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Theory of hole superconductivity in high Tc superconductors: BCS-Gorkov formalism in extended Hubbard model
0038-1098/92 $5.00 + .00 Pergamon Press plc
Solid State Communications, Vol. 81, No. 10, pp. 867-870, 1992. Printed in Great Britain.
T H E O R Y O F H O L E S U P E R C O N D U C T I V I T Y IN H I G H T, S U P E R C O N D U C T O R S : B C S - G O R K O V F O R M A L I S M IN E X T E N D E D H U B B A R D M O D E L Salila Das and N.C. Das Department of Physics, Berhampur University, Berhampur-760007, Orissa, India
(Received 14 October 1991 by E. Burstein) A theory of superconductivity in high T,. oxide superconductors is developed including on site Hubbard interaction and off diagonal Coulomb correlations which give rise to a modified hopping. We have used a recently proposed model Hamiltonian for describing this system and have obtained the energy spectrum in the presence of Coulomb correlations. BCS pairing in this modified energy spectrum has been considered to obtain the superconducting gap function which has explicit dependence on carrier concentration. Calculation of T,. using this gap function confirms to the experimental variation of T, with carrier (hole) concentration. 1. I N T R O D U C T I O N A F T E R T H E discovery of high T, materials theoretical studies have been as numerous as experimental investigations but without a significant break through. At present diverse views [1-7] exist on the mechanism that gives rise to superconductivity in high Tc superconductors. Nevertheless, there are enough indications [8] from the experimental results in support of a BCS type pairing in these systems. Since the materials behave as M o t t - H u b b a r d insulators in their normal phase the majority of the theorists use a Hubbard like model as the starting point of their theoretical study. However, while adopting such a model one should bear in mind that a system with a large on site Coulomb interaction, U, is unlikely to allow any second-order process, of electron-phonon, electronexciton or electron-plasmon origin, to be effective in giving rise to superconductivity. In this context the search for a first-order process as a source of superconductivity becomes a necessary choice. Some recently proposed models [7, 9] deserve detail and careful study as they suggest the possibility of an attractive interaction even in the presence of a large on site Coulomb repulsion U. On the experimental side, the results on various superconducting properties are beset with intriguing features. This makes the job all the more difficult for theorists to establish connection between their theory and experiment. It has now become essential for the theorists to identify the simplest model that can exhibit basic physics and make contact with experiments. The recently proposed model based on hole pairing [7] has achieved some amount of success in its
qualitative description of the variation of T, with charge carrier concentration, n. The essential feature of this model is that holes are the key component of superconductivity in high Tc oxides where superconductivity originates from hole pairing and conduction in oxygen anion network. The Cu dx2 y2 orbital is assumed to play a passive role except for properties involving interplay of magnetism and superconductivity. A satisfactory theory for these materials needs to treat the effect of a large Coulomb correlation U. Simultaneously, it must take care of the fact that an increase of hole concentration by doping drives the system from an insulating to superconducting phase and back to a normal conducting phase at higher hole concentration. It is very essential that the modification in the energy spectrum in the presence of strong correlations be considered before invoking BCS pairing of the charge carriers in the system. In an attempt in this direction we have used a model Hamiltonian [7] and treated the correlation effects following Hubbard procedure to obtain the modified energy spectrum in the presence of strong correlations. The BCS pairing of holes has been considered and G o r k o v formalism based on finite temperature Green's function is used to obtain an energy gap function, which has explicit dependence on charge carrier concentration. We take the inplane model Hamiltonian for the holes as H
=
-~ i,j,a i,j,a
* tijCi~Cja + U E niTnil i i,a
where i,j, are sites on a two-dimensional square lattice
867
868
H O L E S U P E R C O N D U C T I V I T Y I N H I G H T, S U P E R C O N D U C T O R S
and the c operators describe the holes. The interaction described by the third term [7] arises quite generally in electronic energy bands in solids from off-diagonal matrix elements [10] (Hubbard ( i i ] l / r l o ' ) term) of Coulomb interaction and v~j has its origin in the hybrid Coulomb integral [I1] between atomic orbitals on nearest neighbour sites. This term has been identified [7] to be the one which contributes to an increase in hopping probability of a hole between two neighbouring sites when one of the sites has been already occupied by another hole. The Hamiltonian describes the direct hopping of holes from 0 2-. to neighbouring oxygen anion in a two-dimensional CuO2 plane. The present problem of hole propagation has close analogy with the one, discussed by H u b b a r d and co-workers [10] for propagation of an electron of spin through an alloy consisting of two species of sites randomly distributed in the proportions n_.~ and 1 - n ~. On the basis of earlier works for electrons we formulate [12] the problem for holes with appropriate modification. We introduce the notations [10]
dTa
=
nT. aCia'
E
dTa
:
taking into consideration s-wave pairing of holes. We obtain equation of motion for H u b b a r d sub-band operators [I 0] ifib[~(t)
=
EkvD;,~(t) + AkvD~k o(t),
D[~(t) with
1
=
e+
Ek~ - ~ d~,~(t),
=
U , e,
2vkn N~,. flk =
ni~,
n,~
=
1-
T~vNj, v ilk,
(4c)
(Sk ~) + (Pk ~)n.
(4d)
Sk ~ and Pk ~ are Fourier transforms of S~ , and P~ o respectively [12]. Both (Sk ~) and (Pk ~ ) are small [10] and may be neglected in the first approximation Nkll I =
~
(Ek,.
n±
(4e)
~)2,
and the superconducting gap function Ak~ is given by
Cie'
ni~
.
(4f) We define the thermodynamic Green's functions
~ ((ni_~,c,~,; c~,,))E = ~=±
N k v ( D "'~, , D ~~', , )
+ 1
(~ = +_),
and evaluate the propagator
G,~(E) =
(4b)
0,
=
- - - e~
+
(4a)
where v = + refers to the sub-bands,
with ni~ =
Vol. 81, No. 10
~ G77(E), (2)
G~;'(~ - ~') w ''t
~=+
F .o(,
-
,')
=
- (T~[D~(OD['~(z')]),
=
v*
-(T~[D
v 'f
(5a) l
~ ~(z)D~(v
)]),
(5b)
where the component G,j~+(E) is non-vanishing on sites which have a - a spin hole on them and G~- (E) is non-vanishing only on atoms without a - a spin hole. We obtain the energy spectrum of H u b b a r d sub-bands given by
for the system, we obtain their equations of motion and then use Fourier transform and carry on frequency sum [13] over u~. = 2(n + l)~/flt~ to obtain [12]
Ek+
(D"k .D;'.)
=
½(U + ek -- 2 n v , ) + ½[(U -- e~ + 2nvk) 2
-- 4nU(v~ -- e~)
+
4n2v~(vk
-- 2U)]
1/2 - -
/2.
(3a)
Here E,+ and E~ refer to upper and lower sub-bands respectively. The hole kinetic energy ~, is given by 2tA~,
Ak =
cos kx + cos k,.,
Wk,,
~-
(6b)
Using equation (6a) in equation (4f) we have the k dependent energy gap function given by .
.
.
.
k',,' L ( \ ek
(3b) × T~,,
2AtAk,
wk~)la,,,v, (6a)
[/~d "1- [Ak,,[2] I/2.
and 7.) k
Nk~Ak,, 2g~we,, [f(wk,,) -- f ( -
where
:
where
-
2v~ (vk,
ek \v~
_,)
) } Nk~Nk,,,Ak,,,,
(3c)
with t u = t and v u = At for i, j nearest neighbours and we have taken n ~ = n~ = n for the parawhere magnetic case. We obtain a Heisenberg equation of motion for dT~ f ( w k v ) and linearise them using the earlier procedure [10] and
x (f(wk'v)
-
f ( - wk,,.'))/,
(7)
J
(8) -
e~"k" +
1
Voi. 81, No. 10
H O L E S U P E R C O N D U C T I V I T Y IN H I G H T, S U P E R C O N D U C T O R S
869
We ol~tain [12] the average number of charge carriers n (hole) per site consistent with chemical potential/~ n
=
Nkv
N-'Z x[(,
ez + w%W~)
+
(, w%
300
(9) 2. N U M E R I C A L R E S U L T S A N D DISCUSSION As seen from equations (3a), (4c) and (7) the k-dependent gap function has explicit dependence on charge carrier (hole) concentration n given by equation (9). In order to test the validity of our theoretical results we have studied the variation of Tc with hole concentration n. We have calculated the chemical potential # for each value of hole concentration n and used this value of # to calculate Tc from equation (7). As observed earlier [7] the feature Tc vs n is a general property of the hole pairing mechanism and independent of any fine structure in the density of states. In case of a constant density of states we obtain for T = T,. with A(Tc) = O, the relation n~l+4t.
#
(10)
We have made use of this result for calculating # for each n. In order to solve the integral equation (7) involving Ak for obtaining T,, we have assumed the form Ak =
A(T)A,.
(11)
Here A(T,) ~ 0 and Ak is given by equation (3b). In carrying out the summations, i.e. the sum over k in equation (7), we have neglected the interband contribution noting that the separation of bands is large compared to band width and the chemical potential lies in the lower band. The results of our calculation for T,. vs n is shown in Fig. 1. We have used the parameter t = 0.065 eV, At = 0.195 eV and U = 5 eV. Our calculation of T, for values o f n between 0.01 to 0.065 shows that at first T, increases with increase of n and after passing through a maximum near n = 0.05 it starts falling rapidly. Our results are in excellent qualitative agreement with experimental results and confirm the earlier observation [7]. It is to be noted that the range o f n for which T, is non-zero is dependent on the nearest neighbour Coulomb repulsion v which we have not included in the present formulation. The presence of v shifts the range of variation of Tc to higher values of n along with scaling down of the peak value of T,. The study of the feature of T~ vs n in relation to its depen-
rc~
200
lo0
0
I
0,02
t
0.Oz
n
I
006
0.-6~r-
Fig. 1. Variation of T,. with hole concentration, n, with on site Coulomb repulsion U = 5 eV and hopping interaction At = 0.195eV and hopping parameter t = 0.065eV. dence on the parameters t, U, v and At and the formulation of other superconducting properties like Hcz(T) and specific heat jump is in process. In obtaining Fig. 1 we have used in equation (7) the value of Ekv [equation (4c)] which depends on both n and Vk. The v~ dependent terms of Ek+, however, have not been included in numerical calculation for Fig. 1. The n dependence of Ek_+ in equation (3a) are considered through 4n Uek term. Our recent calculation [12] using the same values of parameters t, At and U and including vk dependent terms in Ek+_ indicates that the range of n extends considerably (n = 0.003 to n = 0.44) and the peak of the curve decreases. The curve in this case becomes a little fiat. This points to the fact that the pairing interaction vk in E k +- influences the dependence of T, on n appreciably.
Acknowledgements - This work was supported by the U.G.C. grant. We acknowledge the facilities made available for the work in the computer center of Berhampur University. REFERENCES 1.
P.W. Anderson, Science 235, 1196 (1987); Z. Zou & P.W. Anderson, Phys. Rev. B37, 627 0988); P.W. Anderson, G. Baskaran, Z. Zou &
870
2. 3. 4. 5. 6. 7.
8.
HOLE SUPERCONDUCTIVITY IN HIGH T, SUPERCONDUCTORS T. Hsu, Phys. Rev. Lett. 58, 2790 (1987); G. Baskaran, Z. Zou & P.W. Anderson, Solid State Commun. 63, 973 (1987); P.W. Anderson & Z. Zou, Phys. Rev. Left. 60, 132 (1988) and references therein. V.J. Emery, Phys. Rev. Lett. 58, 2794 (1987). J.R. Schrieffer, X.G. Wen & S.C. Zhang, Phys. Rev. Lett. 60, 944 (1988). J. Ruvalds, Phys. Rev. B35, 8869 (1987). F.C. Zhang & T.M. Rice, Phys. Rev. B37, 3759 (1988). C.M. Varma, S. Schmitt-Rink & E. Abrahams, Solid State Commun. 62, 681 (1987). J.E. Hirsch & S. Tang, Solid State Commun. 69, 987 (1989); J.E. Hirsch & F. Marsiglio, Phys. Rev. B39, 11515 (1989); J.E. Hirsch & F. Marsiglio, Phys. Left. A140, 122 (1989) and references therein. Proceedings of the International Conference on
9.
10. 11. 12. 13.
Vol. 81, No. 10
Materials and Mechanism of Superconductivity, High T, Superconductors II, Stanford, 1989 (Edited by R.N. Shelton, W.A. Harrison & N.E. Phillips), Physiea 162-164e (1989) and references therein. K.P. Jain, R. Ramakumar & C.C. Chancey, Phys. Rev. B (to be published); K.P. Jain, R. Ramakumar & C.C. Chancey, Physica-C (to be published). J. Hubbard, Proc. Roy. Soc. A276, 238 (1963); Proc. Roy. Soc. A281, 401 (1964); J. Hubbard & K.P. Jain, J. Phys. CI, 1650 (1968). J.C. Slater, Quantum Theory of Molecules and Solids, McGraw-Hill, New York (1963), Appendix-6. S. Das & N.C. Das (to be published). A.L. Fetter & J.D. Walecka, in Quantum Theory of Many Particle System, p. 227, McGraw-Hill, New York (1971).
Solid State Communications, Vol. 81, No. 10, pp. 867-870, 1992. Printed in Great Britain.
T H E O R Y O F H O L E S U P E R C O N D U C T I V I T Y IN H I G H T, S U P E R C O N D U C T O R S : B C S - G O R K O V F O R M A L I S M IN E X T E N D E D H U B B A R D M O D E L Salila Das and N.C. Das Department of Physics, Berhampur University, Berhampur-760007, Orissa, India
(Received 14 October 1991 by E. Burstein) A theory of superconductivity in high T,. oxide superconductors is developed including on site Hubbard interaction and off diagonal Coulomb correlations which give rise to a modified hopping. We have used a recently proposed model Hamiltonian for describing this system and have obtained the energy spectrum in the presence of Coulomb correlations. BCS pairing in this modified energy spectrum has been considered to obtain the superconducting gap function which has explicit dependence on carrier concentration. Calculation of T,. using this gap function confirms to the experimental variation of T, with carrier (hole) concentration. 1. I N T R O D U C T I O N A F T E R T H E discovery of high T, materials theoretical studies have been as numerous as experimental investigations but without a significant break through. At present diverse views [1-7] exist on the mechanism that gives rise to superconductivity in high Tc superconductors. Nevertheless, there are enough indications [8] from the experimental results in support of a BCS type pairing in these systems. Since the materials behave as M o t t - H u b b a r d insulators in their normal phase the majority of the theorists use a Hubbard like model as the starting point of their theoretical study. However, while adopting such a model one should bear in mind that a system with a large on site Coulomb interaction, U, is unlikely to allow any second-order process, of electron-phonon, electronexciton or electron-plasmon origin, to be effective in giving rise to superconductivity. In this context the search for a first-order process as a source of superconductivity becomes a necessary choice. Some recently proposed models [7, 9] deserve detail and careful study as they suggest the possibility of an attractive interaction even in the presence of a large on site Coulomb repulsion U. On the experimental side, the results on various superconducting properties are beset with intriguing features. This makes the job all the more difficult for theorists to establish connection between their theory and experiment. It has now become essential for the theorists to identify the simplest model that can exhibit basic physics and make contact with experiments. The recently proposed model based on hole pairing [7] has achieved some amount of success in its
qualitative description of the variation of T, with charge carrier concentration, n. The essential feature of this model is that holes are the key component of superconductivity in high Tc oxides where superconductivity originates from hole pairing and conduction in oxygen anion network. The Cu dx2 y2 orbital is assumed to play a passive role except for properties involving interplay of magnetism and superconductivity. A satisfactory theory for these materials needs to treat the effect of a large Coulomb correlation U. Simultaneously, it must take care of the fact that an increase of hole concentration by doping drives the system from an insulating to superconducting phase and back to a normal conducting phase at higher hole concentration. It is very essential that the modification in the energy spectrum in the presence of strong correlations be considered before invoking BCS pairing of the charge carriers in the system. In an attempt in this direction we have used a model Hamiltonian [7] and treated the correlation effects following Hubbard procedure to obtain the modified energy spectrum in the presence of strong correlations. The BCS pairing of holes has been considered and G o r k o v formalism based on finite temperature Green's function is used to obtain an energy gap function, which has explicit dependence on charge carrier concentration. We take the inplane model Hamiltonian for the holes as H
=
-~ i,j,a i,j,a
* tijCi~Cja + U E niTnil i i,a
where i,j, are sites on a two-dimensional square lattice
867
868
H O L E S U P E R C O N D U C T I V I T Y I N H I G H T, S U P E R C O N D U C T O R S
and the c operators describe the holes. The interaction described by the third term [7] arises quite generally in electronic energy bands in solids from off-diagonal matrix elements [10] (Hubbard ( i i ] l / r l o ' ) term) of Coulomb interaction and v~j has its origin in the hybrid Coulomb integral [I1] between atomic orbitals on nearest neighbour sites. This term has been identified [7] to be the one which contributes to an increase in hopping probability of a hole between two neighbouring sites when one of the sites has been already occupied by another hole. The Hamiltonian describes the direct hopping of holes from 0 2-. to neighbouring oxygen anion in a two-dimensional CuO2 plane. The present problem of hole propagation has close analogy with the one, discussed by H u b b a r d and co-workers [10] for propagation of an electron of spin through an alloy consisting of two species of sites randomly distributed in the proportions n_.~ and 1 - n ~. On the basis of earlier works for electrons we formulate [12] the problem for holes with appropriate modification. We introduce the notations [10]
dTa
=
nT. aCia'
E
dTa
:
taking into consideration s-wave pairing of holes. We obtain equation of motion for H u b b a r d sub-band operators [I 0] ifib[~(t)
=
EkvD;,~(t) + AkvD~k o(t),
D[~(t) with
1
=
e+
Ek~ - ~ d~,~(t),
=
U , e,
2vkn N~,. flk =
ni~,
n,~
=
1-
T~vNj, v ilk,
(4c)
(Sk ~) + (Pk ~)n.
(4d)
Sk ~ and Pk ~ are Fourier transforms of S~ , and P~ o respectively [12]. Both (Sk ~) and (Pk ~ ) are small [10] and may be neglected in the first approximation Nkll I =
~
(Ek,.
n±
(4e)
~)2,
and the superconducting gap function Ak~ is given by
Cie'
ni~
.
(4f) We define the thermodynamic Green's functions
~ ((ni_~,c,~,; c~,,))E = ~=±
N k v ( D "'~, , D ~~', , )
+ 1
(~ = +_),
and evaluate the propagator
G,~(E) =
(4b)
0,
=
- - - e~
+
(4a)
where v = + refers to the sub-bands,
with ni~ =
Vol. 81, No. 10
~ G77(E), (2)
G~;'(~ - ~') w ''t
~=+
F .o(,
-
,')
=
- (T~[D~(OD['~(z')]),
=
v*
-(T~[D
v 'f
(5a) l
~ ~(z)D~(v
)]),
(5b)
where the component G,j~+(E) is non-vanishing on sites which have a - a spin hole on them and G~- (E) is non-vanishing only on atoms without a - a spin hole. We obtain the energy spectrum of H u b b a r d sub-bands given by
for the system, we obtain their equations of motion and then use Fourier transform and carry on frequency sum [13] over u~. = 2(n + l)~/flt~ to obtain [12]
Ek+
(D"k .D;'.)
=
½(U + ek -- 2 n v , ) + ½[(U -- e~ + 2nvk) 2
-- 4nU(v~ -- e~)
+
4n2v~(vk
-- 2U)]
1/2 - -
/2.
(3a)
Here E,+ and E~ refer to upper and lower sub-bands respectively. The hole kinetic energy ~, is given by 2tA~,
Ak =
cos kx + cos k,.,
Wk,,
~-
(6b)
Using equation (6a) in equation (4f) we have the k dependent energy gap function given by .
.
.
.
k',,' L ( \ ek
(3b) × T~,,
2AtAk,
wk~)la,,,v, (6a)
[/~d "1- [Ak,,[2] I/2.
and 7.) k
Nk~Ak,, 2g~we,, [f(wk,,) -- f ( -
where
:
where
-
2v~ (vk,
ek \v~
_,)
) } Nk~Nk,,,Ak,,,,
(3c)
with t u = t and v u = At for i, j nearest neighbours and we have taken n ~ = n~ = n for the parawhere magnetic case. We obtain a Heisenberg equation of motion for dT~ f ( w k v ) and linearise them using the earlier procedure [10] and
x (f(wk'v)
-
f ( - wk,,.'))/,
(7)
J
(8) -
e~"k" +
1
Voi. 81, No. 10
H O L E S U P E R C O N D U C T I V I T Y IN H I G H T, S U P E R C O N D U C T O R S
869
We ol~tain [12] the average number of charge carriers n (hole) per site consistent with chemical potential/~ n
=
Nkv
N-'Z x[(,
ez + w%W~)
+
(, w%
300
(9) 2. N U M E R I C A L R E S U L T S A N D DISCUSSION As seen from equations (3a), (4c) and (7) the k-dependent gap function has explicit dependence on charge carrier (hole) concentration n given by equation (9). In order to test the validity of our theoretical results we have studied the variation of Tc with hole concentration n. We have calculated the chemical potential # for each value of hole concentration n and used this value of # to calculate Tc from equation (7). As observed earlier [7] the feature Tc vs n is a general property of the hole pairing mechanism and independent of any fine structure in the density of states. In case of a constant density of states we obtain for T = T,. with A(Tc) = O, the relation n~l+4t.
#
(10)
We have made use of this result for calculating # for each n. In order to solve the integral equation (7) involving Ak for obtaining T,, we have assumed the form Ak =
A(T)A,.
(11)
Here A(T,) ~ 0 and Ak is given by equation (3b). In carrying out the summations, i.e. the sum over k in equation (7), we have neglected the interband contribution noting that the separation of bands is large compared to band width and the chemical potential lies in the lower band. The results of our calculation for T,. vs n is shown in Fig. 1. We have used the parameter t = 0.065 eV, At = 0.195 eV and U = 5 eV. Our calculation of T, for values o f n between 0.01 to 0.065 shows that at first T, increases with increase of n and after passing through a maximum near n = 0.05 it starts falling rapidly. Our results are in excellent qualitative agreement with experimental results and confirm the earlier observation [7]. It is to be noted that the range o f n for which T, is non-zero is dependent on the nearest neighbour Coulomb repulsion v which we have not included in the present formulation. The presence of v shifts the range of variation of Tc to higher values of n along with scaling down of the peak value of T,. The study of the feature of T~ vs n in relation to its depen-
rc~
200
lo0
0
I
0,02
t
0.Oz
n
I
006
0.-6~r-
Fig. 1. Variation of T,. with hole concentration, n, with on site Coulomb repulsion U = 5 eV and hopping interaction At = 0.195eV and hopping parameter t = 0.065eV. dence on the parameters t, U, v and At and the formulation of other superconducting properties like Hcz(T) and specific heat jump is in process. In obtaining Fig. 1 we have used in equation (7) the value of Ekv [equation (4c)] which depends on both n and Vk. The v~ dependent terms of Ek+, however, have not been included in numerical calculation for Fig. 1. The n dependence of Ek_+ in equation (3a) are considered through 4n Uek term. Our recent calculation [12] using the same values of parameters t, At and U and including vk dependent terms in Ek+_ indicates that the range of n extends considerably (n = 0.003 to n = 0.44) and the peak of the curve decreases. The curve in this case becomes a little fiat. This points to the fact that the pairing interaction vk in E k +- influences the dependence of T, on n appreciably.
Acknowledgements - This work was supported by the U.G.C. grant. We acknowledge the facilities made available for the work in the computer center of Berhampur University. REFERENCES 1.
P.W. Anderson, Science 235, 1196 (1987); Z. Zou & P.W. Anderson, Phys. Rev. B37, 627 0988); P.W. Anderson, G. Baskaran, Z. Zou &
870
2. 3. 4. 5. 6. 7.
8.
HOLE SUPERCONDUCTIVITY IN HIGH T, SUPERCONDUCTORS T. Hsu, Phys. Rev. Lett. 58, 2790 (1987); G. Baskaran, Z. Zou & P.W. Anderson, Solid State Commun. 63, 973 (1987); P.W. Anderson & Z. Zou, Phys. Rev. Left. 60, 132 (1988) and references therein. V.J. Emery, Phys. Rev. Lett. 58, 2794 (1987). J.R. Schrieffer, X.G. Wen & S.C. Zhang, Phys. Rev. Lett. 60, 944 (1988). J. Ruvalds, Phys. Rev. B35, 8869 (1987). F.C. Zhang & T.M. Rice, Phys. Rev. B37, 3759 (1988). C.M. Varma, S. Schmitt-Rink & E. Abrahams, Solid State Commun. 62, 681 (1987). J.E. Hirsch & S. Tang, Solid State Commun. 69, 987 (1989); J.E. Hirsch & F. Marsiglio, Phys. Rev. B39, 11515 (1989); J.E. Hirsch & F. Marsiglio, Phys. Left. A140, 122 (1989) and references therein. Proceedings of the International Conference on
9.
10. 11. 12. 13.
Vol. 81, No. 10
Materials and Mechanism of Superconductivity, High T, Superconductors II, Stanford, 1989 (Edited by R.N. Shelton, W.A. Harrison & N.E. Phillips), Physiea 162-164e (1989) and references therein. K.P. Jain, R. Ramakumar & C.C. Chancey, Phys. Rev. B (to be published); K.P. Jain, R. Ramakumar & C.C. Chancey, Physica-C (to be published). J. Hubbard, Proc. Roy. Soc. A276, 238 (1963); Proc. Roy. Soc. A281, 401 (1964); J. Hubbard & K.P. Jain, J. Phys. CI, 1650 (1968). J.C. Slater, Quantum Theory of Molecules and Solids, McGraw-Hill, New York (1963), Appendix-6. S. Das & N.C. Das (to be published). A.L. Fetter & J.D. Walecka, in Quantum Theory of Many Particle System, p. 227, McGraw-Hill, New York (1971).