Re-entrant ferromagnetism in a two-dimensional mixed spin Ising model with random nearest-neighbour interactions

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Z. Phys. B - Condensed Matter 77, 333-338 (1989) Condensed

Zeitschrift M a t t e r ffJr Physik B

9 Springer-Verlag 1989

Re-entrant ferromagnetism in a two-dimensional mixed

spin Ising model with random nearest-neighbour interactions*

N. Benayad 1, **, A. Kliimper 1, j. Zittartz 1, and A. Benyonssef 2

1 Institut fiir Theoretische Physik, Universitfit zu K61n, Federal Republic of Germany 2 Facult6 des Sciences, Laboratoire de Magn6tisme, Rabat, Morocco

Received June 7, 1989

Using the finite cluster approximation we study a mixed spin model with random interactions on a ferrimagnetic square lattice (with spins o-= 1/2 and S = 1). The interactions are assumed to be independent random variables with distribution, P ( K u ) = p 6 ( K u - K ) + ( 1 - p ) 6(Ku--aK), where K > 0 and N < 1. In a certain range of negative values of e the phase diagrams exhibit re-entrant behaviour. This indicates that re-entrance seems to be a characteristic feature of systems in which both frustration and disorder are present.

I. Introduction

In recent years approximation schemes have been de- veloped which represent remarkable improvements of the traditional mean field theory (MFA) as applied to spin systems. In particular, we refer to the so-called finite cluster approximation (FCA) which is based on the use of some extensions of Callen's identity [1].

This scheme has been successfully applied to a variety of spin-l/2 Ising problems [2-5] as well as the spin- one Ising model [6]. Its application to ferromagnetism in a two dimensional square lattice Ising model with random nearest-neighbour interactions [3] exhibits re-entrant behaviour for a certain range of interactions as conjectured in [7].

The purpose of this paper is to study the phase diagram of a mixed spin Ising system with random nearest-neighbour interactions and to show that this model exhibits the same re-entrant behaviour as the monoatomic spin-l/2 Ising model. Therefore one may conclude that the existence of a re-entrant ferromag- netic phase is a general property of systems in which frustration and disorder are both present.

* Supported by the agreement of cooperation between the D F G and the C N R - M a r o c

** O n leave from Facult6 des Sciences I, Univ6rsit6 H a s s a n II, Casablanca, Morocco; a n d Laboratoire de Magn6tisme, Universit+

de Rabat, Morocco

Mixed spin Ising models have less translational symmetry than their single spin counterparts and are well adapted to study a certain type of ferrimagnetism

[8].

We consider the ferrimagnetic square lattice. The reduced Hamiltonian of our model takes the form:

(1)

(ij>

The underlying lattice is composed of two interpene- trating sublattices, one occupied by spins with spin moment o-= + 1/2 while the other one is occupied by spins with spin moment S = 0, _+ 1. The summation in (1) extends over all pairs of nearest-neighbour sites in the lattice. The reduced nearest-neighbour interactions K u are assumed to be independent random variables with distribution

P (Kq) = p 5 (K u - K) + (1 -- p) 5 (Kq-- eK), (2) where K > 0 , Ic~] < 1, and 0=<p< 1.

The pure case, in which all Kq have the fixed value K, has been studied by renormalization group techniques in two dimensions [9], by high-temperature series expansions [103, and in the free-fermion approximation [11]. The MnNi(EDTA)-6H20 com- plex has been shown to be an example of a mixed-spin system [12].

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II. Finite cluster approximation a. Pure system

We consider a particular spin 0-o(So) and denote by (ao)~((So)~) the mean value of ao(So) while all other spins 0-i and Si(i :# 0) are kept fixed. We have

Tr ao e -p~

O" 0

<0"o>r

Tr e -pg

O" 0

= - tanh 2 1 T (S1+$2+$3+$4 ' (3)

and

Tr So e-r

So

(So)~ Tr e - t~n

So

s i n h [K(0"1 + 0"2 + 0"3 -~- 0-4)]

89 + cosh [K (0"1 + a2 + 0"3 + 0"4)] (4)

The ai (i = 1, 2, 3, 4) are the neighbours of So (Fig. 1 a), the S~ (i = 1, 2, 3, 4) are the neighbours of 0-0 (Fig. 1 b).

Therefore the magnetization per site m~ and m~ for 0-- and S-sublattices, respectively, satisfy the following exact relations:

/1 h [ K + $ 3 + $4)]), (5)

m ~ = k ~ t a n 1~-($1 +$2

m ~ = / s i n h [ K ( a l + a 2 + a 3 + 0 - 4 ) ] \,

\5 + cosh [ K (0-1 + 0-2 + 0"3 + 0"4)]/ (6) where ( . . . ) denotes the full thermodynamic average.

The mean-field approximation (MFA) to (5) and (6) is equivalent to using the simple probability distribution

PMFA ({0-i}) = H (~ ( 0 - i - ms) (7)

i

Si r

Ko2

$2 = vo Ko3

Kol

-'2 $4 o'2= Ko2 ,

Ko~ So"

K0a

K0I

Ko~ ^o',l

~3 ~

( a ) ~ b l

Fig. 1. The central spin ao, or So respectively, with its nearest neighbours and the connecting couplings

on the right hand side. It neglects all spin correlations including self-correlations and therefore leads to the very bad result (0-/2) - - m s 2 rather than the exact result ( a ~ ) = 88 The finite cluster approximation (FCA) [5]

has been designed to treat all spin self-correlations exactly while still neglecting correlations between different spins. For spin-5, a = _+ 89 the appropriate distribution is

~ c . ({ 0-,}) = 1-I ((89 + mJ 6 (0-~ - 1) + ( 89 mJ 6 (0-, + 5)).

i

(8) When calculating the average on the right hand side of (6), it is, however, easier to observe that for 0-i = + 5 any function f(0-) can be written as the linear superposition

f(0-) = f i +f2 0- (9a)

with appropriate coefficients fl, 2. Using (8) and (9 a) the average in (6) then leads to

m~=4Al m~+4 Ba m~ 3 00)

with coefficients 1

A , = ~ 8 9 sinh(2K) 2sinh(K) "(

+ cosh(2K) q 5 ~ ) - f ' sinh (2 K) 2 sinh (K)

B1 = 89 5 +cosh(K)"

Similarly we observe that for spin Si = 0, +_ 1 any function g(S) can be written as the linear superposition

g(S)=g 1+g2S+g3 $2. (9b)

Applying this to all four spins $1, S 2, S 3, S 4 the right hand side of (3) is decomposed as

2 (ao) ~ = A z (S i + $2 + $3 + $4) + B2 [Sa (S 2 + S~ + S])

+ S 2 ( S ~ + S 2 "~- $ 4 ) @ S 3 (81 2 2 -}- S 2 -~- S 2) -~- $4(8 2 @ S 2 2

+ s~)] + c~ [sl(s~ s~ + ~ ~ s2 s , + s3 s~) + s2 (s~ s~

_]_$2S2+ 2 2 ..t_$3(S1S2..~_S1S4_ ~ 2 2 $3 $ 4 ) 2 2 2 2 $ 2 8 4 )

2 2 2 2 2 2

-q- S4 (SI S2 + S1S3.-I- S2 S3)]-~-D 2 IS1 $22S32S42 ..~_82S2S2S24 ~- 9 1 S 2 S 3 S 4 - ~ S 1 S 2 S 3 S 4 ] 2 2 2 2 2 2 -.t.- E2 [S1S2 S3 -.1- $1S2 S4--1- S1S3 S 4--~- $2 S3 S4]

+ F 2 1 8 1 S 2 S 3 S 2 + S 1 S 2 5 3 S 4 ~- S 1 S 2 S 3 S 4 2 2

"[-$2S2S3S4]. (11)

The coefficients are listed in Appendix A. N o w we average (11) over all spin configurations and neglect

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correlations between different spins in the spirit of the FCA. This leads to

m~=2(A2 + 3B2x + 3CzxZ-t-D2x3) ms-F 2(E 2

+ F2 x) m 3 (12)

with the notation: (S)=m~, ( a ) = m , , and ( S 2) = x . In order to evaluate m,, we also have to calculate the quantity x within the frame of the FCA. Similar to (5) and (6) we have the exact relation

/ cosh[K(al+0-2+a3+0"4)] \

X= 1 \ : + cosh [-K(~I +0"2 +0"3 +0"4-)]/ (13) Using the probability distribution (8), we then obtain

2 4

x = A3 +6 B3m~ + C3m,r ⁽¹⁴⁾

with coefficients

A 1 ( c o s h ( 2 K ) cosh(K) } 3 = ~ ~ 89 cosh(2K) + 4 89 cos~(K) f- 2 ,

1 cosh(2K) 1 B 3 = 2 8 9 3'

cosh(2K) cosh(K)

C 3 = 2 8 9 8 8 9

By combining the Eqs. (10), (12), and (14), we obtain an equation for m, of the form:

m~=am~+bma~ + ... (15)

within this approximation. As usual the condition a

= 1 determines the critical temperature of a second order transition. In the present case we have the condition

1 = 8A, (A2 + 3B2A3 + 3 C2A ~ +DzA3). (16) This gives K [ a = 1.298 which is to be compared with the mean-field result (obtained by using (7)) and results by other methods (see Table 1).

Table 1. Transition temperatures K~- ~ for different approximation methods

MFA RG ~ H.T.S.E b FCA

1.633 0.728 0.975 __+ 0.007 1.298

Renormalization group calculation [9]

b High-temperature series expansion [10]

b. Random system

The extension of the approximation method to random mixed spin Ising models is straightforward. The mean values (0"O)c, (So)~, and ($2)~, when all other spins 0"~, Si (i + 0) and all interactions K o have fixed values, are given by

(ao)~ = 89 tanh [89 (K o I $1 -[- Ko2 $2 + Ko3 $3

+ Ko4 $4)], (17)

sinh (Ko 10-1 + Koa 0-2 -~- Ko 3 03 "~- Ko4 0"4)

( S o ) c = 1

+ cosh (Ko 1 ~ -~- K020-2 .qt_ Ko 30- 3 + Ko 4 0-4)

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cosh (Ko i a 1 + Ko 2 0-2 + Ko 3 0-3 + No4 ~ ($2>~ - 89 + cosh (K o i O'1 + K02 0"2 -~- Ko 3 a3 + Kor 0-4)

(19) where the interactions K o are shown in Fig. 1. The two magnetizations m, and m, are now given by m~ = <89 tanh [89 i S 1 + Ko2 $2 + Ko3 $3 + Ko 4 $4)] >,

(20) / sinh(Kolffl+Ko2a2+Ko3a3+Ko4ff4) \

= \ + c sh ( K ) / '

while the parameter x satisfies the equation

(21)

/ c~ \

X = \-21- ~ 0 ~ ff i ~ 2 0 2 b2 ~ K033 ~ ~ 4 ) / "

(22) The angular brackets denote the thermal average and the bar denotes the average over the disorder of bond interactions with the distribution (2). The thermal average of (20), (21), (22) essentially leads to (10), (12), (14) with coefficients A, B, C . . . . which are functions of the four couplings Kol , Ko2, Ko3, Ko4 appearing in (17)(19). Averaging these coefficients with respect to the bond disorder by using the distribution (2) leads to the final set of equations

m, = 441 m, + 4 B 1 m~, (23)

m~ = 2(A z + 3Bzx + 3 C2 X2 +/)2 X3) ms+ 2(Ez

+ F2 x) m 3 , (24)

x = A3 + 6 B 3 m~ + C a m~. (25)

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1 . . . 1 1 I

I

K - - 1

1.2988 1.2

1.0

! ^{ct ~. .5}

.4

( _~ ,2

0 ,2 .4 .6 .8

P

Fig. 2. T h e p h a s e d i a g r a m s of the m i x e d spin s y s t e m s with r a n d o m couplings in p - - K 1 plane for different values of c~

1.0

The averaged coefficients are lengthy expressions and are listed in Appendix B. Then using the Eqs. (23), (25) in (24) we obtain an equation for m~ of the form

me = c/rnr + b-m~ + . . . (26)

where explicitly we have

a(K,p,c~)=SAl(Az + 3B2A3 + 3C2712 + D2~-3), b(K, p, e) = 144 A 1 (B 2 B 3 + 2 C2 B3 A3 + D2 B3 A~)

+ S B~ (A2 + 3Be A3 q- 3 C2.d ~ -t-/)2/~3) + 128 .,4~ (E 2 + F2 A3). (27) To obtain the phase diagram we proceed as in the previous section. Equation (26) is similar to (15) ex- cept that now the coefficients are not only functions of the temperature K - 1, but also depend on the pa- rameters p and c~ of the distribution (2). Phase diagrams in the p - K - 1 plane for different values of c~ are shown in Fig. 2. All the transition lines are second order lines since the quantity m 2 - - ( 1 - ~)/b is always positive in the vicinity of these lines. F o r non-

frustrated models, e > 0, there is always just one phase transition at a finite critical temperature T~(p,e)>0 from the disordered phase to the ordered ferromag- netic phase at low temperature. At e = 0 we have the bond diluted model. At zero temperature this model exhibits a transition at the percolation threshold p*--0.413 which may be compared with the exact value of 0.5 [13]. F o r frustrated models, ~ < 0 , and for certain ranges of c~, we have re-entrant behaviour.

There appear different thresholds as solutions of the equations K~- 1 (p, e) = 0 (~ < 0). The dependence on e- ranges is certainly determined by the approximation.

Our general conclusion is that re-entrant behaviour seems to be a characteristic feature when both disorder and frustration are present in spin systems.

As we have seen the relative complexity of the model already leads to fairly complicated equations with lengthy expressions within the FCA-approximation.

However, the purpose of this paper was not to obtain precise numerical results, but rather to explore the qualitative behaviour of the model in its phase diagram.

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A p p e n d i x A

The coefficients in formula (11) are explicitly:

A 2 = tanh (K),

Bz =1 {tanh (K)- 2 tanh (K)},

C2 = 14 {tanh (~K-) + 5 tanh ( K ) - 4 tank(K)}, D 2 =~ {tanh (2 K) -- 6 tanh (~-) + 14 tanh(K) 1

--14 tank (K)},

E2 = 1 {tank @~K-)- 3 tanh (K)},

F2=l{tanh(2K)-2tanh(~)-2tanh(K)

+6 tanh (K)}.

A p p e n d i x B

For abbreviation we write th(x) instead of tanh(x) and define new functions

f(x)- 1-+ 2coshx ' g(x)=tanh(x)'f(x)"

With q-l-p, K l-K, and K2-c~K the coefficients appearing in (23)-(25) are given by

/[1 =~ {p4 [g(2K0 + 2g(K0] 1

+2p3q[ 2g[3Kl+K2\~ 2 )+g~2[3K1--Kz\)

+ 6p2 q2 [g(K~ + ^{K2) +}g(KO + ^g(Kz)3

+3g(_K1 +K2 \] )]+q [-g(ZK2) + 2g(K2)] ~, 4 a

B1 =p4 [g(2K0_ 2g(K1) ] + 2p3 q

9 [2g[3Kl+K2'~ [3K1-K2\

+ 6p2 q2 [g(K1 + Kz)- g(KO- g (Kz)] + 2p q3

g (3 K22 K 1)] + q4 Fg(2 K2)- 2 g(K2)], A2 = P th (~--~-*) + q th (~--5-2),

2 = ~ P th(K1)+ 2pqth

+ q2 th(K2)-- 2.42},

- 1 3

C2=~{p [th(~-)+th(K~-)]+P2q

9 [ 3 " th~ [2Kl+K2\ ~ - ) + 2 th(~ _ . ~-2 )

+th (2K12K2)]+pq2 [3 th 2-2/ 2 )

+2 th ( ~ ) + th (2K22K1)]

+ q3 [th ( ~ ) + th (~2)]- 4-42 - 8/~2}, 1~ l f 4 2=g~P [th(2K0+2 th(K0]

+2p3q[2th( 3K1+K2\2 )+tll~-[3K1-2---K2\) +3 th (KI ~Kz)]

+ 6p2 q2 [th (K 1 + K2) + th(K2) + th(K0]

+2pq3[2th(Ka2-3K2)+th(-3Kz2 K1)

+ 3 th (.KI 2Kz)I

+ q4 [th (2 K2) + 2 th (K2) ] -- 8 42 -- 24/32 -- 24 C2},

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/~2 = 1 {P3 [th ( ~ ) - th (~--Jk)]

+p2q[3th(3Kx2K2 ) -2th(~-)-th(2K12K2)]

+pq213th(K~22K2)--2th(~-)--th(2K22Ke)]

+ q3 [th (3 2Kz)- th ( ~ 2 ) ] - 2 2 1 - 4 / 3 2 - 2 C2},

1 4

/~ =g{p [ t h ( 2 K 1 ) - 2 th(K1)] + 2 p 3 q

- 3tn~'" [K1 +~ K2-)]+6p2q2[th(K1 + K 2 ) - th(K1) - t h ( K 2 ) ]

+2pq312th(K~?Kz)-th(3Kz2K~ ) -3th(K12Ke)]

+ q4 [th(2 K 2 ) - 2 th (K2) ] -- 8 E2},

1 4

23 = g { P [ - f ( 2 K 0 + 4 f ( / 0 + 3f(0)]

+4p3q[f(-3K~fK2-)+f(3KI2K2 ) -JrBf(gtig2)-l-gf(ge2K2)]

+ 2p2 q2 [3 f(K~ + K2) + 6 f ( K 1 ) + 6 f ( K 2 )

+ 3 f(K 1 - K2) + ^{6 f(0)]}

+4pq3[f(K~23K2)+3f(K12K2 )

-I- q4 [ f ( 2 K 2 ) + 4 f (K2) + 3 f (0)]},

1 ( 4 3

+ 2p2 q2 [ 3 f (Ka + K2)--f(KI -- K2) - ^{2 f (0)]}

+ q4 [f(ZK2)_f(O)]t '

C3 = 2 @ 4 [ f ( 2 K 0 - 4 f ( K 0 + 3 f (0)]

+ 2p2 q2 [ 3 f ( K t + K2 ) _ 6 f ( K 0 - - 6 f (K2) + 3f(K,

+ q4 i f (2K2) _ 4f(K2) + 3 f (0)] }.

R e f e r e n c e s

1. Callen, H.B.: Phys. Lett. 4, 161 (1963)

2. Benayad, N., Benyoussef, A., Boccara, N.: J. Phys. C: Solid State Phys. 20, 2053 (1987)

4. Benyoussef, A., Boccara, N.: J. Phys. 44, 1143 (1983) 5. Boccara, N.: Phys. Lett. 94A, 185 (1983)

7. Wolff, W.F., Zittartz, J.: Z. Phys. B - Condensed Matter 60, 185 (1985)

8. N6el, L.: Ann. Phys. (Paris) 3, 137 (1948)

9. Schofield, S.L., Bowers, R.G.: J. Phys. A: Math. Gen. 13, 3697 (1980)

10. Yousif, B.Y., Bowers, R.G.: J. Phys. A: Math. Gen. 17, 3389 (1984)

11. Kun-Fa Tang: J. Phys. A: Math. Gen. 21, L1097 (1988) 12. Drillon, M., Coronado, E., Beltran, D., Georges, R.: J. Chem.

Phys. 79, 449 (1983)

13. Sykes, M.F., Essam, J.W.: Phys. Rev. 133A, 310 (1964)

N. Benayad, A. Kliimper, J. Zittartz Institut fiir Theoretische Physik Universit/it zu K61n

Z/ilpicher Strasse 77 D-5000 K61n 41

Federal Republic of Germany A. Benyoussef

Facult6 des Sciences Laboratoire de Magn6tism B.P. 1014

Rabat Morocco