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Electronic structure and magnetism of C15-type Laves phase compounds Y(Co, Al)2 and Y(Co, Si)2
PHYSICA Ll
Physica B 177 (1992) 259-261 North-Holland
Electronic structure and magnetism of Cl5type compounds Y(Co, Al), and Y(Co, Si), M. Aoki Department
Laves phase
and H. Yamada of Physics, Gifu University, Gifu 501-11, Japan
Electronic structures of the binary compound YCo, and the ordered ternary compounds Y(Co,_,Al,), and Y(Co,_,Si,), with the cubic Laves structure are calculated within the density functional theory. It is shown that the magnetic properties observed in the recent experiments, such as induced weak ferromagnetism and metamagnetic transitions, can be qualitatively explained by using the calculated results of the density-of-states curves. The role of the p-d hybridisation is discussed.
1. Introduction
Cl5type Laves phase compounds YCo, and LuCo, are known as strongly exchange-enhanced Pauli paramagnets with a maximum in the temof the susceptibility. perature dependence Metamagnetic transition from the paramagnetic to the ferromagnetic state is observed at the critical magnetic fields H, = 69 and 75 T for YCo, and LuCo*, respectively [l]. These magnetic properties are explained by the characteristic double-sharp-peak structure of the nonmagnetic density of states (DOS) formed just below the Fermi level (EF) [2]. In the pseudo-binary compounds Y(Co, Al), and Lu( Co, Al), , the value of W, is found to decrease with increasing x, the concentration of Al [3, 41. Also, weak ferromagnetism is observed in the range of x = 0.13-0.19 for Y(Co,_,Al,), [5] and x = 0.60-0.24 for Lu(Co,_,Al,), [4]. It is said that the volume expansion with X, accompanied with the narrowing in the DOS of the d electrons of Co, would lead to the appearance of the weak ferromagnetism. However, the recently observed results show that the metamagnetic transition occurs even for the pseudo-binary compounds Y(Co, Si), and Lu(Co, Si), where the volume of the compounds is reduced by the substituent Si [6]. Despite the opposite change in the volume, the decrease of H, with increasing x 0921-4526/92/$05.00
0
1992 - Elsevier
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for Y(Co,_,Si,), and Lu(Co,_,Si,), is observed. By calculating the electronic structure of Y(Co,.,, Al,,,,), in the density functional theory, the present authors [7] have shown that the higher energy peak of the characteristic two peaks in the DOS is sensitively destroyed and E, shifts towards the low energy side, when one out of every four Co atoms in a unit cell is replaced by an Al atom. This suggests that the p-d hybridisation between Al and Co atoms and the accompanying shift of E, are responsible for the weak ferromagnetism observed in this system. The purpose of the present work is to understand the magnetic properties of these intermetallic compounds on the basis of calculated electronic structures. In the next section, we show the electronic structures of the binary compound YCo, and the ordered ternary compounds Y,Co,Al and Y,Co,Si calculated within the density functional theory. A physical interpretation of the calculated results and the role of the p-d hybridisation are discussed in section 3, followed by a conclusion in section 4.
2. Calculated
results
We have calculated the electronic structures in the density functional theory, using the selfconsistent augmented plane wave (APW) meth-
B.V. All rights reserved
M. Aoki, H. Yamada I Electronic structures of Y(Co, AQ2, Y(Co, Si),
260
od with the muffin-tin approximation. The detailed procedure of the calculation is described in ref. [7]. We note, however, that an ordered structure of ternary compounds is assumed for Y(C0 0.75A10.25)2and Y(Co,.&.& Randomness in the sites of the Al or Si atoms due to alloying is ignored. Figure 1 shows the calculated local DOS curves projected onto (a) Co d states of YCo,, (b) Co d and (c) Al p states of Y,Co,Al, and (d) Co d and (e) Si p states of Y,Co,Si. We have used for the lattice constant the observed value of 7.217 8, for YCo,, the same value for Y,Co,Si, and an extrapolated value of 7.38OA for Y,Co,Al. For YCo,, E, lies just above the characteristic double-sharp-peak structure of local DOS of Co d states. However, one sharp peak on the higher energy side in the Co d DOS is sensitively destroyed when Al is substituted to form Y(CO,_,A~~)~. At the same time E, shifts, rela-
n
J
e - (c) Al P
0 al
.O
2
0 6.
_
0. I
(d) Co d
0.2
0.3
0.4
Em-w Fig. 1. Projected DOS curves Y2Co,AI, and (d, e) Y2Co,Si,
0.5
0.6
0.7
FIY)
calculated for (a) YCo,, in states/Ry atom.
(b, c)
tive to the Co d band, towards the lower energy side, as shown in fig. l(b). The lowering of E, is consistent with the observed decrease of H, with increasing ‘Al content. A small peak just above E, in fig. l(b) may be a vestige of the destroyed sharp peak. This broad peak arises from some admixtures of the Co 3d and Al 3p states. We have pointed out that the hybridisation between Co d states and Al p states is so strong in this system [7] that the picture based on the rigid band model is insufficient. If we employ the rigid band model, the lowering of E, in YCo, may stabilise the ferromagnetic state because the DOS at E, increases as E, shifts close to the sharp peak. However, for the Y(Co,_,Al,), system, the effect of p-d hybridisation and the resultant broadening of the sharp peak must be taken into account. Thus, in the possible mechanism of the occurrence of weak ferromagnetism proposed in ref. [7], the situation would be at a certain critical condition for the ferromagnetic transition. Actually, the weakly ferromagnetic state of Y(Co,,,, Al,,,,), is observed to collapse under compression at 9 kbar [8]. In the case of Y,Co,Si, E, shifts towards the centre of the Co d band, and one sharp peak of the Co d DOS is destroyed. The relative position of E, is just at the position where the fragile sharp peak in YCo, is located. The vestige of the fragile sharp peak broadened by the p-d hybridisation is a broad peak at E, in the local DOS of Si p states. Here we see similar trends of change in the shift of E, and in the broadening of one sharp peak of Co d DOS, for both Al and Si substitutions. The lowering of E, is also consistent with the observed decrease of H, with increasing Si content. However, it is found that the amount of shift of E, caused by Si substitution is rather small compared with that caused by Al substitution. Also we find that the height of the DOS of Si p states is larger than that of Al p states and most of the occupied Si p states appear in the lower energy region. The difference in the number of occupied p states of the substituent atom is trivial since the Si atom has an extra 3p electron compared with the Al atom. These interesting trends caused by different substituents, Al and Si, are discussed in the next section.
M. Aoki, H. Yamada I Electronic structures of Y(Co, AI),,
3. Discussion
To illustrate the important role of the p-d hybridisation in the calculated results described in the previous section, let us consider a Hiickel model for the linear molecule consisting of three identical atoms with only the d orbitals. Because of an axial symmetry, we can reduce the problem to the case of a single d orbital on each site. Assuming that the hopping integral h,, is limited to the nearest neighbours and setting the atomic d energy level cd to be 0, we get three energy eigenvalues of E = 0, +filhdd]. The number of states projected onto the d orbital of atom 1 at either edge of the molecule is given by NY(O) = 0.5 )
iv; (+vqh,,~)
= 0.25 .
(1)
The eigenstate with E = E,, = 0 is a non-bonding d state, where the centre atom has no amplitude. Other eigenstates are the bonding and anti-bonding states. Next we substitute an atom with only the p orbital for the centre atom in the molecule. Assuming the hopping integral ?fh,, for the nearest neighbours, we get energy eigenvalues of non-bonding d state E = 0, and of p-d bonding/ anti-bonding states E*, with E’ = .spd/2 7 [(.+l 2)2 + 2/~:,]“~, where &rd is the atomic energy level of the p orbital. From the free atom calculations, within the density functional theory, we have Q = 0.4 and 0.3 Ry for Al and Si, respectively. In this case, the numbers of states projected onto the d orbital of the atom 1 and the substituent atom 2 are given by
N;(o) = 0.5
)
N;(E’)
= 0.25(1+
p) ,
(2)
and
N;(O) = 0
)
respectively, E-+E+ P= E-_E+
iI’;
= 0.5(1~
p) ,
(3)
where = (&,,/2)/[(&,,/2)2
+ 2lz;,y
, .\
is positive. It is to be noted that the projected number of the d states on atom 1 for the p-d bonding state, Nf(E+), is greater than 0.25.
Y(Co, Si),
261
Therefore the number of d states below the non-bonding state increases with the substitution of Al and Si atoms. This results in a lowering of E,, being required by the atomic charge neutrality. If h,, is kept constant, the larger the cpd, the lower the shift of E, we can expect. Also we see from eq. (3) that the decrease in the number of p states on the substituent atom 2 for the p-d bonding state follows the increase of .+. Although the three-atom linear molecule model presented here is quite simple, it provides a qualitative interpretation of the trends found in the calculated results of the electronic structure for YCo,, Y,Co,Al and Y,Co,Si. 4. Conclusion In the present work we have shown how the electronic structure changes when Al and Si are substituted for Co in YCo, to form the ordered ternary compounds Y,Co,Al and Y,Co,Si. The substituents Al and Si both introduce the p-d hybridisation in the electronic structure, which results in the lowering of E, and also the broadening of the fragile sharp peak just below the E,. Difference in the effects caused by the different substituents are explained by the different atomic 3p energy levels of the Al and Si atoms. All these results are consistent with the experimental ones. The alloying and other effects are left to future work. References [l] T. Goto, T. Sakakibara, K. Murata, H. Komatsu and K. Fukamichi, J. Magn. Magn. Mater. 90&91 (1990) 700. [2] H. Yamada, Physica B 149 (1988) 390, and references therein. [3] T. Sakakibara, T. Goto, K. Yoshimura, K. Murata and K. Fukamichi, J. Magn. Magn. Mater. 90&91 (1990) 131. [4] K. Iijima, K. Endo, T. Sakakibara and T. Goto, J. Phys. Condens. Mat. 2 (1990) 10069. [5] K. Yoshimura and Y. Nakamura, Solid State Commun. 56 (1985) 767. [6] K. Murata, K. Fukamichi, T. Sakakibara, T. Goto and K. Suzuki, private communication (1991). [7] M. Aoki and H. Yamada, J. Magn. Magn. Mater. 78 (1989) 377. [8] J.G.M. Armitage, R.G. Graham, P.C. Riedi and J.S. Abell, J. Phys. Condens. Mat. 2 (1990) 8779.
Physica B 177 (1992) 259-261 North-Holland
Electronic structure and magnetism of Cl5type compounds Y(Co, Al), and Y(Co, Si), M. Aoki Department
Laves phase
and H. Yamada of Physics, Gifu University, Gifu 501-11, Japan
Electronic structures of the binary compound YCo, and the ordered ternary compounds Y(Co,_,Al,), and Y(Co,_,Si,), with the cubic Laves structure are calculated within the density functional theory. It is shown that the magnetic properties observed in the recent experiments, such as induced weak ferromagnetism and metamagnetic transitions, can be qualitatively explained by using the calculated results of the density-of-states curves. The role of the p-d hybridisation is discussed.
1. Introduction
Cl5type Laves phase compounds YCo, and LuCo, are known as strongly exchange-enhanced Pauli paramagnets with a maximum in the temof the susceptibility. perature dependence Metamagnetic transition from the paramagnetic to the ferromagnetic state is observed at the critical magnetic fields H, = 69 and 75 T for YCo, and LuCo*, respectively [l]. These magnetic properties are explained by the characteristic double-sharp-peak structure of the nonmagnetic density of states (DOS) formed just below the Fermi level (EF) [2]. In the pseudo-binary compounds Y(Co, Al), and Lu( Co, Al), , the value of W, is found to decrease with increasing x, the concentration of Al [3, 41. Also, weak ferromagnetism is observed in the range of x = 0.13-0.19 for Y(Co,_,Al,), [5] and x = 0.60-0.24 for Lu(Co,_,Al,), [4]. It is said that the volume expansion with X, accompanied with the narrowing in the DOS of the d electrons of Co, would lead to the appearance of the weak ferromagnetism. However, the recently observed results show that the metamagnetic transition occurs even for the pseudo-binary compounds Y(Co, Si), and Lu(Co, Si), where the volume of the compounds is reduced by the substituent Si [6]. Despite the opposite change in the volume, the decrease of H, with increasing x 0921-4526/92/$05.00
0
1992 - Elsevier
Science
Publishers
for Y(Co,_,Si,), and Lu(Co,_,Si,), is observed. By calculating the electronic structure of Y(Co,.,, Al,,,,), in the density functional theory, the present authors [7] have shown that the higher energy peak of the characteristic two peaks in the DOS is sensitively destroyed and E, shifts towards the low energy side, when one out of every four Co atoms in a unit cell is replaced by an Al atom. This suggests that the p-d hybridisation between Al and Co atoms and the accompanying shift of E, are responsible for the weak ferromagnetism observed in this system. The purpose of the present work is to understand the magnetic properties of these intermetallic compounds on the basis of calculated electronic structures. In the next section, we show the electronic structures of the binary compound YCo, and the ordered ternary compounds Y,Co,Al and Y,Co,Si calculated within the density functional theory. A physical interpretation of the calculated results and the role of the p-d hybridisation are discussed in section 3, followed by a conclusion in section 4.
2. Calculated
results
We have calculated the electronic structures in the density functional theory, using the selfconsistent augmented plane wave (APW) meth-
B.V. All rights reserved
M. Aoki, H. Yamada I Electronic structures of Y(Co, AQ2, Y(Co, Si),
260
od with the muffin-tin approximation. The detailed procedure of the calculation is described in ref. [7]. We note, however, that an ordered structure of ternary compounds is assumed for Y(C0 0.75A10.25)2and Y(Co,.&.& Randomness in the sites of the Al or Si atoms due to alloying is ignored. Figure 1 shows the calculated local DOS curves projected onto (a) Co d states of YCo,, (b) Co d and (c) Al p states of Y,Co,Al, and (d) Co d and (e) Si p states of Y,Co,Si. We have used for the lattice constant the observed value of 7.217 8, for YCo,, the same value for Y,Co,Si, and an extrapolated value of 7.38OA for Y,Co,Al. For YCo,, E, lies just above the characteristic double-sharp-peak structure of local DOS of Co d states. However, one sharp peak on the higher energy side in the Co d DOS is sensitively destroyed when Al is substituted to form Y(CO,_,A~~)~. At the same time E, shifts, rela-
n
J
e - (c) Al P
0 al
.O
2
0 6.
_
0. I
(d) Co d
0.2
0.3
0.4
Em-w Fig. 1. Projected DOS curves Y2Co,AI, and (d, e) Y2Co,Si,
0.5
0.6
0.7
FIY)
calculated for (a) YCo,, in states/Ry atom.
(b, c)
tive to the Co d band, towards the lower energy side, as shown in fig. l(b). The lowering of E, is consistent with the observed decrease of H, with increasing ‘Al content. A small peak just above E, in fig. l(b) may be a vestige of the destroyed sharp peak. This broad peak arises from some admixtures of the Co 3d and Al 3p states. We have pointed out that the hybridisation between Co d states and Al p states is so strong in this system [7] that the picture based on the rigid band model is insufficient. If we employ the rigid band model, the lowering of E, in YCo, may stabilise the ferromagnetic state because the DOS at E, increases as E, shifts close to the sharp peak. However, for the Y(Co,_,Al,), system, the effect of p-d hybridisation and the resultant broadening of the sharp peak must be taken into account. Thus, in the possible mechanism of the occurrence of weak ferromagnetism proposed in ref. [7], the situation would be at a certain critical condition for the ferromagnetic transition. Actually, the weakly ferromagnetic state of Y(Co,,,, Al,,,,), is observed to collapse under compression at 9 kbar [8]. In the case of Y,Co,Si, E, shifts towards the centre of the Co d band, and one sharp peak of the Co d DOS is destroyed. The relative position of E, is just at the position where the fragile sharp peak in YCo, is located. The vestige of the fragile sharp peak broadened by the p-d hybridisation is a broad peak at E, in the local DOS of Si p states. Here we see similar trends of change in the shift of E, and in the broadening of one sharp peak of Co d DOS, for both Al and Si substitutions. The lowering of E, is also consistent with the observed decrease of H, with increasing Si content. However, it is found that the amount of shift of E, caused by Si substitution is rather small compared with that caused by Al substitution. Also we find that the height of the DOS of Si p states is larger than that of Al p states and most of the occupied Si p states appear in the lower energy region. The difference in the number of occupied p states of the substituent atom is trivial since the Si atom has an extra 3p electron compared with the Al atom. These interesting trends caused by different substituents, Al and Si, are discussed in the next section.
M. Aoki, H. Yamada I Electronic structures of Y(Co, AI),,
3. Discussion
To illustrate the important role of the p-d hybridisation in the calculated results described in the previous section, let us consider a Hiickel model for the linear molecule consisting of three identical atoms with only the d orbitals. Because of an axial symmetry, we can reduce the problem to the case of a single d orbital on each site. Assuming that the hopping integral h,, is limited to the nearest neighbours and setting the atomic d energy level cd to be 0, we get three energy eigenvalues of E = 0, +filhdd]. The number of states projected onto the d orbital of atom 1 at either edge of the molecule is given by NY(O) = 0.5 )
iv; (+vqh,,~)
= 0.25 .
(1)
The eigenstate with E = E,, = 0 is a non-bonding d state, where the centre atom has no amplitude. Other eigenstates are the bonding and anti-bonding states. Next we substitute an atom with only the p orbital for the centre atom in the molecule. Assuming the hopping integral ?fh,, for the nearest neighbours, we get energy eigenvalues of non-bonding d state E = 0, and of p-d bonding/ anti-bonding states E*, with E’ = .spd/2 7 [(.+l 2)2 + 2/~:,]“~, where &rd is the atomic energy level of the p orbital. From the free atom calculations, within the density functional theory, we have Q = 0.4 and 0.3 Ry for Al and Si, respectively. In this case, the numbers of states projected onto the d orbital of the atom 1 and the substituent atom 2 are given by
N;(o) = 0.5
)
N;(E’)
= 0.25(1+
p) ,
(2)
and
N;(O) = 0
)
respectively, E-+E+ P= E-_E+
iI’;
= 0.5(1~
p) ,
(3)
where = (&,,/2)/[(&,,/2)2
+ 2lz;,y
, .\
is positive. It is to be noted that the projected number of the d states on atom 1 for the p-d bonding state, Nf(E+), is greater than 0.25.
Y(Co, Si),
261
Therefore the number of d states below the non-bonding state increases with the substitution of Al and Si atoms. This results in a lowering of E,, being required by the atomic charge neutrality. If h,, is kept constant, the larger the cpd, the lower the shift of E, we can expect. Also we see from eq. (3) that the decrease in the number of p states on the substituent atom 2 for the p-d bonding state follows the increase of .+. Although the three-atom linear molecule model presented here is quite simple, it provides a qualitative interpretation of the trends found in the calculated results of the electronic structure for YCo,, Y,Co,Al and Y,Co,Si. 4. Conclusion In the present work we have shown how the electronic structure changes when Al and Si are substituted for Co in YCo, to form the ordered ternary compounds Y,Co,Al and Y,Co,Si. The substituents Al and Si both introduce the p-d hybridisation in the electronic structure, which results in the lowering of E, and also the broadening of the fragile sharp peak just below the E,. Difference in the effects caused by the different substituents are explained by the different atomic 3p energy levels of the Al and Si atoms. All these results are consistent with the experimental ones. The alloying and other effects are left to future work. References [l] T. Goto, T. Sakakibara, K. Murata, H. Komatsu and K. Fukamichi, J. Magn. Magn. Mater. 90&91 (1990) 700. [2] H. Yamada, Physica B 149 (1988) 390, and references therein. [3] T. Sakakibara, T. Goto, K. Yoshimura, K. Murata and K. Fukamichi, J. Magn. Magn. Mater. 90&91 (1990) 131. [4] K. Iijima, K. Endo, T. Sakakibara and T. Goto, J. Phys. Condens. Mat. 2 (1990) 10069. [5] K. Yoshimura and Y. Nakamura, Solid State Commun. 56 (1985) 767. [6] K. Murata, K. Fukamichi, T. Sakakibara, T. Goto and K. Suzuki, private communication (1991). [7] M. Aoki and H. Yamada, J. Magn. Magn. Mater. 78 (1989) 377. [8] J.G.M. Armitage, R.G. Graham, P.C. Riedi and J.S. Abell, J. Phys. Condens. Mat. 2 (1990) 8779.