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Exchange interactions in Gd1−xCexMn2Ge2 compounds
Journal of Alloys and Compounds 368 (2004) 8–12
Exchange interactions in Gd1−x Cex Mn2 Ge2 compounds S. Kervan∗ Materials Research Department, Ankara Nuclear Research and Training Centre, Turkish Atomic Energy Authority, 06100 Besevler, Ankara, Turkey Received 25 June 2003; received in revised form 30 July 2003; accepted 30 July 2003
Abstract The temperature dependence of the magnetisation and exchange interactions in Gd1−x Cex Mn2 Ge2 compounds have been investigated by the molecular field theory based on two-sublattice model. The molecular field coefficients nMnMn , nGdMn and nGdGd , obeying the hierarchical relation nMnMn > |nGdMn | > nGdGd , were obtained by a numerical fitting process. The exchange coupling constants JMnMn , JGdMn and JGdGd have been calculated by using the molecular field coefficients. The Mn–Mn exchange energy increases with increasing x. JMnMn dominates the three kinds of interactions, implying that the magnetic interactions are dominated by the exchange between 3d electrons. The theoretically calculated Curie temperatures were found to agree well with the experimental results. The exchange fields HMn (T) and HGd (T) were presented. © 2003 Elsevier B.V. All rights reserved. Keywords: Rare-earth intermetallic compounds; Molecular field theory; Curie temperature; Ferrimagnetism
1. Introduction The ternary intermetallic RT2 X2 compounds, where R is a rare earth element, T is a 3d transition metal and X is Si or Ge, crystallize in the body-centred tetragonal ThCr2 Si2 -type crystal structure with the space group I4/mmm, in which R, T and X atoms occupy 2a(0 0 0), 4d(0 1/2 1/4) and 4e(0 0 z) sites, respectively. The unit cell consists of two formula units and may be described as a stacking of atomic layers along the c-axis direction with the sequence T–X–R–X–T. This layer crystal structure leads to a variety of magnetic properties and complex magnetic behaviour [1–9]. In this crystal structure, the Ce moments do not order [6]. The two-sublattice molecular field theory analysis of binary systems based on RFe3 and R6 Fe23 (R = rare earth) was investigated by employing molecular field theory study, demonstrating that two-sublattice molecular field model is capable of rather accurately describing the temperature dependence of the magnetisation for RFe3 and R6 Fe23 [10,11]. Later works have been carried out for other 3d–4f intermetallic compounds [12–25]. In this study, for obtaining more information on the magnetic behaviour of Gd and Mn ion and determining the exchange interactions, the temperature dependence of the ∗
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magnetisation of Gd1−x Cex Mn2 Ge2 compounds has been analysed using the molecular field theory. In addition, for analysing the spin-reorientation phenomenon and the temperature dependence of the magnetocrystalline anisotropy energy in these compounds, the temperature dependence of the exchange fields is needed. Up to now, the relation between the temperature and the exchange field could only be obtained by the molecular field theory analysis of the magnetic experimental data.
2. Two-sublattice molecular field approximation In these calculations, it is assumed that Gd and Mn moments are rigidly antiparallel along the c-axis over the entire temperature range, and the presence of intralayer antiferromagnetic Mn–Mn interactions is neglected. According to molecular field theory based on two-sublattice model, the molecular fields acting on gadolinium and manganese sublattices in an applied magnetic field Ha can be written as follows: HGd (T) = Ha + (1 − x)nGdGd MGd (T) + 2nGdMn MMn (T), (1) HMn (T) = Ha + (1 − x)nMnGd MGd (T) + 2nMnMn MMn (T), (2)
S. Kervan / Journal of Alloys and Compounds 368 (2004) 8–12
where we assume that nGdMn = nMnGd and nMnMn , nGdMn and nGdGd , describing the Mn–Mn, Gd–Mn and Gd–Gd magnetic interactions, are the molecular field coefficients, respectively. Temperature dependence of each sublattice magnetisation can be described by Brillouin function: MGd (0)HGd (T) MGd (T) = MGd (0)BJGd , (3) kB T MMn (0)HMn (T) MMn (T) = MMn (0)BJMn , (4) kB T where MGd (0) and MMn (0) are the zero temperature magnetic moments of the Gd and Mn ions, JGd and JMn are the total angular momenta of Gd and Mn, and Brillouin function has the standard definition 2J + 1 2J + 1 1 1 BJ (x) = coth x − coth x . (5) 2J 2J 2J 2J In calculations, the free ion magnetic moment is used for gadolinium (MGd (0) = gGd JGd ). MMn (0) is derived from the experimental data measured at low temperature by assuming that Gd and Mn moments couple antiferromagnetically. The coefficients nMnMn , nGdMn and nGdGd are determined by numerically solving Eqs. (1)–(4) subject to condition that the calculated total magnetic moments Mtot (T) = |(1 − x)MGd (T) − 2MMn (T)| correspond best with the experimental data. This is done by minimizing the percentage deviation |Mexp (Ti ) − Mtot (Ti )| R = 100 , (6) Mexp (Ti )
Moreover, there are relationships between the exchange energies and the spin coupling constants: aMnMn = ZJMn (JMn + 1)JMnMn , aGdMn = [Z1 Z2 JMn (JMn + 1)(gGd − 1) × JGd (JGd + 1)]1/2 JGdMn , aGdGd = Z3 (gGd − 1)JGd (JGd + 1)JGdGd ,
3JGd kB α= , 2 (0) (1 − x)µB (JGd + 1)MGd 3JMn kB β= . 2 (0) 2µB (JMn + 1)MMn
(8)
On the other hand, molecular field theory provides us with the relationship [19,26] 3kB TC = (aMnMn + aGdGd ) + [(aMnMn + aGdGd )2 2 − 4(aMnMn aGdGd − aGdMn )]1/2 .
(9)
By comparing Eqs. (7) and (9), the exchange energies aMnMn , aGdMn and aGdGd can be obtained from the molecular field coefficients as follows: 3kB nMnMn , 2β 3kB = nGdGd . 2α
aMnMn = aGdGd
3kB aGdMn = √ nGdMn , 2 αβ (10)
(11)
where Z (4) and Z2 (4) are the numbers of Mn and Gd nearest neighbours of the Mn ion, respectively; Z1 (8) and Z3 (4) are the numbers of Mn and Gd nearest neighbours of the Gd ion, respectively. JMnMn , JGdMn and JGdGd are the spin coupling constants between Mn–Mn, Gd–Mn and Gd–Gd. The number of the nearest neighbours was determined by using SHELXL97 [27] program. The values of JMnMn , JGdMn and JGdGd can be obtained by substituting the values of aMnMn , aGdMn and aGdGd into Eq. (11).
3. Results and discussion The magnetic properties of Gd1−x Cex Mn2 Ge2 compounds have been previously studied by means of X-ray diffraction and magnetic measurements [6]. The temperature dependence of magnetisation of Gd1−x Cex Mn2 Ge2 compounds are plotted in Fig. 1. The circles represent the experimental data, while molecular field theory results for the total moment and sublattice magnetizations are indicated by solid line. Table 1 summarizes the experimental information and obtained parameters. The experimental values agree quite
where Mexp (Ti ) is the experimental total moment at temperature Ti . The Curie temperature can be derived from Eqs. (1)–(4) in zero applied field: αnMnMn + βnGdGd + (αnMnMn + βnGdGd )2 − 4αβ(nMnMn nGdGd − n2GdMn ) TC = , 2αβ where
9
(7)
well with the theoretical values (R < 4%), indicating that two-sublattice molecular field theory is successful in describing the temperature dependence of magnetisation for Gd1−x Cex Mn2 Ge2 compounds. The negative values of nGdMn are consistent with the antiferromagnetic interactions between the Gd and Mn moments. The values of aMnMn , aGdMn , aGdGd , JMnMn , JGdMn and JGdGd are listed in Table 2. These values agree well with the exchange interaction parameters calculated by two-sublattice molecular field approximation [12–14,18]. The intersublattice coupling constant JGdMn can be obtained more accurately by means of inelastic neutron scattering (INS) [28]. The JGdMn value for Gd1−x Cex Mn2 Ge2 compounds calculated in the present paper is about 1 × 10−22 J, which is greater than 0.35 × 10−22 J and 0.4 × 10−22 J obtained from inelastic neutron scattering for GdMn2 Ge2 and GdMn2 Si2 compounds [28]. Deviation from the inelastic neutron scattering results seems to be due to the neglecting of the intralayer antiferromagnetic Mn–Mn interactions.
10
S. Kervan / Journal of Alloys and Compounds 368 (2004) 8–12
Fig. 1. Temperature dependence of the magnetisation of Gd1−x Cex Mn2 Ge2 compounds. Circles denote the measured values and solid line represents molecular field theory results.
Table 1 Rare-earth gyromagnetic ratio gGd , total angular momentum JGd and molecular field theory coefficients x
0.17 0.33 0.5 0.67
gGd
JGd
2 2 2 2
7/2 7/2 7/2 7/2
MGd (0) (µB )
7 7 7 7
MMn (0) (µB )
2.08 1.92 1.92 1.69
nMnMn (Gs/µB )
915760 1052230 1072770 1304700
nGdMn (Gs/µB )
−154810 −164270 −131180 −124100
nGdGd (Gs/µB )
45150 72530 9740 0
R (%)
3.9 3.8 2.4 1.1
TC (K) Calc.
Exp.
381 383 373 372
346 348 341 338
Table 2 The exchange energies and coupling constants x
aMnMn (10−22 J)
aGdMn (10−22 J)
aGdGd (10−22 J)
JMnMn (10−22 J)
JGdMn (10−22 J)
JGdGd (10−22 J)
0.17 0.33 0.5 0.67
72.1 73.6 74.9 75.9
−33.2 −33.2 −26.5 −22.9
26.4 42.4 5.7 0
8.49 9.76 9.95 12.10
−1.02 −1.08 −0.86 −0.81
0.42 0.67 0.09 0
S. Kervan / Journal of Alloys and Compounds 368 (2004) 8–12
11
Table 3 Coefficients A, B and C appearing in Eq. (12)
Fig. 2. The variation of JMnMn , JGdMn and JGdGd with Ce concentration.
x
HGd (0) (×106 Gs)
A
B
C
0.17 0.33 0.5 0.67
1.062 1.025 0.989 0.844
0.290 0.410 0.033 −0.038
−0.036 −0.240 0.097 0.303
−1.114 −0.863 −0.762 −0.856
Fig. 2 shows the variation of JMnMn , JGdMn and JGdGd with Ce concentration. The substitution of Ce at Gd site affects the exchange interactions. JMnMn increases with increasing x. Therefore, Ce substitution strengthens the Mn–Mn exchange as a result of increasing interatomic distances. It is interesting to note that the change in the intersublattice coupling constant JGdMn is small irrespective of the degree of dilution with Ce atoms. It has been reported, through band structure calculations, that the rare earth–transition metal coupling is proportional to the ratio of 3d and 5d spins [29,30]. The substitution Ce atoms may result in an increasing of the 3d and 5d spins by the same factor resulting in an aproximately constant JGdMn . The exchange coupling constant JGdGd is the smallest and vanishes with increasing x. The temperature dependence of HMn (T) and HGd (T) for Gd1−x Cex Mn2 Ge2 compounds, calculated by using molec-
Fig. 3. Temperature dependence of HMn (T) and HGd (T).
12
S. Kervan / Journal of Alloys and Compounds 368 (2004) 8–12
ular field coefficients, is plotted in Fig. 3. In the study of the relation between the magneto-crystalline anisotropy energy and the temperature, it is necessary to know the property of HGd (T). For the sake of application, the numerical results of HGd (T) are fitted to analytic form by the least-squares method 2 2 T T T HGd (T) = HGd (0) 1+A +B +C . TC TC TC (12) The parameters A, B and C are listed in Table 3. 4. Conclusions The two-sublattice molecular field theory described successfully the temperature dependence of magnetisation for Gd1−x Cex Mn2 Ge2 compounds. The molecular field coefficients nMnMn , nGdMn and nGdGd have been calculated iteratively. The exchange energies and coupling constants have been obtained from the molecular field coefficients. It has been found that the Ce substitution for Gd in these compounds has little influence on Gd–Mn exchange interaction. References [1] Z. Ban, M. Sikirica, Acta Cryst. 18 (1965) 594. [2] K.S.V.L. Narasimhan, V.U.S. Rao, R.L. Bergner, W.E. Wallace, J. Appl. Phys. 46 (1975) 4957. [3] H. Fujii, T. Okamoto, T. Shigeoka, N. Iawata, Solid State Commun. 53 (1985) 715. [4] A. Szytula, J. Leciejewicz, in: K.A. Gschneidner Jr., L. Eyring (Eds.), Handbook on the Physics and Chemistry of Rare Earths, vol. 12, Elsevier, Amsterdam, 1989, p. 133. [5] R. Welter, G. Venturini, E. Ressouche, B. Malaman, J. Alloys Compd. 218 (1995) 204.
[6] S. Kervan, M. Acet, Y. Elerman, Solid State Commun. 119 (2001) 95. [7] S. Kervan, Y. Elerman, M. Acet, J. Alloys Compd. 321 (2001) 35. [8] S. Kervan, Y. Elerman, A. Elmali, R. Theissmann, J. Alloys Compd. 327 (2001) 27. [9] S. Kervan, Y. Elerman, M. Acet, J. Alloys Compd. 335 (2002) 70. [10] J.F. Herbst, J.J. Croat, J. Appl. Phys. 53 (1982) 4304. [11] J.F. Herbst, J.J. Croat, J. Appl. Phys. 55 (1984) 3023. [12] H.S. Li, Z.W. Zhang, M.Z. Dang, J. Magn. Magn. Mater. 71 (1988) 355. [13] H.S. Li, Z.W. Zhang, M.Z. Dang, Y.L. Liu, J. Magn. Magn. Mater. 75 (1988) 159. [14] W.H. Qiao, Q. Wang, X.P. Zhong, H.L. Luo, J. Magn. Magn. Mater. 110 (1992) 170. [15] R.W. Huang, Z.W. Zhang, B. Hu, J.D. Kang, Z.D. Zang, X.K. Sun, Y.C. Chuang, J. Magn. Magn. Mater. 119 (1993) 180. [16] S.W. Ren, Z.W. Zhang, Y. Liu, J. Magn. Magn. Mater. 139 (1995) 175. [17] Z.G. Zhao, N. Tang, E. Brück, K.H.J. Buschow, F.R. de Boer, J. Appl. Phys. 79 (1996) 6321. [18] M.M. Raja, A. Narayanasamy, J. Voiron, R. Krishnan, J. Magn. Magn. Mater. 184 (1998) 227. [19] Z.Z. Wu, Z.X. Mu, R.S. Wei, H.L. Ping, N.Z. Cheng, L.Z. Yong, J. Magn. Magn. Mater. 248 (2002) 158. [20] Z.W. Zang, S.W. Ren, X.M. Zhang, X.J. Song, L.P. Han, Z.C. Ni, J. Magn. Magn. Mater. 170 (1997) 184. [21] Z. Jin, W. Tang, H. Qin, J. Zhang, Y. Du, J. Magn. Magn. Mater. 177–181 (1998) 1115. [22] T. Fujiwara, H. Fujii, Physica B 300 (2001) 198. [23] X.M. Zhang, R.W. Huang, Z.W. Zhang, J. Magn. Magn. Mater. 241 (2002) 131. [24] H. Zang, S.J. Campbell, J. Appl. Phys. 93 (2003) 9177. [25] S. Kervan, Phys. Status Solidi (b), in press. [26] S. Sinnema, R.J. Radwanski, J.J.M. Franse, D.B. de Mooij, K.H.J. Buschow, J. Magn. Magn. Mater. 44 (1984) 333. [27] G.M. Sheldrick, SHELXL97, Program for Refining Crystal Structures, University of Göttingen, Germany, 1997. [28] P. Tils, M. Loewenhaupt, K.H.J. Buschow, R.S. Eccleston, J. Alloys Compd. 279 (1998) 123. [29] M.S.S. Brooks, L. Nordstrom, B. Johansson, J. Phys.: Condens. Matter 3 (1991) 2357. [30] M.S.S. Brooks, T. Gasche, S. Auluck, L. Nordstrom, L. Severin, J. Trygg, B. Johansson, J. Magn. Magn. Mater. 104–107 (1992) 1381.
Exchange interactions in Gd1−x Cex Mn2 Ge2 compounds S. Kervan∗ Materials Research Department, Ankara Nuclear Research and Training Centre, Turkish Atomic Energy Authority, 06100 Besevler, Ankara, Turkey Received 25 June 2003; received in revised form 30 July 2003; accepted 30 July 2003
Abstract The temperature dependence of the magnetisation and exchange interactions in Gd1−x Cex Mn2 Ge2 compounds have been investigated by the molecular field theory based on two-sublattice model. The molecular field coefficients nMnMn , nGdMn and nGdGd , obeying the hierarchical relation nMnMn > |nGdMn | > nGdGd , were obtained by a numerical fitting process. The exchange coupling constants JMnMn , JGdMn and JGdGd have been calculated by using the molecular field coefficients. The Mn–Mn exchange energy increases with increasing x. JMnMn dominates the three kinds of interactions, implying that the magnetic interactions are dominated by the exchange between 3d electrons. The theoretically calculated Curie temperatures were found to agree well with the experimental results. The exchange fields HMn (T) and HGd (T) were presented. © 2003 Elsevier B.V. All rights reserved. Keywords: Rare-earth intermetallic compounds; Molecular field theory; Curie temperature; Ferrimagnetism
1. Introduction The ternary intermetallic RT2 X2 compounds, where R is a rare earth element, T is a 3d transition metal and X is Si or Ge, crystallize in the body-centred tetragonal ThCr2 Si2 -type crystal structure with the space group I4/mmm, in which R, T and X atoms occupy 2a(0 0 0), 4d(0 1/2 1/4) and 4e(0 0 z) sites, respectively. The unit cell consists of two formula units and may be described as a stacking of atomic layers along the c-axis direction with the sequence T–X–R–X–T. This layer crystal structure leads to a variety of magnetic properties and complex magnetic behaviour [1–9]. In this crystal structure, the Ce moments do not order [6]. The two-sublattice molecular field theory analysis of binary systems based on RFe3 and R6 Fe23 (R = rare earth) was investigated by employing molecular field theory study, demonstrating that two-sublattice molecular field model is capable of rather accurately describing the temperature dependence of the magnetisation for RFe3 and R6 Fe23 [10,11]. Later works have been carried out for other 3d–4f intermetallic compounds [12–25]. In this study, for obtaining more information on the magnetic behaviour of Gd and Mn ion and determining the exchange interactions, the temperature dependence of the ∗
Tel.: +90-312-212-6230; fax: +90-312-223-4439. E-mail address: [email protected] (S. Kervan).
0925-8388/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2003.07.012
magnetisation of Gd1−x Cex Mn2 Ge2 compounds has been analysed using the molecular field theory. In addition, for analysing the spin-reorientation phenomenon and the temperature dependence of the magnetocrystalline anisotropy energy in these compounds, the temperature dependence of the exchange fields is needed. Up to now, the relation between the temperature and the exchange field could only be obtained by the molecular field theory analysis of the magnetic experimental data.
2. Two-sublattice molecular field approximation In these calculations, it is assumed that Gd and Mn moments are rigidly antiparallel along the c-axis over the entire temperature range, and the presence of intralayer antiferromagnetic Mn–Mn interactions is neglected. According to molecular field theory based on two-sublattice model, the molecular fields acting on gadolinium and manganese sublattices in an applied magnetic field Ha can be written as follows: HGd (T) = Ha + (1 − x)nGdGd MGd (T) + 2nGdMn MMn (T), (1) HMn (T) = Ha + (1 − x)nMnGd MGd (T) + 2nMnMn MMn (T), (2)
S. Kervan / Journal of Alloys and Compounds 368 (2004) 8–12
where we assume that nGdMn = nMnGd and nMnMn , nGdMn and nGdGd , describing the Mn–Mn, Gd–Mn and Gd–Gd magnetic interactions, are the molecular field coefficients, respectively. Temperature dependence of each sublattice magnetisation can be described by Brillouin function: MGd (0)HGd (T) MGd (T) = MGd (0)BJGd , (3) kB T MMn (0)HMn (T) MMn (T) = MMn (0)BJMn , (4) kB T where MGd (0) and MMn (0) are the zero temperature magnetic moments of the Gd and Mn ions, JGd and JMn are the total angular momenta of Gd and Mn, and Brillouin function has the standard definition 2J + 1 2J + 1 1 1 BJ (x) = coth x − coth x . (5) 2J 2J 2J 2J In calculations, the free ion magnetic moment is used for gadolinium (MGd (0) = gGd JGd ). MMn (0) is derived from the experimental data measured at low temperature by assuming that Gd and Mn moments couple antiferromagnetically. The coefficients nMnMn , nGdMn and nGdGd are determined by numerically solving Eqs. (1)–(4) subject to condition that the calculated total magnetic moments Mtot (T) = |(1 − x)MGd (T) − 2MMn (T)| correspond best with the experimental data. This is done by minimizing the percentage deviation |Mexp (Ti ) − Mtot (Ti )| R = 100 , (6) Mexp (Ti )
Moreover, there are relationships between the exchange energies and the spin coupling constants: aMnMn = ZJMn (JMn + 1)JMnMn , aGdMn = [Z1 Z2 JMn (JMn + 1)(gGd − 1) × JGd (JGd + 1)]1/2 JGdMn , aGdGd = Z3 (gGd − 1)JGd (JGd + 1)JGdGd ,
3JGd kB α= , 2 (0) (1 − x)µB (JGd + 1)MGd 3JMn kB β= . 2 (0) 2µB (JMn + 1)MMn
(8)
On the other hand, molecular field theory provides us with the relationship [19,26] 3kB TC = (aMnMn + aGdGd ) + [(aMnMn + aGdGd )2 2 − 4(aMnMn aGdGd − aGdMn )]1/2 .
(9)
By comparing Eqs. (7) and (9), the exchange energies aMnMn , aGdMn and aGdGd can be obtained from the molecular field coefficients as follows: 3kB nMnMn , 2β 3kB = nGdGd . 2α
aMnMn = aGdGd
3kB aGdMn = √ nGdMn , 2 αβ (10)
(11)
where Z (4) and Z2 (4) are the numbers of Mn and Gd nearest neighbours of the Mn ion, respectively; Z1 (8) and Z3 (4) are the numbers of Mn and Gd nearest neighbours of the Gd ion, respectively. JMnMn , JGdMn and JGdGd are the spin coupling constants between Mn–Mn, Gd–Mn and Gd–Gd. The number of the nearest neighbours was determined by using SHELXL97 [27] program. The values of JMnMn , JGdMn and JGdGd can be obtained by substituting the values of aMnMn , aGdMn and aGdGd into Eq. (11).
3. Results and discussion The magnetic properties of Gd1−x Cex Mn2 Ge2 compounds have been previously studied by means of X-ray diffraction and magnetic measurements [6]. The temperature dependence of magnetisation of Gd1−x Cex Mn2 Ge2 compounds are plotted in Fig. 1. The circles represent the experimental data, while molecular field theory results for the total moment and sublattice magnetizations are indicated by solid line. Table 1 summarizes the experimental information and obtained parameters. The experimental values agree quite
where Mexp (Ti ) is the experimental total moment at temperature Ti . The Curie temperature can be derived from Eqs. (1)–(4) in zero applied field: αnMnMn + βnGdGd + (αnMnMn + βnGdGd )2 − 4αβ(nMnMn nGdGd − n2GdMn ) TC = , 2αβ where
9
(7)
well with the theoretical values (R < 4%), indicating that two-sublattice molecular field theory is successful in describing the temperature dependence of magnetisation for Gd1−x Cex Mn2 Ge2 compounds. The negative values of nGdMn are consistent with the antiferromagnetic interactions between the Gd and Mn moments. The values of aMnMn , aGdMn , aGdGd , JMnMn , JGdMn and JGdGd are listed in Table 2. These values agree well with the exchange interaction parameters calculated by two-sublattice molecular field approximation [12–14,18]. The intersublattice coupling constant JGdMn can be obtained more accurately by means of inelastic neutron scattering (INS) [28]. The JGdMn value for Gd1−x Cex Mn2 Ge2 compounds calculated in the present paper is about 1 × 10−22 J, which is greater than 0.35 × 10−22 J and 0.4 × 10−22 J obtained from inelastic neutron scattering for GdMn2 Ge2 and GdMn2 Si2 compounds [28]. Deviation from the inelastic neutron scattering results seems to be due to the neglecting of the intralayer antiferromagnetic Mn–Mn interactions.
10
S. Kervan / Journal of Alloys and Compounds 368 (2004) 8–12
Fig. 1. Temperature dependence of the magnetisation of Gd1−x Cex Mn2 Ge2 compounds. Circles denote the measured values and solid line represents molecular field theory results.
Table 1 Rare-earth gyromagnetic ratio gGd , total angular momentum JGd and molecular field theory coefficients x
0.17 0.33 0.5 0.67
gGd
JGd
2 2 2 2
7/2 7/2 7/2 7/2
MGd (0) (µB )
7 7 7 7
MMn (0) (µB )
2.08 1.92 1.92 1.69
nMnMn (Gs/µB )
915760 1052230 1072770 1304700
nGdMn (Gs/µB )
−154810 −164270 −131180 −124100
nGdGd (Gs/µB )
45150 72530 9740 0
R (%)
3.9 3.8 2.4 1.1
TC (K) Calc.
Exp.
381 383 373 372
346 348 341 338
Table 2 The exchange energies and coupling constants x
aMnMn (10−22 J)
aGdMn (10−22 J)
aGdGd (10−22 J)
JMnMn (10−22 J)
JGdMn (10−22 J)
JGdGd (10−22 J)
0.17 0.33 0.5 0.67
72.1 73.6 74.9 75.9
−33.2 −33.2 −26.5 −22.9
26.4 42.4 5.7 0
8.49 9.76 9.95 12.10
−1.02 −1.08 −0.86 −0.81
0.42 0.67 0.09 0
S. Kervan / Journal of Alloys and Compounds 368 (2004) 8–12
11
Table 3 Coefficients A, B and C appearing in Eq. (12)
Fig. 2. The variation of JMnMn , JGdMn and JGdGd with Ce concentration.
x
HGd (0) (×106 Gs)
A
B
C
0.17 0.33 0.5 0.67
1.062 1.025 0.989 0.844
0.290 0.410 0.033 −0.038
−0.036 −0.240 0.097 0.303
−1.114 −0.863 −0.762 −0.856
Fig. 2 shows the variation of JMnMn , JGdMn and JGdGd with Ce concentration. The substitution of Ce at Gd site affects the exchange interactions. JMnMn increases with increasing x. Therefore, Ce substitution strengthens the Mn–Mn exchange as a result of increasing interatomic distances. It is interesting to note that the change in the intersublattice coupling constant JGdMn is small irrespective of the degree of dilution with Ce atoms. It has been reported, through band structure calculations, that the rare earth–transition metal coupling is proportional to the ratio of 3d and 5d spins [29,30]. The substitution Ce atoms may result in an increasing of the 3d and 5d spins by the same factor resulting in an aproximately constant JGdMn . The exchange coupling constant JGdGd is the smallest and vanishes with increasing x. The temperature dependence of HMn (T) and HGd (T) for Gd1−x Cex Mn2 Ge2 compounds, calculated by using molec-
Fig. 3. Temperature dependence of HMn (T) and HGd (T).
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S. Kervan / Journal of Alloys and Compounds 368 (2004) 8–12
ular field coefficients, is plotted in Fig. 3. In the study of the relation between the magneto-crystalline anisotropy energy and the temperature, it is necessary to know the property of HGd (T). For the sake of application, the numerical results of HGd (T) are fitted to analytic form by the least-squares method 2 2 T T T HGd (T) = HGd (0) 1+A +B +C . TC TC TC (12) The parameters A, B and C are listed in Table 3. 4. Conclusions The two-sublattice molecular field theory described successfully the temperature dependence of magnetisation for Gd1−x Cex Mn2 Ge2 compounds. The molecular field coefficients nMnMn , nGdMn and nGdGd have been calculated iteratively. The exchange energies and coupling constants have been obtained from the molecular field coefficients. It has been found that the Ce substitution for Gd in these compounds has little influence on Gd–Mn exchange interaction. References [1] Z. Ban, M. Sikirica, Acta Cryst. 18 (1965) 594. [2] K.S.V.L. Narasimhan, V.U.S. Rao, R.L. Bergner, W.E. Wallace, J. Appl. Phys. 46 (1975) 4957. [3] H. Fujii, T. Okamoto, T. Shigeoka, N. Iawata, Solid State Commun. 53 (1985) 715. [4] A. Szytula, J. Leciejewicz, in: K.A. Gschneidner Jr., L. Eyring (Eds.), Handbook on the Physics and Chemistry of Rare Earths, vol. 12, Elsevier, Amsterdam, 1989, p. 133. [5] R. Welter, G. Venturini, E. Ressouche, B. Malaman, J. Alloys Compd. 218 (1995) 204.
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