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On the magnetic ordering of the high Tc superconductor GdBa2Cu3O7
Volume 129, number
PHYSICS
2
ON THE MAGNETIC
ORDERING
LETTERS
9 May 1988
A
OF THE HIGH T, SUPERCONDUCTOR
GdBa2Cu30,
Harabhusan SEN Department
of Physics, Burdwan University, Burdwan 713 104, West Bengal, India
Received 8 January 1988; accepted Communicated by D. Bloch
for publication
8 February
1988
The magnetically ordered state of GdBazCu307 at low temperatures has been investigated using the extended Luttinger-Tisza method. It is found that the observed ordering of this compound can be explained by a purely dipole-dipole interaction. An estimate has been made of the internal magnetic field produced due to magnetic ordering of Gd metal ions at a copper site and it is found that the values are within the range of superconducting quenching field and may permit the coexistence of magnetic ordering and superconductivity.
After the discovery [ 1 ] of high T,superconductivity in YBa2Cu30, much insight has been gained over the past few months into the physical properties of these compounds. One remarkable feature of these materials is that the superconductivity transition temperature T,is not influenced [2-41 when other trivalent species, especially heavy rare-earth elements, are substituted for Y3+. In some cases T,is even slightly higher than in the case of YBa,Cu,O,, though most of the rare-earth ions possess their own magnetic moments. Hence these are ideal materials to investigate the interplay between magnetism and superconductivity. Subsequent work [ 5-71 further demonstrated that, in fact, superconductivity and magnetic order coexist in one of these compounds GdBa2Cu307 and this was confirmed by the magnetic and heat capacity measurements and muon spin rotation experiments. The magnetic transition temperature of this compound was found to be N 2.3 K in the superconducting state. Such behavior has also been found in certain RMoS, [ 8 ] and RRh4B4 [ 9 ] compounds (R = rare earth) in which the long range antiferromagnetic order of the localized f-electrons left essentially unperturbed superconductivity associated with the transition metal d-band. Similarly a comparable state exists in GdBa,Cu,O,. However, the nature of the interaction leading to magnetic order remains an interesting question. Another interesting aspect of the problem is the
estimation of the magnetic fields at a copper site as produced by the magnetic ordering of Gd ions. Since 3d electrons of Cu atoms are responsible for superconductivity, it is conceivable that if the magnetic field at Cu sites is not sufficiently large, the superconductivity may not be quenched by the magnetic field due to the ordered Gd sublattice, and the coexistence of superconductivity and magnetic ordering will be allowed. In this Letter we report the results of a study on this compound by the extended Luttinger-Tisza (LT) [ lo] method taking rigorously into account the dipole-dipole interaction (the exchange interaction of the rare-earth ions with the superconducting electrons is sufficiently small because the exchange interaction alone can destroy the superconductivity). If the magnetic unit cell of the ordered state at T= 0 is given, energy calculations can be performed as a function of the spin directions of the unit cell. In the LT method these spin-direction variables are combined into a vector in multidimensonal space. The components of the field at various lattice sites, produced by a certain spin configuration, can be represented by a vector in the same space. In practice, the crystal lattice is supposed to be divided into 8 sublattices (since there is one magnetic ion per unit crystallographic cell of GdBa2Cu30,) and each sublattice is generated from it by translation
0375-9601/88/S 03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
131
Volume 129, number 2
PHYSICS LETTERS A
( = 21u+ 2mb+ 2nc; a, b, c are unit cell vectors) to the 8 vertices of the magnetic unit cell [ lo]. The
configuration of spins is thus described by 24 numbers (x, y, z components of 8 spins). These can be considered as components of a vector S in a 24-dimensional vector space. Since the magnetic fields depend linearly on the components of the spins, it may be written F= -A*& where A is a 24x24 matrix. The energy of the configuration per magnetic unit cell is - &S.F= jS-A*S, which is a quadratic form in the components of S. This is to be minimized subject to the strong constraint s$ +s$ +.~a,= 1 (t= 1, .... 8, where s,,, s,,, sZtare the components of a representative spin of the tth sublattice), which implies the weak constraint 5 S:=8. I=1
Now the minimum value of @-A-S subject to the weak constraint is the lowest eigenvalue of iA, and a configuration with corresponds to this value of energy is the corresponding eigenvector and can be expressed as q(k)&(a), k= 1, .... 8, (Y=x, y, z. For the present case the q(k) are 8-dimensional [ 121 and determine the nature of the ordering (ferromagnetic or antiferromagnetic). The particular physical space directions along which the spins of the kth sublattice point are directed are given by & ( (Y) . The & ( (Y) are the eigenvectors of a 3X 3 matrix Lk, whose components are given by (fl=x, y, z) L$“Y= i A$“f”t,(i,j,(k) i=l
(i= 1, *.a)8) 9
(2)
9 May 1988
where the cp(,,,,)(k) are the eigenvalues of q(k) under the permutation group of 8 objects, P( i,j) being the permutation containing the cycle (ij) [ 12 1. The matrix elements A$’ represent the dipole-dipole interaction between an ion situated on sublattice i, and all ions on sublattice j:
=
C
Jr,
/ED’),I# I
for
i=j,
(3)
and J~=tS2~2S[gwgvv(r~~~y-3r~r~)lr:]
,
(4)
where S is the ion spin, pB is the Bohr magneton and g is the g-tensor. The lattice sums required for the present case from which the matrix A is formed are evaluated over a sphere of radius 500 8, and are listed in table 1. The three ferromagnetic configurations ql& (a), a =x, y, z are corrected for demagnetization as the lattice sums are evaluated over a sphere 1111. The atomic arrangement of GdBa2Cu30, in a unit cell is given in fig. 1 of ref. [ 51. Gd and Ba atoms are located at the centers of the cubes whose corners are occupied by Cu06 octahedrons. The unit cell can be understood as the staking of three simple perovskite structures in an ordered manner with highly ordered oxygen vacancies that give rise to an orthorhombic distortion and the unit cell parameters are ~~3.861 A, bc3.912 8, and cc11.715 8, [5]. Hav-
Table 1 Lattice sums for GdBa2Cu30,. The variables r, x, y and z are the displacements from an origin situated in the sublattice 1 to each ion in one of various r* sublattices. The unit of length is 1 A. The x, y and z axes are made coincident with the a, b and c vectors of the unit cell respectively. Sublattice
1 2 3 4 5 6 7 8
132
Lattice sum X(9--3x2)/r5
- 13xylr5
I(+-3y2)/r5
- E3xzlr5
I(?-3z2)/r5
-C3yz/r5
-0.006907 -0.070711 0.02476 1 -0.014228 0.002984 0.002933 0.002984 0.002933
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
-0.006417 0.026254 -0.067974 -0.015029 0.002984 0.002984 0.002933 0.002933
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.013325 0.044456 0.0432 13 0.029257 -0.005971 -0.005971 -0.005971 -0.005868
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Volume
129, number
PHYSICS
2
Table 2 Lattice sums from Gd-ion of table 1. Sublattice
positions
LETTERS
9 May 1988
A
over a sphere of radius 500 A at a Cu site as the origin for GdBa2Cu30,.
For details see the caption
Lattice sum I(?-3x2)/9
- X3xy/r5
I(r’-3y2)/rS
-X3x2/P
I(?-3z2)/r5
- C3yz/r5
-0.008699 -0.008699 -0.008699 -0.008699 0.002958 0.002958 0.002958 0.002958
-0.028171 0.028171 0.028 17 1 -0.028171 0.0. 0.0 0.0 0.0
-0.009331 -0.009331 -0.009331 -0.009331 0.002958 0.002958 0.002958 0.002958
-0.025922 0.025922 -0.025922 0.025922 0.0 0.0 0.0 0.0
0.018029 0.018029 0.018029 0.018029 -0.005918 -0.005918 -0.005918 -0.005918
-0.026424 -0.026424 0.026424 0.026424 0.0 0.0 0.0 0.0
ing zero orbital angular momentum, the Gd ion does not interact to first order with the crystalline electric field, and it retains the degeneracy of its half-integral spin S= 712. Without explicitly taking into account the numerical values of the g-factors the energies per ion for the few lowest-lying states are found in units of K, q( 1):
I?,,= -0.301015g:,
)
q( 1):
E_vy= -0.286064&,
,
of the g-tensor may be the lowest energy configuration or the ordered state will always be antiferromagnetic, which agrees with that observed experimentally. The direction of spins in the ordered state will be determined by the g-values. Now to estimate the magnetic fields due to Gd ions at Cu sites the expression for the magnetic field components for a dipole at a distance r is h,=~uBSgaol(3r,r,-s,,r2)/rs,
pointing in the cy direction and CX,8=x, y, z; r,, rs are the components of r. Taking a Cu site as origin the expression for the component of the total magnetic field hi, due to ordering described by q(k) is
-O.l71994g&,
q(2) orq(7):
E,,=
q( 3) or q(6):
E,, = -0.336208g?jx
,
q(5) orq(8):
E,,,,=-0.326539g&,
,
(5)
where g,,, g,, and g,, are the g-factors along the x, y and z axis respectively. From the above equations we see that whatever the values of the components
(6)
Table 3 Magnetic field components at a Cu site for the various orderings q(k) of the rare-earth ions Gd in GdBaQtsO, and h, means the component of the magnetic field when all the spins point parallel and antiparallel to the a direction ((Y, j?=x, y, z) depending on the particular ordering. The units are such that actual values in gauss can be obtained by multiplying each h, by gee’. Ordering
h,
q(l)
1513 0 0 1513 0 0 0 0
q(2) q(3) q(4) q(5) q(6) q(7) q(8)
0 3656 0 0 0 0 3656 0
0 0 0 0 3359 0 0 -3370
1599 0 0 1595 0 0 0 0
0 0 3423 0 0 3436 0 0
- 804 0 0 -3108 0 0 0 0
133
Volume
129, number
2
PHYSICS
where q(k, j), j= 1, 2, .... 8, describe the particular ordering as dictated by the elements of eigenvectors q(k) [ 121. The subscript i in eq. (6) covers all the dipoles of a sublattice k, and rf is the distance of ion i on sublattice k from the origin. For use in eq. (6 ) the required lattice sums are given in table 2. The lattice sums are corrected for ferromagnetic ordering [131. The values of the magnetic fields (in units of g,,, gYYand g,, gauss) for various orderings q(k), k= 1, ...) 8, at a copper site are given in table 3. As a particular example, the magnetic fields at a copper site are found to be, for q( 3), h,,(max) ~6846 G and for q(7), h,(max) ~7312 G. Values of any where from 1 to 6 kG, or even as large as 12 kG are found for the superconducting quenching field [ 14 ] and so the present field values may permit the coexistence of superconductivity and magnetic ordering. The calculations of the present paper indicate that the observed antiferromagnetic ordering of GdBazCuSO, can be accounted for solely by dipole-dipole interaction between the Gd ions and the results may be helpful in further developing the understanding of the phenomena of superconductivity and magnetic ordering as they occur in GdBazCu30,.
134
LETTERS
A
9 May 1988
References [ I] J.G. Bednorz and K.A. Mtiller, Z. Phys. B 64 ( 1986) 189. [2] P.H. Hor, R.L. Meng, Y.Q. Wang, L. Gao, Z.J. Huang, J. Bechtold, K. Forster and C.W. Chu, Phys. Rev. Lett. 58 (1987) 1891. [ 31 S. Ohshima and T. Wakiyama, Japan. J. Appl. Phys. 26 (1987) L815. [4] D.W. Murphy, S. Sunshine, R.B. van Dover, R.J. Cava, B. Batlogg, S. Zahurak and L.F. Schneemeyer, Phys. Rev. Lett. 58 (1987) 1888. [5 K. Kadowaki, H.P. van der Meulen, J.C.P. Klaasse, M. van Sprang, J.Q.A. Koster, L.W. Roeland, F.R. de Boer, Y.K. Huang, A.A. Menovsky and J.J.M. Franse, Physica B 145 (1987) 260. S.W. Cheong, R.M. ]6 J.O. Willis, Z. Fisk, J.D. Thompson, Aikim, J.L. Smith and E. Zirngiebl, J. Magn. Magn. Mater. 67 (1987) L139. E. Recknagel, M. Ross[7 A. Golnik, Ch. Niedermayer, manith, A. Weidinger, J.I. Budnick, B. Chamberland, M. Filipkowsky, Y. Zhang, D.P. Yang, L.L. Lynds, F.A. Otter and C. Baines, Phys. Lett. A I25 ( 1987) 7 1. [8 0. Fischer, A. Treyvand, R. Chevrel and M. Sergent, Solid State Commun. 17 ( 1985) 72 1. ]9 H.C. Hamaker, L.D. Woolf, H.B. MacKay, Z. Fisk and M.B. Maple, Solid State Commun. 32 ( 1979) 32. [ lo] M. Luttinger and L. Tisza, Phys. Rev. 70 (1946) 954. [ 1 l] Th. Niemeyer, Physica 57 (1972) 281. [ 121 S.K. Misra, Phys. Rev. B 5 ( 1973) 2026. [ 131 C. Kittel, Phys. Rev. 82 (1951) 965. [ 141 H.C. Hamaker, H.B. MacKay, L.D. Woolf, M.B. Maple, W. Odoni and H.R. Ott, Phys. Len. A 8 1 ( 198 1) 9 1.
PHYSICS
2
ON THE MAGNETIC
ORDERING
LETTERS
9 May 1988
A
OF THE HIGH T, SUPERCONDUCTOR
GdBa2Cu30,
Harabhusan SEN Department
of Physics, Burdwan University, Burdwan 713 104, West Bengal, India
Received 8 January 1988; accepted Communicated by D. Bloch
for publication
8 February
1988
The magnetically ordered state of GdBazCu307 at low temperatures has been investigated using the extended Luttinger-Tisza method. It is found that the observed ordering of this compound can be explained by a purely dipole-dipole interaction. An estimate has been made of the internal magnetic field produced due to magnetic ordering of Gd metal ions at a copper site and it is found that the values are within the range of superconducting quenching field and may permit the coexistence of magnetic ordering and superconductivity.
After the discovery [ 1 ] of high T,superconductivity in YBa2Cu30, much insight has been gained over the past few months into the physical properties of these compounds. One remarkable feature of these materials is that the superconductivity transition temperature T,is not influenced [2-41 when other trivalent species, especially heavy rare-earth elements, are substituted for Y3+. In some cases T,is even slightly higher than in the case of YBa,Cu,O,, though most of the rare-earth ions possess their own magnetic moments. Hence these are ideal materials to investigate the interplay between magnetism and superconductivity. Subsequent work [ 5-71 further demonstrated that, in fact, superconductivity and magnetic order coexist in one of these compounds GdBa2Cu307 and this was confirmed by the magnetic and heat capacity measurements and muon spin rotation experiments. The magnetic transition temperature of this compound was found to be N 2.3 K in the superconducting state. Such behavior has also been found in certain RMoS, [ 8 ] and RRh4B4 [ 9 ] compounds (R = rare earth) in which the long range antiferromagnetic order of the localized f-electrons left essentially unperturbed superconductivity associated with the transition metal d-band. Similarly a comparable state exists in GdBa,Cu,O,. However, the nature of the interaction leading to magnetic order remains an interesting question. Another interesting aspect of the problem is the
estimation of the magnetic fields at a copper site as produced by the magnetic ordering of Gd ions. Since 3d electrons of Cu atoms are responsible for superconductivity, it is conceivable that if the magnetic field at Cu sites is not sufficiently large, the superconductivity may not be quenched by the magnetic field due to the ordered Gd sublattice, and the coexistence of superconductivity and magnetic ordering will be allowed. In this Letter we report the results of a study on this compound by the extended Luttinger-Tisza (LT) [ lo] method taking rigorously into account the dipole-dipole interaction (the exchange interaction of the rare-earth ions with the superconducting electrons is sufficiently small because the exchange interaction alone can destroy the superconductivity). If the magnetic unit cell of the ordered state at T= 0 is given, energy calculations can be performed as a function of the spin directions of the unit cell. In the LT method these spin-direction variables are combined into a vector in multidimensonal space. The components of the field at various lattice sites, produced by a certain spin configuration, can be represented by a vector in the same space. In practice, the crystal lattice is supposed to be divided into 8 sublattices (since there is one magnetic ion per unit crystallographic cell of GdBa2Cu30,) and each sublattice is generated from it by translation
0375-9601/88/S 03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
131
Volume 129, number 2
PHYSICS LETTERS A
( = 21u+ 2mb+ 2nc; a, b, c are unit cell vectors) to the 8 vertices of the magnetic unit cell [ lo]. The
configuration of spins is thus described by 24 numbers (x, y, z components of 8 spins). These can be considered as components of a vector S in a 24-dimensional vector space. Since the magnetic fields depend linearly on the components of the spins, it may be written F= -A*& where A is a 24x24 matrix. The energy of the configuration per magnetic unit cell is - &S.F= jS-A*S, which is a quadratic form in the components of S. This is to be minimized subject to the strong constraint s$ +s$ +.~a,= 1 (t= 1, .... 8, where s,,, s,,, sZtare the components of a representative spin of the tth sublattice), which implies the weak constraint 5 S:=8. I=1
Now the minimum value of @-A-S subject to the weak constraint is the lowest eigenvalue of iA, and a configuration with corresponds to this value of energy is the corresponding eigenvector and can be expressed as q(k)&(a), k= 1, .... 8, (Y=x, y, z. For the present case the q(k) are 8-dimensional [ 121 and determine the nature of the ordering (ferromagnetic or antiferromagnetic). The particular physical space directions along which the spins of the kth sublattice point are directed are given by & ( (Y) . The & ( (Y) are the eigenvectors of a 3X 3 matrix Lk, whose components are given by (fl=x, y, z) L$“Y= i A$“f”t,(i,j,(k) i=l
(i= 1, *.a)8) 9
(2)
9 May 1988
where the cp(,,,,)(k) are the eigenvalues of q(k) under the permutation group of 8 objects, P( i,j) being the permutation containing the cycle (ij) [ 12 1. The matrix elements A$’ represent the dipole-dipole interaction between an ion situated on sublattice i, and all ions on sublattice j:
=
C
Jr,
/ED’),I# I
for
i=j,
(3)
and J~=tS2~2S[gwgvv(r~~~y-3r~r~)lr:]
,
(4)
where S is the ion spin, pB is the Bohr magneton and g is the g-tensor. The lattice sums required for the present case from which the matrix A is formed are evaluated over a sphere of radius 500 8, and are listed in table 1. The three ferromagnetic configurations ql& (a), a =x, y, z are corrected for demagnetization as the lattice sums are evaluated over a sphere 1111. The atomic arrangement of GdBa2Cu30, in a unit cell is given in fig. 1 of ref. [ 51. Gd and Ba atoms are located at the centers of the cubes whose corners are occupied by Cu06 octahedrons. The unit cell can be understood as the staking of three simple perovskite structures in an ordered manner with highly ordered oxygen vacancies that give rise to an orthorhombic distortion and the unit cell parameters are ~~3.861 A, bc3.912 8, and cc11.715 8, [5]. Hav-
Table 1 Lattice sums for GdBa2Cu30,. The variables r, x, y and z are the displacements from an origin situated in the sublattice 1 to each ion in one of various r* sublattices. The unit of length is 1 A. The x, y and z axes are made coincident with the a, b and c vectors of the unit cell respectively. Sublattice
1 2 3 4 5 6 7 8
132
Lattice sum X(9--3x2)/r5
- 13xylr5
I(+-3y2)/r5
- E3xzlr5
I(?-3z2)/r5
-C3yz/r5
-0.006907 -0.070711 0.02476 1 -0.014228 0.002984 0.002933 0.002984 0.002933
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
-0.006417 0.026254 -0.067974 -0.015029 0.002984 0.002984 0.002933 0.002933
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.013325 0.044456 0.0432 13 0.029257 -0.005971 -0.005971 -0.005971 -0.005868
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Volume
129, number
PHYSICS
2
Table 2 Lattice sums from Gd-ion of table 1. Sublattice
positions
LETTERS
9 May 1988
A
over a sphere of radius 500 A at a Cu site as the origin for GdBa2Cu30,.
For details see the caption
Lattice sum I(?-3x2)/9
- X3xy/r5
I(r’-3y2)/rS
-X3x2/P
I(?-3z2)/r5
- C3yz/r5
-0.008699 -0.008699 -0.008699 -0.008699 0.002958 0.002958 0.002958 0.002958
-0.028171 0.028171 0.028 17 1 -0.028171 0.0. 0.0 0.0 0.0
-0.009331 -0.009331 -0.009331 -0.009331 0.002958 0.002958 0.002958 0.002958
-0.025922 0.025922 -0.025922 0.025922 0.0 0.0 0.0 0.0
0.018029 0.018029 0.018029 0.018029 -0.005918 -0.005918 -0.005918 -0.005918
-0.026424 -0.026424 0.026424 0.026424 0.0 0.0 0.0 0.0
ing zero orbital angular momentum, the Gd ion does not interact to first order with the crystalline electric field, and it retains the degeneracy of its half-integral spin S= 712. Without explicitly taking into account the numerical values of the g-factors the energies per ion for the few lowest-lying states are found in units of K, q( 1):
I?,,= -0.301015g:,
)
q( 1):
E_vy= -0.286064&,
,
of the g-tensor may be the lowest energy configuration or the ordered state will always be antiferromagnetic, which agrees with that observed experimentally. The direction of spins in the ordered state will be determined by the g-values. Now to estimate the magnetic fields due to Gd ions at Cu sites the expression for the magnetic field components for a dipole at a distance r is h,=~uBSgaol(3r,r,-s,,r2)/rs,
pointing in the cy direction and CX,8=x, y, z; r,, rs are the components of r. Taking a Cu site as origin the expression for the component of the total magnetic field hi, due to ordering described by q(k) is
-O.l71994g&,
q(2) orq(7):
E,,=
q( 3) or q(6):
E,, = -0.336208g?jx
,
q(5) orq(8):
E,,,,=-0.326539g&,
,
(5)
where g,,, g,, and g,, are the g-factors along the x, y and z axis respectively. From the above equations we see that whatever the values of the components
(6)
Table 3 Magnetic field components at a Cu site for the various orderings q(k) of the rare-earth ions Gd in GdBaQtsO, and h, means the component of the magnetic field when all the spins point parallel and antiparallel to the a direction ((Y, j?=x, y, z) depending on the particular ordering. The units are such that actual values in gauss can be obtained by multiplying each h, by gee’. Ordering
h,
q(l)
1513 0 0 1513 0 0 0 0
q(2) q(3) q(4) q(5) q(6) q(7) q(8)
0 3656 0 0 0 0 3656 0
0 0 0 0 3359 0 0 -3370
1599 0 0 1595 0 0 0 0
0 0 3423 0 0 3436 0 0
- 804 0 0 -3108 0 0 0 0
133
Volume
129, number
2
PHYSICS
where q(k, j), j= 1, 2, .... 8, describe the particular ordering as dictated by the elements of eigenvectors q(k) [ 121. The subscript i in eq. (6) covers all the dipoles of a sublattice k, and rf is the distance of ion i on sublattice k from the origin. For use in eq. (6 ) the required lattice sums are given in table 2. The lattice sums are corrected for ferromagnetic ordering [131. The values of the magnetic fields (in units of g,,, gYYand g,, gauss) for various orderings q(k), k= 1, ...) 8, at a copper site are given in table 3. As a particular example, the magnetic fields at a copper site are found to be, for q( 3), h,,(max) ~6846 G and for q(7), h,(max) ~7312 G. Values of any where from 1 to 6 kG, or even as large as 12 kG are found for the superconducting quenching field [ 14 ] and so the present field values may permit the coexistence of superconductivity and magnetic ordering. The calculations of the present paper indicate that the observed antiferromagnetic ordering of GdBazCuSO, can be accounted for solely by dipole-dipole interaction between the Gd ions and the results may be helpful in further developing the understanding of the phenomena of superconductivity and magnetic ordering as they occur in GdBazCu30,.
134
LETTERS
A
9 May 1988
References [ I] J.G. Bednorz and K.A. Mtiller, Z. Phys. B 64 ( 1986) 189. [2] P.H. Hor, R.L. Meng, Y.Q. Wang, L. Gao, Z.J. Huang, J. Bechtold, K. Forster and C.W. Chu, Phys. Rev. Lett. 58 (1987) 1891. [ 31 S. Ohshima and T. Wakiyama, Japan. J. Appl. Phys. 26 (1987) L815. [4] D.W. Murphy, S. Sunshine, R.B. van Dover, R.J. Cava, B. Batlogg, S. Zahurak and L.F. Schneemeyer, Phys. Rev. Lett. 58 (1987) 1888. [5 K. Kadowaki, H.P. van der Meulen, J.C.P. Klaasse, M. van Sprang, J.Q.A. Koster, L.W. Roeland, F.R. de Boer, Y.K. Huang, A.A. Menovsky and J.J.M. Franse, Physica B 145 (1987) 260. S.W. Cheong, R.M. ]6 J.O. Willis, Z. Fisk, J.D. Thompson, Aikim, J.L. Smith and E. Zirngiebl, J. Magn. Magn. Mater. 67 (1987) L139. E. Recknagel, M. Ross[7 A. Golnik, Ch. Niedermayer, manith, A. Weidinger, J.I. Budnick, B. Chamberland, M. Filipkowsky, Y. Zhang, D.P. Yang, L.L. Lynds, F.A. Otter and C. Baines, Phys. Lett. A I25 ( 1987) 7 1. [8 0. Fischer, A. Treyvand, R. Chevrel and M. Sergent, Solid State Commun. 17 ( 1985) 72 1. ]9 H.C. Hamaker, L.D. Woolf, H.B. MacKay, Z. Fisk and M.B. Maple, Solid State Commun. 32 ( 1979) 32. [ lo] M. Luttinger and L. Tisza, Phys. Rev. 70 (1946) 954. [ 1 l] Th. Niemeyer, Physica 57 (1972) 281. [ 121 S.K. Misra, Phys. Rev. B 5 ( 1973) 2026. [ 131 C. Kittel, Phys. Rev. 82 (1951) 965. [ 141 H.C. Hamaker, H.B. MacKay, L.D. Woolf, M.B. Maple, W. Odoni and H.R. Ott, Phys. Len. A 8 1 ( 198 1) 9 1.