- Email: [email protected]
Electronic structure and spontaneous volume magnetostriction of antiferromagnetic YMn2
Volume 104A, number 2
PHYSICS LETTERS
13 August 1984
ELECTRONIC STRUCTURE AND SPONTANEOUS VOLUME MAGNETOSTRICTION OF ANTIFERROMAGNETIC YMn 2
K. TERAO Department of Physics, Shinshu University, Matsumoto 390, Japan and M. SHIMIZU Department of Applied Physics, Nagoya University, Nagoya 464, Japan Received 15 May 1984
The density of states of d-electrons for YMn2 is calculated both in the paramagnetic and antiferromagnetie states by making use of the recursion method. The large and positive value of the observed spontaneous volume magnetostrietion is explained from an increase Of the kinetic energy of d-electrons due to the polarization of Mn atoms.
Recently, Nakamura et al. [ 1] have found that YMn 2 with the cubic Laves structure is anti.ferromagnetic below about 100 K and that the spontaneous volume magnetostriction cos at 0 K is positive and fairly large ("-5%). Gaydukova and Markosyan [2] observed earlier the first.order transition at about 100 K and a small tetragonal distortion in the lowtemperature phase, which is compatible with the sym. metry of the magnetic structure proposed by Nakamura et al. [1]. Cyrot and Lavagna [15] calculatea the density of states (DOS) for YFe2, YCo 2 and YNi 2 with the same structure as that of YMn 2 in the moment method. More recently, Yamada et al. [3] have calculated the DOS of d-electrons for YM 2 (M = Mn, Fe, Co and Ni) with the cubic Laves structure in the tight-binding method. They have calculated the temperature dependence of the paramagnetic spin susceptibility by making use of the calculated DOS and by taking into account the effect of spin fluctuations and have obtained a satisfactory agreement between the calculated and observed results. Terao and Shimizu [4] have shown that the composition dependence of cos for ferromagnetic Y(Fel_xCox) 2 and Zr(Fel_xCox) 2 can be explained on the basis of a simple itinerant electron model by taking into account the volume dependence of the effective interaction between elec0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
trons. The value of cos in YFe 3 has been calculated as ~3% by Moruzzi et al. [16]. Shimizu et al. [5] have calculated the local DOS for the intermetaUic compounds o f Y and Ni using the recursion method with coefficients in 5 X 5 matrices and have successfully explained their magnetic properties from the calculated DOS. The purposes of this paper are to calculate the local DOS on Y and Mn atoms of YMn 2 in the antiferromagnetic and paramagnetic states, to calculate the difference between the energies in the paramagnetic and antiferromagnetic states, and to explain the large and positive observed value of cos on the basis of the Stoner theory. The DOS is calculated in the recursion method [6]. We assume that a magnetic moment on the Mn atom collapses in the paramagnetic state and Y is nonmagnetic. The small distortion observed below T N is neglected in our calculations. The difference AE between the energies in the paramagnetic and antfferromagnetic states and cos, which is calculated from AE, do not depend sensitively on the Free details of DOS as the magnetic moment on Mn (2.7ta B [1]) is large. We tridiagonalize a tight-binding hamfltonian, manipulating 5 X 5 block matrices because of the low symmetry of Mn site in the paramagnetic state and of Y site as well as Mn site in the antiferromagnetic state. 113
Volume 104A, n u m b e r 2
PHYSICS LETTERS
We calculate an approximate DOS of the tight-binding d-band hamiltonian given by
H = ~(~ ~ li, m,s)ti/mnq, m,sl s (i,])m,n + ~ li, m,s)'e~(i, rn, sl), i,m where the indices i and/" refer to sites, [i, m, s) are the o~hogonalized atomic d-orbitals, m and n run from I to 5 denoting xy, yz, zx, x 2 - y2 and 3z 2 _ r 2 symmetries of d-orbitals and s (= +, - ) is the spin index. In the summation with respect to pairs (i, ]) the nearest neighbours of Y - Y , M n - M n and Y - M n pairs are taken into account. We use the same values of ti]mnas given by Yamada et al. [3] in Slater and Koster's scheme [7]. The d-level energy of Y or Mn atom at site i, est, is supposed to be independent of re. We f'tx e si = 0 for Y site both in the paramagnetic and antifer. romagnetic states. In the paramagnetic state e~ = ep at Mn site. In the antiferromagnetic state, Mn sites are classified into two kinds for the direction of the magnetic moment. We represent them by Mn+ and Mn_ and denote as es/ = es+ or % . We define A and ~ for convenience as A = e+_ -- e++= e+ -- e-_ and g -- il (e++ + e * _ ) = ~(e+ ~ - + eL). We construct a new basis set IL, m, s) from the atomic orbitals l i, m, s) using the following recursive formula,
niL, m, s) = ~
[IL - 1, n, s) cSL(n,m) + IL, n, s) a~(n, m)]
n
+ IL + 1, m, s) b]~+l (m), where m and n run from 1 to 5. We take 10, m, s) = Ii0, m, s), where i 0 is the index of the starting site on which the local DOS is calculated and I - 1 , m, s) = 0. IL, m, s) are orthogonal with respect to L, but are not orthogonal with respect to rn in a general symmetry. The coefficient matrices are determined from the equations
~ ( L - 1,n,slL - 1,m,s)c~(m,k) m
= (L, n, slz, k, s) b~(n),
m 114
(L, n, slL,m, s) a~(m, k) = (L, n, slHIL, k, s).
13 August 1984
bSL(m) is the normalization factor of IL, m, s). The re. cursive formula is asymmetric with respect to c~; and b~, that is, an asymmetric vector chain model [5]. (If IL, m, s) are orthogonalized with respect to m through the Schmidt method, b~ is triangular and is the transposed matrix of c~. Then the recursive formula is symmetric [8]. The coefficient matrices obtained by this procedure for the low-symmetry sites are oscillating and ill convergent in the present case.) In a certain symmetry where a~(m, n) is diagonal, c~ (m, n) is also diagonal and equal to b~ (m). The numerical calculations are carried out up to L = 12 using a 5419 atom cluster. The calculated results o f a ~ (m, n), b~(m) and c~(m, n) are well convergent and these coefficient matrices for L > 12 are replaced by diagonal matrices independent of L. The local DOS NA(e) of A (A = Y or Mn) atom at site i 0 is given by
NsA(e) = _ . - 1 X Im ~
(io,m ,s[[e + i O - H ] - l l i 0 , m , s ) .
m
The shape of N (e) depends on
and on :e in the
paramagnetic state and on ~ and A in the antiferromagnetic state. The values of ep and the position of the Fermi level e F in the paramagnetic state are determined as ep = - 1 9 7 . 6 m R y d and eF = - 1 7 1 . 0 mRyd, so that the number of d-electrons n o and n on Y and Mn atoms, which are calculated by integrating the local DOS up to eF, are 1.7 and 6.1 per atom, respectively. The values of ~, A and e F in the antiferromagnetic state are determined as - 2 0 8 . 2 , 147.4 and - 1 6 4 . 5 mRyd, respectively, so that the calculated magnetic moment on a Mn atom is 2.7PB per atom as well as n o = 1.7]atom and n = 6.1]atom. The calculated results of the local DOS are shown in fig. 1. The low- temperature specific heat coefficient is estimated as 13.9 and 13.5 mJ/mol K 2 in the paramagnetic and antfferromagnetic states, respectively. On the estimation of the interaction energy of delectrons on Mn atoms, we adopt a Hubbard type hamiltonian with 5 d-orbitals on each atom. In the H a r t r e e - F o c k approximation, the energy of d-level is given by e s = e 0 + ln_s, where e 0 is the energy without the interaction and I the effective Coulomb integral. 1 1 On Mn atoms, ep = e 0 + ~Ipn, ~ = eo + ~lan and
Volume 104A, number 2
¢O 8O tad I--
CE l---tO
6O
LL
C:) >- 40
PHYSICS LETTERS
! (o) II 111 \"i
80
(b) 60
40
I"M U3
Z ua 2O
0 -0.4
13 August 1984
20
-0.2 ENERGY (RYO}
'
0
.2
-0.4
'P
-0.2
"P
_0+.0
0.2
ENERGY (RYD}
Fig. 1. The local DOS in the unit of states/Ryd.spin.atom for the paramagnetic (a) and antfferromagnetie (b) states of YMn~. The bold curves show the local DOS on Mn and the thin curves on Y. The broken curve is the total DOS]YMn2. The vertical thin line shows the position of the Fermi level and the small arrows show the position of the energy of d-levels of Y and Mn atoms. A = Ia(n+ - n _ ) , wherelp and 1a are the values o f / i n the paramagnetic and antiferromagnetic states. 1a is evaluated from A, and Ip from ep - g --- ~ (lp - Ia) n. The obtained values o f I a and Ip are 54.59 and 58.07 mRyd, respectively, which axe comparable to the value o f 59.8 m R y d obtained by Moruzzi et al. [9] for fcc Mn. The total energy is given by
with
bridization with the orbitals on other atoms as seen from fig. 1. For the main part, the change o f DOS is determined mainly through the change o f ti/mn and LEM is a good approximation. Although LEM may not be satisfactory for the change in the part o f the taft o f DOS which depends sensitively on ep or g as well as ti/mn, a contribution o f this part on the calculated result o f cos is small because o f the small number o f d-electrons occupying the energy levels o f the tail. Therefore, LEM will be a good approximation for the present purpose. Assuming WA- ~ [2 -5/3 [13] a n d I "~ ~2- r , we obtain
eF
e: = f
~ N : ( , ) de
and E: = IAnAnA,
where A = Y, Mn+ or M n and n A are the number o f electrons with spin s on A atom. The difference between the values o f A E in the paramagnetic and anti. ferromagnetic states is evaluated from these formulae as A E = E a - Ep = - 9 8 mRyd/YMn 2. As in the ferromagnetic case [10,11 ], we get the expression cos = -K (aAE/aI2)M for the antiferromagnetic case, where K is the compressibility, I2 the volume and M the sublattice magnetization. The change in the kinetic energy part of A E due to the change in I2 is calculated by making use o f a simple Lang and Ehrenreich model (LEM) [12]. The local DOS on each atom has a main part around each atomic d-level o f that atom and has a tail o f low DOS due to the hy-
where AK A and A E : are the difference between K A
= E A - e : n A arid that between E : in the paramagnetic and antiferromagnetic states, respectively. If we approximate K = 1 X 10 -12 cm2/dyn by referring to the observed values o f K for YFe 2 and YCo 2 [14], we obtain the calculated result o f cos as a function o f P, as shown by the solid line in fig. 2. Now, we consider another case, that is, we assume that the effective Coulomb integral I is independent o f the magnetic state so that n O and n may depend on the magnetic state but the total number n O + 2n is fixed to the same value o f 13.9 as in the paramagnetic state. The calculated results o f DOS in this case are very similar to the previous ones and axe not shown 115
Volume 104A, number 2
PHYSICS LETTERS
References
10
0
0.5
C'
1 0 ~ 5
-5 Fig. 2. The calculated results of the spontaneous volume magnetostriction tos. The solid line shows the result for the case where the values of no and n in the antfferromagnetic state are assumed to be the same as those in the paramagnetie state. The broken line shows the result for the case where the values of I Y and iMn in the antiferromagnetic state are assumed to be the same as those in the paramagnetic state.
here. The resuR on w s is shown in fig. 2 b y the broken line. The value of AE is --34 mRyd and is considerably reduced compared to the previous one. In the antiferromagnetic state, the following numerical values, e F = - 1 5 6 . 9 m R y d , n 0 = 1.782,n = 6.059, = - 2 0 0 . 9 mRyd, A = 145.4 m R y d a n d I Mn = 538.5 m R y d are obtained. The atomic d-level of Y is slightly raised b y 2.00 m R y d , which is estimated from the value o f I Y = 488 m R y d obtained by Moruzzi et al. [9] for fcc Y, due to the increase of n 0. As seen from fig. 2, ff the volume dependence of I is neglected cos becomes about 10%. In conclusion, the experimental value o f 5% [ 1,2] can be explained if we take F "" 0.5 and this value of F is comparable with the values estimated for the Laves structure compounds before [ 11 ].
116
13 August 1984
[ 1 ] Y. Nakamura, M. Shiga and S. Kawano, Physica 120B (1983) 212; M. Shiga, H. Wada and Y. Nakamura, J. Magn. Magn. Mater. 31-34 (1983) 119. [2] I.Yu. Gaydukova and A.S. Markosyan, Phys. Met. Metall. 54 (1982) 168. [3] H. Yamada, J. Inoue, K. Terao, S. Kanda and M. Shimizu, J. Phys. F, to be published. [4] K. Terao and M. Shimizu, Phys. Lett. 95A (1983) 111. [5] M. Shimizu, J. Inoue and H. Nagasawa, J. Phys. F, to be published. [6] R. Haydock, in: Solid state physics, Vol. 30, eds. H. Ehtenreich, F. Seitz and D. Turnbull (Academic Press, New York, 1980) p. 215. [7) J.C. Siater and G.F. Koster, Phys. Rev. 94 (1954) 1498. [8] G. Cubiotti and B. Ginatempo, J. Phys. F8 (1978) 601. [9] V.L. Motuzzi, J.F. Janak and A.R. Williams,Calculated electron properties of metals (Pergamon, New York, 1978). [10] K. Terao and A. Katsuki, J. Phys. Soc. Japan 37 (1974) 828. [11 ] M. Shimizu, Rep. Prog. Phys. 44 (1981) 329. [12] N.D. Lang and H. Ehrenreich, Phys. Rev. 168 (1968) 605. [13] V. Heine, Phys. Rev. 153 (1967) 673. [14] H. Klimker, M. Rosen, M.Y. Dariel and U. Atzmony, Phys. Rev. B10 (1974) 2968; M. Brouha, K.H.L Buschow and A.R. Miedeina, IEEE Trans. Magn. MAG-10 (1974) 182. [15] M. Cyrot and M. Lavagna, J. Phys. (Paris) 40 (1979) 763. [16] V.L. Mo~uzzi, A.R. Williams, A.P. Malozemoff and R.]. Gambino, Phys. Rev. B28 (1983) 5511.
PHYSICS LETTERS
13 August 1984
ELECTRONIC STRUCTURE AND SPONTANEOUS VOLUME MAGNETOSTRICTION OF ANTIFERROMAGNETIC YMn 2
K. TERAO Department of Physics, Shinshu University, Matsumoto 390, Japan and M. SHIMIZU Department of Applied Physics, Nagoya University, Nagoya 464, Japan Received 15 May 1984
The density of states of d-electrons for YMn2 is calculated both in the paramagnetic and antiferromagnetie states by making use of the recursion method. The large and positive value of the observed spontaneous volume magnetostrietion is explained from an increase Of the kinetic energy of d-electrons due to the polarization of Mn atoms.
Recently, Nakamura et al. [ 1] have found that YMn 2 with the cubic Laves structure is anti.ferromagnetic below about 100 K and that the spontaneous volume magnetostriction cos at 0 K is positive and fairly large ("-5%). Gaydukova and Markosyan [2] observed earlier the first.order transition at about 100 K and a small tetragonal distortion in the lowtemperature phase, which is compatible with the sym. metry of the magnetic structure proposed by Nakamura et al. [1]. Cyrot and Lavagna [15] calculatea the density of states (DOS) for YFe2, YCo 2 and YNi 2 with the same structure as that of YMn 2 in the moment method. More recently, Yamada et al. [3] have calculated the DOS of d-electrons for YM 2 (M = Mn, Fe, Co and Ni) with the cubic Laves structure in the tight-binding method. They have calculated the temperature dependence of the paramagnetic spin susceptibility by making use of the calculated DOS and by taking into account the effect of spin fluctuations and have obtained a satisfactory agreement between the calculated and observed results. Terao and Shimizu [4] have shown that the composition dependence of cos for ferromagnetic Y(Fel_xCox) 2 and Zr(Fel_xCox) 2 can be explained on the basis of a simple itinerant electron model by taking into account the volume dependence of the effective interaction between elec0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
trons. The value of cos in YFe 3 has been calculated as ~3% by Moruzzi et al. [16]. Shimizu et al. [5] have calculated the local DOS for the intermetaUic compounds o f Y and Ni using the recursion method with coefficients in 5 X 5 matrices and have successfully explained their magnetic properties from the calculated DOS. The purposes of this paper are to calculate the local DOS on Y and Mn atoms of YMn 2 in the antiferromagnetic and paramagnetic states, to calculate the difference between the energies in the paramagnetic and antiferromagnetic states, and to explain the large and positive observed value of cos on the basis of the Stoner theory. The DOS is calculated in the recursion method [6]. We assume that a magnetic moment on the Mn atom collapses in the paramagnetic state and Y is nonmagnetic. The small distortion observed below T N is neglected in our calculations. The difference AE between the energies in the paramagnetic and antfferromagnetic states and cos, which is calculated from AE, do not depend sensitively on the Free details of DOS as the magnetic moment on Mn (2.7ta B [1]) is large. We tridiagonalize a tight-binding hamfltonian, manipulating 5 X 5 block matrices because of the low symmetry of Mn site in the paramagnetic state and of Y site as well as Mn site in the antiferromagnetic state. 113
Volume 104A, n u m b e r 2
PHYSICS LETTERS
We calculate an approximate DOS of the tight-binding d-band hamiltonian given by
H = ~(~ ~ li, m,s)ti/mnq, m,sl s (i,])m,n + ~ li, m,s)'e~(i, rn, sl), i,m where the indices i and/" refer to sites, [i, m, s) are the o~hogonalized atomic d-orbitals, m and n run from I to 5 denoting xy, yz, zx, x 2 - y2 and 3z 2 _ r 2 symmetries of d-orbitals and s (= +, - ) is the spin index. In the summation with respect to pairs (i, ]) the nearest neighbours of Y - Y , M n - M n and Y - M n pairs are taken into account. We use the same values of ti]mnas given by Yamada et al. [3] in Slater and Koster's scheme [7]. The d-level energy of Y or Mn atom at site i, est, is supposed to be independent of re. We f'tx e si = 0 for Y site both in the paramagnetic and antifer. romagnetic states. In the paramagnetic state e~ = ep at Mn site. In the antiferromagnetic state, Mn sites are classified into two kinds for the direction of the magnetic moment. We represent them by Mn+ and Mn_ and denote as es/ = es+ or % . We define A and ~ for convenience as A = e+_ -- e++= e+ -- e-_ and g -- il (e++ + e * _ ) = ~(e+ ~ - + eL). We construct a new basis set IL, m, s) from the atomic orbitals l i, m, s) using the following recursive formula,
niL, m, s) = ~
[IL - 1, n, s) cSL(n,m) + IL, n, s) a~(n, m)]
n
+ IL + 1, m, s) b]~+l (m), where m and n run from 1 to 5. We take 10, m, s) = Ii0, m, s), where i 0 is the index of the starting site on which the local DOS is calculated and I - 1 , m, s) = 0. IL, m, s) are orthogonal with respect to L, but are not orthogonal with respect to rn in a general symmetry. The coefficient matrices are determined from the equations
~ ( L - 1,n,slL - 1,m,s)c~(m,k) m
= (L, n, slz, k, s) b~(n),
m 114
(L, n, slL,m, s) a~(m, k) = (L, n, slHIL, k, s).
13 August 1984
bSL(m) is the normalization factor of IL, m, s). The re. cursive formula is asymmetric with respect to c~; and b~, that is, an asymmetric vector chain model [5]. (If IL, m, s) are orthogonalized with respect to m through the Schmidt method, b~ is triangular and is the transposed matrix of c~. Then the recursive formula is symmetric [8]. The coefficient matrices obtained by this procedure for the low-symmetry sites are oscillating and ill convergent in the present case.) In a certain symmetry where a~(m, n) is diagonal, c~ (m, n) is also diagonal and equal to b~ (m). The numerical calculations are carried out up to L = 12 using a 5419 atom cluster. The calculated results o f a ~ (m, n), b~(m) and c~(m, n) are well convergent and these coefficient matrices for L > 12 are replaced by diagonal matrices independent of L. The local DOS NA(e) of A (A = Y or Mn) atom at site i 0 is given by
NsA(e) = _ . - 1 X Im ~
(io,m ,s[[e + i O - H ] - l l i 0 , m , s ) .
m
The shape of N (e) depends on
and on :e in the
paramagnetic state and on ~ and A in the antiferromagnetic state. The values of ep and the position of the Fermi level e F in the paramagnetic state are determined as ep = - 1 9 7 . 6 m R y d and eF = - 1 7 1 . 0 mRyd, so that the number of d-electrons n o and n on Y and Mn atoms, which are calculated by integrating the local DOS up to eF, are 1.7 and 6.1 per atom, respectively. The values of ~, A and e F in the antiferromagnetic state are determined as - 2 0 8 . 2 , 147.4 and - 1 6 4 . 5 mRyd, respectively, so that the calculated magnetic moment on a Mn atom is 2.7PB per atom as well as n o = 1.7]atom and n = 6.1]atom. The calculated results of the local DOS are shown in fig. 1. The low- temperature specific heat coefficient is estimated as 13.9 and 13.5 mJ/mol K 2 in the paramagnetic and antfferromagnetic states, respectively. On the estimation of the interaction energy of delectrons on Mn atoms, we adopt a Hubbard type hamiltonian with 5 d-orbitals on each atom. In the H a r t r e e - F o c k approximation, the energy of d-level is given by e s = e 0 + ln_s, where e 0 is the energy without the interaction and I the effective Coulomb integral. 1 1 On Mn atoms, ep = e 0 + ~Ipn, ~ = eo + ~lan and
Volume 104A, number 2
¢O 8O tad I--
CE l---tO
6O
LL
C:) >- 40
PHYSICS LETTERS
! (o) II 111 \"i
80
(b) 60
40
I"M U3
Z ua 2O
0 -0.4
13 August 1984
20
-0.2 ENERGY (RYO}
'
0
.2
-0.4
'P
-0.2
"P
_0+.0
0.2
ENERGY (RYD}
Fig. 1. The local DOS in the unit of states/Ryd.spin.atom for the paramagnetic (a) and antfferromagnetie (b) states of YMn~. The bold curves show the local DOS on Mn and the thin curves on Y. The broken curve is the total DOS]YMn2. The vertical thin line shows the position of the Fermi level and the small arrows show the position of the energy of d-levels of Y and Mn atoms. A = Ia(n+ - n _ ) , wherelp and 1a are the values o f / i n the paramagnetic and antiferromagnetic states. 1a is evaluated from A, and Ip from ep - g --- ~ (lp - Ia) n. The obtained values o f I a and Ip are 54.59 and 58.07 mRyd, respectively, which axe comparable to the value o f 59.8 m R y d obtained by Moruzzi et al. [9] for fcc Mn. The total energy is given by
with
bridization with the orbitals on other atoms as seen from fig. 1. For the main part, the change o f DOS is determined mainly through the change o f ti/mn and LEM is a good approximation. Although LEM may not be satisfactory for the change in the part o f the taft o f DOS which depends sensitively on ep or g as well as ti/mn, a contribution o f this part on the calculated result o f cos is small because o f the small number o f d-electrons occupying the energy levels o f the tail. Therefore, LEM will be a good approximation for the present purpose. Assuming WA- ~ [2 -5/3 [13] a n d I "~ ~2- r , we obtain
eF
e: = f
~ N : ( , ) de
and E: = IAnAnA,
where A = Y, Mn+ or M n and n A are the number o f electrons with spin s on A atom. The difference between the values o f A E in the paramagnetic and anti. ferromagnetic states is evaluated from these formulae as A E = E a - Ep = - 9 8 mRyd/YMn 2. As in the ferromagnetic case [10,11 ], we get the expression cos = -K (aAE/aI2)M for the antiferromagnetic case, where K is the compressibility, I2 the volume and M the sublattice magnetization. The change in the kinetic energy part of A E due to the change in I2 is calculated by making use o f a simple Lang and Ehrenreich model (LEM) [12]. The local DOS on each atom has a main part around each atomic d-level o f that atom and has a tail o f low DOS due to the hy-
where AK A and A E : are the difference between K A
= E A - e : n A arid that between E : in the paramagnetic and antiferromagnetic states, respectively. If we approximate K = 1 X 10 -12 cm2/dyn by referring to the observed values o f K for YFe 2 and YCo 2 [14], we obtain the calculated result o f cos as a function o f P, as shown by the solid line in fig. 2. Now, we consider another case, that is, we assume that the effective Coulomb integral I is independent o f the magnetic state so that n O and n may depend on the magnetic state but the total number n O + 2n is fixed to the same value o f 13.9 as in the paramagnetic state. The calculated results o f DOS in this case are very similar to the previous ones and axe not shown 115
Volume 104A, number 2
PHYSICS LETTERS
References
10
0
0.5
C'
1 0 ~ 5
-5 Fig. 2. The calculated results of the spontaneous volume magnetostriction tos. The solid line shows the result for the case where the values of no and n in the antfferromagnetic state are assumed to be the same as those in the paramagnetie state. The broken line shows the result for the case where the values of I Y and iMn in the antiferromagnetic state are assumed to be the same as those in the paramagnetic state.
here. The resuR on w s is shown in fig. 2 b y the broken line. The value of AE is --34 mRyd and is considerably reduced compared to the previous one. In the antiferromagnetic state, the following numerical values, e F = - 1 5 6 . 9 m R y d , n 0 = 1.782,n = 6.059, = - 2 0 0 . 9 mRyd, A = 145.4 m R y d a n d I Mn = 538.5 m R y d are obtained. The atomic d-level of Y is slightly raised b y 2.00 m R y d , which is estimated from the value o f I Y = 488 m R y d obtained by Moruzzi et al. [9] for fcc Y, due to the increase of n 0. As seen from fig. 2, ff the volume dependence of I is neglected cos becomes about 10%. In conclusion, the experimental value o f 5% [ 1,2] can be explained if we take F "" 0.5 and this value of F is comparable with the values estimated for the Laves structure compounds before [ 11 ].
116
13 August 1984
[ 1 ] Y. Nakamura, M. Shiga and S. Kawano, Physica 120B (1983) 212; M. Shiga, H. Wada and Y. Nakamura, J. Magn. Magn. Mater. 31-34 (1983) 119. [2] I.Yu. Gaydukova and A.S. Markosyan, Phys. Met. Metall. 54 (1982) 168. [3] H. Yamada, J. Inoue, K. Terao, S. Kanda and M. Shimizu, J. Phys. F, to be published. [4] K. Terao and M. Shimizu, Phys. Lett. 95A (1983) 111. [5] M. Shimizu, J. Inoue and H. Nagasawa, J. Phys. F, to be published. [6] R. Haydock, in: Solid state physics, Vol. 30, eds. H. Ehtenreich, F. Seitz and D. Turnbull (Academic Press, New York, 1980) p. 215. [7) J.C. Siater and G.F. Koster, Phys. Rev. 94 (1954) 1498. [8] G. Cubiotti and B. Ginatempo, J. Phys. F8 (1978) 601. [9] V.L. Motuzzi, J.F. Janak and A.R. Williams,Calculated electron properties of metals (Pergamon, New York, 1978). [10] K. Terao and A. Katsuki, J. Phys. Soc. Japan 37 (1974) 828. [11 ] M. Shimizu, Rep. Prog. Phys. 44 (1981) 329. [12] N.D. Lang and H. Ehrenreich, Phys. Rev. 168 (1968) 605. [13] V. Heine, Phys. Rev. 153 (1967) 673. [14] H. Klimker, M. Rosen, M.Y. Dariel and U. Atzmony, Phys. Rev. B10 (1974) 2968; M. Brouha, K.H.L Buschow and A.R. Miedeina, IEEE Trans. Magn. MAG-10 (1974) 182. [15] M. Cyrot and M. Lavagna, J. Phys. (Paris) 40 (1979) 763. [16] V.L. Mo~uzzi, A.R. Williams, A.P. Malozemoff and R.]. Gambino, Phys. Rev. B28 (1983) 5511.