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Magnetic phase diagram of diluted spinel Zn1−xCuxCr2Se4 system
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 320 (2008) 1431–1435 www.elsevier.com/locate/jmmm
Magnetic phase diagram of diluted spinel Zn1xCuxCr2Se4 system M. Hamedouna,, R. Masroura, K. Bouslykhanea, A. Hourmatallaha,b, N. Benzakoura a
Laboratoire de Physique du Solide, Universite´ Sidi Mohammed Ben Abdellah, Faculte´ des sciences, BP 1796, Fes, Morocco b Equipe de Physique du Solide, Ecole Normale Supe´rieure, BP 5206, Bensouda, Fes, Morocco Received 1 August 2007; received in revised form 23 November 2007 Available online 28 November 2007
Abstract By using mean field theory, we have evaluated the nearest-neighbour and the next-neighbour super-exchange J1(x) and J2(x), respectively, for Zn1xCuxCr2Se4 in the range 0pxp1. The intraplanar and the interplanar interactions are deduced. 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. The magnetic phase diagram is deduced. A spin glass phase is predicted for intermediate range of concentration. The results are comparable with those obtained by magnetic measurements. The critical exponents associated with the magnetic susceptibility (g) and the correlation lengths (n) have been deduced. The values are comparable to those of the 3D Heisenberg model, and are insensitive to the dilution x. r 2007 Elsevier B.V. All rights reserved. PACS: 74.25.Ha; 71.71.Gm; 75.40.Cx; 71.10.Hf Keywords: Spinel; Exchange interaction; Critical exponent; Phase diagram
1. Introduction Recently magnetic chalcogenides crystallising in the spinel structure have attracted considerable attention. Within the last few years exotic phenomena and fascinating ground states have been observed in this class of materials: heavy-fermion behaviour [1], complex spin order and spin dimerisation [2–4], spin–orbital liquid [5] and orbital glass [6], as well as coexistence of ferromagnetism and ferroelectricity [7,8]. They are attributed to the cooperativity and competition between charge, spin and orbital degrees of freedom, all of which are strongly coupled to the lattice. In addition, topological frustration due to the tetrahedral arrangement of the magnetic cations and bond frustration due to competing ferromagnetic (FM) and antiferromagnetic (AFM) exchange interactions hamper any simple spin and orbital arrangement in the ground state. The Zn1xCuxCr2Se4 is normal spinel with strong preference Corresponding author.
E-mail address: [email protected] (M. Hamedoun). 0304-8853/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2007.11.023
of Cr ions to locate in the octahedral (B) positions. Substitution of Zn with Cu ions gives a structural solid solution of the normal spinel structure [9] and causes fundamental changes of both magnetic and electrical properties. These systems are still the subject of extensive experimental and theoretical studies [10–14]. The stochiometric compounds ZnCr2Se4 (x=0) and CuCr2Se4 (x=1) are, respectively, semiconductor with a magnetic spiral structure [15,16], (with spiral angle j ¼ 42 1) and metallic ferromagnet [16,17]. The Ne´el temperature of the former is TN=22 K (x=0), while the Curie temperature of the latter is much higher and equal to TC=416 K (x=1). In ZnCr2Se4, with the highest Cr–Cr (dCr–Cr=3.53 A˚) separation, the direct exchange is almost suppressed and the spin arrangement follows from the dominating FM nn (nearestneighbour) 901 Cr–Se–Cr exchange and the additional AFM nnn (next nearest-neighbour) Cr–Se–Zn–Se–Cr and Cr–Se–Se–Cr exchange interactions [18]. In recent works [19], we have used the high-temperature series expansions (HTSE) to study the thermal and disorder variation of the short-range order (SRO) in the
ARTICLE IN PRESS M. Hamedoun et al. / Journal of Magnetism and Magnetic Materials 320 (2008) 1431–1435
1432
Table 1 The Ne´el temperature TN (K), the critical temperature TC (K), the Curie–Weiss temperature yp (K) given by magnetic measurement and the values of the first, second, intra-planar, inter-planar exchange integrals, the ratio of inter-to-intraplanar and the energy of the magnetic structure obtained by mean-field theory of the Zn1xCuxCr2Se4 system in this work x
TN,C (K) [24]
yp (K) [24]
J1/KB (K)
J2/KB (K)
Jaa/KB (K)
Jab/KB (K)
Jac/KB (K)
|(Jab+Jac)/Jaa|
E(K)/KB(S2)
0.00 0.01 0.02 0.025 0.05 0.07 0.1 0.2 0.3 0.5 0.7 0.8 0.9 1
22 20 16 10 14 16 20 377 382 390 395 404 408 416
118 142 188 240 276 296 332 388 391 396 411 418 427 436
7.866 9.466 12.533 16.000 18.400 19.733 22.133 50.633 51.233 52.200 53.200 54.333 55.033 56.133
2.646 3.138 4.090 5.221 5.920 6.260 6.839 12.383 12.583 12.900 12.900 13.233 13.283 13.533
15.732 18.932 25.065 32.000 36.800 39.466 44.266 101.266 102.466 104.400 106.400 108.666 110.066 112.266
10.296 12.760 17.412 22.232 26.240 28.852 33.820 103.468 104.268 105.600 109.600 111.468 113.868 116.268
10.584 12.552 16.360 20.884 23.680 25.040 27.356 49.532 50.332 51.600 51.600 52.932 53.132 54.132
0.0183 0.0109 0.0420 0.0421 0.0695 0.0965 0.1460 0.5320 0.5260 0.5170 0.5450 0.5380 0.5518 0.5530
15.444 19.140 26.118 33.348 39.360 43.278 50.730 155.202 156.402 158.400 164.400 167.202 170.802 174.402
particular B-spinel CdGa22xCr2xSe4 compounds. Three first spin correlation functions have been calculated with the aid of a diagrammatic representation. In this work, by using the mean field theory, we have calculated the nearest-neighbour and the next-neighbour super-exchange J1(x) (nn) and J2(x) (nnn), respectively, for Zn1xCuxCr2Se4 in the range 0pxp1. The values of the intraplanar and interplanar interactions Jaa(x), Jab(x) and Jac(x), respectively, are deduced from the values of J1(x) and J2(x) for 0pxp1. The interaction energy of the magnetic structure is obtained in the range 0pxp1. These values are given in Table 1. The aim of the present paper is to calculate from the results of the random phase approximation (RPA) [20,21], the magnetic susceptibility and the correlation functions for a Heisenberg FM model having both nn and nnn exchange integrals J1(x) and J2(x); respectively. Our work extends by several terms the earlier classic work on this subject by lines [21]. The HTSE of the magnetic susceptibility and correlation functions is given up b ¼ 1/KBT. The theoretical results obtained are then used to study the paramagnetic (PM) region of the spinel Zn1xCuxCr2Se4 systems in the dilution range 0pxp1. In order to determine the critical temperature TC or TN, the critical exponents g and n associated with the magnetic susceptibility w and the correlation length x; we have applied the Pade´ approximate (PA) methods [22]. The results obtained are found to be in agreement with experimental ones and can be compared with other theoretical studies based on the 3D Heisenberg model. 2. Calculation of the values of the exchange integrals Starting with the well-known Heisenberg model, the Hamiltonian of the system is given by X ~i S ~j , H ¼ 2 J ij S (1) i;j
where Jij is the exchange integral between the spins situated ~i is the operator of the spin localised at the at sites i and j. S site i. In this work, we consider the nn and nnn, J1 and J2, respectively: X X ~i S ~j J 2 ~i S ~k . H ¼ J1 (2) S S i;j
i;k
The sums over ij and ij 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 of this expression leads to simple relations between the PM Curie temperature yp and the critical temperature TC or the Ne´el temperature TN, respectively. The angle of helices j is deduced in the range 0pxp1. Following, the method of Holland and Brown [23], the expressions of TC and yp describing the Zn1xCuxCr2Se4 systems are 5 ½2J 1 4J 2 , 2K B
(3)
5 ½6J 1 þ 12J 2 , 2K B
(4)
TC ¼ yp ¼
where KB is the Boltzmann constant. In the AFM region, we have used the formula of Ne´el temperature TN given in Ref. [9]: 2 (5) T N ¼ SðS þ 1ÞlðjÞ, 3 where l(j) is the eigenvalue of the matrix formed by the Fourier transform of the exchange integral. We have used the experimental values of TC or TN, f and yp for the Zn1xCuxCr2Se4, given in Ref. [24], respectively, to determinate the values of exchange integrals J1 and J2 for each composition x. In the same table, we also give the values of the intraplanar and interplanar interactions Jaa ¼ 2J1, Jab ¼ 4J1+8J2 and Jac ¼ 4J2, respectively, and the ratio of inter-to-intraplanar interactions
ARTICLE IN PRESS M. Hamedoun et al. / Journal of Magnetism and Magnetic Materials 320 (2008) 1431–1435
J inter =J intra ¼ jðJ ab þ J ac Þ=J aa j. The interaction energy [25] of the magnetic structure is obtained in the range 0pxp1. These values are given in Table 1. 3. High-temperature series expansions In this section, we shall derive the HTSE for both the zero-field 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 J1 and J2. In the PM region (T4TC), the presence of the field h is necessary for deduced the magnetisation /SzS. We shall explicitly consider the case of a sufficiently small field, and defined the magnetic susceptibility as z hS i wðT Þ ¼ lim . (6) h!0 gmB h In the limit h-0, the quantity gmBh//SzS will approach the inverse susceptibility; w1 will approach the inverse susceptibility, whereas the correlation function approaches its isotropic value S(S+1)/3, therefore the implicit equation for w(T) in the PM phase may be evaluated in the RPA [20,21] as follows: 1 w1 ðT Þ ¼ t (7) 1 þ wj ~q
t¼
3kB T SðS þ 1Þ
where A1 ¼ hji, A2 ¼ hj2 i hji2 , A3 ¼ hj3 i 2hj2 ihji þ hji3 , A4 ¼ hj4 i 3hj3 ihji þ 3hj2 ihji2 þ hji4 . For pX1, a possible expression of Ap can be generalised in the form Ap ¼ ð1Þp1 hjip þ
p2 X
C kp1 ð1Þk hjpk ihjik ,
(14)
k¼0
where C m n are binomial coefficient defined by Cm n ¼
n! ; m!ðn mÞ!
for nXm.
If we take into account all the nn and nnn exchange interactions J1 and J2, j~q takes the form qÞÞ þ z2 J 2 ð1 g2 ð~ qÞÞ, j~q ¼ z1 J 1 ð1 g1 ð~ X
expði~ q~ di Þ,
(15) (16)
di
and
j~q ¼ ðJ 0 J ~q Þ,
where T is the absolute temperature and J q¯ is the Fourier transform of the exchange integrals Jij defined by 1X ~i R ~j Þ. J ij exp½i~ J ~q ¼ qð R (8) N The symbol h i~q is the average value when the wave vector ~ q runs over its N allowed values in the first Brillouin zone. At high-temperature, the magnetic susceptibility goes to zero; we can then expand the relation (7) as 1 w1 ðTÞ ¼ t ¼ th1 jw i~q . (9) 1 þ wj ~q To zero-order approximation, we have wðTÞ ¼ t1 .
After some algebra, we arrive at the following expression of the magnetic susceptibility: " # 1 X ð1Þp Ap 1 , (13) w ¼t 1þ tp p¼1
gi ð~ qÞ ¼ ðzi Þ1
with
1433
(10)
By replacing Eq. (10) into Eq. (7) and expanding this latter to the first-order, we obtain D E hji~q j . (11) ðwtÞ1 ¼ 1 ¼ 1 t t ~ q We replace again Eq. (11) into Eq. (7) and using the same procedure, we get ðj hjiÞ ðj hjiÞ2 ðwtÞ1 ¼ ð1 hjiÞ 1 . t t2 (12)
where di is the vector connecting ith nn, the total number of such neighbours, to a given ion, being zi. The factor gi ð~ qÞ depends on the lattice geometry. In the case of spinel structure, the method should be developed for four interpenetrating sublattices. Unfortunately, mathematical difficulties arise and it becomes progressively more difficult to obtain the analytic solution of the problem, especially when one includes the nnn exchange coupling. The inverse of the 4 4 matrices of the susceptibility is more complicated and no mathematical approximation can be used because of the translational invariance of the green functions. However, one may approximate the spinel structure by a simple cubic lattice, which have the same first and second coordinate numbers as the spinel the factor structure [26]. gi ð~ qÞ is defined by g1 ð~ qÞ ¼ 13½cosðqx aÞ þ cosðqy aÞ þ cosðqz aÞ, g2 ð~ qÞ ¼ 16
X
(17)
½cosððqx þ qy ÞaÞ þ cosððqx þ qz ÞaÞ
¼1
þ cosððqz þ qy ÞaÞ.
ð18Þ
By replacing the summation by an integral over the 3D Brillouin zone, we can write that as ZZZ 1X V jð~ qÞdqx dqy dqz . jð~ qÞ ! 3 (19) hjð~ qÞi~q ¼ N ~q 8p
ARTICLE IN PRESS M. Hamedoun et al. / Journal of Magnetism and Magnetic Materials 320 (2008) 1431–1435
1
Using the previously evaluated series of w , we may readily calculated the three first correlation functions with gaa, gab and gac:
gaa ¼ 2g1 þ 4g3 is the in plane correlation, gab ¼ 4g1 þ 8g2 is the correlation between neighbouring planes, and gac ¼ 4g2 þ 8g3 is the correlation between the secondneighbour planes.
We have computed these correlation functions as a function of temperature T and for different composition x to order 6 in b taking into account parameters which are appropriate for the FM spinel Zn1xCuxCr2Se4 system (S=3/2, z1=6 and z2=12). The experimental values of the (nn) and (nnn) interactions are taken from Table 1. The sums over the Brillouin zones were performed using Gauss approximation quadrature method similar to that given in the appendix of reference [21]. In recent works [27,28], a relation between the correlation length x(T) and the correlation functions is given in the case of the case of diluted magnetic B-spinel lattice with a particular ordering vector ~ qð0; 0; qÞ: 2 x 1 1 g2ab gac þ ¼ , (21) a 8Sð~ q0 Þ 16 gac q¼~ q0 . In the FM case, with Sð~ q0 Þ is the structure factor at ~ we get ~ q0 ¼ ~ 0. The simplest assumption that one can make concerning the nature of the singularity of the magnetic susceptibility w and the correlation length x is that the neighbourhood of the critical point the above two functions exhibit shows an asymptotic behaviour: wðTÞ / ðT C TÞg ,
4. Discussion and conclusion In this work, we have used the experimental data of TC or TN, f and Wp to derive the two first exchange integrals J1(x) and J2(x) for the spinel Zn1xCuxCr2Se4 system in the range 0pxp1. From these values, we have deduced the values of the intraplanar and interplanar interactions Jaa, Jbb and Jac, respectively. The ratio of inter to intraplanar interactions J inter =J intra ¼ jðJ ab þ J ac Þ=J aa j for 0pxp1 is presented in Table 1. The interaction energy of the magnetic structure is obtained in the range 0pxp1. These values are given in Table 1 and increases with the dilution x increases. This increasing is due to the increases of the critical temperature of the Zn1xCuxCr2Se4 system. By employing the results of the HTSE for a randomly diluted Heisenberg magnet, we have deduced the magnetic phase diagram of Zn1xCuxCr2Se4 in the range 0pxp1. The obtained result is presented in Fig. 1. Several thermodynamic phases may appear including the PM, the AFM in the range 0pxp0.1, the spin glass (SG) in the range 0.1oxo0.2 and the FM for 0.2pxp1. The passage of a phase to the other caused by the variation of exchange interactions. This variation is observed in the energy of the magnetic structure (see Table 1). The values of critical 60 420 PM
50
400
FM
40
380
32
TC (K)
~ q
The sequence of [M, N] PA to both the logðwðTÞÞ and logðx2 ðTÞÞ was found to be convergent. The simple pole corresponds to TC or TN and the residues to the critical exponents g and n for different dilutions are given for Zn1xCuxCr2Se4 system.
PM
20
30 TN (K)
The summation is to be taken for qx, qy, qz running independently between p/a and +p/a. V volume of the unit cell and a is the lattice parameter. The relationship between the reduced correlation func~0 S ~n i=SðS þ 1Þ and the inverse magnetic tion gn ¼ hS 1 susceptibility w is given by [20,21] * + ~0 S ~n i ~n R ~0 Þ hS exp½i~ qð R ¼t gn ¼ . (20) SðS þ 1Þ j þ w1
TN (K)
1434
(22)
24
15
SG
(23)
Estimates of TC or TN concentration range 0pxp1 and the critical exponents g and n associated with the magnetic susceptibility magnetic w(T) and correlation length x(T), respectively, for Zn1xCuxCr2Se4 in the region ordered, have been obtained using the PA methods [22]. The [M, N] PA to the series w(b) is a rational PM/QN with PM and QN polynomials of degrees M and N in b satisfying d PM log wðbÞ ¼ þ yðbMþNþ1 Þ. db QN
20
10 0.0
10
16
AFM
5
AFM
x2 ðTÞ / ðT C TÞ2n .
0.00
0.01
0.02 x
0.2
0.4
0.6
0.8
0 1.0
x Fig. 1. Magnetic phase diagram of Zn1xCuxCr2Se4. The various phases are the PM phase, the antiferromagnetic phase (AFM) 0pxp0.1, ferromagnetic (FM) 0.2pxp1 and the spin glass state (SG) 0.1oxo0.2. The solid line is the present results. The solid circles represent the experimental points deduced by Ref. [24].
ARTICLE IN PRESS M. Hamedoun et al. / Journal of Magnetism and Magnetic Materials 320 (2008) 1431–1435
exponents g and n associated to the magnetic susceptibility w(T) and the correlation length x(T), respectively, have been estimated in the region ordered. We have obtained the central values of g and n, g ¼ 1:329 0:002 and n ¼ 0:709 0:001. These values are insensitive to the dilution in ordering phase. To conclude, it would be interesting to compare the critical exponents g with other theoretical values. A lot of methods of extracting critical exponents have been given in the literature. We have selected many of the methods, and summarised our findings below. In the critical region, i.e. 5 104pe ¼ (TCT)/ TCp5 103, Zarek [29] has found experimentally by magnetic balance for CdCr2Se4, g ¼ 1:29 0:02; for HgCr2Se4, g ¼ 1:30 0:02 and for CuCr2Se4, g ¼ 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 FM 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. Acknowledgement This study is a part of the PROTARS III D12/10. References [1] S. Kondo, et al., Phys. Rev. Lett. 78 (1997) 3729; A. Krimmel, et al., Phys. Rev. Lett. 82 (1999) 2919. [2] S.-H. Lee, et al., Nature (London) 418 (2002) 856. [3] P.G. Radaelli, et al., Nature (London) 416 (2002) 155. [4] M. Schmidt, et al., Phys. Rev. Lett. 92 (2004) 056402. [5] V. Fritsch, et al., Phys. Rev. Lett. 92 (2004) 116401; A. Krimmel, et al., Phys. Rev. Lett. 94 (2005) 237402. [6] R. Fichtl, et al., Phys. Rev. Lett. 94 (2005) 027601.
1435
[7] J. Hemberger, et al., Nature (London) 434 (2005) 364; S. Weber, et al., Phys. Rev. Lett. 96 (2006) 157202. [8] Y. Yamasaki, et al., Phys. Rev. Lett. 96 (2006) 207204. [9] M. Hamedoun, A. Rachadi, A. Hourmatallah, D. El Allam, A. Benyoussef, Phys. Status Solidi (b) 191 (1995) 503. [10] S. Juszczyk, Physica B 161 (1989) 183. [11] S. Juszczyk, J. Magn. Magn. Mater. 83 (1990) 473. [12] M.N. Khan, A. Ventkatalachan, A. Ahmud, V.S. Darshane, J. Mater. Sci. 25 (1990) 595. [13] M. Hidaka, N. Tokiwa, T.T. Wai, H. Fujii, K. Ohoyama, T. Mizushima, J. Sakurai, F. Ichikawa, K. Itoh, Phase Transitions 75 (2002) 273; M. Hidaka, N. Tokiwa, T.T. Wai, H. Fujii, M. Mitoh, K. Takeda, Appl. Phys. A Mater. Sci. Process. 74 (2002) 820. [14] N. Tokiwa, M. Hidaka, Phys. Status Solidi (b) 241 (2004) 1910. [15] R. Plumier, C. R. Acad. Sci. 260 (1965) 3348. [16] F.K. Lotgering, in: Proceedings of the International Conference on Magnetizm, Nottingham 1964, Institute of Physics, London, 1965, p. 533. [17] R. Plumier, M. Lecomte, A. Miedan-Gros, M. Sougi, Phys. Lett. A 55 (1975) 239. [18] N. Menyuk, et al., J. Appl. Phys. 37 (1966) 1387; K. Baltzer, et al., Phys. Rev. 151 (1966) 367. [19] H. Bakrim, K. Bouslykhane, M. Hamedoun, A. Hourmatallah, N. Benzakour, J. Magn. Magn. Mater. 285 (2005) 327. [20] A.P. Young, B.S. Shastry, J. Phys. C 15 (1982) 4547. [21] M.E. Lines, Phys. Rev. 139 (1965) A1304. [22] G.A. Backer, P. Graves-Morris (Eds.), Pade´ Approximants, AdissonWesley, London, 1981. [23] W.E. Holland, H.A. Brown, Phys. Status Solidi (a) 10 (1972) 249. [24] J. Krok, J. Spalek, S. Juszczyk, J. Warczewski, Phys. Rev. B 28 (1983) 6499. [25] M. Hamedoun, A. Wiedenmann, J.L. Dormann. M. Nogues, J. Rossat-Mignod, J. Phys. C: Solid State Phys. 19 (1986) 1783. [26] J.M. Weisselinowa, Phys. Status Solidi 120 (1983) 585; J.M. Weisselinowa, A.T. Apostolov, J. Phys.: Condens. Matter 8 (1996) 473. [27] M. Hamedoun, M. Houssa, N.Y. Cherriet, F.Z. Bakkali, Phys. Status Solidi (b) 214 (1999) 403. [28] M. Hamedoun, M. Houssa, N. Benzakour, A. Hourmatallah, Physica B 270 (1999) 384. [29] W. Zarek, Acta Phys. Pol. A 52 (1977) 657.
Journal of Magnetism and Magnetic Materials 320 (2008) 1431–1435 www.elsevier.com/locate/jmmm
Magnetic phase diagram of diluted spinel Zn1xCuxCr2Se4 system M. Hamedouna,, R. Masroura, K. Bouslykhanea, A. Hourmatallaha,b, N. Benzakoura a
Laboratoire de Physique du Solide, Universite´ Sidi Mohammed Ben Abdellah, Faculte´ des sciences, BP 1796, Fes, Morocco b Equipe de Physique du Solide, Ecole Normale Supe´rieure, BP 5206, Bensouda, Fes, Morocco Received 1 August 2007; received in revised form 23 November 2007 Available online 28 November 2007
Abstract By using mean field theory, we have evaluated the nearest-neighbour and the next-neighbour super-exchange J1(x) and J2(x), respectively, for Zn1xCuxCr2Se4 in the range 0pxp1. The intraplanar and the interplanar interactions are deduced. 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. The magnetic phase diagram is deduced. A spin glass phase is predicted for intermediate range of concentration. The results are comparable with those obtained by magnetic measurements. The critical exponents associated with the magnetic susceptibility (g) and the correlation lengths (n) have been deduced. The values are comparable to those of the 3D Heisenberg model, and are insensitive to the dilution x. r 2007 Elsevier B.V. All rights reserved. PACS: 74.25.Ha; 71.71.Gm; 75.40.Cx; 71.10.Hf Keywords: Spinel; Exchange interaction; Critical exponent; Phase diagram
1. Introduction Recently magnetic chalcogenides crystallising in the spinel structure have attracted considerable attention. Within the last few years exotic phenomena and fascinating ground states have been observed in this class of materials: heavy-fermion behaviour [1], complex spin order and spin dimerisation [2–4], spin–orbital liquid [5] and orbital glass [6], as well as coexistence of ferromagnetism and ferroelectricity [7,8]. They are attributed to the cooperativity and competition between charge, spin and orbital degrees of freedom, all of which are strongly coupled to the lattice. In addition, topological frustration due to the tetrahedral arrangement of the magnetic cations and bond frustration due to competing ferromagnetic (FM) and antiferromagnetic (AFM) exchange interactions hamper any simple spin and orbital arrangement in the ground state. The Zn1xCuxCr2Se4 is normal spinel with strong preference Corresponding author.
E-mail address: [email protected] (M. Hamedoun). 0304-8853/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2007.11.023
of Cr ions to locate in the octahedral (B) positions. Substitution of Zn with Cu ions gives a structural solid solution of the normal spinel structure [9] and causes fundamental changes of both magnetic and electrical properties. These systems are still the subject of extensive experimental and theoretical studies [10–14]. The stochiometric compounds ZnCr2Se4 (x=0) and CuCr2Se4 (x=1) are, respectively, semiconductor with a magnetic spiral structure [15,16], (with spiral angle j ¼ 42 1) and metallic ferromagnet [16,17]. The Ne´el temperature of the former is TN=22 K (x=0), while the Curie temperature of the latter is much higher and equal to TC=416 K (x=1). In ZnCr2Se4, with the highest Cr–Cr (dCr–Cr=3.53 A˚) separation, the direct exchange is almost suppressed and the spin arrangement follows from the dominating FM nn (nearestneighbour) 901 Cr–Se–Cr exchange and the additional AFM nnn (next nearest-neighbour) Cr–Se–Zn–Se–Cr and Cr–Se–Se–Cr exchange interactions [18]. In recent works [19], we have used the high-temperature series expansions (HTSE) to study the thermal and disorder variation of the short-range order (SRO) in the
ARTICLE IN PRESS M. Hamedoun et al. / Journal of Magnetism and Magnetic Materials 320 (2008) 1431–1435
1432
Table 1 The Ne´el temperature TN (K), the critical temperature TC (K), the Curie–Weiss temperature yp (K) given by magnetic measurement and the values of the first, second, intra-planar, inter-planar exchange integrals, the ratio of inter-to-intraplanar and the energy of the magnetic structure obtained by mean-field theory of the Zn1xCuxCr2Se4 system in this work x
TN,C (K) [24]
yp (K) [24]
J1/KB (K)
J2/KB (K)
Jaa/KB (K)
Jab/KB (K)
Jac/KB (K)
|(Jab+Jac)/Jaa|
E(K)/KB(S2)
0.00 0.01 0.02 0.025 0.05 0.07 0.1 0.2 0.3 0.5 0.7 0.8 0.9 1
22 20 16 10 14 16 20 377 382 390 395 404 408 416
118 142 188 240 276 296 332 388 391 396 411 418 427 436
7.866 9.466 12.533 16.000 18.400 19.733 22.133 50.633 51.233 52.200 53.200 54.333 55.033 56.133
2.646 3.138 4.090 5.221 5.920 6.260 6.839 12.383 12.583 12.900 12.900 13.233 13.283 13.533
15.732 18.932 25.065 32.000 36.800 39.466 44.266 101.266 102.466 104.400 106.400 108.666 110.066 112.266
10.296 12.760 17.412 22.232 26.240 28.852 33.820 103.468 104.268 105.600 109.600 111.468 113.868 116.268
10.584 12.552 16.360 20.884 23.680 25.040 27.356 49.532 50.332 51.600 51.600 52.932 53.132 54.132
0.0183 0.0109 0.0420 0.0421 0.0695 0.0965 0.1460 0.5320 0.5260 0.5170 0.5450 0.5380 0.5518 0.5530
15.444 19.140 26.118 33.348 39.360 43.278 50.730 155.202 156.402 158.400 164.400 167.202 170.802 174.402
particular B-spinel CdGa22xCr2xSe4 compounds. Three first spin correlation functions have been calculated with the aid of a diagrammatic representation. In this work, by using the mean field theory, we have calculated the nearest-neighbour and the next-neighbour super-exchange J1(x) (nn) and J2(x) (nnn), respectively, for Zn1xCuxCr2Se4 in the range 0pxp1. The values of the intraplanar and interplanar interactions Jaa(x), Jab(x) and Jac(x), respectively, are deduced from the values of J1(x) and J2(x) for 0pxp1. The interaction energy of the magnetic structure is obtained in the range 0pxp1. These values are given in Table 1. The aim of the present paper is to calculate from the results of the random phase approximation (RPA) [20,21], the magnetic susceptibility and the correlation functions for a Heisenberg FM model having both nn and nnn exchange integrals J1(x) and J2(x); respectively. Our work extends by several terms the earlier classic work on this subject by lines [21]. The HTSE of the magnetic susceptibility and correlation functions is given up b ¼ 1/KBT. The theoretical results obtained are then used to study the paramagnetic (PM) region of the spinel Zn1xCuxCr2Se4 systems in the dilution range 0pxp1. In order to determine the critical temperature TC or TN, the critical exponents g and n associated with the magnetic susceptibility w and the correlation length x; we have applied the Pade´ approximate (PA) methods [22]. The results obtained are found to be in agreement with experimental ones and can be compared with other theoretical studies based on the 3D Heisenberg model. 2. Calculation of the values of the exchange integrals Starting with the well-known Heisenberg model, the Hamiltonian of the system is given by X ~i S ~j , H ¼ 2 J ij S (1) i;j
where Jij is the exchange integral between the spins situated ~i is the operator of the spin localised at the at sites i and j. S site i. In this work, we consider the nn and nnn, J1 and J2, respectively: X X ~i S ~j J 2 ~i S ~k . H ¼ J1 (2) S S i;j
i;k
The sums over ij and ij 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 of this expression leads to simple relations between the PM Curie temperature yp and the critical temperature TC or the Ne´el temperature TN, respectively. The angle of helices j is deduced in the range 0pxp1. Following, the method of Holland and Brown [23], the expressions of TC and yp describing the Zn1xCuxCr2Se4 systems are 5 ½2J 1 4J 2 , 2K B
(3)
5 ½6J 1 þ 12J 2 , 2K B
(4)
TC ¼ yp ¼
where KB is the Boltzmann constant. In the AFM region, we have used the formula of Ne´el temperature TN given in Ref. [9]: 2 (5) T N ¼ SðS þ 1ÞlðjÞ, 3 where l(j) is the eigenvalue of the matrix formed by the Fourier transform of the exchange integral. We have used the experimental values of TC or TN, f and yp for the Zn1xCuxCr2Se4, given in Ref. [24], respectively, to determinate the values of exchange integrals J1 and J2 for each composition x. In the same table, we also give the values of the intraplanar and interplanar interactions Jaa ¼ 2J1, Jab ¼ 4J1+8J2 and Jac ¼ 4J2, respectively, and the ratio of inter-to-intraplanar interactions
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J inter =J intra ¼ jðJ ab þ J ac Þ=J aa j. The interaction energy [25] of the magnetic structure is obtained in the range 0pxp1. These values are given in Table 1. 3. High-temperature series expansions In this section, we shall derive the HTSE for both the zero-field 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 J1 and J2. In the PM region (T4TC), the presence of the field h is necessary for deduced the magnetisation /SzS. We shall explicitly consider the case of a sufficiently small field, and defined the magnetic susceptibility as z hS i wðT Þ ¼ lim . (6) h!0 gmB h In the limit h-0, the quantity gmBh//SzS will approach the inverse susceptibility; w1 will approach the inverse susceptibility, whereas the correlation function approaches its isotropic value S(S+1)/3, therefore the implicit equation for w(T) in the PM phase may be evaluated in the RPA [20,21] as follows: 1 w1 ðT Þ ¼ t (7) 1 þ wj ~q
t¼
3kB T SðS þ 1Þ
where A1 ¼ hji, A2 ¼ hj2 i hji2 , A3 ¼ hj3 i 2hj2 ihji þ hji3 , A4 ¼ hj4 i 3hj3 ihji þ 3hj2 ihji2 þ hji4 . For pX1, a possible expression of Ap can be generalised in the form Ap ¼ ð1Þp1 hjip þ
p2 X
C kp1 ð1Þk hjpk ihjik ,
(14)
k¼0
where C m n are binomial coefficient defined by Cm n ¼
n! ; m!ðn mÞ!
for nXm.
If we take into account all the nn and nnn exchange interactions J1 and J2, j~q takes the form qÞÞ þ z2 J 2 ð1 g2 ð~ qÞÞ, j~q ¼ z1 J 1 ð1 g1 ð~ X
expði~ q~ di Þ,
(15) (16)
di
and
j~q ¼ ðJ 0 J ~q Þ,
where T is the absolute temperature and J q¯ is the Fourier transform of the exchange integrals Jij defined by 1X ~i R ~j Þ. J ij exp½i~ J ~q ¼ qð R (8) N The symbol h i~q is the average value when the wave vector ~ q runs over its N allowed values in the first Brillouin zone. At high-temperature, the magnetic susceptibility goes to zero; we can then expand the relation (7) as 1 w1 ðTÞ ¼ t ¼ th1 jw i~q . (9) 1 þ wj ~q To zero-order approximation, we have wðTÞ ¼ t1 .
After some algebra, we arrive at the following expression of the magnetic susceptibility: " # 1 X ð1Þp Ap 1 , (13) w ¼t 1þ tp p¼1
gi ð~ qÞ ¼ ðzi Þ1
with
1433
(10)
By replacing Eq. (10) into Eq. (7) and expanding this latter to the first-order, we obtain D E hji~q j . (11) ðwtÞ1 ¼ 1 ¼ 1 t t ~ q We replace again Eq. (11) into Eq. (7) and using the same procedure, we get ðj hjiÞ ðj hjiÞ2 ðwtÞ1 ¼ ð1 hjiÞ 1 . t t2 (12)
where di is the vector connecting ith nn, the total number of such neighbours, to a given ion, being zi. The factor gi ð~ qÞ depends on the lattice geometry. In the case of spinel structure, the method should be developed for four interpenetrating sublattices. Unfortunately, mathematical difficulties arise and it becomes progressively more difficult to obtain the analytic solution of the problem, especially when one includes the nnn exchange coupling. The inverse of the 4 4 matrices of the susceptibility is more complicated and no mathematical approximation can be used because of the translational invariance of the green functions. However, one may approximate the spinel structure by a simple cubic lattice, which have the same first and second coordinate numbers as the spinel the factor structure [26]. gi ð~ qÞ is defined by g1 ð~ qÞ ¼ 13½cosðqx aÞ þ cosðqy aÞ þ cosðqz aÞ, g2 ð~ qÞ ¼ 16
X
(17)
½cosððqx þ qy ÞaÞ þ cosððqx þ qz ÞaÞ
¼1
þ cosððqz þ qy ÞaÞ.
ð18Þ
By replacing the summation by an integral over the 3D Brillouin zone, we can write that as ZZZ 1X V jð~ qÞdqx dqy dqz . jð~ qÞ ! 3 (19) hjð~ qÞi~q ¼ N ~q 8p
ARTICLE IN PRESS M. Hamedoun et al. / Journal of Magnetism and Magnetic Materials 320 (2008) 1431–1435
1
Using the previously evaluated series of w , we may readily calculated the three first correlation functions with gaa, gab and gac:
gaa ¼ 2g1 þ 4g3 is the in plane correlation, gab ¼ 4g1 þ 8g2 is the correlation between neighbouring planes, and gac ¼ 4g2 þ 8g3 is the correlation between the secondneighbour planes.
We have computed these correlation functions as a function of temperature T and for different composition x to order 6 in b taking into account parameters which are appropriate for the FM spinel Zn1xCuxCr2Se4 system (S=3/2, z1=6 and z2=12). The experimental values of the (nn) and (nnn) interactions are taken from Table 1. The sums over the Brillouin zones were performed using Gauss approximation quadrature method similar to that given in the appendix of reference [21]. In recent works [27,28], a relation between the correlation length x(T) and the correlation functions is given in the case of the case of diluted magnetic B-spinel lattice with a particular ordering vector ~ qð0; 0; qÞ: 2 x 1 1 g2ab gac þ ¼ , (21) a 8Sð~ q0 Þ 16 gac q¼~ q0 . In the FM case, with Sð~ q0 Þ is the structure factor at ~ we get ~ q0 ¼ ~ 0. The simplest assumption that one can make concerning the nature of the singularity of the magnetic susceptibility w and the correlation length x is that the neighbourhood of the critical point the above two functions exhibit shows an asymptotic behaviour: wðTÞ / ðT C TÞg ,
4. Discussion and conclusion In this work, we have used the experimental data of TC or TN, f and Wp to derive the two first exchange integrals J1(x) and J2(x) for the spinel Zn1xCuxCr2Se4 system in the range 0pxp1. From these values, we have deduced the values of the intraplanar and interplanar interactions Jaa, Jbb and Jac, respectively. The ratio of inter to intraplanar interactions J inter =J intra ¼ jðJ ab þ J ac Þ=J aa j for 0pxp1 is presented in Table 1. The interaction energy of the magnetic structure is obtained in the range 0pxp1. These values are given in Table 1 and increases with the dilution x increases. This increasing is due to the increases of the critical temperature of the Zn1xCuxCr2Se4 system. By employing the results of the HTSE for a randomly diluted Heisenberg magnet, we have deduced the magnetic phase diagram of Zn1xCuxCr2Se4 in the range 0pxp1. The obtained result is presented in Fig. 1. Several thermodynamic phases may appear including the PM, the AFM in the range 0pxp0.1, the spin glass (SG) in the range 0.1oxo0.2 and the FM for 0.2pxp1. The passage of a phase to the other caused by the variation of exchange interactions. This variation is observed in the energy of the magnetic structure (see Table 1). The values of critical 60 420 PM
50
400
FM
40
380
32
TC (K)
~ q
The sequence of [M, N] PA to both the logðwðTÞÞ and logðx2 ðTÞÞ was found to be convergent. The simple pole corresponds to TC or TN and the residues to the critical exponents g and n for different dilutions are given for Zn1xCuxCr2Se4 system.
PM
20
30 TN (K)
The summation is to be taken for qx, qy, qz running independently between p/a and +p/a. V volume of the unit cell and a is the lattice parameter. The relationship between the reduced correlation func~0 S ~n i=SðS þ 1Þ and the inverse magnetic tion gn ¼ hS 1 susceptibility w is given by [20,21] * + ~0 S ~n i ~n R ~0 Þ hS exp½i~ qð R ¼t gn ¼ . (20) SðS þ 1Þ j þ w1
TN (K)
1434
(22)
24
15
SG
(23)
Estimates of TC or TN concentration range 0pxp1 and the critical exponents g and n associated with the magnetic susceptibility magnetic w(T) and correlation length x(T), respectively, for Zn1xCuxCr2Se4 in the region ordered, have been obtained using the PA methods [22]. The [M, N] PA to the series w(b) is a rational PM/QN with PM and QN polynomials of degrees M and N in b satisfying d PM log wðbÞ ¼ þ yðbMþNþ1 Þ. db QN
20
10 0.0
10
16
AFM
5
AFM
x2 ðTÞ / ðT C TÞ2n .
0.00
0.01
0.02 x
0.2
0.4
0.6
0.8
0 1.0
x Fig. 1. Magnetic phase diagram of Zn1xCuxCr2Se4. The various phases are the PM phase, the antiferromagnetic phase (AFM) 0pxp0.1, ferromagnetic (FM) 0.2pxp1 and the spin glass state (SG) 0.1oxo0.2. The solid line is the present results. The solid circles represent the experimental points deduced by Ref. [24].
ARTICLE IN PRESS M. Hamedoun et al. / Journal of Magnetism and Magnetic Materials 320 (2008) 1431–1435
exponents g and n associated to the magnetic susceptibility w(T) and the correlation length x(T), respectively, have been estimated in the region ordered. We have obtained the central values of g and n, g ¼ 1:329 0:002 and n ¼ 0:709 0:001. These values are insensitive to the dilution in ordering phase. To conclude, it would be interesting to compare the critical exponents g with other theoretical values. A lot of methods of extracting critical exponents have been given in the literature. We have selected many of the methods, and summarised our findings below. In the critical region, i.e. 5 104pe ¼ (TCT)/ TCp5 103, Zarek [29] has found experimentally by magnetic balance for CdCr2Se4, g ¼ 1:29 0:02; for HgCr2Se4, g ¼ 1:30 0:02 and for CuCr2Se4, g ¼ 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 FM 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. Acknowledgement This study is a part of the PROTARS III D12/10. References [1] S. Kondo, et al., Phys. Rev. Lett. 78 (1997) 3729; A. Krimmel, et al., Phys. Rev. Lett. 82 (1999) 2919. [2] S.-H. Lee, et al., Nature (London) 418 (2002) 856. [3] P.G. Radaelli, et al., Nature (London) 416 (2002) 155. [4] M. Schmidt, et al., Phys. Rev. Lett. 92 (2004) 056402. [5] V. Fritsch, et al., Phys. Rev. Lett. 92 (2004) 116401; A. Krimmel, et al., Phys. Rev. Lett. 94 (2005) 237402. [6] R. Fichtl, et al., Phys. Rev. Lett. 94 (2005) 027601.
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