Physica C 398 (2003) 13–19

www.elsevier.com/locate/physc

Effect of out-of-plane interactions on normal statespectral properties of bilayer cuprate

Govind *, S.K. Joshi

Cryogenics and Superconductivity Group, National Physical Laboratory, Dr. K.S. Krishnan Marg, New Delhi 110012, India

Received 17 December 2002; received in revised form 11 April 2003; accepted 2 May 2003

Abstract

We have investigated the effect of intrabilayer coupling and out-of-plane Coulomb correlation on spectral properties

of bilayer cuprates in their normal state. The electron correlations which exist in the individual CuO2 layers are de-

scribed by tk–t0–t00–U model. The coupling between the two layer of same unit cell is included by hopping matrix element

t? and out-of-plane Coulomb correlation U?. Calculations of the electronic spectral function have been made for

various values of intrabilayer hopping and out-of-plane Coulomb correlation at different k-points of Brillouin zone in

the overdoped regime. It is found through our numerical calculations that the intrabilayer coupling provides favorable

condition for splitting of quasiparticle peak of spectral function, while out-of-plane Coulomb correlation tries to

suppress this effect. Calculations for density of states and hole density have also been made.

� 2003 Elsevier B.V. All rights reserved.

Keywords: Bilayer high-Tc cuprates; Spectral function; Coulomb interaction

1. Introduction

The copper oxides are unusual in that the un-

derdoped materials are antiferromagnetic insu-

lator but doping converts them into high-Tcsuperconductor [1–3]. Knowledge of the electronicstructure is critical step towards a microscopic

understanding of this fascinating behaviour. To

study the electronic structure of these systems in

their normal as well as in superconducting state,

angle resolved photoemission spectroscopy (AR-

PES) has emerged as one of the most informative

* Corresponding author. Tel.: +91-11-25732016; fax: +91-11-

25730593.

E-mail address: govind@csnpl.ren.nic.in (Govind).

0921-4534/$ - see front matter � 2003 Elsevier B.V. All rights reserv

doi:10.1016/S0921-4534(03)01173-0

experimental tool. Recent availability of high res-

olution in the angle resolved photoemission spec-

tra measurements [4–24], revealing the direct

relation between energy and momentum, refresh

experimental as well as theoretical study of the

spectral function Aðk;xÞ of cuprates. The data onAðk;xÞ suggests the presence of strong correlation

in the cuprate systems.

Among the various HTSC system Bi2Sr2CaCu2-

O8þd (Bi2212), contains two CuO2 layers in its unit

cell, is one of the most studied cuprate system

using ARPES technique [10–23]. The ARPES

studies on Bi2212 carried out by Dessau et al.

[11] have suggested that spectral weight transferstrongly depends upon doping concentration. They

observed a peak-dip-hump structure in the spectral

function. Feng et al. [18] have studied electronic

ed.

14 Govind, S.K. Joshi / Physica C 398 (2003) 13–19

structure of heavily overdoped sample of Bi2212

system. They found that the bilayer band splits

into two bands in normal as well as in supercon-

ducting state. The maximum energy splitting oc-

curs around (p; 0) point in the momentum space. A

similar conclusion has also been drawn by Chuanget al. [21] using high resolution ARPES data of the

bilayer Bi2212. Their study suggests doubling of

EF bands in the overdoped regime. They observed

that the splitting of band approaches to zero along

the ð0; 0Þ ! ðp;pÞ nodal line and maximum near

the (p; 0) point.On the theoretical side, lots of work has been

done so far to study the electronic structure ofbilayer system in their normal state [25–30]. Dif-

ferent workers have used different method to ex-

plain the features of spectral function ðAðk;xÞÞ inthese cuprates [25–30]. Some of the researchers

have used t–J model [25–27] while others Hubbard

model [28–30]. In the studies carried out with t–Jmodel, various workers [25–27] have considered

intrabilayer exchange coupling along with intra-bilayer hopping interaction. However these studies

with t–J model are supposed to be very successful

at low doping. On the other hand, these systems

are strongly correlated and it is believed that

Hubbard model could be able to well describe

the physics of these system. However most of the

studies within Hubbard model [28–30] ignores the

presence of out-of-plane Coulomb correlation inthe system which is equivalent to the intrabilayer

exchange coupling in t–J model.Moreover Lal et al.

[31] have proposed a model considering out-of-

plane correlation while studying c-axis resistivity

of the cuprate. They incorporate the importance of

Cu3d3z2�r2 and O2pz orbitals for which the elec-

tronic charge distribution is directed normal to the

CuO2 plane along with the Cu3dx2�y2 and O2px;y

orbitals. Srivastava et al. [32] have also drawn

similar conclusions on the basis of the polariza-

tion-dependent L3-absorption measurement on

Tl2212 system. They suggest that the overlap of

Cu3d3z2�r2 orbital with optical O 2pz is consider-

able and should provide a channel for the out-of-

plane conduction of charge carriers.

In view of the above it is clear that the essentialphysics of these bilayer cuprates will be captured if

we set a model Hamiltonian which includes int-

rabilayer coupling as well as out-of-plane Cou-

lomb correlation. Here we have planned to study

the effect of intrabilayer coupling and out-of-plane

Coulomb correlation on the spectral properties of

bilayer cuprates (Bi2212) in their normal state. We

describe the individual layer by tk–t0–t00–U modeland couple these layers via intrabilayer hopping

ðt?Þ and out-of-plane Coulomb correlation ðU?Þ.We used Green�s function technique within mean

field approximation to obtain the expressions for

spectral function, density of state (DOS) and hole

density. The detail of our theoretical formulation

is presented in Section 2. We have discussed the

results of our numerical analysis in Section 3. Fi-nally, we conclude our results in Section 4.

2. Theoretical calculation

The high-Tc cuprates are strongly correlated

systems [3]. In a strongly correlated system the

Coulomb correlation energy is much larger thanthe hopping integral. To define strong correlation

in a bilayer system, having two CuO2 layers per

unit cell (Bi2212), we consider the Hubbard model.

In a bilayer cuprate system the distance between

two CuO2 layers within unit cell is smaller than

that of single layer cuprate system. Hence the

coupling between the two layers within the same

unit cell of a bilayer system is important comparedto coupling between layers in two neighbouring

cell for single layer system. We have described each

of the individuals layers by a tk–t0–t00–U model and

consider that the two layers within a cell are cou-

pled via hopping of holes from one layer to another

i.e. intrabilayer hopping as well as out-of-plane

Coulomb correlation between the Cu sites of these

two layers. For describing a bilayer cuprate withtwo coupled CuO2 layers, we use the following

Hamiltonian:

H ¼Xi;j;a

ðtij � lÞCþiraCjra þ U

Xi;a;r

nirani�ra

þXi;ra;b

t?CþirbCirb þ U?

Xi;r;a;b

nirani�rb ð1Þ

where tij includes tk (nearest neighbour hoppingmatrix element), t0 (next nearest neighbour hop-

Govind, S.K. Joshi / Physica C 398 (2003) 13–19 15

ping matrix element) and t00 (next next nearest

neighbour matrix element). t? is hopping matrix

element between the planes i.e. intrabilayer hop-

ping interaction. U is the on site Coulomb corre-

lation energy and U? is the out-of-plane Coulombcorrelation energy. CþðCÞ is the creation (annihi-

lation) operator and nr is the number operator. aand b are the layer indices ða 6¼ bÞ. Here l is the

chemical potential i.e. the energy of the top most

filled level at T ¼ 0 K.

The Hamiltonian can be presented in momen-

tum space by performing Fourier transforma-

tion. The Hamiltonian in momentum space can beread as

H ¼Xk;r;a

e0kðCþkraCkraÞ þ U

Xk;k0 ;a;r

nkrank0�ra

þXk;ra;b

ek?ðCþkraCkrbÞ þ U?

Xk;k0r;a;b

nkrank0�rb ð2Þ

where

e0k ¼ ek � l

ek is dispersion along the ab-plane and is given by

ek ¼ �2tkðcosðkxaÞ þ cosðkyaÞÞ� 4t0ðcosðkxaÞ cosðkyaÞÞ� 2t00ðcosð2kxaÞ þ cosð2kyaÞÞ ð3aÞ

In Eq. (2), ek? is the dispersion between the planes

i.e. along c-direction. We have taken the k-dependence of ek? is such a way that this agrees

with the experimental measurements of angle re-

solved photoemission spectra. The experimentsuggests that there is no hopping of hole between

the two layers along kx ¼ ky direction [33]. This

means that ek? will not contribute for the kx ¼ kydirection. A reasonable form of ek? which satisfies

the experimental condition has been suggested by

Chakrabarty et al. [33] as

ek? ¼ �t?ðcosðkxaÞ � cosðkyaÞÞ2=4 ð3bÞIn order to obtain expression for spectral function,

DOS and hole density, we define the following

Green�s function

GðxÞ ¼ hhC1k"jCþ1k0"ii ð4Þ

where x is the energy. GðxÞ corresponds to the

motion of the particle within the same plane. Ap-

plying the Zubarev�s double time Green�s function

technique [34] we obtain the equation of motion

corresponding to GðxÞ, which is given by

xGðxÞ ¼ 12hC1k"Cþ

1k0"i þ hh½C1k"H �jCþ1k0"ii ð5Þ

Solving commutator ½C1k"�H , we get higher order

Green�s function and Green�s function correspond-ing to out-of-plane motion. To linearize these

higher orders and out-of-plane Green�s function

we employed the mean field-decoupling scheme

within Hubbard III technique [34,35]. We solve all

the Green�s function analytically and obtain ex-

pression for Green�s function GðxÞ as

G ¼ 1

x� e0k � Rðk;xÞð Þ 1

(þ Up

Xk

nk#ðx� e0k1Þ

)

ð6Þhere Up ¼ U þ U? and Rðk;xÞ is the self-energy

which can be written as

R ¼ U 2p

Xk

nk#e2k?x2

1x2

( )þ ek?Up

x2

Xk

nk#ek?x2

þ e2k?x2

ð7Þwhere x1 ¼ ðx� e0k1Þ; x2 ¼ ðx� e0kÞ and e0k1 ¼ e0k þUpnk#. Here nk# denotes the number of particle withmomentum k and spin down.

To obtain the expression for spectral function

and DOS of the system we further solve Eq. (6)

analytically using Eq. (7). Finally, we obtain the

expression for the imaginary part of Green�sfunction which can be read as

Im G ¼ R2ðk;xÞx� e0k � R1ðk;xÞð Þ2 � R2

2ðk;xÞh i

� 1

(þ Up

Xk

nk#x1

)ð8Þ

Here R1ðk;xÞ and R2ðk;xÞ are the real and imag-inary parts of self-energy respectively. The ex-

pressions for these real and imaginary parts are

R1ðk;xÞ ¼ 2U 2p

Xk

nk#/e0k1x2

1

þ U 2p

Xk

nk#/x2

þ ek?Up

x2

Xk

nk#ek?x2

þ e2k?x2

ð9Þ

-0.25 -0.15 -0.05 0.05 0.15

Energy relative to Fermi energy

A(k

,w) (

arb,

units

)

( π ,0)

(0,0)

(π/2,0)

-0.25 -0.15 -0.05 0.05 0.15

(π /2 , π /2 )

( π , π )

(0,0)

(a) (b)

Fig. 1. Spectral function Aðk;xÞ for different k-points of

Brillouin zone with tj ¼ 0:4 eV, t0 ¼ �0:3tk, t00 ¼ 0:2tk,t? ¼ 0:3tk, U ¼ 3:2 eV, U? ¼ 1:0 eV, T ¼ 0:5tk, and d ¼ 0:3. (a)

Aðk;xÞ along C–M direction i.e. (0, 0)–(p; 0) and (b) along C–Xdirection i.e. (0, 0)–(p; p).

16 Govind, S.K. Joshi / Physica C 398 (2003) 13–19

R2ðk;xÞ ¼ �2pU 2p

Xk

nk#/e0k1x1

dðx1Þ

� pU 2p

Xk

nk#/dðx2Þ

� pek?Updðx1ÞXk

nk#ek?ðx1Þ

!

� pe2k?dðx1Þ ð10Þ

where

/ ¼ e2k?½e02k1 � 2e0k1e

0k�

ð11Þ

The electronic spectral function of the system is

given by negative of imaginary part of Green�sfunction.

Aðk;xÞ ¼ �ImGðk;xÞ ð12ÞThe DOS of the system can be evaluated by inte-

grating the spectral function for all possible k-values

NðxÞ ¼Z p

�p

Z p

�pAðk;xÞdkx dky ð13Þ

and finally, the hole density ðdÞ is obtained by

integrating the DOS over the occupied energy

states

d ¼Z l

�1NðxÞdx ð14Þ

We solve numerically Eqs. (12)–(14) for different

values of intrabilayer hopping and out-of-plane

Coulomb correlation and present results of our

analysis in the next section.

3. Results and discussion

In the present work, we have calculated spectralfunction, DOS and hole density of bilayer cuprate

for arbitrary values of the parameter t? and U?.

We have taken the data suggested by LDA cal-

culation carried out by Anderson et al. [36] and the

recent calculation of spectral function by Pratap

et al. [25] for bilayer cuprate system. We consider

inplane hopping integral tk ¼ 0:4 eV, and n.n.

hopping t0 ¼ �0:3tk, n.n.n. hopping t00 ¼ 0:2tk andU ¼ 3:2 eV.

We first present our calculation for the spectralfunction for (d ¼ 0:3, overdoped) with t? ¼ 0:3tkand U? ¼ 0:33U . Spectral function Aðk;xÞ is

plotted in two panel in the Fig. 1. In panel 1(a), we

plot the spectral function with energy ðxÞ along

C–M (i.e. (0, 0) to (p; 0)) direction and in panel

1(b) along C–X (i.e. (0, 0) to (p; p)) direction. On

the basis of our calculation, we observe that the

electron like quasiparticle character above Fermienergy increases while going from (0, 0) to (p=2; 0)and further for (p=2; 0) to (p; 0Þ direction (Fig. 1a).

A similar behaviour of spectral function is also

observed along C–X direction. These features are

very much similar to those observed in single layer

cuprates [3,37].

Next, we study the effect of intrabilayer cou-

pling on the spectral function. In Fig. 2, we haveplotted the spectral function for higher values of

intrabilayer hopping integral (i.e. t? ¼ 0:5tk). We

observe from Fig. 2(a) that a splitting of band

appears when we go from (0, 0) direction to

(p=2; 0) direction. The strength of this band split-

ting further increases and becomes maximum at

(p; 0) point. A similar bilayer splitting is observed

by Feng et al. [18] for overdoped bilayer Bi2212system. They have observed that the bilayer band

splits into two band namely antibonding bands

(AB) and bonding bands (BB). Their experimental

observations suggest that Fermi surface of the

system overlap in the nodal region (i.e. along (p; p)

-0.25 -0.15 -0.05 0.05 0.15

Energy relative to Fermi energy

A(k

,w)(a

rb. U

nits

)

( π ,0)

(π/2,0)

(0,0)

-0.25 -0.15 -0.05 0.05 0.15

(0,0)

(π /2 , π /2 )

( π , π )

(a) (b)

Fig. 3. Effect of out-of-plane correlation energy on spectral

function Aðk;xÞ for different k-points of Brillouin zone with

tk ¼ 0:4 eV, t0 ¼ �0:3tk, t00 ¼ 0:2tk, t? ¼ 0:5tk, U ¼ 3:2 eV,

U? ¼ 2:0 eV, T ¼ 0:5tk, and d ¼ 0:3. (a) Aðk;xÞ along C–Mdirection and (b) along C–X direction.

-0.25 -0.15 -0.05 0.05 0.15

Energy relative to Fermi energy

A(k

,w)((

abr.

Uni

ts)

( π ,0)

(π/2,0)

(0,0)

-0.25 -0.15 -0.05 0.05 0.15

(0,0)

(π /2 , π /2 )

( π , π )

(a) (b)

Fig. 2. Effect of intrabilayer coupling on spectral function

Aðk;xÞ for different k-points of Brillouin zone with tk ¼ 0:4 eV,

t0 ¼ �0:3tk, t00 ¼ 0:2tk, t? ¼ 0:5tk, U ¼ 3:2 eV, U? ¼ 1:0 eV,

T ¼ 0:5tk, and d ¼ 0:3. (a) Aðk;xÞ along C–M direction and (b)

along C–X direction.

Govind, S.K. Joshi / Physica C 398 (2003) 13–19 17

direction) and gradually depart from each other

when approaching the (p; 0) region. They observe

that the strength of splitting will be maximum

around (p; 0) point. We do not observe any peak

splitting along in C–X direction. However, we

observe that the spectral function shows a rela-tively broad feature along (p=2; p=2) direction. Toclarify the possibility of peak splitting around

(p=2; p=2) direction, we have performed calcula-

tions for various k-point near (p=2; p=2) point. Our

calculations reveal that there is no splitting present

along (p=2; p=2) direction. The details of these

calculation is beyond the scope of the present pa-

per. Our numerical results are found to be in ac-cord to the ARPES studies carried out by Chuang

et al. [21] where they have observed that the peak

splitting in bilayer band occurs only in C–M di-

rection and no such splitting present along C–Xdirection. Our numerical results are also found in

agreement with the bilayer LDA calculation [3]

and bilayer Hubbard model [30] calculations.

Comparing Figs. 1 and 2 we observe that the shapeof spectral function does not change on increasing

intrabilayer hopping along kx ¼ ky direction i.e. (0,

0), (p=2; p=2) and (p; p) point of Brillouin zone. A

similar feature has also been predicted by Chak-

ravarty et al. [33] where they have suggested that

there is no hole hopping along kx ¼ ky direction.The effect of out-of-plane Coulomb correlation

on spectral function has been presented in Fig. 3.

We observe that the peak height of spectral func-

tion is reduced on increasing the out-of-plane

Coulomb correlations. It can also be seen from Fig.

3 that the bilayer peak splitting disappears on in-

creasing out-of-plane Coulomb correlation. Com-

paring Figs. 1–3, it is also clear that intrabilayer

coupling tries to enhance the strength of spectralfunction while out-of-plane Coulomb correlations

suppress the peak height of the spectral function.

Thus, we found that there is competition between

intrabilayer hopping and out-of-plane coupling i.e.

the effect of intabilayer hopping is counter bal-

anced by the presence of strong out-of-plane

Coulomb correlation. One can infer from the figure

that if the out-of-plane Coulomb correlation isstrong enough then one needs higher values of

intrabilayer coupling to obtain splitting of bilayer

bands. Since intrabilayer hopping depends upon

doping concentration, hence to obtain higher val-

ues of intrabilayer hopping one has to go for over-

doped regime. This addresses the issue of occurrence

of maximum splitting in the overdoped regime.

Next, in Fig. 4 we plot single particle DOS withenergy for different values of out-of-plane Cou-

lomb correlations. For an overdoped system d ¼0:3, we observe that the DOS features two peaks

below and above Fermi level (Fig. 4(a)). On in-

creasing out-of-plane Coulomb correlation (U? ¼2:0 eV), the electronic contribution to DOS de-

creases and the quasiparticle peak broaden in

-1 -0.5 0 0.5Energy Relative to Fermi energy

N(w

)

(a)

(b)

Fig. 4. Single particle density of states for different values of

out-of-plane Coulomb correlation for tk ¼ 0:4 eV, t0 ¼ �0:3tk,t00 ¼ 0:2tk, t? ¼ 0:5tk, U ¼ 3:2 eV, T ¼ 0:5tk, and d ¼ 0:3. Curve

(a) out-of-plane Coulomb correlation U? ¼ 1:0 eV and (b)

U? ¼ 2:0 eV.

0

0.1

0.2

0.3

0.4

0.5

−1.09 −1.06 −1.03 −1 −0.97µ

δ

(a)

(d)

(b)

(c)

Fig. 5. Hole density versus chemical potential for different

values of intrabilayer coupling and out-of-plane Coulomb

correlation with tk ¼ 0:4 eV, t0 ¼ �0:3tk, t00 ¼ 0:2tk, U ¼ 3:2 eV,

T ¼ 0:5tk, and d ¼ 0:3. Curve (a) t? ¼ 0:1tk, U? ¼ 1:0 eV, (b)

t? ¼ 0:3tk, U? ¼ 1:0 eV, (c) t? ¼ 0:5tk, U? ¼ 1:0 eV, and (d)

t? ¼ 0:3tk, U? ¼ 2:0 eV.

18 Govind, S.K. Joshi / Physica C 398 (2003) 13–19

compare to the low value of the out-of-plane

Coulomb correlation (Fig. 4(b)). These results sug-

gest that the Coulomb correlation suppress theavailable electronic state above the Fermi energy.

These results are in accord with that of Jacklic and

Prelovesek [38] for overdoped case where they have

shown that the DOS contains a quasiparticle peak

and broad background.

Finally, we present the results for the hole

density ðdÞ as a function of bilayer hopping and

out-of-plane Coulomb correlations in Fig. 5. Wehave calculated hole density corresponding to the

bilayer hopping matrix element t? ¼ 0:1tk, 0.3tkand 0.5tk and different values of out-of-plane

Coulomb correlation U? ¼ 1:0, 2.0 eV. From this

figure, it is clear that for a given value of tk, t0, t00,U , U?, number of hole increases with intrabilayer

hopping ðt?Þ. This increase in hole density with t?can be explained as on increasing the intrabilayerhopping Fermi energy increases, therefore the

number of available states upto Fermi energy also

increases and hence hole density ðdÞ. In Fig. 5, we

have also plotted the effect of out-of-plane Cou-

lomb correlation on hole density. Comparing

curves 5(b) and (d) it is clear that number of

available holes decreases on increasing out-of-

plane Coulomb correlation.

4. Conclusion

In the conclusion, we observe that the intrabi-layer hopping significantly affect the behaviour of

spectral function. We observed bilayer splitting in

the quasiparticle peak of spectral function along

C–M direction for higher value of intrabilayer

hopping. The strength of these peak splitting is

found maximum around point (p; 0). We do not

find any bilayer splitting along ð0; 0Þ ! ðp; pÞ di-

rection. The out-of-plane Coulomb correlation isfound to suppress the peak height of the spectral

function. We observe that on increasing out-of-

plane Coulomb correlation, bilayer splitting dis-

appears. We also observe that the out-of-plane

Coulomb correlation suppress the electronic part

of the single particle DOS. Finally, we observe that

hole density of the bilayer system increases on in-

creasing intrabilayer hopping and decreases whenout-of-plane Coulomb correlation increases.

Acknowledgements

This work was financially supported by the

Department of Science & Technology (DST),

Govind, S.K. Joshi / Physica C 398 (2003) 13–19 19

Government of India via grant no. SP/S2/M-

32/99.

References

[1] W.E. Pickett, Rev. Mod. Phys. 61 (1989) 433.

[2] B. Keimer, N. Belk, R.J. Birgeneau, A. Cassanho, C.Y.

Chen, M. Greven, M.A. Kastner, A. Aharnoy, Y. Endoh,

R.W. Erwin, G. Shirane, Phys. Rev. B 46 (1992) 14034.

[3] E. Dagatto, Rev. Mod. Phys. 66 (1994) 763.

[4] Z.X. Shen, D.S. Dessau, Phys. Rep. 253 (1995) 1.

[5] T. Sato, T. Kamiyama, Y. Naitoh, T. Takahashi, I. Chong,

T. Terashima, M. Takano, Phys. Rev. B 63 (2001) 13502.

[6] J. Mesot, M. Renderia, M.R. Norman, A. Kaminski, H.M.

Fretwell, J.C. Campuzano, H. Ding, T. Takeuchi, T. Sato,

T. Yokoya, T. Takahashi, I. Chong, T. Terashima, M.

Takano, T. Mochiku, K. Kadowaki, Phys. Rev. B 62

(2001) 224516.

[7] T. Yoshida, X.J. Zhou, M. Nakamura, S.A. Kellar, E.D.

Lu, A. Lanzara, Z. Hussain, A. Ino, T. Mizokawa, A.

Fujimori, H. Eisaki, C. Kim, Z.X. Shen, T. Kakeshita, S.

Uchida, Phys. Rev. B 63 (2001) 220501.

[8] A. Ino, T. Mizokawa, K. Kobayashi, A. Fujimori, T.

Sasagawa, T. Kimura, K. Kishino, K. Tamasaku, H.

Esaki, S. Uchida, Phys. Rev. Lett. 81 (1998) 2124.

[9] A. Ino, C. Kim, M. Nakamura, T. Yosida, T. Mizokawa,

Z.-X. Shen, A. Fujimori, T. Kakeshita, H. Eisaki, S.

Uchida, Phys. Rev. B 62 (2000) 4137.

[10] Y.D. Chuang, A.D. Gromko, D.S. Dessau, Y. Aiura, Y.

Yamaguchi, K. Oka, A.J. Arko, J. Joyce, H. Eisaki, S.I.

Uchida, K. Nakamura, Y. Ando, Phys. Rev. Lett. 83

(1999) 3717.

[11] D.S. Dessau, B.O. Wells, Z.X. Shen, W.E. Spicer, A.J.

Arko, R.S. List, D.B. Mitzi, A. Kapitulnik, Phys. Rev.

Lett. 66 (1991) 2160.

[12] P. Aebi, J. Osterwalder, P. Schwaller, L. Sclapbach, M.

Shimoda, T. Mochiku, K. Kadowaki, Phys. Rev. Lett. 72

(1994) 2757.

[13] H. Ding, A.F. Bellman, J.C. Campuzano, M. Renderia,

M.R. Norman, T. Yokoya, T. Takahashi, H.K. Yoshida,

T. Mochiku, K. Kadowaki, G. Jennings, G.P. Brivio, Phys.

Rev. Lett. 76 (1996) 1533.

[14] N.L. Saini, J. Avila, A. Binaconi, A. Lanzara, M.C.

Asensio, S. Tajima, G.D. Gu, N. Koshizuka, Phys. Rev.

Lett. 79 (1997) 3467.

[15] Z.X. Shen, J.R. Schrieffer, Phys. Rev. Lett. 78 (1997) 1771.

[16] C. Kim, P.J. White, Z.X. Shen, T. Tohyama, Y. Shibata, S.

Meakawa, B.O. Welles, Y.J. Kim, R.J. Birgeneau, M.A.

Kaster, Phys. Rev. Lett. 80 (1998) 4245.

[17] M.S. Golden, S.V. Borisenko, S. Lenger, T. Pichler, C.

Durr, M. Knupfer, J. Fink, G. Yang, S. Abell, G.

Reichaedt, R. Muller, C. Janowitz, Physica C 341–348

(2000) 2099.

[18] D.L. Feng, N.P. Armitage, D.H. Lu, A. Damascelli, J.P.

Hu, P. Bogdanov, A. Lanzara, F. Ronning, K.M. Shen, H.

Esaki, C. Kim, Z.X. Shen, J.J. Shimoyama, K. Kishio,

Phys. Rev. Lett. 86 (2001) 5550.

[19] J. Mesot, M. Boehm, M.R. Norman, M.Renderia, N.

Metoki, A. Kaminki, S. Rosenkranz, A. Hiess, H.M.

Fretwell, J.C. Campuzano and K. Kadowaki, Lanl pre-

print cond-mat/0102339.

[20] Y.D. Chuang, A.D. Gromko, A. Fedorov, D.S. Dessau, Y.

Aiura, K. Oka, Y. Ando, H. Eisaki, S.I. Uchida, Phys.

Rev. Lett. 87 (2001) 117002.

[21] Y.D. Chuang, A.D. Gromko, A. Fedorov, Y. Aiura, K.

Oka, Y. Ando and D.S. Dessau, Lanl preprint cond-mat/

0107002.

[22] A.A. Kordyuk, S.V. Borisenko, M.S. Golden, S. Legner,

K.A. Nenkov, M. Knupfer, J. Fink, H. Berger, L. Forro,

R. Follath, Phys. Rev. B 66 (2002) 014502.

[23] S.V. Borisenko, A.A. Kordyuk, S. Langer, C. Duerr, M.

Knupfer, M.S. Golden, J. Fink, K. Nevkov, D. Eckert, G.

Yang, S. Abell, H. Berger, L. Forro, B. Liang, A. Maliouk,

C.T. Lin, B. Keimer, Phys. Rev. B 64 (2001) 094513.

[24] D.L. Feng, A. Damascelli, K.M. Shen, N. Motoyama,

D.H. Lu, H. Eisaki, K. Shimizu, J.i. Shimoyama, K.

Kishio, N. Kaneko, M. Greven, G.D. Gu, X.J. Zho, C.

Kim, F. Ronning, N.P. Armitage, Z.X. Shen, Phys. Rev.

Lett. 88 (2002) 107001, Lanl preprint.

[25] A. Pratap, R. Lal, Govind, S.K. Joshi, Phys. Rev. B 64

(2001) 224512.

[26] Govind, R. Lal, S.K. Joshi, unpublished.

[27] R. Eder, Y. Ohta, G.A. Sawatzky, Phys. Rev. B 55 (1997)

R3114.

[28] R. Putz, R. Preuss, A. Muramatsu, W. Hanke, Phys. Rev.

B 53 (1996) 5133.

[29] Th.A. Maier, Th. Prushke, M. Jarrell, Phys. Rev. B 66

(2002) 075102.

[30] A.I. Liechtenstein, O. Gunnarson, O.K. Andersen, R.M.

Martin, Phys. Rev. B 54 (1996) 12505.

[31] R. Lal, Ajay, R.L. Hota, S.K. Joshi, Phys. Rev. B 57

(1998) 6126.

[32] P. Srivastava, F. Studer, K.B. Garg, Ch. Gasser, H.

Murray, M. Pompa, Phys. Rev. B 54 (1996) 693.

[33] S. Chakarvarty, A. Sudbo, P.W. Anderson, S. Strong,

Science 261 (1993) 337.

[34] D.N. Zubarev, Sov. Phys. USP 3 (1960) 303.

[35] Ajay, A. Pratap, S.K. Joshi, Physica C 371 (2002) 139.

[36] O.K. Anderson, A.I. Liechtenstein, O. Jepsen, F. Paulson,

J. Phys. Chem. Solids 56 (1995) 1573.

[37] T. Tohyama, S. Meakawa, Supercond. Sci. Technol. 13

(2000) R17.

[38] J. Jacklic, P. Prelovesek, Phys. Rev. B 55 (1997)

R7307.