Journal of Magnetism and Magnetic Materials 191 (1999) 373-379
Effect of biquadratic exchange and crystal field anisotropy
on the Curie temperature of anisotropic ferromagnet
G.S. Chaddha*, A. Sharma
Department of Physics, Punjab Agricultural University, Ludhiana 141004, Punjab, India
Received 28 May 1998
Abstract
A spin-1 anisotropic ferromagnet with crystal field anisotropy parameter D and biquadratic exchange interaction
parameter (0) )1) has been investigated using the method of double time temperature dependent Green functions.
Mixed Callen and RPA decoupling approximations have been utilized. The variations of Curie temperature ą! with for
different values of the exchange anisotropy parameter (0) )1) at D/J"1 and for different values of D/J at "1
have been studied. Also the value of exchange anisotropy parameter for which dependence of ą! on vanishes has been
calculated at finite D. Drawbacks and limitations in the earlier calculations have been pointed out. 1999 Elsevier
Science B.V. All rights reserved.
PACS: 75.10; 75.30; 75.40
Keywords: Crystal field anisotropy; Biquadratic exchange; Curie temperature; Anisotropic ferromagnet
1. Introduction cal point) for which the transition becomes first
order was also determined by Biegala [3].
The effect of biquadratic exchange on magnetic In all these previous studies, the spin system has
properties such as magnetization, Curie temper- been confirmed as being isotropic with respect to
ature, magnetic susceptibility, etc. have been inves- both the bilinear and biquadratic exchange interac-
tigated by Allan and Betts [1] Brown [2], Biegala tions. Due to the failure of this isotropic model to
[3], Chaddha and Singh [4] and Chaddha [5] represent many real magnetic compounds, the in-
using high-temperature series expansion method, clusion of anisotropy in the exchange interaction in
constant coupling molecular field approximation the presence of crystal field anisotropy has been
and Green's function (GF) decoupling theories. The proposed by many authors. The observed data
critical value of (" from magnetic resonance experiments and theoret-
corresponds to the tricriti- ical studies of the crystal field, along with the per-
turbation calculation of the spin-orbit interaction
[6,21] have clearly established the fact that the ex-
* Corresponding author. change anisotropy and the crystal field anisotropy
0304-8853/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved.
PII: S 0 3 0 4 - 8 8 5 3 ( 9 8 ) 0 0 3 4 6 - 1
374 G.S. Chaddha, A. Sharma / Journal of Magnetism and Magnetic Materials 191 (1999) 373-379
are two fundamentally important features existing The GF, 11A
in a wide variety of magnetic substances and thus E; S\
F 22 arising from the biquad-
ratic exchange and crystal field anisotropy terms
the inclusion of the corresponding terms in the was of course treated as such, that is without de-
Hamiltonian is fully justified. coupling.
The variation of Curie temperature ą! with the They concluded that the effects due to the bi-
biquadratic exchange interaction for different quadratic exchange and crystal field anisotropy
values of the exchange anisotropy parameter annul each other at "0.8 and D/J"0.1 (J being
(0) )1, the particular cases "0 and 1 de- the exchange integral) and from then ą
scribe Blume-Emery-Griffiths model [7] and ! is found to
increase with the increase of (up to "1) instead
Heisenberg model that contains in addition to of decreasing with the increase of as expected. The
the bilinear exchange, biquadratic exchange and expected behaviour could be interpreted as follows.
single-ion anisotropy D, respectively) was studied Because of the squared form, that is (S
by Iwashita and Uryu [8] using pair model appro- G ) SH), SG and
S
ximation but the crystal field anisotropy parameter H being the two spin operators at the lattice sites
i and j, respectively, the biquadratic exchange inter-
D was taken to be zero. At the same time only one action is able to stabilize the antiferromagnetic
ordering parameter, viz. "1S82, called dipolar state ( "180°) as well as the ferromagnetic state
ordering parameter was taken into account where- ( "0°). Therefore, the spin-1 Heisenberg system
as due to the presence of the biquadratic exchange with the biquadratic term does become disordered
term, the present problem needs consideration at a temperature lower than the pure Heisenberg
of two ordering parameters, as well as model. However, only ferromagnetic state is stabil-
y"1CE2"61(S8E)2!4, called quadrupolar or- ized in the Ising spin system.
dering parameter. Though the problem was later Their [10] calculations were, however, based on
on studied by Chaddha and Kalsi [9] by taking the approximation D/J;1 whereas the values of
both the said ordering parameters into account D/J used by them ranged from 0-1 in the text.
using simple GF technique but here too the calcu- Though the calculations were later on extended by
lations were restricted to D"0 them [11] by assuming the exchange interaction to
Tiwari and Srivastava [10] considered the com- be anisotropic but with the similar error.
bined effect of biquadratic exchange and crystal The combined effect of biquadratic exchange and
field anisotropy using a simplified form of the ran- crystal field anisotropy was also considered by
dom phase approximation (RPA). Their calcu- Zheng [12] using GF technique but with the follow-
lations were, however, restricted to "1 case only ing new type of Callen decoupling approximation:
and at the same time while linearizing the higher
order Green functions they did not even pay atten- 11S8DAE; S\F22
tion to the fact that there exist the following spin-1 " 11A
identities. E;S\
F 22!y1S\
D S>
E 211AD; S\
F 22, (3a)
11CDAGE;S\F22
S8ECE"S8E+6 (S8E)!4,"2S8E (1a) "y11AGE;S\F22!y1S\DAGE211AD;S\F22,
and (i"1, 2) (3b)
AES8E#S8E AE"AE"S>E (1b) for which there was no physical justification. At the
in which same time, the calculations were restricted to "1
case only. Though the problem was earlier studied
A by Zheng [13] in the case of anisotropic Heisen-
E"![AE, (S8E)]"AES8E#S8EAE (1c)
berg ferromagnet in the presence of crystal field
and used the following linearization: anisotropy but with the same decoupling approxi-
11A mation and at the same time at "0.
E S8E#S8EAE; S\
F 22"21S8211AE; S\
F 22, (2a) Later on, the problem at "1 was also studied
11S8ECEAD; S\F22"1S8E21CE211AD; S\F22. (2b) by Chakraborty [14] using irreducible GF theory
G.S. Chaddha, A. Sharma / Journal of Magnetism and Magnetic Materials 191 (1999) 373-379 375
which he claims, yields different results from those to extend over nearest neighbour pairs i and j. In
reported by GF equation of motion method em- the present paper, we restrict ourselves to the spin-1
ploying various decoupling approximations, at case.
very small. But at the same time there were Using Devlin's [18] notation, we define the fol-
limitations on the calculations and it was not pos- lowing two Green functions:
sible to have systematic quantitative estimates for
the variation of dipolar and quadrupolar ordering GGEF(t!t )"11AGE(t); S\F(t )22 (i"1, 2). (5)
parameters with temperature using the said theory. Each AG
Chakraborty [15] applying the irreducible GF the- E satisfies the relation
ory did study the case of anisotropic Heisenberg [AG
ferromagnet in the presence of crystal field anisot- E, S8E]"!AGE (6a)
ropy but the biquadratic exchange term was taken and also
to be zero. [A
Keeping all these things in mind, we thought to E, (S8E)]"!AE, CE"[AE,S\E]. (6b)
re-study the problem. It was also thought to find The equations of motion of the Fourier trans-
the critical value of the exchange anisotropy para- form of the above Green functions generated by the
meter for which biquadratic exchange interaction Hamiltonian [4] can be written as
has no effect on the Curie temperature ą!. A de-
coupling procedure, the same as that used by wGEF(w)"(2 /2 ) EF
Chaddha and Singh [4] in their earlier work has
been utilized here. That is, the three spin Green #(2! ) JED11S8DAE; S\F22
functions are decoupled using Callen decoupling D
approximation [16] whereas for the still higher-
order Green functions, RPA [17] has been utilized. ! (2! ) JED11S8EAD; S\F22
D
!( /2) J
2. Theory ED11CEAD; S\
F 22
D
We consider a generalized Heisenberg model ex- #( /2) J
pressed as ED11+(4#2 )
D
;(S8D)!4 ,AE; S\F22
H"! JGH[+S8GS8H# (S6GS6H#S7GS7H),
GH #D11AE;S\F22, (7)
# +S8GS8H# (S6GS6H#S7GS7H,] wGEF(w)"(y/2 ) EF#(2! ) JED11S8DAE; S\F22
D
!D (S8G), (4)
G ! (2! ) JED11CEAD; S\F22
where J D
GH is a measure of the exchange force be-
tween the ith and jth lattice sites, SG is the spin ! J
operator associated with the ith lattice site with ED11SXEAD; S\
F 22
D
components S6G, S7G and S8G. measures the strength
of the anisotropic exchange whereas D is the para- #( /2) J
meter measuring strength of the crystal field anisot- ED11+(4#2 )(S8D)!4 ,
D
ropy. the biquadratic exchange parameter is ;A
defined by the ratio of the biquadratic exchange to E; S\
F 22
the bilinear exchange. The summation is assumed #D11AE; S\F22. (8)
376 G.S. Chaddha, A. Sharma / Journal of Magnetism and Magnetic Materials 191 (1999) 373-379
The three spin Green functions are decoupled 1S\
using Callen decoupling approximation [16]: E AE2" !y/2
1 b
" b> # \
11S8 N e@U>!1 e@U\!1 (13)
DAE; S\
F 22D$E" GEF! /21S\
D AE2 GDF (9a) )
and for the still higher-order Green functions, we with the renormalization function,
have chosen the following decoupling approxima- 1
tion: f" J
NJ ) (K) (14)
)
11S8DAE; S\F22D$E"1S8D2 GEF, (9b) and
11CDAE; S\F22D$E"1CD2 GEF. (9c) (C#B)#y(D#F)/2
a ,
The operators S8 $" $
E and CE correspond to the lon- ( (C#B)#w
gitudinal and the operators AG (K)
E to the transverse
motions of the spins, respectively. Since only the (D#G)! y(C#B)/2
b$"y/2$ ,
operators S8E and CE have finite ensemble averages, ( (C#B)#w(K)
the above decoupling represents the same type of A"(2! )J
factorization as in ! J),
B"f+ (2! )J
11S8 !(2! )J),/4,
DAE; S\
F 22D$E" GEF, (10)
C"! (1! )J
which is nothing but RPA [17] and implies that the ),
transverse motion of the spin at site g is completely F" [+(4#2 )m!4 ,J! yJ)]/2,
uncorrelated with the longitudinal motion of the G" +(4#2 )m!4 ,J
spin at site f. This is certainly a reasonable approxi- /2#Cy,
mation as long as fOg.
Using the above decoupling approximations and GLU(K)" GLEF(w)e\ ) E\F , n"1, 2,
assuming the translational invariance, we can solve E\F
for the spatial Fourier transforms of Green func- J J
tions G )" EFe\ ) E\F ,
U(K) and GU(K), i.e. E\F
1 a "1/ką and m"1(S8)2.
G \
U(K)" #
2 a>
w!w ,
> w!w\ It may be noted that the above result reduces to
1 b that of Callen [16] for "1 and "D"0, that is
G \
U(K)" # in the case of the Heisenberg model. Also the result
2 b>
w!w , (11)
> w!w\ reduces to that of Chaddha and Singh [4] obtained
where the poles of the Green functions are given by earlier in the case of isotropic ferromagnets at
D"0.
w!" (A#B)$( (C#B)#w(K) Following Devlin [18] for the accurate deter-
and mination of ą!, Eqs. (12) and (13) are expanded in
the power series of about the point "0. The
w(K)"(D#F)(D#G). coefficients of each power of must be the same on
Making use of spectral theorem and setting g"h each side of each equation. A comparison of the
and t"t , one obtains the following correlation zeroth power of will yield the paramagnetic equa-
functions: tion valid for all ą'ą!. The coefficients of the first
power of in each equation will yield one addi-
8!y
1S\ tional equation. Accordingly, we get the following
E AE2" !
6 two coupled equations,
1 a 8!y 1 (D#F)
" # \ " coth
N a> ! w(K)/2, (15)
) e@U>!1 e@U\!1 , (12) 3y N ) w(K)
G.S. Chaddha, A. Sharma / Journal of Magnetism and Magnetic Materials 191 (1999) 373-379 377
4(2!y) 1 (B!C)
" coth
3y 2N ! w(K)/2
) w(K)
1
#
4N !(A#B)cosech ! w(K)/2
) (16)
with
y J (D#F)
f" ) coth
2N ! w(K)/2. (17)
) J w(K)
The reason why ą! now appears as an unknown
is that the comparison of the coefficients of the first
power of gives equations which are not valid for
ą'ą!, and hence they are valid only at the single
temperature ą!, which is as yet undetermined.
3. Results and discussion
Eqs. (15)-(17) have been solved numerically in
Fig. 1. Variation of ką
the case of spin-1 anisotropic BCC ferromagnet to !/J with on a BCC lattice for D/J"0,
1 and 4 at "1.
study the variation of Curie temperature ą! with
(0) )1) for different values of exchange anisot-
ropy parameter and for different D/J values. The Fig. 2 shows the variation of ą
variation of ą ! with for
! with for D/J"0, 1 and 4 at "1 different values of the exchange anisotropy para-
that is, in the case of a Heisenberg model, has been meter "0, 0.5, 0.65, 0.8 and 1.0 at D/J"1. The
displayed in Fig. 1. ą! is found to be a decreasing results indicate a decrease in ą
function of for all values of D/J. Thus the two ! with an increase of
for "1 spin system whereas ą
effects one due to the crystal field anisotropy and ! is found to
increase with increasing for "0 spin system.
another due to the biquadratic exchange never an- This trend is consistent with the results reported by
nul each other as far as 0) )1 is concerned Iwashita and Uryu [8] in pair model approxima-
which is contrary to the results earlier reported by tion and by Chaddha and Kalsi [9] using simple
Tiwari and Srivastava [10]. Their wrong con- GF technique. However, the results for the inter-
clusions are probably because of the invalid ap- mediate values of differ significantly. The depar-
proximations used by them during the course of ture could be probably due to the fact that Iwashita
calculations. The slopes of our ą! versus curves and Uryu [8] considered only the dipolar ordering
for finite values of the crystal field anisotropy para- parameter whereas the present problem needs con-
meter D are significantly different from those re- sideration of two ordering parameters viz. dipolar
ported by Zheng [12]. This could be due to the fact and as well as the quadrupolar ordering para-
that we have used a decoupling procedure which is meters. The omission of the quadrupolar ordering
physically very sound. However, for D/JPR, the parameter by Brown [19] resulted in two values of
renormalization factor fP0 and the results reduce ą
to those in RPA. Also at D/JPR, one gets y"2, ! for the same value of which is quite unphysical
and also the existence of a tricritical point at '1
implying 1(S8)2"1, which is an expected result which contradicted the results reported in molecu-
because in this limit S8 is effectively restricted to lar field approximation by Chen and Levy [20] and
take only the values $S so that 1(S8)2PS. also by Biegala [3] in RPA.
378 G.S. Chaddha, A. Sharma / Journal of Magnetism and Magnetic Materials 191 (1999) 373-379
Fig. 3. Variation of ą!/ą! with on a BCC lattice at
"D/J"1.
Fig. 2. Variation of ką!/J with on a BCC lattice for "0.0,
0.5, 0.65, 0.8 and 1.0 at D/J"1.
The present results are thought to be certainly
more reliable and at the same time more realistic
compared with the earlier results as they have been
Denoting the Curie temperature in the case obtained for the finite value of D in the presence of
of vanishing as ą! and in order to estimate !, the anisotropic exchange and by assuming that there
value of at which biquadratic exchange inter- are two ordering parameters and y in the bi-
action has no effect on the Curie temperature, quadratic problem and also by using a decoupling
we have studied the variation of ą!/ą! with procedure which is physically very sound. The for-
at "D/J"1. The results so obtained are malism, of course eliminates completely the need to
displayed in Fig. 3. The value of ! determined decouple the higher order GF, 11A
from these observations comes out to be 0.57 in the E; S\
F 22 arising
from the biquadratic exchange and crystal field
case of spin-1 BCC ferromagnet. Though the value anisotropy terms.
of ! has been determined earlier by Iwashita
and Uryu [8] using pair model approximation
and by Chaddha and Kalsi [9] using simple GF Acknowledgements
technique they were both determined at D/J"0.
To the best of our knowledge it is for the first Authors are thankful to Dr. R.K. Jindal and Mr.
time that the value of ! has been found at the finite Banwari Lal for their sincere help at the University
value of D. Computer Centre.
G.S. Chaddha, A. Sharma / Journal of Magnetism and Magnetic Materials 191 (1999) 373-379 379
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