Size of the BCS ratio in strongly underdoped high-Tc cuprates

A. A. AbrikosovMaterials Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439

~Received 8 November 1996!

An explanation is proposed of the huge~over 20! BCS ratio 2D(0)/Tc discovered recently in stronglyunderdoped Bi2Sr2CaCu2O82d. Since no marked shrinking of the Fermi surface was found, our explanation isbased on the idea of disruption of the connection between CuO2 bilayers due to reduction of localized centersproviding resonant tunneling between the layers. An increase of phase fluctuations of the order parameter,mostly due to spontaneous vortex formation, leads to a possibility of a drastic reduction ofTc , which practi-cally coincides withTBKT , the temperature of the Berezinsky-Kosterlitz-Thouless transition. At the same time,the local energy gap is likely to undergo no major changes. An argument is given why at not too smallconcentrations of the localized centers the critical temperature can remain rather stable, despite strong varia-tions in the normalc-axis conductivity.@S0163-1829~97!50810-1#

I. INTRODUCTION

One of the unusual features of high-Tc cuprates is therelatively high BCS ratio, 2D(0)/Tc . Contrary to its conven-tional value 3.52, it is close to 5 in optimally dopedYBa2Cu3O7, and around 7 in Bi2Sr2CaCu2O8. Usually this isexplained by the strong interaction between electrons. On thebasis of Eliashberg theory, within formal limits of its appli-cability ~‘‘Migdal theorem’’! the ratio can reach even highervalues. On the other hand, in the limit of very strong inter-action, ‘‘preformed pairs,’’ or ‘‘bipolarons’’ are formed;then the binding energy of pairs andTc are more or lessdisconnected,Tc being the Bose-condensation temperatureof these preformed pairs. In the bipolaron version the in-crease of the electron-phonon interaction enhances the bind-ing energy and at the same time raises the effective mass ofbipolarons, decreasing their Bose-condensation temperature.In these approaches the essential quantity is the ratio of elec-tron interaction to their Fermi energy: the increase of thisratio moves the electron system in the direction of‘‘preformed pairs’’ and increases the BCS ratio. This allseemed, until recently, very clear but some results of therecent photoemission experiments with underdopedBi2Sr2CaCu2O82d,

1,2 where the so-called ‘‘pseudogap’’ wasdiscovered, cannot be explained in that way. Of course, thepseudogap, i.e., the presence of a gap in the quasiparticlespectrum aboveTc and its amazing thermal stability instrongly underdoped samples, was the most spectacularamong the phenomena. There was, however, also somethingelse~see Ref. 2!: in the same experiments it was found thatthe BCS ratio could reach values over 20, and at the sametime the Fermi surface practically did not change comparedto the optimally doped substance. This made it likely that theFermi energy also remained the same, and since there was noreason to suspect a strong increase of the interaction of elec-trons, we encountered a case where the conventional expla-nation of the large BCS ratio did not work.

What could be then the action of underdoping? In thispaper we propose a different explanation based on destruc-tion of superconductivity by phase fluctuations in low-dimensional systems. In Refs. 3–5 an idea of the mechanism

of c-axis transport in underdoped YBa2Cu3Ox was proposedbased on resonant tunneling between the CuO2 bilayersthrough localized states formed by broken CuO chains. Thesituation in Bi2Sr2CaCu2O8-d is more complicated, sincethere are no chains, and, actually the median plane betweenthe CuO2 bilayers is empty with two BiO layers located sym-metrically around it. Thec-axis conductivity could be due todirect tunneling, the more so that it is much less than inYBa2Cu3Ox , if there were no observations6,7 that the tem-perature dependence ofrc /rab is exponential with the acti-vation energy depending strongly on the sample preparation.This cannot happen in case of direct tunneling and can beconsidered as evidence of resonant tunneling through somecenters formed as defects in the regular crystalline structure.Since they are not originating from a regular structure ele-ment, their amount must be much less than in YBa2Cu3Ox ,and they have to be more vulnerable to the treatment of thesample. This can explain the lowc-axis conductivity and thedependence of the activation energy inrc /rab on samplepreparation. Moreover, it is natural to suspect that these cen-ters are wrongly positioned oxygen atoms, and they are firstto go under oxygen depletion. After that the connection be-tween the CuO2 bilayers is disrupted, and fluctuations of thephase suppress the critical temperature without any essentialchange of the electron concentration, or their interaction.

II. ESTIMATE OF Tc DEPLETION

Our estimate will be based on the method developed byEfetov and Larkin.8 The Ginsburg-Landau free energy wasderived in Ref. 9. In order to make things simple, we willsuppose the parameterh defining the connection between thesingular regions to be large. ThenCa52Cb[C, and wereturn to the isotropic situation in theab plane. In case thereis no magnetic field, and only phase fluctuations are impor-tant, we can write a reduced expression:

F5(nE dx dy$~ns

~0!/8mx!@~]wn /]x!21~]wn /]y!2#

14adns@12cos~wn2wn11!#%, ~1!

PHYSICAL REVIEW B 1 MARCH 1997-IIVOLUME 55, NUMBER 10

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where ns(0) is the superconducting electron density in one

layer,wn is the phase of the order parameter in thenth layer,a;Tc

(0)2/«F , some scaling energy andd is a dimensionlessconstant defining the connection between the layers. In Refs.4 and 9 it was shown thatd5d0cj

2, wherecj is the atomicconcentration of resonant centers andd0!1. Strictly speak-ing, the Ginsburg-Landau~GL! free energy is valid only inthe vicinity ofTc but expression~1! is very general, and it isvery likely to be applicable also beyond this vicinity, at leastfor the purpose of estimates. As in Ref. 8, we will assumethat a long-range order exists, described by an order param-eter ^exp(iw)&. Using the mean-field approximation, we getfrom the last term in Eq.~1!

22adns~0!^exp~ iw!&exp~2 iwn!.

From here it is possible to define the critical temperatureusing the self-consistency condition at infinitesimal^exp(iw)& ~see Ref. 8!:

15~2adns~0!/T!E d2p^exp@ iw~0!2 iw~r!#&. ~2!

The average is taken over one two-dimensional layer and isdefined by fluctuations;ns

(0) means that the superconductingelectron density is defined by the 2D mean-field BCS-typetheory with the critical temperatureTc

(0) .The strongest fluctuations are those associated with spon-

taneous formation of vortices. The ‘‘ordering’’ of these vor-tices, namely, their binding into pairs with opposite orienta-tion leads to the so-called Berezinsky-Kosterlitz-Thouless~BKT! transition.10,11Since this transition can happen only ina superfluid, its critical temperatureTBKT has to be belowTc , the real superconducting critical temperature. A roughestimate ofTBKT is given by the condition11

TBKT5pns2m

. ~3!

This timens is the true superconducting density in two di-mensions. It can be estimated asns5(ns /n)3n, wherens /n;(Tc2T)/Tc , and n can be equal to n(0)

5p1Py0 /p2, the total electron density, associated with a cer-

tain singular region~Py0 is the length of the region in mo-mentum space, andp1 the one-dimensional Fermi momen-tum, see Ref. 9!, but can also be smaller. In our theory, basedon the dominant role of extended saddle points, it is assumedthatPy0@p1, and hence, in casen;n(0)

«05pn~0!/~2m!5p1Py0 /~2pm!@m1@Tc~0!.TBKT ,

~4!

m15p12/2m being the chemical potential with respect to the

saddle point. According to Eq.~3!, this would meanns /n;(Tc2TBKT )/Tc;TBKT /«0!1. ThereforeTBKT must bevery close to the realTc .

The correlator, entering Eq.~2!, according to Refs. 12–14, is equal to

^exp@ iw~0!2 iw~r!#&5A exp~2r/j!, ~5!

whereA;1, and the correlation radiusj is given by

j5j0 exp$b@TBKT /~T2TBKT!#1/2%. ~6!

Herej0 is the usual superconducting coherence length andbsome constant; for qualitative estimates we can assume it isof the order of unity, as in Ref. 12. Substituting Eq.~5! intocondition ~2! defining the critical temperature, we obtain

adns~0!j2/T;1. ~7!

Here we substitute Eq.~6! and a;Tc(0)2/m1, j0

2;v12/Tc

(0)2

;m1 /(mTc(0)2), ns

(0);n(0);«0m ~we assume here thatTc'TBKT!Tc

(0) and obtain an equation for x5(Tc2TBKT)/Tc :

x

d5expS 2bAxD . ~8!

For very smalld we get from herex'@ ln(1/d)#22, or, sinceaccording to Eq.~3!, x;Tc /«0'TBKT /«0,

Tc;«0

@ ln~1/d!#2. ~9!

This relation gives the connection between the realTc andthe concentration of resonant centers in the case ofTc!Tc

(0) . At larger values ofd, when Eq.~7! formally predictsTc*Tc

(0) , the trueTc remains equal toTc(0) .

Since, according to Eq.~4!, Tc(0)!«0, a substantial de-

crease ofTc due to insufficient connection between the lay-ers starts only at rather small concentrations of resonant cen-ters, and at larger concentrationsTc remains close toTc

(0) ; itis very stable and reproducible, despite considerable varia-tions in the normal-statec-axis conductivity.

TheD~0! entering the BCS ratio is obtained as the energygap, or singularity in the density of states from the tunnelingconductance, or from angle-resolved photoemission spec-troscopy~ARPES!. The latter experiments have shown thatthis quantity remains finite high aboveTc , particularly instrongly underdoped samples~‘‘pseudogap’’!. This suggeststhatD, as a feature of a one-particle excitation spectrum, canpersist even without a long-range order, being some sort oflocal characteristic. We have no explanation at present of thepseudogap and its stability with temperature. Our only goalis to draw attention to the fact that the existence of thepseudogap does not necessarily mean the presence of long-range order, just as the presence of the long-range order doesnot necessarily lead to a gap in the spectrum of one-particleexcitations~see Ref. 15!. This means that the gap can appearin tunneling conductance and ARPES and not be diminishedby the disruption of the interlayer connection; this can resultin huge values of the BCS ratio.

ACKNOWLEDGMENTS

I would like to express my gratitude to Dr. B. Ivlevfor numerous discussions. This work was supported by theDepartment of Energy under Contract No. W-31-109-ENG-38.

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55 R6151SIZE OF THE BCS RATIO IN STRONGLY . . .