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Electronic structure and magnetic properties of the ThCo4B compound
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 320 (2008) 36–42 www.elsevier.com/locate/jmmm
Electronic structure and magnetic properties of the ThCo4 B compound D. Beneaa,, V. Popa, O. Isnardb a
Babes Bolyai University, Faculty of Physics, 400084 Cluj-Napoca, Romania Laboratoire de Cristallographie du CNRS, associe´ a` Universite´ Joseph Fourier, BP166X, 38042 Grenoble, France
b
Received 11 January 2007; received in revised form 21 March 2007 Available online 18 May 2007
Abstract Detailed theoretical investigations of the electronic and magnetic properties of the newly discovered ThCo4 B compound have been performed. The influence of the local environment on the magnitude of the Co magnetic moments is discussed by comparing the magnetic and electronic properties in the ThCo4 B, YCo4 B and ThCo5 systems. All theoretical investigations of the electronic and magnetic properties have been done using the Korringa–Kohn–Rostoker (KKR) band-structure method in the ferromagnetic state. Very good agreement of the calculated and the experimental magnetic moments is obtained. Larger exchange-splitting is observed on the 2c site which carries by far the largest magnetic moment. Comparison of the band structure calculation for ThCo5 and ThCo4 B reveals that the presence of boron in the Co 6i site environment induces a broadening of the electronic bands as well as a significant reduction of the exchange-splitting and a diminution of the DOS at the Fermi level. These differences are attributed to the hybridization of the boron electronic states to the cobalt 3d ones. The calculated magnetic moment is 1:94 mB /formula unit. A large difference on the magnetic moment magnitude of the two Co sites is observed since 1.30 and 0:27 mB /atom are calculated for the 2c and 6i sites, respectively. The orbital contribution is found to differ by almost an order of magnitude on both cobalt sites. The Co magnetic moment is much smaller in the ThCo4 B than in the YCo4 B or RCo4 B (where R is a rare earth) isotypes evidencing the major role played by the Th–Co bands on the electronic properties. r 2007 Elsevier B.V. All rights reserved. PACS: 71.15.Mb; 71.20.b; 71.20.Eh; 74.25.H Keywords: Electronic band structure; Magnetic moment; Density of state
1. Introduction Magnetic compounds of elements with 4f or 5f electrons and transition metals have been a challenging class of materials for decades [1–5] with equal interest on theoretical description, experimental investigations of intrinsic physical properties and technological applications. The thorium containing compounds are of particular interest since the tetravalent state of Th may lead to original magnetic behavior. Among the new phases discovered in the last decades are the thorium-transition metal intermetallic phases such as ThFe0:22 Sn2 , Th4 Fe13 Sn5 [6,7] ternary phases and the ThFe11 Cx compounds discovered by Jacobs and co-workers [8]. Th–Co binary intermetallic Corresponding author.
E-mail address: [email protected] (D. Benea). 0304-8853/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2007.05.002
compounds have been described by Buschow et al. [9,10] in their pioneer studies. Four compositions have been studied in the Th–Co system in the range between ThCo5 and ThCo5:7 . This large homogeneity range was found to lead to different magnetic behavior [11] and even to a magnetic phase transition for some Co concentration. In compounds where Co content is larger than ThCo5:1 a normal ferromagnetic behavior is observed. In contrast, a metamagnetic transition is observed at about 10 T in ThCo5 [11], a transition which has been attributed to the 3d band on the 3g site of Co in CaCu5 structure type. The Co 3g site changes its magnetic moment from 1.0 to 1:6 mB by applying the magnetic field [11]. Also, the lattice collapse in YCo5 driven by magnetic interactions [12] adds to the complexity of magnetic properties in RCo5 compounds. A new phase of the Th–Co–B ternary phase diagram which can be described as deriving from the ThCo5 crystal
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structure has been reported by Isnard et al. [13]. The ThCo4 B compound orders ferromagnetically below 303 K, a temperature which is much smaller than those of the RCo4 B isotypic compounds, as a result of the much smaller Co magnetization [13]. We present the band-structure calculations in order to investigate the influence of the local environment on the Co magnetic moments of this new phase. Also, we compare the results of our calculation for ThCo4 B with similar calculations for ThCo5 and YCo4 B in order to evidence the influence of Th–Co and Th–Co–B hybridization on the behavior of the magnetic moments on the two Co sites of the structure. 2. Computational details The electronic structure of the ThCo4 B was calculated self-consistently by means of the spin polarized relativistic Korringa–Kohn–Rostocker (SPR-KKR) method in the atomic sphere approximation (ASA) mode [14–16]. The calculation method is based on the KKR-Green’s function formalism that makes use of multiple scattering theory. The details of the calculation method have been described in details elsewhere [17,18]. The general gradient approximation (GGA) for the exchange-correlation energy using the Perdew–Burke–Ernzerhof (PBE) functional was used [19,20]. Computations were done for 336 k-points in the irreducible wedge of the first Brillouin zone. For integration over the Brillouin zone, the special points method has been used [21]. 3. Results and discussions The details concerning the preparation, the structural and magnetic studies of the ThCo4 B phase were reported by us in a previous paper [13]. The X-ray and neutron diffraction spectra confirm the presence of a single phase compound with the CeCo4 B structure of the space group P6/mmm [13]. The CeCo4 B type of structure can be derived from the CaCu5 structure and has two different crystal sites for the Ce (1a and b), two other sites for Co (2c and 6i) and one site for the B (2d), as can be seen in Fig. 1. First reports [22] show that the B for Co substitution in RCo5 (R is a
1b 6i
2c 1a Fig. 1. Crystal structure of ThCo4 B comprising two types of Co site (6i and 2c), one site for B (2d) and two sites for Th (1a, 1b).
37
rare earth or Y) structure leads to a series of compounds of Rnþ1 Co3nþ5 B2n type. ThCo4 B differ significantly from the RCo4 B where R is a rare earth, having much higher c lattice parameter and an a lattice parameter close to that observed for PrCo4 B and NdCo4 B [23]. For ThCo4 B the room temperature lattice parameters obtained from X-ray ˚ and c ¼ 7:003 A. ˚ There has diffraction are a ¼ 5:088 A been several studies focused on the investigations of the structural and magnetic properties of the RCo4 B phases [23–30]. The SPR-KKR fully relativistic band-structure calculations have been performed for the ThCo4 B in CeCo4 B type of structure with the experimental lattice parameters [13]. The spin resolved total density of states (DOS) for this compound is presented in Fig. 2. The main features of the KKR calculated DOS of this system are similar to the LMTO DOS calculations reported earlier for YCo4 B [24,31,32] with the same type of structure. One should note that the partial DOS of each component is weighted by number of atoms in the unit cell in order to get the total DOS. The origin of the energy scale is the Fermi level, E F . Co 3d has the major DOS contribution in the valence band. B contribution to DOS is small because of the small number of electrons provided to the valence band. In the lower part of the valence band, the exchangesplitting is minor. The higher part of the valence band shows a clear exchange-splitting due to the d states of Co. More pronounced exchange-splitting can be found in the DOS of Co 2c. Also, one should note the very high local DOS in the spin down band at the Fermi level for Co atom at the 2c site. As can be seen from Fig. 2 the band structure of the two Co sites are very different in shape. This is noticeable at the Fermi level but also at about 8 eV where the hybridization with the boron atoms induces a significant DOS for the Co 6i site only. Similarly the Th–B hybridization leads to significant difference of the DOS observed for the two inequivalent Th sites. In particular, the Th 1b site exhibits much larger DOS above 10 eV as a consequence of the hybridization with the s and p electronic states of the boron neighbors. Considering the close relationship between the ThCo4 B and ThCo5 structure types (the ordered substitution of one half of the Co 2c site in the ThCo5 by B leads to ThCo4 B structure, with B located on the 2d site), the Co contributions to DOS in both compounds have been compared. One should note that the ordered substitution of Co by B changes the environment of the Co atoms and the Co 3g site from the ThCo5 structure becomes the 6i site in the ThCo4 B structure. The most important consequence of the substitution is the difference between the nearest neighbor species (B for Co) between the Co 6i site of ThCo4 B and the Co 3g site of ThCo5 . The change of the environment is evidenced by the lower exchange-splitting and enhanced broadening of the Co 6i bands in ThCo4 B (Fig. 2) compared with the bands of Co 3g of ThCo5 (Fig. 3). Also, the DOS at Fermi level for the Co 6i in ThCo4 B is lower in both spin up and spin down channels. This difference has probably originated from the hybridization
ARTICLE IN PRESS
tot Co (2c) s p d
1.6
nCo_6i(E) (sts./eV)
↑
1.6 -4 energy (eV)
tot Th (1a) s p d f
0.8
-12
-8
-4 energy (eV)
0
tot Th (1b) s p d f
0.8 0.4 0
nTh_b(E) (sts./eV)
0.4 0.8
0.4 0.8
↑
↑
nTh_a(E) (sts./eV)
1.6
↓
0.4
-12
-8
-4 energy (eV)
0
tot B s p
0.3
-12
ntot(E) (sts./eV)
0.2 0.1
↓
↓
0.8
0
↓
nTh_a(E) (sts./eV)
-8
nTh_b(E) (sts./eV)
n↑Co_2c(E) (sts./eV)
0.8
0
nB(E) (sts./eV)
0.8 0
0
-12
-8
-4 energy (eV)
0
tot
12 8 4 0
ntot(E) (sts./eV)
0 0.1 0.2
4 8
↑
↑
nB(E) (sts./eV)
tot Co (6i) s p d
1.6
↓
0.8
↓
nCo_2c(E) (sts./eV)
nCo_6i(E) (sts./eV)
D. Benea et al. / Journal of Magnetism and Magnetic Materials 320 (2008) 36–42
38
0.3 -12
-8
-4 energy (eV)
0
12 -12
-8
-4 energy (eV)
0
Fig. 2. Spin and component resolved DOS in ThCo4 B obtained by SPR-KKR GGA calculations.
of p electronic states of boron with the 3d ones of cobalt atoms. Similar reduction of the DOS at E F upon B for Co substitution has been observed previously for the calculations in the YCo4 B isotype compounds [31]. The magnetic measurements and neutron diffraction on ThCo4 B compound [13] show that the Co 2c site carries by far the largest magnetic moment, similar to that observed on the 2c site in the other RCo4 B compounds, where R is a rare earth element or Y. The neutron diffraction measurements performed at 2 K show a magnetic moment of 1:8 0:2 mB on Co 2c site whereas a very small magnetic moment ð0:1
0:2 mB Þ is found on Co 6i site. A similar behavior was also found at room temperature, 1:2 0:2 mB on the Co 2c site and about zero mB on the Co 6i site. According to the neutron diffraction measurements, the Co magnetic moments are aligned along the c-axis of the crystal structure from 2 K up to the Curie temperature. The neutron diffraction investigations show that unlike YCo4 B, ThCo4 B does not exhibit any spin reorientation phenomenon of its easy magnetization direction. The easy magnetization direction is imposed by the preferential anisotropy direction of the Co 2c site, which carries most of
ARTICLE IN PRESS ↑ nCo_(2c) (E) (sts./eV) n↓Co_(2c)(E) (sts./eV)
D. Benea et al. / Journal of Magnetism and Magnetic Materials 320 (2008) 36–42
0.8
Magnetic moments ðmB =atomÞ
0 mspin morbit
0.8
1.6 -12
-8
-4 energy (eV)
0
4
-4 energy (eV)
0
4
-4 energy (eV)
0
4
tot Co (3g) s p d
1.6
0.8
↓
nCo_(3g) (E) (sts./eV)
Table 1 Magnetic moments in the ThCo4 B compound by SPR-KKR GGA calculations
total Co (2c) s p d
1.6
↑
nCo_(3g)(E) (sts./eV)
0 0.8
1.6
↓ ntot (E) (sts./eV)
-12
-8
tot
12 8 4 0
n↑tot(E) (sts./eV)
39
4 8 12 -12
-8
Fig. 3. Spin and component resolved DOS in ThCo5 obtained by SPRKKR GGA calculations. The densities of states for Co 2c and Co 3g are shown separately.
the magnetization according to the measurements. The isothermal magnetization curve of ThCo4 B at 4 K shows a saturation of magnetization of 1:5 0:1 mB /f.u. A remanent magnetization of 1:3 mB /f.u. has been observed by decreasing the external field. According to the SPR-KKR calculations, the total energy of the system is lower for magnetization along the c-axis as for the magnetization in the basal plane, in agreement with experiment. The magnetic moments for ThCo4 B obtained by SPR-KKR
Th 1a
Th 1b
Co 2c
Co 6i
B 2d
Total
0.169 0.043
0.167 0.004
1.106 0.193
0.252 0.021
0.025 0.000
1.670 0.276
band-structure calculations with the magnetization along c-axis are presented in Table 1. The calculated magnetization of 1:94 mB /f.u. is higher than the experimental value, 1:5 0:1 mB /f.u. determined from the saturation of magnetization [13]. The values of the magnetic moments on the Co atoms depend strongly on the local environment. The calculations are in agreement with the experimental measurements which find that only the Co 2c sites give significant contribution to the magnetization. The calculated magnetic moment on the Co 2c sites is 1:30 mB /f.u. atom, which is smaller but reasonably close to the experimental value of 1:8 mB /Co 2c atom determined by neutron diffraction. The magnetic moment calculated on Co 6i sites is 0:27 mB =atom, whilst the neutron diffraction investigations show that Co 6i carry a small magnetic moment ð0:1 0:2 mB =atomÞ. We note also that, despite a smaller contribution to the DOS at the Fermi level coming from Co 6i site, the exchange-splitting (Fig. 2) supports the presence of magnetic moments on Co 6i sites. The discrepancy between the theory and experiment can, at least partially, be attributed to the approximations used in the calculations. It is a well-known experience that the LDA or GGA approximation in ASA, or a similar approximation often gives an inaccuracy of about 5–10% [33,34]. It is also worth to note that powder neutron diffraction technique unlike polarized single crystal neutron diffraction is known to underestimate the negative polarization coming from valence electrons thus leading to slightly overestimated magnetic moment in the case of the Co atoms. This may contribute to the observed difference between experimental and calculated values of the magnetic moments. A remarkable result is that orbital moments on the 2c and 6i sites differ by about one order of magnitude. The small magnetic moment on Co 6i sites was explained to originate from the hybridization of the Co 6i d-states with the p and d states of Th and s and p states of B [13], as can be seen in Fig. 2. The large reduction of the Co 6i site orbital moment is a major change in comparison to the RCo5 related compounds in which both inequivalent Co crystal sites have been reported to exhibit a large orbital magnetic moment of similar magnitude [35–37]. It is worth to note that according to the present calculations the orbital magnetic moment of the Co 6i site is much reduced in ThCo4 B. We got similar results for the YCo4 B isotype phase in contradiction with earlier published results [31,38]. This most probably originates from the different approaches to treat the spin–orbit coupling. We use a fully
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Table 2 Magnetic moments in the ThCo4 B, YCo4 B and ThCo5 compounds Co mag. mom. ðmB =atomÞ
Co mag. mom. ðmB =atomÞ
Magnetization ðmB =f:u:Þ
Calc.
Exp.a
Calc.
Exp.a
Exp.b
1.94c
2.1 [13]
1.5 [13]
d
Co 2c
Co 6i, 3g
Co 2c
Co 6i, 3g
1.30c
0.27c
1.8 [13]
0.1 [13]
YCo4 B
c
1.65 1.43e [24] 1.75e [31] 1.50e [32] 1.63e [38] 1.78f [38]
c
0.41 0.59e [24] 0.55e [31] 0.77e [32] 0.67e[38] 0.77f [38]
d
1.5 [39] 1.6g [48]
d
0.5 [39] 0.6g [48]
2.69 2.99e [24] 3.40e [31] 3.50e [32] 3.48e [38] 3.90f [38]
3.0 [39] 3.4g [48]
2.9 [39]
ThCo5 (HMS)
1.55c 1.30f [33]
1.48c 1.10f [33]
1.6 [40]
1.6 [40]
7.2c 6.6 [33]
8.0
7.3 [41]
ThCo5 (LMS)
1.10f [33]
0.55f [33]
1.2 [40]
1.0 [40]
4.2 [33]
5.4
4.9 [41]
ThCo4 B
c
a
Neutron diffraction. Magnetization measurements. c Present work, fully relativistic SPR-KKR calculations. d At 300 K. e Scalar-relativistic LMTO calculations. f Spin–orbit coupling included as perturbation. g At 4 K. b
relativistic approach [17] which starts from Dirac equation. The approach used to obtain the previous published results [31,38] solve first the Kohn–Sham equation for the scalar relativistic spin-polarized Hamiltonian self-consistently. In the second step, the spin–orbit term is added and the full Hamiltonian is diagonalized non-self-consistently. Experimental results of Co NMR or of polarized neutron diffraction would be necessary for comparison. We should note also the negative polarization of the spin magnetic moment on both B and Th atoms as can be seen in Table 1. The magnetic moments in ThCo4 B are compared in Table 2 with that of the isotypic YCo4 B and with the related ThCo5 compound. The magnetic moments in YCo4 B compounds have been calculated by SPR-KKR method and they are compared in Table 2 with the results of earlier reported calculations [24,31,32,38] and with the experimental results [39]. One should note an improved agreement of the fully relativistic SPR-KKR calculations with the experimental values [39] compared with the other calculations. The difference between the Co 2c and Co 6i magnetic moments is evidenced in both ThCo4 B and YCo4 B compounds. The SPR-KKR calculations confirm the weakness of the magnetic moment on the Co 6i site in both ThCo4 B and YCo4 B compounds. The value of the Co 6i magnetic moment is appreciably reduced in ThCo4 B (with more than 30%) compared with the corresponding value for Co 6i magnetic moment in YCo4 B compound. The decrease of the Co 2c magnetic moment by Th for Y substitution is less pronounced (only about 20%). These differences most probably originate from the different
electron number between Y and Th. Indeed, Y has trivalent character whereas Th is tetravalent. The intermetallic compound ThCo5 is ferromagnetic at room temperature having lower Co moments than other RCo5 compounds. However, experiments have shown that ThCo5 exhibits a metamagnetic transition from a low-moment state (LMS) to a high-moment state (HMS) with an applied field [40,41]. The neutron diffraction experiments [40,41] on ThCo5 show that in the LMS the two cobalt sites have different local magnetic moments, whereas in the HMS the two moments are of the same size (Table 2). It was supposed [42] that the Co atoms exhibiting this transition should be paramagnetic before the transition, with only a small moment induced by the interatomic exchange field from the ferromagnetic Co atoms situated at the other type of sites. The calculations performed by Nordstro¨m et al. [33] using the so-called fixed-spin (FSM) based on the LMTO method are in agreement with the experiment showing that for the LMS state two significantly different local Co magnetic moments are observed. On the contrary, for the HMS, the two Co magnetic moments are nearly of the same magnitude (Table 2). The orbital contributions obtained from the calculations of Nordstro¨m et al. in LMS state are different for the two Co sites, but the difference between the orbital moments is less pronounced in the HMS state. The present fully relativistic SPR-KKR calculations have been performed at the experimental lattice parameters reported for ThCo5 [43]. In fact at the experimental volume of the unit cell, the calculations of Nordstro¨m et al. did not found actually the LMS state,
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only a local minimum indicated the closeness to such an equilibrium state. This is the reason to compare our calculation with the calculations of Nordstro¨m et al. for HMS state (see Table 2). The two Co magnetic moments obtained by SPR-KKR calculations (including the orbital contribution) are very close: 1.55 and 1:48 mB . Also, the orbital contributions are comparable for the two different Co crystal sites in HMS state, 0.16 and 0:12 mB for Co 2c and Co 3g sites, respectively. One should mention the good agreement between the orbital moment determined by SPR-KKR calculations (0:69 mB /unit cell) and the experimental estimate of 0:7 mB /unit cell [36]. The broader bands of Co 6i in ThCo4 B compared with the bands of Co 3g in ThCo5 are partially responsible for the weaker 3d magnetization in ThCo4 B than in ThCo5 . The results of the calculations for ThCo5 and ThCo4 B compounds, combined with the experimental measurements shows that the B for Co substitution greatly affects the Co 6i magnetic moments. The change of the magnetic environment of the Co 6i crystal site is the origin for the dramatic decrease of the magnetization in ThCo4 B. According to our calculations, this occurs via the hybridization of the s, p (B) and d (Co) electronic states. 4. Conclusions The SPR-KKR calculations in ThCo4 B show that the Co 2c site is found to carry most of the magnetic moment, in agreement with experiment [13] and with earlier investigations in isotype RCo4 B compounds. Also, the SPR-KKR calculations are in agreement with experimental neutron investigations showing that the easy magnetization direction of ThCo4 B is along the c-axis. The Co 6i site carries a significantly reduced magnetic moment because of its hybridization with B and Th, as it was evidenced by the DOS calculations. Both inequivalent Co sites have smaller magnetic moment in ThCo4 B compared with YCo4 B. The magnitude of the Co 2c site magnetic moment is moderately affected by the replacement of Y by Th, whereas the Co 6i site magnetic moment is much more sensitive to it. Another interesting result of our calculations is the large difference in the orbital magnetic moment magnitude of the two inequivalent Co sites of ThCo4 B, whereas the RCo5 compounds have been found to exhibit similar orbital moment on both Co sites. The calculations agree with the presence of a very weak magnetic moment on Co 6i site as suggested by the experimental measurements. Furthermore, this magnetic moment is significantly reduced ð0:27 mB Þ compared with the corresponding value for isotypic YCo4 B ð0:6 mB Þ. The presence of tetravalent Th atom induces a reduction of the Co magnetic moment in comparison with isotype RCo4 B compounds containing trivalent Y or rare earth elements. This tendency is confirmed by both experiment and calculation performed on UCo4 B where U can be considered as pentavalent. Indeed this compound has been found not to be magnetically ordered even at the lowest temperature
41
[44–47]. The comparison between the ThCo5 and ThCo4 B magnetic moments on the Co sites shows that substituting B for Co greatly affects the magnetic moment on the Co 6i site, whilst the magnetic moment on the Co 2c site is less affected by this substitution. All these observations lead to conclude that the change in the magnetic environment of Co 6i is the origin of its dramatic decrease of the magnetic moment in ThCo4 B. This occurs via the Co–B hybridization of electronic states. Acknowledgments The authors would like to thanks Prof. Postnikov for interesting discussions. The financial supports of the Romanian Ministry of Education and Research through the CEEX program and of the Re´gion Rhoˆne Alpes through the MIRA program have been greatly appreciated. References [1] K.H.J. Buschow, Rep. Prog. Phys. 40 (1977) 1179. [2] K.H.J. Buschow, Rep. Prog. Phys. 42 (1979) 1373. [3] K.H.J. Buschow, Ferromagnetic Materials, North Holland, Amsterdam, 1980, p. 297. [4] E. Burzo, A. Chelkovski, H. Kirchmayr, Landolt Bo¨rnstein Handbook, vol. 3, Springer, Berlin, 1990, p. 1983. [5] H. Kirchmayr, J. Phys. D: Appl. Phys. 29 (1996) 2763. [6] O. Moze, P. Manfrinetti, F. Canepa, A. Palenzona, M.L. Fornasini, J.R. Rodriguez-Carvajal, Intermetallics 8 (2000) 273. [7] P. Manfrinetti, F. Canepa, F. Palenzona, M. Fornasini, E. Giannini, J. Alloys Compds. 247 (1997) 109. [8] T. Jacobs, G. Long, A. Pringle, F. Grandjean, K. Buschow, J. Appl. Phys. 70 (1991) 5983. [9] K. Buschow, J. Appl. Phys. 42 (1971) 3433. [10] A. van der Groot, K. Buschow, Phys. Stat. Sol. (a) 5 (1971) 665. [11] D. Givord, J. Laforest, R. Lemaire, Physica B 86–88 (1977) 204. [12] H. Rosner, D. Koudela, U. Schwarz, A. Handstein, M. Hanfland, I. Opahle, K. Koepernik, M.D. Kuz’min, K.H. Mu¨ller, J.A. Mydosh, M. Richter, Nat. Phys. 2 (2006) 469. [13] O. Isnard, V. Pop, J. Toussaint, J. Phys. Condens. Matter 15 (2003) 791. [14] P. Weinberger, Electron Scattering Theory for Order and Disordered Matter, University Press, Oxford, 1990. [15] A. Gonis, Green Function for Ordered and Disordered Systems, North-Holland, Amsterdam, 1992. [16] P. Strange, Relativistic Quantum Mechanics, University Press, Cambridge, 1998. [17] H. Ebert, in: H. Dreysse´ (Ed.), Electronic Structure and Physical Properties of Solids, vol. 535, Springer, Berlin, 2000, p. 191. [18] H. Ebert, The Munich SPRKKR package, Version 2.1.1., 2002, hhttp://olymp.cup.uni-muenchen.de/ak/ebert/sprkkri. [19] J. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 78 (1997) 1396. [20] J. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. [21] H. Monkhorst, J. Pack, Phys. Rev. B 13 (1976) 5188. [22] Y.B. Kuzma, N. Bilonizhko, Sov. Phys. Crystallogr. 18 (1974) 447. [23] C. Chacon, O. Isnard, J. Solid State Chem. 154 (2000) 242. [24] P. Vlaic, E. Burzo, Mold. J. Phys. Sci. 2 (2002) 40. [25] E. Burzo, V. Pop, C. Borodi, R. Ballou, IEEE Trans. Magn. 30 (1994) 628. [26] E. Burzo, V. Pop, N. Plugaru, Mater. Sci. Forum 62–64 (1990) 611. [27] E. Burzo, V. Pop, N. Plugaru, I. Creanga, Phys. Stat. Sol. (a) 113 (1989) K253. [28] E. Burzo, M. Ursu, J. Magn. Magn. Mater. 70 (1987) 345.
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[41] D. Givord, J. Laforest, R. Lemaire, Q. Lu, J. Magn. Magn. Mater. 31–34 (1983) 191. [42] M. Shimizu, J. Phys. (Paris) 43 (1982) 155. [43] W. Wallace, E. Ganapathy, R. Craig, J. Appl. Phys. 50 (1979) 2327. [44] G. Chelkowska, A. Kowalczyk, A. Szajek, A. Szlaferek, J. Magn. Magn. Mater. 236 (2001) 243. [45] A. Szajek, L. Smardz, A. Szlaferek, A. Kowalczik, Acta Phys. Pol. A 98 (2000) 599. [46] H. Nakotte, L. Havela, J. Kuang, F. de Boer, K. Buschow, J. Brabers, T. Kuroda, S. Sugiyama, M. Date, in: P. Oppenheimer, J. Lu¨bler (Eds.), Proceedings of International Conference on Physics of the Transition Metals, Darmstadt, 1993. [47] H. Nakotte, L. Havela, J. Kuang, F. de Boer, K. Buschow, J. Brabers, T. Kuroda, S. Sugiyama, M. Date, Int. J. Mod. Phys. B 7 (1993) 838. [48] O. Isnard, unpublished.
Journal of Magnetism and Magnetic Materials 320 (2008) 36–42 www.elsevier.com/locate/jmmm
Electronic structure and magnetic properties of the ThCo4 B compound D. Beneaa,, V. Popa, O. Isnardb a
Babes Bolyai University, Faculty of Physics, 400084 Cluj-Napoca, Romania Laboratoire de Cristallographie du CNRS, associe´ a` Universite´ Joseph Fourier, BP166X, 38042 Grenoble, France
b
Received 11 January 2007; received in revised form 21 March 2007 Available online 18 May 2007
Abstract Detailed theoretical investigations of the electronic and magnetic properties of the newly discovered ThCo4 B compound have been performed. The influence of the local environment on the magnitude of the Co magnetic moments is discussed by comparing the magnetic and electronic properties in the ThCo4 B, YCo4 B and ThCo5 systems. All theoretical investigations of the electronic and magnetic properties have been done using the Korringa–Kohn–Rostoker (KKR) band-structure method in the ferromagnetic state. Very good agreement of the calculated and the experimental magnetic moments is obtained. Larger exchange-splitting is observed on the 2c site which carries by far the largest magnetic moment. Comparison of the band structure calculation for ThCo5 and ThCo4 B reveals that the presence of boron in the Co 6i site environment induces a broadening of the electronic bands as well as a significant reduction of the exchange-splitting and a diminution of the DOS at the Fermi level. These differences are attributed to the hybridization of the boron electronic states to the cobalt 3d ones. The calculated magnetic moment is 1:94 mB /formula unit. A large difference on the magnetic moment magnitude of the two Co sites is observed since 1.30 and 0:27 mB /atom are calculated for the 2c and 6i sites, respectively. The orbital contribution is found to differ by almost an order of magnitude on both cobalt sites. The Co magnetic moment is much smaller in the ThCo4 B than in the YCo4 B or RCo4 B (where R is a rare earth) isotypes evidencing the major role played by the Th–Co bands on the electronic properties. r 2007 Elsevier B.V. All rights reserved. PACS: 71.15.Mb; 71.20.b; 71.20.Eh; 74.25.H Keywords: Electronic band structure; Magnetic moment; Density of state
1. Introduction Magnetic compounds of elements with 4f or 5f electrons and transition metals have been a challenging class of materials for decades [1–5] with equal interest on theoretical description, experimental investigations of intrinsic physical properties and technological applications. The thorium containing compounds are of particular interest since the tetravalent state of Th may lead to original magnetic behavior. Among the new phases discovered in the last decades are the thorium-transition metal intermetallic phases such as ThFe0:22 Sn2 , Th4 Fe13 Sn5 [6,7] ternary phases and the ThFe11 Cx compounds discovered by Jacobs and co-workers [8]. Th–Co binary intermetallic Corresponding author.
E-mail address: [email protected] (D. Benea). 0304-8853/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2007.05.002
compounds have been described by Buschow et al. [9,10] in their pioneer studies. Four compositions have been studied in the Th–Co system in the range between ThCo5 and ThCo5:7 . This large homogeneity range was found to lead to different magnetic behavior [11] and even to a magnetic phase transition for some Co concentration. In compounds where Co content is larger than ThCo5:1 a normal ferromagnetic behavior is observed. In contrast, a metamagnetic transition is observed at about 10 T in ThCo5 [11], a transition which has been attributed to the 3d band on the 3g site of Co in CaCu5 structure type. The Co 3g site changes its magnetic moment from 1.0 to 1:6 mB by applying the magnetic field [11]. Also, the lattice collapse in YCo5 driven by magnetic interactions [12] adds to the complexity of magnetic properties in RCo5 compounds. A new phase of the Th–Co–B ternary phase diagram which can be described as deriving from the ThCo5 crystal
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structure has been reported by Isnard et al. [13]. The ThCo4 B compound orders ferromagnetically below 303 K, a temperature which is much smaller than those of the RCo4 B isotypic compounds, as a result of the much smaller Co magnetization [13]. We present the band-structure calculations in order to investigate the influence of the local environment on the Co magnetic moments of this new phase. Also, we compare the results of our calculation for ThCo4 B with similar calculations for ThCo5 and YCo4 B in order to evidence the influence of Th–Co and Th–Co–B hybridization on the behavior of the magnetic moments on the two Co sites of the structure. 2. Computational details The electronic structure of the ThCo4 B was calculated self-consistently by means of the spin polarized relativistic Korringa–Kohn–Rostocker (SPR-KKR) method in the atomic sphere approximation (ASA) mode [14–16]. The calculation method is based on the KKR-Green’s function formalism that makes use of multiple scattering theory. The details of the calculation method have been described in details elsewhere [17,18]. The general gradient approximation (GGA) for the exchange-correlation energy using the Perdew–Burke–Ernzerhof (PBE) functional was used [19,20]. Computations were done for 336 k-points in the irreducible wedge of the first Brillouin zone. For integration over the Brillouin zone, the special points method has been used [21]. 3. Results and discussions The details concerning the preparation, the structural and magnetic studies of the ThCo4 B phase were reported by us in a previous paper [13]. The X-ray and neutron diffraction spectra confirm the presence of a single phase compound with the CeCo4 B structure of the space group P6/mmm [13]. The CeCo4 B type of structure can be derived from the CaCu5 structure and has two different crystal sites for the Ce (1a and b), two other sites for Co (2c and 6i) and one site for the B (2d), as can be seen in Fig. 1. First reports [22] show that the B for Co substitution in RCo5 (R is a
1b 6i
2c 1a Fig. 1. Crystal structure of ThCo4 B comprising two types of Co site (6i and 2c), one site for B (2d) and two sites for Th (1a, 1b).
37
rare earth or Y) structure leads to a series of compounds of Rnþ1 Co3nþ5 B2n type. ThCo4 B differ significantly from the RCo4 B where R is a rare earth, having much higher c lattice parameter and an a lattice parameter close to that observed for PrCo4 B and NdCo4 B [23]. For ThCo4 B the room temperature lattice parameters obtained from X-ray ˚ and c ¼ 7:003 A. ˚ There has diffraction are a ¼ 5:088 A been several studies focused on the investigations of the structural and magnetic properties of the RCo4 B phases [23–30]. The SPR-KKR fully relativistic band-structure calculations have been performed for the ThCo4 B in CeCo4 B type of structure with the experimental lattice parameters [13]. The spin resolved total density of states (DOS) for this compound is presented in Fig. 2. The main features of the KKR calculated DOS of this system are similar to the LMTO DOS calculations reported earlier for YCo4 B [24,31,32] with the same type of structure. One should note that the partial DOS of each component is weighted by number of atoms in the unit cell in order to get the total DOS. The origin of the energy scale is the Fermi level, E F . Co 3d has the major DOS contribution in the valence band. B contribution to DOS is small because of the small number of electrons provided to the valence band. In the lower part of the valence band, the exchangesplitting is minor. The higher part of the valence band shows a clear exchange-splitting due to the d states of Co. More pronounced exchange-splitting can be found in the DOS of Co 2c. Also, one should note the very high local DOS in the spin down band at the Fermi level for Co atom at the 2c site. As can be seen from Fig. 2 the band structure of the two Co sites are very different in shape. This is noticeable at the Fermi level but also at about 8 eV where the hybridization with the boron atoms induces a significant DOS for the Co 6i site only. Similarly the Th–B hybridization leads to significant difference of the DOS observed for the two inequivalent Th sites. In particular, the Th 1b site exhibits much larger DOS above 10 eV as a consequence of the hybridization with the s and p electronic states of the boron neighbors. Considering the close relationship between the ThCo4 B and ThCo5 structure types (the ordered substitution of one half of the Co 2c site in the ThCo5 by B leads to ThCo4 B structure, with B located on the 2d site), the Co contributions to DOS in both compounds have been compared. One should note that the ordered substitution of Co by B changes the environment of the Co atoms and the Co 3g site from the ThCo5 structure becomes the 6i site in the ThCo4 B structure. The most important consequence of the substitution is the difference between the nearest neighbor species (B for Co) between the Co 6i site of ThCo4 B and the Co 3g site of ThCo5 . The change of the environment is evidenced by the lower exchange-splitting and enhanced broadening of the Co 6i bands in ThCo4 B (Fig. 2) compared with the bands of Co 3g of ThCo5 (Fig. 3). Also, the DOS at Fermi level for the Co 6i in ThCo4 B is lower in both spin up and spin down channels. This difference has probably originated from the hybridization
ARTICLE IN PRESS
tot Co (2c) s p d
1.6
nCo_6i(E) (sts./eV)
↑
1.6 -4 energy (eV)
tot Th (1a) s p d f
0.8
-12
-8
-4 energy (eV)
0
tot Th (1b) s p d f
0.8 0.4 0
nTh_b(E) (sts./eV)
0.4 0.8
0.4 0.8
↑
↑
nTh_a(E) (sts./eV)
1.6
↓
0.4
-12
-8
-4 energy (eV)
0
tot B s p
0.3
-12
ntot(E) (sts./eV)
0.2 0.1
↓
↓
0.8
0
↓
nTh_a(E) (sts./eV)
-8
nTh_b(E) (sts./eV)
n↑Co_2c(E) (sts./eV)
0.8
0
nB(E) (sts./eV)
0.8 0
0
-12
-8
-4 energy (eV)
0
tot
12 8 4 0
ntot(E) (sts./eV)
0 0.1 0.2
4 8
↑
↑
nB(E) (sts./eV)
tot Co (6i) s p d
1.6
↓
0.8
↓
nCo_2c(E) (sts./eV)
nCo_6i(E) (sts./eV)
D. Benea et al. / Journal of Magnetism and Magnetic Materials 320 (2008) 36–42
38
0.3 -12
-8
-4 energy (eV)
0
12 -12
-8
-4 energy (eV)
0
Fig. 2. Spin and component resolved DOS in ThCo4 B obtained by SPR-KKR GGA calculations.
of p electronic states of boron with the 3d ones of cobalt atoms. Similar reduction of the DOS at E F upon B for Co substitution has been observed previously for the calculations in the YCo4 B isotype compounds [31]. The magnetic measurements and neutron diffraction on ThCo4 B compound [13] show that the Co 2c site carries by far the largest magnetic moment, similar to that observed on the 2c site in the other RCo4 B compounds, where R is a rare earth element or Y. The neutron diffraction measurements performed at 2 K show a magnetic moment of 1:8 0:2 mB on Co 2c site whereas a very small magnetic moment ð0:1
0:2 mB Þ is found on Co 6i site. A similar behavior was also found at room temperature, 1:2 0:2 mB on the Co 2c site and about zero mB on the Co 6i site. According to the neutron diffraction measurements, the Co magnetic moments are aligned along the c-axis of the crystal structure from 2 K up to the Curie temperature. The neutron diffraction investigations show that unlike YCo4 B, ThCo4 B does not exhibit any spin reorientation phenomenon of its easy magnetization direction. The easy magnetization direction is imposed by the preferential anisotropy direction of the Co 2c site, which carries most of
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0.8
Magnetic moments ðmB =atomÞ
0 mspin morbit
0.8
1.6 -12
-8
-4 energy (eV)
0
4
-4 energy (eV)
0
4
-4 energy (eV)
0
4
tot Co (3g) s p d
1.6
0.8
↓
nCo_(3g) (E) (sts./eV)
Table 1 Magnetic moments in the ThCo4 B compound by SPR-KKR GGA calculations
total Co (2c) s p d
1.6
↑
nCo_(3g)(E) (sts./eV)
0 0.8
1.6
↓ ntot (E) (sts./eV)
-12
-8
tot
12 8 4 0
n↑tot(E) (sts./eV)
39
4 8 12 -12
-8
Fig. 3. Spin and component resolved DOS in ThCo5 obtained by SPRKKR GGA calculations. The densities of states for Co 2c and Co 3g are shown separately.
the magnetization according to the measurements. The isothermal magnetization curve of ThCo4 B at 4 K shows a saturation of magnetization of 1:5 0:1 mB /f.u. A remanent magnetization of 1:3 mB /f.u. has been observed by decreasing the external field. According to the SPR-KKR calculations, the total energy of the system is lower for magnetization along the c-axis as for the magnetization in the basal plane, in agreement with experiment. The magnetic moments for ThCo4 B obtained by SPR-KKR
Th 1a
Th 1b
Co 2c
Co 6i
B 2d
Total
0.169 0.043
0.167 0.004
1.106 0.193
0.252 0.021
0.025 0.000
1.670 0.276
band-structure calculations with the magnetization along c-axis are presented in Table 1. The calculated magnetization of 1:94 mB /f.u. is higher than the experimental value, 1:5 0:1 mB /f.u. determined from the saturation of magnetization [13]. The values of the magnetic moments on the Co atoms depend strongly on the local environment. The calculations are in agreement with the experimental measurements which find that only the Co 2c sites give significant contribution to the magnetization. The calculated magnetic moment on the Co 2c sites is 1:30 mB /f.u. atom, which is smaller but reasonably close to the experimental value of 1:8 mB /Co 2c atom determined by neutron diffraction. The magnetic moment calculated on Co 6i sites is 0:27 mB =atom, whilst the neutron diffraction investigations show that Co 6i carry a small magnetic moment ð0:1 0:2 mB =atomÞ. We note also that, despite a smaller contribution to the DOS at the Fermi level coming from Co 6i site, the exchange-splitting (Fig. 2) supports the presence of magnetic moments on Co 6i sites. The discrepancy between the theory and experiment can, at least partially, be attributed to the approximations used in the calculations. It is a well-known experience that the LDA or GGA approximation in ASA, or a similar approximation often gives an inaccuracy of about 5–10% [33,34]. It is also worth to note that powder neutron diffraction technique unlike polarized single crystal neutron diffraction is known to underestimate the negative polarization coming from valence electrons thus leading to slightly overestimated magnetic moment in the case of the Co atoms. This may contribute to the observed difference between experimental and calculated values of the magnetic moments. A remarkable result is that orbital moments on the 2c and 6i sites differ by about one order of magnitude. The small magnetic moment on Co 6i sites was explained to originate from the hybridization of the Co 6i d-states with the p and d states of Th and s and p states of B [13], as can be seen in Fig. 2. The large reduction of the Co 6i site orbital moment is a major change in comparison to the RCo5 related compounds in which both inequivalent Co crystal sites have been reported to exhibit a large orbital magnetic moment of similar magnitude [35–37]. It is worth to note that according to the present calculations the orbital magnetic moment of the Co 6i site is much reduced in ThCo4 B. We got similar results for the YCo4 B isotype phase in contradiction with earlier published results [31,38]. This most probably originates from the different approaches to treat the spin–orbit coupling. We use a fully
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Table 2 Magnetic moments in the ThCo4 B, YCo4 B and ThCo5 compounds Co mag. mom. ðmB =atomÞ
Co mag. mom. ðmB =atomÞ
Magnetization ðmB =f:u:Þ
Calc.
Exp.a
Calc.
Exp.a
Exp.b
1.94c
2.1 [13]
1.5 [13]
d
Co 2c
Co 6i, 3g
Co 2c
Co 6i, 3g
1.30c
0.27c
1.8 [13]
0.1 [13]
YCo4 B
c
1.65 1.43e [24] 1.75e [31] 1.50e [32] 1.63e [38] 1.78f [38]
c
0.41 0.59e [24] 0.55e [31] 0.77e [32] 0.67e[38] 0.77f [38]
d
1.5 [39] 1.6g [48]
d
0.5 [39] 0.6g [48]
2.69 2.99e [24] 3.40e [31] 3.50e [32] 3.48e [38] 3.90f [38]
3.0 [39] 3.4g [48]
2.9 [39]
ThCo5 (HMS)
1.55c 1.30f [33]
1.48c 1.10f [33]
1.6 [40]
1.6 [40]
7.2c 6.6 [33]
8.0
7.3 [41]
ThCo5 (LMS)
1.10f [33]
0.55f [33]
1.2 [40]
1.0 [40]
4.2 [33]
5.4
4.9 [41]
ThCo4 B
c
a
Neutron diffraction. Magnetization measurements. c Present work, fully relativistic SPR-KKR calculations. d At 300 K. e Scalar-relativistic LMTO calculations. f Spin–orbit coupling included as perturbation. g At 4 K. b
relativistic approach [17] which starts from Dirac equation. The approach used to obtain the previous published results [31,38] solve first the Kohn–Sham equation for the scalar relativistic spin-polarized Hamiltonian self-consistently. In the second step, the spin–orbit term is added and the full Hamiltonian is diagonalized non-self-consistently. Experimental results of Co NMR or of polarized neutron diffraction would be necessary for comparison. We should note also the negative polarization of the spin magnetic moment on both B and Th atoms as can be seen in Table 1. The magnetic moments in ThCo4 B are compared in Table 2 with that of the isotypic YCo4 B and with the related ThCo5 compound. The magnetic moments in YCo4 B compounds have been calculated by SPR-KKR method and they are compared in Table 2 with the results of earlier reported calculations [24,31,32,38] and with the experimental results [39]. One should note an improved agreement of the fully relativistic SPR-KKR calculations with the experimental values [39] compared with the other calculations. The difference between the Co 2c and Co 6i magnetic moments is evidenced in both ThCo4 B and YCo4 B compounds. The SPR-KKR calculations confirm the weakness of the magnetic moment on the Co 6i site in both ThCo4 B and YCo4 B compounds. The value of the Co 6i magnetic moment is appreciably reduced in ThCo4 B (with more than 30%) compared with the corresponding value for Co 6i magnetic moment in YCo4 B compound. The decrease of the Co 2c magnetic moment by Th for Y substitution is less pronounced (only about 20%). These differences most probably originate from the different
electron number between Y and Th. Indeed, Y has trivalent character whereas Th is tetravalent. The intermetallic compound ThCo5 is ferromagnetic at room temperature having lower Co moments than other RCo5 compounds. However, experiments have shown that ThCo5 exhibits a metamagnetic transition from a low-moment state (LMS) to a high-moment state (HMS) with an applied field [40,41]. The neutron diffraction experiments [40,41] on ThCo5 show that in the LMS the two cobalt sites have different local magnetic moments, whereas in the HMS the two moments are of the same size (Table 2). It was supposed [42] that the Co atoms exhibiting this transition should be paramagnetic before the transition, with only a small moment induced by the interatomic exchange field from the ferromagnetic Co atoms situated at the other type of sites. The calculations performed by Nordstro¨m et al. [33] using the so-called fixed-spin (FSM) based on the LMTO method are in agreement with the experiment showing that for the LMS state two significantly different local Co magnetic moments are observed. On the contrary, for the HMS, the two Co magnetic moments are nearly of the same magnitude (Table 2). The orbital contributions obtained from the calculations of Nordstro¨m et al. in LMS state are different for the two Co sites, but the difference between the orbital moments is less pronounced in the HMS state. The present fully relativistic SPR-KKR calculations have been performed at the experimental lattice parameters reported for ThCo5 [43]. In fact at the experimental volume of the unit cell, the calculations of Nordstro¨m et al. did not found actually the LMS state,
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only a local minimum indicated the closeness to such an equilibrium state. This is the reason to compare our calculation with the calculations of Nordstro¨m et al. for HMS state (see Table 2). The two Co magnetic moments obtained by SPR-KKR calculations (including the orbital contribution) are very close: 1.55 and 1:48 mB . Also, the orbital contributions are comparable for the two different Co crystal sites in HMS state, 0.16 and 0:12 mB for Co 2c and Co 3g sites, respectively. One should mention the good agreement between the orbital moment determined by SPR-KKR calculations (0:69 mB /unit cell) and the experimental estimate of 0:7 mB /unit cell [36]. The broader bands of Co 6i in ThCo4 B compared with the bands of Co 3g in ThCo5 are partially responsible for the weaker 3d magnetization in ThCo4 B than in ThCo5 . The results of the calculations for ThCo5 and ThCo4 B compounds, combined with the experimental measurements shows that the B for Co substitution greatly affects the Co 6i magnetic moments. The change of the magnetic environment of the Co 6i crystal site is the origin for the dramatic decrease of the magnetization in ThCo4 B. According to our calculations, this occurs via the hybridization of the s, p (B) and d (Co) electronic states. 4. Conclusions The SPR-KKR calculations in ThCo4 B show that the Co 2c site is found to carry most of the magnetic moment, in agreement with experiment [13] and with earlier investigations in isotype RCo4 B compounds. Also, the SPR-KKR calculations are in agreement with experimental neutron investigations showing that the easy magnetization direction of ThCo4 B is along the c-axis. The Co 6i site carries a significantly reduced magnetic moment because of its hybridization with B and Th, as it was evidenced by the DOS calculations. Both inequivalent Co sites have smaller magnetic moment in ThCo4 B compared with YCo4 B. The magnitude of the Co 2c site magnetic moment is moderately affected by the replacement of Y by Th, whereas the Co 6i site magnetic moment is much more sensitive to it. Another interesting result of our calculations is the large difference in the orbital magnetic moment magnitude of the two inequivalent Co sites of ThCo4 B, whereas the RCo5 compounds have been found to exhibit similar orbital moment on both Co sites. The calculations agree with the presence of a very weak magnetic moment on Co 6i site as suggested by the experimental measurements. Furthermore, this magnetic moment is significantly reduced ð0:27 mB Þ compared with the corresponding value for isotypic YCo4 B ð0:6 mB Þ. The presence of tetravalent Th atom induces a reduction of the Co magnetic moment in comparison with isotype RCo4 B compounds containing trivalent Y or rare earth elements. This tendency is confirmed by both experiment and calculation performed on UCo4 B where U can be considered as pentavalent. Indeed this compound has been found not to be magnetically ordered even at the lowest temperature
41
[44–47]. The comparison between the ThCo5 and ThCo4 B magnetic moments on the Co sites shows that substituting B for Co greatly affects the magnetic moment on the Co 6i site, whilst the magnetic moment on the Co 2c site is less affected by this substitution. All these observations lead to conclude that the change in the magnetic environment of Co 6i is the origin of its dramatic decrease of the magnetic moment in ThCo4 B. This occurs via the Co–B hybridization of electronic states. Acknowledgments The authors would like to thanks Prof. Postnikov for interesting discussions. The financial supports of the Romanian Ministry of Education and Research through the CEEX program and of the Re´gion Rhoˆne Alpes through the MIRA program have been greatly appreciated. References [1] K.H.J. Buschow, Rep. Prog. Phys. 40 (1977) 1179. [2] K.H.J. Buschow, Rep. Prog. Phys. 42 (1979) 1373. [3] K.H.J. Buschow, Ferromagnetic Materials, North Holland, Amsterdam, 1980, p. 297. [4] E. Burzo, A. Chelkovski, H. Kirchmayr, Landolt Bo¨rnstein Handbook, vol. 3, Springer, Berlin, 1990, p. 1983. [5] H. Kirchmayr, J. Phys. D: Appl. Phys. 29 (1996) 2763. [6] O. Moze, P. Manfrinetti, F. Canepa, A. Palenzona, M.L. Fornasini, J.R. Rodriguez-Carvajal, Intermetallics 8 (2000) 273. [7] P. Manfrinetti, F. Canepa, F. Palenzona, M. Fornasini, E. Giannini, J. Alloys Compds. 247 (1997) 109. [8] T. Jacobs, G. Long, A. Pringle, F. Grandjean, K. Buschow, J. Appl. Phys. 70 (1991) 5983. [9] K. Buschow, J. Appl. Phys. 42 (1971) 3433. [10] A. van der Groot, K. Buschow, Phys. Stat. Sol. (a) 5 (1971) 665. [11] D. Givord, J. Laforest, R. Lemaire, Physica B 86–88 (1977) 204. [12] H. Rosner, D. Koudela, U. Schwarz, A. Handstein, M. Hanfland, I. Opahle, K. Koepernik, M.D. Kuz’min, K.H. Mu¨ller, J.A. Mydosh, M. Richter, Nat. Phys. 2 (2006) 469. [13] O. Isnard, V. Pop, J. Toussaint, J. Phys. Condens. Matter 15 (2003) 791. [14] P. Weinberger, Electron Scattering Theory for Order and Disordered Matter, University Press, Oxford, 1990. [15] A. Gonis, Green Function for Ordered and Disordered Systems, North-Holland, Amsterdam, 1992. [16] P. Strange, Relativistic Quantum Mechanics, University Press, Cambridge, 1998. [17] H. Ebert, in: H. Dreysse´ (Ed.), Electronic Structure and Physical Properties of Solids, vol. 535, Springer, Berlin, 2000, p. 191. [18] H. Ebert, The Munich SPRKKR package, Version 2.1.1., 2002, hhttp://olymp.cup.uni-muenchen.de/ak/ebert/sprkkri. [19] J. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 78 (1997) 1396. [20] J. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. [21] H. Monkhorst, J. Pack, Phys. Rev. B 13 (1976) 5188. [22] Y.B. Kuzma, N. Bilonizhko, Sov. Phys. Crystallogr. 18 (1974) 447. [23] C. Chacon, O. Isnard, J. Solid State Chem. 154 (2000) 242. [24] P. Vlaic, E. Burzo, Mold. J. Phys. Sci. 2 (2002) 40. [25] E. Burzo, V. Pop, C. Borodi, R. Ballou, IEEE Trans. Magn. 30 (1994) 628. [26] E. Burzo, V. Pop, N. Plugaru, Mater. Sci. Forum 62–64 (1990) 611. [27] E. Burzo, V. Pop, N. Plugaru, I. Creanga, Phys. Stat. Sol. (a) 113 (1989) K253. [28] E. Burzo, M. Ursu, J. Magn. Magn. Mater. 70 (1987) 345.
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