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Electronic structure of the ordered ternary compound Y(M0.75Al0.25)2 (M = Co and Fe) with C15-type laves phase structure
Journal of Magnetism and Magnetic North-Holland. Amsterdam
Materials
78 (1989) 377-383
377
ELECTRONIC STRUCTURE OF THE ORDERED TERNARY COMPOUND (M = Co AND Fe) WITH C&TYPE LAVES PHASE STRUCTURE Masato
AOKI
and Hideji
Y(M,,,AIO,LS)Z
YAMADA
Department of Physics, Gifu University, Yanagido, Gifu 501-I I, Japan
Received
20 October
1988
The electronic structures of the binary compound YM, and the ordered ternary compound Y(M,,,,Al,,,), (M = Co and Fe) with the Cl5type Laves phase structure are calculated by the self-consistent APW method. It is shown that the calculated density-of-states (DOS) curves for YM, are very similar to those obtained previously, being characterised by the two sharp peaks mainly of 3d-states of M atoms near the Fermi level. For ordered Y(Mo,,sAlo,zs)z, some 3d-states of the M atom hybridise strongly with 3p-states of the Al atom and only a single sharp peak is shown to appear in the DOS near the Fermi level. Moreover, the position of the Fermi level is shown to shift toward the lower energy side by the substitution of Al atoms. The anomalous magnetic properties observed in the pseudobinary compound Y(M,Al), are discussed on the basis of these calculated electronic structures.
1. Introduction Intermetallic compounds of yttrium and the 3d-transition metal M (= Mn, Fe, Co or Ni), with the cubic MgCu,-type (Cl%type) YM,, Laves phase structure are known to exhibit many interesting magnetic properties. Particularly, YCq shows curious magnetic properties; a maximum in the temperature dependence of the susceptibility [1,2] and anomalous field dependence of the induced magnetic moment [3,4] were observed. Recently, these magnetic properties of YM, have been discussed theoretically from the point of view of the electronic structures [5]. The calculation of the density-of-states (DOS) curve in these compounds was first carried out in the moment method [6]. Subsequently, many calculations have been done in the tight-binding (TB) approximation [7], the augmented spherical wave (ASW) methods [8], the linearised muffin-tin orbital (LMTO) and Korringa-Kohn-Rostoker (KKR) methods [9]. The shapes of the DOS calculated in the various methods are very similar to each other and the characteristic of the DOS for YM, is the existence of two sharp peaks 0304-8853/89/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
mainly of 3d-states of M atoms near the Fermi level E,. The positions of E, for YCo, and YFe, in the non-magnetic state are a little beyond the sharp peak of the DOS in the higher energy side and just on the sharp peak in the lower energy side, respectively. Then, YCo, is the strongly exchange-enhanced paramagnet and YFe, is the ferromagnet, as observed. For YCo, the sharp peak of the DOS in the higher energy side, which is very close to E,, has been shown to be responsible for both observed results of the anomalous temperature and magnetic field dependences of the susceptibility [7]. In a narrow concentration range 0.13 < x < 0.19 for the pseudobinary compounds Y(Co,_,Al,),, a weakly ferromagnetic state has been observed [lo]. The spontaneous magnetisation at low temperature is very small (less than 0.14~~ per Co atom) and the paramagnetic susceptibility obeys the Curie-Weiss law at high temperature, which is a typical magnetic behaviour of a weak itinerant electron ferromagnet [ll]. It is said that the onset of the weakly ferromagnetic moment in this system may be ascribed to a large atomic volume of Al atoms; the narrowed 3d band associated with B.V.
378
M. Aoki, H. Yamada
/ Electronic structure of Y(M,AI),
the lattice expansion by the substitution of Al atoms promotes a tendency to the onset of magnetic moment. On the other hand, in Y(Fe,_,Al,), the concentration dependence of the magnetic moment at low temperature shows the opposite trend to that in the Y(Co,Al), system. The lattice constant increases with increasing x, like in Y(Co,Al),. However, the concentration dependence of the magnetic moment shows a rapid decrease with increasing x [12-161, which cannot be explained by the narrowed 3d band associated with the lattice expansion. Besnus et al. [14] have pointed out that the magnetic moment on the Fe atom is strongly affected by the surrounding atoms, that is, the effect of the local environment is important in this system. It is not clear why the magnetic properties in both systems of Y(Co,Al), and Y(Fe,Al), are so different from each other. The aim of the present work is to study these magnetic properties on the basis of the electronic structures. In section 2, the crystal structure of the ordered ternary compound Y(M,,,,Al,,,,), (M = Co and Fe), for which the electronic structure is calculated in this paper, is shown. In section 3, the details of the band calculation for the present system are given and the calculated results are shown in section 4. In section 5, the appearance of the magnetic moment in a narrow concentration range of Al in Y(Co,AI), and the rapid decrease of the magnetic moment of Y(Fe,Al), with increasing the
(M = Co, Fe)
Al concentration are discussed calculated electronic structures.
2. Crystal structure A projected crystal structure of YM, in the Cl5type Laves phase on the x-y plane is shown in fig. l(a). Large circles represent Y atoms and both small and dotted circles represent M atoms. The number in circles denotes the z component of the atomic position in units of lattice constant a. A unit cell contains two Y atoms and four M atoms. The translational symmetry of this structure is the same as that of the fee structure and the crystal structure can be described as a stacking of the atomic planes (111). These atomic planes are classified into three types of Y,-plane, M,-plane and M,-plane, where the subscript stands for the number of atoms on the plane contained in a unit cell. It is noted that on the M,-plane M atoms form a special two-dimensional lattice, the socalled “KagomC” lattice [17] as shown in fig. l(b). For the ordered ternary compound Y,M, Al in the Cl5type Laves phase, the crystal structure can be constructed from that of YM, by replacing one of the four equivalent M atoms in a unit cell with an Al atom. The Al atoms are indicated by dotted circles in fig. 1. Such a crystal structure has recently been used in the calculation of the magnetic moments in the ordered ternary compound
R
(a)
Fig. 1. (a) Projected crystal structure of YM,
on the basis of the
in the ClS-type Laves phase structure on x-y formed by M atoms on M,-plane.
(b)
plane; (b) so-called “KagomB” lattice
M. Aoki, H. Yamada / Electronic structure of Y(M,AI),
Fig. 2. The first Brillouin zone for the ordered pound with the ClS-type Laves phase structure. represents the l/12 IBZ.
Y,M,Al comA shaded part
Y(Fe,Co), [18]. It is noted that the atomic plane (111) of this structure does not consist only of a single kind of Y or M atom. However, any atomic plane (lli) consists of a single kind of atom, i.e., Y,-plane, M,-plane or Al,-plane. Then, the “Kagomt” lattice of M atoms is formed only on the (117) plane. Since the point group operations which move the Al atom site to the M atom site in fig. l(a) have to be excluded, the space group is restricted to a sub-group for the ClS-type Laves phase structure and the number of the operations of the point group is reduced to 12 from 48. As shown in fig. 2, the irreducible Brillouin zone (IBZ) of this structure then becomes l/12 of the full Brillouin zone; the volume of the IBZ becomes four times larger than that of YM,. The high symmetry points in the IBZ are I’(OOO), X(200), L(lli), L’(lll) and W(210) and the symmetry lines are R(qq?/) with 0 < n < 1 and ~(~OTJ) with 0 < n < 3/2, where the unit is taken as a/a.
(M = Co,Fe)
319
ues of non-overlapping spheres, i.e., (0/8)a for Y and (a/8) a for M and Al [20]. The core electrons are treated in the so-called frozen-core approximation [21]; the Kr core without 4p6 for Y, the Ar core for Co and for Fe and the Ne core for Al are made use of. The maximum angular momentum, I,, , is taken to be 7. The number of APW’s is about 260, that is, ] k + G ] 5 6.3 (2n/a), where k is a wave vector in the first Brillouin zone and G is an fee reciprocal lattice vector. The six symmetry points I, X, L, K, W, U and each midpoint of six symmetry lines A, A, Z, S, Z, Q are taken as the sampling points for the iterative APW calculation for YM, and the nine uniformly distributed k points in the IBZ are taken for the calculation of Y,M,Al. The iterations are carried out until a following practical condition of the self-consistency for the charge density is satisfied in each MT sphere,
/ MT
d3rI Pin(r)- lout
I<
10e2,
(1)
where pi,(r) and p,,,(r) are the input and output charge densities. The eigenvalues have been converged to within the accuracy of less than mRy. The values of the lattice constant for YCo,, YFe, and Y,Fe,Al are taken from the experimental data, a = 7.217 A [l], 7.362 A [14] and 7.481 A [14], respettively. For Y,Co,Al we have estimated u = 7.380 A by the extrapolation of experimental data for Y(Co,_,Al,), [lo] to x = 0.25. The total and Z-decomposed DOS’s are calculated by the tetrahedron method [22]. Here, we use the eigenvalues and I-components of the eigenstates at 89 k points in the IBZ for YM, and at 50 k points in the IBZ for Y,M,Al. The DOS’s are computed at an energy interval of 2 mRy and will be shown in the next section.
3. Method of calculation
4. Calculated results
In the present calculation, the local density approximation [19] and the muffin-tin (MT) approximation are used to determine the charge densities self-consistently. The MT radii of the constituent atoms are taken as the maximum val-
The calculated results of the total DOS and I-decomposed local DOS for YCq, Y,Co, Al, YFe, and Y,Fe,Al are shown in figs. 3(a)-(d), respectively. In each figure, the upper curve denotes the total DOS D(E). The dashed-and-dotted, broken
M. Aoki, H. Yamada / Electronic structure of Y(M,AI),
380
CM = Co.Fe)
4
0.5
0.4
0.6
0.7
E (Ryd)
:
0.5
0.4
E (Ryd) Fig. 3. Calculated
total DOS D(E) atoms for (a) YCo,;
in states/Ry (b) Y,Co,Al;
total
0.2
0.3
(d)
oEf
0.6
0.4
0.7
E (Ry:; atom of the constituent unit cell, I-decomposed local DOS’s D, MT(E) in states/Ry (c) YFe, and (d) Y,Fe,AI, respectively. E, denotes the Fermi level.
M. Aoki, H. Yamada / Electronic structure of Y(M,AI), Table
381
1
Calculated O,“‘(
(M = Co,Fe)
results of the total DOS
E,) (states/Ry
at Fermi Ieve! D(E,)
atom) and the charge content D(G)
MT
D,“‘( S
66.4
YCO, Y&o,
YFe, Y,Fe,Al
Al
62.9
273.5 263.8
(states/Ry
unit cell),
/-decomposed
in each MT sphere Q, MT for YM, E, )
local DOS
and Y,M,AI
for the constituent
atoms
(M = Co and Fe)
Q,“’ P
d
S
P
d
Y
0.1
0.2
1.7
0.314
0.318
1.084
co
0.1
0.8
13.6
0.512
0.406
7.397
Y
0.3
0.6
2.9
0.338
0.338
1.104
co
0.2
1.2
14.0
0.524
0.447
7.449
Al
0.4
0.9
0.6
0.820
0.905
0.139
Y
0.1
0.2
2.8
0.335
0.345
1.078
Fe
0.1
0.7
62.9
0.510
0.396
6,374
Y
0.3
0.6
3.7
0.350
0.349
1.088
Fe
0.2
1.3
77.3
0.507
0.436
6.451
Al
0.4
2.1
1.1
0.830
0.934
0.132
and dotted curves in each figure denote the d, p and s components of the local DOS Df”lT(E) in each MT sphere of the constituent atoms, respectively. The values of D( E,), Df”‘( E,) and charge content in the MT sphere Q,“’ of each atom are listed in table 1. For YCo,, the calculated DOS shown in fig. 3(a) is characterised by two sharp peaks below E,; first sharp peak exists just below E, and second sharp peak with some fine structures is located very closely to the first one. From the Z-decomposed local DOS shown in fig. 3(a), the states in energy range of about 0.3-0.6 Ry are found to be constructed mainly of d-components of Co atom. The present calculated result of the DOS is very similar to that obtained in the TB approximation [7] and in the LMTO method [9]. The origin of such characteristic DOS features can be seen from the energy dispersion curves along the P-L-W line. Around the L point a very flat band exsists just below E,. The first sharp peak of the DOS comes from this particular d-band of Co electrons. Furthermore, below that band, a couple of flat bands also exist along the P-L-W line. The second peak, which has some fine structures, comes from these flat d-bands. From the wave functions at the L point calculated in the TB approximation, we can see that those flat bands are mainly formed by the intraplanar linear combination of the 3dorbitals in the “KagomC” lattice shown in fig. l(b) in the (111) Co,-plane.
On the other hand, the calculated DOS for Y,Co,Al is somewhat different from that for YCq. One sharp peak of the DOS still remains below E,, which is composed mainly of the d-components of Co atoms, as shown in fig. 3(b) This peak is slightly small in width and in height, compared with the second sharp peak of the DOS for YCo,. Another broad peak appears near above E,. The p-component of Al electrons appears throughout the considerably wide energy range of larger than 0.5 Ry around E,, reflecting the free-electron nature. Then, all the dispersive bands running near E, have more or less the character of the admixture between the d-component of the Co atom and the p-component of the Al atom. And the sharp peaks of the DOS become broad by the substitution of Al atoms. Nevertheless, one sharp peak of the DOS remains still just below E, for Y,Co,Al. This is becuase the “KagomC” lattice of only Co atoms on the (111) Co,-plane is not destroyed by the substitution of Al atoms in this ordered structure. Then the intraplanar linear combination of the 3d-orbitals in this plane is formed in the same way as that for YCo,. However, in the (111) M,-plane both Co and Al atoms are distributed and then such an intraplanar linear combination of the 3d-orbitals cannot be formed. This means that some d-bands have weak dispersion in the [lli] direction of the k-space but in the [ill] direction they become dispersive. As the APW method is
382
M. Aoki, H. Yamada
/ Electronic structure of Y(M,AI),
made use of in these calculations, the connection between the energy bands for YCo, and for Y,Co,Al is not so clear. But it can be considered that the energy band, which gives the first sharp peak of the DOS below E, for YCo,, becomes very dispersive but the energy bands giving the second sharp peak are not so dispersive. Then, only the second sharp peak of the DOS is clearly seen in fig. 3(b). The first peak becomes broad and lies above E,. Since the large amount of the d-states of the Co atoms appears in the lower energy region due to the p-d hybridisation, E, shifts toward the lower energy side. Then the first broad peak goes beyond Et. This fact is very important to explain the weakly ferromagnetic state observed in Y(Co,Al),. A similar broad peak to that for Y,Co,Al exists above E, for YCo, as shown in fig. 3(a). However, this broad peak for YCo, does not come from the relatively flat dbands around the L point but comes from the d-bands over the whole Brillouin zone. These dbands are hybridised strongly with the p-states of Al atoms in Y&o, Al and cannot form any distinct peaks as shown in fig. 3(b). The shapes of the DOS for YCo, and YFe, are very similar to each other as shown in figs. 3(a) and (c). Such a similarity is also found in the DOS’s for Y,Co,Al and Y,Fe,Al shown in figs. 3(b) and (d). Klein et al. [20] reported a similar behaviour in the calculated band structures for ZrFe, and ZrCo, with the ClS-type Laves,phase structure. For YFe,, E, lies at the right slope of the second peak of the DOS. For Y,Fe,Al, on the other hand, E, lies at the left slope of the peak, that is, E, shifts toward the lower energy side, compared with that for YFe,, in the same way as Y,Co,Al. The sharp peak in the total DOS for Y,Fe,Al is slightly small in width and in height, compared with the second sharp peak for YFe,. From table 1, we can see that the charge contents of d-electrons within the MT spheres of Co and Fe atoms become a little large when the Al atoms are substituted. The difference between the values of Q,” for YM, and Y,M, Al is partially due to the actual change in the number of 3d electrons on Co or Fe atoms and partially due to the lattice expansion by the substitution of Al atoms. As there exist a few electrons with the
(M = Co,Fe)
d-character outside the MT sphere, then the value of Q,” becomes large when the lattice and then the volume of the MT expands. So, the number of d-electrons themselves on Co and Fe atoms may not be changed so much by the substitution of Al atoms.
5. Conclusions
and discussion
We have carried out the self-consistent APW calculation of the electronic structures for the binary compound YM, (M = Co and Fe) and the ordered ternary compound Y,M,Al with the ClS-type Laves phase structure in the non-magnetic state. The calculated electronic structures of YCo, and YFe, have been found to be very similar with each other and also with the previous results in the TB approximation [7] and in the LMTO method [9]. The calculated band structures of YM, are also similar to those of ZrM, [20], particularly in the energy range near the double peak of the DOS. However, the position of E, for YM, is a little lower than that of ZrM,. Furthermore, the electronic structures of Y,M, Al for M = Co and Fe have also been found to be very similar to each other. E, of Y,Co,Al lies near a dip of the DOS. The value of D( E,) for this ordered ternary compound is smaller than that for YCo, as shown in table 1. Then this compound will not become magnetic. On the other hand, E, for Y,Fe,Al is still on a sharp peak of the DOS. The value of D(E,) is very large as shown in table 1 and this ordered ternary compound will become ferromagnetic. The magnitude of the magnetic moment is estimated roughly as 0.5~~ per Fe atom from the area of the DOS between the positions of E, and a dip of the DOS just below E,. Bulk magnetic measurements show that the disordered Y(Fe,_,Al,), loses its moment at x > 0.22 [15]. However, the Mossbauer measurements show that Fe atoms in the pseudobinary compound at x = 0.25 have still magnetic moments. The strength of the hyperfine field on the Fe atom for Y(Fe,,,,Al,,,,), is about a half of that for YFe, [13]. This is consistent with our calculated result. The disappearance of the bulk moment at x > 0.22 will be attributed to the mic-
M. Aoki, H. Yamada / Electronic structure of Y(M,Al),
tomagnetic or spin glass behaviour as pointed out in refs. [13-161. The hybridisation between 3d states of Co and 3p state of Al has been shown to be so strong that the characteristic shapes of the DOS for YCo, and YFe,, i.e., double sharp peak of the DOS near E,, are destroyed by the substitution of Al atoms. However, a single sharp peak and a small broad peak have been shown to remain near E, for Y,M, Al. Moreover, E, shifts toward the lower energy side and then the small broad peak of the DOS goes beyond E, for Y,Co,Al. At a certain value of x between 0 and 0.25, E, goes across this broad peak and then the weakly ferromagnetic state will be stabilised as observed for 0.13 < x < 0.19 [lo]. In the present calculation, the effect on the DOS of the lattice expansion due to the substitution of Al atoms has been taken into account as we have used the observed lattice constants. The width of the 3d band is certainly narrowed by the lattice expansion but this effect has been shown not to play an essential role in the onset of the magnetic moment for Y(Co,Al),. In the present calculation the ordered structure shown in fig. 1 has been assumed for the ternary Y,M,Al compounds and then the alloying effects have been neglected. Therefore, we have to extend the calculations to the case of actual pseudobinary compounds, by taking into account the local environment effect and other alloying effects [23]. Nevertheless, we have been able to explain qualitatively the two observed results of the appearance of the magnetic moment in Y(Co,Al), and the decrease of the local moment in Y(Fe,Al),, by making use of the calculated results of the electronic structures.
Acknowledgements The present authors would like to thank Professors A. Yanase and S. Asano for giving valuable comments on the APW calculations. The computer program used in the present study was originally made by Professor A. Yanase. One of the present authors (HY) is also grateful to Professors
(M = Co,Fe)
383
M. Shimizu and W. Steiner for informative discussion on the present problem. The numerical calculations were carried out with the FACOM M-780 system in Computer Center of Nagoya University.
References ill R. Lemaire, Cobalt 33 (1966) 201. VI E. Burzo, Int. J. Magn. 3 (1972) 161. M. Shimizu and J. Voiron, J. [31 D. Bloch, D.M. Edwards, Phys. F 5 (1975) 1217. [41 C.J. Schinkel, J. Phys. F 8 (1978) L87. 151 H. Yamada, Physica B 149 (1988) 390. Fl M. Cyrot and M. Lavagna, J. de Phys. 40 (1979) 763. [71 H. Yamada, J. Inoue, K. Terao, S. Kanda and M. Shimizu, J. Phys. F 14 (1984) 1943; and references cited in ref. [5]. PI K. Schwarz and P. Mohn, J. Phys. F 14 (1984) L129. P. Mohn and K. Schwarz, Physica B 130 (1985) 26. [91 S. Asano and S. Ishida, J. Magn. Magn. Mat. 70 (1987) 39; J. Phys. F 18 (1988) 501. and Y. Nakamura, Solid State Commun. 56 UOI K. Yoshimura (1985) 767. M. Takigawa, Y. Takahashi, H. Yasuoka 1111 K. Yoshimura, and Y. Nakamura, J. Phys. Sot. Jpn. 56 (1987) 1138. K. Yoshimura, M. Takigawa, Y. Takahashi, H. Yasuoka, M. Mekata and Y. Nakamura, J. Magn. Magn. Mat. 70 (1987) 11. K. Yoshimura, M. Mekata, M. Takigawa, Y. Takahashi and H. Yasuoka, Phys. Rev. B 37 (1988) 3593. Metals 40 (1975) 361. WI K.H.J. Buschow, J. Less-Common Phys. Stat. Sol. P31 Y. Muraoka, M. Shiga and Y. Nakamura, (a) 42 (1977) 369. P41 M.J. Besnus, P. Bauer and J.M. G&in, J. Phys. F 8 (1978) 191. [I51 W. Steiner, J. Magn. Magn. Mat. 14 (1979) 47. M. Reissner, W. Steiner, J.P. Kappler, P. Bauer and M.J. Besnus, J. Phys. F 14 (1984) 1249. V. Sechovsky and P. Nazar, J. [161 G. Hilscher, R. G&singer, Phys. F 12 (1982) 1209. and Critical Phenomena, 1171 I. Syoji, in: Phase Transitions vol. 1, eds. C. Domb and M.S. Green (Academic Press, London, 1972) p. 269. B. Johansson and M.S.S. Brooks, to be WI 0. Eriksson, published in J. de Phys. Colloque. and B.I. Lundqvist, Phys. Rev. B 13 1191 0. Gunnarsson (1976) 4274. and PO1 B.M. Klein, W.E. Pickett, D.A. Papaconstantpoulos L.L. Boyer, Phys. Rev. B 27 (1983) 6721. WI A. Yanase, J. Phys. F 16 (1986) 1501. and T. Taut, Phys. Stat. Sol. (b) 54 (1972) [221 G. Lehmann 469. to be published v31 H. Akai, P.H. Dederichs and J. Kanamori, in J. de Phys. Colloque.
Materials
78 (1989) 377-383
377
ELECTRONIC STRUCTURE OF THE ORDERED TERNARY COMPOUND (M = Co AND Fe) WITH C&TYPE LAVES PHASE STRUCTURE Masato
AOKI
and Hideji
Y(M,,,AIO,LS)Z
YAMADA
Department of Physics, Gifu University, Yanagido, Gifu 501-I I, Japan
Received
20 October
1988
The electronic structures of the binary compound YM, and the ordered ternary compound Y(M,,,,Al,,,), (M = Co and Fe) with the Cl5type Laves phase structure are calculated by the self-consistent APW method. It is shown that the calculated density-of-states (DOS) curves for YM, are very similar to those obtained previously, being characterised by the two sharp peaks mainly of 3d-states of M atoms near the Fermi level. For ordered Y(Mo,,sAlo,zs)z, some 3d-states of the M atom hybridise strongly with 3p-states of the Al atom and only a single sharp peak is shown to appear in the DOS near the Fermi level. Moreover, the position of the Fermi level is shown to shift toward the lower energy side by the substitution of Al atoms. The anomalous magnetic properties observed in the pseudobinary compound Y(M,Al), are discussed on the basis of these calculated electronic structures.
1. Introduction Intermetallic compounds of yttrium and the 3d-transition metal M (= Mn, Fe, Co or Ni), with the cubic MgCu,-type (Cl%type) YM,, Laves phase structure are known to exhibit many interesting magnetic properties. Particularly, YCq shows curious magnetic properties; a maximum in the temperature dependence of the susceptibility [1,2] and anomalous field dependence of the induced magnetic moment [3,4] were observed. Recently, these magnetic properties of YM, have been discussed theoretically from the point of view of the electronic structures [5]. The calculation of the density-of-states (DOS) curve in these compounds was first carried out in the moment method [6]. Subsequently, many calculations have been done in the tight-binding (TB) approximation [7], the augmented spherical wave (ASW) methods [8], the linearised muffin-tin orbital (LMTO) and Korringa-Kohn-Rostoker (KKR) methods [9]. The shapes of the DOS calculated in the various methods are very similar to each other and the characteristic of the DOS for YM, is the existence of two sharp peaks 0304-8853/89/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
mainly of 3d-states of M atoms near the Fermi level E,. The positions of E, for YCo, and YFe, in the non-magnetic state are a little beyond the sharp peak of the DOS in the higher energy side and just on the sharp peak in the lower energy side, respectively. Then, YCo, is the strongly exchange-enhanced paramagnet and YFe, is the ferromagnet, as observed. For YCo, the sharp peak of the DOS in the higher energy side, which is very close to E,, has been shown to be responsible for both observed results of the anomalous temperature and magnetic field dependences of the susceptibility [7]. In a narrow concentration range 0.13 < x < 0.19 for the pseudobinary compounds Y(Co,_,Al,),, a weakly ferromagnetic state has been observed [lo]. The spontaneous magnetisation at low temperature is very small (less than 0.14~~ per Co atom) and the paramagnetic susceptibility obeys the Curie-Weiss law at high temperature, which is a typical magnetic behaviour of a weak itinerant electron ferromagnet [ll]. It is said that the onset of the weakly ferromagnetic moment in this system may be ascribed to a large atomic volume of Al atoms; the narrowed 3d band associated with B.V.
378
M. Aoki, H. Yamada
/ Electronic structure of Y(M,AI),
the lattice expansion by the substitution of Al atoms promotes a tendency to the onset of magnetic moment. On the other hand, in Y(Fe,_,Al,), the concentration dependence of the magnetic moment at low temperature shows the opposite trend to that in the Y(Co,Al), system. The lattice constant increases with increasing x, like in Y(Co,Al),. However, the concentration dependence of the magnetic moment shows a rapid decrease with increasing x [12-161, which cannot be explained by the narrowed 3d band associated with the lattice expansion. Besnus et al. [14] have pointed out that the magnetic moment on the Fe atom is strongly affected by the surrounding atoms, that is, the effect of the local environment is important in this system. It is not clear why the magnetic properties in both systems of Y(Co,Al), and Y(Fe,Al), are so different from each other. The aim of the present work is to study these magnetic properties on the basis of the electronic structures. In section 2, the crystal structure of the ordered ternary compound Y(M,,,,Al,,,,), (M = Co and Fe), for which the electronic structure is calculated in this paper, is shown. In section 3, the details of the band calculation for the present system are given and the calculated results are shown in section 4. In section 5, the appearance of the magnetic moment in a narrow concentration range of Al in Y(Co,AI), and the rapid decrease of the magnetic moment of Y(Fe,Al), with increasing the
(M = Co, Fe)
Al concentration are discussed calculated electronic structures.
2. Crystal structure A projected crystal structure of YM, in the Cl5type Laves phase on the x-y plane is shown in fig. l(a). Large circles represent Y atoms and both small and dotted circles represent M atoms. The number in circles denotes the z component of the atomic position in units of lattice constant a. A unit cell contains two Y atoms and four M atoms. The translational symmetry of this structure is the same as that of the fee structure and the crystal structure can be described as a stacking of the atomic planes (111). These atomic planes are classified into three types of Y,-plane, M,-plane and M,-plane, where the subscript stands for the number of atoms on the plane contained in a unit cell. It is noted that on the M,-plane M atoms form a special two-dimensional lattice, the socalled “KagomC” lattice [17] as shown in fig. l(b). For the ordered ternary compound Y,M, Al in the Cl5type Laves phase, the crystal structure can be constructed from that of YM, by replacing one of the four equivalent M atoms in a unit cell with an Al atom. The Al atoms are indicated by dotted circles in fig. 1. Such a crystal structure has recently been used in the calculation of the magnetic moments in the ordered ternary compound
R
(a)
Fig. 1. (a) Projected crystal structure of YM,
on the basis of the
in the ClS-type Laves phase structure on x-y formed by M atoms on M,-plane.
(b)
plane; (b) so-called “KagomB” lattice
M. Aoki, H. Yamada / Electronic structure of Y(M,AI),
Fig. 2. The first Brillouin zone for the ordered pound with the ClS-type Laves phase structure. represents the l/12 IBZ.
Y,M,Al comA shaded part
Y(Fe,Co), [18]. It is noted that the atomic plane (111) of this structure does not consist only of a single kind of Y or M atom. However, any atomic plane (lli) consists of a single kind of atom, i.e., Y,-plane, M,-plane or Al,-plane. Then, the “Kagomt” lattice of M atoms is formed only on the (117) plane. Since the point group operations which move the Al atom site to the M atom site in fig. l(a) have to be excluded, the space group is restricted to a sub-group for the ClS-type Laves phase structure and the number of the operations of the point group is reduced to 12 from 48. As shown in fig. 2, the irreducible Brillouin zone (IBZ) of this structure then becomes l/12 of the full Brillouin zone; the volume of the IBZ becomes four times larger than that of YM,. The high symmetry points in the IBZ are I’(OOO), X(200), L(lli), L’(lll) and W(210) and the symmetry lines are R(qq?/) with 0 < n < 1 and ~(~OTJ) with 0 < n < 3/2, where the unit is taken as a/a.
(M = Co,Fe)
319
ues of non-overlapping spheres, i.e., (0/8)a for Y and (a/8) a for M and Al [20]. The core electrons are treated in the so-called frozen-core approximation [21]; the Kr core without 4p6 for Y, the Ar core for Co and for Fe and the Ne core for Al are made use of. The maximum angular momentum, I,, , is taken to be 7. The number of APW’s is about 260, that is, ] k + G ] 5 6.3 (2n/a), where k is a wave vector in the first Brillouin zone and G is an fee reciprocal lattice vector. The six symmetry points I, X, L, K, W, U and each midpoint of six symmetry lines A, A, Z, S, Z, Q are taken as the sampling points for the iterative APW calculation for YM, and the nine uniformly distributed k points in the IBZ are taken for the calculation of Y,M,Al. The iterations are carried out until a following practical condition of the self-consistency for the charge density is satisfied in each MT sphere,
/ MT
d3rI Pin(r)- lout
I<
10e2,
(1)
where pi,(r) and p,,,(r) are the input and output charge densities. The eigenvalues have been converged to within the accuracy of less than mRy. The values of the lattice constant for YCo,, YFe, and Y,Fe,Al are taken from the experimental data, a = 7.217 A [l], 7.362 A [14] and 7.481 A [14], respettively. For Y,Co,Al we have estimated u = 7.380 A by the extrapolation of experimental data for Y(Co,_,Al,), [lo] to x = 0.25. The total and Z-decomposed DOS’s are calculated by the tetrahedron method [22]. Here, we use the eigenvalues and I-components of the eigenstates at 89 k points in the IBZ for YM, and at 50 k points in the IBZ for Y,M,Al. The DOS’s are computed at an energy interval of 2 mRy and will be shown in the next section.
3. Method of calculation
4. Calculated results
In the present calculation, the local density approximation [19] and the muffin-tin (MT) approximation are used to determine the charge densities self-consistently. The MT radii of the constituent atoms are taken as the maximum val-
The calculated results of the total DOS and I-decomposed local DOS for YCq, Y,Co, Al, YFe, and Y,Fe,Al are shown in figs. 3(a)-(d), respectively. In each figure, the upper curve denotes the total DOS D(E). The dashed-and-dotted, broken
M. Aoki, H. Yamada / Electronic structure of Y(M,AI),
380
CM = Co.Fe)
4
0.5
0.4
0.6
0.7
E (Ryd)
:
0.5
0.4
E (Ryd) Fig. 3. Calculated
total DOS D(E) atoms for (a) YCo,;
in states/Ry (b) Y,Co,Al;
total
0.2
0.3
(d)
oEf
0.6
0.4
0.7
E (Ry:; atom of the constituent unit cell, I-decomposed local DOS’s D, MT(E) in states/Ry (c) YFe, and (d) Y,Fe,AI, respectively. E, denotes the Fermi level.
M. Aoki, H. Yamada / Electronic structure of Y(M,AI), Table
381
1
Calculated O,“‘(
(M = Co,Fe)
results of the total DOS
E,) (states/Ry
at Fermi Ieve! D(E,)
atom) and the charge content D(G)
MT
D,“‘( S
66.4
YCO, Y&o,
YFe, Y,Fe,Al
Al
62.9
273.5 263.8
(states/Ry
unit cell),
/-decomposed
in each MT sphere Q, MT for YM, E, )
local DOS
and Y,M,AI
for the constituent
atoms
(M = Co and Fe)
Q,“’ P
d
S
P
d
Y
0.1
0.2
1.7
0.314
0.318
1.084
co
0.1
0.8
13.6
0.512
0.406
7.397
Y
0.3
0.6
2.9
0.338
0.338
1.104
co
0.2
1.2
14.0
0.524
0.447
7.449
Al
0.4
0.9
0.6
0.820
0.905
0.139
Y
0.1
0.2
2.8
0.335
0.345
1.078
Fe
0.1
0.7
62.9
0.510
0.396
6,374
Y
0.3
0.6
3.7
0.350
0.349
1.088
Fe
0.2
1.3
77.3
0.507
0.436
6.451
Al
0.4
2.1
1.1
0.830
0.934
0.132
and dotted curves in each figure denote the d, p and s components of the local DOS Df”lT(E) in each MT sphere of the constituent atoms, respectively. The values of D( E,), Df”‘( E,) and charge content in the MT sphere Q,“’ of each atom are listed in table 1. For YCo,, the calculated DOS shown in fig. 3(a) is characterised by two sharp peaks below E,; first sharp peak exists just below E, and second sharp peak with some fine structures is located very closely to the first one. From the Z-decomposed local DOS shown in fig. 3(a), the states in energy range of about 0.3-0.6 Ry are found to be constructed mainly of d-components of Co atom. The present calculated result of the DOS is very similar to that obtained in the TB approximation [7] and in the LMTO method [9]. The origin of such characteristic DOS features can be seen from the energy dispersion curves along the P-L-W line. Around the L point a very flat band exsists just below E,. The first sharp peak of the DOS comes from this particular d-band of Co electrons. Furthermore, below that band, a couple of flat bands also exist along the P-L-W line. The second peak, which has some fine structures, comes from these flat d-bands. From the wave functions at the L point calculated in the TB approximation, we can see that those flat bands are mainly formed by the intraplanar linear combination of the 3dorbitals in the “KagomC” lattice shown in fig. l(b) in the (111) Co,-plane.
On the other hand, the calculated DOS for Y,Co,Al is somewhat different from that for YCq. One sharp peak of the DOS still remains below E,, which is composed mainly of the d-components of Co atoms, as shown in fig. 3(b) This peak is slightly small in width and in height, compared with the second sharp peak of the DOS for YCo,. Another broad peak appears near above E,. The p-component of Al electrons appears throughout the considerably wide energy range of larger than 0.5 Ry around E,, reflecting the free-electron nature. Then, all the dispersive bands running near E, have more or less the character of the admixture between the d-component of the Co atom and the p-component of the Al atom. And the sharp peaks of the DOS become broad by the substitution of Al atoms. Nevertheless, one sharp peak of the DOS remains still just below E, for Y,Co,Al. This is becuase the “KagomC” lattice of only Co atoms on the (111) Co,-plane is not destroyed by the substitution of Al atoms in this ordered structure. Then the intraplanar linear combination of the 3d-orbitals in this plane is formed in the same way as that for YCo,. However, in the (111) M,-plane both Co and Al atoms are distributed and then such an intraplanar linear combination of the 3d-orbitals cannot be formed. This means that some d-bands have weak dispersion in the [lli] direction of the k-space but in the [ill] direction they become dispersive. As the APW method is
382
M. Aoki, H. Yamada
/ Electronic structure of Y(M,AI),
made use of in these calculations, the connection between the energy bands for YCo, and for Y,Co,Al is not so clear. But it can be considered that the energy band, which gives the first sharp peak of the DOS below E, for YCo,, becomes very dispersive but the energy bands giving the second sharp peak are not so dispersive. Then, only the second sharp peak of the DOS is clearly seen in fig. 3(b). The first peak becomes broad and lies above E,. Since the large amount of the d-states of the Co atoms appears in the lower energy region due to the p-d hybridisation, E, shifts toward the lower energy side. Then the first broad peak goes beyond Et. This fact is very important to explain the weakly ferromagnetic state observed in Y(Co,Al),. A similar broad peak to that for Y,Co,Al exists above E, for YCo, as shown in fig. 3(a). However, this broad peak for YCo, does not come from the relatively flat dbands around the L point but comes from the d-bands over the whole Brillouin zone. These dbands are hybridised strongly with the p-states of Al atoms in Y&o, Al and cannot form any distinct peaks as shown in fig. 3(b). The shapes of the DOS for YCo, and YFe, are very similar to each other as shown in figs. 3(a) and (c). Such a similarity is also found in the DOS’s for Y,Co,Al and Y,Fe,Al shown in figs. 3(b) and (d). Klein et al. [20] reported a similar behaviour in the calculated band structures for ZrFe, and ZrCo, with the ClS-type Laves,phase structure. For YFe,, E, lies at the right slope of the second peak of the DOS. For Y,Fe,Al, on the other hand, E, lies at the left slope of the peak, that is, E, shifts toward the lower energy side, compared with that for YFe,, in the same way as Y,Co,Al. The sharp peak in the total DOS for Y,Fe,Al is slightly small in width and in height, compared with the second sharp peak for YFe,. From table 1, we can see that the charge contents of d-electrons within the MT spheres of Co and Fe atoms become a little large when the Al atoms are substituted. The difference between the values of Q,” for YM, and Y,M, Al is partially due to the actual change in the number of 3d electrons on Co or Fe atoms and partially due to the lattice expansion by the substitution of Al atoms. As there exist a few electrons with the
(M = Co,Fe)
d-character outside the MT sphere, then the value of Q,” becomes large when the lattice and then the volume of the MT expands. So, the number of d-electrons themselves on Co and Fe atoms may not be changed so much by the substitution of Al atoms.
5. Conclusions
and discussion
We have carried out the self-consistent APW calculation of the electronic structures for the binary compound YM, (M = Co and Fe) and the ordered ternary compound Y,M,Al with the ClS-type Laves phase structure in the non-magnetic state. The calculated electronic structures of YCo, and YFe, have been found to be very similar with each other and also with the previous results in the TB approximation [7] and in the LMTO method [9]. The calculated band structures of YM, are also similar to those of ZrM, [20], particularly in the energy range near the double peak of the DOS. However, the position of E, for YM, is a little lower than that of ZrM,. Furthermore, the electronic structures of Y,M, Al for M = Co and Fe have also been found to be very similar to each other. E, of Y,Co,Al lies near a dip of the DOS. The value of D( E,) for this ordered ternary compound is smaller than that for YCo, as shown in table 1. Then this compound will not become magnetic. On the other hand, E, for Y,Fe,Al is still on a sharp peak of the DOS. The value of D(E,) is very large as shown in table 1 and this ordered ternary compound will become ferromagnetic. The magnitude of the magnetic moment is estimated roughly as 0.5~~ per Fe atom from the area of the DOS between the positions of E, and a dip of the DOS just below E,. Bulk magnetic measurements show that the disordered Y(Fe,_,Al,), loses its moment at x > 0.22 [15]. However, the Mossbauer measurements show that Fe atoms in the pseudobinary compound at x = 0.25 have still magnetic moments. The strength of the hyperfine field on the Fe atom for Y(Fe,,,,Al,,,,), is about a half of that for YFe, [13]. This is consistent with our calculated result. The disappearance of the bulk moment at x > 0.22 will be attributed to the mic-
M. Aoki, H. Yamada / Electronic structure of Y(M,Al),
tomagnetic or spin glass behaviour as pointed out in refs. [13-161. The hybridisation between 3d states of Co and 3p state of Al has been shown to be so strong that the characteristic shapes of the DOS for YCo, and YFe,, i.e., double sharp peak of the DOS near E,, are destroyed by the substitution of Al atoms. However, a single sharp peak and a small broad peak have been shown to remain near E, for Y,M, Al. Moreover, E, shifts toward the lower energy side and then the small broad peak of the DOS goes beyond E, for Y,Co,Al. At a certain value of x between 0 and 0.25, E, goes across this broad peak and then the weakly ferromagnetic state will be stabilised as observed for 0.13 < x < 0.19 [lo]. In the present calculation, the effect on the DOS of the lattice expansion due to the substitution of Al atoms has been taken into account as we have used the observed lattice constants. The width of the 3d band is certainly narrowed by the lattice expansion but this effect has been shown not to play an essential role in the onset of the magnetic moment for Y(Co,Al),. In the present calculation the ordered structure shown in fig. 1 has been assumed for the ternary Y,M,Al compounds and then the alloying effects have been neglected. Therefore, we have to extend the calculations to the case of actual pseudobinary compounds, by taking into account the local environment effect and other alloying effects [23]. Nevertheless, we have been able to explain qualitatively the two observed results of the appearance of the magnetic moment in Y(Co,Al), and the decrease of the local moment in Y(Fe,Al),, by making use of the calculated results of the electronic structures.
Acknowledgements The present authors would like to thank Professors A. Yanase and S. Asano for giving valuable comments on the APW calculations. The computer program used in the present study was originally made by Professor A. Yanase. One of the present authors (HY) is also grateful to Professors
(M = Co,Fe)
383
M. Shimizu and W. Steiner for informative discussion on the present problem. The numerical calculations were carried out with the FACOM M-780 system in Computer Center of Nagoya University.
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