- Email: [email protected]
NdNd exchange interaction in Nd2Fe14B
EISEVIER
Journal of Magnetism
Nd-Nd
and Magnetic
Materials
164 ( 1996) XII-207
exchange interaction in Nd, Fe ,4B
Yan Yu
“,
Jin Han-min, Zhao Tie-song
Abstract A series of intrinsic magnetic properties of Nd,Fe,,B measured by previous authors are fitted by the calculations of a combined molecular-field and crystalline electric field approximation. Three sets of fitted parameters are obtained. In addition to the isotropic Nd-Fe exchange interaction. the NddNd exchange interaction and the anisotropy of the Nd-Fe exchange interaction are taken into account in set 1, the Nd-Nd exchange interaction is taken into account by neglecting the anisotropy of the Nd-Fe interaction in set 2. and both are neglected in set 3. The Nd-Nd exchange interaction is comparable with the Nd-Fe exchange interaction for sets I and 2. The value of &/nNdFc is nearly the same as that obtained for Gd,Fe,,B in our previous work, thus supporting the argument that the exchange interaction occur via the 4ff5d interaction. The results show that both the Nd-Nd exchange interaction and the anisotropy of the Nd-Fe exchange interaction need to be taken into account to obtain a reasonable explanation of the experimental results for Nd?Fe,,B. Ke~uord.c:
Exchange
interaction:
Crystalline
electric
field:
Nd?Fe,,B
1. Introduction As Nd,Fe,,B is the main phase of the outstanding Nd-Fe-B permanent magnets, its intrinsic magnetic properties have been studied most extensively among the R 2Fe, 1B compounds (R = rare-earth) [ II I]. This has also prompted analyzation of the experimental data within a combined molecular field and crystalline electric field (CEF) approximation [4,121.51. The calculations in which the CEF parameters A II,,, are set invariant with temperature do not reproduce the experimental magnetization curves at different temperatures very well [ 12,131. It was pointed out that for satisfactory reproduction of the curve at
Corresponding
0304-X853/96/$15.00
author.
Copyright
PI1 SO304-8853(96)00397-6
0
1996 Elsevier
Science B.V.
275 K, the values of AZ!,) (m = 0, f 2) should be reduced by 15% from those at 4.2 K [4]. In this work. the calculations were made by neglecting the mixing of the excited /-multiplet, while it has been reported that the effect of the mixing is not negligible for the light rare-earth ions [ 12.161. Our recalculations. using the parameters of Ref. [4], show that A 211,should be reduced by 22% if the mixing of the first excited J-multiplet is taken into account. Such a large decrease seems unreasonable. A Miissbauer study of “‘Cd in the Gd,Fe ,.rB compound has shown that the electric field gradients at the Gd ion sites. which are proportional to AzO. vary by f.5% on increasing the temperature from 4.2 to 120 K [17]. In all of these calculations, both the Nd-Nd exchange interaction and the anisotropy of the Nd-Fe exchange interaction have been neglected as is commonly adopted. All rights reserved
Subsequent work, however, has suggested that the R-R exchange interaction would not be negligibly small for R2Fe,,B. For the compound with R = Gd, experiments on the inelastic neutron scattering, the temperature dependence of the magnetic moment of the Gd ion, and the magnetization curves at different temperatures could be fitted satisfactorily only if the Gd-Gd exchange interaction in addition to the GdFe exchange interaction is taken into account [18]. The temperature dependence of paramagnetic susceptibility could be reproduced well by taking into account the R-R exchange interactions in addition to R-Fe and Fe-Fe ones for the compounds with R = Gd, Dy and Er [l&19]. The large K,(O K) = 0.9 K/(Fe atom) magnetocrystalline anisotropy of Y,Fe,,B and the large difference between the values of K,(OK) = 9.6 K/f.u. and 12.8 K/f.u. for Gd,Fe,,B and Y?Fe,,B [ 1,201 suggest that the anisotropy of the R-Fe exchange interaction is also not negligible for R2 Fe ,J B
neutron scattering experiments can not distinguish the difference between the exchange fields acting on the Nd ions at the f and g sites [7], and the calculation of a magnetization curve using the averaged CEF parameters gives rise to the same result as that obtained using different values for the sites [22]. Assuming that the R ions are trivalent, the Hamiltonian of the R ions at the two magnetically inequivalent sites R(I) (2 = 1, 2) are represented as H(l)
= AL .S + 2Pa(%+
+&XX,)
+ C A~~,~z( I) C ‘,,,,,( 11.m / + pB( L + 2s)
‘S
',y+j)
(1)
H,
where
(2)
1211. In this paper we analyze a series of experimental data for Nd,Fe,,B by taking into account the mixing of the first excited J-multiplet. Three sets of parameters are used for fitting the calculations to the experimental data. In addition to the isotropic Nd-Fe exchange interaction, the Nd-Nd exchange interaction and the anisotropy of the Nd-Fe exchange interaction are taken into account in set 1, the Nd-Nd exchange interaction is taken into account by neglecting the anisotropy of the Nd-Fe interaction in set 2, and both are neglected in set 3 as usual. The magnetization curves of single-crystal (Nd,,,Y,,,),Fe,,B at a series of temperatures are analyzed using the fitted parameters of set 1.
2. Method of analysis
H exFe+
HcxNd = nNdFe(l + P sin",,) +
MFe
2%livdMrd s> )
(3)
Y,l,,l(e,,$j> are the spherical harmonics, 0, and $, are the polar and azimuthal angles of the position vector of the jth 4f electron, h = 536 K [23], M,, is the magnetic moment of the Fe sublattice per f.u., M,,(S) = - 2 pa(S) is the mean spin magnetic moment of the Nd ions. tlNdFc and nNdNd are the corresponding molecular field coefficients, p is the anisotropy of the Nd-Fe exchange interaction, and 8,, is the angle between M,, and the c-axis. The following relations hold between the CEF parameters for the R(1) and R(2) sites in the coordinate system with the z- and x-axes along the c- and [loo] axes, respectively: A,,,,,(2) = ( - l)““‘A,,_,,,(2)
= ( - l)““‘A,,,,(
1), (4)
R,Fe,,B has the tetragonal crystal structure belonging to the space group P4,/mnm. The crystal cell consists of four R,Fe,,B formula units (f.u.). Each of the rare-earth ion crystallographic sites f and g has two magnetically inequivalent sites. Averaged over f and g sites exchange fields and CEF parameters were used in the calculations. In fact, inelastic
with n = 2, 4, 6; m = 0, L- 2, f 4, f 6; Irnl 5 n. The matrix elements of Eq. (1) were calculated using the irreducible operator technique [24]. At a given temperature T, by introducing the input parameters into Eq. (1) and diagonalizing the 22 X 22 matrix, the eigenstates Ii(Z)) and eigenvalues E,(l) (i = 1, 2, . . . .
Y. Yu et cd. / Journnl
of Mqnrtism
22; I= 1, 2) were obtained. The free energy (Y,_.Nd.,12Fe,,B per f.u. is given by
nnd Magnrtic Materials 164 (19961 201-207
of
F( $+J) = -xkT
c
In Z(I)
+ K,(T)
sir&r,
- M,,
H
The correct input value and direction of M,,(S) in the Hamiltonian of Eq. (1) should satisfy self-consistency, i.e. it should be equal to that calculated by Eq. (7). The correct direction of MFe, which is an input parameter for a given applied field, is obtained from minimization of the free energy as a function of its direction under the condition of selfconsistency of MNd(S). The magnetic moment of (Y,-.,Nd,),Fe,,B (X = 0.9) is calculated as M(T)
x
exd -N WT) Z(U
(5)
1’
Z(l) = C ev( -E;( I)/@),
(6)
K,(T) is the magnetocrystalline anisotropy constant of the Fe sublattice per f.u. The values of M,,(T)/M,,(O) and K,(T)/K,(O) were taken as those for Y2 Fe,,B normalized to the Curie temperature Tc = 586 K of NdzFe,,B, with M,,(O) = 31.0 pa/f.u. and K,(O) = 12.8 K/f.u., respectively [I]. Here, M,,(O) = 3 1.0 pn/f.u. was obtained by subtracting the calculated magnetic moment of the Nd’ ’ ion, 6.2 Fa/f.u., from the average of M,(O K) = 37.1 pa/f.u. [lo] and 37.3 pn/f.u. [5], and includes the negative polarization of the conduction band. The last term on the right side of Eq. (5), which represents the negative value of the Nd-Nd exchange interaction, is added because the Nd-Nd exchange interaction is accounted for twice in the first term on the right side of the equation. M,,(S) and the mean magnetic moment of the Nd ions are calculated as
203
=2xM,,
+M,,(T).
(9)
Since x is small, the fitted parameters for Nd,Fe,,B were used for the calculation. The values of the fitted parameters were obtained from best fits of the calculations to a series of experimental data; these include the magnetization curves along the [loo], [1 lo] and [OOl] axes at 4.2, 100, 150 and 275 K. The spin reorientation temperature (SRT), 8(T), MNd(T), and the temperature dependence of energy levels of the first and second excited eigenstates in reference to the ground level E,-E, and E,-E,. A,, and A,, were neglected. The ratio I A?>I/AZ0 was fixed to 0.53, which coincides with the value of 0.52 obtained from a ‘“%d Miissbauer study of Gd?Fe,,B [25]. A,, and A,, were allowed to vary with temperature by maintaining the ratio. The higher-order CEF parameters were treated as invariant since the CEF interaction terms are less important and become increasingly insignificant with increasing temperature. The hyperfine fields of the Nd ions at 0 K were calculated using the fitted parameters, as follows [26]: H,,( 1) = H,,(free) N,( /) + H,,( transf) ,
(10)
in which MNd(S)
= -;c
pa c I
x
(i(1)12S1i(l)) i
N,.(9=
w-E,(/)/@) Z(l)
1’
(7) N=C
x
exp(-WWT) Z(‘)
(8) 1
j
O( qNI’( I)) I(J,J,NpJ)I
(I= ‘,2),
(1’)
zj-S,+3(s,TJ)r,
If
(‘2)
H,,(free) = 4280 kOe is the hyperfine field for the free Nd ion obtained from experiment 1271, Zj and s, are the angular moments of the jth 4f electron, and J = 9/2 is the total angular moment for the ground
multiplet. The following calculation:
=
-s,,
J3 [ (cjy +
relations
were used in the
))i’)- (c;?yil))y),] .
sublattice magnetic moment by the Gd spins. Taking into account the polarization [30]. the transferred hyperfine field for the Nd ion is estimated to be about 470 kOe.
(13)
3. Results and discussion
3(y,)?‘, Three sets of the fitted parameters are listed in Table I. The values of HcsFe and A,,,,, for set 3 are quite different from those of sets I and 2. while those for sets 1 and 2 are close. For set I, p = 0.25% is much smaller than the I .2S-2.5% for RCo, [2 1.3 I]. This result is expected from the following experimental facts: At 4.2 K. the anisotropy of Y, Fe,,B, 0.9 K/(Fe atom) [1.20]. is much smaller than 8.8 K/(Fe atom) of YCo, [3l]; The difference of the anisotropy between Y?Fe,,B and Gd,Fe,,B, 3 K/f.u. [ 11, is much smaller that between YCo, and GdCo,. 21 K/f.u. [3 I]. Fig. I shows the temperature dependence of the fitted A ?,, The solid, dashed and dotted curves are for sets I. 2 and 3. respectively. With an increase in temperature from 0 to 275 K. the decreases in A,,, are I I%. 147~ and 218 for sets I. 2 and 3, respectively. The decrease in A,,, for set 3 is unreasonably large. Figs. 2-6 compare the results of the calculations (curves) and experiments (symbols). Essentially the same results are obtained using the parameters of different sets, and in general, the agreement with the experiments is satisfactory. Fig. 2 shows the magnetization curves along the [ 1001, [I IO] and [OOl] axes at 4.2 K. The solid and dashed curves were calculated by taking into account and neglecting the mixing of the first excited J-multiplet, respectively. They are obviously different, thus demonstrating the non-negligible mixing of the excited multiplet. The solid curves reproduce well the experiments. includ-
r,’
Here (C/‘)sj’)):‘) is the q-component of the irreducible operator of rank 1 constructed by multiplying the irreducible operators of rank 2 Ci” and rank 1 .rl”. The transferred hyperfine field H,,(transf) is from the Fe spins. The contribution from other rareearth ions can be neglected [28]. The direction of H,,(transf) is opposite to that of M,, [25]. The value of H,,(transf) is estimated as follows. Known values of H,,(transf) are 640 and 230 kOe for the Lu and Y ions in Lu,Fe,,B and Y?Fe,,B, respectively [8]. The ratio of these transferred hyperfine fields, 640/240 = 2.78, is consistent with that of the hyperfine coupling parameters of the ions, 4.8/1.7 = 2.8 [29], and the transferred hyperfine field for the La ion with the hyperfine coupling parameter 3.1 is estimated to be (3.1,’ 1.7) X 230 = 420 kOe. By linearly extrapolating the results, they are 465 and 525 kOe for the Nd and Gd ions in their compounds. The experimental value for the Gd ion averaged over the f and g sites in GdzFe,,B is 28 kOe larger than the 525 kOe [8] that might result from the polarization of the Fe
I Fitted values of p. 2 kLuH,,,,(O K). 2 pR ffcxvd(O K) and CEF parameters T=OK Table
A,,,,, for Nd,Fe,,B
(all in K). The values of AZ,) and A,,
are for
Set
2Pu f&,(O)
2/.+ H&(O)
p
A 12
A??/i
A JO
A J-l
A 60
Ad
A,,
I
460 460 630
315 315 0
0.0025 0 0
552 545 600
294 290 320
~ 175 - 175 - 260
-60 -55
-700 - 70s - 550
60 60 0
-47s -475 - 400
2 3
0
30
Fig. I. Temperature dependence Nd,Fe,,B. - set I; --- set 2;
of the fitted parameter
A2(, for
E iOOl1
I1001 Ill01
20
set3.
10
ing the first-order magnetization process along the [loo] axis. Fig. 3 shows the magnetization curves along the [ IOO], [ 1 lo] and [OOI] axes at 100, 150 and 275 K. The agreement between the calculations and experiments is very good. Fig. 4 shows the temperature dependence of the cone angle. Fig. 5 presents the temperature dependence of the first and second excited energy levels of the Nd ions averaged over the different sites in reference to the ground energy level, E,-E, and E,-E,. The quite different temperature dependences of the two levels are reproduced fairly well.
10
I.-
0
I 100
I 200 H (kOe)
OO
50
100 H ikOe)
Fig. 3. Magnetization curves of Nd,Fe,,B experimental data from Ref. [a]; -
275K 150
200
at 100. 150 and 275 K. computed.
Fig. 6 shows the temperature dependence of the mean magnetic moment of the Nd ions. The experimental data were obtained by subtracting the M,,(T) [2] from the product of M,(OK) = 37.3 pu/f.u. [5] and M,(T)/M,(O) [2]. The kink at the SRT is simulated well. For reference. the calculations by using the parameters of Cadogan et al. [4] are also presented in Fig. 5 and 6. The dashed and dotted curves
4. 2K 300
Fig. 2. Magnetization curves of Nd?Fe,,B at 1.2 K. mental data from Ref. [5]: - computed: - computed ing the mixing of the first excited J-multiplet.
400 0 experiby neglect-
50
100
150
2
TiK)
Fig. 3. Temperature dependence of the cone angle for Nd,Fe,,B. experimental data from Ref. [4]; - computed.
206
Y. Yu et al. /Journul
of Mqnetisrrt
and Magnetic Materials 164 (I9961 201L207
4
] 2 2 L: -S
4. 2K
;4
Fig. 5. Temperature dependence of the first and second excited energy levels E? - E, and E3 - E, for the Nd ions. I: experimental data from Ref. [7]; - computed; - - computed using the parameters of Cadogan et al. [4] by taking into account the mixing of the first excited J-multiplet; computed using the parameters of Cadogan et al. [4] by neglecting the mixing of the first excited J-multiplet.
2
200
100
3
H (hoe)
represent the calculations by taking into account and by neglecting the first excited J-multiple& respectively. The neglect results in an evident departure of the calculation from experiment above 125 K. The hyperfine field is calculated to be 3790 kOe, which is somewhat larger than the experimental values of 3530-3540 kOe [6,8,9]. Fig. 7 shows the magnetization curves along the [loo], [llO] and [OOl] axes at 4.2 and 290 K, and
3.a
1
...*....
Fig. 7. Magnetization K. experimental
Fig. 8 those along the [ 1001 axis at 100, 130 and 160 K for (Nd,,,Y,,),Fe,,B. The solid curves represent the calculations using the fitted parameters of set 1, and the symbols the experiments. The agreement between the calculations and experiments is excellent.
I
. .
curves of (Ya ,Nd,,),Fe,,B at 4.2 and 290 data from Ref. [32]; - computed.
40
2. 42. 2 0
I 50
I 100 T IK)
I 150
200
Fig. 6. Temperature dependence of the mean magnetic moment M,,(T) for Nd?Fe,,B. W experimental data from Ref. [2]; computed; --- computed using the parameters of Cadogan et al. [4] by taking into account the mixing of the first excited J-multiplet; computed using the parameters of Cadogan et al. [4] by neglecting the mixing of the first excited J-multiplet.
Ol Fig. 8. Magnetization
H (kOe1 curves
of (Y,, ,Nd, g)zFe,,B experimental
[ IOO] axis at 100, 130 and 160 K. Ref. [32]; -
computed.
along the data from
Y. YLIet al. / Jounml
of Magnrtism and Mqnetic’ Matrrials 164 (19961 201-207
4. Conclusions This work has shown that it is necessary to take into account the Nd-Nd exchange interaction and the anisotropy of the Nd-Fe exchange interaction in addition to the isotropic Fe-Nd exchange interaction to obtain a reasonably satisfactory explanation of the magnetic properties of Nd,Fe,,B. The size of the Nd-Nd exchange interaction is comparable with that of the Nd-Fe exchange interaction in Nd?Fe,,B as for the R-R exchange interaction in the compounds of Gd, Dy and Er [I 8,191. The value of = 0.038 is nearly the same as the l”‘n,,,d/%,,, 0.040 obtained for Gd,Fe,,B in our previous publication, thus supporting the idea that the exchange interactions are mediated via 5d conduction electrons [33], as originally suggested by Campbell [34].
Acknowledgements This work was supported by the National Science Foundation of China.
Natural
References 111S. Hirosawa, Y. Matsuura. H. Yamamoto,
S. Fujimura and M. Sagawa, J. Appl. Phys. 59 (1986) 873. L21K. Tokuhara, Y. Ohtsu, F. Ono, 0. Yamada, M. Sagawa and Y. Matsuura, Solid State Commun. 56 (1985) 333. [31 F. Bolzoni, 0. Moze and L. Pareti, J. Appl. Phys. 62 (1987) 615. 141 J.M. Cadogan, J.P. Gavigan, D. Givord and H.S. Li, J. Phys. F: Metal Phys. 18 (1988) 779. 151 R. Verhoef, J.J.M. Frame, A.A. Menovsky, R.J. Radwanski. Ji Song-quan. Yang Fuming. H.S. Li and J.P. Gavigan, J. Physique 49 (1988) C8-565. 161I. Nowik, K. Muraleedharan, G. Wortmann, B. Perscheid, G. Kaindl and N.C. Koon, Solid State Commun. 76 (1990) 967. [71 M. Loewenhaupt, M. Prager, A.P. Maruni and H.E. Hoenig, J. Magn. Magn. Mater. 76-77 (1988) 408. iSI Y. Berthier, M. Boge, G. Czjzek. D. Givord, C. Jeandey, H.S. Li and J.L. Oddou, J. Magn. Magn. Mater. 54-57 (1986) 589. [91 E. Jedryka, M. Wojcik and P. Panissod, J. Magn. Magn. Mater. 104-107 (1992) 1405.
201
[lo] D. Givord, H.S. Li and R. Perrier de la Bathie, Solid State Commun. 88 (1993) 907. [I I] G. Marusi, N.V. Mushnikov, L. Pareti, M. Solai and A.E. Ermakov, J. Phys.: Condens. Matter 2 (1990) 7317. 121 M. Yamada, H. Kato, H. Yamamoto and Y. Nakagawa, Phys. Rev. B 38 (1988) 620. 131 Y. Nakagawa, H. Kato, D.W. Lim, G. Kido and M. Yamada, in: Proc. 6th Int. Symp. on Magnetic Anisotropy and Coercivity in Rare-Earth Transition Metal Alloys, Carnegie-Mellon University, Pittsburgh, PA, 1990, p. 12. 141 R.J. Radwanski and J.J.M. Franse, J. Magn. Magn. Mater. 80 (1989) 14. [15] H.S. Li and J.M. Cadogan, Solid State Commun. 81 (1992) 283. [16] Yan Yu, Zhao Tie-song and Jin Han-min, J. Phys.: Condens. Matter 3 (1991) 195. [17] G. Czjzek and H.J. Bornemann, J. Magn. Magn. Mater. 80 (1989) 23. [18] Yan Yu, Zhao Xiao-hong and Jin Han-min, J. Magn. Magn. Mater. 127 (1993) Ll. [19] E. Burzo, J. Appl. Phys. 57 (1985) 3232. [20] D. Givord, H.S. Li and R. Perrier de la Bathie, Sol. State Commun. 51 (1984) 857; M. Borg, J.M.D. Coey, G. Czjzek, D. Givord, C. Jeandey, H.S. Li and J.L. Oddou, Solid State Commun. 55 (1985) 295. [21] Jin Han-min and Yan Yu, Phys. Rev. B 48 (1993) 1022. [22] Han Xiu-feng, Jin Han-mitt, Wang Zi-jun and T.S. Zhao, Phys. Rev. B 47 (1993) 3253. [23] S. Hufner. Optical Spectra of Transparent Rare-earth Compounds (Academic Press, London, 1978) p. 34. [24] M.E. Rose, Elementary Theory of Angular Momentum (Interscience, New York. 1957). [25] M. Boge. G. Czjzek, D. Givord, C. Jeandey, H.S. Li and J.L. Oddou, J. Phys. F: Metal Phys. 16 (1986) L67. [26] Jin Han-mitt, K. Shimizu, Han Xiu-feng, Yan Yu and Zhao Tiesong, J. Phys.: Condens. Matter 4 (1992) 8609. [27] M.A.H. McCausland and IS. Mackenzie, Adv. Phys. 28 (1979) 305. [28] K. Shim& in: Proc. 2nd Int. Symp. on Physics of Magnetic Materials, Beijing, 1992, p. 707. [29] LA. Campbell, J. Phys. C: Solid State Phys. 2 (1969) 1338. [30] R. Fruchart, P. L’Heritier, P. Dalmas de Reotier, D. Fruchart, P. Wolfers, J.M.D. Coey, L.P. Ferreira, R. Guillen, P. Vulliet and A. Yaouanc. J. Phys. F: Metal Phys. 17 (1987) 483. [31] D. Ballou, J. Deportes, B. George, R. Lemaires and J.C. Ousset, J. Magn. Magn. Mater. 54-57 (1986) 465. [32] D.W. Lim, H. Kato, M. Yamada, G. Kido, Y. Nakagawa and S. Hirosawa, J. Magn. Magn. Mater. 90-91 (1990) 72. [33] E. Belorisky, J.P. Gavigan, D. Givord and H.S. Li. Europhys. Lett. 5 (1988) 349. [34] LA. Campbell, J. Phys. F: Metal Phys. 2 (1972) L47.
Journal of Magnetism
Nd-Nd
and Magnetic
Materials
164 ( 1996) XII-207
exchange interaction in Nd, Fe ,4B
Yan Yu
“,
Jin Han-min, Zhao Tie-song
Abstract A series of intrinsic magnetic properties of Nd,Fe,,B measured by previous authors are fitted by the calculations of a combined molecular-field and crystalline electric field approximation. Three sets of fitted parameters are obtained. In addition to the isotropic Nd-Fe exchange interaction. the NddNd exchange interaction and the anisotropy of the Nd-Fe exchange interaction are taken into account in set 1, the Nd-Nd exchange interaction is taken into account by neglecting the anisotropy of the Nd-Fe interaction in set 2. and both are neglected in set 3. The Nd-Nd exchange interaction is comparable with the Nd-Fe exchange interaction for sets I and 2. The value of &/nNdFc is nearly the same as that obtained for Gd,Fe,,B in our previous work, thus supporting the argument that the exchange interaction occur via the 4ff5d interaction. The results show that both the Nd-Nd exchange interaction and the anisotropy of the Nd-Fe exchange interaction need to be taken into account to obtain a reasonable explanation of the experimental results for Nd?Fe,,B. Ke~uord.c:
Exchange
interaction:
Crystalline
electric
field:
Nd?Fe,,B
1. Introduction As Nd,Fe,,B is the main phase of the outstanding Nd-Fe-B permanent magnets, its intrinsic magnetic properties have been studied most extensively among the R 2Fe, 1B compounds (R = rare-earth) [ II I]. This has also prompted analyzation of the experimental data within a combined molecular field and crystalline electric field (CEF) approximation [4,121.51. The calculations in which the CEF parameters A II,,, are set invariant with temperature do not reproduce the experimental magnetization curves at different temperatures very well [ 12,131. It was pointed out that for satisfactory reproduction of the curve at
Corresponding
0304-X853/96/$15.00
author.
Copyright
PI1 SO304-8853(96)00397-6
0
1996 Elsevier
Science B.V.
275 K, the values of AZ!,) (m = 0, f 2) should be reduced by 15% from those at 4.2 K [4]. In this work. the calculations were made by neglecting the mixing of the excited /-multiplet, while it has been reported that the effect of the mixing is not negligible for the light rare-earth ions [ 12.161. Our recalculations. using the parameters of Ref. [4], show that A 211,should be reduced by 22% if the mixing of the first excited J-multiplet is taken into account. Such a large decrease seems unreasonable. A Miissbauer study of “‘Cd in the Gd,Fe ,.rB compound has shown that the electric field gradients at the Gd ion sites. which are proportional to AzO. vary by f.5% on increasing the temperature from 4.2 to 120 K [17]. In all of these calculations, both the Nd-Nd exchange interaction and the anisotropy of the Nd-Fe exchange interaction have been neglected as is commonly adopted. All rights reserved
Subsequent work, however, has suggested that the R-R exchange interaction would not be negligibly small for R2Fe,,B. For the compound with R = Gd, experiments on the inelastic neutron scattering, the temperature dependence of the magnetic moment of the Gd ion, and the magnetization curves at different temperatures could be fitted satisfactorily only if the Gd-Gd exchange interaction in addition to the GdFe exchange interaction is taken into account [18]. The temperature dependence of paramagnetic susceptibility could be reproduced well by taking into account the R-R exchange interactions in addition to R-Fe and Fe-Fe ones for the compounds with R = Gd, Dy and Er [l&19]. The large K,(O K) = 0.9 K/(Fe atom) magnetocrystalline anisotropy of Y,Fe,,B and the large difference between the values of K,(OK) = 9.6 K/f.u. and 12.8 K/f.u. for Gd,Fe,,B and Y?Fe,,B [ 1,201 suggest that the anisotropy of the R-Fe exchange interaction is also not negligible for R2 Fe ,J B
neutron scattering experiments can not distinguish the difference between the exchange fields acting on the Nd ions at the f and g sites [7], and the calculation of a magnetization curve using the averaged CEF parameters gives rise to the same result as that obtained using different values for the sites [22]. Assuming that the R ions are trivalent, the Hamiltonian of the R ions at the two magnetically inequivalent sites R(I) (2 = 1, 2) are represented as H(l)
= AL .S + 2Pa(%+
+&XX,)
+ C A~~,~z( I) C ‘,,,,,( 11.m / + pB( L + 2s)
‘S
',y+j)
(1)
H,
where
(2)
1211. In this paper we analyze a series of experimental data for Nd,Fe,,B by taking into account the mixing of the first excited J-multiplet. Three sets of parameters are used for fitting the calculations to the experimental data. In addition to the isotropic Nd-Fe exchange interaction, the Nd-Nd exchange interaction and the anisotropy of the Nd-Fe exchange interaction are taken into account in set 1, the Nd-Nd exchange interaction is taken into account by neglecting the anisotropy of the Nd-Fe interaction in set 2, and both are neglected in set 3 as usual. The magnetization curves of single-crystal (Nd,,,Y,,,),Fe,,B at a series of temperatures are analyzed using the fitted parameters of set 1.
2. Method of analysis
H exFe+
HcxNd = nNdFe(l + P sin",,) +
MFe
2%livdMrd s> )
(3)
Y,l,,l(e,,$j> are the spherical harmonics, 0, and $, are the polar and azimuthal angles of the position vector of the jth 4f electron, h = 536 K [23], M,, is the magnetic moment of the Fe sublattice per f.u., M,,(S) = - 2 pa(S) is the mean spin magnetic moment of the Nd ions. tlNdFc and nNdNd are the corresponding molecular field coefficients, p is the anisotropy of the Nd-Fe exchange interaction, and 8,, is the angle between M,, and the c-axis. The following relations hold between the CEF parameters for the R(1) and R(2) sites in the coordinate system with the z- and x-axes along the c- and [loo] axes, respectively: A,,,,,(2) = ( - l)““‘A,,_,,,(2)
= ( - l)““‘A,,,,(
1), (4)
R,Fe,,B has the tetragonal crystal structure belonging to the space group P4,/mnm. The crystal cell consists of four R,Fe,,B formula units (f.u.). Each of the rare-earth ion crystallographic sites f and g has two magnetically inequivalent sites. Averaged over f and g sites exchange fields and CEF parameters were used in the calculations. In fact, inelastic
with n = 2, 4, 6; m = 0, L- 2, f 4, f 6; Irnl 5 n. The matrix elements of Eq. (1) were calculated using the irreducible operator technique [24]. At a given temperature T, by introducing the input parameters into Eq. (1) and diagonalizing the 22 X 22 matrix, the eigenstates Ii(Z)) and eigenvalues E,(l) (i = 1, 2, . . . .
Y. Yu et cd. / Journnl
of Mqnrtism
22; I= 1, 2) were obtained. The free energy (Y,_.Nd.,12Fe,,B per f.u. is given by
nnd Magnrtic Materials 164 (19961 201-207
of
F( $+J) = -xkT
c
In Z(I)
+ K,(T)
sir&r,
- M,,
H
The correct input value and direction of M,,(S) in the Hamiltonian of Eq. (1) should satisfy self-consistency, i.e. it should be equal to that calculated by Eq. (7). The correct direction of MFe, which is an input parameter for a given applied field, is obtained from minimization of the free energy as a function of its direction under the condition of selfconsistency of MNd(S). The magnetic moment of (Y,-.,Nd,),Fe,,B (X = 0.9) is calculated as M(T)
x
exd -N WT) Z(U
(5)
1’
Z(l) = C ev( -E;( I)/@),
(6)
K,(T) is the magnetocrystalline anisotropy constant of the Fe sublattice per f.u. The values of M,,(T)/M,,(O) and K,(T)/K,(O) were taken as those for Y2 Fe,,B normalized to the Curie temperature Tc = 586 K of NdzFe,,B, with M,,(O) = 31.0 pa/f.u. and K,(O) = 12.8 K/f.u., respectively [I]. Here, M,,(O) = 3 1.0 pn/f.u. was obtained by subtracting the calculated magnetic moment of the Nd’ ’ ion, 6.2 Fa/f.u., from the average of M,(O K) = 37.1 pa/f.u. [lo] and 37.3 pn/f.u. [5], and includes the negative polarization of the conduction band. The last term on the right side of Eq. (5), which represents the negative value of the Nd-Nd exchange interaction, is added because the Nd-Nd exchange interaction is accounted for twice in the first term on the right side of the equation. M,,(S) and the mean magnetic moment of the Nd ions are calculated as
203
=2xM,,
+M,,(T).
(9)
Since x is small, the fitted parameters for Nd,Fe,,B were used for the calculation. The values of the fitted parameters were obtained from best fits of the calculations to a series of experimental data; these include the magnetization curves along the [loo], [1 lo] and [OOl] axes at 4.2, 100, 150 and 275 K. The spin reorientation temperature (SRT), 8(T), MNd(T), and the temperature dependence of energy levels of the first and second excited eigenstates in reference to the ground level E,-E, and E,-E,. A,, and A,, were neglected. The ratio I A?>I/AZ0 was fixed to 0.53, which coincides with the value of 0.52 obtained from a ‘“%d Miissbauer study of Gd?Fe,,B [25]. A,, and A,, were allowed to vary with temperature by maintaining the ratio. The higher-order CEF parameters were treated as invariant since the CEF interaction terms are less important and become increasingly insignificant with increasing temperature. The hyperfine fields of the Nd ions at 0 K were calculated using the fitted parameters, as follows [26]: H,,( 1) = H,,(free) N,( /) + H,,( transf) ,
(10)
in which MNd(S)
= -;c
pa c I
x
(i(1)12S1i(l)) i
N,.(9=
w-E,(/)/@) Z(l)
1’
(7) N=C
x
exp(-WWT) Z(‘)
(8) 1
j
O( qNI’( I)) I(J,J,NpJ)I
(I= ‘,2),
(1’)
zj-S,+3(s,TJ)r,
If
(‘2)
H,,(free) = 4280 kOe is the hyperfine field for the free Nd ion obtained from experiment 1271, Zj and s, are the angular moments of the jth 4f electron, and J = 9/2 is the total angular moment for the ground
multiplet. The following calculation:
=
-s,,
J3 [ (cjy +
relations
were used in the
))i’)- (c;?yil))y),] .
sublattice magnetic moment by the Gd spins. Taking into account the polarization [30]. the transferred hyperfine field for the Nd ion is estimated to be about 470 kOe.
(13)
3. Results and discussion
3(y,)?‘, Three sets of the fitted parameters are listed in Table I. The values of HcsFe and A,,,,, for set 3 are quite different from those of sets I and 2. while those for sets 1 and 2 are close. For set I, p = 0.25% is much smaller than the I .2S-2.5% for RCo, [2 1.3 I]. This result is expected from the following experimental facts: At 4.2 K. the anisotropy of Y, Fe,,B, 0.9 K/(Fe atom) [1.20]. is much smaller than 8.8 K/(Fe atom) of YCo, [3l]; The difference of the anisotropy between Y?Fe,,B and Gd,Fe,,B, 3 K/f.u. [ 11, is much smaller that between YCo, and GdCo,. 21 K/f.u. [3 I]. Fig. I shows the temperature dependence of the fitted A ?,, The solid, dashed and dotted curves are for sets I. 2 and 3. respectively. With an increase in temperature from 0 to 275 K. the decreases in A,,, are I I%. 147~ and 218 for sets I. 2 and 3, respectively. The decrease in A,,, for set 3 is unreasonably large. Figs. 2-6 compare the results of the calculations (curves) and experiments (symbols). Essentially the same results are obtained using the parameters of different sets, and in general, the agreement with the experiments is satisfactory. Fig. 2 shows the magnetization curves along the [ 1001, [I IO] and [OOl] axes at 4.2 K. The solid and dashed curves were calculated by taking into account and neglecting the mixing of the first excited J-multiplet, respectively. They are obviously different, thus demonstrating the non-negligible mixing of the excited multiplet. The solid curves reproduce well the experiments. includ-
r,’
Here (C/‘)sj’)):‘) is the q-component of the irreducible operator of rank 1 constructed by multiplying the irreducible operators of rank 2 Ci” and rank 1 .rl”. The transferred hyperfine field H,,(transf) is from the Fe spins. The contribution from other rareearth ions can be neglected [28]. The direction of H,,(transf) is opposite to that of M,, [25]. The value of H,,(transf) is estimated as follows. Known values of H,,(transf) are 640 and 230 kOe for the Lu and Y ions in Lu,Fe,,B and Y?Fe,,B, respectively [8]. The ratio of these transferred hyperfine fields, 640/240 = 2.78, is consistent with that of the hyperfine coupling parameters of the ions, 4.8/1.7 = 2.8 [29], and the transferred hyperfine field for the La ion with the hyperfine coupling parameter 3.1 is estimated to be (3.1,’ 1.7) X 230 = 420 kOe. By linearly extrapolating the results, they are 465 and 525 kOe for the Nd and Gd ions in their compounds. The experimental value for the Gd ion averaged over the f and g sites in GdzFe,,B is 28 kOe larger than the 525 kOe [8] that might result from the polarization of the Fe
I Fitted values of p. 2 kLuH,,,,(O K). 2 pR ffcxvd(O K) and CEF parameters T=OK Table
A,,,,, for Nd,Fe,,B
(all in K). The values of AZ,) and A,,
are for
Set
2Pu f&,(O)
2/.+ H&(O)
p
A 12
A??/i
A JO
A J-l
A 60
Ad
A,,
I
460 460 630
315 315 0
0.0025 0 0
552 545 600
294 290 320
~ 175 - 175 - 260
-60 -55
-700 - 70s - 550
60 60 0
-47s -475 - 400
2 3
0
30
Fig. I. Temperature dependence Nd,Fe,,B. - set I; --- set 2;
of the fitted parameter
A2(, for
E iOOl1
I1001 Ill01
20
set3.
10
ing the first-order magnetization process along the [loo] axis. Fig. 3 shows the magnetization curves along the [ IOO], [ 1 lo] and [OOI] axes at 100, 150 and 275 K. The agreement between the calculations and experiments is very good. Fig. 4 shows the temperature dependence of the cone angle. Fig. 5 presents the temperature dependence of the first and second excited energy levels of the Nd ions averaged over the different sites in reference to the ground energy level, E,-E, and E,-E,. The quite different temperature dependences of the two levels are reproduced fairly well.
10
I.-
0
I 100
I 200 H (kOe)
OO
50
100 H ikOe)
Fig. 3. Magnetization curves of Nd,Fe,,B experimental data from Ref. [a]; -
275K 150
200
at 100. 150 and 275 K. computed.
Fig. 6 shows the temperature dependence of the mean magnetic moment of the Nd ions. The experimental data were obtained by subtracting the M,,(T) [2] from the product of M,(OK) = 37.3 pu/f.u. [5] and M,(T)/M,(O) [2]. The kink at the SRT is simulated well. For reference. the calculations by using the parameters of Cadogan et al. [4] are also presented in Fig. 5 and 6. The dashed and dotted curves
4. 2K 300
Fig. 2. Magnetization curves of Nd?Fe,,B at 1.2 K. mental data from Ref. [5]: - computed: - computed ing the mixing of the first excited J-multiplet.
400 0 experiby neglect-
50
100
150
2
TiK)
Fig. 3. Temperature dependence of the cone angle for Nd,Fe,,B. experimental data from Ref. [4]; - computed.
206
Y. Yu et al. /Journul
of Mqnetisrrt
and Magnetic Materials 164 (I9961 201L207
4
] 2 2 L: -S
4. 2K
;4
Fig. 5. Temperature dependence of the first and second excited energy levels E? - E, and E3 - E, for the Nd ions. I: experimental data from Ref. [7]; - computed; - - computed using the parameters of Cadogan et al. [4] by taking into account the mixing of the first excited J-multiplet; computed using the parameters of Cadogan et al. [4] by neglecting the mixing of the first excited J-multiplet.
2
200
100
3
H (hoe)
represent the calculations by taking into account and by neglecting the first excited J-multiple& respectively. The neglect results in an evident departure of the calculation from experiment above 125 K. The hyperfine field is calculated to be 3790 kOe, which is somewhat larger than the experimental values of 3530-3540 kOe [6,8,9]. Fig. 7 shows the magnetization curves along the [loo], [llO] and [OOl] axes at 4.2 and 290 K, and
3.a
1
...*....
Fig. 7. Magnetization K. experimental
Fig. 8 those along the [ 1001 axis at 100, 130 and 160 K for (Nd,,,Y,,),Fe,,B. The solid curves represent the calculations using the fitted parameters of set 1, and the symbols the experiments. The agreement between the calculations and experiments is excellent.
I
. .
curves of (Ya ,Nd,,),Fe,,B at 4.2 and 290 data from Ref. [32]; - computed.
40
2. 42. 2 0
I 50
I 100 T IK)
I 150
200
Fig. 6. Temperature dependence of the mean magnetic moment M,,(T) for Nd?Fe,,B. W experimental data from Ref. [2]; computed; --- computed using the parameters of Cadogan et al. [4] by taking into account the mixing of the first excited J-multiplet; computed using the parameters of Cadogan et al. [4] by neglecting the mixing of the first excited J-multiplet.
Ol Fig. 8. Magnetization
H (kOe1 curves
of (Y,, ,Nd, g)zFe,,B experimental
[ IOO] axis at 100, 130 and 160 K. Ref. [32]; -
computed.
along the data from
Y. YLIet al. / Jounml
of Magnrtism and Mqnetic’ Matrrials 164 (19961 201-207
4. Conclusions This work has shown that it is necessary to take into account the Nd-Nd exchange interaction and the anisotropy of the Nd-Fe exchange interaction in addition to the isotropic Fe-Nd exchange interaction to obtain a reasonably satisfactory explanation of the magnetic properties of Nd,Fe,,B. The size of the Nd-Nd exchange interaction is comparable with that of the Nd-Fe exchange interaction in Nd?Fe,,B as for the R-R exchange interaction in the compounds of Gd, Dy and Er [I 8,191. The value of = 0.038 is nearly the same as the l”‘n,,,d/%,,, 0.040 obtained for Gd,Fe,,B in our previous publication, thus supporting the idea that the exchange interactions are mediated via 5d conduction electrons [33], as originally suggested by Campbell [34].
Acknowledgements This work was supported by the National Science Foundation of China.
Natural
References 111S. Hirosawa, Y. Matsuura. H. Yamamoto,
S. Fujimura and M. Sagawa, J. Appl. Phys. 59 (1986) 873. L21K. Tokuhara, Y. Ohtsu, F. Ono, 0. Yamada, M. Sagawa and Y. Matsuura, Solid State Commun. 56 (1985) 333. [31 F. Bolzoni, 0. Moze and L. Pareti, J. Appl. Phys. 62 (1987) 615. 141 J.M. Cadogan, J.P. Gavigan, D. Givord and H.S. Li, J. Phys. F: Metal Phys. 18 (1988) 779. 151 R. Verhoef, J.J.M. Frame, A.A. Menovsky, R.J. Radwanski. Ji Song-quan. Yang Fuming. H.S. Li and J.P. Gavigan, J. Physique 49 (1988) C8-565. 161I. Nowik, K. Muraleedharan, G. Wortmann, B. Perscheid, G. Kaindl and N.C. Koon, Solid State Commun. 76 (1990) 967. [71 M. Loewenhaupt, M. Prager, A.P. Maruni and H.E. Hoenig, J. Magn. Magn. Mater. 76-77 (1988) 408. iSI Y. Berthier, M. Boge, G. Czjzek. D. Givord, C. Jeandey, H.S. Li and J.L. Oddou, J. Magn. Magn. Mater. 54-57 (1986) 589. [91 E. Jedryka, M. Wojcik and P. Panissod, J. Magn. Magn. Mater. 104-107 (1992) 1405.
201
[lo] D. Givord, H.S. Li and R. Perrier de la Bathie, Solid State Commun. 88 (1993) 907. [I I] G. Marusi, N.V. Mushnikov, L. Pareti, M. Solai and A.E. Ermakov, J. Phys.: Condens. Matter 2 (1990) 7317. 121 M. Yamada, H. Kato, H. Yamamoto and Y. Nakagawa, Phys. Rev. B 38 (1988) 620. 131 Y. Nakagawa, H. Kato, D.W. Lim, G. Kido and M. Yamada, in: Proc. 6th Int. Symp. on Magnetic Anisotropy and Coercivity in Rare-Earth Transition Metal Alloys, Carnegie-Mellon University, Pittsburgh, PA, 1990, p. 12. 141 R.J. Radwanski and J.J.M. Franse, J. Magn. Magn. Mater. 80 (1989) 14. [15] H.S. Li and J.M. Cadogan, Solid State Commun. 81 (1992) 283. [16] Yan Yu, Zhao Tie-song and Jin Han-min, J. Phys.: Condens. Matter 3 (1991) 195. [17] G. Czjzek and H.J. Bornemann, J. Magn. Magn. Mater. 80 (1989) 23. [18] Yan Yu, Zhao Xiao-hong and Jin Han-min, J. Magn. Magn. Mater. 127 (1993) Ll. [19] E. Burzo, J. Appl. Phys. 57 (1985) 3232. [20] D. Givord, H.S. Li and R. Perrier de la Bathie, Sol. State Commun. 51 (1984) 857; M. Borg, J.M.D. Coey, G. Czjzek, D. Givord, C. Jeandey, H.S. Li and J.L. Oddou, Solid State Commun. 55 (1985) 295. [21] Jin Han-min and Yan Yu, Phys. Rev. B 48 (1993) 1022. [22] Han Xiu-feng, Jin Han-mitt, Wang Zi-jun and T.S. Zhao, Phys. Rev. B 47 (1993) 3253. [23] S. Hufner. Optical Spectra of Transparent Rare-earth Compounds (Academic Press, London, 1978) p. 34. [24] M.E. Rose, Elementary Theory of Angular Momentum (Interscience, New York. 1957). [25] M. Boge. G. Czjzek, D. Givord, C. Jeandey, H.S. Li and J.L. Oddou, J. Phys. F: Metal Phys. 16 (1986) L67. [26] Jin Han-mitt, K. Shimizu, Han Xiu-feng, Yan Yu and Zhao Tiesong, J. Phys.: Condens. Matter 4 (1992) 8609. [27] M.A.H. McCausland and IS. Mackenzie, Adv. Phys. 28 (1979) 305. [28] K. Shim& in: Proc. 2nd Int. Symp. on Physics of Magnetic Materials, Beijing, 1992, p. 707. [29] LA. Campbell, J. Phys. C: Solid State Phys. 2 (1969) 1338. [30] R. Fruchart, P. L’Heritier, P. Dalmas de Reotier, D. Fruchart, P. Wolfers, J.M.D. Coey, L.P. Ferreira, R. Guillen, P. Vulliet and A. Yaouanc. J. Phys. F: Metal Phys. 17 (1987) 483. [31] D. Ballou, J. Deportes, B. George, R. Lemaires and J.C. Ousset, J. Magn. Magn. Mater. 54-57 (1986) 465. [32] D.W. Lim, H. Kato, M. Yamada, G. Kido, Y. Nakagawa and S. Hirosawa, J. Magn. Magn. Mater. 90-91 (1990) 72. [33] E. Belorisky, J.P. Gavigan, D. Givord and H.S. Li. Europhys. Lett. 5 (1988) 349. [34] LA. Campbell, J. Phys. F: Metal Phys. 2 (1972) L47.