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Electronic structure and magnetism of YCo4B and YCo3B2
Journal of Magnetism and Magnetic Materials 183 (1998) 94—100
Electronic structure and magnetism of YCo B and YCo B 4 3 2 H. Yamada!,*, K. Terao!, H. Nakazawa!, I. Kitagawa", N. Suzuki", H. Ido# ! Faculty of Science, Shinshu University, Matsumoto 390, Japan " Faculty of Engineering Science, Osaka University, Toyonaka 560, Japan # Department of Applied Physics, Tohoku Gakuin University, Tagajo 985, Japan Received 3 June 1997; received in revised form 16 September 1997
Abstract Electronic structures of YCo B and YCo B are calculated in a self-consistent linear muffin-tin orbital method and the 4 3 2 atomic sphere approximation. The effects of B atom in these compounds are discussed, by comparing the calculated local density-of-states curves with those of YCo obtained previously. A good agreement between the calculated and observed 5 values of the low-temperature specific heat coefficient for YCo B is obtained. Spin-polarized band calculations are also 3 2 carried out and the local spin and orbital magnetic moments are evaluated at each Co atom site in YCo B. It is found 4 that the Co moment on 6i site is small, while the Co moment on 2c site is large. On the other hand, YCo B is found to be 3 2 non-magnetic. These results are consistent with the observed ones and can be explained by the mixing model between 3d-states of Co and 2p-states of B. ( 1998 Elsevier Science B.V. All rights reserved. PACS: 75.30.Cr; 75.50.Cc; 75.10.Lp Keywords: Rare-earth ternary compounds; Itinerant electron magnetism; Electronic structure; Magnetic moments
1. Introduction Ternary compounds of rare-earth (R), cobalt and boron atoms show a lot of interesting magnetic properties [1—4]. RCo B and RCo B are deriva4 3 2 tives of RCo with the CaCu -type structure. 5 5 Nevertheless, the magnetic moments of Co atoms in these compounds are very different from those in RCo . For instance, YCo B is a ferromagnet with 5 4 small average Co moment 0.63 l /atom, while the B * Corresponding author. Tel.: #81 263 37 2461; fax :#81 263 37 2438; e-mail: [email protected].
average Co moment in YCo is 1.5—1.7 l /atom. 5 B Moreover, YCo B is paramagnetic even at low 3 2 temperature. The crystal structure of these compounds is basically the hexagonal one composed of the Co and B atom layers, as shown in Fig. 1a and Fig. 1b, where Y atoms are denoted by large spheres and Co and B atoms are small spheres. B atoms occupy 2d and 2c sites in YCo B and YCo B , respectively. 4 3 2 Co atom sites are classified into three kinds of sites denoted by Co(N) with N"0, 1 and 2, where the Co atom has N layers of B atoms in the nearestneighbour upper and/or lower layers. For example,
0304-8853/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 0 4 - 8 8 5 3 ( 9 7 ) 0 1 0 6 2 - 7
H. Yamada et al. / Journal of Magnetism and Magnetic Materials 183 (1998) 94—100
Fig. 1. Lattice structures of (a) YCo B and (b) YCo B . 4 3 2
Co atoms at the 2c and 6i sites in YCo B are Co(0) 4 and Co(1), respectively. On the other hand, the Co atom at 3g site in YCo B is Co(2). Co moments at 3 2 the different sites were estimated from the observed bulk moments as m "1.55 l /atom, m " C0(0) B C0(1) 0.33 l /atom and m "0, by assuming that B C0(2) m ,m and m have common values in the C0(0) C0(1) C0(2) Y—Co—B system [5,6]. The aim of the present paper is to study the role of B atoms in the instability of Co moments in Y—Co—B system, by calculating electronic structures of YCo B and YCo B in the linear muffin4 3 2 tin orbital (LMTO) method and the atomic sphere approximation (ASA). As far as the authors know, no band calculation for these compounds has been performed so far except for Y Co B [7]. Details of 2 14 the band calculation are given in Section 2. In Section 3 the calculated results of the densityof-states curves (DOS) in the non-magnetic state are given and compared with those for YCo [8]. 5 Local spin and orbital moments on each Co sites are obtained in Section 4 by the spin polarized band calculations. Our conclusions and discussion are given in Section 5.
2. Band calculation YCo B with the CeCo B-type structure contains 4 4 two Y atoms at 1a and 1b sites, eight Co atoms at 6i and 2c sites and two B atoms at 2d site in a hexagonal unit cell, as shown in Fig. 1a. We first estimate the radii of the touching rigid spheres of atoms in the lattice. The radii of the touching
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spheres of atoms in YCo B are determined so that 4 (i) Co atom at 6i site touches the nearest-neighbour Co atoms at 6i and 2c sites, (ii) Co atom at 2c site touches the nearest-neighbour Y atoms at 1a site, (iii) Co atom at 6i site touches the nearest-neighbour Y atoms at 1b site and (iv) B atom at 2d site touches the nearest-neighbour Co atoms at 6i site. The ideal ratio c/a between the lattice constants a and c constructed from these touching spheres is 2/J6#1/J3("1.394), being almost the same as the observed one 1.366. On the other hand, YCo B with CeCo B -type 3 2 3 2 structure contains one Y atom at 1a site, three Co atoms at 3g site and two B atoms at 2c site in a unit cell, as shown in Fig. 1b. The radii of the touching spheres of atoms can be determined in a similar way as that in YCo B. However, in this case, 4 Y atom at 1a site touches the Y atom itself on the nearest upper and lower layers. The B atom at 2c site touches the nearest-neighbour Co atoms at 3g site. The ideal ratio c/a constructed from these touching spheres is 2/J3!1/2("0.655), which is rather larger than the observed one 0.599. The electronic structures were calculated in the local-density functional formalism with the LMTO-ASA. The radii of the atomic spheres used in the ASA were determined in such a way that the ratio is kept equal to the ideal value of the touching spheres estimated above. In most of the present calculations, the spin—orbit interaction was not included although the calculations are scalar relativistic, including mass velocity and Darwin terms. Self-consistent calculations for YCo B and 4 YCo B are carried out at 150 and 216 k-points in 3 2 the irreducible 1 Brillouin zone, respectively. The 24 convergence in charge density was achieved so that the root mean square of moments of the occupied partial density of states becomes smaller than 10~5. The basic set of functions with angular momenta upto l"3 on all atoms was adopted.
3. DOS in the non-magnetic state Fig. 2 shows the local DOSs for Y atoms at 1a and 1b sites, Co atoms at 2c and 6i sites and B atom at 2d site in YCo B calculated at the observed 4
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Fig. 2. Calculated local DOSs for Y, Co and B atoms at each site in YCo B. 4
Fig. 3. Calculated local DOSs for Y, Co and B atoms at each site in YCo B . 3 2
lattice constant a"5.035 A_ . The position of the Fermi level is shown by E . It can be seen clearly F that the height of the local DOS at E for Co atom F at 2c site is very high. The hybridizations with the B atom states are seen at the lower energy side in the local DOSs for Y atom at 1b site and Co atom at 6i site, as these Y and Co atoms are positionally close to the B atom site. In Fig. 3 the local DOSs for Y atom at 1a site, Co atom at 3g site and B atom at 2c site in YCo B 3 2 calculated at the observed a"5.05 A_ are shown.
Fig. 4. Local DOSs for Co atoms at 3g site in (a) YCo [8], (b) 5 6i site in YCo B and (c) 3g site in YCo B . 4 3 2
The shape of the local DOS for Co atom at 3g site is very similar to that for Co atom at 6i site in YCo B. 4 This is because the 6i site in YCo B is crystallo4 graphically similar position to the 3g sites in YCo 5 and YCo B . Calculated value of the total DOS 3 2 for YCo B is 74.9 states/Ry f.u. at E . This gives 3 2 F the low-temperature specific heat coefficient c" 13.0 mJ/mol K2, which is very close to the observed one, 13.9 mJ/mol K2 [9]. Moreover, the value of the susceptibility observed recently for this compound is 1.9]10~6 emu/g [6], which is rather smaller than the observed one by Ballou et al. [4]. The non-interacting (unenhanced) spin susceptibility at 0 K estimated from the DOS at E is 1.24] F 10~6 emu/g. Comparing with our observed susceptibility, the enhancement factor of the susceptibility is found to be about 1.5. In Fig. 4 the calculated local DOSs of Co atoms at 3g site in YCo [8], 6i site in YCo B and 3g site 5 4 in YCo B are compared with each other. It can be 3 2 seen that the portion of the relatively higher local DOS above E decreases with increasing concenF tration of B. That is, the number of holes seems to decrease with increasing concentration of B. This fact may mean that the magnetic moment of Co atom at 3g and 6i sites decreases with increasing concentration of B when the system is magnetized. On the other hand, the shapes of local DOS of Co atom at 2c site in YCo and YCo B are very similar 5 4
H. Yamada et al. / Journal of Magnetism and Magnetic Materials 183 (1998) 94—100
Fig. 5. Local DOSs for Co atoms at 2c site in (a) YCo [8] and 5 (b) YCo B. 4
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Fig. 6. Calculated results of total energies per unit cell in the paramagnetic and ferromagnetic states (upper panel) and Co moments in the unit of l /atom on 2c and 6i sites in YCo B B 4 (lower panel) as a function of the unit-cell volume.
to each other, as shown in Fig. 5. Moreover, E is F found to lie at a very similar position on the local DOSs. This is because 2c sites of Co atom in YCo 5 and YCo B are crystallographically same position 4 with each other, being surrounded by six Co atoms at 3g and 6i sites, respectively.
4. Local Co moments Spin-polarized band calculations for YCo B and 4 YCo B are carried out in the LMTO-ASA with3 2 out the spin—orbit interaction for several lattice constants. The upper part in Fig. 6 shows the total energy for YCo B in the ferro- and para-magnetic 4 states as a function of the unit-cell volume. The minimum energy for YCo B is achieved in the 4 ferromagnetic state near the unit-cell volume 144 A_ 3, which is a little smaller than the observed one. Calculated Co moments at 2c and 6i sites are shown in the lower part in Fig. 6. It is found for YCo B that the Co at 2c site has a large moment 4 m "1.75l /atom, while the Co on 6i-site has C0(0) B a small moment m "0.55 l /atom at the unitC0(1) B cell volume 144 A_ 3. Small negative moments !0.11 and !0.15 l /atom are induced on Y atoms at 1a B and 1b sites, respectively. The negative-induced moments on Y come from the mixing between the 3d-states of Co and 4d-states of Y [10]. On the B atom site, no local moment is obtained.
Fig. 7. Majority and minority spin sub-bands for Y, Co and B atoms at each site in the ferromagnetic YCo B calculated at 4 the unit-cell volume 144 A_ 3.
The calculated value of the bulk moment 3.4 l /f.u. for YCo B is larger than the observed B 4 one 2.54 l /f.u. [6]. However, it has been confirmed B that the Co moment at 6i site becomes small in YCo B as predicted in Refs. [5,6]. Fig. 7 shows the 4
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local DOSs in the majority and minority spin subbands for Y atoms at 1a and 1b sites, Co atoms at 2c and 3g sites and B atom at 2d site in YCo B 4 calculated at the unit-cell volume 144 A_ 3. On the other hand, we have obtained no selfconsistent solution for any ferromagnetic state in YCo B near the observed lattice constant. This 3 2 fact means that m "0, which is consistent with C0(2) the experimental result that this compound is paramagnetic even at low temperature. Finally, the LMTO-ASA calculations with inclusion of the spin—orbit interaction have been done for ferromagnetic YCo B at the equilibrium lattice 4 constant. The actual calculation procedures are as follows: the Kohn—Sham equations for the scalar relativistic spin-polarized Hamiltonian are first solved self-consistently, and then the spin—orbit coupling term is added to and the full Hamiltonian is diagonalized non-self-consistently. The total orbital moment of YCo B has been obtained as 4 0.45 l /f.u. which is smaller than that of YCo , B 5 0.61 l /f.u., estimated by Daalderop et al. [11] in B the non-self-consistent calculation. The orbital moments on Co atoms at 2c and 6i sites are m03" " C0(0) 0.15 l /atom and m03" "0.10 l /atom, respecC0(1) B B tively. The ratio between the orbital and spin moments on Co atom at 2c site in YCo B is almost the 4 same as that in YCo obtained in Ref. [11]. On the 5 other hand, the ratio between the orbital and spin moments on Co atom at 6i site in YCo B is twice as 4 large as that on Co atom at 3g site in YCo . 5 5. Conclusions and discussion In this paper, the electronic structures of YCo B 4 and YCo B compounds have been calculated by 3 2 the self-consistent LMTO-ASA. By comparing the local DOSs with those of YCo obtained in Ref. 5 [8], the effect of B atom has been discussed. It was found that the shape of local DOS for Co at 6i site in YCo B is very similar to that at 3g sites in 4 YCo B and YCo . However, the position of E is 3 2 5 F different from each other. The effective number of hole at 6i or 3g site seems to decrease with increasing concentration of B. Then, the local Co moments at 6i and 3g sites were predicted to decrease with increasing concentration of B. On the other hand,
Fig. 8. Schematic view of the mixings between 3d-states of Co and 2p-states of B in YCo B and YCo B . The energies at the 4 3 2 centre of gravity of the local DOSs for 3d-states of Co atoms at 6i, 2c and 3g sites and 2p-states of B are shown by 3d(6i), 3d(2c), 3d(3g) and 2p, respectively.
the local DOSs for Co at 2c site in YCo B and 4 YCo were found to be very similar to each other. 5 E was found to lie near a sharp peak of the local F DOS. Then, the local Co moment on 2c site was expected not to be affected so much by B atoms. These facts have been confirmed by the spin-polarized calculation in the present paper. Moreover, the orbital moments on Co at 2c and 6i sites have been evaluated as 0.15 and 0.10 l /atom, which are simB ilar to the calculated ones at 2c and 3g sites in YCo 5 [11], respectively. Kanamori [12] has previously shown that the effect of B in R—Fe—B system can be explained by the mixing between the 3d-states of Fe and 2pstates of B. Our results in Y—Co—B system are also explained by the mixing model between the 3dstates of Co and 2p-states of B. The importance of the mixing between them in Y Co B is also em2 14 phasized by Coehoorn [7]. As seen in Fig. 1a, the atomic layer of Co at 6i site in YCo B lies between 4 the atomic layers of B at 2d site and Co at 2c site. Fig. 8 gives a schematic view of the mixings between 3d-states of Co and 2p-states of B in YCo B 4 and YCo B . The energies at the centre of gravity 3 2 of the local DOSs for 3d-states of Co at 6i, 2c and 3g sites and 2p-states of B are shown by 3d(6i), 3d(2c), 3d(3g) and 2p, respectively. By the mixing between 3d-states of Co at 6i site and 2p-states of B at 2d site in YCo B, the centres of gravity for the 4 local DOSs of Co at 6i site and B at 2d site shift towards the lower- and higher-energy sides, respectively. Moreover, the centre of gravity for the local DOS of Co at 6i site shifts much more towards the lower energy side by the mixing with 3d-states of
H. Yamada et al. / Journal of Magnetism and Magnetic Materials 183 (1998) 94—100
99
Table 1 The decreased and increased total numbers of electrons on each atom from the neutral one in YCo B and YCo B calculated at the 4 3 2 unit-cell volume 144 and 60 A_ 3, respectively, in the non-magnetic state YCo B 4
YCo B 3 2
Y (1a)
Y (1b)
Co (2c)
Co (6i)
B (2d)
Y (1a)
Co (3g)
B (2c)
!0.06
0.67
!0.27
0.37
!1.15
0.53
0.29
!0.70
Co at 2c site. In this case, some electrons flow into Co at 6i site from the neighbouring B, because the 2p-states of B shift towards higher energy side as shown in Fig. 8. Therefore, the position of E to F the local DOS of Co at 6i site shifts towards the higher-energy side and effective number of holes on Co at 6i site decreases. In YCo B , on the other hand, both upper and 3 2 lower sides of the atomic layer of Co atoms at 3g site are B atom layers at 2c site. As shown in Fig. 8, the centre of gravity of the local DOS of Co at 3g site shifts towards lower energy side by the hybridization between 3d-states of Co and 2p-states of B, as well as that of Co at 6i site in YCo B. Then, some 4 electrons flow into the Co at 3g site from the neighbouring B and the position of E to the local DOS F of Co at 3g site shifts towards higher-energy side. The increased and decreased total numbers of electrons on each atom from the neutral one in YCo B 4 and YCo B are listed in Table 1. However, such 3 2 a change in the number of electrons on atoms comes from the use of ASA in the present calculation and simply means the tails of the wave function spread on the neighbouring sites [13]. That is, it does not mean the real charge transfers between the atoms. The present mixing model also explains the magnitude of local moments of Co. The local moment of Co at 6i site in YCo B becomes small as shown 4 in Section 4. By the hybridization with the 3dstates of Co at 2c site and 2p-states of B at 2d site, the centre of gravity of the 3d-states of Co at 6i site shifts towards lower-energy side. Then some electrons flow into this site from the neighbouring Co at 2c site and B at 2d site. As the majority spin band is filled up by electrons in the magnetized state, the number of electrons with minority spin will in-
crease and, then, the magnetic moment of Co atoms at 6i site becomes small. On the other hand, the local moment of Co at 2c site (1.75 l /atom) in B YCo B is larger than that in YCo (1.52 l /atom) 4 5 B [8]. The centre of gravity for the minority spin band of Co at 2c site shifts towards higher energy side, by the hybridization with 3d-states of Co at 3g site, of which energy shifts also towards the lower energy side by the hybridization with 2p-states of B at 2d site, as shown in Fig. 8. Then, some electrons of Co at 2c site flow out. As the majority spin band of Co at 2c site is filled up by electrons, the number of electrons with minority spin on Co at 2c site will decrease. Therefore, the magnetic moment of Co at 2c site in YCo B becomes larger than that 4 in YCo . 5 For YCo B , we have obtained a good agree3 2 ment between the calculated and observed values of the low temperature specific heat coefficient. By the comparison between the observed susceptibility and the calculated DOS at E , the enhancement F factor of the susceptibility has been estimated as about 1.5. In this compound, no self-consistent solution for the ferromagnetic state has been obtained. This is because Co atoms at 3g site in YCo B are put between the B atom layers and the 3 2 d-band of Co is almost filled by the electrons flowed out from B atoms. In this way, it is concluded that the effect of B atom on the Co moment observed in YCo B and YCo B can be explained by the 3d—2p 4 3 2 mixing model. References [1] H. Ido, Kotai Butsuri 30 (1995) 875 (in Japanese). [2] H. Ido, H. Yamauchi, S.F. Cheng, S.G. Sankar, W.E. Wallace, J. Appl. Phys. 70 (1991) 6540.
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[3] N.P. Thuy, N.M. Hong, J.P. Liu, X. Li, J.J.M. Franse, F.R. de Boer, Physica B 177 (1992) 270. [4] R. Ballou, E. Burzo, V. Pop, J. Magn. Magn. Mater. 140—144 (1995) 945. [5] H. Ogata, H. Ido, H. Yamauchi, J. Appl. Phys. 73 (1993) 5911. [6] H. Ido, unpublished work. [7] R. Coehoorn, J. Magn. Magn. Mater. 99 (1991) 55. [8] I. Kitagawa, K. Terao, H. Yamada, J. Phys.: Condens. Matter 9 (1997) 231.
[9] W. Perthold, N.M. Hong, H. Michor, G. Hilscher, H. Ido, H. Asano, J. Magn. Magn. Mater. 157/158 (1996) 649. [10] M.S.S. Brooks, B. Johansson, in: K.H.J. Buschow (Ed.), Handbook of Magnetic Materials, vol. 7, Elsevier, Amsterdam, 1993, p. 139. [11] G.H.O. Daalderop, P.J. Kelly, M.F.H. Schuurmans, Phys. Rev. B 53 (1996) 14415. [12] J. Kanamori, Proc. 10th Int. Workshop on Rare-Earth Magnets, Kyoto, 1989, p. 1. [13] S. Asano, private communication.
Electronic structure and magnetism of YCo B and YCo B 4 3 2 H. Yamada!,*, K. Terao!, H. Nakazawa!, I. Kitagawa", N. Suzuki", H. Ido# ! Faculty of Science, Shinshu University, Matsumoto 390, Japan " Faculty of Engineering Science, Osaka University, Toyonaka 560, Japan # Department of Applied Physics, Tohoku Gakuin University, Tagajo 985, Japan Received 3 June 1997; received in revised form 16 September 1997
Abstract Electronic structures of YCo B and YCo B are calculated in a self-consistent linear muffin-tin orbital method and the 4 3 2 atomic sphere approximation. The effects of B atom in these compounds are discussed, by comparing the calculated local density-of-states curves with those of YCo obtained previously. A good agreement between the calculated and observed 5 values of the low-temperature specific heat coefficient for YCo B is obtained. Spin-polarized band calculations are also 3 2 carried out and the local spin and orbital magnetic moments are evaluated at each Co atom site in YCo B. It is found 4 that the Co moment on 6i site is small, while the Co moment on 2c site is large. On the other hand, YCo B is found to be 3 2 non-magnetic. These results are consistent with the observed ones and can be explained by the mixing model between 3d-states of Co and 2p-states of B. ( 1998 Elsevier Science B.V. All rights reserved. PACS: 75.30.Cr; 75.50.Cc; 75.10.Lp Keywords: Rare-earth ternary compounds; Itinerant electron magnetism; Electronic structure; Magnetic moments
1. Introduction Ternary compounds of rare-earth (R), cobalt and boron atoms show a lot of interesting magnetic properties [1—4]. RCo B and RCo B are deriva4 3 2 tives of RCo with the CaCu -type structure. 5 5 Nevertheless, the magnetic moments of Co atoms in these compounds are very different from those in RCo . For instance, YCo B is a ferromagnet with 5 4 small average Co moment 0.63 l /atom, while the B * Corresponding author. Tel.: #81 263 37 2461; fax :#81 263 37 2438; e-mail: [email protected].
average Co moment in YCo is 1.5—1.7 l /atom. 5 B Moreover, YCo B is paramagnetic even at low 3 2 temperature. The crystal structure of these compounds is basically the hexagonal one composed of the Co and B atom layers, as shown in Fig. 1a and Fig. 1b, where Y atoms are denoted by large spheres and Co and B atoms are small spheres. B atoms occupy 2d and 2c sites in YCo B and YCo B , respectively. 4 3 2 Co atom sites are classified into three kinds of sites denoted by Co(N) with N"0, 1 and 2, where the Co atom has N layers of B atoms in the nearestneighbour upper and/or lower layers. For example,
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H. Yamada et al. / Journal of Magnetism and Magnetic Materials 183 (1998) 94—100
Fig. 1. Lattice structures of (a) YCo B and (b) YCo B . 4 3 2
Co atoms at the 2c and 6i sites in YCo B are Co(0) 4 and Co(1), respectively. On the other hand, the Co atom at 3g site in YCo B is Co(2). Co moments at 3 2 the different sites were estimated from the observed bulk moments as m "1.55 l /atom, m " C0(0) B C0(1) 0.33 l /atom and m "0, by assuming that B C0(2) m ,m and m have common values in the C0(0) C0(1) C0(2) Y—Co—B system [5,6]. The aim of the present paper is to study the role of B atoms in the instability of Co moments in Y—Co—B system, by calculating electronic structures of YCo B and YCo B in the linear muffin4 3 2 tin orbital (LMTO) method and the atomic sphere approximation (ASA). As far as the authors know, no band calculation for these compounds has been performed so far except for Y Co B [7]. Details of 2 14 the band calculation are given in Section 2. In Section 3 the calculated results of the densityof-states curves (DOS) in the non-magnetic state are given and compared with those for YCo [8]. 5 Local spin and orbital moments on each Co sites are obtained in Section 4 by the spin polarized band calculations. Our conclusions and discussion are given in Section 5.
2. Band calculation YCo B with the CeCo B-type structure contains 4 4 two Y atoms at 1a and 1b sites, eight Co atoms at 6i and 2c sites and two B atoms at 2d site in a hexagonal unit cell, as shown in Fig. 1a. We first estimate the radii of the touching rigid spheres of atoms in the lattice. The radii of the touching
95
spheres of atoms in YCo B are determined so that 4 (i) Co atom at 6i site touches the nearest-neighbour Co atoms at 6i and 2c sites, (ii) Co atom at 2c site touches the nearest-neighbour Y atoms at 1a site, (iii) Co atom at 6i site touches the nearest-neighbour Y atoms at 1b site and (iv) B atom at 2d site touches the nearest-neighbour Co atoms at 6i site. The ideal ratio c/a between the lattice constants a and c constructed from these touching spheres is 2/J6#1/J3("1.394), being almost the same as the observed one 1.366. On the other hand, YCo B with CeCo B -type 3 2 3 2 structure contains one Y atom at 1a site, three Co atoms at 3g site and two B atoms at 2c site in a unit cell, as shown in Fig. 1b. The radii of the touching spheres of atoms can be determined in a similar way as that in YCo B. However, in this case, 4 Y atom at 1a site touches the Y atom itself on the nearest upper and lower layers. The B atom at 2c site touches the nearest-neighbour Co atoms at 3g site. The ideal ratio c/a constructed from these touching spheres is 2/J3!1/2("0.655), which is rather larger than the observed one 0.599. The electronic structures were calculated in the local-density functional formalism with the LMTO-ASA. The radii of the atomic spheres used in the ASA were determined in such a way that the ratio is kept equal to the ideal value of the touching spheres estimated above. In most of the present calculations, the spin—orbit interaction was not included although the calculations are scalar relativistic, including mass velocity and Darwin terms. Self-consistent calculations for YCo B and 4 YCo B are carried out at 150 and 216 k-points in 3 2 the irreducible 1 Brillouin zone, respectively. The 24 convergence in charge density was achieved so that the root mean square of moments of the occupied partial density of states becomes smaller than 10~5. The basic set of functions with angular momenta upto l"3 on all atoms was adopted.
3. DOS in the non-magnetic state Fig. 2 shows the local DOSs for Y atoms at 1a and 1b sites, Co atoms at 2c and 6i sites and B atom at 2d site in YCo B calculated at the observed 4
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H. Yamada et al. / Journal of Magnetism and Magnetic Materials 183 (1998) 94—100
Fig. 2. Calculated local DOSs for Y, Co and B atoms at each site in YCo B. 4
Fig. 3. Calculated local DOSs for Y, Co and B atoms at each site in YCo B . 3 2
lattice constant a"5.035 A_ . The position of the Fermi level is shown by E . It can be seen clearly F that the height of the local DOS at E for Co atom F at 2c site is very high. The hybridizations with the B atom states are seen at the lower energy side in the local DOSs for Y atom at 1b site and Co atom at 6i site, as these Y and Co atoms are positionally close to the B atom site. In Fig. 3 the local DOSs for Y atom at 1a site, Co atom at 3g site and B atom at 2c site in YCo B 3 2 calculated at the observed a"5.05 A_ are shown.
Fig. 4. Local DOSs for Co atoms at 3g site in (a) YCo [8], (b) 5 6i site in YCo B and (c) 3g site in YCo B . 4 3 2
The shape of the local DOS for Co atom at 3g site is very similar to that for Co atom at 6i site in YCo B. 4 This is because the 6i site in YCo B is crystallo4 graphically similar position to the 3g sites in YCo 5 and YCo B . Calculated value of the total DOS 3 2 for YCo B is 74.9 states/Ry f.u. at E . This gives 3 2 F the low-temperature specific heat coefficient c" 13.0 mJ/mol K2, which is very close to the observed one, 13.9 mJ/mol K2 [9]. Moreover, the value of the susceptibility observed recently for this compound is 1.9]10~6 emu/g [6], which is rather smaller than the observed one by Ballou et al. [4]. The non-interacting (unenhanced) spin susceptibility at 0 K estimated from the DOS at E is 1.24] F 10~6 emu/g. Comparing with our observed susceptibility, the enhancement factor of the susceptibility is found to be about 1.5. In Fig. 4 the calculated local DOSs of Co atoms at 3g site in YCo [8], 6i site in YCo B and 3g site 5 4 in YCo B are compared with each other. It can be 3 2 seen that the portion of the relatively higher local DOS above E decreases with increasing concenF tration of B. That is, the number of holes seems to decrease with increasing concentration of B. This fact may mean that the magnetic moment of Co atom at 3g and 6i sites decreases with increasing concentration of B when the system is magnetized. On the other hand, the shapes of local DOS of Co atom at 2c site in YCo and YCo B are very similar 5 4
H. Yamada et al. / Journal of Magnetism and Magnetic Materials 183 (1998) 94—100
Fig. 5. Local DOSs for Co atoms at 2c site in (a) YCo [8] and 5 (b) YCo B. 4
97
Fig. 6. Calculated results of total energies per unit cell in the paramagnetic and ferromagnetic states (upper panel) and Co moments in the unit of l /atom on 2c and 6i sites in YCo B B 4 (lower panel) as a function of the unit-cell volume.
to each other, as shown in Fig. 5. Moreover, E is F found to lie at a very similar position on the local DOSs. This is because 2c sites of Co atom in YCo 5 and YCo B are crystallographically same position 4 with each other, being surrounded by six Co atoms at 3g and 6i sites, respectively.
4. Local Co moments Spin-polarized band calculations for YCo B and 4 YCo B are carried out in the LMTO-ASA with3 2 out the spin—orbit interaction for several lattice constants. The upper part in Fig. 6 shows the total energy for YCo B in the ferro- and para-magnetic 4 states as a function of the unit-cell volume. The minimum energy for YCo B is achieved in the 4 ferromagnetic state near the unit-cell volume 144 A_ 3, which is a little smaller than the observed one. Calculated Co moments at 2c and 6i sites are shown in the lower part in Fig. 6. It is found for YCo B that the Co at 2c site has a large moment 4 m "1.75l /atom, while the Co on 6i-site has C0(0) B a small moment m "0.55 l /atom at the unitC0(1) B cell volume 144 A_ 3. Small negative moments !0.11 and !0.15 l /atom are induced on Y atoms at 1a B and 1b sites, respectively. The negative-induced moments on Y come from the mixing between the 3d-states of Co and 4d-states of Y [10]. On the B atom site, no local moment is obtained.
Fig. 7. Majority and minority spin sub-bands for Y, Co and B atoms at each site in the ferromagnetic YCo B calculated at 4 the unit-cell volume 144 A_ 3.
The calculated value of the bulk moment 3.4 l /f.u. for YCo B is larger than the observed B 4 one 2.54 l /f.u. [6]. However, it has been confirmed B that the Co moment at 6i site becomes small in YCo B as predicted in Refs. [5,6]. Fig. 7 shows the 4
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H. Yamada et al. / Journal of Magnetism and Magnetic Materials 183 (1998) 94—100
local DOSs in the majority and minority spin subbands for Y atoms at 1a and 1b sites, Co atoms at 2c and 3g sites and B atom at 2d site in YCo B 4 calculated at the unit-cell volume 144 A_ 3. On the other hand, we have obtained no selfconsistent solution for any ferromagnetic state in YCo B near the observed lattice constant. This 3 2 fact means that m "0, which is consistent with C0(2) the experimental result that this compound is paramagnetic even at low temperature. Finally, the LMTO-ASA calculations with inclusion of the spin—orbit interaction have been done for ferromagnetic YCo B at the equilibrium lattice 4 constant. The actual calculation procedures are as follows: the Kohn—Sham equations for the scalar relativistic spin-polarized Hamiltonian are first solved self-consistently, and then the spin—orbit coupling term is added to and the full Hamiltonian is diagonalized non-self-consistently. The total orbital moment of YCo B has been obtained as 4 0.45 l /f.u. which is smaller than that of YCo , B 5 0.61 l /f.u., estimated by Daalderop et al. [11] in B the non-self-consistent calculation. The orbital moments on Co atoms at 2c and 6i sites are m03" " C0(0) 0.15 l /atom and m03" "0.10 l /atom, respecC0(1) B B tively. The ratio between the orbital and spin moments on Co atom at 2c site in YCo B is almost the 4 same as that in YCo obtained in Ref. [11]. On the 5 other hand, the ratio between the orbital and spin moments on Co atom at 6i site in YCo B is twice as 4 large as that on Co atom at 3g site in YCo . 5 5. Conclusions and discussion In this paper, the electronic structures of YCo B 4 and YCo B compounds have been calculated by 3 2 the self-consistent LMTO-ASA. By comparing the local DOSs with those of YCo obtained in Ref. 5 [8], the effect of B atom has been discussed. It was found that the shape of local DOS for Co at 6i site in YCo B is very similar to that at 3g sites in 4 YCo B and YCo . However, the position of E is 3 2 5 F different from each other. The effective number of hole at 6i or 3g site seems to decrease with increasing concentration of B. Then, the local Co moments at 6i and 3g sites were predicted to decrease with increasing concentration of B. On the other hand,
Fig. 8. Schematic view of the mixings between 3d-states of Co and 2p-states of B in YCo B and YCo B . The energies at the 4 3 2 centre of gravity of the local DOSs for 3d-states of Co atoms at 6i, 2c and 3g sites and 2p-states of B are shown by 3d(6i), 3d(2c), 3d(3g) and 2p, respectively.
the local DOSs for Co at 2c site in YCo B and 4 YCo were found to be very similar to each other. 5 E was found to lie near a sharp peak of the local F DOS. Then, the local Co moment on 2c site was expected not to be affected so much by B atoms. These facts have been confirmed by the spin-polarized calculation in the present paper. Moreover, the orbital moments on Co at 2c and 6i sites have been evaluated as 0.15 and 0.10 l /atom, which are simB ilar to the calculated ones at 2c and 3g sites in YCo 5 [11], respectively. Kanamori [12] has previously shown that the effect of B in R—Fe—B system can be explained by the mixing between the 3d-states of Fe and 2pstates of B. Our results in Y—Co—B system are also explained by the mixing model between the 3dstates of Co and 2p-states of B. The importance of the mixing between them in Y Co B is also em2 14 phasized by Coehoorn [7]. As seen in Fig. 1a, the atomic layer of Co at 6i site in YCo B lies between 4 the atomic layers of B at 2d site and Co at 2c site. Fig. 8 gives a schematic view of the mixings between 3d-states of Co and 2p-states of B in YCo B 4 and YCo B . The energies at the centre of gravity 3 2 of the local DOSs for 3d-states of Co at 6i, 2c and 3g sites and 2p-states of B are shown by 3d(6i), 3d(2c), 3d(3g) and 2p, respectively. By the mixing between 3d-states of Co at 6i site and 2p-states of B at 2d site in YCo B, the centres of gravity for the 4 local DOSs of Co at 6i site and B at 2d site shift towards the lower- and higher-energy sides, respectively. Moreover, the centre of gravity for the local DOS of Co at 6i site shifts much more towards the lower energy side by the mixing with 3d-states of
H. Yamada et al. / Journal of Magnetism and Magnetic Materials 183 (1998) 94—100
99
Table 1 The decreased and increased total numbers of electrons on each atom from the neutral one in YCo B and YCo B calculated at the 4 3 2 unit-cell volume 144 and 60 A_ 3, respectively, in the non-magnetic state YCo B 4
YCo B 3 2
Y (1a)
Y (1b)
Co (2c)
Co (6i)
B (2d)
Y (1a)
Co (3g)
B (2c)
!0.06
0.67
!0.27
0.37
!1.15
0.53
0.29
!0.70
Co at 2c site. In this case, some electrons flow into Co at 6i site from the neighbouring B, because the 2p-states of B shift towards higher energy side as shown in Fig. 8. Therefore, the position of E to F the local DOS of Co at 6i site shifts towards the higher-energy side and effective number of holes on Co at 6i site decreases. In YCo B , on the other hand, both upper and 3 2 lower sides of the atomic layer of Co atoms at 3g site are B atom layers at 2c site. As shown in Fig. 8, the centre of gravity of the local DOS of Co at 3g site shifts towards lower energy side by the hybridization between 3d-states of Co and 2p-states of B, as well as that of Co at 6i site in YCo B. Then, some 4 electrons flow into the Co at 3g site from the neighbouring B and the position of E to the local DOS F of Co at 3g site shifts towards higher-energy side. The increased and decreased total numbers of electrons on each atom from the neutral one in YCo B 4 and YCo B are listed in Table 1. However, such 3 2 a change in the number of electrons on atoms comes from the use of ASA in the present calculation and simply means the tails of the wave function spread on the neighbouring sites [13]. That is, it does not mean the real charge transfers between the atoms. The present mixing model also explains the magnitude of local moments of Co. The local moment of Co at 6i site in YCo B becomes small as shown 4 in Section 4. By the hybridization with the 3dstates of Co at 2c site and 2p-states of B at 2d site, the centre of gravity of the 3d-states of Co at 6i site shifts towards lower-energy side. Then some electrons flow into this site from the neighbouring Co at 2c site and B at 2d site. As the majority spin band is filled up by electrons in the magnetized state, the number of electrons with minority spin will in-
crease and, then, the magnetic moment of Co atoms at 6i site becomes small. On the other hand, the local moment of Co at 2c site (1.75 l /atom) in B YCo B is larger than that in YCo (1.52 l /atom) 4 5 B [8]. The centre of gravity for the minority spin band of Co at 2c site shifts towards higher energy side, by the hybridization with 3d-states of Co at 3g site, of which energy shifts also towards the lower energy side by the hybridization with 2p-states of B at 2d site, as shown in Fig. 8. Then, some electrons of Co at 2c site flow out. As the majority spin band of Co at 2c site is filled up by electrons, the number of electrons with minority spin on Co at 2c site will decrease. Therefore, the magnetic moment of Co at 2c site in YCo B becomes larger than that 4 in YCo . 5 For YCo B , we have obtained a good agree3 2 ment between the calculated and observed values of the low temperature specific heat coefficient. By the comparison between the observed susceptibility and the calculated DOS at E , the enhancement F factor of the susceptibility has been estimated as about 1.5. In this compound, no self-consistent solution for the ferromagnetic state has been obtained. This is because Co atoms at 3g site in YCo B are put between the B atom layers and the 3 2 d-band of Co is almost filled by the electrons flowed out from B atoms. In this way, it is concluded that the effect of B atom on the Co moment observed in YCo B and YCo B can be explained by the 3d—2p 4 3 2 mixing model. References [1] H. Ido, Kotai Butsuri 30 (1995) 875 (in Japanese). [2] H. Ido, H. Yamauchi, S.F. Cheng, S.G. Sankar, W.E. Wallace, J. Appl. Phys. 70 (1991) 6540.
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[3] N.P. Thuy, N.M. Hong, J.P. Liu, X. Li, J.J.M. Franse, F.R. de Boer, Physica B 177 (1992) 270. [4] R. Ballou, E. Burzo, V. Pop, J. Magn. Magn. Mater. 140—144 (1995) 945. [5] H. Ogata, H. Ido, H. Yamauchi, J. Appl. Phys. 73 (1993) 5911. [6] H. Ido, unpublished work. [7] R. Coehoorn, J. Magn. Magn. Mater. 99 (1991) 55. [8] I. Kitagawa, K. Terao, H. Yamada, J. Phys.: Condens. Matter 9 (1997) 231.
[9] W. Perthold, N.M. Hong, H. Michor, G. Hilscher, H. Ido, H. Asano, J. Magn. Magn. Mater. 157/158 (1996) 649. [10] M.S.S. Brooks, B. Johansson, in: K.H.J. Buschow (Ed.), Handbook of Magnetic Materials, vol. 7, Elsevier, Amsterdam, 1993, p. 139. [11] G.H.O. Daalderop, P.J. Kelly, M.F.H. Schuurmans, Phys. Rev. B 53 (1996) 14415. [12] J. Kanamori, Proc. 10th Int. Workshop on Rare-Earth Magnets, Kyoto, 1989, p. 1. [13] S. Asano, private communication.