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Magnetic phase diagrams of the spinels AB2xGa2−2xO4 (A = Zn, Co; B = Al, Cr) systems
Journal of Alloys and Compounds 462 (2008) 125–128
Magnetic phase diagrams of the spinels AB2xGa2−2xO4 (A = Zn, Co; B = Al, Cr) systems M. Hamedoun a,∗ , R. Masrour a , K. Bouslykhane a , A. Hourmatallah a,b , N. Benzakour a a
Laboratoire de Physique du Solide, Universit´e Sidi Mohammed Ben Abdellah, Facult´e des sciences Dhar Mahraz, BP 1796 Fes, Morocco b Equipe de Physique du Solide, Ecole Normale Superieure, BP 5206 Bensouda, Fes, Morocco Received 23 May 2007; accepted 24 June 2007 Available online 27 July 2007
Abstract The magnetic properties of the spinels CoAl2x Ga2−2x O4 and ZnCr2x Ga2−2x O4 systems in the range 0 ≤ x ≤ 1 have been studied by mean field theory and high-temperature series expansions. By using the first theory, we have evaluated the nearest neighbour and the next-neighbour superexchange interaction J1 (x) and J2 (x), respectively, for the first systems in the range 0 ≤ x ≤ 1 and J1 (x = 1), J2 (x = 1) for the second system. The intra-planar and the inter-planar interactions are deduced. The corresponding classical exchange energy for magnetic structure is obtained for the first system. The second theory have been applied in the spinels CoAl2x Ga2−2x O4 and ZnCr2x Ga2−2x O4 systems, combined with the Pad´e approximants method, we have obtained the magnetic phase diagrams (TN versus dilution x) in the range 0 ≤ x ≤ 1. The obtained theoretical results are in agreement with experimental ones obtained by magnetic measurements and M¨ossbauer spectroscopy. The threshold percolation in the second system is xp ≈ 0.4. The critical exponents associated with the magnetic susceptibility (γ) and the correlation lengths (ν) are deduced in the range 0 ≤ x ≤ 1. © 2007 Elsevier B.V. All rights reserved. Keywords: Spinel; Critical temperature; Exchange interactions; High-temperature series expansions; Pad´e approximants; Critical exponents; Magnetic phase diagram; Percolation threshold
1. Introduction Materials with spinel structures, with the formula AB2 X4 are of continuing interest because of their wide variety of physical properties and potential applications in nanoscience and technology. 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 flexibility of the structure in hosting various metal ions, differently distributed 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 composition [1–6]. In the CoAl2x Ga2−2x O4 system, the Co2+ site distribution varies with dilution x, exhibiting the same trend as was observed
∗
Corresponding author. E-mail address: [email protected] (M. Hamedoun).
0925-8388/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2007.06.112
for the previous system, i.e., the Co2+ octahedral occupation increases with the gallium content, giving rise to concomitant AA, BB and AB interactions. The antiferromagnetic spinels CoAl2 O4 (TN = 5 K) and CoGa2 O4 (TN = 10 K) have studied extensively [7]. In the system ZnCr2x Ga2−2x O4 magnetic Cr3+ ions occupy octahedral sites only and the interactions are strongly antiferromagnetic ZnCr2 O4 (TN = 13 K; θ P = −392 K) [8]. A spin glass phase is predicted for such an antiferromagnetic octahedral spinel sublattice when a sufficient number of non-magnetic substitutional impurities are introduced [9]. This related to the removal of the high-ground state degeneracy present in the pure lattice itself in which nearest neighbour interactions prevail. We have calculated the first and the second nearest neighbours exchange interactions J1 (x) and J2 (x) on the basis of magnetic results in CoAl2x Ga2−2x O4 for 0 ≤ x ≤ 1 [7] and J1 (x = 1), J2 (x = 1) for ZnCr2x Ga2−2x O4 system [10]. The values of the intra-plane and inter-plane interactions Jaa , Jab and Jac ; respectively, are deduced from the values of J1 (x) and J2 (x) for 0 ≤ x ≤ 1. The interaction energy of the magnetic structure is
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M. Hamedoun et al. / Journal of Alloys and Compounds 462 (2008) 125–128
obtained for the first system. In recent works [1,2], we have used the high-temperature series expansions (HTSE) to study the thermal and disorder variation of the short-range order (SRO) in the particular B-spinel ZnCr2x Ga2−2x S4 compound. The Pad´e approximant (PA) [11] analysis of the HTSE of the correlation length has been shown to be a useful method for the study of the critical region [12,13]. The model that we used in this present work is valid in the case of a spinel structure. The percolation threshold is deduced in the ZnCr2x Ga2−2x O4 system. We have used the HTSE technique to determine the N´eel temperature TN or the freezing temperature TSG and the critical exponents γ and ν associated with the magnetic susceptibility χ and the correlation length ξ, respectively, in the range 0 ≤ x ≤ 1 for the two systems.
Using the experimental values of TN and ϕ obtained by magnetic measurement for the spinel CoAl2x Ga2−2x O4 [7] and ZnCr2x Ga2−2x O4 [10]. We have deduced the values of exchange integrals J1 (x) and J2 (x). 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 spinel CoAl2x Ga2−2x O4 system with 0 ≤ x ≤ 1. The values obtained of the exchange integrals are J1 (x = 1) = −1.3 K, J2 (x = 1) = −0.65 K for the ZnCr2x Ga2−2x O4 system. The obtained optimums values of J1 (x) and J2 (x) are given in Table 1. The values of corresponding classical exchange energy for the magnetic structure [14] and the values of the intraplane and inter-plane interactions Jaa = 2J1 , Jab = 4J1 + 8J2 and Jac = 4J2 , respectively, are given in the same table in the range 0 ≤ x ≤ 1 for the first system.
2. Theoretical method 2.2. High-temperature series expansions 2.1. Calculation of the values of the exchange integrals from mean field approximation Starting with the well-known Heisenberg model, the Hamiltonian of the system is given by: H = −2 Jij Si Sj (1) i,j
where Jij is the exchange integral between the spins situated at sites i and j. Si is the spin operator of the spin localised at the site i. In this work we consider the nearest neighbour (nn) and next nearest neighbour (nnn) interactions J1 and J2 , respectively. H = J1 Si Sj − J2 Si Sk (2) i,j
i,k
The sums over ij and ik include all (nn) and (nnn) pair interactions, respectively. In the case of spinels containing the magnetic moment only in the octahedral sublattice, the mean field approximation leads to a simple relations between the ϕ angle of helimagnetic ordering and the N´eel temperature TN , respectively, and the considered two exchange integrals J1 and J2 . The N´eel temperature TN and the ϕ angle of helimagnetic ordering are given by Hamedoun et al. [14]: 5 TN = − [2J1 + 4J2 ] 2KB 1 J1 + 2J2 cos(ϕ) = − 4 J2
In this section we shall derive the high-temperature series expansions (HTSE) for both the zero field magnetic susceptibility χ to order six in β. The relation ship between the magnetic susceptibility per spin and the correlation functions may be expressed as follows: β Si Sj (5) χ(T ) = N ij
where β = 1/KB T and N is the number of magnetic ion and TrSi Sj e−βH Tr e−βH is the correlation function between spins at sites i and j. In ref. [1], a relation between the susceptibility and the three first correlation functions is given in the case of the Bspinel lattice with a particular ordering vector Q = (0,0,k). In the ferromagnetic case we get k = 0. The high temperature series expansion of χ(T) gives the function: Si Sj =
n 6
χ(T ) =
a(m, n)ym τ n
(6)
m=−n n=1
The high temperature series expansion of ξ 2 gives the function:
(3) ξ (T ) = 2
n 6
b(m, n)ym τ n
(7)
m=−n n=1
(4)
where y = J2 /J1 and τ = 2S(S + 1)J1 /KB T. The series coefficients a(m,n) and b(m,n) are given in ref. [15].
where KB is the Boltzmann constant.
Table 1 The Curie–Weiss temperature θ P (K), the N´eel temperature TN (K), the values of the first, second, intra-plane, inter-plane exchange integrals and the energy of CoAl2x Ga2−2x O4 as a function of dilution x x
θ P (K) [7]
1.00 0.62 0.50 0.25 0.00
97 70 65 55 65
TN (K) [7] 5 8 8 8 10
J1 /KB (K)
J2 /KB (K)
Jaa /KB (K)
Jab /KB (K)
Jac /KB (K)
|E|/KB S2 (K)
−0.50 −0.80 −0.80 −0.80 −1.00
−0.25 −0.40 −0.40 −0.40 −0.50
−1.0 −1.6 −1.6 −1.6 −2.0
−4.00 −6.40 −6.40 −6.40 −8.00
−1.0 −1.6 −1.6 −1.6 −2.0
6.0 9.6 9.6 9.6 12.00
M. Hamedoun et al. / Journal of Alloys and Compounds 462 (2008) 125–128
127
In spin-glasses (SG) critical behaviour near the TSG SG transition, it is expected not in the linear part χ0 of the dc susceptibility χ, but in the nonlinear susceptibility χs = χ − χ0 . This is due to the fact that the order parameter q in the spin glass state is not the magnetization but the quantity q=
1 [Si 2 ]av N i
As suggested by Edwards and Anderson [16], leading to an associated susceptibility χs =
1 [Si Sj 2 ]av , NT 3 ij
where the correlation length of the correlation function [Si Sj 2 ] possibly diverges at T = TSG . The behaviour of the nonlinear susceptibility has been already extensively studied theoretically and experimentally [17,18]. We have used the expression of χs , to determinate the critical temperature in the region of spin glass for ZnCr2x Ga2−2x O4 system in the range 0 ≤ x ≤ 1. Figs. 1 and 2 show magnetic phase diagrams of spinels CoAl2x Ga2−2x O4 and ZnCr2x Ga2−2x O4 systems, respectively. We can see the good agreement between the magnetic phase diagrams obtained by the HTSE technique and the experimental ones, in particular in the case of the last systems of which the phase diagrams have been established well by different methods [19–22]. The percolation threshold in the second system is xp ≈ 0.4. This value is comparable with that given by Fiorani et al. [10]. The simplest assumption that one can make concerning the nature of the singularity of the magnetic susceptibility χ(T) is that at the neighbour hood of the critical point the above two
Fig. 2. Magnetic phase diagram of ZnCr2x Ga2−2x O4 . The various phases are the paramagnetic phase (PM), antiferromagnetic phase (AFM) (0.85 ≤ x ≤ 1) and Spin glass phase (SG) (0.4 ≤ x < 0.85). The solid squares are the present results. The open circles (deduced by M¨ossbauer) and the open triangle up (deduced by magnetic measurement) [10].
functions exhibit an asymptotic behaviour: χ(T ) ∝ (T − TN )−γ
(8)
ξ 2 (T ) ∝ (TN − T )−2ν
(9)
Estimates of TN or the freezing temperature, γ and ν for CoAl2x Ga2−2x O4 and ZnCr2x Ga2−2x O4 have been obtained using the Pad´e approximate method (PA) [11]. The simple pole corresponds to TN or the freezing temperature and the residues to the critical exponents γ and ν. The obtained central values are γ = 1.380 ± 0.004, ν = 0.70 ± 0.01 and γ = 1.360 ± 0.02, ν = 0.67 ± 0.020. These values of γ and ν are nearest to the one of Heisenberg model and insensitive to the dilution. 3. Discussion and conclusion
Fig. 1. Magnetic phase diagram of CoAl2x Ga2−2x O4 . The various phases are the paramagnetic phase (PM) and antiferromagnetic phase (AFM) (0 ≤ x ≤ 1). The solid squares are the present results. The open circles represent the experimental points deduced by magnetic measurements [7].
J1 (x) and J2 (x) have been determined from mean field theory, using the experimental data of TN [6] for each dilution (see Table 1). From these values we have deduced the values of the intra-plane and inter-plane interactions Jaa , Jbb and Jac , respectively, the energy of the magnetic structure is given in the same table for the spinel CoAl2x Ga2−2x O4 system with 0 ≤ x ≤ 1. From Table 1 on can see that J1 (x) and J2 (x) increases with the absolute value when x decreases. The sign of J1 (x) and J2 (x) are negative in the whole range of concentration. The value of exchange integrals are J1 (x = 1) = −1.3 K, J2 (x = 1) = −0.65 K for the ZnCr2x Ga2−2x O4 system deduced of TN [10]. For x = 1 a long-range antiferromagnetic order is observed in the two systems. In the system ZnCr2x Ga2−2x O4 , the introduced of non-magnetic impurities perturbs this antiferromagnetic order, by consequent the magnetic spin glass phase is produced. The concentration x = 0.85 therefore corresponds to the limit
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above which long-range antiferromagnetism is detected. In the CoAl2x Ga2−2x O4 , the introduced of non-magnetic impurities in the non-magnetic do not perturbs this antiferromagnetic order for the whole range of dilution x. The high-temperature series expansion (HTSE) extrapolated with Pad´e approximants method 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 χ(T) we have estimated the critical temperature TN (in order phase) or the freezing temperature (in disorder phase) for each dilution x. The obtained magnetic phase diagrams of spinel CoAl2x Ga2−2x O4 and ZnCr2x Ga2−2x O4 systems are presented in Figs. 1 and 2, respectively. Several thermodynamic phases may appear including the paramagnetic (PM) and the antiferromagnetic (AFM) in the range 0 ≤ x ≤ 1 for the first system. The antiferromagnetic (AFM) in the range 0.85 ≤ x ≤ 1 and spin glass in the range 0.4 ≤ x < 0.85 for the second system. In these figures we have included, for comparison, the experimental results obtained by magnetic measurement. From this figure one can see good agreement between the theoretical phase diagram and experimental results. In this figure the initial increases TN with increases in the number of AB interactions suggests that these interactions stabilize a long-range order. This order is not simple and would take the form of an antiferromagnetic interaction between small clusters. The higher TN value for CoGa2 O4 than for the other samples could be due to a new kind of antiferromagnetic ordering, resulting from the new balancing between the different exchange interactions. The percolation threshold in the ZnCr2x Ga2−2x O4 system is xp ≈ 0.4. This value is comparable with that given by Fiorani et al. [10]. In the other hand, the value of critical exponents γ and ν associated to the magnetic susceptibility χ(T) and the correlation length ξ(T), have been estimated in the range of the composition 0 ≤ x ≤ 1. The sequence of [M, N] PA to series of χ(T) and ξ(T) has been evaluated. By examining the behaviour of these PA, the convergence was found to be quite rapid. Estimates of the critical exponents associated with magnetic susceptibility and the correlation length for the first and the second systems are found to be γ = 1.380 ± 0.004, ν = 0.70 ± 0.01 and γ = 1.360 ± 0.02,
ν = 0.67 ± 0.020, respectively. In the magnetic order the values of γ and ν are nearest to the one of Heisenberg model and insensitive to the dilution. References [1] M. Hamedoun, M. Houssa, N. Benzakour, A. Hourmatallah, J. Phys.: Condens. Matter 10 (1998) 3611. [2] M. Hamedoun, M. Houssa, N. Benzakour, A. Hourmatallah, Physica B 270 (1999) 384. [3] Y. Cherriet, M. Hamedoun, J. Magn. Magn. Mater. 106 (2001) 224; F. Mahjoubi, M. Hamedoun, F.Z. Bakkali, A. Hourmatallah, A. Benyoussef, J. Magn. Magn. Mater. 221 (2000) 359. [4] A.I. Abramovich, L.I. Koreleva, L.N. Lukina, Phys. Solid State 41 (1999) 73. [5] A. Rachadi, M. Hamedoun, D. Allam, Physica B 222 (1996) 160. [6] J. Hemberger, H.-A. Krug von Nidda, V. Tsurkan, A. Loidl, Phys. Rev. Lett 97 (8) (2006) (id. 087204). [7] D. Fiorani, S. Viticoli, J. Solid State Chem. 26 (1978) 107–110. [8] F. Hartmann-Boutron, A. Gerard, P. Imbert, R. Kleinberger, F. Varret, C. R. Acad. 268 (1969) 906; R. Plumier, C. R. Acad. Sci. (France) 267 (1968) 98; A. Oles, Phys. Status Solidi (a) 3 (1970) 569. [9] J. Vallain, Z. Phys. B 33 (1979) 31. [10] D. Fiorani, J.L. Dormann, J.L. Tholence, J.L. Soubeyroux, J. Phys. C: Solid State Phys. 18 (1985) 3053–3063. [11] G.A. Baker, P. Graves-Morris (Eds.), Pad´e Approximants, Addison-Wesley, London, 1981. [12] R. Navaro, in: L.J. DE Jonsgh (Ed.), Magnetic Properties of Layered Transition Metal Compounds, Kluwer, Daventa, 1990, p. 105. [13] M.C. Moron, J. Phys.: Condens. Matter 8 (1996) 11141. [14] M. Hamedoun, A. Wiedenmann, J.L. Dormann, M. Nogues, J. RossatMignod, J. Phys. C: Solid State Phys. 19 (1986) 1783–1800. [15] N. Benzakour, M. Hamedoun, M. Houssa, A. Hourmatallah, F. Mahjoubi, Phys. Status Solidi (b) 212 (1999) 335. [16] S.F. Edwards, P.W. Anderson, J. Phys. F 5 (1975) 965. [17] S. Katsura, Prog. Theo. Phys. 55 (1976) 1049. [18] G. Toulouse, M. Gabay, J. Phys. Lett. 42 (1981) L103. [19] M. Alba, J. Hammann, M. Nougues, J. Phys. C 15 (1982) 5441. [20] K. Afif, A. Benyoussef, M. Hamedoun, A. Hourmatallah, Phys. Status Solidi (b) 219 (2000) 383. [21] M. Hamedoun, R. Masrour, K. Bouslykhane, A. Hourmatallah, N. Benzakour, Phys. Status Solidi (c) 3 (9) (2006) 3307–3310. [22] M. Hamedoun, R. Masrour, K. Bouslykhane, F. Talbi, A. Hourmatallah, N. Benzakour, M. J. Condens. Matter 8 (2007) 1.
Magnetic phase diagrams of the spinels AB2xGa2−2xO4 (A = Zn, Co; B = Al, Cr) systems M. Hamedoun a,∗ , R. Masrour a , K. Bouslykhane a , A. Hourmatallah a,b , N. Benzakour a a
Laboratoire de Physique du Solide, Universit´e Sidi Mohammed Ben Abdellah, Facult´e des sciences Dhar Mahraz, BP 1796 Fes, Morocco b Equipe de Physique du Solide, Ecole Normale Superieure, BP 5206 Bensouda, Fes, Morocco Received 23 May 2007; accepted 24 June 2007 Available online 27 July 2007
Abstract The magnetic properties of the spinels CoAl2x Ga2−2x O4 and ZnCr2x Ga2−2x O4 systems in the range 0 ≤ x ≤ 1 have been studied by mean field theory and high-temperature series expansions. By using the first theory, we have evaluated the nearest neighbour and the next-neighbour superexchange interaction J1 (x) and J2 (x), respectively, for the first systems in the range 0 ≤ x ≤ 1 and J1 (x = 1), J2 (x = 1) for the second system. The intra-planar and the inter-planar interactions are deduced. The corresponding classical exchange energy for magnetic structure is obtained for the first system. The second theory have been applied in the spinels CoAl2x Ga2−2x O4 and ZnCr2x Ga2−2x O4 systems, combined with the Pad´e approximants method, we have obtained the magnetic phase diagrams (TN versus dilution x) in the range 0 ≤ x ≤ 1. The obtained theoretical results are in agreement with experimental ones obtained by magnetic measurements and M¨ossbauer spectroscopy. The threshold percolation in the second system is xp ≈ 0.4. The critical exponents associated with the magnetic susceptibility (γ) and the correlation lengths (ν) are deduced in the range 0 ≤ x ≤ 1. © 2007 Elsevier B.V. All rights reserved. Keywords: Spinel; Critical temperature; Exchange interactions; High-temperature series expansions; Pad´e approximants; Critical exponents; Magnetic phase diagram; Percolation threshold
1. Introduction Materials with spinel structures, with the formula AB2 X4 are of continuing interest because of their wide variety of physical properties and potential applications in nanoscience and technology. 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 flexibility of the structure in hosting various metal ions, differently distributed 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 composition [1–6]. In the CoAl2x Ga2−2x O4 system, the Co2+ site distribution varies with dilution x, exhibiting the same trend as was observed
∗
Corresponding author. E-mail address: [email protected] (M. Hamedoun).
0925-8388/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2007.06.112
for the previous system, i.e., the Co2+ octahedral occupation increases with the gallium content, giving rise to concomitant AA, BB and AB interactions. The antiferromagnetic spinels CoAl2 O4 (TN = 5 K) and CoGa2 O4 (TN = 10 K) have studied extensively [7]. In the system ZnCr2x Ga2−2x O4 magnetic Cr3+ ions occupy octahedral sites only and the interactions are strongly antiferromagnetic ZnCr2 O4 (TN = 13 K; θ P = −392 K) [8]. A spin glass phase is predicted for such an antiferromagnetic octahedral spinel sublattice when a sufficient number of non-magnetic substitutional impurities are introduced [9]. This related to the removal of the high-ground state degeneracy present in the pure lattice itself in which nearest neighbour interactions prevail. We have calculated the first and the second nearest neighbours exchange interactions J1 (x) and J2 (x) on the basis of magnetic results in CoAl2x Ga2−2x O4 for 0 ≤ x ≤ 1 [7] and J1 (x = 1), J2 (x = 1) for ZnCr2x Ga2−2x O4 system [10]. The values of the intra-plane and inter-plane interactions Jaa , Jab and Jac ; respectively, are deduced from the values of J1 (x) and J2 (x) for 0 ≤ x ≤ 1. The interaction energy of the magnetic structure is
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M. Hamedoun et al. / Journal of Alloys and Compounds 462 (2008) 125–128
obtained for the first system. In recent works [1,2], we have used the high-temperature series expansions (HTSE) to study the thermal and disorder variation of the short-range order (SRO) in the particular B-spinel ZnCr2x Ga2−2x S4 compound. The Pad´e approximant (PA) [11] analysis of the HTSE of the correlation length has been shown to be a useful method for the study of the critical region [12,13]. The model that we used in this present work is valid in the case of a spinel structure. The percolation threshold is deduced in the ZnCr2x Ga2−2x O4 system. We have used the HTSE technique to determine the N´eel temperature TN or the freezing temperature TSG and the critical exponents γ and ν associated with the magnetic susceptibility χ and the correlation length ξ, respectively, in the range 0 ≤ x ≤ 1 for the two systems.
Using the experimental values of TN and ϕ obtained by magnetic measurement for the spinel CoAl2x Ga2−2x O4 [7] and ZnCr2x Ga2−2x O4 [10]. We have deduced the values of exchange integrals J1 (x) and J2 (x). 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 spinel CoAl2x Ga2−2x O4 system with 0 ≤ x ≤ 1. The values obtained of the exchange integrals are J1 (x = 1) = −1.3 K, J2 (x = 1) = −0.65 K for the ZnCr2x Ga2−2x O4 system. The obtained optimums values of J1 (x) and J2 (x) are given in Table 1. The values of corresponding classical exchange energy for the magnetic structure [14] and the values of the intraplane and inter-plane interactions Jaa = 2J1 , Jab = 4J1 + 8J2 and Jac = 4J2 , respectively, are given in the same table in the range 0 ≤ x ≤ 1 for the first system.
2. Theoretical method 2.2. High-temperature series expansions 2.1. Calculation of the values of the exchange integrals from mean field approximation Starting with the well-known Heisenberg model, the Hamiltonian of the system is given by: H = −2 Jij Si Sj (1) i,j
where Jij is the exchange integral between the spins situated at sites i and j. Si is the spin operator of the spin localised at the site i. In this work we consider the nearest neighbour (nn) and next nearest neighbour (nnn) interactions J1 and J2 , respectively. H = J1 Si Sj − J2 Si Sk (2) i,j
i,k
The sums over ij and ik include all (nn) and (nnn) pair interactions, respectively. In the case of spinels containing the magnetic moment only in the octahedral sublattice, the mean field approximation leads to a simple relations between the ϕ angle of helimagnetic ordering and the N´eel temperature TN , respectively, and the considered two exchange integrals J1 and J2 . The N´eel temperature TN and the ϕ angle of helimagnetic ordering are given by Hamedoun et al. [14]: 5 TN = − [2J1 + 4J2 ] 2KB 1 J1 + 2J2 cos(ϕ) = − 4 J2
In this section we shall derive the high-temperature series expansions (HTSE) for both the zero field magnetic susceptibility χ to order six in β. The relation ship between the magnetic susceptibility per spin and the correlation functions may be expressed as follows: β Si Sj (5) χ(T ) = N ij
where β = 1/KB T and N is the number of magnetic ion and TrSi Sj e−βH Tr e−βH is the correlation function between spins at sites i and j. In ref. [1], a relation between the susceptibility and the three first correlation functions is given in the case of the Bspinel lattice with a particular ordering vector Q = (0,0,k). In the ferromagnetic case we get k = 0. The high temperature series expansion of χ(T) gives the function: Si Sj =
n 6
χ(T ) =
a(m, n)ym τ n
(6)
m=−n n=1
The high temperature series expansion of ξ 2 gives the function:
(3) ξ (T ) = 2
n 6
b(m, n)ym τ n
(7)
m=−n n=1
(4)
where y = J2 /J1 and τ = 2S(S + 1)J1 /KB T. The series coefficients a(m,n) and b(m,n) are given in ref. [15].
where KB is the Boltzmann constant.
Table 1 The Curie–Weiss temperature θ P (K), the N´eel temperature TN (K), the values of the first, second, intra-plane, inter-plane exchange integrals and the energy of CoAl2x Ga2−2x O4 as a function of dilution x x
θ P (K) [7]
1.00 0.62 0.50 0.25 0.00
97 70 65 55 65
TN (K) [7] 5 8 8 8 10
J1 /KB (K)
J2 /KB (K)
Jaa /KB (K)
Jab /KB (K)
Jac /KB (K)
|E|/KB S2 (K)
−0.50 −0.80 −0.80 −0.80 −1.00
−0.25 −0.40 −0.40 −0.40 −0.50
−1.0 −1.6 −1.6 −1.6 −2.0
−4.00 −6.40 −6.40 −6.40 −8.00
−1.0 −1.6 −1.6 −1.6 −2.0
6.0 9.6 9.6 9.6 12.00
M. Hamedoun et al. / Journal of Alloys and Compounds 462 (2008) 125–128
127
In spin-glasses (SG) critical behaviour near the TSG SG transition, it is expected not in the linear part χ0 of the dc susceptibility χ, but in the nonlinear susceptibility χs = χ − χ0 . This is due to the fact that the order parameter q in the spin glass state is not the magnetization but the quantity q=
1 [Si 2 ]av N i
As suggested by Edwards and Anderson [16], leading to an associated susceptibility χs =
1 [Si Sj 2 ]av , NT 3 ij
where the correlation length of the correlation function [Si Sj 2 ] possibly diverges at T = TSG . The behaviour of the nonlinear susceptibility has been already extensively studied theoretically and experimentally [17,18]. We have used the expression of χs , to determinate the critical temperature in the region of spin glass for ZnCr2x Ga2−2x O4 system in the range 0 ≤ x ≤ 1. Figs. 1 and 2 show magnetic phase diagrams of spinels CoAl2x Ga2−2x O4 and ZnCr2x Ga2−2x O4 systems, respectively. We can see the good agreement between the magnetic phase diagrams obtained by the HTSE technique and the experimental ones, in particular in the case of the last systems of which the phase diagrams have been established well by different methods [19–22]. The percolation threshold in the second system is xp ≈ 0.4. This value is comparable with that given by Fiorani et al. [10]. The simplest assumption that one can make concerning the nature of the singularity of the magnetic susceptibility χ(T) is that at the neighbour hood of the critical point the above two
Fig. 2. Magnetic phase diagram of ZnCr2x Ga2−2x O4 . The various phases are the paramagnetic phase (PM), antiferromagnetic phase (AFM) (0.85 ≤ x ≤ 1) and Spin glass phase (SG) (0.4 ≤ x < 0.85). The solid squares are the present results. The open circles (deduced by M¨ossbauer) and the open triangle up (deduced by magnetic measurement) [10].
functions exhibit an asymptotic behaviour: χ(T ) ∝ (T − TN )−γ
(8)
ξ 2 (T ) ∝ (TN − T )−2ν
(9)
Estimates of TN or the freezing temperature, γ and ν for CoAl2x Ga2−2x O4 and ZnCr2x Ga2−2x O4 have been obtained using the Pad´e approximate method (PA) [11]. The simple pole corresponds to TN or the freezing temperature and the residues to the critical exponents γ and ν. The obtained central values are γ = 1.380 ± 0.004, ν = 0.70 ± 0.01 and γ = 1.360 ± 0.02, ν = 0.67 ± 0.020. These values of γ and ν are nearest to the one of Heisenberg model and insensitive to the dilution. 3. Discussion and conclusion
Fig. 1. Magnetic phase diagram of CoAl2x Ga2−2x O4 . The various phases are the paramagnetic phase (PM) and antiferromagnetic phase (AFM) (0 ≤ x ≤ 1). The solid squares are the present results. The open circles represent the experimental points deduced by magnetic measurements [7].
J1 (x) and J2 (x) have been determined from mean field theory, using the experimental data of TN [6] for each dilution (see Table 1). From these values we have deduced the values of the intra-plane and inter-plane interactions Jaa , Jbb and Jac , respectively, the energy of the magnetic structure is given in the same table for the spinel CoAl2x Ga2−2x O4 system with 0 ≤ x ≤ 1. From Table 1 on can see that J1 (x) and J2 (x) increases with the absolute value when x decreases. The sign of J1 (x) and J2 (x) are negative in the whole range of concentration. The value of exchange integrals are J1 (x = 1) = −1.3 K, J2 (x = 1) = −0.65 K for the ZnCr2x Ga2−2x O4 system deduced of TN [10]. For x = 1 a long-range antiferromagnetic order is observed in the two systems. In the system ZnCr2x Ga2−2x O4 , the introduced of non-magnetic impurities perturbs this antiferromagnetic order, by consequent the magnetic spin glass phase is produced. The concentration x = 0.85 therefore corresponds to the limit
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above which long-range antiferromagnetism is detected. In the CoAl2x Ga2−2x O4 , the introduced of non-magnetic impurities in the non-magnetic do not perturbs this antiferromagnetic order for the whole range of dilution x. The high-temperature series expansion (HTSE) extrapolated with Pad´e approximants method 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 χ(T) we have estimated the critical temperature TN (in order phase) or the freezing temperature (in disorder phase) for each dilution x. The obtained magnetic phase diagrams of spinel CoAl2x Ga2−2x O4 and ZnCr2x Ga2−2x O4 systems are presented in Figs. 1 and 2, respectively. Several thermodynamic phases may appear including the paramagnetic (PM) and the antiferromagnetic (AFM) in the range 0 ≤ x ≤ 1 for the first system. The antiferromagnetic (AFM) in the range 0.85 ≤ x ≤ 1 and spin glass in the range 0.4 ≤ x < 0.85 for the second system. In these figures we have included, for comparison, the experimental results obtained by magnetic measurement. From this figure one can see good agreement between the theoretical phase diagram and experimental results. In this figure the initial increases TN with increases in the number of AB interactions suggests that these interactions stabilize a long-range order. This order is not simple and would take the form of an antiferromagnetic interaction between small clusters. The higher TN value for CoGa2 O4 than for the other samples could be due to a new kind of antiferromagnetic ordering, resulting from the new balancing between the different exchange interactions. The percolation threshold in the ZnCr2x Ga2−2x O4 system is xp ≈ 0.4. This value is comparable with that given by Fiorani et al. [10]. In the other hand, the value of critical exponents γ and ν associated to the magnetic susceptibility χ(T) and the correlation length ξ(T), have been estimated in the range of the composition 0 ≤ x ≤ 1. The sequence of [M, N] PA to series of χ(T) and ξ(T) has been evaluated. By examining the behaviour of these PA, the convergence was found to be quite rapid. Estimates of the critical exponents associated with magnetic susceptibility and the correlation length for the first and the second systems are found to be γ = 1.380 ± 0.004, ν = 0.70 ± 0.01 and γ = 1.360 ± 0.02,
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