Spin glass in the site-diluted antiferromagnetic FCC

1457

Journal of Magnetism and Magnetic Materials 31-34 (1983) 1457-1458 SPIN GLASS S. F U J I K I ,

IN THE SITE-DILUTED

T. S U E N A G A

ANTIFERROMAGNETIC

FCC

a n d S. K A T S U R A

Department of Apphed Physics, Faculty of Engineering, Tohoku Umversit), Sendal 980, Japan

The diluted spin glass on the frustrated face-centered cubic lattice with the first and second nelghbour interactions are considered by the square cactus approximation Phase boundaries between the paramagnetlc and ordered (ferromagnetic, superantiferromagnettc and spin glass) phases are obtained The results qualitatively explain the experiments on EupSr I pS and MnpCd I pTe

M u c h a t t e n t i o n has been paid to diluted spin-glass materials such as E u p S r l _ p S [1] a n d M n p C d I eTe [2] At each lattice point, the magnetic atom A or the n o n - m a g n e t i c atom B is located with the p r o b a b l h t y p or l-p, respectively The ground state of the pure (nondiluted) fcc lattice with the first a n d second neighbour interactions J and J ' are k n o w n to be in the ferromagnetic state a n d the antlferromagnetlc states of the first, second and third kinds (F, AF1, AF2, AF3) F r o m fig. 1 we can see that A F 2 of the fcc lattice is represented by the superantiferromagnetlc (SAF) state in the square cactus lattice We consider the F phase, antiferromagnetlc (AF) phase, and S A F phase of the square (sq) lattice, as well as the F and A F 2 phases of the fcc lattice, in a general way These phases can be described as states m the square cactus lattice We show the presentations of these ordered phases on the square cluster m the square cactus lattice m fig 2 We denote the n u m b e r of square clusters which are connected at each vertex by z~. We take z~ = 2 (z, = 4, z 2 = 2) for the square lattice, and z~ = 6 (z 1 = 12, z 2 = 6) for the fcc lattice Here z j a n d z: denote the n u m b e r of the first a n d second nearest nelghbours, respectively. The AF1 and A F 3 states in the fcc lattice are not represented m the present scheme. We consider the site-diluted Ising model by the square cactus a p p r o x i m a t i o n (cf refs [3] [5]) We take

/

I/ -

,:- --

/ S,

I J-

i

~

J

I

j

(b) SqAF

(c) sqSAF fccAF2 Fig 2 The sublattice structure of several ordered states represented by the square cluster of the square cactus lattice

the square a n d vertex clusters m the system Let H} l) a n d H,(4) be the effecuve fields acting on the site ~ of the vertex and the square cluster, respectively, c o n t r i b u t e d from the outside of the respective cluster, a n d J~, be the exchange interaction between sites IX and v, and l} ])-= thflH(,), l}4)= thflH~4), t~, -= th(flJ~,J2) The density matrices of the clusters are then given by O~i) = A'l)(1 + l}')a,), ,4, Ijkl -_ - za(4) *

I~

(1) (1 + t . . % o . )

b t p = tJ , J k , ]~l

l-I

(1 + 114)%)

I . t - - t , J , ,{ , l

h ,tA ,Jl

(2) We require the reduciblhty condlUon between the denstty matrices of the square cluster and the vertex, trek,C3~)k, = 16~I),

(3)

where ~ denotes the normalized density matrix F r o m the reducibility condition we get a relation for the effective field up to the relevant order of l, -.I- ~ / ( 4 ) n -~,t'I .

(4)

where the bar means r a n d o m average over the atomic configuration The r a n d o m averaged quantities in eq (4) are given by (see ref. [5]) ~_

_v"

3(

I- P

.! /:,

i Fig 1 Face centered cubic (fcc) lattice and its representation by the square cactus tree lattice The antiferromagnetic state of the second kind (AF2) is represented by the square cactus tree lattice as the superantiferromagnetic (SAF) state 0304-8853/83/0000-0000/$03.00

I

l( i )n __/(4)n ..~ ~ ] ( 4 ) n -4- ~ / ( 4 ) n ' -" -~'h -T ,,.,

I

\' X / i \ ' ~ , ~ - I - - ) / ,

I

(a) sqF

© 1983 N o r t h - H o l l a n d

(l+t')(t+t3) )" l + 2t2(2t' + t'2) + t 4

+2pZ(l_p)

(

t(l+t')

+p(1-p)Zt",

1 + t2t '

~3(t'(l+t)+2t2(l+t'+t'2)) ,~=P l+t2(4t,+2t,2)+t4

n

(5)

S FuJlkl et a l /

1458

( t2+t' +2pZ(l--P)~l+t2t

ln '] +p(l-p~

,2

Spm y;la~s m the site-dduted anttferromagnett¢ f~ ~ pn

t ,

(6) 4O

/(I). t

~+

--~t

Z~)kt"

l,(4)n -__c , n _{_ (~,-

~/~ 1)~.,.

(8)

w h e r e t =- t h ( f l J / 2 ) , t' ==-t h ( , S J ' / 2 ) , c, --- thflH,, H, is the e x t e r n a l field a p p h e d o n t h e site t, a n d X ~s t h e e f f e c t w e field at t h e site ~ c o n t r i b u t e d f r o m o n e s q u a r e c l u s t e r o u t s i d e the p a r t i c i p a n t cluster. ~-,~ ts g w e n by t h e rep l a c e m e n t j ~ l m eq (5) G e n e r a h z e d s u s c e p t i b i l i t i e s are t h e n g~ven by

(/,T)"X

,,+ n , , , + ,, ,, 1 + c:'r,: ~ " a, ~:-r: ,,

1<1~,,

~"

2 0 3l

(v)

-,,,

1 - (z

-

1)[c/T,:±

c/'r ,A

2.0

10

I I

oc ~

o5 tap

p

10 o~ g

(3'5

p

J

lo

(b)

fig 3 The phase boundaries of the fcc lamce (a) J > 0 J ' < 1) t. The ferromagnetic (F) and the spin glass (G) states appear (b) J < O. J' < O. J ' / J = i~> The antfferromagnetlc state of the second kind (AF2) and the spin glass (G) state appear Broken curves are expected phase boundaries between the G and the other ordered states (F or AF2) Inserts are the phase diagrams obtained from the experiments on (a) EupSr I pS [1] and (b) MnpCd I pTe [2]

.l'/J = (IT, l )

(9) w h e r e the set (~,, % ~ , ¢/) refers to the k i n d of o r d e r e d s t a t e s as follows t h e set (1, 1, l, 1) with n = 1 refers to t h e F order, the set (1, - 1, 1, - 1) w~th n = 1 to t h e A F order, t h e set (1. l, - 1, - 1) with n = 1 to the S A F or A F 2 order, a n d the set ( ± 1, ± 1, ± 1, ± 1) with n = 2 to the spin-glass (G) order The phase boundaries between t h e p a r a m a g n e t t c p h a s e a n d the o r d e r e d p h a s e s are obtained from the condmon that the corresponding s u s c e p t ~ b l h t t e s diverge. F i g s 3a a n d b e x p l a i n q u a h t a tlvely t h e e x p e r i m e n t s o n E u e S r I _t,S [1] ( J > 0, J ' < 0), a n d o n M n p G d I pTe [2] ( J < 0, J ' < 0). r e s p e c t w e l y In t h e s e figures, b r o k e n h n e s m e a n e x p e c t e d p h a s e b o u n d a r i e s b e t w e e n t h e F or A F 2 p h a s e a n d t h e G p h a s e T h e A F 2 a n d G s t a t e s are o b t a i n e d in t h e fcc lattice w h e n J, J ' < 0. It is n o t e d t h a t t h e S A F s t a t e d o e s n o t a p p e a r in the s q u a r e lattice, b e c a u s e the ex-

p r e s s l o n of t h e s u s c e p t l b l h t y c o r r e s p o n d i n g to t h e S A F s t a t e ~s r e d u c e d to t h a t o f t h e o n e - d ~ m e n s ~ o n a l s y s t e m w h e n we p u t z~ = 2 m eq (9)

Reference,,

[1] H Maletta and W Felsch, Phys Rev B20 (1979) 1245 [2] R R Galazka. S Nagata and P H Keesom, Phy,, Rev B22 (1980) 3344 [3] S Katsura and I Nagahara. Z Phys B41 (1981) 349, 42 (1981) 1980 [4] S Katsura and S Fujlkl. J Phy,, C 13 (1980) 4711 [5] S Katsura S Fujikl, T Suenaga and I Nagahara. Phy,, Stat Sol (b) 111 (1982)83