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An effective field theory for dilute anisotropic Heisenberg ferromagnets
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Journal of Magnetism and Magnetic Materials 154 (1996) 221-230
Jeurnalof magnetism and magatlc materials
An effective field theory for dilute anisotropic Heisenberg ferromagnets T. Idogaki a,1, y. Miyoshi b, J.W. Tucker a,* a Department of Physics, The University of Sheffield, Sheffield $3 7RH, UK b Department of Applied Science, Kyushu University, Fukuoka 812-81, Japan
Received 8 June 1995
Abstract
Dilute anisotropic Heisenberg ferromagnets have been studied by an effective field theory based on a two-site cluster approximation with an Ising character assumed for the surrounding spins. The results for the Curie temperature, To, and the critical concentration for magnetic ordering, are improvements on those of a single-site approximation where the exchange anisotropy cannot be accounted for. The initial reduction in Tc with dilution is confirmed to decrease with increases in both the lattice coordination number and exchange anisotropy. Finally, by utilising its connection with the zero-temperature magnetization for the Ising system, the percolation probability is examined.
1. Introduction
During the last decade a new effective field theory (NEFT) based on the use of identities of the type introduced by Callen [1] and Suzuki [2] for classical spin correlations has been employed extensively in studies of localised Ising spin systems. Apart from pure, dilute, random bond, and amorphous Ising models, the transverse Ising model, semi-infinite systems with a variety of surfaces, and thin films have also been studied. Although mathematically simple, the approach is superior to the standard mean-field approximation since it correctly accounts for all the single-site kinematic relations of the spin operators. In principle, NEFF can be developed within the framework of an n-site cluster approximation, although for mathematical simplicity and numerical tractability, most studies have adopted a single-site approximation (see, for example, Ref. [3]). A two-site cluster approximation introduced by Bobfik and J a ~ u r [4] has been used in a study of the diluted spin 1 / 2 Ising model [5]. It has also been extended to the spin 1 Ising model [6,7]. Some works have also used even larger clusters [8-12], and general expansion techniques to enable this to be done for both the spin 1 / 2 [9] and spin 1 [11] Ising models have been presented. A further development has been the application of the two-site approximation to the quantum Heisenberg
* Corresponding author. Fax: + 44-114-272-8079; email: [email protected]. i Permanent address: Department of Applied Science, Faculty of Engineering, Kyushu University, Fukuoka 812-81, Japan. 0304-8853f96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0304-8853(95)00598-6
T. Idogaki et al. / Journal of Magnetism and Magnetic Materials 154 (1996) 221-230
222
system [13]. The interaction between the pair of quantum spins forming the cluster is treated exactly, but the surrounding spins with which the pair directly interacts are assumed to be Ising in character. Idogaki and Ury~ [14] developed this approach to study in detail the anisotropic Heisenberg ferromagnet. In the present paper we extend that work to a study of dilute systems. The study includes, as special cases, the dilute Ising, Heisenberg and X - Y models. For a selection of two- and three- dimensional lattices the concentration dependence of the critical temperature, Tc, is studied, and the critical concentration for magnetic ordering to occur is determined. In particular, the anisotropy dependence of the initial reduction in Tc with dilution is investigated, and the percolation probability as determined from the zero temperature magnetization is examined. The magnetization curves for selected values of dilution are also obtained.
2. Two-site cluster theory
We consider a dilute anisotropic Heisenberg ferromagnet of spin S = 1/2, described by the Hamiltonian
= -J
E cicj(~SxS) " -.k ~s~'sf ~- ~ s { s 1 ) ,
(l)
where c i is a random variable which takes the value 1 or 0 according to whether the site i is occupied by a spin S i, or not. The parameters ~, 77 and ( control the anisotropy of the exchange interaction J. For some special values of ~, rl and ( one recovers the well-known models, namely, the Ising model [I] (~:=~7 = 0), the isotropic Heisenberg model [H] ( ~: = ~ = ~") and the X - Y model [XY] ( ~ = r/, ~"= 0). In this paper we report results for the two-site cluster approximation. The following notation will be adopted throughout. The two nearest-neighbouring sites forming the pair cluster are denoted by f and g. Ai (i = 1 to zf) denote the nearest-neighbouring sites of f (excluding g), while /*i (i = 1 to Zg) those of g (excluding f). Some lattices have sites common to both the sets {Ai} and {/zi}. These are denoted by v i (i = 1 to Zrg). {A'i} and {/x~} denote the sets {Ai} and { tz~} when the sets {v i} have been removed. The starting point for the two-site cluster approximation is to split the Hamiltonian into the following terms: H = ~Ufg + ~"r + Xg +.yz~, _ Z o + y , ,
(2)
~,.~ = -Jcrcg ( ~ S ; S ; + rlS~Sg' + CS?Sg),
(3)
with
~,Ur= - J c r ~S~' ~_, ca Sx~, + rlS¢' ~_~ c;~ S~,' + ( S ( ~ i=1
Yfg
-JCg
~Sg
i=1
cu S~,' i=1
rlS~'
ca Sf, ' ,
(4)
i=1
cmS~, ,+ i=1
~ • i=1
Allowing for the fact that PU0 and ~ ' do not commute, the thermal average of S~, for example, can be written as
(6) where a = Tr ° exp( -/3~-/'o),
B = Tr°[ S~ exp( - fl"~0)],
a = 1 - exp( -/3~Y 0) exp( - f l Y ' ) exp(/3~g'~).
(7) (8)
In the above, Tr ° means the partial trace with respect to the states of the cluster spins Sf and Sg. Eq. (6) is an
T. ldogaki et al. / Journal of Magnetism and Magnetic Materials 154 (1996) 221-230
223
exact relation, but is difficult to deal with owing to the presence of the second thermal average on the right-hand side. Following Ref. [13], we avoid this difficulty by replacing the quantum spins surrounding the two-site cluster by Ising spins. With this replacement, ~T(0 and ~ ' commute and zl ~ 0, so we are left with the simple relation
X(_ + (S()---\
TrOexp[_/3(Xfg+Xfz+Xgz) ]
(9)
/,
where
(a - ~ ~_." c~ S~,),
~'~ = -aJcfS~
(10)
i=1
(
•g = - bJcgS~
b- ~
)
c. S~, . i=
(11)
Remembering that cf2 = cf, etc., on effecting the partial traces in Eq. (9), one has the following result:
(S~) = CfCg[(F3(a, b)) + (F2(a, b))] + 2ce(1 - Cg)(Fl(a)>,
(12)
F,(a) = ¼tanh(½Ka).
(13)
where
F2(a'b)= F3(a'b) =
a+ b
sinh(¼ KX )
X
cosh(¼KX) +exp(-½K~)cosh(¼KY) '
(14)
a- b
exp( - ½K~" ) sinh(¼ KY ) cosh(¼KX) +exp(-½K~)cosh(¼KY) '
(15)
Y
with
X=~/4(a+b)Z+(~-'q)
2,
Y=~/4(a-b)2+(~+~)
2,
(16),(17)
and K = flJ ( fl - 1/(KBT)). To proceed with the calculation, use is made of generalised Van der Waerden operators [15] to expand the functions Fl(a), F2(a,b) and F3(a,b) as finite polynomials of {Sz} in such a way that the single-site kinematic relations of the spin operators are accounted for. Thus one has
ots Fz(a,b)=
c,
O(S,~,,ca, ) ~
,c,,> O(S~,:,c~,:) f2({ xJ,{ /J,{cx,I,{cd,}),
(19)
and likewise for F 3, where
O(S,(,ca, ) = ½[(1 + 2S,(16(S~,,1/2)+ ( 1 - 2S,~,)6(S~,,-1/2)] [ca 6(ca,, 1) + (1-ca~)6(c~,,O)].
(20) fl and f2 are the functions Fl(a) and Fz(a,b) regarded as functions of {SO, {ca,}, etc., and g is the forward Kronecker delta function that operates only on a function to the fight.
T. Idogakiet al./ Journalof MagnetismandMagneticMaterials154(1996)221-230
224
Eq. (12) is for a fixed configuration of occupied sites. If one now performs a random configurational averaging, denoted by ( - .. }r, one has m = ((SfZ}}r = ((CfCgF2(a,b)}} r + 2 ( ( c f ( 1 - Cg)F,(a)}}~,
(21)
where the term in ~(a,b) has disappeared on symmetry grounds since we assume f and g of the pair cluster are equivalent sites of the lattice (any other choice, if possible, would be unreasonable in any case). The thermal and random average ( ( • • - })r of F,(a) and F2(a,b) are then carried out in the simple approximation in which the correlations belonging to quantities pertaining to different sites are neglected. If this is done, one finds m = c2(( F2( a,b)}}r + 2c(1 - c ) ( ( F , ( a ) )}r,
(22)
((Ft(a)}}r =
(23)
with
Y'~ i=1
nl= z, (
1/2
((F2(a,b))}r= I-I
I
E
i=l
X
a(n,,n2)8(S~,nl)~(cAi, n2) f,({Sf~,},{c~,}),
1/2 n2=0
E a(n,,n2)6(S;i,n,)6(c~,,n2)
n,=-1/2
H
i= I
)
n2=O
E
nl= - 1/2 n2=0
a(nl,n2)(~(S~;,nl)~(cix'i,n2
X f2({ S.~,}, {S~, }, { c,~,}, {c d,} ),
(24)
where
a(+½,1)=½(c+2m),
a(+ ½,0)=½(1-c).
(25),(26)
From Eqs. (23)-(26) it is seen that Eq. (22) has the form
m = E A,(c,T)m",
(27)
n where n is an odd integer satisfying 1 < n < zf + z g - zfg. This equation may be solved directly to find the magnetization curves by one of the standard numerical techniques that finds the roots of a function f(x) = 0. It is noticed that the double product in Eq. (24) is over all the spins (counted only once) that are the nearest-neighbours to the pair cluster. Thus in the numerical calculation, no special attention has to be paid to sites that are common nearest-neighbours to both f and g, as was necessary in the analytic approach used in Ref.
[14]. The critical temperature Tc(c) can be obtained by taking the limit m ~ 0 in Eq. (27). That is, it is given by the condition
Z,(c, Tc(c)) = 1.
(28)
Further, in the limit of Tc = 0, the critical concentration, c 0, for ferromagnetic ordering to occur is given as the solution of
al(co,O ) = 1.
(29)
To obtain a,(c, Tc(c)) we may rewrite, for example, ((F~(a))), as
fi ( 1~2
~
i=, k,t,, = ,/2 ,~"=0"
,~(i)(rl(i) vl(i)~(S~ nli ) t'°' ,"2 !
,,
)6(c.~,n(2 i,
))f,({S;,},{c,~,}),
(30)
T. ldogaki et al. / Journal of Magnetism and Magnetic Materials 154 (1996) 221-230
225
with rj(i)'~] ]] , a(i)(n~ i), n (i)) ---- l [ 1 -- c + 2cn (i) - n (i) + 4n~i'n(i)m] = ½b(i'(n(i)) [ 1 + -Arm(i)r~(i)~,,/l~(i)[ , . , "2 , , , / v 1,,.2
(31)
where b(i)(rl.(2i,)
=--
((Fl(a)))r is
Thus the coefficient of m in 1
2zf-2
(32)
1 - c + ,~,...(i) ~ ' ~ 2 __ re, "2 i) -
1/2
1
E
1/2
E
"'"
,?)=-1/2 ,~)=0
l
E
E
[ ~--1 [b(i)(n~i))~(S~"n~i))~(cA"nC2i))]
4~o=-1/2 ,~o=0
zf ~(i)~(i)/l~(i)[~(i)~ ] X i=IE "l
"2
I,'~2
/~
(33)
]]fl({Sx,},{cx,}) •
((F2(a,b))~r may be treated in a similar fashion. 3. Results
Our results for the temperature and concentration dependence of the magnetization m(c, T) are shown in Fig. 1 for the square and simple cubic lattices. Apart from the expected suppression of the long-range ordering with magnetic atom dilution, these results also exhibit suppression of m(c, T) with decreasing exchange anisotropy in going from the Ising to Heisenberg system. This trend is supported in the higher-temperature region by the XY curve which lies intermediate between those for the Heisenberg and Ising systems. However, at lower temperatures the XY and Heisenberg curves unexpectedly cross because of the failure of the theory to correctly predict the zero temperature saturation magnetization for the XY model [14]. The same situation is found in other traditional pair-spin theories, for example, in the Oguchi approximation [16], and in the effective Hamiltonian method [17]. For the two-dimensional magnetic systems, an absence of spontaneous magnetization
t
0.5
i -
m,c,
~ -
I
I
t
i
.0
;,, 1
I
t
(a)
~ 1 . 0
!i/
I
m(c,T)
I
I
I
I
,
(b)
0.8 00
.',\ '
t ~\
h\
"~
iq 0
,
, il 1
,
iil 2 4kBT ~'j
3
!!1
q
I 0
2
4kBT
i|
4
~'j
Fig. 1. Thermal dependence of the magnetization, m(c, T), on a square (a) and simple cubic (b) lattice, for selected values of the magnetic site concentration. The solid, dashed and dashed-dot lines are for the Ising, XY and Heisenberg models, respectively.
226
T. Idogaki et al. / Journal of Magnetism and Magnetic Materials 154 (1996) 221-230
Table 1 Curie temperature, 4KBTc(1)/~J Lattice hc Ka sq pt sc bcc fcc
1-site
2-site
2.1038 3.0898 3.0898 5.0733 5.0733 7.0606 11.0446
' Exact'
[I]
[XY]
[H]
[I]
[XY]
[H]
1.9870 2.9235 3.0250 4.9504 5.0392 7.0394 10.9863
1.8369 2.8238 2.9286 4.8901 4.9801 6.9961 10.9582
1.7182 2.7179 2.8147 4.8065 4.8910 6.9250 10.9095
1.519 2.143 2.269 3.641 4.511 6.353 9.795
0 0 0 0 4.04 5.814 9.050
0 0 0 0 3.36 5.06 8.04
at finite temperature in the Heisenberg system, or the occurrence of a topological Kosterlitz-Thouless transition without long-range order in the XY model, should be expected [18,19]. However, for these systems our calculation gives a non-zero magnetization, as seen in Fig. l(a). This shows that the lattice dimensionality and the quantum nature of the spins are not fully accounted for by this two-site cluster approximation. The Curie temperatures Tc(1) for the pure system are given in Table 1 for several lattice structures. Comparison with 'exact' results (exact, series expansion [18]) show that our results give fair improvement for all lattices over those of the one-site approximation. This is due to the fact that proper account has been taken of the spin correlations between the pair of spins defining the cluster. The improvement is less marked the greater the lattice coordination number since the fraction of intra-cluster bonds, properly accounted for, is proportional to 1 / ( z f + Zg + I). Also, in our two-site cluster theory the kinematic relation for the spin operators is correctly accounted for, and thus our results are an improvement on those of the two-site Oguchi approximation [16]. The results of Table 1 exhibit the expected anisotropy dependence (34)
Tc (I) > Tc ( X Y ) > Tc ( H ) .
.
1(]
(a)
10
.
.
i
.
-I .0-.
1 . 0 "~.
4kBTc(c)
.
0.8-,"'-, 0 . 6 , -.. " 0.4\ -., 0.2\"\ x\" 0.0 ,,\,, " ~ " ~
' 015
C
4kaTc(c) J
.
.
///.;
-0.8.'-.
-0.6,,"','', 5
.
(b)
/ / /
/
-0.4,", ", " / / / / / J -0.2,,","-
-
"
o.o ,,,
\\\
. . . .
0.5
C
Fig. 2. Concentration dependence of the Curie temperature for the simple cubic lattice. The crossovers from the Heisenberg to Ising model and from the Heisenberg to XY model are shown in (a) and (b), respectively. The number accompanying each curve is the value of the parameter 6 as defined in Eqs. (35) and (36).
T. Idogaki et al. / Journal of Magnetism and Magnetic Materials 154 (1996) 221-230 .
.
.
.
i
,
,
,
227
j ,
/ Tc(C)
bcc - .
T~Oi 0.~
sc - .
,g,p~'//' ""-~4,p////
sq ... " " - J 2 ~ / / / /
"° - "-.Z/'/,' /,' /,' "\
0
0.5 c
Fig. 3. Concentration dependence of the reduced Curie temperature for some typical two- and three-dimensional lattices. The solid and broken curves refer to the Ising and Heisenberg models, respectively.
The variation of the Curie temperature with the concentration of magnetic atoms is shown in Fig. 2 for the simple cubic lattice. There, a single anisotropy parameter 6, defined through the relations sc=r/=l-8,
~=1+8
(0_<8_<1),
(35)
sc=~'=l-8,
rt=l+6
(-1<8<0),
(36)
has been introduced [14]. The crossover between Ising and Heisenberg symmetry can be described by the former definition, and that between XY and Heisenberg symmetry by the latter. Reflecting the non-saturation of m(c, 0), the Curie temperature Tc(c) of the XY system drops rapidly at low temperatures and gives a higher critical concentration than that of the Heisenberg system. For several lattices, the concentration dependence of the reduced temperature Tc(c)/Tc(1) for the Ising and Heisenberg systems are plotted in Fig. 3. The critical concentrations c o, for the onset of long-range order, are listed in Table 2. For comparison with the results for the Ising model, the threshold concentrations c* expected from percolation theory [20,21] are shown in the last column. It is seen for this model, that compared to the single-site approximation, the two-site cluster theory has the tendency to improve the value of c o for some lattices, but not for others. This can be explained as follows. In the derivation of Eq. (22), all the spin correlations between the spins that are nearest-neighbours to the pair defining the cluster have been neglected, even when they are nearest neighbours to each other. The number of these latter nearest-neighbour pairs in one- (and two-) site cluster theory is 0 (0), 2
Table 2 Critical concentrations for magnetic ordering Lattice
hc Ka sq pt sc bcc fcc
1-site
0.5575 0.4284 0.4284 0.2929 0.2929 0.2225 0.1502
2-site
Percolation
[I]
[XY]
[H]
0.5706 0.4654 0.4294 0.3101 0.2902 0.2200 0.1536
0.6419 0.5081 0.4668 0.3283 0.3066 0.2292 0.1591
0.5901 0.4707 0.4426 0.3126 0.2974 0.2244 0.1545
0.6962 0.6527 ] 0.5927 0.5 0.3117 0.2460 0.198
T. ldogaki et al. / Journal of Magnetism and Magnetic Materials 154 (1996) 221-230
228
Table 3 Values of I S = (d/dc)(Tc(c)/Tc(1)),.= 1 for several lattices Lattice
1-site
hc Ka sq pt sc bcc fcc
2-site
1.537 ! .345 1.345 1.200 1.200 1.142 1.090
[I]
[XY]
[H]
1.537 1.384 1.318 1.209 1.183 1.131 1.090
1.746 1.455 1.380 1.228 1.201 1.139 1.093
1.786 1.489 1.423 1.247 1.222 1.151 1.098
(2), 0 (2), 6 (8), 0 (4), 0 (12), 24 (40) for the honeycomb, Kagom& square, plane triangular, simple cubic, body-centred cubic and face-centred cubic lattices, respectively. Therefore, with respect to the decoupling of these spins the present two-site cluster theory is less refined than that of the one-site approximation (apart from the honeycomb and Kagom6 lattices), particularly in the cases of the square, simple cubic and body-centred cubic lattices, where the number of neglected nearest-neighbour pairs in the single-site theory is zero. In the low concentration region the neglect of these correlations is more serious because fluctuations in the ratio of nearest-neighbour occupied sites to non-nearest-neighbour occupied sites occur in the sites bordering the cluster pair. For some configurations this ratio is greater than for the pure case. For the simple cubic and body-centred cubic lattices, having large zf and Zg, the drawback in neglecting such correlations (that only appear in the pair theory) more than offsets the improvement gained by including the spin correlations of the pair defining the cluster. This results in a somewhat smaller value of c 0 in the two-site cluster theory than in the one-site approximation. In the two-site cluster theory, however, c o is no longer degenerate for different lattices with the same coordination number, but splits into separate values in the direction predicted by percolation theory [20,21]. See, for example, the values given for the Kagom6 and square lattices (both with z = 4), and those for the plane triangular and simple cubic lattices (z = 6). From both the experimental and theoretical viewpoints, it is valuable to estimate the initial slope in the reduction of Tc(c) with dilution, defined by I x = (d/dc)(Tc(c)/Tc(1))c= i [22]. The numerical results are shown in Table 3. It is observed that the value of I s decreases with increases in the lattice coordination number z, and also with increases in the exchange anisotropy, i.e. Is(I) < I S(XY) < I s(H).
(37)
The former variation is consistent with the fact that for large z the system approaches the molecular field limit, ,(, = 1. The latter variation reflects the stronger stability of long-range order for anisotropic spin system. These results are consistent with previous predictions [23,24]. For the Ising square lattice, the exact result, I s = 1.565, is known [25]. Compared with the one-site cluster result (I S = 1.345), the present result, I s = 1.318, is slightly worse. This means that even though sometimes both Tc and c o are improved in the two-site cluster theory, their degrees of improvement are different. Noting that m(c, T) is the magnetization per site (not per spin), the total magnetization M(c, T) of N sites is
M( c, T) = Nm( c, T) = Ucm¢Sl( c, T) = Uc~m~S)(c, T),
(38)
where m~S)(c,T) = m(c, T ) / c is the magnetization per spin. c~ is the fraction of spins in the infinite cluster and m~)(c, T) is their magnetization. Since m~~s)(c, T) --+ 1 / 2 at T = 0 K for the Ising spin system, the percolation probability P(c), given by P(c) = c~/c [26], can be expressed as follows: c:~ P ( c ) = - - = 2m{S)(c,0) c
2m(c,0) c
(39)
T. Idogaki et al. / Journal of Magnetism and Magnetic Materials 154 (1996) 221-230 .
.
bcc
.
~
.
i
.
.
.
.
.
.
.
.
229
i
-
/ / / / / - - . "-sc P(c)
hc.
P(c)
0.5
0
[.
0.5
-..
"sq
05"'!"'''~q
(b)
o
c
i
i
i
i
0.5 I
l
l
l
l
(C-Co)/O -Co)
Fig. 4. Concentration dependence of the percolation probability for some two- and three-dimensional lattices; (a) linear plot, (b) with the concentration scaled and normalised with respect to the critical concentration, Co, for each lattice.
The value of P(c) estimated from Eq. (39) using re(c, 0) for the Ising spin system is shown in Fig. 4. One can see that the growth of the infinite cluster is promoted for three-dimensional lattices, compared with that in the two-dimensional case. In conclusion, we have developed a two-site cluster theory for the dilute anisotropic Heisenberg ferromagnet with spin S -- 1/2. For some typical two- and three-dimensional lattices it is found that the values of the Curie temperature Tc(c), and the critical concentration c 0, obtained in pair theory, show improvements over those deduced by the single-site approximation, which does not distinguish between Heisenberg, XY, and Ising symmetries. The predictions of the exchange anisotropy dependence of Tc(c), and its initial reduction with dilution at c = 1, are found to be consistent with previous studies. With the aid of generalised Van der Waerden identities [15], the present pair-cluster theory may be extended to studies on pure and dilute quantum spin systems with S >_ 1. A further extension that increases the size of the cluster (particularly in a way that reflects the symmetry of the lattice) would be welcome, in order to give more information concerning the roles of lattice dimensionality and the quantum nature of the spins.
Acknowledgements One of us (TI) wishes to thank the Ministry of Education, Science and Culture (Japan) for financial support during his stay at The University of Sheffield. He also wishes to express his cordial thanks to Dr M. Saber for valuable discussions in the early stage of this work.
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T. Idogaki et al. / Journal of Magnetism and Magnetic Materials 154 (1996) 221-230
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Journal of Magnetism and Magnetic Materials 154 (1996) 221-230
Jeurnalof magnetism and magatlc materials
An effective field theory for dilute anisotropic Heisenberg ferromagnets T. Idogaki a,1, y. Miyoshi b, J.W. Tucker a,* a Department of Physics, The University of Sheffield, Sheffield $3 7RH, UK b Department of Applied Science, Kyushu University, Fukuoka 812-81, Japan
Received 8 June 1995
Abstract
Dilute anisotropic Heisenberg ferromagnets have been studied by an effective field theory based on a two-site cluster approximation with an Ising character assumed for the surrounding spins. The results for the Curie temperature, To, and the critical concentration for magnetic ordering, are improvements on those of a single-site approximation where the exchange anisotropy cannot be accounted for. The initial reduction in Tc with dilution is confirmed to decrease with increases in both the lattice coordination number and exchange anisotropy. Finally, by utilising its connection with the zero-temperature magnetization for the Ising system, the percolation probability is examined.
1. Introduction
During the last decade a new effective field theory (NEFT) based on the use of identities of the type introduced by Callen [1] and Suzuki [2] for classical spin correlations has been employed extensively in studies of localised Ising spin systems. Apart from pure, dilute, random bond, and amorphous Ising models, the transverse Ising model, semi-infinite systems with a variety of surfaces, and thin films have also been studied. Although mathematically simple, the approach is superior to the standard mean-field approximation since it correctly accounts for all the single-site kinematic relations of the spin operators. In principle, NEFF can be developed within the framework of an n-site cluster approximation, although for mathematical simplicity and numerical tractability, most studies have adopted a single-site approximation (see, for example, Ref. [3]). A two-site cluster approximation introduced by Bobfik and J a ~ u r [4] has been used in a study of the diluted spin 1 / 2 Ising model [5]. It has also been extended to the spin 1 Ising model [6,7]. Some works have also used even larger clusters [8-12], and general expansion techniques to enable this to be done for both the spin 1 / 2 [9] and spin 1 [11] Ising models have been presented. A further development has been the application of the two-site approximation to the quantum Heisenberg
* Corresponding author. Fax: + 44-114-272-8079; email: [email protected]. i Permanent address: Department of Applied Science, Faculty of Engineering, Kyushu University, Fukuoka 812-81, Japan. 0304-8853f96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0304-8853(95)00598-6
T. Idogaki et al. / Journal of Magnetism and Magnetic Materials 154 (1996) 221-230
222
system [13]. The interaction between the pair of quantum spins forming the cluster is treated exactly, but the surrounding spins with which the pair directly interacts are assumed to be Ising in character. Idogaki and Ury~ [14] developed this approach to study in detail the anisotropic Heisenberg ferromagnet. In the present paper we extend that work to a study of dilute systems. The study includes, as special cases, the dilute Ising, Heisenberg and X - Y models. For a selection of two- and three- dimensional lattices the concentration dependence of the critical temperature, Tc, is studied, and the critical concentration for magnetic ordering to occur is determined. In particular, the anisotropy dependence of the initial reduction in Tc with dilution is investigated, and the percolation probability as determined from the zero temperature magnetization is examined. The magnetization curves for selected values of dilution are also obtained.
2. Two-site cluster theory
We consider a dilute anisotropic Heisenberg ferromagnet of spin S = 1/2, described by the Hamiltonian
= -J
E cicj(~SxS) " -.k ~s~'sf ~- ~ s { s 1 ) ,
(l)
where c i is a random variable which takes the value 1 or 0 according to whether the site i is occupied by a spin S i, or not. The parameters ~, 77 and ( control the anisotropy of the exchange interaction J. For some special values of ~, rl and ( one recovers the well-known models, namely, the Ising model [I] (~:=~7 = 0), the isotropic Heisenberg model [H] ( ~: = ~ = ~") and the X - Y model [XY] ( ~ = r/, ~"= 0). In this paper we report results for the two-site cluster approximation. The following notation will be adopted throughout. The two nearest-neighbouring sites forming the pair cluster are denoted by f and g. Ai (i = 1 to zf) denote the nearest-neighbouring sites of f (excluding g), while /*i (i = 1 to Zg) those of g (excluding f). Some lattices have sites common to both the sets {Ai} and {/zi}. These are denoted by v i (i = 1 to Zrg). {A'i} and {/x~} denote the sets {Ai} and { tz~} when the sets {v i} have been removed. The starting point for the two-site cluster approximation is to split the Hamiltonian into the following terms: H = ~Ufg + ~"r + Xg +.yz~, _ Z o + y , ,
(2)
~,.~ = -Jcrcg ( ~ S ; S ; + rlS~Sg' + CS?Sg),
(3)
with
~,Ur= - J c r ~S~' ~_, ca Sx~, + rlS¢' ~_~ c;~ S~,' + ( S ( ~ i=1
Yfg
-JCg
~Sg
i=1
cu S~,' i=1
rlS~'
ca Sf, ' ,
(4)
i=1
cmS~, ,+ i=1
~ • i=1
Allowing for the fact that PU0 and ~ ' do not commute, the thermal average of S~, for example, can be written as
(6) where a = Tr ° exp( -/3~-/'o),
B = Tr°[ S~ exp( - fl"~0)],
a = 1 - exp( -/3~Y 0) exp( - f l Y ' ) exp(/3~g'~).
(7) (8)
In the above, Tr ° means the partial trace with respect to the states of the cluster spins Sf and Sg. Eq. (6) is an
T. ldogaki et al. / Journal of Magnetism and Magnetic Materials 154 (1996) 221-230
223
exact relation, but is difficult to deal with owing to the presence of the second thermal average on the right-hand side. Following Ref. [13], we avoid this difficulty by replacing the quantum spins surrounding the two-site cluster by Ising spins. With this replacement, ~T(0 and ~ ' commute and zl ~ 0, so we are left with the simple relation
X(_ + (S()---\
TrOexp[_/3(Xfg+Xfz+Xgz) ]
(9)
/,
where
(a - ~ ~_." c~ S~,),
~'~ = -aJcfS~
(10)
i=1
(
•g = - bJcgS~
b- ~
)
c. S~, . i=
(11)
Remembering that cf2 = cf, etc., on effecting the partial traces in Eq. (9), one has the following result:
(S~) = CfCg[(F3(a, b)) + (F2(a, b))] + 2ce(1 - Cg)(Fl(a)>,
(12)
F,(a) = ¼tanh(½Ka).
(13)
where
F2(a'b)= F3(a'b) =
a+ b
sinh(¼ KX )
X
cosh(¼KX) +exp(-½K~)cosh(¼KY) '
(14)
a- b
exp( - ½K~" ) sinh(¼ KY ) cosh(¼KX) +exp(-½K~)cosh(¼KY) '
(15)
Y
with
X=~/4(a+b)Z+(~-'q)
2,
Y=~/4(a-b)2+(~+~)
2,
(16),(17)
and K = flJ ( fl - 1/(KBT)). To proceed with the calculation, use is made of generalised Van der Waerden operators [15] to expand the functions Fl(a), F2(a,b) and F3(a,b) as finite polynomials of {Sz} in such a way that the single-site kinematic relations of the spin operators are accounted for. Thus one has
ots Fz(a,b)=
c,
O(S,~,,ca, ) ~
,c,,> O(S~,:,c~,:) f2({ xJ,{ /J,{cx,I,{cd,}),
(19)
and likewise for F 3, where
O(S,(,ca, ) = ½[(1 + 2S,(16(S~,,1/2)+ ( 1 - 2S,~,)6(S~,,-1/2)] [ca 6(ca,, 1) + (1-ca~)6(c~,,O)].
(20) fl and f2 are the functions Fl(a) and Fz(a,b) regarded as functions of {SO, {ca,}, etc., and g is the forward Kronecker delta function that operates only on a function to the fight.
T. Idogakiet al./ Journalof MagnetismandMagneticMaterials154(1996)221-230
224
Eq. (12) is for a fixed configuration of occupied sites. If one now performs a random configurational averaging, denoted by ( - .. }r, one has m = ((SfZ}}r = ((CfCgF2(a,b)}} r + 2 ( ( c f ( 1 - Cg)F,(a)}}~,
(21)
where the term in ~(a,b) has disappeared on symmetry grounds since we assume f and g of the pair cluster are equivalent sites of the lattice (any other choice, if possible, would be unreasonable in any case). The thermal and random average ( ( • • - })r of F,(a) and F2(a,b) are then carried out in the simple approximation in which the correlations belonging to quantities pertaining to different sites are neglected. If this is done, one finds m = c2(( F2( a,b)}}r + 2c(1 - c ) ( ( F , ( a ) )}r,
(22)
((Ft(a)}}r =
(23)
with
Y'~ i=1
nl= z, (
1/2
((F2(a,b))}r= I-I
I
E
i=l
X
a(n,,n2)8(S~,nl)~(cAi, n2) f,({Sf~,},{c~,}),
1/2 n2=0
E a(n,,n2)6(S;i,n,)6(c~,,n2)
n,=-1/2
H
i= I
)
n2=O
E
nl= - 1/2 n2=0
a(nl,n2)(~(S~;,nl)~(cix'i,n2
X f2({ S.~,}, {S~, }, { c,~,}, {c d,} ),
(24)
where
a(+½,1)=½(c+2m),
a(+ ½,0)=½(1-c).
(25),(26)
From Eqs. (23)-(26) it is seen that Eq. (22) has the form
m = E A,(c,T)m",
(27)
n where n is an odd integer satisfying 1 < n < zf + z g - zfg. This equation may be solved directly to find the magnetization curves by one of the standard numerical techniques that finds the roots of a function f(x) = 0. It is noticed that the double product in Eq. (24) is over all the spins (counted only once) that are the nearest-neighbours to the pair cluster. Thus in the numerical calculation, no special attention has to be paid to sites that are common nearest-neighbours to both f and g, as was necessary in the analytic approach used in Ref.
[14]. The critical temperature Tc(c) can be obtained by taking the limit m ~ 0 in Eq. (27). That is, it is given by the condition
Z,(c, Tc(c)) = 1.
(28)
Further, in the limit of Tc = 0, the critical concentration, c 0, for ferromagnetic ordering to occur is given as the solution of
al(co,O ) = 1.
(29)
To obtain a,(c, Tc(c)) we may rewrite, for example, ((F~(a))), as
fi ( 1~2
~
i=, k,t,, = ,/2 ,~"=0"
,~(i)(rl(i) vl(i)~(S~ nli ) t'°' ,"2 !
,,
)6(c.~,n(2 i,
))f,({S;,},{c,~,}),
(30)
T. ldogaki et al. / Journal of Magnetism and Magnetic Materials 154 (1996) 221-230
225
with rj(i)'~] ]] , a(i)(n~ i), n (i)) ---- l [ 1 -- c + 2cn (i) - n (i) + 4n~i'n(i)m] = ½b(i'(n(i)) [ 1 + -Arm(i)r~(i)~,,/l~(i)[ , . , "2 , , , / v 1,,.2
(31)
where b(i)(rl.(2i,)
=--
((Fl(a)))r is
Thus the coefficient of m in 1
2zf-2
(32)
1 - c + ,~,...(i) ~ ' ~ 2 __ re, "2 i) -
1/2
1
E
1/2
E
"'"
,?)=-1/2 ,~)=0
l
E
E
[ ~--1 [b(i)(n~i))~(S~"n~i))~(cA"nC2i))]
4~o=-1/2 ,~o=0
zf ~(i)~(i)/l~(i)[~(i)~ ] X i=IE "l
"2
I,'~2
/~
(33)
]]fl({Sx,},{cx,}) •
((F2(a,b))~r may be treated in a similar fashion. 3. Results
Our results for the temperature and concentration dependence of the magnetization m(c, T) are shown in Fig. 1 for the square and simple cubic lattices. Apart from the expected suppression of the long-range ordering with magnetic atom dilution, these results also exhibit suppression of m(c, T) with decreasing exchange anisotropy in going from the Ising to Heisenberg system. This trend is supported in the higher-temperature region by the XY curve which lies intermediate between those for the Heisenberg and Ising systems. However, at lower temperatures the XY and Heisenberg curves unexpectedly cross because of the failure of the theory to correctly predict the zero temperature saturation magnetization for the XY model [14]. The same situation is found in other traditional pair-spin theories, for example, in the Oguchi approximation [16], and in the effective Hamiltonian method [17]. For the two-dimensional magnetic systems, an absence of spontaneous magnetization
t
0.5
i -
m,c,
~ -
I
I
t
i
.0
;,, 1
I
t
(a)
~ 1 . 0
!i/
I
m(c,T)
I
I
I
I
,
(b)
0.8 00
.',\ '
t ~\
h\
"~
iq 0
,
, il 1
,
iil 2 4kBT ~'j
3
!!1
q
I 0
2
4kBT
i|
4
~'j
Fig. 1. Thermal dependence of the magnetization, m(c, T), on a square (a) and simple cubic (b) lattice, for selected values of the magnetic site concentration. The solid, dashed and dashed-dot lines are for the Ising, XY and Heisenberg models, respectively.
226
T. Idogaki et al. / Journal of Magnetism and Magnetic Materials 154 (1996) 221-230
Table 1 Curie temperature, 4KBTc(1)/~J Lattice hc Ka sq pt sc bcc fcc
1-site
2-site
2.1038 3.0898 3.0898 5.0733 5.0733 7.0606 11.0446
' Exact'
[I]
[XY]
[H]
[I]
[XY]
[H]
1.9870 2.9235 3.0250 4.9504 5.0392 7.0394 10.9863
1.8369 2.8238 2.9286 4.8901 4.9801 6.9961 10.9582
1.7182 2.7179 2.8147 4.8065 4.8910 6.9250 10.9095
1.519 2.143 2.269 3.641 4.511 6.353 9.795
0 0 0 0 4.04 5.814 9.050
0 0 0 0 3.36 5.06 8.04
at finite temperature in the Heisenberg system, or the occurrence of a topological Kosterlitz-Thouless transition without long-range order in the XY model, should be expected [18,19]. However, for these systems our calculation gives a non-zero magnetization, as seen in Fig. l(a). This shows that the lattice dimensionality and the quantum nature of the spins are not fully accounted for by this two-site cluster approximation. The Curie temperatures Tc(1) for the pure system are given in Table 1 for several lattice structures. Comparison with 'exact' results (exact, series expansion [18]) show that our results give fair improvement for all lattices over those of the one-site approximation. This is due to the fact that proper account has been taken of the spin correlations between the pair of spins defining the cluster. The improvement is less marked the greater the lattice coordination number since the fraction of intra-cluster bonds, properly accounted for, is proportional to 1 / ( z f + Zg + I). Also, in our two-site cluster theory the kinematic relation for the spin operators is correctly accounted for, and thus our results are an improvement on those of the two-site Oguchi approximation [16]. The results of Table 1 exhibit the expected anisotropy dependence (34)
Tc (I) > Tc ( X Y ) > Tc ( H ) .
.
1(]
(a)
10
.
.
i
.
-I .0-.
1 . 0 "~.
4kBTc(c)
.
0.8-,"'-, 0 . 6 , -.. " 0.4\ -., 0.2\"\ x\" 0.0 ,,\,, " ~ " ~
' 015
C
4kaTc(c) J
.
.
///.;
-0.8.'-.
-0.6,,"','', 5
.
(b)
/ / /
/
-0.4,", ", " / / / / / J -0.2,,","-
-
"
o.o ,,,
\\\
. . . .
0.5
C
Fig. 2. Concentration dependence of the Curie temperature for the simple cubic lattice. The crossovers from the Heisenberg to Ising model and from the Heisenberg to XY model are shown in (a) and (b), respectively. The number accompanying each curve is the value of the parameter 6 as defined in Eqs. (35) and (36).
T. Idogaki et al. / Journal of Magnetism and Magnetic Materials 154 (1996) 221-230 .
.
.
.
i
,
,
,
227
j ,
/ Tc(C)
bcc - .
T~Oi 0.~
sc - .
,g,p~'//' ""-~4,p////
sq ... " " - J 2 ~ / / / /
"° - "-.Z/'/,' /,' /,' "\
0
0.5 c
Fig. 3. Concentration dependence of the reduced Curie temperature for some typical two- and three-dimensional lattices. The solid and broken curves refer to the Ising and Heisenberg models, respectively.
The variation of the Curie temperature with the concentration of magnetic atoms is shown in Fig. 2 for the simple cubic lattice. There, a single anisotropy parameter 6, defined through the relations sc=r/=l-8,
~=1+8
(0_<8_<1),
(35)
sc=~'=l-8,
rt=l+6
(-1<8<0),
(36)
has been introduced [14]. The crossover between Ising and Heisenberg symmetry can be described by the former definition, and that between XY and Heisenberg symmetry by the latter. Reflecting the non-saturation of m(c, 0), the Curie temperature Tc(c) of the XY system drops rapidly at low temperatures and gives a higher critical concentration than that of the Heisenberg system. For several lattices, the concentration dependence of the reduced temperature Tc(c)/Tc(1) for the Ising and Heisenberg systems are plotted in Fig. 3. The critical concentrations c o, for the onset of long-range order, are listed in Table 2. For comparison with the results for the Ising model, the threshold concentrations c* expected from percolation theory [20,21] are shown in the last column. It is seen for this model, that compared to the single-site approximation, the two-site cluster theory has the tendency to improve the value of c o for some lattices, but not for others. This can be explained as follows. In the derivation of Eq. (22), all the spin correlations between the spins that are nearest-neighbours to the pair defining the cluster have been neglected, even when they are nearest neighbours to each other. The number of these latter nearest-neighbour pairs in one- (and two-) site cluster theory is 0 (0), 2
Table 2 Critical concentrations for magnetic ordering Lattice
hc Ka sq pt sc bcc fcc
1-site
0.5575 0.4284 0.4284 0.2929 0.2929 0.2225 0.1502
2-site
Percolation
[I]
[XY]
[H]
0.5706 0.4654 0.4294 0.3101 0.2902 0.2200 0.1536
0.6419 0.5081 0.4668 0.3283 0.3066 0.2292 0.1591
0.5901 0.4707 0.4426 0.3126 0.2974 0.2244 0.1545
0.6962 0.6527 ] 0.5927 0.5 0.3117 0.2460 0.198
T. ldogaki et al. / Journal of Magnetism and Magnetic Materials 154 (1996) 221-230
228
Table 3 Values of I S = (d/dc)(Tc(c)/Tc(1)),.= 1 for several lattices Lattice
1-site
hc Ka sq pt sc bcc fcc
2-site
1.537 ! .345 1.345 1.200 1.200 1.142 1.090
[I]
[XY]
[H]
1.537 1.384 1.318 1.209 1.183 1.131 1.090
1.746 1.455 1.380 1.228 1.201 1.139 1.093
1.786 1.489 1.423 1.247 1.222 1.151 1.098
(2), 0 (2), 6 (8), 0 (4), 0 (12), 24 (40) for the honeycomb, Kagom& square, plane triangular, simple cubic, body-centred cubic and face-centred cubic lattices, respectively. Therefore, with respect to the decoupling of these spins the present two-site cluster theory is less refined than that of the one-site approximation (apart from the honeycomb and Kagom6 lattices), particularly in the cases of the square, simple cubic and body-centred cubic lattices, where the number of neglected nearest-neighbour pairs in the single-site theory is zero. In the low concentration region the neglect of these correlations is more serious because fluctuations in the ratio of nearest-neighbour occupied sites to non-nearest-neighbour occupied sites occur in the sites bordering the cluster pair. For some configurations this ratio is greater than for the pure case. For the simple cubic and body-centred cubic lattices, having large zf and Zg, the drawback in neglecting such correlations (that only appear in the pair theory) more than offsets the improvement gained by including the spin correlations of the pair defining the cluster. This results in a somewhat smaller value of c 0 in the two-site cluster theory than in the one-site approximation. In the two-site cluster theory, however, c o is no longer degenerate for different lattices with the same coordination number, but splits into separate values in the direction predicted by percolation theory [20,21]. See, for example, the values given for the Kagom6 and square lattices (both with z = 4), and those for the plane triangular and simple cubic lattices (z = 6). From both the experimental and theoretical viewpoints, it is valuable to estimate the initial slope in the reduction of Tc(c) with dilution, defined by I x = (d/dc)(Tc(c)/Tc(1))c= i [22]. The numerical results are shown in Table 3. It is observed that the value of I s decreases with increases in the lattice coordination number z, and also with increases in the exchange anisotropy, i.e. Is(I) < I S(XY) < I s(H).
(37)
The former variation is consistent with the fact that for large z the system approaches the molecular field limit, ,(, = 1. The latter variation reflects the stronger stability of long-range order for anisotropic spin system. These results are consistent with previous predictions [23,24]. For the Ising square lattice, the exact result, I s = 1.565, is known [25]. Compared with the one-site cluster result (I S = 1.345), the present result, I s = 1.318, is slightly worse. This means that even though sometimes both Tc and c o are improved in the two-site cluster theory, their degrees of improvement are different. Noting that m(c, T) is the magnetization per site (not per spin), the total magnetization M(c, T) of N sites is
M( c, T) = Nm( c, T) = Ucm¢Sl( c, T) = Uc~m~S)(c, T),
(38)
where m~S)(c,T) = m(c, T ) / c is the magnetization per spin. c~ is the fraction of spins in the infinite cluster and m~)(c, T) is their magnetization. Since m~~s)(c, T) --+ 1 / 2 at T = 0 K for the Ising spin system, the percolation probability P(c), given by P(c) = c~/c [26], can be expressed as follows: c:~ P ( c ) = - - = 2m{S)(c,0) c
2m(c,0) c
(39)
T. Idogaki et al. / Journal of Magnetism and Magnetic Materials 154 (1996) 221-230 .
.
bcc
.
~
.
i
.
.
.
.
.
.
.
.
229
i
-
/ / / / / - - . "-sc P(c)
hc.
P(c)
0.5
0
[.
0.5
-..
"sq
05"'!"'''~q
(b)
o
c
i
i
i
i
0.5 I
l
l
l
l
(C-Co)/O -Co)
Fig. 4. Concentration dependence of the percolation probability for some two- and three-dimensional lattices; (a) linear plot, (b) with the concentration scaled and normalised with respect to the critical concentration, Co, for each lattice.
The value of P(c) estimated from Eq. (39) using re(c, 0) for the Ising spin system is shown in Fig. 4. One can see that the growth of the infinite cluster is promoted for three-dimensional lattices, compared with that in the two-dimensional case. In conclusion, we have developed a two-site cluster theory for the dilute anisotropic Heisenberg ferromagnet with spin S -- 1/2. For some typical two- and three-dimensional lattices it is found that the values of the Curie temperature Tc(c), and the critical concentration c 0, obtained in pair theory, show improvements over those deduced by the single-site approximation, which does not distinguish between Heisenberg, XY, and Ising symmetries. The predictions of the exchange anisotropy dependence of Tc(c), and its initial reduction with dilution at c = 1, are found to be consistent with previous studies. With the aid of generalised Van der Waerden identities [15], the present pair-cluster theory may be extended to studies on pure and dilute quantum spin systems with S >_ 1. A further extension that increases the size of the cluster (particularly in a way that reflects the symmetry of the lattice) would be welcome, in order to give more information concerning the roles of lattice dimensionality and the quantum nature of the spins.
Acknowledgements One of us (TI) wishes to thank the Ministry of Education, Science and Culture (Japan) for financial support during his stay at The University of Sheffield. He also wishes to express his cordial thanks to Dr M. Saber for valuable discussions in the early stage of this work.
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