Spin-glass in the site diluted ising ferromagnet: Octahedron approximation in the FCC lattice

Spin-glass in the site diluted ising ferromagnet: Octahedron approximation in the FCC lattice

Computer Physics Communications 27 (1982) 113—117 North-Holland Publishing Company 113 SPIN-GLASS IN THE SITE DILUTED ISING FERROMAGNET: OCTAHEDRON ...

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Computer Physics Communications 27 (1982) 113—117 North-Holland Publishing Company

113

SPIN-GLASS IN THE SITE DILUTED ISING FERROMAGNET: OCTAHEDRON APPROXIMATION IN THE FCC LATFICE Izuru NAGAHARA, Shigeru HOSAKA and Shigetoshi KATSURA Department of Applied Physics, Tohoku University, Sendai, Japan

Received 1 February 1982

Eu~Sr

1_~Sis regarded as a random site Heisenberg model on the fcc lattice where the first neighbour Eu—Eu exchange energy is ferromagnetic, the second neighbour Eu—Eu exchange energy is antiferromagnetic, and all other exchange energies (Eu—Sr, Sr—Sr) are nonmagnetic. The material is considered as a random site Ising model on an octahedron cactus tree lattice, z,~octahedrons of which are connected at each vertex. The partial trace of the density matrices of the octahedron is calculated by using the formula manipulating program REDUCE. The phase boundaries between the paramagnetic and ferromagnetic phases, and between the paramagnetic and spin-glass phases are obtained. They represent qualitatively well the experimentally observed phase boundaries by Maletta and Convert.

4z~,z

Much attention has been paid to Eu~Sr1_~S which is regarded as a dilute spin-glass (Maletta and Convert [1], Maletta and Felsch [21, Binder et al. [3], Katsura and Nagahara [4] (hereafter referred to a KN) and references therein), The material is regarded as a random site Ising model. At each lattice point of the face-centred cubic lattice (fig. 1), Eu or Sr is located with the probabilityp or 1 —p, respectively. First neighbour and second neighbour interactions are denoted by J~and J~8,where a and $ denote Eu or Sr. The exchange energy ~Eu-Eu is positive while ~Eu.-Eu is negative, and all other J~and J~are zero [1—4]. J~U_EU/JEU_EU is reported to be —0.5, and the spin glass is observed [1,2] as shown in fig. 2. In the previous paper [4] we considered a random site Ising model on a checkerboard lattice where the second neighbour interactions are spanned on the red squares of the two-dimensional square lattice as a simple model of the material, and we approximated it as a square cactus tree lattice. Thus, we obtained a phase diagram showing the spin-glass phase. In the present paper we consider a model which reflects the fcc structure more truly. The model is an octahedron cactus tree lattice with first and second neighbour interactions, z~octahedrons of

which are connected at each vertex (z3 = 2= z~where z1 and z2 are the number of first and second neighbours). When we assign the number of first neighbours to be equal to that of the fcc, z~ is taken to be 3, while when we assign one octahedron in each unit cube of the fcc lattice, z~is taken to be 2. Notation in the present paper is the same as in KN if not otherwise stated. We take a master cluster 123456 as in fig. 1 and denote the external field by H, the effective field

- -

- - - -

I

.

Fig. 1. Crystal structure of Eu Sr1 — S and a unit octahedron in the fcc lattice. White circle: Eu orSr, shaded circle: S. Thick line: first neighbour interaction in the octahedron, dotted line: second neighbour interaction in the octahedron.

001 0-4655/82/0000—0000/$02.75 © 1982 North-Holland

114

1. Nagahara et al.

/

Eu,,, Sr,, —Ps regarded as a random site Ising model

The density matrix of the octahedron up to i9(l)

I K I5

p(6)(aia

~

2a6m2m2

1(6)~

‘6

2

10

H

/

(1+im1m~a,,~)

and its 12,13,14, partial15trace .23 24,26 denoted 35 .36,45.46 by.56p”, are given by

_~_~____,__~/////

[I

x

(l+t’mkm,aka/)

kI= 16,25,34 5

x

fi

(l+m~1~6)a~)l

1,2

fl

j a’

6

S 0

05

p

1.0

p*(ai; m1,m2

..-...........

m 6’

6

(5)

~(6) ~(6) 1 ‘‘2

‘6

tr23456p

Fig. 2. Observed phase diagram of Eu~Sr1~S [1,2].

p(6)(Go

0,...,0; m 1,m2,...,m6

at the vertex i by II,(1) and that at the vertex i of the cluster 123456 contributed from the outside of that cluster by ~ Let 1~ ~6) th /3H~61, $H~~ c1~1) thth f~H, th(13.J, 1/2). (1) The averages of the one-body quantity ~ 1) and the clustercluster quantity 1(6)X,are in terms of the single field by expressed (see ref. [7])

~(6)

Here (9(12) = 0 means that the second and higher orderinterms of 1 are [discarded the right-hand side the brackets ] in (5) isafter calculated. The coefficients of 1~6)in p* are denoted by D and D., i.e., 6~ +D 6~+ +D661(6)) (7) P” = D + (D1l~ 21~ where ...

DD(m 2 0,0

in lowest order, require theone-body reducibility of the cluster-densityWe matrix to the density matrix, that is, “the simple version of the octahedron approxima-

D,.p*(a1 6) =

D and D. are functions of m

tion” (cf. refs. [6,7]):

The relation between

j(

~,

p*(o; 1,m2,m3

m6)

(1’)

~r(6)n~i+

(zr— 1) A”,

(6)

~(6) i6) 2 ‘“~,‘6

0), 1; =m10(j l,m2,m3 1, 2,..., 6)),m6

=

~~‘~(i)

(2)

(8)

j(~’)

2, m3 tr23456p(l, 2, 3,4,5,6)

m6

/,~)‘~and

(

given by

m6,

l~6)~(n

t

=

and t’. 1, 2) is

6)~~+P~

(~denotes the normalized density matrix). Since we are treating the randomly diluted site model, weput =

~ l( ~)+(D DD 2 \“~

+

tm 11m~, t

th(/3J/2)

(3)

~

D

I n— (9)

/(6)n

~D /

6

for ~v = 12, 13, 14, 15, 23, 24, 26, 35, 36, 45, 46, 56, and

From (1) we see that the uniform susceptibility X(i) and the spin-glass susceptibility X(2) are given by

tMPt’mPm~, t’=th(/3J’/2)

(kT)”x~-~--—

for ,~tv= 16, 25, 34.

(4)

~

l+B~

n1,2,

(10)

/

I. Nagahara et a!.

where B~=

-~) ( _~)

115

Eu,, Sr, —PS regarded as a random site Ising model

The calculations and the expressions of D and D, are straightforward but tedious. The authors (11)

calculated them manipulating system by REDUCE the use [8] of asthe a generalizaformula-

The Curie (n = 1) and spin glass (n = 2) transition temperatures are given by

tion of the method in ref. [9] to the dilute system. Examples of nonvanishing D and D, are listed in

~ID

‘ “+



+~

+

Bn=l/(Zc~l).

(12) kTIJ

__________

Here (D1/D )“ is the average over the distribution of the sites, and is calculated‘nto be 6 (D

D(m2,...,m6)

H

f(D(m2,...,m6))

~=1

~“

D/

I

)) J

D~(l,l,l,l,l) D(l,l,l,1,l) I D(0,l,l,l,l) \ “1 +51~I D(l,1,l,l,0) D (1,1,1,1,0) D(0,l,l,l,l) 3( I —p +p

)2[4

[



( D~(0,0,1,1,1) D(0,0,l,l,1)

0

)fl

a

-

P(m,) dm,

-

/~~w SG

0.5

,,,~‘<40

1.0

p —÷ Fig. 3. Phase diagram obtained from z~= 3, (z~= 12, z

)

2 3). (a) 1/2, P-F boundary; (a’) J’/J= —1/2, P-SO boundary; (b)J’/Js —3/2, P—F boundary; (b’)J’/J= —3/2,

J’/J

+21 D,(o,l,1,o,1) \“

P—SO boundary.

~D(0,l,l,0,l))

+4(Di(0~l~1~l~0) )nl D(0,l,l,l,0)

j

2(l _p)314( D,(0,0,0,l,l) \ D(0,0,0,l,l))

4.0

+p

+41I D,(0,0,l,l,0) D(0,0,l,l,0)

+21 D,(0,l,1,0,0) \~1 D(0,l,l,0,0)) j [1D,(0,0,0,0,l))n +p(l _~)4[~D(0,0,0,0,l) 0

+41ID(0,0,0,l,0))nl D(0,0,0,l 0)

j’

where P(m,)=p~(m1—l)+(l —p)ô(m1).

(13)

SG

0.5

1.0

Fig. 4. Phase diagram obtained from z,, =2 (z 1 ‘8, Z2 ~r2). (a) boundary. = — 1/2; (b) J’/J P-F = — boundary. 3/2, P—F boundary. (a’) J’/J (b’) = —f/f 1/2,= —P-SO 3/2, P—SGboundary.

Table 1 REDUCE program and result for calculating c2L3, c

2L4, c2L5, C2L6 and c2,,

RESTORE(30) BEGIN NIL LET SI *51 = l,S3~S3 1,S4~S4 l,S5~S5 1,S6*S6=1; LET L*L~r0; RHO3: = (I + S3 * S5 * X) * (I + S3 * S6 * X) * (1 + S3 * L3 * L) * (I + S 1 * S3 * X) * (I + S3 * S4 * Y)$ RH033: = SUB(S3 = 0,RHO3)$ CLEAR RHO3; RHO4: RHO33s(l +S4~S5* X)*(l ±S4*S6* X)*(1 +S4~L4* L)*(1 +S1 * S4*X)$ CLEAR RH033; RH044: = SUB(S4 = 0,RHO4)$ CLEAR RHO4; RHO5: RHO44*(1 +S5 * 56*X)*(l +S5 * L5 *L)*(l +S1 * S5 sX)$ CLEAR RHO44; RH055: = SUB(S5 rO,RHO5)$ CLEAR RHO5; RHO6: RH055 *(l +S6*L6 * L)*(l +Sl * S6 *Y)$ CLEAR RH055; RH066: = SUB(S6 = O,RHO6)$ CLEAR RHO6; RHO7: RHO66~(l+Sl *Ll * L)*(1 +S2*L2*L)$ CLEAR RHO66; ORDER L3,L4,L5,L6; FACTOR L3,L4,L5,L6; C2C: ‘SUB(Sl ~‘O,S2=O,RHO7)$ RHO!: =SUB(Ll 0,S20,L’ !,Sl = 1,RHO7)$ CLEAR RHO7; RHO2: = RHO 1-C2C$ CLEAR RHOI; C2L3: = SUB(L3 = l,L4 O,L5 ‘O,L6 0,RHO2)$ C2L4: = SUB(L3 = O,L4=1 ,L5 = O,L6 = 0,RHO2)$ C2L5: = SUB(L3 = O,L4 = 0,L5 = I,L6 = O,RHO2)$ C2L6: = SUB(L3 =O,L40,L5 =O,L6 = I,RHO2)$ OFF NAT; SYMBOLIC WRS(16); C2L3: =C2L3; C2L4: = C2L4; C2L5: =C2L5; C2L6: =C2L6; C2C: C2C; END; C2L3=X~(Y** 2*X* *6+Y* *2tX~ * 5+3 *Y* ~2*X* *4± 6*Y~~ ~3+3*Y* *2~X**2+Y* *2~X+Y* *2-)-2*Y*X* ~6+ 2~Y~X~ ~5+6~Y~X* *4+12*Y*X* *3+6sY*Xs *2+2~Y*X+ 2sY+X* s6+Xs s5+3*X* *4+6*Xs s3+3*X* *2+X±1) C2L4=X~(Y*s2*X* s6+Y* s2*X* ~5+3~Y~ ~ ~4+ 6*Y~*2*X* s3+3*Y* *2~X**2+Y* *2sX+Ys *2+2~Y*X~*6+ ~ 2*Y+X* ~6+X* *5+3~X* *4+6.X* *3+3~X~~2+X+1) C2L5 = X * (Y * * 2 * X * * 6 + 2 * Y * * 2 * X ~ * 5 + 2 * Y * * 2 * X * * 4 + 4*Y~~2*X* ~3+5~Y* ~2*X* *2+2sY* ~2~X±Y~X~ *6+ 4*YsX* ~5+7*Y*X* s4+8*Y*X* *3+7*YsX* ~2+4~Y*X+Y+ 2sX* ~5+5*X* ~4+4*X* *3+2*Xs *2+2*X+ I) C2L6=3~Y**2~X**6+4*Ys s2*X* ~5+2.Y* ~2*X~ *4± 4~Y~ *2*X* *3+3sYs ~2~X* *2+Y~X* s8+2*Y*X* *6+ 8*Y~X~ ~5+IO~Y~X~ *4+8*Y~Xs ~3+2~Y~X* ~2+Y+ 3*X~s6+4sX* ~5+2~Xs *4+4sX* *3+3*Xs *2 C2C=Y~*2*X* ~8+4*Y~ ~2*X* ~5+5~Y~ *2*X* s4+4*Y* ~ *3+ ~ 8*Y*Xs ~3+6*Y~X~ ~2+2~X~ s6+4*X* ~5+5*X* ~4+ 4*X~*3+!

I. Nagahara et a!.

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Eu,,, Sr,, ~S regarded as a random site Ising model

table I in the notation D4(0,0,1,1,0) by C236L4, D(0,I,0,0,I) by c245~. Figs. 3 and 4 show the phase diagrams obtained from the models z~= 3 and z~= 2, respectively, both for J’/J= 1/2 and J’/J= —3/2. They are all (in particular the case J’/J = 3/2) qualitatively similar to the experimentally observed phase diagram. In this model z1 = 4z~and z2 = z~, and hence z2/z1 is twice as large as that of the fcc lattice. That is why J’/J = 3/2 is more similar to the experiment than J’/J = — 1/2 for z~ 2 (fig. 4) (in these figures the dotted lines are continuations of the solid lines. The boundaries between the spin-glass phase and the ferro-magnetic phase are to be calculated separately, see for example ref. [10]). A qualitative difference between the experiment and the present results is [dTG/dp]T0, i.e., it is finite in the former and infinite in the latter. It seems to be due to the difference between the Heisenberg model and the Ising model. A similar problem in the site-diluted Ising antiferromagnet has been discussed on a square cactus tree lattice and will be published in another occasion [11].

117

A.C. Hearn, Professor E. Goto, M. Suzuki and H. Saijô for the implementation of REDUCE system in Tohoku University.







Acknowledgements The authors wish to acknowledge valuable discussions with S. Fujiki. They also thank Professor

References [I] H. Maletta and P. Convert, Phys. Rev. Lett. 42 (1979) 108. [2] H. Maletta and W. Felsch, Z. Phys. B 37 (1980) 55. [3] K. Binder, W. Kinzel, H. Maletta and D. Stauffer, J. Magn. Magn. Mat. 15—18 (1980) 189. [4] S. Katsura and I. Nagahara, Z. Phys. 41(1981) 349, 42 (1981) 190. [5] T. Kasuya, 1MB J. Res. Develop. 14 (1970) 214. [6] S. Katsura and S. Fujiki, J. Phys. C 13 (1980) 4711. [7] I. Nagahara, S. Fujiki and S. Katsura, J. Phys. C 14 (1981) 3781. [8] A.C. Hearn, REDUCE 2 user’s manual (University of Utah, 1973). W.S. Brown and A.C. Hearn, Comput. Phys. Commun. 17 (1979) 207. [9] 5. Fujiki, Y. Abe and S. Katsura, Comput. Phys. Cornmun. 25 (1982) 119. [10] 5. Katsura, S. Inawashiro and S. Fujiki, Physica 99A (1979) 193. [11] S. Katsura, S. Fujiki, T. Suenaga and I. Nagahara, Phys. Stat. Sol. (b) 111 (1) (1982) in press.