Curie temperatures for site-diluted Ising ferromagnets

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Curie temperatures for site-diluted Ising ferromagnets

Z. Néda

To cite this version:

Z. Néda. Curie temperatures for site-diluted Ising ferromagnets. Journal de Physique I, EDP Sciences,

1994, 4 (2), pp.175-179. �10.1051/jp1:1994130�. �jpa-00246895�

Classification Physics Abstracts

75.10H

Short Communication

Curie temperatures for Site.diluted Ising ferromagnets

Z. Néda

Babes-Bôlyai University, Dept. of Physics, str.

Kogalniceanu

1, ^RO-3400 Clujj Romania and

University of Bergenj Dept. of Physics, Allégaten 55, N-5007 Bergenj Norway

(Received

10 November 1993, accepted 14 December

1993)

Abstract. Trie dilution dependence of trie Curie temperature _is studied in quenchedj

ran-

domly site-diluted Ising ferromagnets for _square- and simple-cubic Iattices by trie Swensen and

Wang Monte Carlo method. In trie Iimit of small dilutions and at trie critical dilution trie _re- sults _are discussed in comparison with other available approaches. We obtain good agreement with accepted theoretical results

m three dimensions, and found discrepancies with mean-field Iike methods for trie square Iattice. Trie critical dilutions _are found _to be _very close to trie

site-percolation thresholds.

Introduction.

Quenched

site- and bond-diluted

Ising

or

Heisenberg

models _are often used to describe the

magnetic properties

of diiferent materials. As

examples

_{one can} mention the

ferromagnetism

of the

alloys

of

magnetic

and

nonmagnetic

metals _as Fe-AI

il,

_{2], or} the

antiferromagnetism

in the

Kmnpmgi-pFe3

_13] or

CopZni-pCs2C15

(4, _5]

compounds.

Beside their

practical

_use these mortels _are

important

as

purely

theoretical _ores, and

usually

studied in the _more

general

context of the

randomly

^diluted Potts mortel [6].

In this _{paper we propose} to

study

the

quenched, randomly

site-diluted

Ising ferromagnets

_on

square and

simple-cubic

Iattices

by

the Swensen and

Wang

Monte Carlo method [7]. ^{The main}

problems

of interest for this model _are the dilution

dependence

of the Curie temperature and the critical behaviour _neon the

magnetic phase-transition

point. These mortels _are also charac-

terized

by

the critical concentration of

holes,

_q~, above which _no

long-range ferromagnetic

order is

possible

_[8]. In this context _an

interesting

topic is the nature of

phase-transition

_{near q~} when trie hole-concentration

(dilution),

_q, varies. Due to the fact that in two and three dimensions

no exact solution is known for the

model,

the

presently

available results _are aII based _on theo- reticaI

approximations

or computer simulations. For the theoretical

approaches

_we mention the usual molecular-field approximation [9], variational

techniques

and the

constant-couphng

_ap- proximation

iii,

mean-field Iike renormalization

approach

_[2,

10],

^the series

expansion

method

176 JOURNAL DE PHYSIQUE I N°2

[11, 12],

one and

two-spin

cluster

approximations [13,

14] or the random-field method [15].

Computer

simulations _were

performed by

the usual Monte Carlo

algorithms _[16],

vectorized Monte Carlo methods [17], ^{Monte Carlo}

renormalization-group

methods [18] ^{and with the}

Swensen and

Wang algorithm

[19]. ^{Most of the} computer simulations deal with the

study

of the critical behaviour _or the nature of the

phase

transition _{near q~.} In this _way trie atm of

ouf paper is to

complete

this

picture by studying

the Curie temperature _{as a} function of the hole-concentration _on

relatively large

lattices

by high-accuracy

computer simulations. We also

give

_an estimate for _q~, and _{compare our} results in the limit of Iow dilution and in the

vicinity

of the critical dilution with available theoretical

approaches.

The method.

For

simulating

the

proposed magnetic

systems we considered the Swensen and

Wang

Monte Carlo method _[7] with _an

original recursion-type algorithm.

In the two-dimensional

(2d)

_case

we considered

usually

_square lattices of 21~0 _x 201~ sites. In the

neighbourhood

of the critical concentration, _q~j where the results became _very sensitive of the

specific

distribution of

hales,

we considered lattices _up to 41~l~ x 400 Iattice sites. In the three-dimensional

(3d)

_{case we}

made ail _our simulations _{on a}

simple-cubic

lattice of 60 x 60 _x 60 Iattice sites. The critical temperature was found

by detecting

the maximum in trie fluctuation of trie absolute value of

magnetization.

For

achieving

statistical

equilibrium

we considered _up to 100G Swensen and

Wang

_steps and then studied the fluctuation for 2000 _more iterations. The

sensitivity

in the determination of trie critical temperature _was varied between _l~.1~1~1 T~(l~) ^and l~.1~1

T~(0) (we

denote

by T~(q)

^{trie cntical} temperature at hole-concentration

q).

The _{programme was} written in C and the simulations _were

performed

_{on a} CRAY Y-

MP4D

/464

_computer and IBM R-61~l~l~ RISC workstation.

Results.

Our results _are

plotted

in

figure

1 for the _square lattice and

figure

2 for the

simple-cubic

_one. As

an immediate conclusion _{one can} observe in the limit of small dilutions the linear

dependence

of the Curie temperature as a function of _the hole-concentration. To determine the

right

Iimit for the

slope (denoted by, ai)

when _{q - l~,} this _was calculated first _{on two} diiferent scales for the 2d _{case as} is shown in

figure

1. These intervals _were

(l~.01-0.14)

and (0-0.1~1),

getting

almost the _same value for the

slope,

_a. The best fit indicates _a

= -1.695 for trie first interval and _a

= -1.71~3 for the second _one. Due to this result for the 3d _case

(Fig. 2)

_we determined the

slope only

in the (l~.1~1-0.2)

interval, getting

_{a =} -1.1303. The results obtained in the

vicinity

of _{q~ are} presented in

figure

3. As _we stated for this

region

_we worked _on

relatively large lattices,

but

despite

this within the demain

represented by

_gray in

figure

3 _we do _not have accurate results. As mentioned earlier this situation is due to the

sensitivity

of the system on

the

specific configuration

of the holes. As

indicated,

the results _are weII described

by

trie

Tc(qj

K

w

^"

_{In(qc q)}

^(~

equation.

We fitted this _curve for trie

points (dark circles), representing

noie concentrations Iower than the values where _we have _no definite data. The values obtained for _q~ from these lits _{are q~}

= 0.413 for the 2d _case and _q~

= l~.662 for 3d. In

good

accordance with these the simulations indicate that _{at q}

= l~.41 for 2d and _q

= l~.67 for

3d,

trie

long-range magnetic

order

disappears (black

_squares in

Fig. 3).

.o

0.6

0.2

~'$.0

_{o-1 0.3}

0.5

1.0

j

0.9 £

0.8

Îi

0.02 0.10 0.14 ~

l.000

0.994

, 0.988

0.002 0.006 0.010

--

~

--

Fig. l. Variation of trie reduced Curie temperaturej

@

j as a function of dilution, _{qj on} the _square

c

Iattice. For trie small dilution hmit trie best-fit indicates

@

= 1+ _{a q} with _{a =} -1.695 for trie

c

q E [0.01j 0.14] ^{interval and} a = -1.703 for trie _{q E} [0, 0.01] one.

i.o 0.8

0.6 ^{' °}

.

~

0.4 ^°

.

0.2

, ~

~$.0

0.2 0.4 0.6

0.81

~

~

~ ~

Î

0.8

~

O.O o-1 0.2

Fig. 2. Variation of trie reduced Curie temperaturej

@j

as a function of dilution, _{qj on} trie

simple-cubic Iattice. For trie small dilution hmit trie

best-fiÎ

_indicates

@

= 1.1303 _q.

Comparison

with other results.

Our results _are

qualitatively

in

good agreement

with aII the earlier _ones. In this _context the Iinear

regime

for small dilutions and the inverse

Ioganthmic behaviour, (1),

_{near q~ are}

theoretically

well

argumented by

several authors

using

^diiferent

approaches il,

11, 14, 15, 21~-

178 JOURNAL DE PHYSIQUE I N°2

0.4

°.~ 2d

j

~Î.30

_{0.34 0.38 0.42 J~}

~

0A

ÎÎ

°.~ 3d

~'~0.4 _{015 016 0.7}

Fig. 3. Variation of the reduced Curie temperaturej

@j

as a function of dilution, _q, for trie

c

q - 0 Iimit. Fitting with

@

= -~~ ~~_~ ^{indicates K =} 0.983j _qc = 0.413 for trie _square Iattice and K

= 0.734j _{qc =} 0.662

flr

trie

simplelcubic

_one.

22]. However,

^{for the}

slope

of the _curve in the _q

- l~ Iimit and for the value of _q~ earlier results

are not

quite

_so

unambiguous.

We summarize earlier results _m

comparison

with _our data in the next tables:

SQUARE LATTICE.

References this

study

_{[15] [11 [21~] [14]}

-a 1.7 1.29 1.1~8 1.66

q~ 0.413 0.57 0.436 l~.57

SIMPLE-CUBIC LATTICE.

References this

study jlsj iii

_{j8j j12j}

jsj

i14j

-a 1.13 1.18 1.04 1.06 1.04 1.09

q~ 0.662 0.71 0.72 0.69 l~.71~8

One _can leam from here that in 3d trie

problem

_seems to be weII

understood,

and also the mean-field Iike

approaches il, 8,

14, 15]

give acceptable

results.

Despite this,

_as

expected,

m the low-dimensional _case

(2d)

the mean-field type methods

il,

_{14, 15] are}

quantitatively

wrong. The

only acceptable

results _on the _square lattice in this _{sense are} those obtained

by

series expansion or correlated eifective-field

theory

_[21~]. ^It is also important to note that the

site-percolation

Iimit is _q

= 0.41 _on the _square Iattice and _q

= 0.688 for the

simple-cubic

_one [6], very close to the critical dilutions obtained

by

_us.

Conclusions.

The results obtained

by

_our Monte-Carlo simulations for the Curie temperature ^of dilute

Ising ferromagnets

_are in

good

_agreement with aII the available theoretical data _{on a}

simple-cubic Iattice,

and show strong

discrepancies

with the mean-field Iike results in _two dimensions. The values obtained for the critical dilution _{are very} close to the

site-percolation

threshold

indicating

the

possibility

of

using

this _{as q~.} This

study completes

earlier _ones in the

problem by giving

_an

accurate

study

of the critical temperature _{as a} function of dilution and

discussing

the results in _a review context.

Acknowledgements.

This

study

was finished

dunng

_a

bursary

oifered

by

the

Norwegian

^{Research Council.} We thank Y.

Brechet,

A.

Coniglio,

L.

Csemai,

and L. Peliti for their continuons

help

and useful

discussions.

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