Physica 120B (1983) 193-196 North-Hol land Publishing Company

MAGNETI C EXCITATIONS IN TWO-DIMENSIONAL ANTIFERROMAGNETS Rb2CocNil-cF4

H. I K E D A , Y. SOMEYA*, Y. E N D O H * * , Y. NODA** t and F. S H I B A T A Department of Physics, Ochanomizu University, Bunkyo-ku, Tokyo 112, Japan **Department of Physics, Tohoku University, Sendai 980, Japan.

Results of an investigation of the magnet ic excitations in randomly mixed ant i ferromagnets Rb2CocNil oF4 by means of neutron scattering are presented. Two bands of excitations are observed over a whole Brillouin zone. In Rb2Coo.sNio.sF4, a simple Ising cluster model using pure crystal exchange and anisotropy constants is found to fail in predicting the measured zone-boundary excitations. A considerable reduction of the exchange parameters in the mixed system yields satisfactory agreement between experimental and calculated spin-wave dispersion relations. A discussion is also given on the calculation of concentrat ion dependence of the N6el temperature .

1. Introduction

During the last decade, excitations in the ran- dom magnetic systems have received a con- siderable amount of attention [1]. As neutron scattering provides invaluable information on excitations in these systems throughout the Bril- louin zone, many experimental studies were focussed on the determinat ion of spin-wave dis- persion relations in certain insulating magnetic systems, especially the transition-metal fluoride compounds such as KCocMnl-cF3 [2], Mn~Col_cF2 [2], KMncNil_cF3 [3], Rb2Mn0.sNi0.sF4 [4,5], RbzCo0.sMn0.sF4 [6] etc. In these com- pounds, two well-defined bands of excitations with large dispersion compared with the resolu- tion width are observed throughout the Brillouin zone. To date, it is believed that the response function at the zone boundary should exhibit structure originating f rom the multiple Ising cluster modes. In the two-dimensional mixed systems with the so-called K2NiF4-structure, zone-boundary magnon energies, in general, split into ten Ising energies which are determined by the exchange constants J~gA, J~CB and J ~ and

spin values SA and SB where A and B are different kinds of magnetic species. Thus, the two bands of excitations observed result f rom a degeneracy of the Ising energies of each mag- netic atom. It should be noted that the cal- culations using essentially the same exchange constants as in the pure crystal f rom the A - A and B - B interactions and also a geometrical mean of JAB = (JAAJBB) m for the A - B inter- actions are in agreement with experiments.

On the other hand, in Rb2CocNil-cF4, whose magnetic excitations are repor ted below, a sim- ple Ising cluster calculation using pure-crystal parameters for the interactions suggests that the zone-boundary energy splits into ten energies and it will be possible to resolve the predicted fine structure. In the present paper, we report the results on Rb2Co0.sNi0.sF4 only; a full dis- cussion including the results on different con- centrations and a detailed analysis will be repor- ted elsewhere. The appropr ia te anisotropy and exchange constants for the pure crystals Rb2CoF4 and Rb2NiF4 are given in table 1 together with the lattice constants and N6el temperatures .

*Present address: Toshiba Research and Development Center , Saiwai-ku, Kawasaki 210, Japan.

t Pe rmanen t address: Sendai College of Radio Technology, Miyagi 939-31, Japan.

2. Experiments

The experiments were per formed on a triple axis spect rometer of Tohoku University (TUNS) installed at JRR-2, Tokai Establishment JAERI .

0378-4363/83/0000-0000/$03.00 © 1983 North-Hol land and Yamada Science Foundation

194 H. lkeda et al. / Magnetic excitations in Rb2C'o,.Ni~<F4

Tab le 1 La t t i ce cons tan ts , Ndel t empe ra tu r e s , and in te rac t ion cons tan t s in the pure and mixed crystals

Rb?CoF4 Rb2NiF4 RbeCo~ 5Ni,~ sF4

a (A) at 78 K 5.817 5.733 5.770 c (,~) at 78 K 13.615 13.550 13.734

TN (K) 101 98 115 S 1/2 I J (meV) jzz = 7.73 J~; = 4.2

jx~ = 4.25 jxx = 4.2

g#BHA 0.0 0.28

(meV) Re fe r ence [8] [4, 5, 91

JUo-Co - 7.3, J~:i si = 3.3, J~.~, Ni = 4.9

J~'~o-Co = 4.0, J~'~ Ni -- 3.3,

J[~, N i - 3.6 0.(I (Co), 0.28 (Ni)

Conventional cons tant -O techniques were used with the incoming energy being varied with the scattered neutron energy fixed at 14meV. The single crystals used in the present experiments were prepared by a Bridgmann method. Special care was taken to grow the homogeneously mixed crystals with large volume ( - 4 cm3). The critical behavior of these compounds was recently investigated by using elastic and quasi- elastic neutron scattering techniques and an ac calorimetric method [7]. The results do not reveal any smearing of the transitions. Typical scans for the zone-boundary response at (3.5, 0, 0) are shown in fig. 1 together with the Ising cluster excitations calculated by using the pure-crystal parameters (dotted lines). From fig. 1 it is evident that the system exhibits two well- defined peaks, which are quite inconsistent with the prediction based on the Ising cluster model. The resultant dispersion relations in the qa- direction are shown in fig. 2 together with the spin-wave dispersion relations for the pure materials. It is remarkable that the two bands of excitation with less dispersion and large excita- tion widths than in the pure materials have also been observed in many earlier works ment ioned above.

It is now of some interest to see if the main feature of the observed zone-boundary excita- tions and the overall dispersions shown in figs. 1 and 2 emerge from the simple model. It is evident from fig. 1 that the spectrum is far from

5O0

c

E 4oo 2 \ cn 300 LId

z 200

z

100

O = (3.5 0 0 ) Rb2Coo.sNi0.5F 4 I i I i ; i

~o °°

o ° o

20 30 40 50 ~ W ( m e V )

Fig. 1. E x p e r i m e n t a l z o n e - b o u n d a r y r e sponse at (3.5, (I, 0) (open circles). D o t t e d l ines are the Co ++ and Ni** Ising c lus te r m o d e s ca lcu la ted using the pure crystal pa rame te r s . Ver t i ca l bars are the resul ts of f i t t ing the obse rva t i ons using the modi f i ed in te rac t ion cons tan t s g iven in tab le I. P lease no te tha t the h i s t og ram shown by the do t t ed l ines is upside- down. The sol id l ine is on ly a gu ide to the eye.

the one expected from the pure crystal parameters . Instead we obtain bet ter fit to the experiments, using the following parameters:

J~fo-co = 7.3 meV,

J~:Zo_Ni = 4.9 m e V .

J~-Ni = 3.3 meV,

The calculated Ising cluster energies using the above parameters are shown by vertical lines in fig. 1. It is reasonable that the fitted value for

H. lkeda et al. / Magnetic excitations in Rb2CocNi>cF4 195

40

> O E30

B

20

] I I I

Rb 2 Co0.5 Ni0.5 F 4

10

• /

Rb2CoF 4 /

!

/

// Rb2NiF 4 /

/ /

[ I I 1 00 0.1 0.2 0.3 0.4 0.5

WAVE VECTOR qa

Fig. 2. Dispersion relations in the qa-direction for RbaCoF4, Rb2NiF4 and Rb2Coo.sNio.sF4. Solid lines are the results of spin-wave calculations, eq. (1), with the parameters given in table I.

J~2%-Ni satisfies an empi r i ca l r e l a t ion JCo-Ni= (Jco_CoJNi_Ni) I/z. T h e resul ts ind ica te tha t the in- t e rac t ion cons tan t s J~{,-Co and Jf~-N~ in RbzCo0.sNi0.sF4 dec rease to 0.94 and 0.79 t imes sma l l e r than those in pu re mate r i a l s , r e spec- t ively.

3. Discuss ion and conclus ion

W e cons ide r the overa l l d i spe r s ions in the mixed crystal . W e ex t end W a l k e r ' s mean-c rys t a l m o d e l [4] to sys tems with an an i so t rop ic exchange H a m i l t o n i a n and an a rb i t r a ry a tomic c oncen t r a t i on c. W e t ake a s t a n d a r d four - sub la t - t ice m o d e l and t ake a conf igura t iona l ave rage ove r all poss ib le local e n v i r o n m e n t s of a specific type of ion. T h e e q u a t i o n of m o t i o n based on a G r e e n - f u n c t i o n a l f o rma l i sm with a T y a b l i k o v

a p p r o x i m a t i o n is so lved to give the express ion

(.0 4 - - (H2A~ + H ~ - h2AA -- 2hABhBA -- h2B)to 2

+ ( ( H ~ . - 2 2 h AA)(HB~ -- h 2B) - 2hABhsAHA,,HB~

_1_ 2 2 9 2 2 h AB h BA - - , h AA h BB -- 2hAAhABhuAhBB) = 0 , (1)

where

HA,~ = --2Z(JXASA + ]SfBSB)- gAIXBHA

HBa = --2Z(]}~.SA + J}~BSB) - gBI.tBHA

hAA = 2ZJXASA3'~

hA~ = 2ZJ~SA3'k

hBA = 2Z]ffASBTk

h~B = 2ZY'ffBSB3'k

Jie o = JieoC o (P, Q = A , B and i = zz , x x )

Yk = z l .~, ei k .o (z = 4)

It shou ld be n o t e d tha t this express ion r educes to a resul t of W a l k e r ' s fou r - sub la t t i ce m o d e l in the l imit of c = 0.5 and J.~o = J~'}).

W e shou ld also no te tha t in f i t t ing the overa l l o b s e r v e d d i spe rs ion to the ca lcu la t ion as is d e p i c t e d in fig. 2 bo th b r a n c h e s are given with the same reduc t ion fac tors of 0.94 and 0.79 on J~',,-co and J~%Ni and the f ixed a n i so t ropy fields of gAP, BHA (Nil = 0.28 m e V and gBl~BHA (Co) = 0.(I.

W e next cons ide r the effect of the dev ia t ion of the exchange cons tan t s f rom those in pu re crystals . W e have ca lcu la t ed the NOel t e m p e r a t u r e and the sub la t t i ce ma gne t i z a t i on using the same G r e e n - func t iona l fo rma l i sm in the f r a m e w o r k of the mean- f ie ld a p p r o x i m a t i o n . T h e p a r a m e t e r s of the exchange cons tan t s necessa ry in the ca lcu la t ion have been f ixed as o b t a i n e d in the p resen t e x p e r i m e n t s (see tab le I). H e r e we discuss only the concen t r a t i on d e p e n d e n c e of the NOel t em- pe ra tu r e . Resu l t s a re s u m m a r i z e d in fig. 3, which shows the va r ia t ion of TN with respec t to the al loy concen t r a t i on c t o g e t h e r with the m e a s u r e d up- wa rd concave charac te r i s t i c [7]. A l t h o u g h the quan t i t a t ive a g r e e m e n t be tw e e n ca lcula t ion and e x p e r i m e n t is, as expec ted , not excel lent , one

196 H. l k e d a et al. / Magnetic excitations in Rb2CocNiFcF4

200

v I ,L

23 F--

150 LLI [3_

DJ t '--

z lOOd ~

R b 2 C o c N i l _ c F 4 I I I F I I I I

-" "o

] r F l [ [ l l ] 0.5 1.0

C

Fig. 3. Concentration dependence of the N6el temperature. The solid line (c - 0.5) is a result from the calculations based on a linear spin-wave theory with the exchange parameters of RbeCo05Ni0.sF4. Open circles are observed N6el temperatures [7].

important experimental fact, that TN deviates more than the geometrical mean, seems to be reproduced in this calculation. Consequently, the observed modification of the exchange constants seems to play an essential role in the concentration dependence of TN. As a matter of fact, we do not think that the thermodynamic quantities in these random mixtures obey well in the mean-field ap- proximation but we trust that the physical realiza- tion is well understood in this approximation.

In conclusion, inelastic neutron scattering measurements on two-dimensional mixed antifer- romagnet Rb2Co0sNi0 5F4 show two bands of exci- tation throughout the Brillouin zone. The excita- tion spectrum is well explained within the linear spin-wave theory with considerably modified exchange parameters from the pure materials. Thus, it is important to take a resulting change of the superexchange constants owing to the modification of the lattice parameters, if any, into consideration of the magnetic excitations in ran- domly mixed magnetic systems.

References

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