Molecular-field theory analysis of RFe2 intermetallic compounds

Journal of Magnetism and Magnetic Materials 127 (1993) 378-382 North-Holland

Molecular-field theory analysis of RFe, intermetallic compounds Y.J. Tang, X.P. Zhong and H.L. Luo Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China

Received 22 September 1992; in final revised form 5 March 1993

On the basis of the two sublattice model the temperature dependence of magnetization for RFe, compounds (R = Tb, Dy, Ho, Er and Tm) is analyzed. The molecular-field coefficients have been derived by fitting the experimental data, and they are transformed into the spin-exchange parameters. The results show that the spin-exchange interactions in RFe, are stronger than those in RaFer7, R,Fe, and RFes compounds, which makes their Curie temperatures the highest ones in the corresponding R-Fe binary compounds.

1. Introduction

The cubic Laves phase RFe, compounds are attracting great interest caused not only by their application as giant magnetostrictive materials but also as opportunities for extended studies of magnetic properties of the 3d-4f elements. In these compounds the strong 3d-3d exchange interaction (T-T) is combined with large magnetostriction of 4f elements, which brings their Curie temperatures well above room temperature, and the strong 3d-4f interaction (R-T) could maintain the existing large natural magnetostriction of the 4f ions to the highest temperature [l]. The magnetic properties of rare earth-transition-metal (R-T) intermetallics usually can be described by a two-sublattice model [2] in which the total contribution to the magnetic properties can be simply treated as the sum of the contributions from both the rare earth and the transition metal sublattices. The interactions in R-T compounds comprise three different types: the R-R,

R-T and T-T interactions. Of the interactions involving rare earths (R-R, and R-T), the R-T is the most important and essentially determines the magnetic behavior of the rare earth sublattice. Thus the parameters which represent the strength of R-T coupling, such as Rk the intersublattice molecular (ISM) field acting on the R moment by the Fe surroundings, are crucial for both theoretical and experimental studies of the 3d-4f compounds. The strength of R-Fe spin-exchange interactions in RFe, compounds has been derived in several ways, such as from Mossbauer effects [31, neutron inelastic scattering 141, and high-field magnetization measurements 151. In this paper the temperature dependence of magnetization for RFe, (R = Tb, Dy, Ho, Er, Tm) is analyzed with the two-sublattice model of the molecular-field theory. The values for the molecular-field coefficients and spin exchange parameters are derived and compared with those achieved previously.

2. Theory outline Correspondence to: Dr. He-he Luo, Institute of Physics, Chinese Academy of Sciences, Beijing 10080, China. Tel: + 86-1255 9131, ext. 263; fax: +86-l-256 2605. 0304-8853/93/$06.00

According to the molecular field theory based on the two-sublattice model 121, the molecular

0 1993 - Elsevier Science Publishers B.V. All rights reserved

YJ. Tang

et al. /

Molecular-fieldtheory

fields of the rare earth and iron sublattices RFe, compounds can be expressed as: &(T)

=H+d[nR,

H,(T)=H+d[2

Ma(T)

%&f,(T)

in

hF&(~)I 9 (1)

+

of RFe, compounds

analysis

379

magnetic interactions, can be derived by numerically solving eqs. (l)-(5), making M,, compatible with the experimental data by minimizing the percentage deviation:

+ kFWJT)17

i

i

(2)

where H represents the applied M,(T) the magnetic moments earth ion at temperature T, factor d converts the moment pn into Gauss:

field, M,(T) and per Fe and rare respectively. The per RFe, unit in

d = N,P.,P/A,

(8)

The Curie temperature T, can be obtained from (4) and (5) in zero applied field, assuming an antiparallel coupling between the Fe and R moments (ferrimagnetic) [21: (in

- T,a)(n,

- T,P) -&r

T,=

(n,~+n,aP) 1

= 0,

(9)

(3)

where NA is Avogadro’s number, p is the density of RFe,, and A is the RFe, formula weight. The Brillouin function controls the temperature dependence of each sublattice moment: WC(T) = MR(0)BJRIMR(O)HR(T)/kBTl

%(T) =&Wh,[ ~Fu9HFvvwl~

7 (4) (5)

where M,(O) and M,(O) are the magnetic moments of R and Fe ions at zero temperature, respectively. Assuming Mn(O) = gnJ+n, Mr(O) can be derived from the experimental data at low temperature:

- 4aP( nFF%ux - &F )]1’2]/2us,

-&t(O)]

/2.

(10)

111 bb3/ClBdl/~i(O)~ (11)

a = [3JR/(JR

+

B = [3J,/(J,

+ I)]

bWwW2jW’)-

(12)

Furthermore, the spin-coupling constants 4 ji can be derived from the molecular-field coefficients

bybl:

3k,n,/2p=Z,S,(S,+l)A,, 9kgr&,/4ap

MF(0) = [&p(O)

+ [(w++RP)~

= Z,,Z,S,(

(13)

SF + 1)

(6)

x(g,

Under the condition:

- l)zJR(JR + 1)A2m, (14)

W,,=&(T)

+2%(T),

(7)

the molecular-field coefficients nFF, nRF and I?~, which describe the T-T, R-T and R-R

3kanmJ2a

=&Jn(J~

+ l)(g,

- +%w (15)

Table 1 Saturation magnetization M(O), moments of the rare earth M,(O) and iron Mn(O), density of the RFe, p Ill, some of the characteristic parameters of the trivalent rare earth ion (spin Sa, total angular momentum JR and Land6 factor gn) used in the molecular-field theory calculation of the RFe, compounds Compound

h!t(O) (ClR/f.u.)

Ma(O) (wn /ion)

M,(O) (cln /Fe)

P (P/cm31

5,

JR

gR

TbFe, @Fe, HoFe, ErFe, TmFe,

5.81 6.87 6.70 5.79 3.72

9 10 10 9 7

1.60 1.57 1.65 1.61 1.64

9.06 9.28 9.44 9.62 9.79

3 512 2 312 1

6 15/2 8 15/2 6

312 4/3 5/4 6/5 7/6

380

Xl. Tang et al. / Molecular-fEld theory analysis of RFe, compoun~?~

3. Results and discussion

calculation of the RFe, compounds [lo], we take S, = J, = 0.75 in the calculation. The moment of the rare earth sublattice in RFe, is larger than that of iron sublattice. For the sake of simplifying the equations, we take positive and negative values for the R and Fe moments, which are parallel and antiparallel to the resultant moment, respectively. Figures l(a)-(e) show that the calculated temperature dependence of the magnetization for RFe, from 0 K to the Curie temperature is consistent with the experimental data taken from Clark et al. [7,8], Abbundi 191and Ikeda et al. Cl11 measured on single crystals, with all the percent-

The saturation magnetization, rare earth and iron moments, and some parameters used in the calculation are listed in table 1. Values for M(O), iWn(O), Mr(O) and p are cited in ref. [ll, which were taken from single-crystal data [7-91. The values for M,(O) in table 1 are around 1.6pr,, which yields a value of 0.8 for the pseudo-spin of the iron S, (assuming M,(O) = -g&~n and g, = 2). Because the magnitude of the total angular momentum of the iron J, influences the calculated magnetization curve slightly [2], and S, = Jr = 0.75 has been commonly used in the

C1 TbFes 00000

HOFQ

10

ooooo Expenment[7]

Experiment[7]

C

ooooo Expenment[O] -

Calculated

Temperature

(k)

Mno

a 6

-%’

Temperature _

(k)

):;c

400 200 Temperature

I

(k)

-SW

TmFe? ooooo Experiment[B.l _ Calculated

I]

e

~OOOO Experiment [7] Calculated

0 Temperature

(k)

Temperature

(k)

Fig. 1. Temperature dependence of the magnetization of We,: (a) TbFe2, (b) DyFe,, (c) HoFe,, Cd)ErFe, and (e) TrnFe*. Solid lines: theoretical curves; points: experimental data.

381

YJ. Tang et al. / Molecular-fieldtheory analysis of RFe, compounds

Table 2 Molecular-fieldcoefficients nij, spin-exchangeparameters A,, Curie temperature T, and ISM field B& Z,=12, Z,=6, Z,=6and Z,,= 4 were used in the calculations Compound TbFe, &Fez HoFe s ErFe, TmFe,

nFF 10950 10760 9710 10540 9710

nm

nRR

AFF (10-u J)

Anr (lo-= J)

-2300 - 1810 - 1310 -1140 -980

270 180 130 110 50

140 132 138 143 140

-21.0 - 21.7 -21.2 -21.7 - 22.6

age deviations R < 6% (see table 2). The calculation for TbFe, has taken account of the experiment results on polycrystalline sample [12] in which no compensation temperature was found below the Curie temperature for the lack of the experimental data on single crystals over 300 K

111. Values for the molecular-field coefficient nFF in table 2 are about 10000, which is larger than those of 3200 in Er,Fe,, 161, 4000 in R,Fe,,C 1131, 5500 in R,Fe, [2], and 8000 in RFe, [141. According to (lo), the molecular-field coefficient nFF can be directly derived from the Curie temperature of R-Fe compounds where R is a nonmagnetic rare earth by nFF = BT,. Because the Curie temperature in the R-Fe compounds increases with the rare earth concentration from R,Fe,, to RFe,, it is reasonable that the value for nFF in RFe, is the largest one in the R-Fe binary compounds. The value of A, derived from nFF is about 14 X lo-‘* J, which is in good agreement with the value of 14.2 x lo-** J for RFe, obtained by Gubbens et al. [15]. Values for the nRF in table 2 are from -980 in TmFe, to -2300 in TbFe,. Their absolute values are larger than those of 600 in Er,Fe,, [6], 700 + 100 in R,Fe,, [2], 761 in TmFe, and 1000 in TbFe, 1141,showing that the R-Fe interaction in RFe, is stronger than those in R2Fe1,, R,Fe, and RFe, compounds. The magnitude of nRF in RFe, series varies monotonously from TbFe, to TmFe,, which is different from the cases of R,Fe, [2] and RFe, 1141 series. It has been confirmed by experiments that the spin-exchange parameter A,, or JRFe in the R-Fe compounds decreases slightly when the atomic number of the

AR,

(lo-” 5.3 6.3 7.2 8.8 5.5

J)

BL CT)

CR)

136 105 82 70 63

704 637 607 591 555

for RFe,. The values

TF'

R (o/o) 697 635 606 590 560

3.0 2.0 1.5 2.5 5.7

R component goes from Gd to Tm [5,161. From eqs. (10, (12) and (141, noting Mn(O) =g,J,pLe, one can conclude that the magnitude of nRF is approximately proportional to the factor of (ga - 1)/g,. Because the value for this factor decreases from l/3 for TbFe, to l/7 for TmFe,, the magnitude of nRF in TbFe, is expected to be over twice as large as that in TmFe,. Because the spin coupling between the rare earth and iron is antiparallel, we take negative values for the spin-exchange parameter A, listed in table 2. The values for this parameter A, calculated from the nRF are about 2.2 x lOmu J, which is in good agreement with 153 K (yields a value of 2.3 x lOmu J) estimated from Miissbauer spectra of TmFe, [3] and somewhat smaller than 19.1 K for J,,,,/k (yields a value of 2.6 X lo-** J) derived from high-field measurements on Eri_, Y,Fe, compounds [5]. Differing from the experimental results [5,16], the A,, derived here does not decrease when the R component goes from Tb to Tm, but keeps nearly constant. If the A, in RFe, varies with the R component slightly as the case of R,Fe,, compounds 1171,this variation is within the fitting errors. The R-R interaction, which is the smallest one in the three interactions, has been derived as (0.7 & 0.2) x lo-** J for the spin-exchange parameter A,,. It is one-third of the R-Fe spin-exchange interaction. Because this interaction influences the temperature dependence of magnetization slightly, the values for the nRR and A,, derived from the temperature dependence of magnetization are not so accurate. In the calculation, we found that the fitting deviation R is sensitive to the n, and nnr, but

382

IX Tang et al. / Molecular-jield theory analysis of RFe, compouna’s

not to the naa. From fig. 1 and table 2, one can see that the calculated curves fit the experimental data very well and all the percentage deviations R < 6%. This means that on the one hand the values for the molecular-field coefficients nrF and nRF together with the spin-exchange parameters nFF and nZRFderived by this method are reliable, and on the other hand, the two-sublattice molecular-field model describing the temperature dependence of magnetization for RFe, is successful.

4. Conclusion In conclusion, we would like to point out that: (1) The two-sublattice molecular field model successfully describes the temperature dependence of magnetization for RFe,. The values for the molecular-field coefficients nFF and nRF, as well as the spin-exchange parameters A, and A RF derived with this model are reliable. The derived values for the nRR and A,, are not accurate. (2) Values for the spin-exchange parameters A, and A, in RFe, are equal to 14 X lo-‘* and 2.2 x lo-** J, respectively, which are in agreement with those derived from Mijssbauer effects and high-field magnetization measurements. They are larger than those in R2Fei7, R,Fe, and RFe, compounds, which makes the

Curie temperatures in RFe, the highest ones in the corresponding R-Fe binary compounds.

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Amsterdam, 1980) p. 531.

121J.F. Herbst and J.J. Croat, J. Appl. Phys. 55 (1984) 3023. [31 B. Bleaney, G.J. Bowden, J.M. Cadogan, R.K Day and

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