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Electron-phonon interaction and superconductivity in strongly correlated systems
Physics Letters A 172 ( 1993) 467-470 North-Holland
P H Y51C S k ETT ER$ A
Electron-phonon interaction and superconductivity in strongly correlated systems J. Zielifiski ~ and M. Matlak Institute of Physics. Silesian University. 40-007 Katowice, Poland Received 27 July 1992; accepted for publication 19 November 1992 Communicated by J. Flouquet
We have addressed the question of the significance of electronic correlations present in the Hubbard model for superconductivity induced by strong electron-phonon coupling. An interplay of Coulomb correlations and momentum dependence of electronphonon interaction can result in an enhancement of the superconductivity transition temperature for finite doping with respect to half-filling.
The Hubbard model [ 1 ] and originating from it the t - J model [ 2 ] can be supposed to provide support for a purely electronic mechanism o f superconductivity, a problem originating from the discovery o f copper-oxide [ 3 ] and doped C6o superconductors [4 ]. In particular, one can argue in favour o f a tendency towards superconductivity for strongly correlated electrons in the case o f a more than half filled lower Hubbard subband [ 5,6 ]. N o w there is fairly much known about the properties o f these models but there is still no consensus that such a phononfree mechanism is possible. On the other hand, one can observe an increasing evidence for the important role played by phonons in high-temperature superconductors [ 7 - l 0 ]. To be more precise, one should consider the picture o f strong electron-phonon coupling which from the theoretical point o f view is intimately related to the Eliashberg equations [ 11 ]. Because the electrons in the systems o f interest are genuinely strongly correlated, one faces a very difficult problem: H o w to incorporate electronic correlations in the Eliashberg formulation on an advanced level? This could give an answer to the question about the effect o f electronic correlations on superconductivity originating from a strong elect r o n - p h o n o n interaction. We have already ad-
dressed such a problem for the two-site Hubbard model coupled to phonons [ 12 ] with the help o f an interpolation formula for the superconducting transition temperature Tc derived by Kresin [ 13 ]. An exact treatment o f the electronic correlations shows a general tendency towards an enhancement of Tc by on-site Coulomb repulsion [ 12 ]. In this article we will be concerned with the finite band width version o f the Hubbard model coupled to phonons,
H = ~. Ckc~,oc~,o+U~, c,,+ ci, c,,+ c~ k,o
i
+ Z gk.k+qCk+~..Ck..(bq+b-~) •
+
+
k , q , cr
+ ~, toob+bq.
(1)
q
where we restrict ourselves to the single, Einstein phonon branch tOo. The exact expression for the phonon propagator reads 2tOo
D(q, to)= to2_too2_2?(q, t o ) ,
(2)
where 27(q, tO)=2tOo//(q, tO) and 11(¢, tO) is given by
' Supported by the State Committee for Scientific Research, Grant No. 2 2405 9203. 0375-9601/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All fights reserved.
467
Volume 172, number 6
PHYSICS LETTERSA
analytical evaluation of the q-average of (4). We finally obtain
//(q, o9)= Y g k . + . g , , , . - . k,k' a, v +
x (
(3)
To evaluate/-/(q, to) we approximate the electronic part of (1) by the one-particle Hamiltonian originating from the mean-field-like approximation within the slave boson technique [ 14-16 ], ek--, gk = qek -fl, where electronic correlations are taken into account in q and//. They can be evaluated integrating out the Grassmann variables (saddle point approximation) or minimizing the free-energy functional [ 16 ]. This leads us to g,) -fig,+,) • II(q, o))=2 ~ gk.k+,gk+,.k f (og--_g,+------~
(4)
The evaluation of (4) is a non-trivial problem. One of the possible approaches could be to average H(q, o)) and q and replace gk,k+~ by the momentum independent quantity g. It has been argued that the q dependence of the electron-phonon interaction can be of importance, for the two-dimensional lattice this interaction is maximal near the M point and vanishes at the zone center [ 17 ]. We convert the q-average of (4) into a double integral with the density of states function and the above feature of gk,k+ q can be simulated by the additional distribution function p=p+p_, where 1 aF
p+ = ~ - ~ [ ( E - e ' + b ) 2 + V 21 - '
18 January 1993
(5)
i m / L (- o g ) = g 2 ~zJ a/'~[og( ~ ~i+ + g-q-) -2LI( ~r+ + 7"r --2~r) ] X [ (O9-- [7)2 +/'~2) ] [ (O92+ 4z~2)O91 - ' ,
a=II[b2+ ( F + 2 A ) 2 ] ( F + 2A) -~
(9)
where g"i{r) stands for the imaginary (real) part of the digamma function g-' [ 11 ] and
)
2--~ - 2~ (og-T-fl) ,
fl= ( k T ) - '
(10a)
g"r = gJ+ ( o 9 = 0 ) .
(10b)
We have introduced the notation A = qA where A = A, a, b, f'. For the sake of brevity we do not present the fairly long expression for Re H(o9). However, it can be directly obtained from ReH(o9)=
-n
I p i dzImH(z) o9-z
(11)
-oo
To calculate Tc we use an interpolation formula derived by Kresin [ 13 ]:
kTc = 0 . 2 5 ~ (e 2/~- 1 )-1/2,
(12)
where 2 = ~ do9ot2F(m)m -l .
and
(8)
H ( o g ) = H + (og) + H _ (og) ,
(13a)
(6)
originates from the normalization condition
~2=~--1
; do9aEF(o9)o9
(13b)
--oo
S
S po(,)po(,')p(,-,')d,d,'=l,
(7)
--oo --oo
with the Lorentzian model density of states function Po ( ~ ) = n - ~A( ~z + Az ) - ~. b stands for the measure of the magnitude of q for which gk,k+ q approaches its maximal value. The introduction of the model Lorentzian density of states function seems to be relatively consistent with the spectral features of a two-dimensional lattice (a weak logarithmic singularity) and enables an 468
otEF(og)
and stands for the Eliashberg function [ 11,13 ]. In our case a2F(og) = _g2 1 Im D(o9) .
(14)
Figure 1 shows some results for Tc as a function of the occupation number n normalized to the value obtained with the bare phonon propagator ( Z ( o g ) = 0 ) . We take A as the energy unit and our choice of o9o and g leads to Toc= 31.6 K. To simplify
Volume 172, number 6
PHYSICS LETTERS A
1.s
r /roc
:001Aoo 0.5
I
0.8
I
0.6
0.2
0J I
I
0.6
0.2
n
0
Fig. 1. Superconducting transition temperature normalized to the value obtained with the bare phonon propagator (Z(og)=0) as a function of the occupation number n. The dashed line corresponds to the uncorrelated case for b= 0.12, too= 0.1, g= 0. ! 5 with the model density of states half-width A as the energy unit. In the insert we show the spectral function of the phonons (continuous line) for b=0.12 and T= 36.8 and the dotted line represents the approximate formula given by ( 15 ). the numerical calculations we a p p r o x i m a t e 1 - -ImD(co)
(
,
c (co-~)2+c~
1
)
( CO.~_(j);) 2 ..[_C2 ,
18 January 1993
ues o f n close to 1 where the stabilization o f the antiferromagnetic isolating phase could act to the d e t r i m e n t o f the superconducting condensation. It is r e m a r k a b l e that the qualitative features o f this beh a v i o u r do not d e p e n d on the origin o f the decrease o f q when approaching n = 1. T h e results presented in fig. 1 correspond to q evaluated as a function o f n for the two-dimensional H u b b a r d m o d e l [ 16 ] ( U equal to its critical value) but a similar b e h a v i o u r can be observed for the limit U--,oo when q = 1 - n . Generally, T¢ is mostly enhanced for values o f b o f the o r d e r o f 090 a n d any p r o n o u n c e d increase o f b causes that the corresponding curves b e c o m e m o r e a n d m o r e fiat. A n increase (decrease) in T¢ originates from the softening ( h a r d e n i n g ) o f the effective p h o n o n frequency 030. Figure 2 d e m o n s t r a t e s the role o f the correlations for the actual value o f 030 for the half-filled case. There are regions o f q for which the solutions for 030 do not exist (for larger values o f F ) . The cod e p e n d e n c e o f the p h o n o n self-energy is shown in the insert. One can note that our m o d e l evaluation o f 27(co) correctly reproduces the features typical for this q u a n t i t y [ 19 ]. The results presented so far show the role o f the electronic correlations for superconductivity ind u c e d by e l e c t r o n - p h o n o n interaction. O u r approx-
(15a)
°I. /" ~--..." ..-' ---.-r(,,vz~z,.~ o.2 1
where 27(03o) I c= IIm 2090
(15b)
a n d 030 is the effective p h o n o n frequency given by the solution o f the equation 092 - co2o+ Re X ( c o ) = 0. The quality o f this a p p r o x i m a t i o n is d e m o n s t r a t e d in the insert in fig. 1. W i t h respect to Tc one can see that correlations lead to a definitely different behaviour o f the transition t e m p e r a t u r e when c o m p a r e d to the uncorrelated case ( d a s h e d c u r v e ) . Finite values o f b (simulating the q-dependence o f the e l e c t r o n p h o n o n i n t e r a c t i o n ) enhance Tc for finite doping. T¢ suddenly decreases in the vicinity o f half-filling. The following increase o f T¢ with n ~ 1 ( q ~ 0 ) is brought about by the B r i n k m a n - R i c e transition [ 18 ] a n d should be u n d e r s t o o d as a result o f our approxim a t i o n scheme. Therefore, we do not consider val-
-5 iy "" ."
.---T
---o
~o -10
~.~
o,15~"
, ~ ~
o.2
0.5 Oil
0.6
, 0.~
, 0.2
q
0
Fig. 2. Solutions of the equation to2 - too2 - Re 27(o9)= 0 versus the renormalization factor q for the half-filled case o9o=0.1, g--0.15. The insert shows the real (dashed line) and imaginary (dotted line) parts of the phonon self-energy for F= 0.5, q = 0.1, T= 100 K. This corresponds to O)o-- 0.125. 469
Volume 172, number 6
PHYSICS LETTERS A
i m a t i o n t r e a t m e n t o f C o u l o m b c o r r e l a t i o n s allows us to r e p r o d u c e t h e fact t h a t the t r a n s i t i o n t e m p e r a t u r e a c h i e v e s its m a x i m a l v a l u e for finite d o p i n g , a feature w h i c h is o b s e r v e d e x p e r i m e n t a l l y . T h e lack o f c o r r e l a t i o n s s h o w s up in a c o m p l e t e l y d i f f e r e n t beh a v i o u r - the t r a n s i t i o n t e m p e r a t u r e is m a x i m a l at half-filling. We h a v e s i m u l a t e d t h e m o m e n t u m dep e n d e n c e o f t h e e l e c t r o n - p h o n o n i n t e r a c t i o n in a s i m p l y way. T h e m a g n i t u d e o f Tc can strongly dep e n d on the r e g i o n o f the B r i l l o u i n z o n e in w h i c h gk,k * q a c h i e v e s its m a x i m a l value. We c o n c l u d e t h a t the c o m b i n a t i o n o f e l e c t r o n i c c o r r e l a t i o n s a n d a strong e l e c t r o n - p h o n o n i n t e r a c t i o n can serve as o n e o f the i m p o r t a n t m e c h a n i s m s resulting in h i g h - t e m perature superconductivity.
References [ 1 ] J. Hubbard, Proc. R. Soc. A 276 ( 1963 ) 238. [2] K.A. Chao, J. Spatek and A.M. Oleg, J. Phys. C l0 (1977) L271; J.E. Hirsch, Phys. Rev. Lett. 54 ( 1986 ) 1317. [3] J.G. Bednorz and K.A. Miiller, Z. Phys. B 64 (1986) 989.
470
18 January 1993
[4] A.F. Hebard et al., Nature 350 ( 1991 ) 600; M.J. Rosseinsky et al., Phys. Rev. Lett. 66 ( 1991 ) 2830; K. Holczer et al., Science 252 ( 1991 ) 1154. [5] R.O. Zaitsev, Phys. Lett. A 134 (1988) 199. [6]J. Zielifiski, P. Entel and M. Matlak, Phys. Lett. A 136 (1989) 441; J. Zielifiski and R. Grabowski, Phys. Lett. A 154 (1991) 457. [7] R. Cohen, W.E. Pickett, L.L. Bayer and H. Krakauer, Phys. Rev. Lett. 60 (1988) 817. [ 8 ] R. Zeyher and G. Zwicknagl, Z. Phys. B 78 (1990) 175. [9] J. Ranninger, Z. Phys. B 84 ( 1991 ) 167. [10] C. Falter, M. Klenner and W. Ludwig, Phys. Lett. A 165 ( 1992 ) 260. [ 11 ] P.B. Allen and B. Mitrovi~, Solid State Phys. 37 (1992) 1. [12]J. Zielifiski, M. Matlak and P. Entel, Phys. Lett. A 165 (1992) 285. [ 13 ] V.Z. Kresin, Phys. Lett. A 122 ( 1987 ) 434. l 14 ] G. Kotliar and A.E. Ruckenstein, Phys. Rev. Lett. 57 (1986) 1362. [ 15l M. Lavagna, int. J. Mod. Phys. B 5 ( 1991 ) 885. [ 16] P. Entel, S.N. Behera, J. Zielifiski and E. Kaufmann, Int. J. Mod. Phys. B 5 (1991) 271. [17] C.J. Ventura, A.A. Aligia and B. Alascio, Solid State Commun. 71 (1989) 957; A.A. Aligia, Phys. Rev. B 39 (1988) 6700. [ 18 ] W.F. Brinkman and T.M. Rice, Phys. Rev. B 2 (1970) 4302. [ 19 ] F.C. Khanna and H.R. Glyde, Can. J. Phys. 54 ( 1976 ) 648.
P H Y51C S k ETT ER$ A
Electron-phonon interaction and superconductivity in strongly correlated systems J. Zielifiski ~ and M. Matlak Institute of Physics. Silesian University. 40-007 Katowice, Poland Received 27 July 1992; accepted for publication 19 November 1992 Communicated by J. Flouquet
We have addressed the question of the significance of electronic correlations present in the Hubbard model for superconductivity induced by strong electron-phonon coupling. An interplay of Coulomb correlations and momentum dependence of electronphonon interaction can result in an enhancement of the superconductivity transition temperature for finite doping with respect to half-filling.
The Hubbard model [ 1 ] and originating from it the t - J model [ 2 ] can be supposed to provide support for a purely electronic mechanism o f superconductivity, a problem originating from the discovery o f copper-oxide [ 3 ] and doped C6o superconductors [4 ]. In particular, one can argue in favour o f a tendency towards superconductivity for strongly correlated electrons in the case o f a more than half filled lower Hubbard subband [ 5,6 ]. N o w there is fairly much known about the properties o f these models but there is still no consensus that such a phononfree mechanism is possible. On the other hand, one can observe an increasing evidence for the important role played by phonons in high-temperature superconductors [ 7 - l 0 ]. To be more precise, one should consider the picture o f strong electron-phonon coupling which from the theoretical point o f view is intimately related to the Eliashberg equations [ 11 ]. Because the electrons in the systems o f interest are genuinely strongly correlated, one faces a very difficult problem: H o w to incorporate electronic correlations in the Eliashberg formulation on an advanced level? This could give an answer to the question about the effect o f electronic correlations on superconductivity originating from a strong elect r o n - p h o n o n interaction. We have already ad-
dressed such a problem for the two-site Hubbard model coupled to phonons [ 12 ] with the help o f an interpolation formula for the superconducting transition temperature Tc derived by Kresin [ 13 ]. An exact treatment o f the electronic correlations shows a general tendency towards an enhancement of Tc by on-site Coulomb repulsion [ 12 ]. In this article we will be concerned with the finite band width version o f the Hubbard model coupled to phonons,
H = ~. Ckc~,oc~,o+U~, c,,+ ci, c,,+ c~ k,o
i
+ Z gk.k+qCk+~..Ck..(bq+b-~) •
+
+
k , q , cr
+ ~, toob+bq.
(1)
q
where we restrict ourselves to the single, Einstein phonon branch tOo. The exact expression for the phonon propagator reads 2tOo
D(q, to)= to2_too2_2?(q, t o ) ,
(2)
where 27(q, tO)=2tOo//(q, tO) and 11(¢, tO) is given by
' Supported by the State Committee for Scientific Research, Grant No. 2 2405 9203. 0375-9601/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All fights reserved.
467
Volume 172, number 6
PHYSICS LETTERSA
analytical evaluation of the q-average of (4). We finally obtain
//(q, o9)= Y g k . + . g , , , . - . k,k' a, v +
x (
(3)
To evaluate/-/(q, to) we approximate the electronic part of (1) by the one-particle Hamiltonian originating from the mean-field-like approximation within the slave boson technique [ 14-16 ], ek--, gk = qek -fl, where electronic correlations are taken into account in q and//. They can be evaluated integrating out the Grassmann variables (saddle point approximation) or minimizing the free-energy functional [ 16 ]. This leads us to g,) -fig,+,) • II(q, o))=2 ~ gk.k+,gk+,.k f (og--_g,+------~
(4)
The evaluation of (4) is a non-trivial problem. One of the possible approaches could be to average H(q, o)) and q and replace gk,k+~ by the momentum independent quantity g. It has been argued that the q dependence of the electron-phonon interaction can be of importance, for the two-dimensional lattice this interaction is maximal near the M point and vanishes at the zone center [ 17 ]. We convert the q-average of (4) into a double integral with the density of states function and the above feature of gk,k+ q can be simulated by the additional distribution function p=p+p_, where 1 aF
p+ = ~ - ~ [ ( E - e ' + b ) 2 + V 21 - '
18 January 1993
(5)
i m / L (- o g ) = g 2 ~zJ a/'~[og( ~ ~i+ + g-q-) -2LI( ~r+ + 7"r --2~r) ] X [ (O9-- [7)2 +/'~2) ] [ (O92+ 4z~2)O91 - ' ,
a=II[b2+ ( F + 2 A ) 2 ] ( F + 2A) -~
(9)
where g"i{r) stands for the imaginary (real) part of the digamma function g-' [ 11 ] and
)
2--~ - 2~ (og-T-fl) ,
fl= ( k T ) - '
(10a)
g"r = gJ+ ( o 9 = 0 ) .
(10b)
We have introduced the notation A = qA where A = A, a, b, f'. For the sake of brevity we do not present the fairly long expression for Re H(o9). However, it can be directly obtained from ReH(o9)=
-n
I p i dzImH(z) o9-z
(11)
-oo
To calculate Tc we use an interpolation formula derived by Kresin [ 13 ]:
kTc = 0 . 2 5 ~ (e 2/~- 1 )-1/2,
(12)
where 2 = ~ do9ot2F(m)m -l .
and
(8)
H ( o g ) = H + (og) + H _ (og) ,
(13a)
(6)
originates from the normalization condition
~2=~--1
; do9aEF(o9)o9
(13b)
--oo
S
S po(,)po(,')p(,-,')d,d,'=l,
(7)
--oo --oo
with the Lorentzian model density of states function Po ( ~ ) = n - ~A( ~z + Az ) - ~. b stands for the measure of the magnitude of q for which gk,k+ q approaches its maximal value. The introduction of the model Lorentzian density of states function seems to be relatively consistent with the spectral features of a two-dimensional lattice (a weak logarithmic singularity) and enables an 468
otEF(og)
and stands for the Eliashberg function [ 11,13 ]. In our case a2F(og) = _g2 1 Im D(o9) .
(14)
Figure 1 shows some results for Tc as a function of the occupation number n normalized to the value obtained with the bare phonon propagator ( Z ( o g ) = 0 ) . We take A as the energy unit and our choice of o9o and g leads to Toc= 31.6 K. To simplify
Volume 172, number 6
PHYSICS LETTERS A
1.s
r /roc
:001Aoo 0.5
I
0.8
I
0.6
0.2
0J I
I
0.6
0.2
n
0
Fig. 1. Superconducting transition temperature normalized to the value obtained with the bare phonon propagator (Z(og)=0) as a function of the occupation number n. The dashed line corresponds to the uncorrelated case for b= 0.12, too= 0.1, g= 0. ! 5 with the model density of states half-width A as the energy unit. In the insert we show the spectral function of the phonons (continuous line) for b=0.12 and T= 36.8 and the dotted line represents the approximate formula given by ( 15 ). the numerical calculations we a p p r o x i m a t e 1 - -ImD(co)
(
,
c (co-~)2+c~
1
)
( CO.~_(j);) 2 ..[_C2 ,
18 January 1993
ues o f n close to 1 where the stabilization o f the antiferromagnetic isolating phase could act to the d e t r i m e n t o f the superconducting condensation. It is r e m a r k a b l e that the qualitative features o f this beh a v i o u r do not d e p e n d on the origin o f the decrease o f q when approaching n = 1. T h e results presented in fig. 1 correspond to q evaluated as a function o f n for the two-dimensional H u b b a r d m o d e l [ 16 ] ( U equal to its critical value) but a similar b e h a v i o u r can be observed for the limit U--,oo when q = 1 - n . Generally, T¢ is mostly enhanced for values o f b o f the o r d e r o f 090 a n d any p r o n o u n c e d increase o f b causes that the corresponding curves b e c o m e m o r e a n d m o r e fiat. A n increase (decrease) in T¢ originates from the softening ( h a r d e n i n g ) o f the effective p h o n o n frequency 030. Figure 2 d e m o n s t r a t e s the role o f the correlations for the actual value o f 030 for the half-filled case. There are regions o f q for which the solutions for 030 do not exist (for larger values o f F ) . The cod e p e n d e n c e o f the p h o n o n self-energy is shown in the insert. One can note that our m o d e l evaluation o f 27(co) correctly reproduces the features typical for this q u a n t i t y [ 19 ]. The results presented so far show the role o f the electronic correlations for superconductivity ind u c e d by e l e c t r o n - p h o n o n interaction. O u r approx-
(15a)
°I. /" ~--..." ..-' ---.-r(,,vz~z,.~ o.2 1
where 27(03o) I c= IIm 2090
(15b)
a n d 030 is the effective p h o n o n frequency given by the solution o f the equation 092 - co2o+ Re X ( c o ) = 0. The quality o f this a p p r o x i m a t i o n is d e m o n s t r a t e d in the insert in fig. 1. W i t h respect to Tc one can see that correlations lead to a definitely different behaviour o f the transition t e m p e r a t u r e when c o m p a r e d to the uncorrelated case ( d a s h e d c u r v e ) . Finite values o f b (simulating the q-dependence o f the e l e c t r o n p h o n o n i n t e r a c t i o n ) enhance Tc for finite doping. T¢ suddenly decreases in the vicinity o f half-filling. The following increase o f T¢ with n ~ 1 ( q ~ 0 ) is brought about by the B r i n k m a n - R i c e transition [ 18 ] a n d should be u n d e r s t o o d as a result o f our approxim a t i o n scheme. Therefore, we do not consider val-
-5 iy "" ."
.---T
---o
~o -10
~.~
o,15~"
, ~ ~
o.2
0.5 Oil
0.6
, 0.~
, 0.2
q
0
Fig. 2. Solutions of the equation to2 - too2 - Re 27(o9)= 0 versus the renormalization factor q for the half-filled case o9o=0.1, g--0.15. The insert shows the real (dashed line) and imaginary (dotted line) parts of the phonon self-energy for F= 0.5, q = 0.1, T= 100 K. This corresponds to O)o-- 0.125. 469
Volume 172, number 6
PHYSICS LETTERS A
i m a t i o n t r e a t m e n t o f C o u l o m b c o r r e l a t i o n s allows us to r e p r o d u c e t h e fact t h a t the t r a n s i t i o n t e m p e r a t u r e a c h i e v e s its m a x i m a l v a l u e for finite d o p i n g , a feature w h i c h is o b s e r v e d e x p e r i m e n t a l l y . T h e lack o f c o r r e l a t i o n s s h o w s up in a c o m p l e t e l y d i f f e r e n t beh a v i o u r - the t r a n s i t i o n t e m p e r a t u r e is m a x i m a l at half-filling. We h a v e s i m u l a t e d t h e m o m e n t u m dep e n d e n c e o f t h e e l e c t r o n - p h o n o n i n t e r a c t i o n in a s i m p l y way. T h e m a g n i t u d e o f Tc can strongly dep e n d on the r e g i o n o f the B r i l l o u i n z o n e in w h i c h gk,k * q a c h i e v e s its m a x i m a l value. We c o n c l u d e t h a t the c o m b i n a t i o n o f e l e c t r o n i c c o r r e l a t i o n s a n d a strong e l e c t r o n - p h o n o n i n t e r a c t i o n can serve as o n e o f the i m p o r t a n t m e c h a n i s m s resulting in h i g h - t e m perature superconductivity.
References [ 1 ] J. Hubbard, Proc. R. Soc. A 276 ( 1963 ) 238. [2] K.A. Chao, J. Spatek and A.M. Oleg, J. Phys. C l0 (1977) L271; J.E. Hirsch, Phys. Rev. Lett. 54 ( 1986 ) 1317. [3] J.G. Bednorz and K.A. Miiller, Z. Phys. B 64 (1986) 989.
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