,d a
CNRS, Institut Néel, 25 rue des Martyrs, BP166, 38042 Grenoble, France University Grenoble Alpes, Institut Néel, 38042 Grenoble, France c Advanced Material Engineering Division, Toyota Motor Corporation, Susono 410-1193, Japan d Instituto de Fisica, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil b
art ic l e i nf o
a b s t r a c t
Article history: Received 22 June 2015 Received in revised form 11 August 2015 Accepted 14 August 2015 Available online 19 August 2015
The hard magnetic properties of “R–M–B” (R ¼rare earth, M ¼mainly Fe) magnets derive from the specific intrinsic magnetic properties encountered in Fe-rich R2M14B compounds. Exchange interactions are dominated by the 3d elements, Fe and Co, and may be modeled at the macroscopic scale with good accuracy. Based on classical formulae that relate the anisotropy coefficients to the crystalline electric field parameters and exchange interactions, a simple numerical approach is used to derive the temperature dependence of anisotropy in various R2Fe14B compounds (R¼ Pr, Nd, Dy). Remarkably, a unique set of crystal field parameters give fair agreement with the experimentally measured properties of all compounds. This implies reciprocally that the properties of compounds that incorporate a mixture of different rare-earth elements may be predicted accurately. This is of special interest for material optimization that often involves the partial replacement of Nd with another R element and also the substitution of Co for Fe. & 2015 Elsevier B.V. All rights reserved.
Keywords: Molecular field calculations Crystalline-electric field interactions R2M14B intermetallic compounds
1. Introduction The hard magnetic properties of “R–M–B” (R ¼rare earth, M ¼mainly Fe) magnets derive from the specific intrinsic magnetic properties encountered in Fe-rich R2M14B compounds [1,2]. These are large magnetization and high magnetocrystalline uniaxial anisotropy occurring at room temperature and above. The reference ternary phase used in these magnets is Nd2Fe14B. Material optimization very often involves the partial substitution of another R element for Nd and that of Co for Fe. It is of special interest in such cases to be able to predict what will be the impact of a given atom substitution on the values of magnetization and anisotropy and on their temperature dependence. The magnetic properties of most R2Fe14B compounds across the rare-earth series (lanthanides þY) [3–7], as well as those of a number of R2Co14B compounds [8–10], have been measured, in most cases using single crystal samples. The analysis developed involved a molecular field approach for the exchange interactions and a single-ion model for the crystalline-electric field (CEF) interactions [3,11,12]. From the individual temperature dependence n Corresponding author at: CNRS, Institut Néel, 25 rue des Martyrs, BP166, 38042 Grenoble, France. E-mail address: [email protected] (M. Ito).
http://dx.doi.org/10.1016/j.jmmm.2015.08.065 0304-8853/& 2015 Elsevier B.V. All rights reserved.
of the rare-earth and transition metal magnetic moments, the temperature dependence of the compound’s spontaneous magnetization can be easily derived. The connection between temperature independent crystalline-electric field parameters and temperature dependent anisotropy coefficients and anisotropy constants is more subtle. The latter are intrinsically related to coercivity and, for this reason, knowledge of them is very important from the point of view of applications. In the present study, the model used to derive the molecular field and CEF parameters in R2M14B compounds is recalled. A classical approach is used to macroscopically derive the anisotropy coefficients and anisotropy constants describing the material’s magnetocrystalline anisotropy from the CEF parameters. The calculated temperature dependence of the magnetization and anisotropies are compared to experimental values from literature.
2. Molecular field and CEF analysis of R2M14B compounds Due to the itinerant character of 3d electrons, a realistic description of the 3d magnetism in R–M compounds is much more difficult than that of the rare-earth ions, in which the magnetic 4f electrons are localized. In previous studies, two complementary expressions were used to analyse the magnetic properties of these
380
M. Ito et al. / Journal of Magnetism and Magnetic Materials 400 (2016) 379–383
Table 1 Curie temperature, R-M molecular field coefficients (nRM and n0 RM) and CEF parameters for various R2Fe14B compounds. nRM and n0 RM are related by nRM ¼ (2(gJ–1)/gJ) n0 RM where gJ is the Landé factor. R2Fe14B
nRM×μ0 (SI)
TC (K)
Ce Pr Nd Gd Dy
427 565 592 664 589
0 138.9 205.0 149.1 67.1
n0 RM×μ0 (SI)
0 277.7 273.3 149.1 134.2
Bn0(K /R‐ion)
B20
B40
B60
A20
A 40
A60
0 7.7 2.2 ― 1.5
0 3.4×10 2 1.1×10 2 ― 1.2×10 3
0 2.5×10 3 1.3×10 3 ― 8.2×10 6
0 305 305 ― 293
0 13.5 13.5 ― 13.1
0 2.2 2.2 ― 1.3
materials [4]. The state of the transition metal moment was expressed phenomenologically as:
EM = KM sin2 ϑ − ( nRM < MR >T + μ 0 Happ ) < MM >T
(1)
In this expression, KM is the second order anisotropy constant, usually assumed equal to the transition metal anisotropy ( KFe or KCo ) in La2M14B or Y2M14B (neither La nor Y are magnetic), ϑ is the angle between the M moment and the c-axis of the tetragonal structure, nRM is the molecular field coefficient between the R and M moments (discussed below), Happ is the applied field,
5
m (τ ) = [1 − sτ 2 − ( 1 − s ) τ 2 ]1/3
(2)
where m (τ )=Ms (T ) /Ms (0) is the normalized magnetization ( Ms (T ) is the spontaneous magnetization at temperature T and Ms (0) is the magnetization at absolute saturation). Expression (2) reproduces closely the real temperature dependence of the compounds, and in particular describes the low temperature dependence of the magnetization governed by the 3/2 spin-wave coefficient as well as the critical exponent 1/3 close to the Curie temperature. Expression (2) was used in the present study, with s = 0.6 in all cases [14]. The only free parameter in Expression (2) is the Curie temperature, which was taken from [15]. The state of the R ion is described with the Hamiltonian representing the combined action of the exchange interactions and CEF interactions on the rare-earth J multiplet:
/R =
∑ n, m
Bnm Onm
− nRM ⟨MM ⟩T MR
An0 (Ka0−n)
(3)
in which Onm represent the Stevens coefficients and Bnm are CEF parameters. The parameter nRM represents indirect coupling between the 3d and the 4f electrons. It should be written more rigorously as nRM = n3d − 5d ·n5d − 4f , where the first term represents the coupling between the M 3d electrons and the R 5d ones and the second term represents the on-site R coupling between the 5d and 4f electrons. The coefficient n5d − 4f depends on the nature of the R element considered: (i) the coupling is not between the R moments but between the spins, thus leading to the so-called de Gennes law, (ii) the distance between the 4f and 5d shells varies strongly across the R series and thus also the associated exchange interactions between the electrons involved [15]. The Onm Stevens operators in expression (3) are function of J and Jz, where J is the total angular momentum of the considered R ion and Jz the projection of J along the quantization axis, which corresponds in the present study to the c-axis of the tetragonal structure. The coefficient n (¼2, 4, or 6) is related to the order of the anisotropy term considered and the integer parameter m obeys the rules m ≤ n and m ≤ 4 in tetragonal symmetry.
From expression (3), the energy splitting of the rare-earth J multiplet may be obtained, with each state corresponding to a given Jz value. The population of each state is then simply given by Boltzmann statistics. The thermal average of the R magnetic moments,
3. From CEF parameters to anisotropy constants and anisotropy coefficients For practical purposes, the anisotropy energy of hard magnetic materials is often expressed as:
EA = K1 sin2 ϑ + K2 sin4 ϑ + K3 sin6 ϑ
(4)
The Ki’s are the anisotropy constants of different orders and ϑ is the angle between the magnetization direction and the c-axis of the tetragonal structure. Each anisotropy constant in Eq. (4) may be the sum of a contribution due to the M element and another contribution due to the R element. Only axial terms in the development of the anisotropy energy are considered here. Non-axial terms that represent in-plane anisotropy are smaller than the axial terms. Besides this, the easy magnetization axis of hard materials involved in magnet applications must be aligned along the c-axis, thus in-plane anisotropy effects are not relevant. The functions involved in the expansion of the anisotropy energy in terms of the powers of sin ϑ are not orthogonal. It results that the strength of the constants derived in the analysis depends on the value of the order n, at which the expansion is limited. An expansion in terms of the Legendre polynomials, Pn0 , that form a set of orthogonal functions, is more appropriate. The anisotropy energy is written:
Ea = k20 P20 (cos ϑ) + k 40 P40 (cos ϑ) + k60 P60 (cos ϑ)
(5)
where the k n0 are the so-called anisotropy coefficients. The anisotropy constants, Ki (T ) and coefficients are related through [16]
M. Ito et al. / Journal of Magnetism and Magnetic Materials 400 (2016) 379–383
K1= −
1 (3k20+10k 40+21k60 ) 2
1 K2= (35k 40+189k60 ) 8 K3= − (
231 0 k6 ) 16
381
(6a)
(6b)
(6c)
Expression (6) is valid for both the M and R anisotropy constants. However, the contribution of the M elements to anisotropy is reasonably well described within a development to first order in Ki (or 2nd order in k i0 ). Thus, the distinction between anisotropy constants and anisotropy coefficients become irrelevant. Concerning the R contribution to anisotropy, the anisotropy coefficients may be related to the thermal average values of the Stevens operators [16]
k n0 (T ) = Bn0
(7)
Expression (7) illustrates the fact that the anisotropy is not directly linked to temperature but indirectly via the temperature dependence of the R magnetic moment [17]. This suggests a procedure to derive the temperature dependence of anisotropy. At each temperature, the Boltzmann population of the split states of the J multiplet can be calculated, and the thermal average of the R magnetic moment and Stevens operators of various orders are directly deduced. Assuming that Bn0 is temperature independent, which is certainly valid to first order, this gives access to k n0 (T ) and through relations (6) to the temperature dependent anisotropy constants that can be compared to experimental values. The present approach does not offer the inherent elegance of analytical expressions, that have been proposed to describe the temperature dependence of anisotropy [11,17] but it has the advantage that it involves simple numerical calculations.
4. Temperature dependence of the anisotropy in the R2M14B compounds Using values given in Table 1, the calculated temperature dependences of the spontaneous magnetization in all compounds are compared in Fig. 1 to the experimental temperature dependences [10]. The calculated temperature dependences of the anisotropy coefficients of order 2, 4 and 6 are shown in Fig. 2 for Nd2Fe14B. n (n + 1)
They follow approximately the power law k n0 (T )∝m (T ) 2 as predicted by Akulov [18]. From these coefficients, the anisotropy constants Ki (T ) in Nd2Fe14B were derived using expression (6). They are compared in Fig. 3a to experimental constants taken from
Fig. 2. Continuous line: temperature dependence of the anisotropy coefficients, kn (T ) , of order n¼ 2, 4 and 6 in Nd2Fe14B, derived from the CEF parameters known for this compound. Dots: temperature dependence of mn (n + 1) /2 , where m is the reduced magnetization. Each curve is normalized to the zero temperature value of the anisotropy coefficient to which it should be compared.
[4,10]. The calculated temperature dependences reproduce qualitatively experimental values but not quantitatively, in particular at low temperatures. The Fe contribution to the 2nd order anisotropy (inset to Fig. 3b) [10] was not included in the calculated K1 (T ). It represents a small but sizeable contribution to anisotropy at room temperature but it is negligible at low temperature and in no case can explain the disagreement between experiment and calculation. However, the calculated temperature dependence of the total anisotropy energy ET (T )=K1 (T )+K2 (T )+K3 (T ) is in excellent agreement with experimental anisotropy (Fig. 3b). This suggests that part of the disagreement comes from the difficulty in separating experimental anisotropy constants of various orders from each other. The field dependence of the magnetization M (Happ ) under Happ perpendicular to the easy axis, was calculated in a classical approach, in which the total magnetic energy was written as:
ET = KFe sin2 ϑ Fe + K1 sin2 ϑ Nd − nRFe ⟨MNd ⟩T ⟨MFe ⟩T cos ( ϑ Fe − ϑ Nd ) −μ 0 Happ ⟨MNd ⟩T sin ( ϑ Nd ) − μ 0 Happ ⟨MFe ⟩T sin ( ϑ Fe)
At each field value, the energy minimum was determined, considering the angles ϑ Fe and ϑNd as independent variables. Remarkably, the calculated magnetization curve M (Happ ) closely follow the experimental curve (Fig. 4 at 300 K) [19]. During reversal the angles ϑ Fe and ϑNd may differ by at most 4° (Fig. 4 inset). The calculated magnetization curve obtained when forcing collinearity ( ϑ Fe ¼ ϑNd ) is shown also in Fig. 4 for comparison. The anisotropy coefficients and constants for Pr2Fe14B and Dy2Fe14B were derived using the exact same An0 values as in Nd2Fe14B. The calculated temperature dependence K1 (T ) are shown in Fig. 5. No experimental temperature dependence exists for either of these compounds in the entire temperature range studied. The calculated K1 (300) in Pr2Fe14B is to be compared to the experimental value K1 = 5. 5 MJ/m3 at the same temperature, derived from μ0 HA = 8. 7 T assuming collinearity between the moments and that only second order anisotropy is present [10]. The anisotropy constant K1 (T ) derived from the slope, α, of the field dependence of magnetization in Dy2Fe14B using the classical expression α =
Fig. 1. Temperature dependence of the spontaneous magnetization in a series of R2Fe14B compounds. Dots: experimental data, continuous line: calculation.
(8)
μ0 Ms2 2K1
is systematically much smaller than the cal-
culated anisotropy constant. This discrepancy illustrates the fact that in ferrimagnetic compounds where the R and M moments are antiparallel, non-collinearity effects under a field applied perpendicular to the easy direction, drastically modify the magnetization variation, thus invalidating the usual procedure to derive
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Fig. 3. (a) Temperature dependence of the anisotropy constants in Nd2Fe14B. Dots: experimental data. Continuous lines: variations derived from the values of anisotropy coefficients. (b) Temperature dependence of the total anisotropy energy in Nd2Fe14B. Dots: experimental data. Continuous line: derived from calculated anisotropy constants Ki’s. Inset: Experimental temperature dependence of anisotropy in Y2Fe14B.
Fig. 4. Field dependence of magnetization under an applied field perpendicular to the easy c-axis in Nd2Fe14B at 300 K. Blue dots: experiment, red curve: field dependence of the magnetization in a classical model, allowing for non-collinearity between the Nd and Fe moments. Green curve: same as red curve except that collinearity is imposed ( ϑFe ¼ ϑNd ). Inset: scheme illustrating a non-collinear magnetization process. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 6. Field dependence of the magnetization under an applied field perpendicular to the easy c-axis in Dy2Fe14B at 4.2 K. Blue dots: experiment, red curve: field dependence of the magnetization in a classical model, allowing for non-collinearity. Green curve: same as red curve except that collinearity is imposed ( ϑFe ¼ ϑDy ). Inset: scheme illustrating a non-collinear magnetization process. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
agreement is excellent thus confirming the relevance of the analysis here proposed.
5. Conclusions
Fig. 5. Temperature dependence of the anisotropy constants in Pr2Fe14B and Dy2Fe14B compared to Nd2Fe14B. These constants were derived by assuming that their CEF parameters are identical to those of Nd2Fe14B.
Using the known exchange parameters in R2Fe14B and the crystal-field parameters determined previously in Nd2Fe14B, we have derived the temperature dependence of the magnetization and of the anisotropy constants in a series of R2Fe14B compounds. The results are in good agreement with experiment. A classical model then permits calculations of the field dependence of the magnetization, which reproduce experimental data rather reasonably. This approach may now be extended to the calculation of the properties of R–M–B compounds including a partial substitution of another R element for Nd and of Co for Fe. This will contribute to material optimization, often required during magnet fabrication for specified applications.
Acknowledgments anisotropy constants experimentally. The calculated field dependence of the magnetization at 4.2 K in Dy2Fe14B using expression (8) is compared in Fig. 6 to experimental data [5,20]. The
This work was carried out in the framework of the Magnetic Materials for High-Efficient Motors (MagHEM) project.
M. Ito et al. / Journal of Magnetism and Magnetic Materials 400 (2016) 379–383
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