Magnetic coupling in the rare-earth compounds RFe2 and RFe3

Journal of Magnetism North-Holland

Magnetic

and Magnetic

Materials

coupling

98 (1991) 291-297

291

in the rare-earth

compounds

RFe,

and RFe,

J.P. Liu, F.R. de Boer Van der Waals - Zeeman

Laboratorium,

Uniuerslteit van Amsterdam,

Valckemerstraat

65, 1018 XE Amsterdam,

Netherlands

and K.H.J.

Buschow

Philips Research Laboratories, Received

17 December

P.O. Box 80000, 5600 JA Eindhouen. Netherlands

1990

The field dependence (up to 35 T) at 4.2 K of the magnetization of several compounds of the type R, _ ,Y,Fe, was studied. The magnetic isotherms at 4.2 K were analyzed with a mean-field model. The magnetic coupling constant J,,, between the moments of the rare earth (R) and Fe in the Hamiltonian H = C RFe2JKFeSRSFr were compared with the coupling constants derived from a mean-field analysis of the Curie temperatures. From a comparison of the magnetic coupling constants found by means of high-field measurements in the compounds RFe2, RFe,, R,Fe,, and R*Fe,, it follows that the R-Fe coupling constants have a pronounced tendency to decrease with Fe concentration

1. Introduction Considerable efforts have been made recently to study the magnetic coupling strengths for rareearth compounds, including those for which single crystals could not be prepared [l]. Such experimental studies have focused mainly on intermetallics of relatively high 3d atom concentration [2,3] and there is a general lack of such coupling constants for compounds of lower 3d atom content. In the present investigation, we have concentrated therefore on rare-earth Fe compounds of the compositions RFe, and RFe, in order to be able to compare the coupling constants in these compounds with those determined earlier for compounds of the type R,Fe,, and R,Fe,, [l-3]. Owing to the strongly localized character of the 4f electrons, the coupling between the 4f and 3d moments has to be indirect. On the basis of the large amount of experimental data regarding the type of coupling in the many R-3d compounds, it 0304~8853/91/$03.50

0 1991 - Elsevier Science Publishers

was suggested many years ago by Campbell that the 4f-3d coupling proceeds via the rare-earth 5d electrons [4]. Recent self-consistent band-structure calculations [5] have confirmed the importance of the 5d electrons in this respect. These band-structure calculations have furthermore offered a deeper insight into the mechanism of the indirect 4f-3d coupling, which has been shown to be due to 3d-5d hybridization and local 4f-5d exchange interactions. It is mainly the availability of these calculations of the 4f-3d coupling strength in the RFe, compound which has triggered us to study these materials in more detail by high-field measurements.

2. Experimental It was already shown in several previous reports [l-3] that it is possible to determine values of the intersublattice magnetic coupling constant

B.V. (North-Holland)

from high-field measurements on compounds in which the 4f- and 3d-sublattice magnetizations cancel or nearly cancel. For this reason we have performed our investigations on pseudobinary compounds that have a reduced R-sublattice magnetization (R, ,Y,Fe, and R, ,Y,Fe,). using Y concentrations that are higher the larger the R moment. The various R , ,Y, Fe, and R, ,Y, Fe, compounds were prepared by arc melting, followed by vacuum annealing at 1050 o C for 2-3 weeks. From X-ray diffraction measurements it was derived that all samples investigated were approximately single phase, the main phase having the cubic Cl 5 structure and rhombohedral PuNi ?-type structure. respectively. The magnetic isotherms at 4.2 K of all these compounds were measured in the high-field installation at the University of Amsterdam [6] in fields up to 3X T. The samples consisted of finely powdered material, the individual particles being free to rotate into their minimum-energy direction during the measurements. Results of these measurements are shown in figs. l-3.

I-lg. 7. Flrld-dependence R, ,Y, Fe, compounda tained on fine powder

I,

of the magnetic moment in LL~~IOU\ (R = Gd, Th. Ho. Er) at 4.2 K. ,Ihparttck free to rotate in Ihr field apphed

-

,,a : 0:.

Fig. I. Field-dependence of the magnetic moment in various Fr, ,Y,Fe, compounds at 4.2 K, obtained on fine powder particles free to rotate in the field applied.

Fig. 3. Fteld-dependence R, ,Y,Fe, compounds tained on fine powder

of the magnetic moment tn var~ou\ (R = tid. Dy. Ho, Er) at 4.2 K. ohparticles free to rotate in the field applied.

J.P. Liu ei al. / Magnetic coupling m RFe, and RFe, compounds

293

3. Discussion In the same way as in previous investigations we have analyzed the magnetic isotherms shown in figs. l-3 by means of a two-sublattice meanfield model [l]. Since the powder particles used for the magnetization measurements were fairly small, they may be regarded as monocrystalline particles (particle size -C 40 pm). These particles will orient their magnetic moments parallel to the external field as they are able to rotate freely in the sample holder, since the field strengths used in this investigations are generally much larger than the stray fields of adjacent particles. The equilibrium state is found by minimizing the free-energy expression E = MFeMRnRFe cos a - B[ M& + Mi + 2M,,M, cos 0~1~” with respect to the angle (Y between the R and Fe sublattice magnetizations, n are representing the intersublattice-coupling constant. It can be shown that the following behaviour is expected as a function of the applied field B: For relatively low fields the moment configuration is strictly antiparallel and the magnetization is equal to the values Ms = 1M, - M,, I. Beyond a critical field strength (B,,,, mo= i”RM,e inRFe ). the exactly antiparallel ments start to bend towards each other. From there on, the magnetic moment is described by M = B/n.,, and thus dM,‘dB=

[nRFc]-‘.

(1)

The parameter n,,, can therefore be derived straightforwardly from the high-field slopes obtained for B > B,,,,. The behavior sketched above is an ideal one. In a schematical way, it is illustrated in the lower part of fig. 4 where M, represents A4s = 1 M, MF,

I.

The behavior observed in most of the practical cases is represented schematically in the upper part of fig. 4. The obvious difference with the behavior shown in the lower part of the same figure is the nonzero vertical intercept M3 and the nonzero slope of the low-field part (B c B,,,,) of the M(B) curve. Both features can be attributed to the presence of small amounts of impurity phases in the samples. Owing to the fact that we

0

/ / BC,ll

B(T)

Fig. 4. Field dependence of the magnetic moment expected for fine powder particles of ferrimagnetic R-T compounds able to rotate freely in the external field: (A) in the presence of small amounts of impurities; (b) ideal behavior. EC,,, = nRT ( M, MT I. The high field slope is determined by the relation d M/d B = l/n RT. The meaning of the moments M,, M, and M, is discussed in the main text.

have deliberately chosen the composition of most of the samples so that there is almost complete cancellation between the R- and Fe-sublattice magnetizations, one finds that the relative error is fairly large when M2 in fig. 4A is used to calculate %= i"Fe-MRi. A better estimate of MS can be obtained, however, from the difference M, - M3 (fig. 4A) where M, corresponds to the magnetic moment reached at the critical field B,,,, and where M3 represents the magnetic moment by extrapolating the highfield slope to B = 0. All of the isotherms shown for Er, __XYXFe2(x > 0) correspond to the situation shown in fig. 4A. The values of M, and M, - M3 derived from these isotherms have been listed in table 1. Note that in the case of ErFe, where the net moment is sufficiently large our value of MS = M2 = 5.85p, is in excellent agreement with the value MS = 5.79~~ obtained from measurements on a single crystal [7]. The values of M2 and M, - M3, both of which are a measure of ( M, - M,, 1, have been plotted as a function of concentration in fig.

7ahle

1

Intzr\uhlatt~ce

couphng

the R- and Fe-suhlattlcrs ment.\

constant

/ ‘M, ~ M,, 1 are given

p,,,,‘f.u.)

,I~, ~ I” Tf.u.,

/L” hetwew

,Y, Fe?_ The saturation

in Er,

by M,

or

mo-

M, - ,MM, (both

III

as defined in fig. 4. The values of !I,,,, (in T) are thaw

of the hreak points in fig. 1 Cr,


M, - I+43

L,

nKtc

M,

,. = 0.75

30.‘)

0.70

0.50

\ = 0.70

2Y.O

0.45

0.00

0

1 = O.fl5

21.2

0.41

0.50

15

\ = 0 60

2x.3

0.X6

0.76

I6

77

‘* 3s

5.x5

\ --0

5. The concentration dependence may be described by the relation

_ 2[pLlc+

of

(1 -.u)

/ M, - M, (’ )

ApI.

(2)

where pFc represents the Fe moment in the ahsence of any polarizing influence due to the rareearth spin. The quantity &I is the excess moment due the presence of a rare-earth spin. The value fo p,_< equals 1.5p”. as may be derived from the saturation moment of YFe, and LuFe, [8]. Using this latter value in conjunction with gJpB = 9~,,, and MS = 5.85~~. for ErFe,. one finds Ap = 0.15~~~. The continuous line in fig. 5 represents

tIn,e =

X-

Fig. 5. Concentration tr, vu

dependence

,Y, Fe, compounds the relatmn

of the saturatmn

obtained

moment

m

at 4.2 K. Open circles: data ohtalnrd

MS = M, - M? (fig. 4A).

1M, - M,, / calculated by means of eq. (2) after substituting gJpH = 9p13, Pa,. = 1.5~” and Ap = 0.15~~. It may be seen that the experimental vaues of M, - M3 (and also those of M,) arc in satisfactory agreement with expectations. The remainder of the R, ,Y, Fe, and R, ,Y, Fe, compounds were studied in the same way. The 11,~~~values derived from the high-field slopes have been listed in table 2. In order to make it possible to compare the intersublattice magnetic-coupling strengths 111 RFe, compounds with those in RFe, and other rare-earth Fe compounds. one may derive coupling constants from the rlK, c values listed that are independent of the number ZK,+ of Fe neighbours surrounding the R atoms and that are expressed per (unit moment)’ of a pair of R and Fe atoms. In fact. the intersublattice molecularfield coefficients rrR, c are related to the R -Fe exchange constants JKtc appearing in the correinteraction Hamiltonian IIC._,, cm sponding C2J,,,S,S,,. via the expression:

filled

by using MS = Mz (fig. 4A).

circles:

data

-J

weZwc(

g,

-

1 )/N,,Pl,RK~

l.3)

In the crystal structure of the RFe, and RFc, compounds the average number Z,, e of nearest transition-metal atom neighbours to an R atom

J. P. idu et al. / Magnetic coupling tn RFe2 and RFe

equals 12 and 14, respectively. The number of transition-metal atoms N,, per formula unit equals 2 in RFe, and 3 in RFe,. Only nearest-neighbour interactions are taken into account in eq. (3). The values of the exchange-coupling constants the J RFe obtained from eq. (3), after substituting Z RFc, N,, and the n RFe values given in table 2, have been listed in the same table in column 3. Inspection of the JRFe values listed in table 2 reveals the surprising fact that the JRFe values decrease with atomic number of the R component in the RFe, compounds whereas in the RFe, series they have a tendency to increase. In the second column of table 3, we have compared the JErFe values of all the existing 4 types of binary R-Fe compounds. We have restricted ourselves here to R = Er, since only for the Er compounds a complete set of high-field data is available. It follows from the values listed in the table that there is a fairly strong decrease of the intersublattice-coupling strengths when going from RFe, to R,Fe,,. This feature is also illustrated in fig. 6. An alternative method that can be used for obtaining experimental values of the intersublattice-coupling constant is based on the meanfield expression, ( J,,,/k

)’ = 9( c - T,’ >T,/4Z,,,Z,& XSl=,(S,,

+ I)(&

- U2JV+

I>> (4)

where T, and T,’ represent the Curie temperatures of rare-earth transition metal compounds in which R is magnetic (J # 0) or R is nonmagnetic (J = 0), respectively. The magnetic-coupling constants derived from the parameters listed for the four types

I compounds

295

Er2 Fe1 71

01 60

I

I

I

I

70

80

90

100

at % Fe Fig. 6. Concentration dependence of the magnetic JErFe of Er and Fe atoms.

of binaries in table 3, using expression (4), have been listed in column 8 of the same table. In view of the limited experimental accuracy of the T, values and in view of the fact that the determination of JRFe via eq. (4) involves the difference between two large numbers, we have limited ourselves here to compounds of Gd for which the difference (T, - T,‘) is a maximum. Comparison of the data listed for JGdFe in table 3 with those derived from high-field measurements (column 2) shows that the low-temperature (or high-field) mean-field approach and the high-temperature approach lead to slightly different numerical results, although the overall agreement is still satisfactory. In fact, we consider the high-field data as the most reliable ones, the comparison with the data derived from the Curie temperature analysis serving only to demonstrate that the high-field data are compatible with the overall magnetic behavior of

Table 3

Magnetic and structural data of R-Fe compounds. values of JCidFc/k were derived from the parameters

The values of JErFr/k (in K) are derived listed using relation (4)

from high field measurements.

Compound

Jme /k

r, (R = Gd)

r, (R=Y)

PFC

Z RFe

Z FCR

RFe,

- 19.1

795

542

1.46

12

6

- 17.8

RFe,

- 16.7

129

569

1.68

14

4.7

- 12.8

fG,Fez

- 12.0 a)

659

481

1.89

13

3.4

- 14.2

476

324

1.95

19

2.2

-11.3

RIFeI,

‘I Taken

-7.0

from ref. (31.

‘I’

-&Fe/k

The

these compounds within the mean-field model. It is gratifying to observe that the strong tendency of the high-field values of JRE.e to decrease with Fe concentration is also displayed by the magneticcoupling constants derived from the Curie temperatures. Finally, we wish to compare our data with results obtained by means of inelastic neutronscattering experiments and with ab initio calculations of the intersublattice interaction obtained from density functional theory. Both types of data are available only for the cubic RFe, compounds. It follows from the band-structure calculations of Brooks et al. [5] that the Fe-3d and R-5d bands in RFe, hybridize to form bonding and antibonding bands. Owing to the formation of a moment at the Fe sites. there is a general lowering of the spin up 3d states leading to a reduction of the 3d-5d hybridization for the spin up states. Since less hybridization implies less transfer of charge from the 3d character to 5d character there is a concomitant reduction of the occupation of the 5d spin up states. For the spin-down states the effect is opposite. Consequently, the induced 5d moment is antiparallel to the 3d moment, even when the 4f moment is zero. It also follows from the calculations of Brooks et al. that the R-4f and R-5d spins couple parallel by means of a local exchange interaction. The interaction between the R-4f and Fe-3d spins is therefore mediated entirely by the R-5d spin. Results of the calculations of Brooks et al. (51 have been reproduced in fig. 7 where their values of the interaction constant

are represented by the full curve. The quantity lidlid in eq. (5) represents the 4f-5d exchange integral and $,, and T,+ denote average values of the 5d and 3d spins, respectively. The filled circles in fig. 7 are R-Fe coupling constants derived from inelastic neutron-scattering experiments carried out on several of the RFe, compounds by Nicklow et al. [9] and Koon and Rhyne [lo]. The open circles are the values of ( g, - 1 )J,,, derived from the high-field data listed in table 2. The agreement between all these data is quite satisfactory, although the fact that the data points are below the

01

I

Gd

J Tb

Dy

Ho

Er

Tm

Yh

Fig. 7. Magnrtlc-couplmg constant of the R and Fe magneuc moment interaction in KFe, compounds calculated hv Brook\ et al. [5] (full curve). The filled circles represent exprnmental data derived from lnelastlc neutron-scattering experiment\ [9.10]. The open circles were dewed from the high-field dau (table 2)

calculated curve suggests that JK,.c derived from the band structure calculation varies somewhat stronger with the R atomic number than the cxperimental data.

Acknowledgement

The present investigation has been carried out within the scientific exchange program between China and Netherlands.

References

III K. Verhoef. K.J. Radwahskl

and J.J.M. Franw. J. Magn Magn. Mater. X9 (1990) 176. 121R.J. Radwatiski. X.P. Zhong. F.K. de Boer and K.H.J Buschow, PhyGca B 164 (1990) 131. PI F.R. de Boer. X.P. Zhong. K.H.J. Buschow and r H Jacobs. J. Magn. Magn. Mater. 90 ci: 91 (1990) ‘5. [41 LA. Campbell, J. Phys F 2 (1972) L47. and B. Johan&~~~n. to hc [51 M.S.S. Brook.\. L. Nordstriim published. F.R. de Boer, J C‘. Wolfrat. f-.A Muller and [61 R. Grrsdorf. L.W. Roeland, m: High-Field Magnetlhm. ed M. I)ate (North-Holland. Amsterdam. 19X?) p. 277.

J. P. Liu ef al. / Magnetic coupling in RFe, [7] A.E. Clark, in: Ferromagnetic Materials, vol. 1, ed. E.P. Wohlfarth (North-Holland, Amsterdam, 1980) p. 531. [8] K.H.J. Buschow, in: Ferromagnetic Materials, vol. 1, ed. E.P. Wohlfarth (North-Holland, Amsterdam, 1980). [9] R.M. Nicklow, N.C. Koon, C.M. Williams and J.B. Milstein, Phys. Rev. Lett. 36 (1976) 532.

and RFe,

compounds

291

[lo] N.C. Koon and J.J. Rhyne, in: Crystalline Electric Field and Structure Effects, eds. J.E. Crow et al. (Plenum, New York, 1980) p. 125.