Phase diagrams of the mixed spin Ising ferromagnet with a ferrimagnetic surface

(1)

ELSEVIER Journal of Magnetism and Magnetic Materials 168 (1997) 105 120

,•1•

^{Inml el}^~enoUsm

A 4 mBou, materials

Phase diagrams of the mixed spin Ising ferromagnet with a ferrimagnetic surface

N. Benayad*, A. D a k h a m a

Laboratoire de Physique Thkorique, Facult~ des Sciences A~n Chock, B.P. 5366, Maarif, Casablanca, Morocco

Received 21 June 1996; revised 4 October 1996

Abstract

The three-dimensional mixed spin-½ and spin-I Ising model with competing surface and bulk exchange interactions is studied. Within the framework of the mean-field theory and the finite cluster approximation, the phase diagrams are investigated and show qualitatively interesting features. The effects of the surface and the bulk crystal field interactions on the phase diagrams and in particular for surface ordering, are also examined. It is also shown that such system can exhibit a variety of phase transitions and multicritical points.

PACS: 05.50. + q; 75.10.Hk; 75.30.Pd; 75.50.Gg

Keywords: Mixed spins; Ising ferromagnet; Ferrimagnet; Competing interactions; Surface magnetism

1. Introduction

The problems related to the nature of phase transitions at surfaces of magnetic materials have generated considerable interest both theoretically and experimentally [1, 2]. One of the simplest three-dimensional semi-infinite models is the spin-½ simple cubic Ising ferromagnet with a free (1, 0, 0) surface. It has been extensively studied using a variety of approximations and mathematical techniques, such as mean-field theory [3, 4], cluster variation method [5], effective field theory [6], finite cluster approximation [7, 8], renormalization-group methods [8-12], Monte Carlo techniques [13, 14], and series expansions [15]. The above simple example includes two ferromagnetic exchange interactions Js and JB, where any two nearest- neighbour spins on the free surface interact with one another via the coupling constant Js which is not necessarily equal to the bulk one JB. It exhibits four different types of phase transitions associated with the surface. If the ratio R = JB/Js is greater than a critical value Rc, the surface orders at the bulk transition temperature. This is the ordinary phase transition. If R is less than Rc, the system may order on the surface at

* Coresponding author.

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106 N. Benayad, A. Dakhama / Journal o f Magnetism and Magnetic Materials 168 (1997) 105 120

a temperature higher than the bulk, followed by the ordering of the bulk at the bulk transition temperature.

These two successive transitions are called the surface and the extraordinary transitions, respectively. If R = Rc, the system orders at the bulk transition temperature, but in this case the critical exponents differ from those of the ordinary transition. This is the special phase transition. Using finite cluster approximation [,16] and real space renormalization group methods [17], similar transitions have been found for the three-dimensional semi-infinite spin-1 ferromagnetic Ising model with crystal field interaction. As a function of the ratio R of the bulk and surface interactions and the ratio D of the bulk and surface crystal-fields, it has been shown that the system exhibits a variety of phase transitions and multicritical points.

The simple cubic Ising ferromagnet with a ferromagnetic exchange interaction (JB > 0) in the bulk and an antiferromagnetic exchange interaction (Js < 0) between surface spins has been studied using mean-field approximation 1-15], renormalization group method [18], and Monte Carlo simulations (MC) [,19, 20]. In this case, the surface layer behaves roughly like an Ising antiferromagnet in a (temperature-dependent) field.

Below the bulk critical temperature, the bulk is ordered ferromagnetically for all Js. For surface exchange greater than some temperature dependent value of Js the surface is also in an ordered ferromagnetic state, but for more negative values of Js the surface is antiferromagnetic instead. As the temperature increases, the bulk disorders, but for strongly negative Js the surface remains ordered up to some higher temperature. The phase boundaries for the bulk and the surface transitions cross at a tetracritical point.

In recent years, some interest has developed in the study of the semi-infinite simple cubic consisting of two sublattices of spin-½ and spin-1 Ising alloy with a free (1, 0, 0) surface. It has been studied using effective field theory with correlations [,21, 22] and real-space renormalization group method [23]. Attention has been devoted to the case where the two exchange interactions, Js on the surface and JB in the bulk, are both positive (or negative). In this case, the only possible states of the surface and the bulk are paramagnetic and ferromagnetic (or ferrimagnetic) ordering. However, no attention has been devoted to the case of the Ising ferromagnet (JB > 0) with a ferrimagnetic surface exchange interaction (Js < 0). Such systems may exhibit very interesting behaviour.

The purpose of this work is to investigate the phase diagram of a semi-infinite simple cubic consisting of two interpenetrating sub-lattices, of a spin-½ and spin-1 alloy, with a ferrimagnetic surface coupling and a ferromagnetic bulk coupling. We also investigate the influence of bulk and surface single-ion crystal-field interactions on the phase diagram. For this purpose, we use the mean-field theory (MFT) and the finite cluster approximation (FCA). We note that FCA and effective field theory with correlations give the same results [24] by two different mathematical techniques.

Our presentation is as follows. In Section 2 we give a description of the model and the theoretical framework. In Section 3, we investigate the phase diagrams and we present our concluding remarks in Section 4.

2. Theoretical framework

Let us consider a semi-infinite simple cubic two-sublattices spin-½ and spin-1 mixed lsing system. The ferromagnetic exchange interaction between all nearest-neighbours is JB, except for spins on the (1, 0, 0) surface, they interact with one another with a ferrimagnetic exchange parameter - J s (Js > 0). The Hamiltonian of our system takes the form

H = Js + DsZ ( S f - J . Z + DBZ (S,) (1)

<i, j> j <k, I> I

where S takes the values + 1 and O, a can be + i or -½. The first and second summations are carried out over nearest-neighbour sites and single sites located on the free surface, respectively. The third and fourth

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N. Benayad, A. Dakhama / Journal of Magnetism and Magnetic Materials 168 (1997) 105-120 107

summations run over all pairs of remaining nearest-neighbour sites and single sites, respectively. Ds and Da, denote the surface and the bulk single-ion crystal field interactions, respectively.

In the framework of the FCA [25-29], we consider a particular spin ao (So) and denote by <ao>c (<So>c) the mean value of ao (So) while all other spins ai and Si (i # 0) are kept in a fixed state. Applying this for a spin aos (Sos) located on the surface and an other one ao. (So,) located in the nth layer (n ~> 1) of the bulk, we obtain

(aos)c = ~ tanh ~', S~s + S51 , (2)

i = l

(Sos>c = 2 sinh [ - Ks ~ L lals + KBasl] (3)

- - g 4

2 cosh [ s ~i= lais + KBa51] + exp (riDs)'

<ao.>c = ~ tanh Si, "3v S5(n+ 1) AV S6(n- I) , (4)

i = l

and

@=, i. + ~ ( . + , ~ + ~ 6 ( . - , ) ] (5)

(So.>c = 2 sinh [KB • a

2 cosh [KB ( ~ = l a i , + crst.+l) + cr6t.- 1))] + exp (flDB)'

where K , = riJ., Ks = riJs and /3 = ( k . T ) - ' . The {a,s, as,} and {Sis, Ssl} (i = 1, 2, 3, 4) are the nearest- neighbours of Sos and aos, respectively (Fig. la). The {a,., ast.+ ,), a6(.- 1)} and {S,,, Ss~.+ ~), S6t,-1)} are the nearest-neighbours of So. and ao,, respectively (Fig. lb). Therefore the surface magnetizations per site

#s = (ais>, ms = (Sis) and the magnetizations p. and m. of the nth layer (Fig. lc) satisfy the following exact relations:

Ps = ~ t a n h L Sis + $51 ,

i = l

( 2sinh [ - K s E L lO'/s "~ KBO'51 ] ;, ms = 2 cosh [ - - - - -- Ks Y',i= 1iris + KBtr51] + ~-4---- . . . . exp (riDs)/

# , = l ~ t a n h [ ? ( i = ~ S i , + S s ( . + ~ ) + S 6 ( . - 1 ) ) l ) ,

2 sinh [ K . (~4= 1¢7i n + aS(n + 1) -t- 0"6( n - 1))l

m. = 2 c o s h [KB (~',/%1ai, + as(,+n + a6(,+1))] + exp(riDB)/'

(6) (7)

(8)

(9) where < -.- ) denotes the full thermodynamic average.

2.1. The mean field approximation

It is a formidable task to average the right-hand side of Eqs. (6)-(9) over all configurations. The mean-field approximation (MFA) to Eqs. (6)-(9) is equivalent to the simple probability distributions:

PMFA({O'i}) = H (~(O'i -- ~)' (I0)

i

PMFA({Sj}) = ~I 6(Sj -- m), (11)

J

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1 0 8 N. Benayad, A. Dakhama / Journal o f Magnetism and Magnetic Materials 168 (1997) 105-120

~ , , ~ S2S

S3Ss~sJ ~Sls G3s: S°s-~ = Ols

~4s S

$51 ~sl

Se(n-1)

$5(n*1)

S2n -~Sln

(a)

(b)

Ge(n-1)

S 0 n ~ G2n

O'3n • ~ • 0"ln

~ J

(~5(n+1)

--I

JB JB, F Ja

l-

m

D

~S ^' mS

~1,1 m 1 JA 2 , m 2

i n

(c)

Fig. l. (a) Nearest neighbours of spins Sos and aos, located on the surface. (b) Nearest neighbours of spins So, and ao,, located on the nth layer. (c) Part of a two-dimensional cross section of a semi-infinite mixed Ising lattice with a (1, 0, 0) free surface. • and

• correspond to S and a-sublattice sites, respectively.

w h i c h n e g l e c t all s p i n c o r r e l a t i o n s i n c l u d i n g s e l f - c o r r e l a t i o n s . W e o b t a i n /~s = ½ t a n h [~( - 4 K s m s + KBml)],

2 s i n h [ - 4Ks/~s + Kepl]

(12)

(13)

m S

2 c o s h [ - 4Ks/zs + KB/al] + e x p (riDs)'

(5)

#. = ~ t a n h (4m. + m(. + 1) + m(._ 1)) , (14)

m. = 2 sinh [Ks(4#. + ~t(. + 1) + #(.- 1))] (15)

2 cosh [Ks(4~. + #(.+ 1) + #t.- 1))] + exp (flDs)'

with/~o = Ps, rno = ms. The bulk magnetizations #B and ms are determined by setting in Eqs. (14) and (15)

#.-1 =/~. = #. + 1 = #B and m._ 1 = m. = m. + 1 = ms. So, #s and ms are given by

Ps = ½ tanh [3KBmB], (16)

2 sinh [6KB#B] (17)

ms = 2 cosh [6Kskts] + exp (flOs)"

2.2. The finite cluster approximation

The finite cluster approximation (FCA) [25] has been designed to treat all spin self-correlations exactly while still neglecting correlations between different spins. For our mixed spin-½ and spin-1 Ising system described by (1), the appropriate distributions are

PvcA({a~}) = [ I ((½ + p)6(a, -- ½) + (½ -- p)b(a~ + ½)),

i

(18)

P F C A ( { S j } ) ---- 1--[ (~(m + x)6(Sg - 1) + (1 - x)6(Sj) + ~( - m + x)6(Sj + 1)), (19)

J

where x = ($2).

When calculating the average on the right-hand side of (6)-(9) it is, however, easier to observe that any function f(a) and 9(S) of a or S can be written as the linear superpositions

f(a) = f l + f2a, (20)

g(S) = gl + g2 S + g3 $ 2 , (21)

with appropriate coefficients fl,2 and gl,2,a. Applying this to all spins tri and Sj in Eqs. (2)-(5), their right-hand sides are decomposed as

d- 4--q

(aos)c = ~ ~ {S~, Ss}p,q'{Ap,q(ns) + Bp.q(Ks, K,)'$51 + Cp,q(Ks, Ks). S~1}, (22)

q=O p = 0 4

<Sos>c = ~ {as}q'{Dq(Ks, KB) + Eq(Ks, KB)" a51}, (23)

q = 0 4 4 - q

<ao.)c = E E { S.2, S.}p,q" {Fp,q(KB) + Gp,q(KB)'(Ss(.+ 1) + S6(n- 1))

q=O p=O

2 2

+ Hp,q(KB)'(S5(.+ n + $6(.- 1)) + Ip,q(Ks)'(Ss~.+ 1)' $6(.- 1))

+ Jp q(KB)" 2 _, (Ss(n+l).S6(n_l) + S5(n + 1)" S6tn - 1)) -~- 2 Kp,q(KB). ( S S ( . + 1) " S6(n - 1))}, 2 2 (24) 4

<So.)c = ~ {a.}q" {Lq(Ks) + Mq(KB)" (ast.+ 1)+ a6(.-1)) + Nq(Ks)'(as(.+ x)'a6(.-1))}, (25)

q=O

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110 N. Benayad, A. Dakhama / Journal o f Magnetism and Magnetic Materials 168 (1997) 1 0 5 - 1 2 0

where {Ss 2, Ss}p,q denote the superposition of all the terms containing p different factors of S}s and q different factors of Sj,s, with j # j ' . These factors are selected from the set {Sis, Szs, S3s, Sgs, St2s, z Szs, S3s, S,~s}. For 2 example, if p = 2 and q -- 1, then

2 2 . S2s.(S s + Sx s • + S 3 s . S 4 . s )

{S 2, S}2.1 = $1S.(S2S .$2S + $2 s.S4.S _[_ $23s $2s) + . 2 2

2 2 2 2 2 2

--F S3s" (S12s" $22s -[- S i s "$4. s Jr- S2s " $2s) -Jr- $4. s " ( S I s "S2s -[- S2s " $2s -[- $2s "$3s ) (26) {as}q denote the superposition of all the terms containing q different factors of als, selected from the set {alS, tr2s, a3s, ¢r4s}. {S 2, S.}p, o is the superposition of all the terms containing p different factors of S}. and q different factors of Sj,., selected from the set {$1., $2., $3., $4.., S~., 2 $2., $2., $42.}, w i t h j # j ' . {a.}q is the superposition of all the terms containing q different factors of tri., selected from {al., a2., a3., a4.}. The non-zero coefficients Ap,q, Bp,q . . . Mq and Nq are listed in the Appendix.

Using the probability distributions (18) and (19) to treat Eqs. (6)-(9), or averaging (22)-(25) over all spin configurations, and neglecting correlations between different spins, we obtain

4- 4.-q

[AS = E E f~4.t~4. - q " " P " ~ q " '-.q'.-p ~s ,,,s {Ap,q(Ks) + Bv.q(Ks, K . ) ml + Cp,q(Ks, KB) xl}, " " (27) q=0 p=0

4-

ms = ~ C*qjA~" {Dq(Ks, Ka) + Eq(Ks, K~)'JA1}, (28)

q=0 4- 4 - q

"x. "m." {Fp,q(KB) + Gp4(KB)'(m.+ t + m._ 1) q=0 p=0

4- Hp,q (KB)'(X.+ 1 + x . - 1) "]- Ip,q(KB)" mn+ i " m._ a

+ Jp,q (KB)" (x. + 1" m . _ 1 + m. + 1" x . _ 1) + Kp,q(KB)" x . + 1" x . _ 1 }, ( 2 9 ) 4.

r a n = E CqjA. {Lq(K.) + 4. q" Mq(K.).(#.+ 1 ~- ]An- 1) Jr- Nq(K.).JA.+ 1" j a n - 1}, (30)

q=0

with jAo = jAs, mo = ms, Xo = Xs and C~ = a!/b!(a - b)! is a combinatorial factor. The parameters Xs and x. are defined by Xs = (S~s) and x, = (S~,), respectively. They have to be evaluated in the spirit of the FCA. Similar to (7) and (9) we have the exact relations

2 cosh [ - Ks Z~= ltrls + KBa51] \ (31)

Xs = 2 c o s h - [ ~ k s ~ 4 - ~ s + Kacr51] + exp ( f l D s ) / '

/- -2-c°shCK"(E4:la"+--ffs'"+" +-- a*~7 t')] \ (32)

x. = \ 2 cosh I-K. (Z~= lai. + as~.+ 1)+ a6~,-1))] + exp ( f l D O / "

Expanding the right-hand side of (31) and (32), by using (20) and neglecting correlations between different spins, or using the probability distribution (18), we then obtain

4-

Xs = ~, C4qjA~'{Oq(Ks, K O + Pq(Ks, K B ) ' P l } , (33)

q=O 4.

x,, = Z C~, u.q" {Qq(Ka) + Rq(Ka)'(#.+l + JA.-1) + Sq(Ka)'JA.+I 'JA.-1} • (34)

q = 0

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N. Benayad, A. Dakhama / Journal o f Magnetism and Magnetic Materials 168 (1997) 105-120 111

The coefficients Oq,Pq,... and Sq are given in the Appendix. We note that the bulk magnetizations #a, ms, and the bulk p a r a m e t e r x~ are determined by setting #._ 1 = #, = #,+ ~ = #s, m,_ x = m, = m,+ t = ma and x,_ ~ = x. = x.+ 1 = xB in Eqs. (29), (30) and (34). Thus #B, ms and xB are solutions of the following equations:

# B ~ -

4. 4 - q

~, C~ ,.,pc'4-q" -~B~P " ',,s~q " {Fp,q(Ka) + 2Gp,q(Ks)" ms + 2Hp.q(Ka)" Xs

q = O p = 0

+ Ip.q(KB)" m 2 + 2Jp,q(Ks)'Xa "ms + Kp,q(Ka)" x2},

m B 4 q = O 2

4 q

Cq#s" {Lq(Ka) + 2Mq(K.)'MB + Nq(Ks)" #~},

(35) (36)

4 4 q

Cq#B" {Qq(KB) + 2Rq(KB)" #B + Sq(KB)" #~}. (37)

X B = q = O

3. Phase diagrams and discussions

In order to investigate the magnetic behaviour of the surface and each layer, we have to solve the M F A coupled Eqs. (12)-(15) and the coupled Eqs. (27)-(30), (33) and (34) obtained in the framework of FCA.

However, we are unable to solve them analytically. Even if we use a numerical method, they must be terminated at a certain layer. N o t e that as n goes to infinity, the magnetizations #,, m, and the p a r a m e t e r x, should a p p r o a c h the bulk values #B, ma and XB. F o r this purpose, let us assume that the magnetizations remain unaltered after the third layer, i.e.

]-13 ---~ [ 2 4 = . . . . # B , m3 = m4 = "- ⁼ ^{m s ,} ^{X 3 =} ^{X 4 =} " ' " = X B , (38)

which m a y be called the four-layer approximation. Analysing the Eqs. (27)-(30), (33)-(37), the surface and/or the bulk m a y exihibit a first order transition, when D~/JB and Ds/Js belong to narrow ranges. We limit this study to the case of second-order transitions, since we are m o r e interested in the region of the phase diagram where the bulk is already ordered.

3.1. Bulk and surface order-disorder transition temperatures

We are first concerned with the evaluations of the o r d e r - d i s o r d e r transition temperatures for the bulk and the surface ordering based on the four-layer approximation. In the framework of MFA, the bulk reduced critical temperature (KaC) - 1 is determined from the coupled Eqs. (16) and (17), as

18K 2

1 - 2 + exp (riDs)" (39)

The corresponding equation in F C A is obtained, from the coupled Eqs. (35)-(37), as

1 = 2 CpOo'(Gp,o+Jp,o'Qo)+4 ⁴ ^p CpQo (Fp, I + 2 H p , I " Q o + K p , I " Q 2) ³ ^{p .} ( 2 M o + 4 L 1 ) .

0 p = O

(40)

Thus, Ka c is a function of Ds. F o r DB = 0, the value of K c is 0.4736 to be c o m p a r e d with the mean-field result 0.4082. One should note that for Ds = -- ~ , the spin configurations in the bulk are completely dominated

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1 1 2 N. Benayad, A. Dakhama / Journal o f Magnetism and Magnetic Materials 168 (1997) 105-120

by S = + 1 on each site of the S-sublattice. Therefore the bulk corresponds to the three-dimensional spin-½ Ising model. Comparing with Monte Carlo critical value Ka c = 0.4433 [30], the FCA result 0.3942 improves the M F A value 0.3333.

In order to obtain the MFA temperature of surface ordering, we have to expand the right-hand side of Eqs. (12)-(15). Within the four-layer approximation (38), we obtain

#s = 4alms + bxma, (41)

ms = 4a2its + b2itl, (42)

It~ = aam s 4- 463ml 4- clm2, (43)

ml = a4its + 4b4itl + c2it2, (44)

It2 = aaml 4- 4b3m2 + ClmB, (45)

me = a,,itl + 4b4it2 + C2]/B, (46)

where the coefficients ai, bl and c~ are given in the appendix.

On the other hand, in the framework of FCA, the surface order-disorder critical temperature is obtained by linearizing Eqs. (27)-(30), (33) and (34). Then, we obtain

4- 3

Its = Z C4"Bp, o'O~)'ml 4- 4 ~ C3"OP'(Ap,~ + Cp,~ "Qo)'ms, (47)

p = O p = 0

ms = 4Dlits + Eoitl, (48)

It1 =

y 4 p = O

4 p , ,

Cv'Qo'{(Gp,o + Jp o'Qo)'ms + (Gp,o + Jp o'Oo)'m2}

3

+ 4 ~ C3"Qg'{Fp,, + Hp, l"(Oo + Oo) + K v , , - O o ' Q o ) } ' m l ,

p = O

(49)

ml = Mo(its + It2) @ 4Lilt1, (50)

4 3

It2 = 2 C4"Q~'(Gp, 0 -[- Jp,o'Qo)'(ml + m,) + 4 ~, C3"Qp'{Fp.I + 2Hp, t'Qo + Kp, l" Qg}'m2, (51)

p = O p = 0

m2 = M0(itl 4- ItB) + 4Ld~z. (52)

Therefore, the critical ordering frontiers (for KB ~< K c) is analytically obtained through a determinantal equation both in the MFA and FCA. In Figs. 2-4 which corresponds to (DB/JB, Ds/Js) = (0, 0), (0.5, -- 1) and (0.5, + 1), respectively; we represent the critical line of ferrimagnetic-paramagnetic surface transition (S). As is seen from these figures, if the ratio R = KB/Ks is less than a critical value R o the surface may ferrimagnetically order at a temperature (Ks c)- a higher than the bulk. K c and Rc depend on the values of the surface and the bulk anisotropies Ds and DB. In particular, for Ds = D~ = 0, the FCA critical value of Rc is 0.6125, to be compared with the mean-field result 0.6781.

We notice that for KB = 0, the system reduces to the two-dimensional mixed spin-½ and spin-1 Ising model with single ion-anisotropy on square lattice. In the case of FCA, its critical line is obtained from Eqs. (47) and

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Ks 2.0

1.0

.7699

0.0 0.0 BP SFi

(S)

BP SP

BF SFi-÷

:E) /

/ / / /" /

:o) BF

SF / / (Q),"

/ / / / / J /

BF SFi ÷-

6.0

4.0

I / K B

2.0

BP SP

(o)

BF //

SF

i

1.0

/

BF SFi-+

i J i 0.0 i ~ , i i

.47~ 1.0 2.0 0.0 2.0 3.0 4,0

Ks IlR

Fig. 2. (a) The phase diagram of the FCA in Ks-KB plane for the three-dimensional semi-infinite mixed spin-½ and spin-I Ising model with competing surface and bulk exchange interactions, (DB, Ds) = (0, 0). (b) The phase diagram of the FCA in 1/KB-1/R plane for the three-dimensional semi-infinite mixed spin-½ and spin-1 Ising model with competing surface and bulk exchange interactions, (DB, Ds) = (0, 0).

Ks 2.0

BP (E) SFi

1.0

7117 ~ P~

BP SP ol )

0,0 L

0.0 .4926 1 O

B E /

SFi-+ .."

/ "

/ / (a),"

/

/ BF

." SFi +- / /

BF SF

6.0

4.0

I/K B

2.0

BP SP

BP SFi

(00 ( O = ) ~

/ / P 2 Sp

B F 0 - /

S F / B F

=,~PI S F i ' +

. . . . . . .

1.0 2.0

I/R

(E)

= J 0.0 I

2.0 0.0 3.0 4.0

K B

Fig. 3. (a) The phase diagram of the FCA in Ks-KB plane for the three-dimensional semi-infinite mixed spin-2 and spin-1 Ising model ± with competing surface and bulk exchange interactions, (Da/Ja, Ds/Js) = (0.5, - 1). (b) The phase diagram of the FCA in 1/KB-1/R plane for the three-dimensional semi-infinite mixed spin~ and spin-1 Ising model with competing surface and bulk exchange interactions, (Da/JB, Ds/Js) = (0.5, - 1).

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114 N. Benayad, A. Dakhama / Journal o f Magnetism and Magnetic Materials 168 (1997) 105-120

Ks

/ /

20 BF /iQ)

SFi-+ / / (E 2) /

/

B P /

SFi

P3 ' [ CEO

1.o 9058

S P ~ ° 0

0 0 .4926 1.0

BF SFi +-

Ka B F SF

i 2.0

6.0

BP SP

l l K a

SFi

(0~) (0' 2 (E z)

, Y / /

BF

/ . F ! sF.

t / / <°,

0.0 1 0 2,0 3,0 4.0

1/R

Fig. 4. (a) The phase diagram of the FCA in Ks-KB plane for the three-dimensional semi-infinite mixed spin-½ and spin-1 Ising model with competing surface and bulk exchange interactions, (DB/JB, Ds/Js) = (0.5, + 1). (b) The phase diagram of the FCA in 1/KB-1/R plane for the three-dimensional semi-infinite mixed spin-½ and spin-1 Ising model with competing surface and bulk exchange interactions, (DR/JB, Os/Js) = (0.5, + 1).

(48). We obtain

3

1 = 16/), ~, Cap OP.Av,,, (53)

p = O

where the coefficents/51 and Oo are equal to D1 and Oo, respectively, with KB = 0. The critical temperature (K c)- 1 as a function of Ds, obtained from Eq. (53), is in good agreement with our very recent exact result [31]

as is shown in Fig. 5. Note, however, that both MFA and FCA predict a tricritical point.

3.2. Other surface transitions

The steps described before are not sufficient to obtain the remaining part of the phase diagram. In fact, any two nearest-neighbours on the surface interact via a ferrimagnetic coupling. At the ground state of the Hamiltonian (1), the system makes a first order transition from a ferrimagnetically ordered state for R < 4, to a ferromagnetically ordered state for R > 4. The critical value 4 does not depend on the anisotropies. In order to obtain the rest of phases and transitions in the framework of FCA, we must solve numerically the coupled Eqs. (27)-(30), (33) and (34) within the four-layer approximation scheme (38). The analysis of the above equations leads to very interesting surface phenomena. The surface behaviours and their dependencies on the anisotropies are shown in Figs. 2 4 corresponding to the values of (DB/JB, Ds/Js) = (0, 0), (0.5, -- 1) and (0.5, + 1), respectively. As is shown in Fig. 2a and Fig. 2b, five physically different phases are identified. These phases are indicated on the phase diagrams by the following symbols

SP, BP: Surface and bulk paramagnetic

SFi, BP: Surface ferrimagnetic and bulk paramagnetic

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N. Benayad, A. Dakhama / Journal of Magnetism and Magnetic Materials 168 (1997) 105-120 115

~.o 1.o 2.0

1.5 ~- FA

0.0

0.0 1.0 2.0

Ds/J s

Fig. 5. The phase diagram of the mixed spin-½ and spin-I lsing model on square lattice, by mean-field approximation (MFA), finite cluster approximation (FCA), and exact calculation [31]. The MFA and FCA critical lines terminate at a tricritical point.

SF, BF: Surface and bulk ferromagnetic

SFi + -, BF: Surface ferrimagnetic and bulk ferromagnetic with #s > 0 and ms < 0.

S F i - +, BF: Surface ferrimagnetic and bulk ferromagnetic with/~s < 0 and ms > 0.

F o r the system under study, we extend the accepted terminology used in the semi-infinite Ising models with no competing exchange interactions [16, 23, 33, 34].

As is seen from Fig. 2a and Fig. 2b, the above phases are separated by different transition lines. Among them, we find all critical lines obtained in the semi-infinite simple cubic ferromagnetic Ising model [8, 14, 15].

They correspond to the surface (S), the extraordinary (E) and the ordinary (O) transitions. When the bulk is ferromagnetically ordered, the surface exhibits, at finite temperature, a second-order transition (L) from the ferromagnetic state (SF, BF) to the ferrimagnetic state (SFi, BF). One of the most interesting features of the surface is the existence of two possible ferrimagnetic orderings (SFi ÷ - and SFi-+), which are separated by a first order transition line (Q). This surface behaviour does not occur in three-dimensional semi-infinite m o n o a t o m i c Ising models with competing surface and bulk exchange interactions [15, 18-20]. One can note that the surface layer behaves roughly like a ferrimagnetic mixed spin-½ and spin-1 Ising model in a non-uniform (temperature-dependent) field [32]. At T --- 0, this field is equal to Ja and ½JB acting on a and S-sublattices, respectively; and then, the configurations S F i - + and SFi + - have the same energy. However, at a finite temperature (Ka < oo and R < 4), and due to the entropy and the bulk boundary conditions, the stability of each one of them depends on the ratio R and Ks as is shown in Fig. 2a and Fig. 2b. In this work, we choose positive boundary conditions (i.e./~B > 0 and mB> 0). As can be seen from the Eqs. (12)-(15), (27)-(30), changing/~B to -/Za and mB to --mB is equivalent to changing/~s to --/~s and ms to - ms. If we choose negative boundary conditions, the phase diagrams are the same as those in Fig. 2a and Fig. 2b, except that the locations of the two phases SFi + - and S F i - + are interchanged.

In the absence of anisotropies (DB = Ds = 0) and within an appropriate range of R (R < Rc = 0.6125), the system exhibits two successive transitions: the surface orders (SFi- +) at a temperature higher than the bulk, followed by the bulk ordering (BF) at the bulk transition temperature (KB c)- 1. F o r 4 > R > Re, the surface and the bulk order ferromagnetically at (KC) - 1; and at a lower temperature (which depends on the value of R), the surface undergoes a second-order transition from the ferromagnetic ordered state (SF) to the

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116 N. Benayad, A. Dakhama / Journal of Magnetism and Magnetic Materials 168 (1997) 105 120

ferrimagnetic ordered state (SFi +-). Furthermore, for certain values of R the surface may undergo a transition from SFi ÷ - to S F i - ÷ phase, when the temperature is decreased. F r o m the figure, we also point out that for R > 4, the bulk promotes its order to the surface in such a way that the bulk and the surface are both paramagnetic or ferromagnetic for KB < K c and KB > K c, respectively.

Let us now investigate the influence of the surface and the bulk crystal field interactions (Ds and DB) on the phase diagram (Fig. 2a and Fig. 2b) of the mixed spin system with competing surface and bulk coupling constants. As is expected, the value of DB has a significant effect on the domain of the bulk ferromagnetic ordering which disappears at a critical value DB/J~ = 3. Therefore, in the absence of the surface crystal field (Ds = 0), the phase diagrams are qualitatively similar to Fig. 2a and Fig. 2b. As is shown in Figs. 3 and 4, Ds has an important influence on the surface behaviour. First, let us consider a value of the anisotropy Ds which reinforces the surface ordering (Fig. 3a and Fig. 3b) (DB/JB = 0.5, Ds/Js = -- 1). In addition to the phases and transitions obtained in the case Ds = DB = 0, the surface exhibits ordinary transition S P B P / S F i - +BF (02) in a certain range of the ratio R. The multicritical point (P) becomes a special point transition (Sp), and gives rise to two new multicritical points P1 and P2, where the second-order transition line (L) meets the first-order transition line (Q) and the ordinary transition lines (O1.2), respectively. In contrast to the case Ds = DB = 0, it is also shown in the figure, that when the bulk is ordered and for certain range of R, the surface may undergo a second-order transition from the ferromagnetic ordered state (SF) to the ferrimagnetic ordered state (SFi- +). We note that the latter transition exists for any Ds < 0, and it can be (for certain values of R) followed by a first order transition from S F i - +/BF to SFi + - / B F phases, when the temperature is decreased.

Secondly, suppose that the surface ordering is disturbed by the presence of a surface anisotropy (0 < Ds/Js < 2). In addition to transitions S F i - + B F / S F i + - B F (Q), SFi + - B F / S F B F (L) and S P B P / S F i B P (S) found in the case Ds = DB = 0, the system exhibits different types of phase transitions at the bulk critical temperature (KC) - 1 (Fig. 4a and Fig. 4b) (D~/J B = 0.5, Ds/J s = + 1). If 0 < R < Rc = 0.5432, the system undergoes extraordinary phase transitions (El) and (E2). As the temperature is increased, the surface layer (according to the value of R) keeps its ordering. P3 is a new multicritical point where two second-order transition lines (El) and (E2) meet a first-order transition line (Q). If Rc < R < oe, and at the bulk critical temperature, the system exhibits ordinary phase transitions (O1) and (O~) which correspond to S P B P / S F B F and SPBP/SFi + - B F second-order transition lines, respectively. These lines intersect with the second-order transition line (L) at a new multicritical point P~. As is seen from the figure, we should note that for positive value of the surface anisotropy (0 < Ds/Js < 2), no transitions occur between the surface ferromagnetic (SF) and the surface ferrimagnetic (SFi- +) phases, when the bulk is magnetically ordered. On the other hand, we should note that for increasing values of the surface anisotropy the domain of SFi + - B F phase becomes wider than the domain of S F i - +BF phase. The latter phase disappears when the system does not exhibit a surface transition (S), for any value of the ratio R. This means that the appearance of the S F i - +BF phase, in the phase diagram, is related to the existence of the surface transition (S).

In the framework of M F A and for the same values of the anisotropies Ds and DB, we have solved numerically the coupled Eqs. (12~(17) within the four-layer approximation (38). The obtained phase diagrams are qualitatively similar to those shown in Figs. 2-4. Since the FCA gives better values in comparaison with the MFA, we reported the FCA results only.

4. Conclusions

In this work we have studied the semi-infinite simple cubic mixed lsing system with competing surface and bulk exchange interactions, denoted by - Js and JB respectively. The phase diagrams, obtained both in the mean-field approximation and the finite-cluster approximation, are qualitatively similar and show very interesting features. Let us summarize by stating the main results of this investigation. We identified five

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physically different phases. When the bulk is disordered (BP), if the ratio R = JB/Js is less than a critical value Rc, the surface may ferrimagnetically order at a temperature higher than the bulk. When the bulk is ferromagnetically ordered (BF), the surface may undergo, at finite temperature and for Rc < R < 4, a second order transition from the ferromagnetic phase (SF) to the ferrimagnetic one (SFi). One of the most qualitatively interesting feature of the surface is the existence of two possible ferrimagnetic orders (SFi + - / B F and S F i - +/BF), where the signs + , - (or - , + ) correspond to the signs of a and S-sublattice magnetizations, respectively. These phases are separated by a first-order transition line.

Finally, the effect of the bulk and the surface crystal-field interactions (DB and Ds) on the phase diagram for surface ordering have been investigated and some characteristic behaviours for the surface magnetism have been found. In particular, for a positive value of Ds, the domain of S F i - +/BF phase becomes narrow and disappears with the surface transition. It is also shown that such a system exhibits other varieties of phase transitions and multicritical points.

Let us conclude the present work by saying that the framework used here, suggests very interesting phase diagrams for the three dimensional semi-infinite mixed spin-½ and spin-1 model with crystal-field and competing surface and bulk exchange interactions. These phase diagrams are qualitatively different from those obtained for the three dimensional semi-infinite monoatomic Ising model.

Appendix

Let us define the functions f ( x ) = ~ tanh , gs(x) :

2 sinh (x) gB(x) =

2 cosh (x) + e p°"'

2 sinh (x) ~ 2 cosh (x)

2 cosh ( x ) + e ~°¢ h s t x ) = 2 cosh ( x ) + e pDs' hB(x) =- 2 cosh(x)

2 cosh ( x ) + e ~°""

The non-zero coefficients of Eqs. (22)-(25), (33) and (34) are given by

A o , 1 = f ( -- Ks),

AI,1 ⁼ ½f( - 2Ks) -- Ao.,,

Az,~ = ¼f( - 2Ks) - 2A,,~ - 3 Ao.,,

A3., = -~f( - 4Ks) - 3A2,~ - ~ Ax,, - ½ Ao,,, Ao,3 = A2.1 + 2A1,1, A1.3 = A3,1 + 2A2,b Bo.o = f(KB),

Bl,o ⁼ ½{T(KB + Ks) + f ( K . - Ks)} - Bo.o,

B2,o = ¼ { f ( K B + 2Ks) + f ( K ~ -- 2Ks)} - 2BLo -- ½ Bo,o, S3,o = ~ { f ( K B + 3Ks) + f ( K B - - 3Ks)} - - 3B2 o , - - -94B, o , - - ZB 4 0 , 0 ,

B4 o = 1A~{f(KB + , 4Ks) + f ( K B -- 4Ks)} - - 4B3.o - - 5B2.o - - 2B1 , ^o- - ^±8 B 0 , 0 ,

Bo,2 = B2.o + 2Bl,o, Bx.2 = B3,o + 2Bz,o, B2,: = B4,o + 2B3,o, Bo,4 = B2,: + 2BI,z,

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118

D1 D 3 Eo E 2 E 4 Oo 0 2 0 4 P1 P 3 F o , 1

E l , 1 F2,1 F 3 , 1 1o,1 12,1 13,1

N. Benayad, A. Dakhama / Journal of Magnetism and Magnetic Materials 168 (1997) 105-120

Co,1 = ½ { f ( K ~ - Ks) - - f ( K B + Ks)} - Ao, b

C,,a = ¼ { f ( K B -- 2 K s ) - f ( K a + 2 K s ) } - C o , , - A I , , - A o , 1 ,

62,1 = ~ { f ( K . - 3Ks) - f ( K a + 3Ks)} - 2 C l , 1 - - -~ Co, 1 - - Ao.3 - ¼ Ao,a,

C31, ⁼ ~ { f ( K B -- 4Ks) - - f ( K B + 4Ks)} - 3C2,1 - - 5 C1.1 - - ½ Cot, - A13, - A03, _ ~1 A1,1 - ±A2 0,1, C o . 3 = C2,1 -~ 2C1,1, C 1 , 3 = C3,1 + 2 C 2 , 1 ,

= ~ { g s ( K B - - 4Ks) - gs(KB + 4Ks)} + ¼{gs(KB -- 2Ks) - gs(KB + 2Ks)},

= 4D1 - 2{gs(KB -- 2Ks) - gs(KB + 2Ks)},

= I{gs(KB + 4Ks) + gs(KB -- 4Ks)} + ½{gs(Ka + 2Ks) + gs(KB -- 2Ks)} + 39s(Ks),

= 4Eo - 2{gs(KB + 2Ks) + g s ( K B - - 2Ks)} - 4gs(KB),

= 8 E 2 - 16Eo + 32gs(Ks),

= ~6{hs(KB + 4Ks) + hs(KB - 4Ks)} + ¼{hs(Ks + 2Ks) + hs(KB - 2Ks)} + ~- hs(KB),

= 4 0 0 - {hs(KB + 2Ks) + hs(KB - 2Ks)} - 2 h s ( / ~ ) ,

= 802 - 16Oo + 16hs(KB),

= ¼{hs(Ka - 4Ks) - hs(Ka + 4Ks)} + ½{hs(KB - 2Ks) - hs(Ka + 2Ks)},

= 4P1 - 4{hs(KB - 2Ks) - hs(KB + 2Ks)},

= f ( K s ) ,

= ½f(2KB) -- Fo.1,

= ¼ f ( 3 K B ) -- 2F1 1 , - - ~ F o . 1 , 3

~ - f ( 4 K a ) - 3 F 2 1 5 ± F

= , - - ~ F l , 1 - - 2 0,D

= F 2 , 1 + 2 F l . 1 , / l , t = F 3 . 1 + 2F2.1,

~ 6 f ( 5 K B ) 2111 5 ~ F

= - - , - - g l o , 1 - - 16 o,D

= a f ( 6 K B ) - - 312 • 1 - - 311 . 1 - - I o , 1 - - Y - 6 F t . 1 3 - - _ 3 F 1 6 o , 1 ,

10, 3 = 12.1 + 211,1, 11,3 = 13,1 + 212.1, J o . o = G I , 0 = H o . 1 = F I , t ,

J l . o = G2.0 = H I , I = K o . I = F 2 , 1 , J 2 , 0 = G3.0 = H 2 , 1 = K I . 1 = F 3 , 1 , J 3 . 0 = 10,3 -- 4 F 3 , 1 - - 4 F e . b J 4 . o = 13,1 - - 2IzA + 4 F 3 . 1 ,

G o , o = F o , 1 , Go,2 = F o . 3 = 10,1, G2,2 = J1,2 = H 1 , 3 = K o . 3 = 12,1, G o . 4 = 10,3, J o , 4 = 11.3,

G1,2 = J o , 2 = H o , 3 = F 1 , 3 -{- 11.1, G4,0 = H3.1 = K 2 . 1 = J 3 , 0 ,

J 2 . 2 = K 1 . 3 = 13,1, K 3 . 1 = J 4 . 0 ,

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N. Benayad, A. Dakhama / Journal o f Magnetism and Magnetic Materials 168 (1997) 105-120

Mo = ~6{ga(3K.) + 4 9 , ( 2 K . ) + 5 g . ( K 0 } , M2 = ¼{gB(3K,) -- 3g,(K,)},

M , = 16Mo - 8gB(2KB),

L1 ---- Mo, L3 = N 1 = M 2 , N3 = M4, Qo = ~ 2 { h n ( 3 K 0 + 6 h B ( 2 K 0 + 1 5 h B ( K 0 + 10hB(0)}, Q2 = ~{hB(3KB) + 2h~(2KB) - hB(Kr~) - 2hB(0)}, So = R1 = Q2,

$2 -- 4Q2 - 2hB(2Ka) + 2hB(0),

$4 = 64Qo - 24hB(2KB) - 40hB(0), Q4 = R3 = 32.

T h e coefficients of Eqs. (41)-(46) are given by

- Ks - 2Ks K a

a l - - 4 ' a 2 - - 2 + e p D ~ ' a 3 = b3 = c l = b l = -~--,

2KB 2KB

a4 = b4 = c2 ^-^- 2 + e a°"' b 2 2 + e pos"

119

R e f e r e n c e s

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