Magnetic properties of amorphous metallic alloys containing rare-earth and transition-metal components

179 MAGNETIC PROPERTIES OF A M O R P H O U S METALLIC ALLOYS CONTAINING RAREEARTH AND T R A N S I T I O N - M E T A L C O M P O N E N T S A.K. B H A T T A C H A R J E E , B. C O Q B L I N , R. J U L L I E N and M.J. Z U C K E R M A N N * Laboratoire de Physique des Solides L Universitk Paris-Sud, Centre d'Orsay 91405 Orsay, France

A review of the properties of a m o r p h o u s rare-earth transition-metal alloys is given and the theoretical model of Harris, Plischke and Z u c k e r m a n n is s h o w n to be related to s o m e of the magnetic properties of such alloys. We present an extension of this model to describe the magnetic properties of RCo3, alloys, where R is a rare-earth atom. The d-band is treated in terms of a H u b b a r d hamiltonian and the R m o m e n t s , which are coupled with the d-band through a local e x c h a n g e interaction are subject to a random anisotropy. The hamiltonian is treated in the mean field approximation. A fit of the magnetization versus temperature curve of RCo~, is presented and the magnetization curves of DyNL4 and DyFe~, are discussed in the light of the same theoretical model.

A m o r p h o u s alloys containing rare-earth impurities were first investigated experimentally by Rhyne and co-workers [1]. In particular a bulk sample of RFe2 was prepared by fast sputtering. H e r e R was one of the following rare earth ions: Gd, Sin, Tb, Dy. The alloy containing Gd was found to be a soft ferrimagnet with properties similar to other a m o r p h o u s magnetic alloys whereas the alloys with non-S state rareearth ions exhibited a square hysteresis loop with an initially c o n c a v e B-H curve at temperatures T ~< 100K. The Curie temperatures were found to be of the order of 700-800K. F u r t h e r m o r e , the coercive force of the hysteresis loop at 4 . 2 K was of the order of 20 kG which is an order of magnitude a b o v e that of crystalline cubic L a v e s phase ferro- and ferrimagnets of type RFe2. At the same time as the experimental work was in progress, Harris et ai. [2], hereafter referred as H P Z , proposed that random magnetic anisotropy could exist in a m o r p h o u s alloys containing rare-earth ions. Their model is described below in detail and their results are based on the assumption of a random distribution of easy axes. N e x t Cochrane et al. [3], constructed a cluster of 1300 spheres of two sizes by computer using a procedure due to Polk and Bennett [4]. This cluster described a random close packed metal with atoms of two different sizes and it was shown that the resultant radial distribution functions were in qualitative agreement with neutron scattering experiments on a m o r p h o u s TbFe2. This analysis has recently * O n sabbatical leave from McGill University, Montr6al, Canada. t Laboratoire associ6 au C N R S .

Physica 91B (1977) 179-184 (~ North-Holland

been improved by O ' L e a r y [5]. The direction of the easy axes was then obtained at each rare earth site using a point charge model based on the effective charges found f r o m analysis of M6ssbauer experiments for crystalline RFez compounds. The resulting distribution of easy axes was found to be completely random and, furthermore, the dominant term in the local electric field at the rare-earth site was shown to be hexagonal and an order of magnitude greater than the cubic terms. This was felt by Cochrane et al. [3] to be a qualitative explanation of the magnitude of the coercive force in a m o r p h o u s RFe2 alloys. Further Cochrane et al. [6] extended the cluster theory for the random anisotropy to quantum mechanics and were able to obtain a general form for the hexagonal term. Their analysis showed that (i) there is a broad distribution of the hexagonal anisotropy field p a r a m e t e r D [see eq. (2)] and (ii) this does not smear out Schottky anomalies in the specific heat. There were at that stage no m e a s u r e m e n t s that constituted a direct confirmation for the existence of random magnetic anisotropy (RMA). This situation was changed by the recent M 6 s s b a u e r experiments of Coey et al. [7] on a m o r p h o u s DyCo3.4 and D y C o F e alloys. The measurements, p e r f o r m e d on 7 ~ films, show that the Co m o m e n t s saturate in a direction in the plane of the film by applying a 100G magnetic field and that the Dy m o m e n t s are in the opposite direction to the Co moments, their directions being randomly distributed in a cone of half angle qJ0-70 ° [see the left part of fig. l(a)]. The measured resultant m o m e n t for DyCo34 is 2.3 t~B per DyCo34 unit. The net

180

M , ~%%& ,, 2Wo

-

r a n d o m f e r r o m a g n e t ( " a s p e r o m a g n e t " ) [see fig. l(a)] in which there is p r o b a b l y no magnetic m o m e n t on the Ni a t o m s and DyFe~.4 is a more c o m p l e x sperimagnet. The curves M(T) versus I" are s h o w n in fig. l(b) for these c o m p o u n d s . In this paper we use a simple model to investigate the magnetic properties of these alloys. As m e n t i o n e d a b o v e the c o n c e p t of r a n d o m magnetic a n i s o t r o p y was introduced by H P Z [2] who p r o p o s e d the following model for an asp e r o m a g n e t . The angular m o m e n t a J, of the rare-earth 4f electrons are coupled by a H e i s e n b e r g interaction

M ~ . 2~o

¢

<

;,,, a)

6_

4,

0

0.2

0.4

0.6

0.8

1

i.i

T / Tc

b) Fig. I. Experimental results from Coey et al. [7] and Arrese-Boggiano et al. [9]. (a) On the left the magnetic structure of a " s p e r i m a g n e t " , (i.e. random ferrimagnet) and, on the right, the magnetic structure of an " a s p e r o m a g n e t " (i.e. random ferromagnet); (b) the s p o n t a n e o u s magnetization M in Bohr m a g n e t o n s versus T/T, for DyFe,~ (1), DyCo,~ (ll) and D y N i , , (Ill).

m o m e n t of the cobalt is 1.3 --+0.4 #B . Magnetization m e a s u r e m e n t s confirm that D y C o ~ 4 is a r a n d o m f e r r i m a g n e t ( " s p e r i m a g n e t " ) . The experimental c u r v e of the s p o n t a n e o u s magnetization M(T) [see fig. l(b)] as a f u n c t i o n of t e m p e r a t u r e exhibits a c o m p e n s a t i o n point at T=230K. More recently, this experimental study was e x t e n d e d by J o u v e et al. [8] to other RCo~ s y s t e m s with R = Gd, Dy, Ho, Er. The results f o u n d for DYC0344, H o C o ~ > and ErCox44 are summarized in table I. Further m e a s u r e m e n t s on DyNi34 and DyFex.4 are reported by A r r e s e - B o g g i a n o et al. [9]. DyNi34 is a

b e t w e e n nearest neighbouring rare-earth a t o m s on sites i and j, .i is the coupling constanl. Then, the R M A is introduced by a s s u m i n g a r a n d o m distribution of easy axes directions for the rare earths and a s s u m i n g uniaxial anisotropy. T h e model form of the R M A hamiltonian of H P Z is written

i

where z; is the rare-earth site H~o,+ H~,,,~ was approximation

direction of the easy axis at the i. Next, the total hamiltonian treated in the molecular field ( M F A ) and the molecular field

ho=<~

(3)

was a s s u m e d to be the same on each rare-earth site. z is an arbitrary but fixed direction and v is the a v e r a g e n u m b e r of rare-earth nearest neighbours. L e t 0~ be the angle b e t w e e n z'~ and z at the ith site. Then, since the M F A r e d u c e s the total hamiltonian to a one-ion hamiltonian in a self-consistent molecular field a0, the thermal a v e r a g e (J:(O;))a,,r can be evaluated numerically.

Table I Experimental wdues of M(O), F,.,,e (compensation Icmperature) and 1" for R('o, s y s t e m s from Jouve et al. [8] and corresponding values of ~ and D c h o s e n for the fits of fig. 3

M(0) DyCo, ~, HoCo, 2~ ErCo, 4~

2.36 ~*, 3.38 ~ . 2.13 p,,

Experimental T,.,.,(K) 228 190 94

TJK) 930 900 940

Theoretical ~(K) D(K) 35.8 23.7 15.1

64.4 12.3 ÷ 2.9

181 If the easy axes are distributed randomly according to a probability distribution P(O) then (J~)~,,r is given by

(J:)*o.r = 2

P (0) sin 0 dO(J:(O)),o,r

(4)

for an asperomagnet. The natural p a r a m e t e r of this theory is a = D/vJ which specifies the degree of competition between the alignment effect of the exchange interaction (~ > 0 ) and the disalignment effect of the RMA. The zero temperature spontaneous magnetization M(0) and the Curie t e m p e r a t u r e T c were calculated and was shown to be reduced c o m p a r e d to their crystalline values. No experimental fits were possible since no asperomagnetic systems were in existence when the H P Z theory appeared. H o w e v e r , Arrese-Boggiano et al. [9] point out that the H P Z model qualitatively describes the behaviour of DyNi34. There are two initial drawbacks to this model vis-a-vis experimental data for a m o r p h o u s metals. First of all, for asperomagnetic metals a nearest neighbour Heisenberg interaction is not suitable since the rare-earths interact by an R K K Y interaction via the s - p band. Secondly, for sperimagnetic metals a second magnetic species has to be taken into account. In the particular case of DyCo34, Coey et al. [7] have shown that the Dy m o m e n t s interact weakly with the cobalt d electrons and that the direct D y - D y interaction via the R K K Y can be neglected. Since it is our primary purpose to describe the magnetic properties of a m o r p h o u s DyCo34, we propose a model in which the magnetic m o m e n t s on the Co atoms are treated in terms of the H u b b a r d model for itinerant delectron ferromagnetism and the m o m e n t s on the Dy atoms interact with the polarized d-band by a local exchange interaction. Furthermore, the Dy m o m e n t s are subject to a random magnetic uniaxial anisotropy with a hamiltonian given by eq. (2) and with D < 0 for Dy since the anisotropy forces the Dy m o m e n t s to lie in the " b a s a l " plane. This model was previously used in the absence of anisotropy effects by Bloch and Lemaire [10] and by BIoch et al. [11] for rare earth-transition metal intermetallic compounds and by Z u c k e r m a n n [12] in conjunction with a magneto-crystalline field for PrCo> The hamiltonian described a b o v e can now be

written H,,,, = H},~.d + Half + Hani~,

(5)

where H~nd describes the kinetic energy and H u b b a r d correlations in the d-band of the a m o r p h o u s alloy and is given by Hb,,,0 : Z tii'Ci+Cr~ + I Z ni ~ni 4, i.i'

(6)

i

where C~+~creates a d-band conduction electron of spin o- at every site i, ni,~ = Ci+C~, t,, is a band transfer matrix, and I is the strength of the H u b b a r d interaction. In order to make the problem tractable, t,, and I are assumed to be quantities which describe an averaged d-band in the a m o r p h o u s alloy. Also the degeneracy of the d-electron is suppressed. The exchange interaction between the rare-earth m o m e n t s and the d-electrons is given by: Hdf ~----- °~ Z JJ" ~], i

(7)

where j is a rare-earth site and dri is the Pauli spin matrix for the electrons in the d-band. The coupling constant J is negative for sperimagnets ( J < 0 ) . The random anisotropy is described by H~ni~ given by eq. (4). We begin by analyzing Hto, in the MFA. In the case of DyCo34 we have no difficulty in establishing an external direction z for the molecular field. This is simply taken as the direction of the spontaneous magnetization of the Co m o m e n t s as saturated by a small external magnetic field. Using the same assumption as H P Z , the M F A hamiltonian is given by HMF = H d +

Hf,

(8)

where Ho describes the magnetic behaviour of the d-electrons and is given by Hd = Z

(~k -- ½0" A)Ck+Ck~,

(9)

k

ek is the Fourier transform of t,, and A is the molecular field of the d-electrons given by A = t(o-~) + 2 ( g / x ) U ~ )

(lO)

in which x is the cobalt concentration (according to the formula DyCox) and where o-z = n T - n ~ . Hf describes the rare-earths and is

182 given by H,

- Z (a./# + DJ~j), i

II 11

w h e r e a is the m o l e c u l a r field at the r a l c - c a r t h sites and is given by a .... j(cr:).

<12)

From eq. (9) the self-consistent equalion for the polarization (G:) of the d-band is given by the set of e q u a t i o n s (or:) - ~_, [f(+~ - ~k) - f / e ~ + '~)1,

113,,~

k

w h e r e f ( e ) is the F e r m i fuYiclion and N~ the total n u m b e r of d - e l e c t r o n s inside the d - b a n d . In f a c l , for a given A, (13b) d e t e r m i n e s the p o s i t i o n of the F e r m i - l e v e l and then (13a) d e t e r m i n e ' , (or:). Eqs. (10)-(13) c h a r a c t e r i z e the s y s t e m and

permit numerical computation. T h e s o l u t i o n s (or+) and (J:) are c a l c u l a t e d n u m e r i c a l l y using a p r o c e d u r e similar to H P Z for fixed v a l u e s o f I, ~/ and D. T h e d - h a n d was a s s u m e d to be f r e e - e l e c t r o n - l i k e for the purp o s e s of c a l c u l a t i o n s and this r e q u i r e s a n o t h e r p a r a m e t e r w h i c h is N ( 0 h the e l e c t r o n d e n s i t y of s t a t e s at the F e r m i level, alld NIt thc hand occupancy. In practice in addition to N . three independent dimensionless parameters i ::: I N ( 0 ) , .,,g = . ¢ N ( 0 ) and / ) - D N ( O ) can be used. First the h a m i l t o n i a n H , - i s d i a g o n a l i z e d for a given v a l u e o f {or,) and for a fixed angle 0. T h e d i a g o n a l i z a t i o n p r o c e d u r e yields a set of e n e r g y levels f r o m w h i c h the t h e r m a l a v e r a g e C.l:(Ol), : can be c a l c u l a t e d . Use of eq. (41 with P(O) 1 g i v e s a v a l u e for (J:),.,. T h e n eq. (10) with the c h o s e n v a l u e f o r {o-:), a l l o w s us to c a l c u l a t e ,:X. N e x t the use of eqs. (13a) and (13b) gives a new v a l u e for (o':). T h e p r o c e s s is then r e p e a t e d until the p r o g r a m c o n v e r g e s . W e then a r r i v e al selfc o n s i s t e n t v a l u e s f o r (G:) and (J:) and the e n e r g y e i g e n v a l u e s E,(O) o f He in eq. (I I) for e a c h 0. W e are then able to e v a l u a t e the a b s o l u t e magn e t i z a t i o n M at e a c h t e m p e r a t u r e T per D y C o , given, in B o h r m a g n e t o n s , by M = .,c(o-:) + g(J:),

< 14)

w h e r e g is the l+and6 f a c t o r (g --~ for Dy). F i r s t , the a b o v e n u m e r i c a l p r o c e d u r e was

used to p e r f o r m a fit of the s p o n t a n e o u s Inag+ n e t i z a t i o n M I T ) for Dy('o~4 for all t e m p e r a t u r e 7 ' < 7 as r e p o r t e d b,, ( ' ~ e y el al. [ 7 ! l ' h c results are s h o w n in fig. 2. T h e p a r a m e l e r r e s u l t i n g f r o m the fit are: N(0) : 3.44 , 10 + K ] 35.8K, I) 64.4K I D / J : t.S), [ 320(I K ( c o r r e s p o n d i n g to /- : I.I), %~++ I z,, ami v :: 3.4. T h e fit was o b t a i n e d as f o l l o w s . I)/J ( c o h e r e n t l y with /-) was c h o s e n 1o fix M(0* at it-, e x p e r i m e n t a l v a l u e s (i.e. 2.~ #~). T h e absoltltC ~alue o f ~/ w a s c h o s e n to fix the c o m p e n s a t i o n t e m p e r a t u r e at the e x p e r i m e n t a l w d u e of 230 K F i n a l l y , I - IN(O) was c h o s e n to fit the o ; e r a l l s h a p e and the m a x i m t u n of the M ( T ) c u r v e a , well as the o r d e r of m a g n i t u d e of 7'. Fig. 2 s h o w s that the fit is very g o o d e x c e p t for / w h i c h is t h e o r e t i c a l l y e s t i m a t e d to be 9 8 0 K . with the p a r a m e t e r s used here. while it is ex+ p e r i m e n t a l l y e s t i m a t e d to be of o r d e r 900 K b} ( ' o c y et al. 171. This is h a r d l } s u r p r i s i n g since the M F A u s u a l l y g i v e s wtttles for "/" w h i c h arc too high. This is f u r t h e r reflected in the value of N(0) w h i c h is u n r e a l i s t i c a l l y htrge sincc a frec e l e c t r o n a p p r o x i m a t i o n was c h o s e n . F i n a l l y , it is i m p o r t a n t to note that we ha',c not i n c l u d e d the t e m p e r a t u r e d e p e n d e n c e of the a n i s o t r o p y p a r a m e t e r since we feel it will onl} have a small effect on the M ( T ) c u r v e in the M F A . This is b e c a u s e the m a g n e t i z a t i o n c u r v e is initially d o m i n a t e d by the s t r e n g t h of the c o u p l i n g c o n s t a n t . H o w e v e r , it is c e r t a i n that such a t e m p e r a t u r e d e p e n d e n c e lllUSt t-~C inc l u d e d in a n y c a l c u l a t i o n i n v o l v i n g t e m p e r a t u r e e f f e c t s in the spin d y n a m i c s .

\ 20it

t0H

t,00

8tl/!

T(K) Fig. 2. Theoreliu'al fit of the spontaneous magm_'lization ~d I)yCo,~. The experimental points I~ } are those of Coo'. ,.g el. 17] and the theoretical cur\c has been drawn wilh: N(I}) 3.44× 10 aK ', j 3s.g K. I) f,4.4 K. / ~200K (i.e. 1 I.Ii+ N+, I.S and ~ ~4.

183

F u r t h e r m o r e , we p r e s e n t in fig. 3 different fits for the e x p e r i m e n t a l M ( T ) c u r v e s of J o u v e et al. [8] for R C % with R = Dy, H o a n d Er. T h e fit for DyCo3.44 is the s a m e as in fig. 2 a n d in the two o t h e r cases we have not c h a n g e d the p a r a m e t e r s NB, N(0) and f related with the t r a n s i t i o n e l e m e n t while we have t a k e n different J a n d D p a r a m e t e r s . T h e values of ~ a n d D u s e d for these fits are r e p o r t e d in table I. Let us notice that the v a l u e s c h o s e n for J do not vary greatly with R w h i c h s e e m s a priori r e a s o n a b l e . Also, while the o r d e r of m a g n i t u d e of the absolute v a l u e s of D are a l w a y s v e r y large if we c o m p a r e with c r y s t a l l i n e pure r a r e - e a r t h s , the relative o r d e r s of m a g n i t u d e a n d p a r t i c u l a r l y the signs of the a n i s o t r o p y are r e s p e c t e d [13]. M(J~ B )

M

(pB) I

/

TITc

Fig. 4. Theoretical curves giving the magnetization in Bohr magneton units as a function of TIT, by taking the same values of D, I and x as in fig. 2 but by taking different values of NB and D[~. Curve (I) corresponds to NB -- 2.2, D/~ = 6; (I1) corresponds to NB = 1.5, DlJ= 1.8; and (Ill) corresponds to N, = 0.6, D/~ = 1.2 [the values of N(0) are deduced from N. as for a parabolic band]. The absolute values of T~ are given in table II.

Table I1 Experimental values of M(0) and T, for DyFe~,, DyCo~4 and DyNi3, and theoretical values of T, Nc,,(O),where N,,,,(O) is the density of states of the d-band in the case of DyCos,, obtained for the calculations of fig 4

200

400

........ 600

800

l

IO00 T~)

Fig. 3. Theoretical fits of the spontaneous magnetization of DyCos 44, HoC032~, ErCo3~. The experimental points © for Dy, • for Ho, A for Er are those of Jouve et al. [8]. The parameters N(0), I and NB are the same as in fig. 2 while the parameters /9, ~ and x are reported in table I.

In o r d e r to c o m p a r e the fit for DyC03.4 with the m a g n e t i z a t i o n c u r v e s of DyNi3. 4 a n d DyFe3.4 we h a v e p e r f o r m e d f u r t h e r n u m e r i c a l calc u l a t i o n s u s i n g the s a m e a b s o l u t e v a l u e s of D a n d I. T h e v a l u e of ~ was c h o s e n to fix M(0) in each alloy w h e r e a s NB was c h o s e n to fix the o c c u p a n c y of the d - b a n d for Fe, Co and Ni, r e s p e c t i v e l y . Fig. 4 s h o w s the results for M ( T ) in all three Dy alloys as a f u n c t i o n of T/T~. T h e r e s u l t a n t t h e o r e t i c a l v a l u e s of T~ are given in table II w h e r e these are c o m p a r e d with the e x p e r i m e n t a l v a l u e s of A r r e s e - B o g g i a n o et al. [9]. T h e r e s u l t s for M ( T ) in the case of DyNi3.4 and DyC03. 4 are in good q u a n t i t a t i v e a g r e e m e n t

Fe(1) Co(H) Ni(lII)

M(0)

TLexp

7~ x N,,,(0)

1.3 ttB 2.3 # . 5/x,

350 K -900 K 47 K

0.9 (I.34 0.0075

relative to o n e a n o t h e r and both have the correct q u a n t i t a t i v e c u r v e s h a p e s f o u n d by exp e r i m e n t [fig. (lb)]. T h e s a m e is true for the Curie t e m p e r a t u r e s of these alloys (see table II). T h e a b s e n c e of a c o m p e n s a t i o n p o i n t in the case of Ni is due to the lack of s p o n t a n e o u s d - m a g n e t ism in this case (there r e m a i n s o n l y a small polarization), w h i c h also e x p l a i n s the low T~ value. In the case of DyFe3.4 o n l y the c u r v e shape is q u a l i t a t i v e l y r e p r o d u c e d , b u t both the o r d e r of m a g n i t u d e of M ( T ) a n d T~ are too large w h e n c o m p a r e d with e x p e r i m e n t s . This is p r o b a b l y due to the l o c a l i z a t i o n of the Fe m o m e n t s w h i c h are b a d l y d e s c r i b e d by an i t i n e r a n t model. W e t h e r e f o r e feel that the a b o v e model gives a good d e s c r i p t i o n of the m a g n e t i c b e h a v i o r of DyC03.4, HOC03.26, ERC03.44 a n d DyNi3. 4 alloys.

1~4 However, the details of the d-f exchange interaction and the nature of the amorphous dhand need further examinations. We arc a! present using the model presented in this coinmunication to predict the nature of lhe dynamical spin excitations in these systems. One of us (M.J.Z.) wishes to thank Dr. J.M.D. Coey for fruitful and inspiring discussions. References III J..l. Rhyne, S.J. Pickarl and H.H. Alpcrm, Phi,,. Roy. l,ett. 29 (1972) 1562. J.J. Rhync..I.M. £chcllcng and N.('. Koon, Phys. Rev. 13 111(1974) 4672. 12] R. Harris. M. Plischkc and M.J. Zuckcrm:mn. l'hv,,. Re,,'. l,ett. 31 (1973) 16(t. 131 R.W. ('ochrane, R. Harris and M. Plischkc. ,I. N~m(ry,d. Solids lg (1974) 239..

14] I).E. Polk, J. Non-(Trysl. Solids I1 (1973) 381. ('.H. Bennett, J. Appl. Phys. 43 (1~)72) 2727. [5] W.P. ()'l.eary, .I. Phys. F. ':, (1975) I,. 17 "~ [~] [,{.W. ('ochranc, R. Harris, I). Zobin ~lnd M.J. Zm.'kcr mann, J. Phy'~. [: 5 (1975) 763. 171 J.M.D. ('ocy, J. ('happerl, J.P. Rebouillat and t ' N Wang. Phys. Re,,. l.etl. 36 (197~) 1061. Igl H. Jouve, .I.P. Rebouillat and R. Meyer, to be pub lished. [~)l R. Arrese-l{oggiano. .I. ('happerl, J.M.I). ('oc'~, \ l,ienard and .l.P. Rebouillat, 1o be published in J. Phi, ~, (Vrance) ( P r o c ( ' o n f ~m M~sshauer speclroscop~, ('orfu. 1976). Ill)} 1). [~loch and F. l,emaire. Phy',. Rev. B 2 11970) 264S. {ll] D. Bloch. D.M. Edwards, M. Shimizu and J. Voiron,,~. Phys. F ~, (1975) 1217. [12l M.J. Zuckcrmann, .1. Phys. F 4 (1974) IgO(L [1~) R. ,,\leonard, P. []oulrou and D. Bloch, .I.P.(~.S. 30 (1969) 2277