Magnetic interactions in RECu2Si2 compounds (RE = Tb-Tm)

316

Journal of Magnetism and Magnetic Materials 67 (1987) 316-322 North-Holland. Amsterdam

MAGNETIC INTERACTIONS IN RECu2Si 2 C O M P O U N D S (RE = T b - T m )

A. B U D K O W S K I , J. L E C I E J E W I C Z 1 and A. S Z Y T U L A Institute of Pf~t,sics, Jagellonian Uni~,ersio', 30-059 Cracow, Poland 1 Institute of Nuclear Uhemistr3' and TeehnoloRv, 03-195 Warsau, Poland Received 4 February 1986

The RKKY theory and the molecular field model including CEF effects (by Noakes and Shenoy) are applied for the explanation of magnetic interactions in the RECu2Si : family (RE = T b - T m ) . The negative sign of the obtained exchange integral J~f may point to the presence of interband mixing. The predicted magnetic anisotropy and enhancement of ordering temperatures T x - 5 above the de Gennes values are consistent with experimental ones only on the basis of the B{~ model with parameter A ° as for CdCu2Si 2, They are not consistent if one uses the full CEF Hamiltonian with 5 A~" parameters as for TmCu2Si 2. A short discussion about the role of magnetic interactions other than the simple RKKY exchange is also given.

1. Introduction

2. Results

The RECu2Si x compounds crystallize in a simple structure of ThCr2Si 2 type, which is tetragonal with the I 4 / m m m space group [1]. In recent years an extensive work is carried out on the magnetic properties of these systems. Some of them such as EuCu~Si~ or YbCu2Si 2 exhibit the valence fluctuation phenomenon [2] while CeCu2Si: is a heavy fermion system [3]. The magnetometric measurement indicates the antiferromagnetic ordering at low temperatures [4,5]. The magnetic ordering temperatures T N for RE atoms with L 4= 0 are larger than one expected on the basis of the T N value for GdCuzSi 2 (only spin interaction). Schlabitz et al. [4] claim that a slight deviation of the magnetic susceptibility from the Curie Weiss law showed by RECu2Si 2 ( R E = T b - T m ) below 100 K is apparently due to the Crystal Electric Field (CEF). The magnetic unit cell for heavy Rare Earth is described by a wavevector k = (1/2, 0, 1/2). The magnetic order can be described as ferromagnetic (101) layers stacked antiferromagnetically [6]. In the present paper we report the results of numerical calculations performed to explain the behaviour of the parameters of magnetic interactions in R E C u z S i : compounds (RE = Tb Tm).

Our considerations can be divided into two steps. Firstly, we tried to estimate the consistent value of the exchange integral J~f using the R K K Y model and different experimental data [7,8] and hence to estimate the magnetic transition temperatures T,,c in the absence of CEF. Next we looked how addition of CEF interactions [9] changes the Ta(' •

The R K K Y model contains a number of sin> plifying assumptions (e.g. spherical Fermi surface and isotropic J,f ), so it is easy to operate. Thanks to this predictions for the behaviour of different quantities upon the same model [10-15] have been given. Among them the following are of our interest: The additional Knight shift K seen by a nonmagnetic atom as a result of the conduction-electron's spin polarisation P [12,13,15]: K = Ko 1 +

3"~Z(g- l ) 2 x f gjt~2

j, r

x y" F ( 2 k F ( R - R , ) ) I ,

(1)

t 4- 0

where K 0 alone represents a term due to Pauli

0304-8853/87/$03.50 ~ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

A. Budkowski et aL / Magnetic interactions in RECu :Si 2 compounds

paramagnetism only, [2 is the atomic volume, Z is the number of conduction-electrons per RE atom and the sum of the oscillating function F ( x ) = (x cos x - sin x ) / x 4 is taken for a nonmagnetic site R over RE-sites Ri; - The effective magnetic moment #eff is derived from the susceptibility X f of bare-ion clothed by polarization P [14,15]: bteff g j C J ( J + 1)

1 + J s f U ( E F ) gJ --

],

3~rNm * 2 " ~ J s f ( g J - 1)2j( J + 1),

(3)

3"rrZ2 j ~ ( g j - 1) 2

EvkBg-------Z xJ(J +

1) E

F(2kvR,)"

(4)

i=~0

We estimated the J~f values using the eqs. (1)-(4). The R K K Y E F sums taken over all REsites inside the spheres of radius k given in tables 1, 3, 4 were convergent. The k F values were calculated considering that only RE-ions contribute effectively to the conduction-electron density. Such an assumption has been used in previous studies [16] giving well-predicted magnetic structures. Also the recently obtained Fourier map of electron density for RECoESi 2 [17] shows that the inner part Si-Co-Si of element layers stacking ( R E - S i - C o - S i - R E ) along the c axis forms an independent sandwich with a saturated electric charge. The dependence of the RKKY sum for TbCuzSi2 on the k F value is shown in fig. 1. In

1.7 kF [~"~]

1.3 I

I

3 2 1 nt"

0

U. I,'4 3 0

gJ

where m* is the effective electronic mass, N is the number of RE-ions per unit volume and E v is the Fermi energy; - The paramagnetic Curie temperature 0p obtained using the molecular field model [10,12,13] has the following form:

Op

I

(2)

where N ( E v ) is the conduction-electron Density Of States (DOS) at the Fermi surface per atom, for two spin directions; - Ospd is the contribution to the resistivity due to conduction-electron scattering in a spin-disordered paramagnetic system [10,11]: Pspd - -

0.9

317

TbCu25i 2

-5 -8 I

2.4

I

I

5.9 11.5 20.1 n . 1 0 2 [ ~-33

Fig. 1. The dependence of the RKKY E F sum for TbCu2Si 2 on the Fermi wavevector k F and on the electronic density n (under the assumption that Z = 3 -~ k v = 1.041 A - ] and n = 3.8x 1 0 - 2 ~ 3).

table 1 are given the resulting Jsf calculated using eq. (1) and the Knight shift data [7] from experiments performed on 29Si in the 4 (e) site for Er and Tm compounds (K 0 = 0.08%). The electronic specific heat coefficient y and DOS at the Fermi surface for different REs are nearly constant [14]. Thus knowing two Y values (ya(La)= 3 [18] and 72(Lu) = 2.8 m J / m o l K 2 [19]) and using eq. (2) we can evaluate two series of Jsf (see table 2). By solving eqs. (3) and (4) simultaneously (using Pspd [8] and 0p-data) one can obtain not only Jsf but also the m* values [10,13]. The resulting m * ' s (table 3 give us information rather about the order of this quantity than about its exact value. The mean value of m* ( = 0.49m) is comparable with m * = 0.5m obtained from EF(La ) = 8.28 eV [18] (with parabolic conduction band). Calculating Jsf Table 1 Values of the exchange integral Jsf calculated on the basis of the Knight shift data [7] RE

Kf (%) [7]

~ i * o F ( 2 k F ( R - Ri)) /~ = 6 nm

-- Jsf (eV)

Er Tm

-0.61 -0.25

0.0186 0.0190

0.036 0.028

A. Budkowski et al. / Magnetic interactions in RECu 2Si 2 compounds

318

transferred hyperfine field can be explained by d-electron's spin polarisation [20] or by interband mixing [21]. However the negative Jsf also from the effective magnetic moment excludes the first explanation (Jsd should also be positive). The second explanation is reasonable because of the high conduction-electron concentration in the RECu2Si 2 system (the electronic specific heat coefficient ~, is comparable with those for metals). The interband mixing model [21] predicts also different values of the conduction-electron's spin polarisation been by an atom and by its nucleus at the same place. In order to predict T~c values in the absence of CEF we have chosen from the J,~r values the value 0.046 eV, treating it as an approximative one. For this purpose the F F sum in eq. (4) should be changed into F ~ F ( 2 k F R i ) c o s ( k . R ~ ) , where k is the wavevector describing the magnetic order [6]. In table 4 the Tac values are compared with those from the de Gennes scaling Tdo (for TN(Gd ) = 12.5 K).

Table 2 The J~f values o b t a i n e d on the basis of the e x p e r i m e n t a l eff effective m o m e n t s #exp RE Tb Dy Ho Er Tm

/~e~p (I~B)

-1)×10~

(eV)

(eV)

9.28 9.33 10.3 10.3 9.4 7.39

-4.53 -4.01 - 3.10 -2.65 - 1.98 - 2.27

0.12 0.11 0.108 0.115 0.103 0.139

0.128 0.119 0.116 0.123 0.11 0.149

we made an assumption that E v is constant for all REs (EF(CE) differs from EF(La) by 3% only [18]. The mean Jsf = 0.08 eV (strict unit is eV × the volume of one formula unit). It should be pointed out that the validity of the above procedure depends on the quality of the 0~pa-data because it is difficult to extract them correctly for materials ordered antiferromagnetically at 0 K [10]. There are a few other necessary comments. The magnitudes of J~e calculated in different ways are not equal to one another. The origins of these differences are as follows. The Jsf from magnetometric data reflects the total electron's spin polarisation P, while this from the Knight shift is only due to s-electron's polarisation seen at the site of the nucleus and that from the (O,~pd, 0p) method may differ from the real one because of the above mentioned superzone effects in AF materials [10]. The sign of the obtained effectiveexchange integral Jsr is negative and is incompatible with the sign of the diagonal-exchange integral which is positive naturally. The negative value of the effective J~f from the Knight shift or from the

The CEF Hamiltonian RECu2Si: has the form:

for

the

RE-site

HcE v = BoO 0 + B ° O °4 + B4404 + Bg06° + B6402,

(5) where (6)

B[" = AT'( rt)4fOl(1 - or)

and O~ is a Stevens operator [22], 01 ( 0 2 = O~j) is a Stevens factor [22,23], ( r / ) 4 f i s the lth moment of the 4f radial function [24], o t is a shielding factor [25,26] and A 7' is a CEF parameter [22].

Table 3 The values of Jsr and effective mass rn * c a l c u l a t e d from 0p- a n d Pspd-data [8] RE

Tb Dy Ho Er Tm M e a n values

in

Ospd/(gJ -- 1)z J ( J + 1) ( ~ cm) [8]

0p

[ 2 , . 0 F ( 2 k FRi) × 103

(K)

k = 5 nm

0.423 0.5 0.52 0.639 1.915!

- 12 - 4 - 3 - 3 - 2

1.25 1.25 1.30 1.31 1.40

m */m

I "lsf I (eV)

0.23 0.55 0.51 0.36 0.79 0.49

0.087 0.061 0.065 0.086 0.101 0.08

A. Budkowski et al. / Magnetic interactions in RECu2Si 2 compounds Table 4 The predicted values of m a g n e t i c t r a n s i t i o n t e m p e r a t u r e s in the absence of C E F (Jsf = 0.046 eV) given by: m o d i f i e d e q . ( 4 ) - Z~c and the de G e n n e s scaling - Td6 RE

Y~i.oF(2kFRi)cos(k'g,)×lO 3 = 4 nm

Tb Dy Ho Er Tm

-3.12 - 3.10 - 3.09 - 3.07 - 3.05

Tac (K)

TdG (K)

8.2 5.5 3.5 2.0 0.9

8.3 5.6 3.6 2.0 0.9

The full HCEF is very often restricted to the B ° term (so called the B ° model). This procedure is valid if B°O° is dominant in HCEF. For RECu2Si 2 the situation is as follows. On the one hand it is forbidden because B2/B 0 40 = 10 [19,27-29] in como o parison to B2/B,~ = 103 [30,31] when this model is valid. On the other hand it seems that quadrupole interactions are responsible for the lattice constants anomaly A(c/a) [4] and even for the TN enhancement above the Td~ value. Namely the ratio (TN -- Td~)/,~(c/a) is constant for all heavy REs [41. The way of finding these enhanced TN values, if CEF splittings are significant was given by Noakes and Shenoy [9]. When the 2 axis of magnetic ordering is also the ~ axis defined by CEF, TN fulfils the relation [9]:

TN= 2SEx (g , -- 1)2(jz2(TN))CEF ,

(7)

where (JzZ/TN/)CEF is the expectation value of Jz2 under influence of HCEv alone (or the B ° term) at the temperature TN, and the exchange parameter JEX can be calculated knowing Jsf, but because Jsf is only an informative value we calculated it from Ty(Gd) = 12.5 K. The T y value can be numerically evaluated using eq. (7). The information about CEF in the RECu2Si 2 family was given by: M~Sssbauer spectroscopy [27,32-34], inelastic neutron scattering [28,29] and Schottky specific heat measurements [19]. These measurements have not yet covered all the members of the family, hence we had to scale B~-parameters from these of the known CEF levels

319

structure to those with unknown ones. The scaling procedure using eq. (6) assumes constant values of AT'-parameters across the series. The available data for systems with unstable 4d shell (CeCu2Si 2 [28] or YbCu2Si 2 [29,34] are not useful for such transformation to compounds with stable 4f shell. For example CEF levels in TmCuzSi 2 obtained from the experiment [19,27] are not consistent with those found by scaling B~ 's from YbCu 2 8 i 2 [29] and only the ground states are similar. Also the picture of CEF in "normal" RECu 2Si 2 systems given by experiments performed on them only is not so clear. The M~Sssbauer measurement of the Electric Field Gradient (EFG) in GdCu 2Si 2 [32] gives the negative value of the second-order CEF parameter A ° ( = - 307.3 K/a.u.), which was believed to be constant and used to determine B ° in DyCuzSi 2 [33] during discussing its spectrum. But the value of A ° corresponding to B ° from the full set of 5 B~'-parameters for TmCuzSi 2 [19,27] has a positive value ( = +136.4 K/a.u.). These two M~Sssbauer experiments, for GdCuzSi 2 [32] and TmCuzSi 2 [27] determine unquestionably signs of EGFs and hence A ° 's, which appeared to differ not only in value but also in sign. This cells the scaling procedure of the B~ 's for their lowest second-order parameter in question. However Bonville [34] noticed for TmCu2Si 2 that the temperature dependence of E F G on the Tm nucleus can be equivalently described by the simpler CEF expressed in terms of only one parameter B °, instead of the above mentioned CEF with 5 parameters. And this B ° is negative which is consistent with negative A ° from GdCuzSi 2. We believe that the scaling procedure for second-order parameters with A ° as for GdCuzSi 2 is a valid one in the sense of the simpler CEFs describing the real CEFs for different RECuzSi z systems equivalently (to some extension) and not in the sense of simple cut of the full CEF Hamiltonian to the lowest dominant term. This remark refers also to the B ° model. Because RE-ions are located at sites with a high symmetry ( 4 / m m m ) the ~ axis with maximal E F G is the fourfold c axis. The CEF splittings produce single-ion magnetic anisotropies at low temperatures. In the B ° model the ~ (c) axis is magnetically easy and the basal plane magnetically hard if

320

A. B u d k o w s k i et al. / Magnetic interactions in R E C u , S i e compounds

B ° < 0. Otherwise (if B2° > 0) the c axis is hard and basal plane easy. This correlation [30,35] was presented as a valid one, e.g. for REFe2Si 2 [36], Er(Cu,Co)2Ge 2 [37] or DyMe2Si 2 [33]. From comparison of magnetostriction- and a(c/a)-data [4] for T m and Er one can deduce that the magnetic moment ~ is parallel to the c axis. The same can be concluded from known C E F levels scheme for T m [19]. The sign of B ° depends on the signs of A ° and c9. Because for these elements oLj > 0 so on the base of the B~) model one can anticipate that a correct value of A ° should be negative. This is consistent with what was stated above. The by the B ° model predicted values and orientations of ~ as well as values of TN'S are compared with experimental data in table 5, T N's are also shown in fig. 2. We tried also to predict C E F levels schemes and other correlated properties using full CEF Hamiltonian with sets of B~' 's scaled from the set for TmCuzSi 2 [19]. The results are listed in table 6. All the lowest levels are doublets or 2-singlet systems [27,40] and are well isolated from higher levels. The T N values for Er and T m were calculated directly for scaled HcEv's, whereas for Tb, Dy and Ho scaled HCEF'S were first transformed (acc. to ref. [22]) to the quantization axis lying in the basal plane. These T N's are also presented in fig. 2.

I

10~_ • ',,,\ ', \\

i \

,,

\ \, •

i

,\ <, ,,, ,, '

iil

\

i

5 " ~5,

GId

Ttb

Dy

HO

Er

Tm

RE

Fig. 2. Comparison of experimental (solid triangles) and calculated magnetic transition temperatures T N for the RECu2Si 2 family. Dashed smooth line represents the de Gennes rule. Broken lines are trends obtained on the basis of the molecular field model [9] including CEF effects. Broken solid line (open circles) indicates TN'S predicted by the B ° model with A2° as for GdCu2Si 2 [32]. Broken dashed line (open squares) represents calculations with full CEF Hamiltonian (5 A~' parameters are; acc. to ref. [27] - [] and acc. to ref. [19] - ~ ) .

We had anticipated that the use of the full

HcE v should give better results than the use of B ° term only, however calculations showed that the

Table 5 The properties of systems with H c ~ v = B°20 ° and A ° = - 307.3 K/a.u. [32] cD

RE

Tb Dy <0 Ho Er

B2° (K)

Ground state (J:)

Pred. T N (K)

g J Jr

0.8 (-0.4) J

0 (_+6) d

15.7

(9) d

0.501 (-0.25) 0.175 (-0.088)

_+1 / 2 (_+15/2) 0 (_+8)

1t.7

(10)

6.5

(10)

-0.199

_+15/2

5.0

9

>o

"

Exp. data ~xp (#B)

q~b

Dominant ground state ~

8.5[38,6]

23.4[38,61

(+4, +5) d

8.3 [39]

90 [39]

( _+ 13/2)

8.2 [38,6]

7.5 [38,6]

( _+7, _+8)

/z±c

#xl[c[4,191

_+9/2

P,11c[4,19]

+6

,~llc Tm

-0.789

+_6

2.4

7

a Predicted magnetic anisotropy. b q0 is the canting angle between /~ and a for/~ 3_ c. c (./:2) were extracted from experimental T N values, then dominant ground states were calculated under assumption about the isolation of the lowest CEF level. d Values in brackets are for the quantization axis lying in the basal plane, others for the c axis as a quantization axis.

321

A. Budkowski et al. / Magnetic interactions in RECueSi e compounds

scaling with 5 - p a r a m e t e r s set gives a d i s c r e p a n c y of p r e d i c t e d a n d e x p e r i m e n t a l l y o b s e r v e d d a t a in c o n t r a s t to the B ° model. T h e o b t a i n e d orientations of m a g n e t i c m o m e n t s are consistent with e x p e r i m e n t a l ones o n l y for starting T m C u 2 S i 2 a n d evaluated TN's are b e l o w the de G e n n e s scaling. N o r m a l l y while the B ° t e r m d o m i n a t e s in HCEF, for instance in RER_haB 4 [9,30,31], 5p a r a m e t e r s scaled C E F gives n o t a b e t t e r b u t also n o t a worse p r e d i c t i o n of m a g n e t i c t r a n s i t i o n temp e r a t u r e s t h a n the simple B ° model. T h e r e are 3 m a i n m e c h a n i s m s which can p r o duce changes in exchange s - f interactions a n d h e n c e variations of T N a c r o s s the series others t h a n the de G e n n e s scaling. Firstly, there are changes of lattice p a r a m e t e r s when going from one R E to a n o t h e r which are negligible in our case. Secondly, it is C E F , c o n s i d e r e d above. A n d at last b u t in some cases n o t at least, there are c o n t r i b u t i o n s to exchange interactions others t h a n the simple R K K Y model. As a r e p r e s e n t a t i o n of 3rd m e c h a n i s m we t o o k into a c c o u n t the indirect exchange via s p i n - o r b i t c o u p l e d states [41]. U s i n g a p r o g r a m consistent with one m e n t i o n e d in ref. [41] we have checked that an a d d i t i o n of effects p r e d i c t e d b y Levy [41] to C E F effects does n o t i m p r o v e the consistency of theoretical a n d experi-

m e n t a l T N values. H o w e v e r it seems that, e.g. i n t e r b a n d m i x i n g (strong a n i s o t r o p y of effective exchange p a r a m e t e r [21]) m a y b e r e s p o n s i b l e for such b e h a v i o u r of T y ' s . The d i p o l e - d i p o l e intera c t i o n evaluated b y s u m m i n g over RE-sites is of the o r d e r of 3 - 4 K, so it can also have its own c o n t r i b u t i o n to the m e n t i o n e d p h e n o m e n o n .

3. Summary I n this p a p e r we have tried to systematize the available d a t a characterizing the m a g n e t i c interactions in the R E C u 2 S i 2 family ( R E = T b - T m ) . It was shown that the m a g n e t i c p r o p e r t i e s could be generally e x p l a i n e d b y the R K K Y m o d e l a n d the effects of C E F . O n the b a s e of the R K K Y theory o n l y an i n f o r m a t i v e value of the exchange integral Jsr is possible to obtain. The negative sign of Jsf seems to p o i n t at the presence of the i n t e r b a n d mixing. T h e r o u g h d e t e r m i n a t i o n of the T N values, as well as the precise one of m a g n e t i c anis o t r o p y across the series, is possible using the m o l e c u l a r field a p p r o a c h of N o a k e s a n d S h e n o y [9] a n d the B ° m o d e l with p a r a m e t e r s scaled from G d C u 2 S i 2 [32]. T h e same is i m p o s s i b l e if one uses the full C E F H a m i l t o n i a n with 5 B~'-parameters

Table 6 Properties predicted by the full HCEF with B~'-set from ref. [19] RE

Dominant ground state 10) (Jz) a

1st excited state 11) (Jz) a

1st and 2nd excited level (K)

Predicted TN (K)

Others predicted

Tb

6, - 6 S

6, - 6 S

0.1,95

1.9

Dy

+11/2 D

+13/2 D

105, 134

4.5

Ho

4, - 4(8, -- 8) S + 1/2 D ( -i-7/2) 6, - 6 S (2, - 2 )

4, - 4 (8, - 8) S + 13/2 D + 5/2) 6, - 6 S (2, - 2 )

4, 174

3.4

22, 78

0.04

7.2, 91.5 [19]

1.9

a = 5.94 b t~IIz c /% = 7.29 d /zx = 0.1 d a = 4.85 tLIIz /~z = 0.55 /.tx = 4.75 a = 5.17 [19] ~. = 3.2 [19] ~x = 0 [191

Er Tm

S - singlet; D - doublet. a In brackets are significant secondaries, if any. b a = (0 [Jz [1) is an electronic matrix element [40]. d /% = gJ#B( + 0 [Jz I0 + ), /~x = gJ#B( + 0 IJx I0 - ), I0 - ) is time conjugation to 10+ ). c Because ( + 0 ' l J ± 10' - ) = 0 where I0 ' ) consist of I0) and admixed by Weiss field I1).

322

A. Budkowski et al. / Magnetic interactions in RECu 2Si : compounds"

scaled from TmCu2Si 2 [19,27]. Further experiments determining full sets of B f " s for others REs are needed to explain the behaviour of CEF across the series. Addition the effects from exchange interactions via spin-orbit coupled states [41], to those described by the B ° model does not help in the explanation of TN behaviour for different REs.

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