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Theoretical study of magnetism in RM12B6 compounds (R=Y, La or Ce; M=Fe, Co)
Journal of Magnetism and Magnetic Materials 323 (2011) 2340–2347
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Theoretical study of magnetism in RM12B6 compounds (R ¼ Y, La or Ce; M ¼Fe, Co) ˇ Blazˇina G.I. Miletic´ , Z. Laboratory for Solid State and Complex Compounds Chemistry, Division of Materials Chemistry, Institute Rudjer Boˇskovic´, P.O. Box 180, 10002 Zagreb, Croatia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 9 July 2009 Received in revised form 4 March 2011 Available online 13 April 2011
Electronic structure study for RM12B6 intermetallics (R ¼ Y, La or Ce; M ¼Fe, Co) was performed. Fixed spin moment calculations for different volumes of unit cell were used to find low and high moment states in LaFe12B6. Obtained results are in agreement with previously obtained experimental and theoretical results. Total magnetic moments obtained for YCo12B6 and LaCo12B6 are in fair agreement with experimental values. In CeCo12B6 discrepancy between theory and experiment seems to be more pronounced. It seems that the calculated Co magnetic moments could be overestimated in the studied RCo12B6 compounds. Present calculations indicate that Fe (Co) atoms prefer 18(h) (18(g)) atomic positions what is in agreement with experiment. & 2011 Elsevier B.V. All rights reserved.
Keywords: Electronic structure Itinerant magnetism Phase transitions
1. Introduction Previous studies have shown that the stability and magnetic properties of the RM12B6, (R¼ rare earth, M¼ Fe, Co) intermetallic compounds (structure type SrNi12B6; space group R3m) strongly depend on their composition. It was found that majority of 4f elements are able to form RCo12B6 compounds [1,2], while in case of RFe12B6 compounds only LaFe12B6 can be prepared with the use of standard arc melting and annealing techniques [3]. Metastable NdFe12B6 was prepared with the melt spinning technique [3,4]. Later technique was also used in preparation of alloy with nominal composition PrFe12B6 [5]. According to experimental results all known RCo12B6 compounds order magnetically [1,2,6–9]. Nuclear magnetic resonance (NMR) study was performed on YCo12B6 and GdCo12B6 [7]. Two different interpretations were offered to explain the obtained spectra [7]. In one of them (which was later also used to explain magnitude of the 3d magnetic moment at the 18(g) and 18(h) crystallographic positions in the R(Fe,Co)12B6 compounds [3]) values of magnetic moments of the Co atoms residing at the two different crystallographic positions (18(g) and 18(h)) were suggested [7]. In the RFe12B6 compounds 3d sublattice exhibits significantly different behaviour if compared to the RCo12B6 compounds in the sense that it clearly shows ability to form two distinct magnetic
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E-mail address: [email protected] (G.I. Miletic´). 0304-8853/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2011.03.035
states—low moment and high moment states [3,10]. Possibly initially antiferromagnetic LaFe12B6 [3,10] shows metamagnetic transition [11,12] from the ferromagnetic (FM) low moment (LM) to the ferromagnetic high moment (HM) state [10]. Metamagnetic transition takes place at the critical magnetic field of 8.6 T and is accompanied with a very large jump of total magnetic moment: 4:6mB =f:u:20:0mB =f:u: or 0:38mB =Featom-1:67mB =Featom [10]. Recent fixed spin moment (FSM) calculations with the tightbinding linear muffin-tin orbital (TB-LMTO) method in atomic sphere approximation (ASA) within local spin density approximation (LSDA; von Barth–Hedin parametrization (BH) [13]) for exchange-correlation energy confirmed the existence of both LM and HM states in LaFe12B6 with calculated total spin magnetic moments of 4:73ð15:32ÞmB =f:u: for LM (HM) state [14]. It was also found that in the LM state magnetic moment of Fe(18(g)) atoms is larger than that of Fe(18(h)) atoms while reverse is true for the HM state [14]. Observed metamagnetic transition in LaFe12B6 demonstrates that low moment state of Fe-3d sublattice is unstable with respect to the LM-HM transition which in LaFe12B6 occurs under the influence of external magnetic field [10]. From the earlier work on the 4f–3d intermetallic compounds follows that in compounds where 3d sublattice is close to transition from nonmagnetic ðNMÞ-ferromagnetic (FM) state (or, as in present case, FM LM-FM HM state) these transitions could be also induced by 4f spin magnetic moments through the 4f–5d–3d interaction [15–17]. This mechanism, to be more precise, includes intraatomic 4f–5d exchange interaction and 5d–3d hybridization [15,16]. The well known example of systems where NM-FM transition can happen as a consequence of mentioned 4f–5d–3d interaction are RCo2 compounds [17,18].
ˇ Blazˇina / Journal of Magnetism and Magnetic Materials 323 (2011) 2340–2347 G.I. Miletic´, Z.
Since in RFe12B6 compounds Fe-3d sublattice shows instability with respect to the LM-HM transition, it can be expected that in RFe12B6 compounds with magnetic 4f element Fe-3d sublattice could be, due to 4f–5d–3d interaction, found in HM state. Magnetization measurements on the La1 xGdxFe12B6 ð0:0 o x r0:5Þ system [10] where it was found that partial substitution La-Gd strongly enhances Fe-3d magnetic moments [10] confirm this expectation. Analogous results were found in magnetization measurements performed on NdFe12B6 where an average Fe magnetic moment of about 1:4mB was extracted [3,4,10]. More recent measurements of ¨ Mossbauer spectra on NdFe12B6 also indicated that Fe magnetic moments are strongly enhanced, the average value of Fe magnetic moment being 1:2mB [19]. Since in NdFe12B6 Fe-3d sublattice is in state of strongly enhanced magnetic moments [3,4,10,19], it is desirable to compare magnetic moments of Fe atoms obtained in mentioned TB-LMTO-ASA-LSDA-BH-FSM calculations for LaFe12B6 in HM ¨ state [14] with Fe magnetic moments extracted from Mossbauer spectra obtained on NdFe12B6 [19]. TB-LMTO-ASA-LSDA-BH-FSM calculations resulted in magnetic moments of 1.13 ð1:54ÞmB for Fe atoms on 18(g) (18(h)) sites for LaFe12B6 in HM state [14], while ¨ Fe magnetic moments in NdFe12B6 extracted from Mossbauer spectra were 1:1ð1:3ÞmB for 18(g) (18(h)) sites [19]. Fair agreement between two results indicates similarity between magnetic state of Fe-3d sublattice in HM state in LaFe12B6 and Fe-3d sublattice in NdFe12B6. The subjects of the present work are the electronic structure and magnetic properties of the RM12B6 compounds (R ¼Y, La or Ce; M¼Fe, Co). Full-potential electronic structure calculations were performed for the RM12B6 compounds with the emphasis on the metamagnetism in case of LaFe12B6 as well as magnetism in RCo12B6. In addition, the site preference of Fe (Co) impurities in the Co (Fe) host lattice was also investigated and compared with experimental data.
2. Computational details Full-potential (FP) electronic structure calculations were performed with WIEN2k package [20] with the augmented planewave (APW)þ local orbitals (l.o.) [21] used as a basis set with the corresponding cut-off parameter RminKmax ¼7.0. Exchangecorrelation energy was treated within the local spin density approximation as given by Perdew and Wang (PW) [22]. Muffin-tin radii were 2.2 a.u. for R, 2.02 a.u. for M and 1.79 a.u. for B atoms. Atomic positions used in the present study were that from SrNi12B6 [23] and together with lattice parameters of studied compounds are given in Table 1. Table 1 Atomic positions in the SrNi12B6 structural type [23] and the corresponding lattice constants of studied compounds. Atom
Position
x
y
z
R M M B
3(a) 18(g) 18(h) 18(h)
0 0.3684 0.4238 0.1912
0 0 0.5762 0.8088
0 0.5 0.0355 0.0421
Compound
˚ a (A)
˚ c (A)
Ref.
LaFe12B6 YCo12B6 LaCo12B6 CeCo12B6
9.610 9.453 9.513 9.515
7.604 7.450 7.504 7.458
[3] [1] [1] [1]
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It should be noted that although there is a possibility of existence of antiferromagnetic state in LaFe12B6, only ferromagnetic arrangement of 3d sublattice is considered in spin-polarized calculations for all studied compounds. Impurities were treated by simply replacing one host atom with one impurity atom within unit cell which will then lead to the symmetry lowering. Such unit cells were then used, as in the case of parent compounds, for standard periodic solid state calculation.
3. Results and discussion 3.1. Non-spinpolarized calculations for RM12B6 compounds Projected densities of states (PDOS) obtained from non-spinpolarized calculations on RM12B6 compounds are given in Fig. 1. It is known that this PDOS contributions could be used to (at least roughly) predict behaviour of different atoms in possible transition from NM state to magnetic state. From Fe-3d-PDOS contributions in the close vicinity of Fermi energy (interval of about 70.25 eV about Fermi energy (EF)) shown in Fig. 1 it could be expected that for small total magnetic moments (as in, e.g., LM state) Fe-(18(g)) atoms will exhibit larger magnetic moments than Fe-(18(h)) atoms. This is in accord with results of Ref. [14] where calculations were performed with the more approximate TB-LMTO-ASA method. In Ref. [14] Fe3d-PDOS contributions were also analysed and similar conclusion was drawn regarding the relative sizes of Fe(18(g)) and Fe(18(h)) magnetic moments in LM state. Fe-3d-PDOS calculated in Ref. [14] with TB-LMTO-ASA method and Fe-3d-PDOS shown in Fig. 1 (calculated with FP-APWþlo method) are similar. However, contributions of two Fe sites in interval of about 0.25 eV below Fermi energy as shown in Ref. [14] (Fig. 2. in Ref. [14]) look identical and, consequently, in Ref. [14] higher magnetic moments of Fe(18(g)) atoms in LM state were alluded to come from the higher contributions of those atoms to the empty states in a small interval (about 0.25 eV) above the Fermi energy [14]. Another important group of states are those centered at about 0.5 eV above the Fermi energy. Contribution of Fe-(18(h)) atoms to these states is considerably higher than that of Fe-(18(g)) atoms. Similar result was obtained in our previous work where TB-LMTO-ASA method was employed [14] (Fig. 2. in Ref. [14]). This difference in contributions was furthermore used in Ref. [14] to explain larger increase of magnetic moments of Fe-(18(h)) atoms compared to Fe(18(g))-atoms during metamagnetic LM-HM transition in LaFe12B6 [14]. Another intermetallic compound in the series of presently investigated compounds is LaCo12B6 and corresponding 3d-PDOS contributions are shown in Fig. 1. The most evident difference compared to LaFe12B6 is additional filling of 3d states. Fermi energy is shifted and now falls at the region (peak) of states which were in NM state of LaFe12B6 placed at 0.5 eV above the Fermi energy. These states were previously identified in Ref. [14] as being responsible for larger increase of Fe-(18(h)) magnetic moments when compared to Fe-(18(g)) magnetic moments during the LM-HM transition in LaFe12B6. Analogously to LaFe12B6, Co-(18(h)) atoms contribute more to this group of states than Co-(18(g)) atoms. Therefore, it could be expected that Co(18(h)) magnetic moments would be higher than the Co(18(g)) magnetic moments if transition from NM state to magnetic state takes place. In Fig. 1 3d-PDOS contributions to the density of states in YCo12B6 are also shown. Since main features of the corresponding Co contributions in LaCo12B6 and YCo12B6 are similar, it could be
ˇ Blazˇina / Journal of Magnetism and Magnetic Materials 323 (2011) 2340–2347 G.I. Miletic´, Z.
DOS (states/(eV cell))
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3.5 3 2.5 2 1.5 1 0.5 0
LaFe12B6: Fe (18 (g))-3d LaFe12B6: Fe (18 (h))-3d
DOS (states/(eV cell))
-6
-2
-1
0
1
2
3
4
-2
-1
0
1
2
3
4
-2
-1
0
1
2
3
4
3.5 CeCo12B6: Co (18 (g))-3d 3 CeCo12B6: Co (18 (h))-3d 2.5 2 1.5 1 0.5 0 -6 -5 -4 -3 -2 70 LaFe12B6: La-4f 60 LaCo12B6: La-4f 2 50 CeCo12B6: Ce-4f 40 1 30 20 0 10 -1 0 -6 -5 -4 -3 -2
-1
0
1
2
3
4
0
1
2
3
4
3.5 3 2.5 2 1.5 1 0.5 0
DOS (states/(eV cell)) DOS (states/(eV cell))
-4
-3
LaCo12B6: Co (18 (g))-3d LaCo12B6: Co (18 (h))-3d
-6
DOS (states/(eV cell))
-5
3.5 3 2.5 2 1.5 1 0.5 0 -6
-5
-4
-3
YCo12B6: Co (18 (g))-3d YCo12B6: Co (18 (h))-3d
-5
-4
-3
Ce-4f
0
1
-1 E-EF (eV)
Fig. 1. 3d-PDOS and 4f-PDOS contributions in investigated RM12B6 compounds obtained from non-spinpolarized calculations. In the inset Ce-4f-PDOS contribution in the interval of 7 1 eV about the Fermi energy is given.
expected that corresponding Co atoms will behave similarly in these two compounds. PDOS contributions for CeCo12B6 are also shown in Fig. 1. Main features of Co-3d-PDOS contributions in CeCo12B6 are similar to the corresponding Co contributions in LaCo12B6 and YCo12B6. In Fig. 1 4f-PDOS contributions for LaFe12B6, LaCo12B6 and CeCo12B6 are shown. As expected, in La 4f states are (almost) empty and their peaks are placed around 3 eV above the Fermi energy (in fact, present non-spinpolarized calculations resulted in a small occupancy of 4f states: in LaFe12B6 and LaCo12B6 there are 0.11 and 0.12 4f electrons, respectively.) It is known that starting with cerium filling of 4f states takes place in the lanthanide series (present non-spinpolarized calculation resulted in 0.89 4f electrons within the Ce muffin-tin sphere). From the inset of Fig. 1 where Ce-4f-PDOS is shown in the interval of 71 eV around the
Fermi energy, it is visible that Ce-4f-PDOS at the Fermi energy is comparable to that of the Co(18(g))-3d states.
3.2. Spin-polarized calculations for RM12B6 compounds without spin–orbit coupling Spin magnetic moments obtained from spin-polarized calculations are collected in Table 2. Values are given for two different setups of the corresponding basis sets (see the notes in Refs. [25,26]). In general, if not stated otherwise, results presented in this article are those obtained with the default setup. If different lattice parameters than those determined from experiment were used in calculations, the initial ratio c=a was always kept constant.
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Table 2 Magnetic moments in studied RM12B6 compounds from spin-polarized calculations without spin–orbit coupling. Magnetic moments are given in mB . (Corresponding values in the parentheses are obtained with non-default setup of the basis set. See the notes in Refs. [25,26].) Explanation of labels: ms-spin magnetic moment, IS-interstitial. R
M18(g)
M18(h)
B
0.68 0.74 0.99 1.15 1.30
0.27 0.29 1.10 1.46 1.63
0.03 0.04 0.07 0.09 0.10
IS
f.u.
Compound
V/Vexp
LaFe12B6
0.94 0.96 0.96 0.98 1.0
ms ms ms ms ms
0.02 0.02 0.04 0.05 0.05
LaCo12B6
0.96 0.98 1.0
ms ms ms
0.04 ( 0.03) 0.04 ( 0.04) 0.04 ( 0.04)
0.32 (0.32) 0.36 (0.33) 0.36 (0.34)
0.76 (0.73) 0.78 (0.76) 0.81 (0.79)
0.04 ( 0.03) 0.04 ( 0.04) 0.04 ( 0.04)
0.30 ( 0.27) 0.35 ( 0.30) 0.37 ( 0.32)
5.94 (5.77) 6.21 (6.00) 6.40 (6.19)
YCo12B6
1.0
ms
0.03 ( 0.04)
0.34 (0.34)
0.79 (0.79)
0.04 ( 0.04)
0.29 ( 0.29)
6.22 (6.23)
CeCo12B6
1.0
ms
0.51 ( 0.49)
0.42 (0.39)
0.87 (0.84)
0.04 ( 0.04)
0.46 ( 0.41)
6.51 (6.25)
( 0.02) ( 0.02) ( 0.03) ( 0.04) ( 0.05)
(0.69) (0.75) (0.98) (1.14) (1.28)
0
-5
Δ E (V, M) (mRy)
-10
-15
-20 V/Vexp = 1.0 -25
V/Vexp = 0.98 V/Vexp = 0.96
-30
V/Vexp = 0.94 -35 0
2
4
6
8
10
12
14
16
18
20
M (μB) Fig. 2. Results of fixed spin moment calculations for LaFe12B6 at different volumes of the unit cell.
Results of FSM calculations for different volumes of LaFe12B6 unit cell are shown in Fig. 2. Present calculations do not find LM state at the observed lattice parameters (Fig. 2). It is known that LSDA in general overestimates stability of spin magnetic moments: magnetic vs. nonmagnetic states [27,28] or HM vs. LM states [14,29]. When lattice parameters are decreased, HM state is destabilized and LM state appears. At V¼0.96Vexp LM and HM states are of almost equal energies (HM is more stable for about 0.2 mRy), while at V ¼0.94Vexp only LM state remains. In a previous work [14] TB-LMTO-ASA-FSM calculations with the two different versions of LSDA (von Barth–Hedin (BH) [13] and Vosko–Wilk–Nusair (VWN) [24]) were performed using the experimentally observed lattice parameters. In fixed spin moment calculations performed with the BH-LSDA parametrization both states (LM and HM) were found [14]. Similarly to the present results, in previous FSM calculations performed with the VWNLSDA parametrization only HM state was found at the experimentally determined lattice constants [14]. In Fig. 3 are shown results which correspond to the minima on Fig. 2.
(0.28) (0.29) (1.08) (1.44) (1.62)
( 0.03) ( 0.04) ( 0.07) ( 0.09) ( 0.10)
0.03 0.03 0.04 0.06 0.08
(0.04) (0.03) ( 0.03) ( 0.05) ( 0.07)
5.56 5.98 12.02 15.01 16.85
(5.61) (6.03) (11.88) (14.86) (16.73)
In LM state magnetic moments of Fe(18(h)) atoms are significantly lower than those of Fe(18(g)) atoms. During LM-HM transition spin magnetic moments of the Fe atoms show (Fig. 3(c)) similar behaviour to that found in [14]: formation of the HM state takes place after a large jump of spin magnetic moment of Fe(18(h)) atoms, while at the same time spin magnetic moments of Fe(18(g)) atoms show significantly smaller change. Finally, in HM state magnetic moments of Fe(18(h)) atoms are higher than magnetic moments of Fe(18(g)) atoms. For illustration (Fig. 3c), Table 2.: at volume V¼0.96Vexp where both magnetic states coexist, in LM state spin magnetic moment of Fe(18(g)) atoms is 0:74mB , that of Fe(18(h)) atoms is 0:29mB ; while in the HM state spin magnetic moment of Fe(18(g)) atoms is 0:99mB , and that of Fe(18(h)) atoms amounts to 1:10mB . It was mentioned above that if the 3d system is unstable with respect to NM-FM transition, or FM LM-FM HM transition, 4f spin moments could induce transition through the 4f–5d–3d interaction [15–18]. Experimental results on NdFe12B6 [3,4,10,19] and La1 xGdx Fe12B6 ð0:0 o x r0:5Þ system [10] confirmed this expectation. ¨ In recent work on NdFe12B6 where Mossbauer spectroscopy was employed [19], magnetic moments of Fe(18(g)) (Fe(18(h))) atoms were determined to be 1:1mB ð1:3mB Þ [19] thus confirming more directly that 4f spin moments enhance Fe-3d spin moments. In addition, this result [19] indicates that magnetic moments of Fe(18(h)) atoms will be higher than magnetic moments of Fe(18(g)) atoms in HM state in agreement with previous [14] and present theoretical results. Fig. 3(a) shows relative energies of the minima found in the present calculations. In Fig. 3(a) another known drawback of LSDA is visible, i.e., volumes belonging to the total energy minima will be too small when compared to experimental values (overbinding) [30,31]. As mentioned above, in Ref. [14] relative magnitudes of 3d magnetic moments in LaFe12B6 in different magnetic states were indicated from 3d-PDOS. In case of RCo12B6 compounds spin-polarized calculations show that the relative magnitude of spin magnetic moments could be indicated from 3d-PDOS obtained from non-spinpolarized calculations. In all investigated compounds contributions (Fig. 1) from Co(18(h)) atoms to the states at EF are significantly larger than that from Co(18(g)) atoms. This is reflected in relative sizes of spin magnetic moments on Co(18(g)) and Co(18(h)) atoms in all investigated RCo12B6 compounds: spin magnetic moments of Co(18(h)) atoms are always higher than that of Co(18(g)) atoms (Table 2). Although experiments do not indicate that in LaCo12B6 compound 3d sublattice will exhibit itinerant band metamagnetism as in LaFe12B6 compound, FSM calculations were performed for
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ˇ Blazˇina / Journal of Magnetism and Magnetic Materials 323 (2011) 2340–2347 G.I. Miletic´, Z.
mFe (μB)
1.50
1.00 Fe (18g) 0.50 Fe (18h)
M (μB)
16.0
12.0
8.0
60.0 40.0
LM 0.2 mRy
Erel (mRy)
80.0
HM
20.0 0.0 0.94
0.96
0.98
1
V/Vexp Fig. 3. Results obtained by ordinary spin-polarized calculations for LaFe12B6. Results correspond to the minima on Fig. 2. Erel is the difference between total energies of different minima and the minimum corresponding to V/Vexp ¼ 0.94.
different volumes of LaCo12B6 and results are shown in Fig. 4. (These calculations were performed with non-default basis set setup. See the notes in Refs. [25,26].) As can be seen from Table 2, Figs. 2–4, volume dependence of magnetic properties differs in LaFe12B6 and LaCo12B6. In LaFe12B6 there are two distinct magnetic states (LM and HM) and HM state exhibits strong volume dependence of both magnetic moments and relative stability vs. NM state. Finally, at V¼0.96Vexp LM state emerges and at V ¼0.94Vexp HM state disappears. In LaCo12B6 there is only one magnetic state which slowly changes both in magnetic moment and relative stability vs. NM state. In LaCo12B6 total spin magnetic moment calculated at experimental lattice constants (Table 2) is 6:40mB =f:u: while magnetization obtained from experiment is 5:35mB =f:u [8]. Another choice of basis set somewhat improves agreement ð6:19mB =f:uÞ but this is obviously also too high. As mentioned above, spin magnetic moments on Co(18(g)) atoms are lower than that of the Co(18(h)) atoms amounting to 0.36 and 0:81mB , respectively. In YCo12B6 situation is analogous to that in LaCo12B6. Spin magnetic moments of Co atoms are 0:34ð0:79ÞmB for Co(18(g)) (Co(18(h))) atoms and total spin magnetic moment is 6:22mB =f:u while experimentally determined values are 6:5 [1], 5:2 [2], 5:32 [6], 5:1 [8] and 5:34mB =f:u [9]. It appears that magnetic moment of 6:5mB =f:u found in Ref. [1] is rather high when compared to the other values which are found in the narrow interval of 5:125:34mB =f:u [2,6,8,9]. It is interesting to note that in case of YCo12B6 both choices of basis set resulted in almost identical spin magnetic moments.
In addition, total spin magnetic moments obtained for YCo12B6 and LaCo12B6 with non-default basis set are very close. Spin magnetic moments obtained for atoms are (within precision used) identical, small difference coming only from interstitial region. It seems that our calculations, to some extent, overestimate total magnetic moments in YCo12B6 and LaCo12B6. As mentioned above, in [7] two different interpretations were given for NMR spectra in YCo12B6 and GdCo12B6. According to the one of them (which was later also employed in [3]) two subspectra could be attributed to two different crystallographic positions of Co atoms in the RCo12B6 lattice [7]. Spin magnetic moments estimated within this interpretation were 0:30 7 0:09mB for the 18(g) site and 0:69mB for the 18(h) site [7] and one can say that they are in fair agreement with the present computational results. Additionally, they indicate that theoretical values of Co magnetic moments, as obtained in the present work, could be too high. However, it should also be mentioned that authors in [7] concluded that further investigations are necessary before one can make any definitive choice between two proposed interpretations. In case of CeCo12B6 present calculations resulted in spin magnetic moment of 6:51mB =f:u: (or 6:25mB =f:u: with non-default basis set) while experimental values are 5:4 [1], 4:61 [8], 4:75 [9] and 4:8mB =f:u [32]. On atoms following spin magnetic moments were obtained in our calculations: 0:42ð0:87ÞmB for Co atoms at 18(g) (18(h)) positions (corresponding values obtained with nondefault basis set are somewhat lower: 0:39ð0:84ÞmB for Co atoms at 18(g) (18(h)) sites).
ˇ Blazˇina / Journal of Magnetism and Magnetic Materials 323 (2011) 2340–2347 G.I. Miletic´, Z.
20 15
V/Vexp = 1.0
10
V/Vexp = 0.98 V/Vexp = 0.96
Δ E (V, M) (mRy)
5 0 -5 -10 -15 -20 -25 0
1
2
3
4
5
6 7 M (μB)
8
9
10
11
12
Fig. 4. Results of fixed spin moment calculations for LaCo12B6 at different volumes of the unit cell. (These calculations were performed with non-default basis set setup. See the notes in Refs. [25,26].)
Results for CeCo12B6 also indicate that Co magnetic moments are too high in investigated RCo12B6 compounds. In all studied compounds there are small spin magnetic moments ð0:0220:10mB Þ induced on non-3d atoms which are oriented opposite to the 3d spin magnetic moments. In CeCo12B6 there is, in addition, a contribution of 4f spin magnetic moment ð0:46mB Þ which is also oriented antiparallel to the 3d spin magnetic moments. 3.3. Spin-polarized calculations for RM12B6 compounds with spin–orbit coupling To determine contribution of orbital moments in RM12B6 compounds, calculations with spin–orbit coupling (SOC) were also performed and corresponding magnetic moments are given in Table 3. In LaFe12B6 obtained orbital moments are very small and they exhibit small change during metamagnetic transition. In LM state at V ¼0.96Vexp orbital moments at Fe(18(g)) (Fe(18(h)) site were 0:03mB ð0:01mB Þ. At V¼0.96Vexp, where, in present calculations, transition occurs, a small change during the transition appears only on Fe(18(h)) site ð0:01mB -0:03mB Þ while orbital moment at Fe(18(g)) remains (within precision used here) constant ð0:03mB Þ. At Vexp, where only HM state exists, orbital moments at both Fe-sites are 0:05mB . In LaCo12B6, small orbital moments are induced on Co atoms after the inclusion of SOC (0:02ð0:03ÞmB for 18(g) (18(h)) site) what gives total magnetic moment of 6:66mB =f:u (6:50mB =f:u with non-default basis set) to be compared with experimental value of 5:35mB =f:u [8]. Induced orbital moments are also very small in YCo12B6: 0:02ð0:03ÞmB for Co(18(g)) (Co(18(h))) atoms while the total magnetic moment has raised to 6:53mB =f:u: Values of magnetic moments determined from experiment for YCo12B6 are 6:5 [1], 5:2 [2], 5:32 [6], 5:1 [8] and 5:34mB =f:u [9]. As in LaCo12B6 and YCo12B6 small orbital moments are induced on Co atoms (0:02ð0:03ÞmB for 18(g) (18(h)) position) in CeCo12B6. However, in CeCo12B6 there is an additional contribution coming
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from orbital moment of Ce-4f electrons which is 0:27mB (0:26mB for non-default basis set) and it is oriented (Hund third rule) opposite to the 4f-spin magnetic moment. Since 3d and 4f spin magnetic moments couple ferrimagnetically, Ce orbital moment additionally increases total magnetic moment in unit cell. In CeCo12B6 dependence of magnetic moments on basis set choice is more pronounced than in the other compounds. For default basis set total magnetic moment is 7:12mB =f:u: while for nondefault basis set it is 6:70mB =f:u: Corresponding experimental values are 5:4 [1], 4:61 [8], 4:75 [9] and 4:8mB =f:u [32]. Present calculations seem to overestimate Co magnetic moments in RCo12B6 compounds. Disagreement is even more pronounced when spin–orbit coupling is included in calculations. Disagreement between experiment and theory deserves additional comment. We believe that one of the reasons for discrepancy between our calculations and experimental results could be the atomic positions used in our calculations. As mentioned above, atomic positions used here are those from SrNi12B6 compound [23]. In this structure only 3(a) position (R atoms) does not contain variable parameters so that other atomic positions in studied compounds could differ from the corresponding positions in the SrNi12B6. 3.4. Calculations for RM11MimpB6 compounds (M¼ Fe (Co); Mimp ¼Co (Fe)): Previous experimental work also included studies of distribution of the particular 3d atoms in RFe12 xCoxB6 compounds over the two available crystallographic positions (18(g) or 18(h)). It was found that Fe atoms prefer 18(h) and Co atoms 18(g) sites [3,33–35]. Present calculations were performed on LaFe12B6, LaCo12B6 and YCo12B6 compounds where one Fe (Co) atom was substituted by Co (Fe) atom on either 18(g) or 18(h) position. Thus, the final studied systems were of the LaFe11CoB6, LaCo11FeB6 and YCo11FeB6 composition. In the case of non-spinpolarized calculations performed with the experimentally observed lattice constants it was found that Co atom in LaFe11CoB6 prefers 18(g) position with an energy difference of 8.2 mRy. In LaCo11FeB6 Fe atom prefers 18(h) site the energy difference being 7.9 mRy. In YCo11FeB6 Fe atom also prefers 18(h) position with the energy difference of 8.6 mRy. This results are in accordance with experimental data [3,33–35]. Spin-polarized calculations with the experimental lattice parameters were also performed for LaFe11CoB6, LaCo11FeB6 and YCo11FeB6 compositions. In the case of LaFe11CoB6 one should note that at Vexp Fe-sublattice is in HM state. Again Co impurity prefers 18(g) position but now the energy difference is 1.9 mRy. In LaCo11FeB6 Fe atom prefers 18(h) crystallographic position the energy difference being 10.6 mRy. In YCo11FeB6, as in LaCo11FeB6, configuration with Fe impurities in 18(h) site is again more stable with the corresponding energy difference being 11.0 mRy. One could therefore say that computational results presented here, also in agreement with the experimental findings [3,33–35], indicate that Fe and Co atoms will prefer 18(h) and 18(g) positions, respectively.
4. Conclusion Full-potential electronic structure calculations were performed for the RM12B6 intermetallic compounds (R¼Y, La or Ce; M¼ Fe, Co). In the case of LaFe12B6 fixed spin moment calculations performed for various volumes resulted in findings of low moment and high moment states in agreement with previous experimental and theoretical results. Low moment state was found at reduced lattice constants with respect to the experimental ones. In LM
ˇ Blazˇina / Journal of Magnetism and Magnetic Materials 323 (2011) 2340–2347 G.I. Miletic´, Z.
2346
Table 3 Magnetic moments in studied RM12B6 compounds from spin-polarized calculations with spin–orbit coupling. Magnetic moments are given in mB . (Corresponding values in the parentheses are obtained with non-default setup of the basis set. See the notes in Refs. [25,26].) Explanation of labels: ms-spin magnetic moment, mo-orbital magnetic moment, mt-total magnetic moment, IS-interstitial. Compound
V/Vexp
LaFe12B6
0.94
0.96
0.96
0.98
1.0
LaCo12B6
0.96
0.98
1.0
M18(g)
R
M18(h)
B
IS
ms mo mt ms mo mt ms mo mt ms mo mt ms mo mt
0.02 0.00 0.01 0.02 0.00 0.02 0.04 0.00 0.03 0.05 0.00 0.04 0.05 0.00 0.05
( 0.02) (0.00) ( 0.01) ( 0.02) (0.00) ( 0.02) ( 0.03) (0.00) ( 0.03) ( 0.04) (0.00) ( 0.04) ( 0.05) (0.00) ( 0.05)
0.68 0.03 0.71 0.74 0.03 0.77 0.98 0.03 1.01 1.14 0.04 1.18 1.29 0.05 1.34
(0.68) (0.03) (0.71) (0.75) (0.03) (0.78) (0.97) (0.03) (1.00) (1.14) (0.04) (1.18) (1.28) (0.05) (1.33)
0.27 0.01 0.28 0.29 0.01 0.30 1.09 0.03 1.12 1.45 0.04 1.49 1.63 0.05 1.69
(0.27) (0.01) (0.28) (0.30) (0.01) (0.31) (1.06) (0.03) (1.09) (1.43) (0.04) (1.48) (1.62) (0.05) (1.68)
0.03 0.00 0.03 0.04 0.00 0.04 0.07 0.00 0.07 0.09 0.00 0.09 0.10 0.00 0.10
( 0.03) (0.00) ( 0.03) ( 0.04) (0.00) ( 0.04) ( 0.07) (0.00) ( 0.07) ( 0.09) (0.00) ( 0.09) ( 0.10) (0.00) ( 0.10)
ms mo mt ms mo mt ms mo mt
0.04 0.00 0.03 0.04 0.00 0.04 0.04 0.00 0.04
( 0.03) (0.00) ( 0.03) ( 0.04) (0.00) ( 0.03) ( 0.04) (0.00) ( 0.04)
0.32 0.01 0.33 0.36 0.02 0.37 0.36 0.02 0.38
(0.31) (0.01) (0.33) (0.33) (0.02) (0.35) (0.34) (0.02) (0.36)
0.77 0.03 0.79 0.79 0.03 0.82 0.81 0.03 0.84
(0.73) (0.03) (0.75) (0.76) (0.03) (0.79) (0.79) (0.03) (0.82)
0.04 0.00 0.04 0.04 0.00 0.04 0.04 0.00 0.04
( 0.03) (0.00) ( 0.03) ( 0.04) (0.00) ( 0.04) ( 0.04) (0.00) ( 0.04)
0.03 (0.04) 0.03 (0.04) 0.03 (0.03) 0.03 (0.03) 0.04 ( 0.03) 0.04 ( 0.03) 0.06 ( 0.05) 0.06 ( 0.05) 0.07 ( 0.06) 0.07 ( 0.06) 0.30 ( 0.27) 0.30 ( 0.27) 0.34 ( 0.29) 0.34 ( 0.29) 0.36 ( 0.32) 0.36 ( 0.32)
YCo12B6
1.0
ms mo mt
0.03 ( 0.03) 0.00 (0.00) 0.03 ( 0.03)
0.34 (0.34) 0.02 (0.02) 0.35 (0.35)
0.79 (0.79) 0.03 (0.03) 0.82 (0.82)
0.04 ( 0.04) 0.00 (0.00) 0.04 ( 0.04)
0.29 ( 0.29)
CeCo12B6
1.0
ms mo mt
0.48 ( 0.46) 0.27 (0.26) 0.21 ( 0.19)
0.42 (0.38) 0.02 (0.02) 0.44 (0.40)
0.87 (0.82) 0.03 (0.03) 0.90 (0.85)
0.04 ( 0.04) 0.00 (0.00) 0.04 ( 0.04)
0.45 ( 0.40)
state Fe(18(g)) atoms show larger magnetic moments than the Fe(18(h)) atoms, while reverse is true in HM state. The same was found in past theoretical work where both LM and HM states were found while previous experimental work also indicates that in HM state Fe(18(h)) atoms have higher magnetic moments than Fe(18(g)) atoms. In RCo12B6 compounds with R¼ Y, La fair agreement was found between theoretical and experimental total magnetic moments. Previous experimental investigation from which values of Co atomic magnetic moments were proposed also indicates that calculated Co magnetic moments could be in fair agreement with experimental values. However, it seems that present calculations to some extent overestimate Co magnetic moments. In CeCo12B6 the disagreement between theory and experiment seems to be more notable. Total energy calculations for LaFe11CoB6, LaCo11FeB6 and YCo11FeB6 show that the Fe (Co) atoms prefer 18(h) (18(g)) sites what is in agreement with the previous experimental findings.
Acknowledgement The support for this research by the Ministry of Science, Education and Sport of the Republic of Croatia under Project no. 098-0982904-2941 is highly appreciated. Calculations were performed at the University Computing Centre of the University of Zagreb. References [1] M. Jurczyk, A.T. Pedziwiatr, W.E. Wallace, J. Magn. Magn. Mater. 67 (1987) L1–L3.
0.29 ( 0.29)
0.45 ( 0.40)
f.u. 5.51 0.21 5.72 6.00 0.24 6.24 11.92 0.37 12.28 14.92 0.51 15.43 16.80 0.63 17.43 5.95 0.26 6.21 6.23 0.27 6.50 6.37 0.28 6.66
(5.56) (0.22) (5.78) (6.10) (0.25) (6.35) (11.75) (0.36) (12.12) (14.82) (0.51) (15.33) (16.70) (0.63) (17.33) (5.73) (0.26) (5.99) (6.01) (0.28) (6.29) (6.21) (0.29) (6.50)
6.25 (6.25) 0.28 (0.28) 6.53 (6.53) 6.52 (6.11) 0.60 (0.59) 7.12 (6.70)
[2] G.F. Zhou, X. Li, F.R. de Boer, K.H.J. Buschow, Physica B 177 (1992) 286–290. [3] M. Rosenberg, T. Sinnemann, M. Mittag, K.H.J. Buschow, J. Alloys Compd. 182 (1992) 145–156. [4] K.H.J. Buschow, D.B. de Mooij, H.M. van Noort, J. Less-Common Met. 125 (1986) 135 (cited in [3]). [5] W. Fang, S. Bao-Gen, Z. Jian, S. Ji-Rong, M. Fan-Bin, L. Yang-Xian, Chin. Phys. B 19 (2010) 067501–1-5 (published after submission of our paper). [6] M. Rosenberg, M. Mittag, K.H.J. Buschow, J. Appl. Phys. 63 (1988) 3586–3588. [7] K. Erdmann, M. Rosenberg, K.H.J. Buschow, J. Appl. Phys. 63 (1988) 4113–4115. [8] M. Mittag, M. Rosenberg, K.H.J. Buschow, J. Magn. Magn. Mater. 82 (1989) 109–117. [9] Z. Arnold, O. Isnard, H. Mayot, M. Mı´sˇ ek, J. Kamara´d, J. Magn. Magn. Mater 322 (2010) 1117–1119 (published after submission of our paper). [10] Q.-A. Li, C.H. de Groot, F.R. de Boer, K.H.J. Buschow, J. Alloys Compd. 256 (1997) 82–85. [11] E.P. Wohlfarth, P. Rhodes, Philos. Mag. 7 (1962) 1817–1824. [12] M. Shimizu, J. Phys. 43 (1982) 155–163. [13] U. von Barth, L. Hedin, J. Phys. C: Solid State Phys. 5 (1972) 1629–1642. ˇ Blazˇina, J. Alloys Compd. 430 (2007) 9–12. [14] G.I. Miletic´, Z. [15] M.S.S. Brooks, O. Eriksson, B. Johansson, J. Phys.: Condens. Matter 1 (1989) 5861–5874. ¨ [16] M.S.S. Brooks, L. Nordstrom, B. Johansson, Physica B 172 (1991) 95–100. ¨ [17] L. Nordstrom, M.S.S. Brooks, B. Johansson, J. Appl. Phys. 70 (1991) 6583–6585. [18] E. Gratz, A.S. Markosyan, J. Phys.: Condens. Matter 13 (2001) R385–R413. [19] Y.-J. Huang, Z.-L. Shi, C.-L. Zhang, S.-Z. Li, N. Chen, Z.-D. Han, Y.-F. Hsia, Mater. Lett. 62 (2008) 85–87. [20] P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, J. Luitz, WIEN2k: An Augmented Plane Waveþ Local Orbitals Program for Calculating Crystal ¨ Wien, Austria, Properties, 2002, Karlheinz Schwarz, Techn. Universitat 2001, ISBN 3-9501031-1-2. ¨ ¨ [21] E. Sjostedt, L. Nordstrom, D.J. Singh, Solid State Commun. 114 (2000) 15–20. [22] J.P. Perdew, Y. Wang, Phys. Rev. B 45 (1992) 13244–13249. [23] P. Villars, Pearson’s Handbook Desk Edition; Crystallographic Data for Intermetallic Phases, vol. 1, ASM International, 1997. [24] S.H. Vosko, L. Wilk, M. Nusair, Can. J. Phys. 58 (1980) 1200–1211. [25] More precisely, non-default basis set means optimized linearization energies and (possibly) additional basis set functions. They were chosen with the help of automatic procedure implemented in WIEN2k. More information can be found in WIEN2k User’s Guide.
ˇ Blazˇina / Journal of Magnetism and Magnetic Materials 323 (2011) 2340–2347 G.I. Miletic´, Z.
[26] As can be seen from Table 2, the highest differences between atomic spin magnetic moments for two different basis sets are 0:03mB /atom, while maximum variation of interstitial spin magnetic moments is 0:05mB . On the other hand, if one uses default settings for basis set in calculations of total energies in FSM calculations in LaCo12B6, one will find counterintuitive results: one would expect that relative stability of FM vs. NM state will decrease with decreasing of volume, but that was not the case with results obtained with default basis set for LaCo12B6. Therefore, non-default basis set was used for FSM calculations in LaCo12B6 and results are shown in Fig. 4. [27] S. Khmelevskyi, I. Turek, P. Mohn, J. Phys.: Condens. Matter 13 (2001) 8405–8414. [28] S. Khmelevskyi, I. Turek, P. Mohn, J. Phys.: Condens. Matter 14 (2002) 13799–13811.
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¨ [29] L. Nordstrom, B. Johansson, O. Errikson, M.S.S. Brooks, Phys. Rev. B 42 (1990) 8367–8374. ¨ [30] J. Kubler, Theory of Itinerant Electron Magnetism, International Series of Monographs on Physics, Clarendon Press, Oxford, 2000. [31] R.O. Jones, O. Gunnarsson, Rev. Mod. Phys. 61 (1989) 689–746. [32] M. Mı´sˇ ek, Z. Arnold, O. Isnard, H. Mayot, Y. Skorokhod, J. Kamara´d, Acta Phys. Pol. A 113 (2008) 263–266. [33] S.J. Campbell, J.M. Cadogan, X.L. Zhao, E. Wu, Hyperfine Interact. 78 (1993) 463–467. [34] E. Wu, J.M. Cadogan, S.J. Campbell, X.L. Zhao, Hyperfine Interact. 94 (1994) 1903–1908. [35] X. Zhao, J.M. Cadogan, S.J. Campbell, J. Magn. Magn. Mater. 140–144 (1995) 959–960.
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Theoretical study of magnetism in RM12B6 compounds (R ¼ Y, La or Ce; M ¼Fe, Co) ˇ Blazˇina G.I. Miletic´ , Z. Laboratory for Solid State and Complex Compounds Chemistry, Division of Materials Chemistry, Institute Rudjer Boˇskovic´, P.O. Box 180, 10002 Zagreb, Croatia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 9 July 2009 Received in revised form 4 March 2011 Available online 13 April 2011
Electronic structure study for RM12B6 intermetallics (R ¼ Y, La or Ce; M ¼Fe, Co) was performed. Fixed spin moment calculations for different volumes of unit cell were used to find low and high moment states in LaFe12B6. Obtained results are in agreement with previously obtained experimental and theoretical results. Total magnetic moments obtained for YCo12B6 and LaCo12B6 are in fair agreement with experimental values. In CeCo12B6 discrepancy between theory and experiment seems to be more pronounced. It seems that the calculated Co magnetic moments could be overestimated in the studied RCo12B6 compounds. Present calculations indicate that Fe (Co) atoms prefer 18(h) (18(g)) atomic positions what is in agreement with experiment. & 2011 Elsevier B.V. All rights reserved.
Keywords: Electronic structure Itinerant magnetism Phase transitions
1. Introduction Previous studies have shown that the stability and magnetic properties of the RM12B6, (R¼ rare earth, M¼ Fe, Co) intermetallic compounds (structure type SrNi12B6; space group R3m) strongly depend on their composition. It was found that majority of 4f elements are able to form RCo12B6 compounds [1,2], while in case of RFe12B6 compounds only LaFe12B6 can be prepared with the use of standard arc melting and annealing techniques [3]. Metastable NdFe12B6 was prepared with the melt spinning technique [3,4]. Later technique was also used in preparation of alloy with nominal composition PrFe12B6 [5]. According to experimental results all known RCo12B6 compounds order magnetically [1,2,6–9]. Nuclear magnetic resonance (NMR) study was performed on YCo12B6 and GdCo12B6 [7]. Two different interpretations were offered to explain the obtained spectra [7]. In one of them (which was later also used to explain magnitude of the 3d magnetic moment at the 18(g) and 18(h) crystallographic positions in the R(Fe,Co)12B6 compounds [3]) values of magnetic moments of the Co atoms residing at the two different crystallographic positions (18(g) and 18(h)) were suggested [7]. In the RFe12B6 compounds 3d sublattice exhibits significantly different behaviour if compared to the RCo12B6 compounds in the sense that it clearly shows ability to form two distinct magnetic
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E-mail address: [email protected] (G.I. Miletic´). 0304-8853/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2011.03.035
states—low moment and high moment states [3,10]. Possibly initially antiferromagnetic LaFe12B6 [3,10] shows metamagnetic transition [11,12] from the ferromagnetic (FM) low moment (LM) to the ferromagnetic high moment (HM) state [10]. Metamagnetic transition takes place at the critical magnetic field of 8.6 T and is accompanied with a very large jump of total magnetic moment: 4:6mB =f:u:20:0mB =f:u: or 0:38mB =Featom-1:67mB =Featom [10]. Recent fixed spin moment (FSM) calculations with the tightbinding linear muffin-tin orbital (TB-LMTO) method in atomic sphere approximation (ASA) within local spin density approximation (LSDA; von Barth–Hedin parametrization (BH) [13]) for exchange-correlation energy confirmed the existence of both LM and HM states in LaFe12B6 with calculated total spin magnetic moments of 4:73ð15:32ÞmB =f:u: for LM (HM) state [14]. It was also found that in the LM state magnetic moment of Fe(18(g)) atoms is larger than that of Fe(18(h)) atoms while reverse is true for the HM state [14]. Observed metamagnetic transition in LaFe12B6 demonstrates that low moment state of Fe-3d sublattice is unstable with respect to the LM-HM transition which in LaFe12B6 occurs under the influence of external magnetic field [10]. From the earlier work on the 4f–3d intermetallic compounds follows that in compounds where 3d sublattice is close to transition from nonmagnetic ðNMÞ-ferromagnetic (FM) state (or, as in present case, FM LM-FM HM state) these transitions could be also induced by 4f spin magnetic moments through the 4f–5d–3d interaction [15–17]. This mechanism, to be more precise, includes intraatomic 4f–5d exchange interaction and 5d–3d hybridization [15,16]. The well known example of systems where NM-FM transition can happen as a consequence of mentioned 4f–5d–3d interaction are RCo2 compounds [17,18].
ˇ Blazˇina / Journal of Magnetism and Magnetic Materials 323 (2011) 2340–2347 G.I. Miletic´, Z.
Since in RFe12B6 compounds Fe-3d sublattice shows instability with respect to the LM-HM transition, it can be expected that in RFe12B6 compounds with magnetic 4f element Fe-3d sublattice could be, due to 4f–5d–3d interaction, found in HM state. Magnetization measurements on the La1 xGdxFe12B6 ð0:0 o x r0:5Þ system [10] where it was found that partial substitution La-Gd strongly enhances Fe-3d magnetic moments [10] confirm this expectation. Analogous results were found in magnetization measurements performed on NdFe12B6 where an average Fe magnetic moment of about 1:4mB was extracted [3,4,10]. More recent measurements of ¨ Mossbauer spectra on NdFe12B6 also indicated that Fe magnetic moments are strongly enhanced, the average value of Fe magnetic moment being 1:2mB [19]. Since in NdFe12B6 Fe-3d sublattice is in state of strongly enhanced magnetic moments [3,4,10,19], it is desirable to compare magnetic moments of Fe atoms obtained in mentioned TB-LMTO-ASA-LSDA-BH-FSM calculations for LaFe12B6 in HM ¨ state [14] with Fe magnetic moments extracted from Mossbauer spectra obtained on NdFe12B6 [19]. TB-LMTO-ASA-LSDA-BH-FSM calculations resulted in magnetic moments of 1.13 ð1:54ÞmB for Fe atoms on 18(g) (18(h)) sites for LaFe12B6 in HM state [14], while ¨ Fe magnetic moments in NdFe12B6 extracted from Mossbauer spectra were 1:1ð1:3ÞmB for 18(g) (18(h)) sites [19]. Fair agreement between two results indicates similarity between magnetic state of Fe-3d sublattice in HM state in LaFe12B6 and Fe-3d sublattice in NdFe12B6. The subjects of the present work are the electronic structure and magnetic properties of the RM12B6 compounds (R ¼Y, La or Ce; M¼Fe, Co). Full-potential electronic structure calculations were performed for the RM12B6 compounds with the emphasis on the metamagnetism in case of LaFe12B6 as well as magnetism in RCo12B6. In addition, the site preference of Fe (Co) impurities in the Co (Fe) host lattice was also investigated and compared with experimental data.
2. Computational details Full-potential (FP) electronic structure calculations were performed with WIEN2k package [20] with the augmented planewave (APW)þ local orbitals (l.o.) [21] used as a basis set with the corresponding cut-off parameter RminKmax ¼7.0. Exchangecorrelation energy was treated within the local spin density approximation as given by Perdew and Wang (PW) [22]. Muffin-tin radii were 2.2 a.u. for R, 2.02 a.u. for M and 1.79 a.u. for B atoms. Atomic positions used in the present study were that from SrNi12B6 [23] and together with lattice parameters of studied compounds are given in Table 1. Table 1 Atomic positions in the SrNi12B6 structural type [23] and the corresponding lattice constants of studied compounds. Atom
Position
x
y
z
R M M B
3(a) 18(g) 18(h) 18(h)
0 0.3684 0.4238 0.1912
0 0 0.5762 0.8088
0 0.5 0.0355 0.0421
Compound
˚ a (A)
˚ c (A)
Ref.
LaFe12B6 YCo12B6 LaCo12B6 CeCo12B6
9.610 9.453 9.513 9.515
7.604 7.450 7.504 7.458
[3] [1] [1] [1]
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It should be noted that although there is a possibility of existence of antiferromagnetic state in LaFe12B6, only ferromagnetic arrangement of 3d sublattice is considered in spin-polarized calculations for all studied compounds. Impurities were treated by simply replacing one host atom with one impurity atom within unit cell which will then lead to the symmetry lowering. Such unit cells were then used, as in the case of parent compounds, for standard periodic solid state calculation.
3. Results and discussion 3.1. Non-spinpolarized calculations for RM12B6 compounds Projected densities of states (PDOS) obtained from non-spinpolarized calculations on RM12B6 compounds are given in Fig. 1. It is known that this PDOS contributions could be used to (at least roughly) predict behaviour of different atoms in possible transition from NM state to magnetic state. From Fe-3d-PDOS contributions in the close vicinity of Fermi energy (interval of about 70.25 eV about Fermi energy (EF)) shown in Fig. 1 it could be expected that for small total magnetic moments (as in, e.g., LM state) Fe-(18(g)) atoms will exhibit larger magnetic moments than Fe-(18(h)) atoms. This is in accord with results of Ref. [14] where calculations were performed with the more approximate TB-LMTO-ASA method. In Ref. [14] Fe3d-PDOS contributions were also analysed and similar conclusion was drawn regarding the relative sizes of Fe(18(g)) and Fe(18(h)) magnetic moments in LM state. Fe-3d-PDOS calculated in Ref. [14] with TB-LMTO-ASA method and Fe-3d-PDOS shown in Fig. 1 (calculated with FP-APWþlo method) are similar. However, contributions of two Fe sites in interval of about 0.25 eV below Fermi energy as shown in Ref. [14] (Fig. 2. in Ref. [14]) look identical and, consequently, in Ref. [14] higher magnetic moments of Fe(18(g)) atoms in LM state were alluded to come from the higher contributions of those atoms to the empty states in a small interval (about 0.25 eV) above the Fermi energy [14]. Another important group of states are those centered at about 0.5 eV above the Fermi energy. Contribution of Fe-(18(h)) atoms to these states is considerably higher than that of Fe-(18(g)) atoms. Similar result was obtained in our previous work where TB-LMTO-ASA method was employed [14] (Fig. 2. in Ref. [14]). This difference in contributions was furthermore used in Ref. [14] to explain larger increase of magnetic moments of Fe-(18(h)) atoms compared to Fe(18(g))-atoms during metamagnetic LM-HM transition in LaFe12B6 [14]. Another intermetallic compound in the series of presently investigated compounds is LaCo12B6 and corresponding 3d-PDOS contributions are shown in Fig. 1. The most evident difference compared to LaFe12B6 is additional filling of 3d states. Fermi energy is shifted and now falls at the region (peak) of states which were in NM state of LaFe12B6 placed at 0.5 eV above the Fermi energy. These states were previously identified in Ref. [14] as being responsible for larger increase of Fe-(18(h)) magnetic moments when compared to Fe-(18(g)) magnetic moments during the LM-HM transition in LaFe12B6. Analogously to LaFe12B6, Co-(18(h)) atoms contribute more to this group of states than Co-(18(g)) atoms. Therefore, it could be expected that Co(18(h)) magnetic moments would be higher than the Co(18(g)) magnetic moments if transition from NM state to magnetic state takes place. In Fig. 1 3d-PDOS contributions to the density of states in YCo12B6 are also shown. Since main features of the corresponding Co contributions in LaCo12B6 and YCo12B6 are similar, it could be
ˇ Blazˇina / Journal of Magnetism and Magnetic Materials 323 (2011) 2340–2347 G.I. Miletic´, Z.
DOS (states/(eV cell))
2342
3.5 3 2.5 2 1.5 1 0.5 0
LaFe12B6: Fe (18 (g))-3d LaFe12B6: Fe (18 (h))-3d
DOS (states/(eV cell))
-6
-2
-1
0
1
2
3
4
-2
-1
0
1
2
3
4
-2
-1
0
1
2
3
4
3.5 CeCo12B6: Co (18 (g))-3d 3 CeCo12B6: Co (18 (h))-3d 2.5 2 1.5 1 0.5 0 -6 -5 -4 -3 -2 70 LaFe12B6: La-4f 60 LaCo12B6: La-4f 2 50 CeCo12B6: Ce-4f 40 1 30 20 0 10 -1 0 -6 -5 -4 -3 -2
-1
0
1
2
3
4
0
1
2
3
4
3.5 3 2.5 2 1.5 1 0.5 0
DOS (states/(eV cell)) DOS (states/(eV cell))
-4
-3
LaCo12B6: Co (18 (g))-3d LaCo12B6: Co (18 (h))-3d
-6
DOS (states/(eV cell))
-5
3.5 3 2.5 2 1.5 1 0.5 0 -6
-5
-4
-3
YCo12B6: Co (18 (g))-3d YCo12B6: Co (18 (h))-3d
-5
-4
-3
Ce-4f
0
1
-1 E-EF (eV)
Fig. 1. 3d-PDOS and 4f-PDOS contributions in investigated RM12B6 compounds obtained from non-spinpolarized calculations. In the inset Ce-4f-PDOS contribution in the interval of 7 1 eV about the Fermi energy is given.
expected that corresponding Co atoms will behave similarly in these two compounds. PDOS contributions for CeCo12B6 are also shown in Fig. 1. Main features of Co-3d-PDOS contributions in CeCo12B6 are similar to the corresponding Co contributions in LaCo12B6 and YCo12B6. In Fig. 1 4f-PDOS contributions for LaFe12B6, LaCo12B6 and CeCo12B6 are shown. As expected, in La 4f states are (almost) empty and their peaks are placed around 3 eV above the Fermi energy (in fact, present non-spinpolarized calculations resulted in a small occupancy of 4f states: in LaFe12B6 and LaCo12B6 there are 0.11 and 0.12 4f electrons, respectively.) It is known that starting with cerium filling of 4f states takes place in the lanthanide series (present non-spinpolarized calculation resulted in 0.89 4f electrons within the Ce muffin-tin sphere). From the inset of Fig. 1 where Ce-4f-PDOS is shown in the interval of 71 eV around the
Fermi energy, it is visible that Ce-4f-PDOS at the Fermi energy is comparable to that of the Co(18(g))-3d states.
3.2. Spin-polarized calculations for RM12B6 compounds without spin–orbit coupling Spin magnetic moments obtained from spin-polarized calculations are collected in Table 2. Values are given for two different setups of the corresponding basis sets (see the notes in Refs. [25,26]). In general, if not stated otherwise, results presented in this article are those obtained with the default setup. If different lattice parameters than those determined from experiment were used in calculations, the initial ratio c=a was always kept constant.
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Table 2 Magnetic moments in studied RM12B6 compounds from spin-polarized calculations without spin–orbit coupling. Magnetic moments are given in mB . (Corresponding values in the parentheses are obtained with non-default setup of the basis set. See the notes in Refs. [25,26].) Explanation of labels: ms-spin magnetic moment, IS-interstitial. R
M18(g)
M18(h)
B
0.68 0.74 0.99 1.15 1.30
0.27 0.29 1.10 1.46 1.63
0.03 0.04 0.07 0.09 0.10
IS
f.u.
Compound
V/Vexp
LaFe12B6
0.94 0.96 0.96 0.98 1.0
ms ms ms ms ms
0.02 0.02 0.04 0.05 0.05
LaCo12B6
0.96 0.98 1.0
ms ms ms
0.04 ( 0.03) 0.04 ( 0.04) 0.04 ( 0.04)
0.32 (0.32) 0.36 (0.33) 0.36 (0.34)
0.76 (0.73) 0.78 (0.76) 0.81 (0.79)
0.04 ( 0.03) 0.04 ( 0.04) 0.04 ( 0.04)
0.30 ( 0.27) 0.35 ( 0.30) 0.37 ( 0.32)
5.94 (5.77) 6.21 (6.00) 6.40 (6.19)
YCo12B6
1.0
ms
0.03 ( 0.04)
0.34 (0.34)
0.79 (0.79)
0.04 ( 0.04)
0.29 ( 0.29)
6.22 (6.23)
CeCo12B6
1.0
ms
0.51 ( 0.49)
0.42 (0.39)
0.87 (0.84)
0.04 ( 0.04)
0.46 ( 0.41)
6.51 (6.25)
( 0.02) ( 0.02) ( 0.03) ( 0.04) ( 0.05)
(0.69) (0.75) (0.98) (1.14) (1.28)
0
-5
Δ E (V, M) (mRy)
-10
-15
-20 V/Vexp = 1.0 -25
V/Vexp = 0.98 V/Vexp = 0.96
-30
V/Vexp = 0.94 -35 0
2
4
6
8
10
12
14
16
18
20
M (μB) Fig. 2. Results of fixed spin moment calculations for LaFe12B6 at different volumes of the unit cell.
Results of FSM calculations for different volumes of LaFe12B6 unit cell are shown in Fig. 2. Present calculations do not find LM state at the observed lattice parameters (Fig. 2). It is known that LSDA in general overestimates stability of spin magnetic moments: magnetic vs. nonmagnetic states [27,28] or HM vs. LM states [14,29]. When lattice parameters are decreased, HM state is destabilized and LM state appears. At V¼0.96Vexp LM and HM states are of almost equal energies (HM is more stable for about 0.2 mRy), while at V ¼0.94Vexp only LM state remains. In a previous work [14] TB-LMTO-ASA-FSM calculations with the two different versions of LSDA (von Barth–Hedin (BH) [13] and Vosko–Wilk–Nusair (VWN) [24]) were performed using the experimentally observed lattice parameters. In fixed spin moment calculations performed with the BH-LSDA parametrization both states (LM and HM) were found [14]. Similarly to the present results, in previous FSM calculations performed with the VWNLSDA parametrization only HM state was found at the experimentally determined lattice constants [14]. In Fig. 3 are shown results which correspond to the minima on Fig. 2.
(0.28) (0.29) (1.08) (1.44) (1.62)
( 0.03) ( 0.04) ( 0.07) ( 0.09) ( 0.10)
0.03 0.03 0.04 0.06 0.08
(0.04) (0.03) ( 0.03) ( 0.05) ( 0.07)
5.56 5.98 12.02 15.01 16.85
(5.61) (6.03) (11.88) (14.86) (16.73)
In LM state magnetic moments of Fe(18(h)) atoms are significantly lower than those of Fe(18(g)) atoms. During LM-HM transition spin magnetic moments of the Fe atoms show (Fig. 3(c)) similar behaviour to that found in [14]: formation of the HM state takes place after a large jump of spin magnetic moment of Fe(18(h)) atoms, while at the same time spin magnetic moments of Fe(18(g)) atoms show significantly smaller change. Finally, in HM state magnetic moments of Fe(18(h)) atoms are higher than magnetic moments of Fe(18(g)) atoms. For illustration (Fig. 3c), Table 2.: at volume V¼0.96Vexp where both magnetic states coexist, in LM state spin magnetic moment of Fe(18(g)) atoms is 0:74mB , that of Fe(18(h)) atoms is 0:29mB ; while in the HM state spin magnetic moment of Fe(18(g)) atoms is 0:99mB , and that of Fe(18(h)) atoms amounts to 1:10mB . It was mentioned above that if the 3d system is unstable with respect to NM-FM transition, or FM LM-FM HM transition, 4f spin moments could induce transition through the 4f–5d–3d interaction [15–18]. Experimental results on NdFe12B6 [3,4,10,19] and La1 xGdx Fe12B6 ð0:0 o x r0:5Þ system [10] confirmed this expectation. ¨ In recent work on NdFe12B6 where Mossbauer spectroscopy was employed [19], magnetic moments of Fe(18(g)) (Fe(18(h))) atoms were determined to be 1:1mB ð1:3mB Þ [19] thus confirming more directly that 4f spin moments enhance Fe-3d spin moments. In addition, this result [19] indicates that magnetic moments of Fe(18(h)) atoms will be higher than magnetic moments of Fe(18(g)) atoms in HM state in agreement with previous [14] and present theoretical results. Fig. 3(a) shows relative energies of the minima found in the present calculations. In Fig. 3(a) another known drawback of LSDA is visible, i.e., volumes belonging to the total energy minima will be too small when compared to experimental values (overbinding) [30,31]. As mentioned above, in Ref. [14] relative magnitudes of 3d magnetic moments in LaFe12B6 in different magnetic states were indicated from 3d-PDOS. In case of RCo12B6 compounds spin-polarized calculations show that the relative magnitude of spin magnetic moments could be indicated from 3d-PDOS obtained from non-spinpolarized calculations. In all investigated compounds contributions (Fig. 1) from Co(18(h)) atoms to the states at EF are significantly larger than that from Co(18(g)) atoms. This is reflected in relative sizes of spin magnetic moments on Co(18(g)) and Co(18(h)) atoms in all investigated RCo12B6 compounds: spin magnetic moments of Co(18(h)) atoms are always higher than that of Co(18(g)) atoms (Table 2). Although experiments do not indicate that in LaCo12B6 compound 3d sublattice will exhibit itinerant band metamagnetism as in LaFe12B6 compound, FSM calculations were performed for
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mFe (μB)
1.50
1.00 Fe (18g) 0.50 Fe (18h)
M (μB)
16.0
12.0
8.0
60.0 40.0
LM 0.2 mRy
Erel (mRy)
80.0
HM
20.0 0.0 0.94
0.96
0.98
1
V/Vexp Fig. 3. Results obtained by ordinary spin-polarized calculations for LaFe12B6. Results correspond to the minima on Fig. 2. Erel is the difference between total energies of different minima and the minimum corresponding to V/Vexp ¼ 0.94.
different volumes of LaCo12B6 and results are shown in Fig. 4. (These calculations were performed with non-default basis set setup. See the notes in Refs. [25,26].) As can be seen from Table 2, Figs. 2–4, volume dependence of magnetic properties differs in LaFe12B6 and LaCo12B6. In LaFe12B6 there are two distinct magnetic states (LM and HM) and HM state exhibits strong volume dependence of both magnetic moments and relative stability vs. NM state. Finally, at V¼0.96Vexp LM state emerges and at V ¼0.94Vexp HM state disappears. In LaCo12B6 there is only one magnetic state which slowly changes both in magnetic moment and relative stability vs. NM state. In LaCo12B6 total spin magnetic moment calculated at experimental lattice constants (Table 2) is 6:40mB =f:u: while magnetization obtained from experiment is 5:35mB =f:u [8]. Another choice of basis set somewhat improves agreement ð6:19mB =f:uÞ but this is obviously also too high. As mentioned above, spin magnetic moments on Co(18(g)) atoms are lower than that of the Co(18(h)) atoms amounting to 0.36 and 0:81mB , respectively. In YCo12B6 situation is analogous to that in LaCo12B6. Spin magnetic moments of Co atoms are 0:34ð0:79ÞmB for Co(18(g)) (Co(18(h))) atoms and total spin magnetic moment is 6:22mB =f:u while experimentally determined values are 6:5 [1], 5:2 [2], 5:32 [6], 5:1 [8] and 5:34mB =f:u [9]. It appears that magnetic moment of 6:5mB =f:u found in Ref. [1] is rather high when compared to the other values which are found in the narrow interval of 5:125:34mB =f:u [2,6,8,9]. It is interesting to note that in case of YCo12B6 both choices of basis set resulted in almost identical spin magnetic moments.
In addition, total spin magnetic moments obtained for YCo12B6 and LaCo12B6 with non-default basis set are very close. Spin magnetic moments obtained for atoms are (within precision used) identical, small difference coming only from interstitial region. It seems that our calculations, to some extent, overestimate total magnetic moments in YCo12B6 and LaCo12B6. As mentioned above, in [7] two different interpretations were given for NMR spectra in YCo12B6 and GdCo12B6. According to the one of them (which was later also employed in [3]) two subspectra could be attributed to two different crystallographic positions of Co atoms in the RCo12B6 lattice [7]. Spin magnetic moments estimated within this interpretation were 0:30 7 0:09mB for the 18(g) site and 0:69mB for the 18(h) site [7] and one can say that they are in fair agreement with the present computational results. Additionally, they indicate that theoretical values of Co magnetic moments, as obtained in the present work, could be too high. However, it should also be mentioned that authors in [7] concluded that further investigations are necessary before one can make any definitive choice between two proposed interpretations. In case of CeCo12B6 present calculations resulted in spin magnetic moment of 6:51mB =f:u: (or 6:25mB =f:u: with non-default basis set) while experimental values are 5:4 [1], 4:61 [8], 4:75 [9] and 4:8mB =f:u [32]. On atoms following spin magnetic moments were obtained in our calculations: 0:42ð0:87ÞmB for Co atoms at 18(g) (18(h)) positions (corresponding values obtained with nondefault basis set are somewhat lower: 0:39ð0:84ÞmB for Co atoms at 18(g) (18(h)) sites).
ˇ Blazˇina / Journal of Magnetism and Magnetic Materials 323 (2011) 2340–2347 G.I. Miletic´, Z.
20 15
V/Vexp = 1.0
10
V/Vexp = 0.98 V/Vexp = 0.96
Δ E (V, M) (mRy)
5 0 -5 -10 -15 -20 -25 0
1
2
3
4
5
6 7 M (μB)
8
9
10
11
12
Fig. 4. Results of fixed spin moment calculations for LaCo12B6 at different volumes of the unit cell. (These calculations were performed with non-default basis set setup. See the notes in Refs. [25,26].)
Results for CeCo12B6 also indicate that Co magnetic moments are too high in investigated RCo12B6 compounds. In all studied compounds there are small spin magnetic moments ð0:0220:10mB Þ induced on non-3d atoms which are oriented opposite to the 3d spin magnetic moments. In CeCo12B6 there is, in addition, a contribution of 4f spin magnetic moment ð0:46mB Þ which is also oriented antiparallel to the 3d spin magnetic moments. 3.3. Spin-polarized calculations for RM12B6 compounds with spin–orbit coupling To determine contribution of orbital moments in RM12B6 compounds, calculations with spin–orbit coupling (SOC) were also performed and corresponding magnetic moments are given in Table 3. In LaFe12B6 obtained orbital moments are very small and they exhibit small change during metamagnetic transition. In LM state at V ¼0.96Vexp orbital moments at Fe(18(g)) (Fe(18(h)) site were 0:03mB ð0:01mB Þ. At V¼0.96Vexp, where, in present calculations, transition occurs, a small change during the transition appears only on Fe(18(h)) site ð0:01mB -0:03mB Þ while orbital moment at Fe(18(g)) remains (within precision used here) constant ð0:03mB Þ. At Vexp, where only HM state exists, orbital moments at both Fe-sites are 0:05mB . In LaCo12B6, small orbital moments are induced on Co atoms after the inclusion of SOC (0:02ð0:03ÞmB for 18(g) (18(h)) site) what gives total magnetic moment of 6:66mB =f:u (6:50mB =f:u with non-default basis set) to be compared with experimental value of 5:35mB =f:u [8]. Induced orbital moments are also very small in YCo12B6: 0:02ð0:03ÞmB for Co(18(g)) (Co(18(h))) atoms while the total magnetic moment has raised to 6:53mB =f:u: Values of magnetic moments determined from experiment for YCo12B6 are 6:5 [1], 5:2 [2], 5:32 [6], 5:1 [8] and 5:34mB =f:u [9]. As in LaCo12B6 and YCo12B6 small orbital moments are induced on Co atoms (0:02ð0:03ÞmB for 18(g) (18(h)) position) in CeCo12B6. However, in CeCo12B6 there is an additional contribution coming
2345
from orbital moment of Ce-4f electrons which is 0:27mB (0:26mB for non-default basis set) and it is oriented (Hund third rule) opposite to the 4f-spin magnetic moment. Since 3d and 4f spin magnetic moments couple ferrimagnetically, Ce orbital moment additionally increases total magnetic moment in unit cell. In CeCo12B6 dependence of magnetic moments on basis set choice is more pronounced than in the other compounds. For default basis set total magnetic moment is 7:12mB =f:u: while for nondefault basis set it is 6:70mB =f:u: Corresponding experimental values are 5:4 [1], 4:61 [8], 4:75 [9] and 4:8mB =f:u [32]. Present calculations seem to overestimate Co magnetic moments in RCo12B6 compounds. Disagreement is even more pronounced when spin–orbit coupling is included in calculations. Disagreement between experiment and theory deserves additional comment. We believe that one of the reasons for discrepancy between our calculations and experimental results could be the atomic positions used in our calculations. As mentioned above, atomic positions used here are those from SrNi12B6 compound [23]. In this structure only 3(a) position (R atoms) does not contain variable parameters so that other atomic positions in studied compounds could differ from the corresponding positions in the SrNi12B6. 3.4. Calculations for RM11MimpB6 compounds (M¼ Fe (Co); Mimp ¼Co (Fe)): Previous experimental work also included studies of distribution of the particular 3d atoms in RFe12 xCoxB6 compounds over the two available crystallographic positions (18(g) or 18(h)). It was found that Fe atoms prefer 18(h) and Co atoms 18(g) sites [3,33–35]. Present calculations were performed on LaFe12B6, LaCo12B6 and YCo12B6 compounds where one Fe (Co) atom was substituted by Co (Fe) atom on either 18(g) or 18(h) position. Thus, the final studied systems were of the LaFe11CoB6, LaCo11FeB6 and YCo11FeB6 composition. In the case of non-spinpolarized calculations performed with the experimentally observed lattice constants it was found that Co atom in LaFe11CoB6 prefers 18(g) position with an energy difference of 8.2 mRy. In LaCo11FeB6 Fe atom prefers 18(h) site the energy difference being 7.9 mRy. In YCo11FeB6 Fe atom also prefers 18(h) position with the energy difference of 8.6 mRy. This results are in accordance with experimental data [3,33–35]. Spin-polarized calculations with the experimental lattice parameters were also performed for LaFe11CoB6, LaCo11FeB6 and YCo11FeB6 compositions. In the case of LaFe11CoB6 one should note that at Vexp Fe-sublattice is in HM state. Again Co impurity prefers 18(g) position but now the energy difference is 1.9 mRy. In LaCo11FeB6 Fe atom prefers 18(h) crystallographic position the energy difference being 10.6 mRy. In YCo11FeB6, as in LaCo11FeB6, configuration with Fe impurities in 18(h) site is again more stable with the corresponding energy difference being 11.0 mRy. One could therefore say that computational results presented here, also in agreement with the experimental findings [3,33–35], indicate that Fe and Co atoms will prefer 18(h) and 18(g) positions, respectively.
4. Conclusion Full-potential electronic structure calculations were performed for the RM12B6 intermetallic compounds (R¼Y, La or Ce; M¼ Fe, Co). In the case of LaFe12B6 fixed spin moment calculations performed for various volumes resulted in findings of low moment and high moment states in agreement with previous experimental and theoretical results. Low moment state was found at reduced lattice constants with respect to the experimental ones. In LM
ˇ Blazˇina / Journal of Magnetism and Magnetic Materials 323 (2011) 2340–2347 G.I. Miletic´, Z.
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Table 3 Magnetic moments in studied RM12B6 compounds from spin-polarized calculations with spin–orbit coupling. Magnetic moments are given in mB . (Corresponding values in the parentheses are obtained with non-default setup of the basis set. See the notes in Refs. [25,26].) Explanation of labels: ms-spin magnetic moment, mo-orbital magnetic moment, mt-total magnetic moment, IS-interstitial. Compound
V/Vexp
LaFe12B6
0.94
0.96
0.96
0.98
1.0
LaCo12B6
0.96
0.98
1.0
M18(g)
R
M18(h)
B
IS
ms mo mt ms mo mt ms mo mt ms mo mt ms mo mt
0.02 0.00 0.01 0.02 0.00 0.02 0.04 0.00 0.03 0.05 0.00 0.04 0.05 0.00 0.05
( 0.02) (0.00) ( 0.01) ( 0.02) (0.00) ( 0.02) ( 0.03) (0.00) ( 0.03) ( 0.04) (0.00) ( 0.04) ( 0.05) (0.00) ( 0.05)
0.68 0.03 0.71 0.74 0.03 0.77 0.98 0.03 1.01 1.14 0.04 1.18 1.29 0.05 1.34
(0.68) (0.03) (0.71) (0.75) (0.03) (0.78) (0.97) (0.03) (1.00) (1.14) (0.04) (1.18) (1.28) (0.05) (1.33)
0.27 0.01 0.28 0.29 0.01 0.30 1.09 0.03 1.12 1.45 0.04 1.49 1.63 0.05 1.69
(0.27) (0.01) (0.28) (0.30) (0.01) (0.31) (1.06) (0.03) (1.09) (1.43) (0.04) (1.48) (1.62) (0.05) (1.68)
0.03 0.00 0.03 0.04 0.00 0.04 0.07 0.00 0.07 0.09 0.00 0.09 0.10 0.00 0.10
( 0.03) (0.00) ( 0.03) ( 0.04) (0.00) ( 0.04) ( 0.07) (0.00) ( 0.07) ( 0.09) (0.00) ( 0.09) ( 0.10) (0.00) ( 0.10)
ms mo mt ms mo mt ms mo mt
0.04 0.00 0.03 0.04 0.00 0.04 0.04 0.00 0.04
( 0.03) (0.00) ( 0.03) ( 0.04) (0.00) ( 0.03) ( 0.04) (0.00) ( 0.04)
0.32 0.01 0.33 0.36 0.02 0.37 0.36 0.02 0.38
(0.31) (0.01) (0.33) (0.33) (0.02) (0.35) (0.34) (0.02) (0.36)
0.77 0.03 0.79 0.79 0.03 0.82 0.81 0.03 0.84
(0.73) (0.03) (0.75) (0.76) (0.03) (0.79) (0.79) (0.03) (0.82)
0.04 0.00 0.04 0.04 0.00 0.04 0.04 0.00 0.04
( 0.03) (0.00) ( 0.03) ( 0.04) (0.00) ( 0.04) ( 0.04) (0.00) ( 0.04)
0.03 (0.04) 0.03 (0.04) 0.03 (0.03) 0.03 (0.03) 0.04 ( 0.03) 0.04 ( 0.03) 0.06 ( 0.05) 0.06 ( 0.05) 0.07 ( 0.06) 0.07 ( 0.06) 0.30 ( 0.27) 0.30 ( 0.27) 0.34 ( 0.29) 0.34 ( 0.29) 0.36 ( 0.32) 0.36 ( 0.32)
YCo12B6
1.0
ms mo mt
0.03 ( 0.03) 0.00 (0.00) 0.03 ( 0.03)
0.34 (0.34) 0.02 (0.02) 0.35 (0.35)
0.79 (0.79) 0.03 (0.03) 0.82 (0.82)
0.04 ( 0.04) 0.00 (0.00) 0.04 ( 0.04)
0.29 ( 0.29)
CeCo12B6
1.0
ms mo mt
0.48 ( 0.46) 0.27 (0.26) 0.21 ( 0.19)
0.42 (0.38) 0.02 (0.02) 0.44 (0.40)
0.87 (0.82) 0.03 (0.03) 0.90 (0.85)
0.04 ( 0.04) 0.00 (0.00) 0.04 ( 0.04)
0.45 ( 0.40)
state Fe(18(g)) atoms show larger magnetic moments than the Fe(18(h)) atoms, while reverse is true in HM state. The same was found in past theoretical work where both LM and HM states were found while previous experimental work also indicates that in HM state Fe(18(h)) atoms have higher magnetic moments than Fe(18(g)) atoms. In RCo12B6 compounds with R¼ Y, La fair agreement was found between theoretical and experimental total magnetic moments. Previous experimental investigation from which values of Co atomic magnetic moments were proposed also indicates that calculated Co magnetic moments could be in fair agreement with experimental values. However, it seems that present calculations to some extent overestimate Co magnetic moments. In CeCo12B6 the disagreement between theory and experiment seems to be more notable. Total energy calculations for LaFe11CoB6, LaCo11FeB6 and YCo11FeB6 show that the Fe (Co) atoms prefer 18(h) (18(g)) sites what is in agreement with the previous experimental findings.
Acknowledgement The support for this research by the Ministry of Science, Education and Sport of the Republic of Croatia under Project no. 098-0982904-2941 is highly appreciated. Calculations were performed at the University Computing Centre of the University of Zagreb. References [1] M. Jurczyk, A.T. Pedziwiatr, W.E. Wallace, J. Magn. Magn. Mater. 67 (1987) L1–L3.
0.29 ( 0.29)
0.45 ( 0.40)
f.u. 5.51 0.21 5.72 6.00 0.24 6.24 11.92 0.37 12.28 14.92 0.51 15.43 16.80 0.63 17.43 5.95 0.26 6.21 6.23 0.27 6.50 6.37 0.28 6.66
(5.56) (0.22) (5.78) (6.10) (0.25) (6.35) (11.75) (0.36) (12.12) (14.82) (0.51) (15.33) (16.70) (0.63) (17.33) (5.73) (0.26) (5.99) (6.01) (0.28) (6.29) (6.21) (0.29) (6.50)
6.25 (6.25) 0.28 (0.28) 6.53 (6.53) 6.52 (6.11) 0.60 (0.59) 7.12 (6.70)
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