Physica B 304 (2001) 382}388
Exchange integrals and magnetic short range order in the system CdCr S Se \V V M. Hamedoun *, Y. Cherriet , A. Hourmatallah, N. Benzakour
Laboratoire de Physique du Solide, Universite& Sidi Mohammed Ben Abdellah, Faculte& des Sciences Dhar Mahraz, B.P. 1796, Fe% s Atlas, Fe& s, Morocco International Centre for Theoretical Physics, P.O. Box 586, 34100 Trieste, Italy Ecole Normale Supe& rieure Bensouda, Fe% s, Morocco Received 28 February 2000; received in revised form 20 November 2000
Abstract High-temperature series expansions are derived for the magnetic susceptibility and two-spin correlation functions for a Heisenberg ferromagnetic model on the B-spinel lattice. The calculations are developed in the framework of the random phase approximation and are given for both nearest and next-nearest neighbour exchange integrals J and J , respectively. Our results are given up to order 6 in "(k ¹)\ and are used to study the paramagnetic region of the ferromagnetic spinel CdCr S Se . The critical temperature ¹ and the critical exponents and associated with \V V the magnetic susceptibility (¹) and the correlation length (¹) respectively are deduced by applying the PadeH approximant methods. The results as a function of the dilution x obtained by the present approach are found to be in excellent agreement with the experimental ones. 2001 Elsevier Science B.V. All rights reserved. PACS: 75.10.!b; 75.30.Et; 75.40.Cx Keywords: Heisenberg model; Critical temperature; Susceptibility; Correlation; Critical exponents
1. Introduction Materials with spinel structures are of continuing interest because of their wide variety of physical properties. This is essentially related to: (i) the existence of two types of crystallographic sublattices, tetrahedral (A) and octahedral (B), available for the metal ions; (ii) the great #exibility of the structure in hosting various metal ions, di!erently distributed
* Corresponding author. Fax: #212-5573-3349. E-mail address:
[email protected] (M. Hamedoun).
between the two sublattices, with a large possibility of reciprocal substitution between them. Solid solutions of thiospinels and selenospinels have received considerable attention for their interesting electrical and magnetic properties, which can vary greatly as a function of the composition [1}8]. In the spinel solid solution AB X X , the magnetic B ion \V V is located in the tetrahedral sites of the cubic spinel lattice. The A ions are divalent metal ions. The X and X ions can be anions of the chalcogenide group. The weakly diluted magnetic system CdCr S Se is a particular example of this \V V family. In a previous works [3,5], interest has been
0921-4526/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 0 2 8 1 - 2
M. Hamedoun et al. / Physica B 304 (2001) 382}388
given to the magnetic properties and critical behaviour of such system in the range of concentration 0)x)1. The main feature of these investigations can brie#y summarised as follows: For x"0, 0.25, 0.50, 0.75 and 1.0 the member of this family exhibit semiconducting and ferromagnetic behaviour. The "rst section of this work is devoted to the calculation of the exchange integrals in the system CdCr S Se by employing the molecular \V V "eld approximation to the B-spinel lattice [9]. In the second one high temperature series expansions (HTSE) are derived for the magnetic susceptibility and the correlation functions for a Heisenberg ferromagnetic model having both nearest-neighbour (n.n.) and next-nearest-neighbour (n.n.n.) exchange integrals J and J , respectively. The calculations are developed in the framework of the random phase approximation (RPA) (or spherical approximation) [10,11]. Our work extends by several terms the earlier classic work on this subject by Lines [11]. The HTSE of the magnetic susceptibility and correlation functions is given up to order 6 in "(k ¹)\. The theoretical results obtained are then used to study the paramagnetic region of the spinel CdCr S Se in the dilution range \V V 0)x)1. In order to determine the critical temperature ¹ , the critical exponents and associated with the magnetic susceptibility (¹) and the correlation length (¹), we have applied the PadeH approximant (PA) methods. The results obtained are found to be in excellent agreement with experimental ones. The paper is organised as follows: In Section 2, we brie#y describe the Hamiltonian and the procedure for deriving the exchange interactions. In Section 3, we write down the results of the RPA and we give our numerical calculations and "nally we discuss and present our conclusions.
2. Calculation of the values of the exchange integrals Starting with the well known Heisenberg model, the Hamiltonian of the system is given by 1 H"! J S S !g h SX, GH G H G 2 G H G
(1)
383
where S is the operator of the spin localised at the G lattice R and the exchange interaction J is a funcG GH tion of the lattice constant. The "rst summation is over all spins pair nearest-neighbour, the second one is over all site of the lattice, g is the gyromagnetic ratio, is the Bohr magneton and h is the external magnetic "eld. A "rst usual simpli"cation consists in a restriction of the interaction to (n.n.) and (n.n.n.) H"!J S S !J S S !g h SX. (2) G H G I G G H G I G The sums over ij and ik include all (n.n.) and (n.n.n.) pair interactions, respectively. In the case of spinels containing the magnetic moment only in the octahedral sublattice, the mean "eld approximation of this expression leads to simple relations between the paramagnetic Curietemperature and the critical temperature ¹ , respectively, and the two exchange integrals J and J . These can be used to derive numerical values for the exchange constants. Following the method of Holland and Brown [9], the expression of ¹ and that can describe the system CdCr S Se are: \V V 5 [2J !4J ], (3) ¹ " 2k 5 " [6J #12J ], 2k
(4)
where k is the Boltzmann constant. Using Eqs. (3) and (4) together with the experimental values of ¹ and [5], the values of J and J have been determined for each composition of the system. The optimum values are given in Table 1. In the same table we give also the values of the intra-plane and inter-plane interactions J "2J , ?? J "4J #8J and J "4J , respectively. ?@ ?A Fig. 1 shows the variation of J , J and J with ?? ?@ ?A the concentration x in the range 0)x)1. We see that in all compounds J '0 indicating a fer romagnetic coupling between nearest neighbours, which increases steadily from CdCr S (x"0) to CdCr Se (x"1). On the other hand, the coupling J between next-nearest neighbours seems to be antiferromagnetic.
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value S(S#1)/3, therefore the implicit equation for (¹) in the paramagnetic phase may be evaluated in the RPA [10}12] as follows:
\(¹)"
1 1#
(6) q
with "3k ¹/S(S#1) and q "(J !Jq ). ¹ is the absolute temperature and Jq is the Fourier transform of the exchange integrals J de"ned by GH 1 Jq " J exp [iq ) (R !R )]. (7) GH G H N H The 2 q is the average value when the wavevector q runs over its N allowed values in the "rst Brillouin zone. At high temperature, the magnetic susceptibility goes to zero, we can then expand the relation (6) as Fig. 1. Variation of the intra-plane J and inter-plane J and ?? ?@ J interactions with the ratio x in CdCr S Se in the ?A \V V range 0)x)1.
\(¹)"
1 1#
" 1!!2 q .
To zero-order approximation, we have (¹)" \.
The obtained values of the exchange integrals (J and J ) will be used in the following section. 3. High-temperature series expansions In this section we shall derive the HTSE for both the zero "eld magnetic susceptibility and two-spin correlation functions. The calculations are developed in the framework of the RPA up to order 6 in inverse temperature and with arbitrary values of J and J . In the paramagnetic region (¹'¹ ), the pres ence of a "eld h is essential to establish a "nite magnetisation SX . We shall explicitly consider the case of a su$ciently small "eld, and de"ned the magnetic susceptibility as
SX
. (5) (¹)"lim g h F In the limit hP0, the quantity g h/ SX will approach the inverse susceptibility \, whereas the correlation function approaches its isotropic
(8)
q
(9)
By replacing Eq. (9) into Eq. (6) and expanding this latter to the "rst-order, we obtain
( )\" 1! !2
q "1! !2, (10) q
we replace again Eq. (10) into Eq. (3) and using the same procedure, we get: ( )\"(1! )
(! ) (! ) ! !2 , ; 1! (11) after some algebra we arrive at the following expression of the magnetic susceptibility:
(!1)NA N , \" 1# N N where A " , A " ! , A " !2 # ,
(12)
M. Hamedoun et al. / Physica B 304 (2001) 382}388
A " !3 #3 ! , A " !4 #6 !4 # , A " !5 #10 !10 #5 ! ,
For p*1, a possible expression of A can be generN alised in the form A "(!1)N\ N N N\ # CI (!1)I N\I I, (13) N\ I where CK is the binomial coe$cient de"ned by L
(14)
(15) (q)"(z )\ exp (iq ) ), G G G BG is the vector connecting ith nearest-neighbours, G the total number of such neighbours, to a given ion, being z . The factor (q) depends on the lattice G G geometry and is de"ned for the various types cubic lattice as follows:
#cos
q a q a W cos X 2 2
1 q #q q !q W a #cos W Xa (q)" cos V 4 4 3 q !q V Xa 4
(for spinel cubic lattice).
1 < . (q) q " (q)P N q 8
(16)
The summation is to be taken for q , q , q runV W X ning independently between ! /a and /a, V is the volume of the unit cell and a is the lattice parameter. The relationship between the correlation function " S ) S /S(S#1) and the inverse magL L netic susceptibility \ is given by Refs. [10}12]:
If we take into account all the (n.n.) and (n.n.n.) exchange interactions J and J , respectively, q takes the form
(q)"cos(q a)cos(q a)cos(q a) (for BCC), V W X 1 q a q a (q)" cos V cos W 3 2 2
(for FCC),
S ) S
exp [iq ) (R !R )] L " L " L S(S#1) #\
n! , n*m. CL " K m!(n!m)!
(q)" [cos (q a)#cos (q a)#cos (q a)] V W X (for spinel cubic lattice),
q a q a X cos V 2 2
By replacing the summation by an integral over the three-dimensional Brillouin zone, we can write that
!6 # .
q "z J (1! (q))#z J (1! (q)),
#cos
#cos
A " !6 #15 !20 #15
385
,
(17)
q
using the previously evaluated series of \ we may readily calculated the three "rst correlation functions , and . ?? ?@ ?A with "2 #4 is the in plane correlation. ?? "4 #8 is the correlation between neigh?@ bouring planes. "4 #8 is the correlation between the ?A second-neighbour planes. We have computed these correlation functions as a function of temperature ¹ and for di!erents composition x to order 6 in taking into account parameters which are appropriate for the ferromagnetic spinel CdCr S Se (S" , z "6 and \V V z "12). The experimental values of the (n.n.) and (n.n.n.) interactions are taken from Table 1. The sums over the Brillouin zones were performed using Gauss' approximate quadrature method similar to that given in the appendix of Ref. [10]. Figs. 2}4 show the evolution of the "rst, second and third n.n. correlation functions with temperatures for x"1, 0.75, 0.50, 0.25 and 0 in diluted CdCr S Se system. The main feature of \V V
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M. Hamedoun et al. / Physica B 304 (2001) 382}388
Table 1 Critical temperature ¹ , Curie}Weiss temperature , values of the "rst, second, intra-plane and inter-plane exchange integrals of CdCr S Se as a function of dilution \V V x
¹ (K)
(K)
J /k
J /k
J /k ??
J /k ?@
J /k ?A
0.0 0.25 0.50 0.75 1.0
84.5 84.7 93.1 110 129.5
152 165 181 191 204
13.52 13.97 15.34 17.37 19.75
!1.69 !1.48 !1.64 !2.32 !3.07
27.04 27.94 30.68 34.74 39.50
40.56 44.04 48.24 50.92 54.44
!6.76 !5.92 !6.56 !9.28 !12.28
Fig. 2. The "rst n.n spin correlation plotted against the ?? temperature for the system CdCr S Se for x"1, 0.75, \V V 0.50, 0.25 and 0.
Fig. 3. The temperature dependence of the second n.n spin correlation for the system CdCr S Se for x"1, 0.75, ?@ \V V 0.50, 0.25 and 0.
these curves is the decrease with ¹ and x, i.e. the short range order is destroyed by the thermal disorder and the non-magnetic dilution. The results of the computation show that for all composition and temperature the correlation functions and are positive and persist far into the ?? ?@ paramagnetic region whereas the correlation function is negative for all composition and temper?A ature, this is may be linked to the sign of J , ?? J and J . ?A In a recent work [1], a relation between the correlation length (¹) and the correlation functions is given in the case of the B, spinel lattice with a particular ordering vector Q"[0, 0, k]
1 " ! # ?@ . (18) ?A 16 a 8S(k ) ?A S(k ) is the Fourier transform of the correlation function at k"k . In the ferromagnetic case we get k "0. The simplest assumption that one can make concerning the nature of the singularity of (¹) and (¹) is that in the neighbourhood of the critical point the two functions exhibit an asymptotic behaviour:
(¹)J(¹ !¹)\A, (¹)J(¹ !¹)\J.
(19) (20)
M. Hamedoun et al. / Physica B 304 (2001) 382}388
387
4. Conclusions
Fig. 4. The third n.n spin correlation function as a function ?A of temperature for the system CdCr S Se for x"1, 0.75, \V V 0.50, 0.25 and 0.
Estimates of ¹ , and for CdCr S Se \V V have been obtained using the PA methods [13]. The [M, N] PA to the series F() is a rational fraction P /Q , with P and Q , polynomials, of + , + , degree M and N in , satisfying: F()"P /Q #O(+>,>). The sequence of + , [M, N] PA to both the Log ((¹)) and Log ((¹)) was found to be convergent. The simple pole corresponds to ¹ and the residues to the critical exponents and . The obtained values of and are presented in Table 2.
In the present work, we have used the experimental values of ¹ and to derive the two "rst exchange integrals J and J . From these values, we have derived the variation of the intra-plane coupling and the coupling between nearest and next-nearest plane with the concentration x in the ferromagnetic spinel CdCr S Se . \V V By employing the results of the RPA, we have derived the HTSE for the magnetic susceptibility and the correlation functions. The formulas derived are valid at high temperature and can be applied to any ferromagnet which can be described by the Heisenberg model. The results are given for both (n.n.) and (n.n.n.) exchange integrals and up to higher order in inverse temperature. We have in particular applied the theoretical results to study the thermal and disorder variation of the shortrange order (SRO) in the paramagnetic region of the B-spinel CdCr S Se . \V V The HTSE extrapolated with PadeH approximant methods is shown to be a convenient method to provide valid estimations of the critical temperatures for real system. By applying this method to the magnetic susceptibility and the correlation functions we have estimated the critical temperature ¹ against the dilution x. In the hole range of the dilution 0)x)1 the obtained values are in good agreement with the previous works [3,4]. The values of critical exponents and have been estimated in the range of the composition 0)x)1. We have obtained the central values of and , "1.387$0.002 and "0.696$0.001. To conclude, it would be interesting to compare the critical exponents with other theoretical values.
Table 2 The theoretical critical temperature ¹ ($2.5 K) and the critical exponents ($0.002) for the magnetic susceptibility and ($0.001) for the correlation length deduced by applying di!erent [M, N] PadeH approximant method. The values are calculated by considering the values of J and J given in Table 1 x
¹ (K)
[1,4]
[2,4]
[2,2]
[2,3]
[3,3]
[1,4]
[2,4]
[2,2]
[2,3]
[3,3]
0.0 0.25 0.50 0.75 1.0
84.22 84.45 90.25 112 132
1.398 1.395 1.391 1.386 1.385
1.396 1.394 1.388 1.384 1.382
1.391 1.390 1.384 1.382 1.381
1.388 1.386 1.385 1.383 1.382
1.383 1.382 1.376 1.375 1.372
0.728 0.717 0.705 0.704 0.702
0.722 0.712 0.694 0.678 0.703
0.694 0.701 0.693 0.686 0.704
0.699 0.693 0.691 0.675 0.705
0.695 0.691 0.685 0.672 0.708
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M. Hamedoun et al. / Physica B 304 (2001) 382}388
In the critical region, i.e. 5;10\) " (¹ !¹)/¹ )5;10\ Zarek [14] has found ex perimentally by magnetic balance for CdCr Se "1.29$0.02; for HgCr Se "1.30$0.02 and for CuCr Se "1.32$0.02. Basing on the above considerations, it may be concluded that the results of the RPA describe very well the critical properties of the ferromagnetic chalcogenide spinels and that the method and the approximation applied here give values for the critical temperatures and critical exponents, which are in very good agreement with the experimental data. Acknowledgements One of the authors (M.H.) would like to thank the International Atomic Energy Agency and UNESCO for hospitality at the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. This work is supported by the ComiteH International Universitaire Maroc-Espagnol under a grant no: 62/SEE/98.
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