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Thermodynamic properties of the ferromagnetic spinel ZnpCd1−pCr2Se4
Journal of Magnetism and Magnetic Materials 247 (2002) 242–248
Thermodynamic properties of the ferromagnetic spinel ZnpCd1pCr2Se4 Y. Cherrieta,b, M. Hamedounb,*, A. Chatwitic b
a International Centre for Theoretical Physics, P.O. Box 586, 34100, Trieste, Italy Laboratoire de Physique du Solide, Univercite! Mohammed Ben Abdellah, Faculte! des Science Dhar Mahraz, B.P. 1796, " Atlas, Morocco Fes c ! Laboratoire de Physique Theorique et des Plasmas, Univercite! Mohammed Ben Abdellah, ! Atlas, Morocco Faculte! des Science Dhar Mahraz, B.P. 1796, Fes
Received 8 November 2000
Abstract The magnetic properties of the B-spinel ferromagnetic ZnpCd1pCr2Se4 compound are studied via a cluster series expansion approximation with nearest and next-nearest exchange integrals J1 and J2, respectively. Using the minimization of the free-energy expressions, the magnetization, the magnetic susceptibility, the two-spin correlation functions and the specific heat are obtained and computed numerically as a function of temperature and for each composition of the system. The magnetization curves are used to determine the critical temperatures Tc : Using the power laws in the vicinity of the critical regions the critical exponents b; g and a associated, respectively, with the magnetization, the magnetic susceptibility and the specific heat are numerically calculated. The critical temperatures Tc obtained are in very good agreement with those predicted by the magnetic measurements and the values of the critical exponents may be compared with other theoretical results based on the 3D Heisenberg model. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Spin; Spinel; Heisenberg model; Critical temperature; Critical exponents
1. Introduction Material 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, the tetrahedral (A) and the 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, *Corresponding author. Fax.: +2125-64-25-00. E-mail address: [email protected] (M. Hamedoun).
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 [1–10]. In the spinel solution ApA0 1pB2X4, the magnetic B ion is located in the tetrahedral sites of the cubic spinel lattice. The A and A0 ions are divalent metal ions. The X ions can be anions of the chalcogenide group. The weakly diluted magnetic system ZnpCd1pCr2Se4 is a particular example of this family. In a previous work [7–9], attention has
0304-8853/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 0 1 3 5 - 4
Y. Cherriet et al. / Journal of Magnetism and Magnetic Materials 247 (2002) 242–248
been given to the magnetic properties and critical behaviour of the system ZnpCd1pCr2Se4 in the range of concentration 0ppp1. The main feature of these investigations can be briefly summarized as follows: For 0ppp0.45 the members of this family exhibit semiconducting and ferromagnetic behaviour. For 0.60ppp1, the samples behave antiferromagnetically and in an intermediate range of composition 0.45ppp0.49, a spin glass behaviour has been observed. On the other hand, the two-spin cluster expansion approximation with nearest and next-nearest exchange integrals J1 and J2 ; respectively [11,12], has proved to be a very useful tool for investigating a variety of the magnetic properties of the Europium chalcogenides such as EuX (X O, S, Se, Te, etc). The method is considered as a modification of the mean field theory (MFT) for three-dimensional (3D) Heisenberg ferromagnets with arbitrary spin values, but it takes into account the correlation effect and the calculations include contributions energy up to two-spin clusters. The method has been applied to analyse some magnetic properties of the Europium chalcogenides and has been shown to give correct prediction of the Europium thermodynamic properties. We have used this technique to compute numerically the spontaneous magnetization, the magnetic susceptibility, the two-spin correlation functions and the specific heat as a function of the temperature in the case of the compositional system ZnpCd1pCr2Se4. The magnetization curves are used to determine the critical temperatures Tc : Using the power laws in the vicinity of the critical points, the critical exponents b; g and a associated, respectively, with the magnetization, magnetic susceptibility and the specific heat have been numerically calculated. The critical temperatures Tc obtained are in very good agreement with those predicted by the magnetic experimental results and the critical exponents may be compared with other theoretical approaches based on the 3D Heisenberg model. The paper is organized as follows: In Section 2, we present the solution of the two-spin cluster system by the partition function method with particular numerical application to our system. In Section 3, we discuss and present our conclusions.
243
2. The two-spin cluster approximation Starting from the well-known Heisenberg model, the Hamiltonian of the system is given by X X ~j gmB h Jij S~i S Siz ð1Þ H ¼ 2 hij i
i
~i is the operator of the spin localized at the where S lattice site ~ ri ; Jij is the exchange integral between ~i and S ~j corresponding to the localized spins S lattice sites i and j; g is the gyromagnetic ratio, mB is the Bohr magneton and h is the external magnetic field. The first summation is over all nearest-neighbour spin pairs, the second one is over all sites of the lattice. The /ijS denotes a pair of nearestneighbour sites. Following Strieb et al. [11,12], we introduce the spin deviation operator at each site and divide the Hamiltonian into two parts, an unperturbed term and a perturbed term. The unperturbed term Hamiltonian consists of only those terms which are linear in the spin operator. si ¼ S Siz
ð2Þ
In terms of si ; the Hamiltonian H becomes H ¼ H0 þ V with H0 ¼ gmB hNS 2S 2
ð3Þ X
Jij
hi; j i
þ
X gmB h þ 2SJi ð0Þ si
ð4Þ
i
and V ¼ 2
X
~i S~j S z S z þ si sj ; Jij S i j
ð5Þ
hi; j i
P where Ji ð0Þ ¼ j Jij : The average unperturbed free energy is given by bF0 ¼ lnðTr expðbH0 ÞÞ
ð6Þ
where b ¼ 1=kB T; kB is the Boltzmann constant and T is the absolute temperature. The perturbed free energy is given by Tr expðbH0 ÞexpðbV Þ bF 0 ¼ bF þ bF0 ¼Ln Tr expðbH0 Þ ¼ LnhexpðbV Þi ð7Þ
244
Y. Cherriet et al. / Journal of Magnetism and Magnetic Materials 247 (2002) 242–248
Strieb et al. [8,9] made a cluster expansion of Eq. (7); the two-spin cluster result is given by bFð02Þ ¼
X
Ln exp bVij
hi; j i
¼
X
Ln Tr 2bJij Hij expðbH0 Þ
hi; j i
Ln Tr expðbH0 Þ;
ð8Þ
where ~j S% S z þ S z þ S%2 : Hij ¼ S~i S i j
ð9Þ
Following the method of Callen and Callen [12] in which two-spin clusters have been selected and interacted between nearest and nextnearest neighbours, and on elaborating Eq. (8) in the space in which the total spin number ~1 þ S ~2 and its zth component Rz ¼ S z þ Sz ~¼ S R 1 2 are diagonal, the statistical average /Sz S is found to be hS z i ¼
z1 J1 hSz i12 þz2 J2 hS z i13 ; J0
ð10Þ
where J0 ¼ z1 J1 þ z2 J2 and z1 and z2 are the coordination numbers. The two-particle averages /S z S12 and /S z S13 may be derived from the two-particle partition functions Z12 and Z13 which have been given explicitly in detail in the original paper of the authors of Refs. [8,9,12]. All the derived equations consist of a set of selfconsistent equations which must be solved numerically. We have computed numerically the spontaneous magnetization given by Eq. (10) taking into account parameters which are appropriate for the ferromagnetic spinel CdCr2Se4 (S ¼ 3=2; J1 =kB ¼ 14:00 K and J2 =kB ¼ 0:10 K) [13]. The results of the computation are shown in Fig. 1 where we plot /S z S=S as a function of the temperature T between absolute zero and the critical point. From the figure it is seen that we obtain Tc =132 K, while the experimental value for this substance is 129.5 K [13]. This is in very good agreement with our numerical calculations.
Fig. 1. The numerical results for the reduced magnetization /Sz S for different compositions: (a) p ¼ 0; (b) p ¼ 0:35; (c) p ¼ 0:41 (d) p ¼ 0:45:
In order to extend the above method to include mixed compositional systems ZnpCd1pCr2Se4, we will assume that magnetic properties of each mixed composition can be described in terms of the spin Hamiltonian of Eq. (1), but having compositiondependent effective exchange interaction parameters J1 ðpÞ and J2 ðpÞ which are obtained on the basis of the magnetic results combined with the MFT [7]. The critical temperatures obtained for p=0, 0.35, 0.41 and 0.45 are also presented in Table 1 with the experimental data for comparison. The magnetization curves obtained are typical for ferromagnets and are plotted versus temperature for the four compositions (see Fig. 1). Fig. 2 shows the thermal variation of the magnetic susceptibility wðTÞ defined by q z wð T Þ ¼ hS i qh h-0 for different compositions of the system. Below Tc the susceptibility increases with the temperature, diverges at T ¼ Tc and follows the Curie–Weiss law at high temperature.
Y. Cherriet et al. / Journal of Magnetism and Magnetic Materials 247 (2002) 242–248
245
Table 1 Critical temperature Tc obtained from the numerical calculations of the cluster series expansion approximation together with those of the experimental data for comparison [7,9] p
J1 =kB ðKÞ [7,9]
J2 =kB ðKÞ [7,9]
Tc (72 K) experience [7,9]
Tc ðKÞ present work
0 0.35 0.41 0.45
14.00 12.36 11.38 10.66
0.10 2.90 0.91 0.36
129.5 66.0 58.0 52.0
132.0 68.5 60.0 54.0
Fig. 3. The first nearest neighbour correlation g1 ¼ ~1 S~2 i=S2 plotted against the temperature for different hS concentrations of the system ZnpCd1pCr2Se4.
These correlation functions are also of direct physical interest in other connections such as the specific heat and the diffuse magnetic scattering of neutrons. We have computed g1 and g2 for different concentrations of the system; the results of the computation are shown graphically in Figs. 3 and 4. Another important quantity, which can be derived from (16), is the specific heat Cv and it is given by Cv ðT Þ ¼ Nz1 J1 Fig. 2. Plots of the magnetic susceptibility wðTÞ versus temperature for various values of the concentration.
The first and second correlation functions are given by ~1 S ~2 i hS 1 qLnðZ12 Þ ¼ 2 ; 2 S 4S qLnx1 ~1 S ~3 i hS 1 qLnðZ13 Þ g2 ¼ ¼ 2 : 2 S 4S qLnx2
g1 ¼
ð11Þ
qg1 qg Nz2 J2 2 ; qT qT
ð12Þ
where N is the number of magnetic ions per mole. Fig. 5 shows the thermal dependence of the specific heat Cv for various values of the concentration p: This latter quantity presents a pronounced l-like discontinuity at the Curietemperature. As the concentration p decreases, the peak becomes better defined and decreases in temperature. The physical interpretation of the discontinuity observed in the specific heat at the Curie-temperature has been explained by the quantum fluctuation of the spin [14].
246
Y. Cherriet et al. / Journal of Magnetism and Magnetic Materials 247 (2002) 242–248 Table 2 Optimum values of the critical exponents p
b (70.030)
g (70.025)
a (70.011)
0 0.35 0.41
0.394 0.397 0.399
1.405 1.409 1.411
0.122 0.125 0.128
0.45
0.421
1.448
0.142
asymptotic behaviour: hSz iEðTc T Þb ;
wðT ÞEðT Tc Þg
and Cv ðT ÞEðT Tc Þa : we have calculated numerically b; g and a for each composition p from: ~1 S~3 i=S2 Fig. 4. Next-nearest neighbour correlation g2 ¼ hS plotted against the temperature for different concentrations of the system ZnpCd1pCr2Se4.
lnhS z i ¼ A þ b lnðTc T Þ;
ð13Þ
ln wðT Þ ¼ B g lnðT Tc Þ;
ð14Þ
ln Cv ðT Þ ¼ C a lnðT Tc Þ;
ð15Þ
where A; B and C are constants. The obtained optimum values of the critical exponents are given in Table 2.
3. Conclusion
Fig. 5. Magnetic contribution to the specific heat Cv ðTÞ as a function of temperature for various values of the concentration.
The simplest assumption that one can make concerning the nature of the singularity of the magnetization, the magnetic susceptibility and the specific heat is that in the neighbourhood of the critical points the above functions exhibit an
By applying the formalism of the two-spin cluster technique to the diluted spinel ferromagnetic ZnpCd1pCr2Se4 compound, we have studied the behaviour of the spontaneous magnetization, magnetic susceptibility, two-spin correlation functions and the specific heat as a function of temperature and nonmagnetic dilution. Our numerical calculations show that in the close vicinity of the critical points Tc ; the magnetization curves change slope and tend vertically to the temperature axis; therefore, they provide much better estimation of the critical temperatures. The obtained values of Tc in the range of concentration 0ppp0.45 are in very good agreement with the experimental data [7]. The main feature of the first and second correlation function curves is the decrease with
Y. Cherriet et al. / Journal of Magnetism and Magnetic Materials 247 (2002) 242–248
temperature T and site dilution p; i.e. the order is destroyed by the thermal disorder and the nonmagnetic dilution. The first correlation function decreases with temperature, abruptly changes slope at the Curietemperature and persists far in the paramagnetic region. The second correlation function reduces to a negative value at Tc ; abruptly changes slope and persists in the paramagnetic region except for p=0.45 which becomes positive; this may be linked to the sign of J2. The values of critical exponents b; g and a have been estimated in the range of the composition 0ppp0.45. These values may be compared with those of the 3D Heisenberg model namely, b=0.367, g=1.386 and a=0.121 [15–17]. For p=0.45, the exponents deviate from this model and approach those found in the reentrant system [18]. The same analysis has been applied to the ferromagnetic spinel CdCr2pIn22pS4, in the hole of the composition 0ppp1, and the critical exponents obtained are also close to those of the 3D Heisenberg model; we may hence conclude that the magnetic or nonmagnetic dilution in the spinel ferromagnetic compounds does not modify considerably the values of the critical exponents. To conclude, it would be interesting to compare the critical exponents with other theoretical values. Several methods of extracting critical exponents have been given in the literature. We have selected many of these methods, and summarize our findings below. In the critical region, i.e. 5 104pe=(TcT)/ Tcp5 103 Zarek [19] has found experimentally by magnetic balance for CdCr2Se4: b=0.3470.02, and g=1.2970.02; for HgCr2Se4: b=0.3470.02, and g=1.3070.02 and for CuCr2Se4: b= 0.3770.02, and g=1.3270.02. Using the macroscopic magnetization measurements and the neutron diffraction experiments, Pouget et al. [10] obtained for CdCr2S4 b= 0.3270.03 and for CdCr1.9In0.1S4 b=0.3070.03. Using the high-temperature series expansions (HTSE) Hamedoun et al. [8] have found the central values of g; g=1.38170.014 for compounds p=0.35 and 0.41.
247
Le Guillou et al. [20] calculated the critical exponents of the n-vector model through fieldtheoretical methods and presented new and more precise values. In the case of n ¼ 3 (isotropic Heisenberg model) they are: b=0.364770.0012 and g=1.386670.0012. In the mean field theory, we have b ¼ 0:5 and g ¼ 1: This failure of the mean field approximation in the critical region is well known in the theory of phase transition. Although the co-ordination numbers, the spin value and the exchange integrals for the Europium chalcogenides are quite different from those of the spinels, our numerical calculation curves are qualitatively similar to those of Europium. These remarks have also been noted by Callen when he applied the same analysis to the ferromagnetic spinel CdCr2Se4 [21]. Based on the above considerations, it may be concluded that the results of the two-particle cluster approximation describe not only the magnetic properties of the Europium chalcogenides (EuS, EuOy) but also the properties of ferromagnetic chalcogenide spinels such as ZnpCd1pCr2Se4.
Acknowledgements One of the authors (Y.C) 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 Comite! International Universitaire Maroc-Espagnol under grant No: 62/SEE/98.
References [1] M. Hamedoun, M. Houssa, N. Benzakour, A. Hourmatallah, J. Phys: Condens. Matter 10 (1998) 3611. [2] M. Hamedoun, M. Houssa, Y. Cherriet, F.Z. Bakkali, Phys. Stat. Sol. B 214 (1999) 403. [3] M. Hamedoun, M. Houssa, N. Benzakour, A. Hourmatallah, Physica B 270 (1999) 384. [4] M. Lubecka, L.J. Maksymowicz, Phys. Rev. B 44 (1991) 10106.
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[5] J. Krok-Kowalski, J. Warczewski, T. Midlarz, A. Pacyna, I. Okonska-Kozlowska, J. Magn. Magn. Mater. 166 (1997) 172. [6] A.I. Abramovich, L.I. Koroleva, L.N. Lukina, Phys. Solid State 41 (1999) 73. [7] M. Hamedoun, A. Hourmatallah, A. Zerhouni, N. Benzakour, A. Belayachi, M. Nogues, Phys. Stat. Sol. B 203 (1997) 521. [8] N. Benzakour, M. Hamedoun, M. Houssa, A. Hourmatallah, F. Mahjoubi, Phys. Stat. Sol. B 212 (1999) 335. [9] M. Nogues, J.L. Dormann, J. Magn. Magn. Mater. 54 (1986) 87. [10] S. Pouget, M. Alba, J. Phys. Condens. Matter 7 (1995) 4739. [11] B. Strieb, H. Callen, G. Horwitz, Phys. Rev. 130 (1963) 1798.
[12] H.B. Callen, E. Callen, Phys. Rev. 136 (1964) A1675. [13] P.K. Baltzer, P.J. Wojtowicz, M. Robbins, E. Lopatin, Phys. Rev. 38 (1966) 953. [14] R.S. Fishmann, H.S. Liu, Phys. Rev. B 40 (1989) 11028. [15] For a review, see M.F. Collins, Magnetic Critical Scattering, Oxford University Press, Oxford, 1989. [16] G.A. Baker Jr., B.G. Nickel, D.I. Meiron, Phys. Rev. B 17 (1978) 1365. [17] M.G. George, J.J. Rehr, Phys. Rev. Lett. 53 (1984) 2063. [18] K. Westerholt, H. Bachem, R. Romer, J. Magn. Magn. Mater. 45 (1982) 252. [19] W. Zarek, Acta. Phys. Polon. A 52 (1977) 657. [20] J.C. Le Guillou, J. Zinn-Justin, Phys. Rev. Lett. 2 (1977) 95. [21] E. Callen, Phys. Rev. Lett. 20 (1968) 1045.
Thermodynamic properties of the ferromagnetic spinel ZnpCd1pCr2Se4 Y. Cherrieta,b, M. Hamedounb,*, A. Chatwitic b
a International Centre for Theoretical Physics, P.O. Box 586, 34100, Trieste, Italy Laboratoire de Physique du Solide, Univercite! Mohammed Ben Abdellah, Faculte! des Science Dhar Mahraz, B.P. 1796, " Atlas, Morocco Fes c ! Laboratoire de Physique Theorique et des Plasmas, Univercite! Mohammed Ben Abdellah, ! Atlas, Morocco Faculte! des Science Dhar Mahraz, B.P. 1796, Fes
Received 8 November 2000
Abstract The magnetic properties of the B-spinel ferromagnetic ZnpCd1pCr2Se4 compound are studied via a cluster series expansion approximation with nearest and next-nearest exchange integrals J1 and J2, respectively. Using the minimization of the free-energy expressions, the magnetization, the magnetic susceptibility, the two-spin correlation functions and the specific heat are obtained and computed numerically as a function of temperature and for each composition of the system. The magnetization curves are used to determine the critical temperatures Tc : Using the power laws in the vicinity of the critical regions the critical exponents b; g and a associated, respectively, with the magnetization, the magnetic susceptibility and the specific heat are numerically calculated. The critical temperatures Tc obtained are in very good agreement with those predicted by the magnetic measurements and the values of the critical exponents may be compared with other theoretical results based on the 3D Heisenberg model. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Spin; Spinel; Heisenberg model; Critical temperature; Critical exponents
1. Introduction Material 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, the tetrahedral (A) and the 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, *Corresponding author. Fax.: +2125-64-25-00. E-mail address: [email protected] (M. Hamedoun).
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 [1–10]. In the spinel solution ApA0 1pB2X4, the magnetic B ion is located in the tetrahedral sites of the cubic spinel lattice. The A and A0 ions are divalent metal ions. The X ions can be anions of the chalcogenide group. The weakly diluted magnetic system ZnpCd1pCr2Se4 is a particular example of this family. In a previous work [7–9], attention has
0304-8853/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 0 1 3 5 - 4
Y. Cherriet et al. / Journal of Magnetism and Magnetic Materials 247 (2002) 242–248
been given to the magnetic properties and critical behaviour of the system ZnpCd1pCr2Se4 in the range of concentration 0ppp1. The main feature of these investigations can be briefly summarized as follows: For 0ppp0.45 the members of this family exhibit semiconducting and ferromagnetic behaviour. For 0.60ppp1, the samples behave antiferromagnetically and in an intermediate range of composition 0.45ppp0.49, a spin glass behaviour has been observed. On the other hand, the two-spin cluster expansion approximation with nearest and next-nearest exchange integrals J1 and J2 ; respectively [11,12], has proved to be a very useful tool for investigating a variety of the magnetic properties of the Europium chalcogenides such as EuX (X O, S, Se, Te, etc). The method is considered as a modification of the mean field theory (MFT) for three-dimensional (3D) Heisenberg ferromagnets with arbitrary spin values, but it takes into account the correlation effect and the calculations include contributions energy up to two-spin clusters. The method has been applied to analyse some magnetic properties of the Europium chalcogenides and has been shown to give correct prediction of the Europium thermodynamic properties. We have used this technique to compute numerically the spontaneous magnetization, the magnetic susceptibility, the two-spin correlation functions and the specific heat as a function of the temperature in the case of the compositional system ZnpCd1pCr2Se4. The magnetization curves are used to determine the critical temperatures Tc : Using the power laws in the vicinity of the critical points, the critical exponents b; g and a associated, respectively, with the magnetization, magnetic susceptibility and the specific heat have been numerically calculated. The critical temperatures Tc obtained are in very good agreement with those predicted by the magnetic experimental results and the critical exponents may be compared with other theoretical approaches based on the 3D Heisenberg model. The paper is organized as follows: In Section 2, we present the solution of the two-spin cluster system by the partition function method with particular numerical application to our system. In Section 3, we discuss and present our conclusions.
243
2. The two-spin cluster approximation Starting from the well-known Heisenberg model, the Hamiltonian of the system is given by X X ~j gmB h Jij S~i S Siz ð1Þ H ¼ 2 hij i
i
~i is the operator of the spin localized at the where S lattice site ~ ri ; Jij is the exchange integral between ~i and S ~j corresponding to the localized spins S lattice sites i and j; g is the gyromagnetic ratio, mB is the Bohr magneton and h is the external magnetic field. The first summation is over all nearest-neighbour spin pairs, the second one is over all sites of the lattice. The /ijS denotes a pair of nearestneighbour sites. Following Strieb et al. [11,12], we introduce the spin deviation operator at each site and divide the Hamiltonian into two parts, an unperturbed term and a perturbed term. The unperturbed term Hamiltonian consists of only those terms which are linear in the spin operator. si ¼ S Siz
ð2Þ
In terms of si ; the Hamiltonian H becomes H ¼ H0 þ V with H0 ¼ gmB hNS 2S 2
ð3Þ X
Jij
hi; j i
þ
X gmB h þ 2SJi ð0Þ si
ð4Þ
i
and V ¼ 2
X
~i S~j S z S z þ si sj ; Jij S i j
ð5Þ
hi; j i
P where Ji ð0Þ ¼ j Jij : The average unperturbed free energy is given by bF0 ¼ lnðTr expðbH0 ÞÞ
ð6Þ
where b ¼ 1=kB T; kB is the Boltzmann constant and T is the absolute temperature. The perturbed free energy is given by Tr expðbH0 ÞexpðbV Þ bF 0 ¼ bF þ bF0 ¼Ln Tr expðbH0 Þ ¼ LnhexpðbV Þi ð7Þ
244
Y. Cherriet et al. / Journal of Magnetism and Magnetic Materials 247 (2002) 242–248
Strieb et al. [8,9] made a cluster expansion of Eq. (7); the two-spin cluster result is given by bFð02Þ ¼
X
Ln exp bVij
hi; j i
¼
X
Ln Tr 2bJij Hij expðbH0 Þ
hi; j i
Ln Tr expðbH0 Þ;
ð8Þ
where ~j S% S z þ S z þ S%2 : Hij ¼ S~i S i j
ð9Þ
Following the method of Callen and Callen [12] in which two-spin clusters have been selected and interacted between nearest and nextnearest neighbours, and on elaborating Eq. (8) in the space in which the total spin number ~1 þ S ~2 and its zth component Rz ¼ S z þ Sz ~¼ S R 1 2 are diagonal, the statistical average /Sz S is found to be hS z i ¼
z1 J1 hSz i12 þz2 J2 hS z i13 ; J0
ð10Þ
where J0 ¼ z1 J1 þ z2 J2 and z1 and z2 are the coordination numbers. The two-particle averages /S z S12 and /S z S13 may be derived from the two-particle partition functions Z12 and Z13 which have been given explicitly in detail in the original paper of the authors of Refs. [8,9,12]. All the derived equations consist of a set of selfconsistent equations which must be solved numerically. We have computed numerically the spontaneous magnetization given by Eq. (10) taking into account parameters which are appropriate for the ferromagnetic spinel CdCr2Se4 (S ¼ 3=2; J1 =kB ¼ 14:00 K and J2 =kB ¼ 0:10 K) [13]. The results of the computation are shown in Fig. 1 where we plot /S z S=S as a function of the temperature T between absolute zero and the critical point. From the figure it is seen that we obtain Tc =132 K, while the experimental value for this substance is 129.5 K [13]. This is in very good agreement with our numerical calculations.
Fig. 1. The numerical results for the reduced magnetization /Sz S for different compositions: (a) p ¼ 0; (b) p ¼ 0:35; (c) p ¼ 0:41 (d) p ¼ 0:45:
In order to extend the above method to include mixed compositional systems ZnpCd1pCr2Se4, we will assume that magnetic properties of each mixed composition can be described in terms of the spin Hamiltonian of Eq. (1), but having compositiondependent effective exchange interaction parameters J1 ðpÞ and J2 ðpÞ which are obtained on the basis of the magnetic results combined with the MFT [7]. The critical temperatures obtained for p=0, 0.35, 0.41 and 0.45 are also presented in Table 1 with the experimental data for comparison. The magnetization curves obtained are typical for ferromagnets and are plotted versus temperature for the four compositions (see Fig. 1). Fig. 2 shows the thermal variation of the magnetic susceptibility wðTÞ defined by q z wð T Þ ¼ hS i qh h-0 for different compositions of the system. Below Tc the susceptibility increases with the temperature, diverges at T ¼ Tc and follows the Curie–Weiss law at high temperature.
Y. Cherriet et al. / Journal of Magnetism and Magnetic Materials 247 (2002) 242–248
245
Table 1 Critical temperature Tc obtained from the numerical calculations of the cluster series expansion approximation together with those of the experimental data for comparison [7,9] p
J1 =kB ðKÞ [7,9]
J2 =kB ðKÞ [7,9]
Tc (72 K) experience [7,9]
Tc ðKÞ present work
0 0.35 0.41 0.45
14.00 12.36 11.38 10.66
0.10 2.90 0.91 0.36
129.5 66.0 58.0 52.0
132.0 68.5 60.0 54.0
Fig. 3. The first nearest neighbour correlation g1 ¼ ~1 S~2 i=S2 plotted against the temperature for different hS concentrations of the system ZnpCd1pCr2Se4.
These correlation functions are also of direct physical interest in other connections such as the specific heat and the diffuse magnetic scattering of neutrons. We have computed g1 and g2 for different concentrations of the system; the results of the computation are shown graphically in Figs. 3 and 4. Another important quantity, which can be derived from (16), is the specific heat Cv and it is given by Cv ðT Þ ¼ Nz1 J1 Fig. 2. Plots of the magnetic susceptibility wðTÞ versus temperature for various values of the concentration.
The first and second correlation functions are given by ~1 S ~2 i hS 1 qLnðZ12 Þ ¼ 2 ; 2 S 4S qLnx1 ~1 S ~3 i hS 1 qLnðZ13 Þ g2 ¼ ¼ 2 : 2 S 4S qLnx2
g1 ¼
ð11Þ
qg1 qg Nz2 J2 2 ; qT qT
ð12Þ
where N is the number of magnetic ions per mole. Fig. 5 shows the thermal dependence of the specific heat Cv for various values of the concentration p: This latter quantity presents a pronounced l-like discontinuity at the Curietemperature. As the concentration p decreases, the peak becomes better defined and decreases in temperature. The physical interpretation of the discontinuity observed in the specific heat at the Curie-temperature has been explained by the quantum fluctuation of the spin [14].
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Y. Cherriet et al. / Journal of Magnetism and Magnetic Materials 247 (2002) 242–248 Table 2 Optimum values of the critical exponents p
b (70.030)
g (70.025)
a (70.011)
0 0.35 0.41
0.394 0.397 0.399
1.405 1.409 1.411
0.122 0.125 0.128
0.45
0.421
1.448
0.142
asymptotic behaviour: hSz iEðTc T Þb ;
wðT ÞEðT Tc Þg
and Cv ðT ÞEðT Tc Þa : we have calculated numerically b; g and a for each composition p from: ~1 S~3 i=S2 Fig. 4. Next-nearest neighbour correlation g2 ¼ hS plotted against the temperature for different concentrations of the system ZnpCd1pCr2Se4.
lnhS z i ¼ A þ b lnðTc T Þ;
ð13Þ
ln wðT Þ ¼ B g lnðT Tc Þ;
ð14Þ
ln Cv ðT Þ ¼ C a lnðT Tc Þ;
ð15Þ
where A; B and C are constants. The obtained optimum values of the critical exponents are given in Table 2.
3. Conclusion
Fig. 5. Magnetic contribution to the specific heat Cv ðTÞ as a function of temperature for various values of the concentration.
The simplest assumption that one can make concerning the nature of the singularity of the magnetization, the magnetic susceptibility and the specific heat is that in the neighbourhood of the critical points the above functions exhibit an
By applying the formalism of the two-spin cluster technique to the diluted spinel ferromagnetic ZnpCd1pCr2Se4 compound, we have studied the behaviour of the spontaneous magnetization, magnetic susceptibility, two-spin correlation functions and the specific heat as a function of temperature and nonmagnetic dilution. Our numerical calculations show that in the close vicinity of the critical points Tc ; the magnetization curves change slope and tend vertically to the temperature axis; therefore, they provide much better estimation of the critical temperatures. The obtained values of Tc in the range of concentration 0ppp0.45 are in very good agreement with the experimental data [7]. The main feature of the first and second correlation function curves is the decrease with
Y. Cherriet et al. / Journal of Magnetism and Magnetic Materials 247 (2002) 242–248
temperature T and site dilution p; i.e. the order is destroyed by the thermal disorder and the nonmagnetic dilution. The first correlation function decreases with temperature, abruptly changes slope at the Curietemperature and persists far in the paramagnetic region. The second correlation function reduces to a negative value at Tc ; abruptly changes slope and persists in the paramagnetic region except for p=0.45 which becomes positive; this may be linked to the sign of J2. The values of critical exponents b; g and a have been estimated in the range of the composition 0ppp0.45. These values may be compared with those of the 3D Heisenberg model namely, b=0.367, g=1.386 and a=0.121 [15–17]. For p=0.45, the exponents deviate from this model and approach those found in the reentrant system [18]. The same analysis has been applied to the ferromagnetic spinel CdCr2pIn22pS4, in the hole of the composition 0ppp1, and the critical exponents obtained are also close to those of the 3D Heisenberg model; we may hence conclude that the magnetic or nonmagnetic dilution in the spinel ferromagnetic compounds does not modify considerably the values of the critical exponents. To conclude, it would be interesting to compare the critical exponents with other theoretical values. Several methods of extracting critical exponents have been given in the literature. We have selected many of these methods, and summarize our findings below. In the critical region, i.e. 5 104pe=(TcT)/ Tcp5 103 Zarek [19] has found experimentally by magnetic balance for CdCr2Se4: b=0.3470.02, and g=1.2970.02; for HgCr2Se4: b=0.3470.02, and g=1.3070.02 and for CuCr2Se4: b= 0.3770.02, and g=1.3270.02. Using the macroscopic magnetization measurements and the neutron diffraction experiments, Pouget et al. [10] obtained for CdCr2S4 b= 0.3270.03 and for CdCr1.9In0.1S4 b=0.3070.03. Using the high-temperature series expansions (HTSE) Hamedoun et al. [8] have found the central values of g; g=1.38170.014 for compounds p=0.35 and 0.41.
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Le Guillou et al. [20] calculated the critical exponents of the n-vector model through fieldtheoretical methods and presented new and more precise values. In the case of n ¼ 3 (isotropic Heisenberg model) they are: b=0.364770.0012 and g=1.386670.0012. In the mean field theory, we have b ¼ 0:5 and g ¼ 1: This failure of the mean field approximation in the critical region is well known in the theory of phase transition. Although the co-ordination numbers, the spin value and the exchange integrals for the Europium chalcogenides are quite different from those of the spinels, our numerical calculation curves are qualitatively similar to those of Europium. These remarks have also been noted by Callen when he applied the same analysis to the ferromagnetic spinel CdCr2Se4 [21]. Based on the above considerations, it may be concluded that the results of the two-particle cluster approximation describe not only the magnetic properties of the Europium chalcogenides (EuS, EuOy) but also the properties of ferromagnetic chalcogenide spinels such as ZnpCd1pCr2Se4.
Acknowledgements One of the authors (Y.C) 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 Comite! International Universitaire Maroc-Espagnol under grant No: 62/SEE/98.
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