Ž .Physica C 317–318 1999 304–311

The non-Fermi liquid behavior and diagonal lattice stripes incuprate superconductors

A. Bianconi a,), N.L. Saini a, A. Lanzara a, J. Avila b,c, M.C. Asensio b,c, S. Tajima d,G.D. Gu d, N. Koshizuka d

a Unita INFM and Dipartimento di Fisica, UniÕersita di Roma ‘La Sapienza’, P. le Aldo Moro 2, 00185 Rome, Italy´ `b LURE, Bat 209D, UniÕersite Paris-Sud, F-91405 Orsay, France´

c Instituto de Ciencia de Materiales de Madrid, CSIC, 28049 Madrid, Spaind SuperconductiÕity Research Laboratory, ISTEC, Shinonome 1-10-13 Koto-ku, Tokyo 135, Japan

Abstract

We have studied momentum distribution of photoemission intensity around the Fermi surface of Bi Sr CaCu O2 2 2 8qd

Ž .Bi2212 high T superconductor by angle-resolved photoemission. The line shape of the energy distribution curves show acŽ .spectral function A k,v expected for a non-Fermi liquid behaviour. The Fermi surface contour in the ‘even symmetry’

Ž .reveals the suppression of spectral weight due to coupling with joint incommensurate charge density wave ICDW atŽ . Ž . Ž .q y0.4p ,0.4p in the diagonal direction and the spin density wave SDW at G p ,p . Additional spectral weight is foundŽ .at 0.6p ,0.6p due to scattering of electrons with SDW at optimum doping as revealed by constant energy contour in ‘odd

symmetry’. q 1999 Published by Elsevier Science B.V. All rights reserved.

Keywords: Electronic structure; Incommensurate charge density wave; Spin density wave; Stripe structure

1. Introduction

Ž .Bi Sr CaCu O Bi2212 is a unique system2 2 2 8qd

that could be studied to investigate the lattice versuselectronic structure of the high T superconductors.c

Ž .In fact the lattice distortions due to the 1 mismatchbetween the CuO layers and the intercalated rock-2

Ž . Ž .salt layers and 2 one-dimensional 1D ordering ofinterstitial oxygen, usually difficult to resolve inother superconducting families, give a 1D long-rangesuperstructure. The resulting structure is formed bywires of stoichiometric Bi2212 perovskite of sections

) Corresponding author. Tel.: q39-649-914-405; Fax: q39-649-576-97; E-mail: antonio.bianconi@roma1.infn.it

˚ ˚;15 A=10 A embedded in a strongly distortedw xlattice 1 . In this heterogeneous structure, two com-

ponents of doped charges have been observed in theŽ .CuO planes using a fast and local short range2

probe as EXAFS and X-ray diffuse scattering experi-Ž . Ž .ments: 1 pseudo Jahn–Teller polarons and 2 an

w xitinerant electron gas 2–6 forming a polaronic gen-eralized Wigner charge density wave coexisting withthe electron gas. This charge segregation arises fromthe instability of a polaron gas near the criticaldensity for a polaronic generalized Wigner charge

Ž . w xdensity wave CDW , at doping 0.125 7–9 . Theordering of polarons gives a particular segregation ofthe two components in alternate parallel stripes. Thisphase is quite stable in the Bi2212 near the optimumdoping with the ‘short-range’ lattice-charge modula-

0921-4534r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0921-4534 99 00073-8

( )A. Bianconi et al.rPhysica C 317–318 1999 304–311 305

tion, due to ordering of polarons in the CuO plane,2

and the crystalline ‘long range’ superstructure due tothe dopants and the mismatch have the same incom-mensurate in-plane wavevector q;0.21bU.

In this particular phase the electron gas is pushedtowards an intermediate regime between the 1DŽ . Ž .Luttinger liquid and a 2D Fermi liquid electrongas. Therefore, the electron gas is expected to showa breakdown of the one-electron picture with typical

w xfeatures of a marginal Fermi liquid 10,11 , wherethe dispersing quasiparticles are strongly coupled

w xwith collective lattice, spin and charge 12 excita-tions. Indications of collective spin excitation or spin

Ž .density wave SDW are given by a number ofw xinelastic neutron scattering experiments 13 . In such

a situation the photoemission experiments are ex-pected to show features similar to that observed in

w x1D conductors 14–16 where the spin–charge sepa-ration and collective excitations play important roles.

Here we have addressed the problem of implica-tion of the collective modes of excitations and per-formed a series of angle-resolved photoemissionŽ .ARPES measurements on a representative Bi2212system to investigate electronic states near the Fermilevel. We have combined the two modes of ARPESto study the representative Bi2212 system at theoptimum doping. The first one is the conventionalmode, based on the measurement of energy distribu-

Ž .tion curves EDC while the second one is theunconventional mode, based on measuring distribu-tion of spectral weight near the k-space within a

Ž .narrow energy window at the Fermi energy EFw x17–19 . The k-distribution of the electronic statesnear the Fermi level in an energy range of 50 meV ismeasured by the angle scanning mode, consideringthe fact that the high T superconductivity involvesc

the pairing of the electrons near the Fermi level,interacting with collective excitations within a char-acteristic cut-off energy of the order of about 50meV. This measurement is combined with conven-tional mode of ARPES, based on EDC. This ap-proach provides clear identification to the key fea-tures related with the collective excitations in thehigh T superconductors. In this paper we will dis-c

Ž .cuss two main features: 1 an asymmetric suppres-Ž .sion of spectral weight around p ,0 and equivalent

Ž . Ž .locations; and 2 shadow bands at 0.5p ,0.5p andequivalent locations. The results demonstrate that, in

the normal phase, the electrons move in a 1D super-lattice of quantum stripes of distorted and undis-torted lattice and they lose their single particle char-acter.

2. Experimental

Ž .A single crystal T ;91 K of size 6=6=1c

mm3, grown by the floating zone method was usedw xfor the experiments 20,21 . The crystal was aligned

by the standard method using specular laser reflec-tion from the crystal surface. The experiments werecarried out at the Laboratoire pour l’Utilisation du

Ž . ŽRayonnement Electromagnetique LURE Orsay,.France on the SU6 beamline. The clean and flat

surface was obtained by cleaving the crystal at roomtemperature. The experiments were performed in an

Ž . ŽUltra-High Vacuum UHV chamber base pressurey11 .7=10 mbar equipped with an angle-resolved

hemispherical analyzer and a high precision manipu-lator which permits rotation in the full 3608 azimuth

Ž . Ž .emission angle f and 908 polar emission angle u

w xrelative to the surface normal 22 . The photoelectronintensities at the Fermi level and below E wereF

recorded along a series of azimuth scans. The samplewas rotated around its normal and the intensity wasrecorded every 1.58 at fixed u , with an absoluteangular precision better than 0.58. The experimentwas repeated with different cleavage, each cleavagegave the same constant energy contour proving thehigh quality of the crystal, and the surface was stablefor several days.

The constant energy contours were obtained bycentering the electron energy window at the E andF

collecting the electrons within an energy window ofŽ .the order of spectrometer resolution ;50 meV

using a photon energy of 32 eV. The polarizationvector of the synchrotron light, the direction of thephoton beam and the normal surface were kept in thesame horizontal plane, called ‘scattering plane’, forall the measurements. The ‘mirror’ plane is definedby the sample normal and the direction of the emit-ted photoelectron selected by the detector position.The detector is moved in the fixed ‘mirror’ plane, bychanging the polar angle, in order to select differentvalues of the wavevector in the superconducting

Ž .CuO plane k . The direction of the initial state k2 5 5

( )A. Bianconi et al.rPhysica C 317–318 1999 304–311306

is selected by rotating the sample around its normal,that is collinear with the crystallographic c-axis.

We have measured the constant energy contoursin two different experimental geometries namely‘even’ and ‘odd’. In the ‘even’ geometry, the scatter-ing plane is coplanar with the ‘mirror plane’ while inthe ‘odd’ geometry the ‘mirror’ plane is orthogonal

w xto the scattering plane 23–28 .The photoemission signal is proportional to the

photoionization cross–cross section given by theFermi Golden rule

ds 2² :A C k APp C k d E yE yhnŽ . Ž . Ž .f i f idV

where p is the momentum operator and A the vectorpotential of the photon beam collinear with the pho-

Ž .ton electric field of the energy hn , C k is the finalfŽ .state at energy E , C k is the initial state at energyf i

E . The matrix element must be invariant underi

symmetry operations, and so only those combina-tions of initial state, final state and vector potentialthat leave the matrix element invariant contribute to

Ž .the photoemission signal. The final state C k mustf

be even with respect to the ‘mirror plane’ to detect asignal.

The momentum operator APp has the same sym-Žmetry as APr r being the direction of the emitted

. Ž .photoelectron . In the ‘even’ ‘odd’ experimentalŽ .geometry the momentum operator has even odd

symmetry with respect to the mirror plane, in fact theŽ .electric field lies within out the mirror plane.

Ž .Therefore in the ‘even’ ‘odd’ geometry only initialŽ .state wave functions of even odd symmetry with

respect to the mirror plane are selected.ŽThe initial states of b symmetry formed by a1

.2 2mixing of Cu 3d and O 2p orbitals are evenx yy x , y

with respect to the mirror plane for k parallel to thei

Cu–O–Cu bonds while they are odd with respect tothe mirror plane for k at 458 to the Cu–O–Cui

bonds. Thus in the ‘even’ experimental geometry,the transition from these states forming the conduc-

Ž .tion band is fully allowed for k in the G–M 0,pi

and equivalent directions and forbidden for k in theiŽ .G–X p ,p and equivalent directions. On the con-

trary in the ‘odd’ geometry the transition from theseŽ .states is fully allowed for k in the G–X p ,p andi

equivalent directions and forbidden for k in theiŽ .G–M 0,p and equivalent directions.

3. Results and discussion

In Fig. 1 we have plotted the energy distributioncurves in the G–X and G–M directions as 2D pic-ture where the photointensity is represented by

Ž .colours. We have plotted the band dispersion E kin the G–X and G–M directions, obtained by plot-ting the energy positions of the dispersing peaks of

Ž .EDC dotted lines . Similar band dispersions on theBi2212 system have been measured by many groupsand interpreted in terms of dispersing quasiparticlesand fitted with band structure models. However, thedirect view of the raw data show that the standarddata analysis neglects key features of the electronicstructure in this material.

It is clear from Fig. 1 that the dispersing broadpeaks show energy dispersion for k -k-k is ofm F

the order of 300 meV and the peaks disappear for˚Ž . Ž .k-k s0.2 2pra as5.4 A in the G–X andm

˚Ž .k-k s0.2prd ds3.8 A in the G–M direction.m

In addition, the line shape of the EDCs are stronglyasymmetric relative to the broad maxima and thisasymmetry is strongly k dependent. The low bindingenergy and the high binding energy parts have differ-ent dispersions. While the low binding energy part

Fig. 1. 3D view of EDC measured in the normal state of Bi2212sample along two high symmetry directions, G – X and G – M.The intensity is highest in the center of the figures. The EDCshow dispersing broad peaks extending up to a few hundreds ofmeV. Dotted lines indicate the standard one electron dispersion.The solid lines indicate the dispersion of spinon-like and holon-likebranches.

( )A. Bianconi et al.rPhysica C 317–318 1999 304–311 307

follows qualitatively the standard dispersion, the highbinding energy part has an opposite dispersion withlong tails, extending up to very high binding energyŽ .)600 meV that is well below the commonly

Žassumed bottom of the conduction band about 300.meV .

These anomalous features are quite similar to thatw xobserved in the 1D ladder systems 14–16 . In fact,

the EDC on the 1D systems exhibit long tails extend-ing to high binding energy and interpreted in term of

w xdispersion of spinons and holons 16 . The shape ofcurves shown in Fig. 1 indicate directly that theelectrons in the conduction band have a non-Fermiliquid behavior. We have plotted the expected dis-

Žpersion of holon-like and spinon-like branches solid. w xlines following the argument of Laughlin 29 . The

maximum dispersion for the spinon-like branch isabout p Jr2 while the maximum dispersion for the

Žholon-like branch is about 2 t J;125 meV and.trJ;3 . The dispersion of the spinon-like branch

Ž .more coherent follows the low energy part of theEDC curves while the dispersion of the holon-like

Ž .branch more incoherent follows the high energypart of the EDC curves. Therefore the anomalousdispersion shown by the raw data in Fig. 1 could bea consequence of the 1D-like character of the cupratesystems induced by the modulation of the lattice,

w xforming a superlattice of quantum stripes 1–6 , thatreduces the dimensionality of the CuO plane from2

2D toward the 1D.Fig. 2 shows the global view of the constant

energy contour measured near the chemical potentialof the Bi2212 superconductor in the ‘even’ geome-try. The dark regions indicate higher intensity ofemitted photoelectrons excited from the initial statehaving constant energy E and in-plane wavevectorFŽ .k spanned over reciprocal space of the 2D CuO5 2

plane. The Fermi surface image shows broken seg-ment with hot spots, i.e., missing parts of weight,

w xindicating charge stripes in the CuO plane 30 . We2

could also identify a new set of electronic states withŽ . Ž .a 1D-like dispersion in the Gs 0,0 to M s p ,0 ,1

direction crossing the Fermi level at one point k sFw x0.2"0.03p 31 .

The clear experimental evidence for stripes isŽ .given by the large suppression about 70% of spec-

Ž .tral weight on the Fermi surface FS in the firstBrillouin zone. After a careful analysis we have

Fig. 2. Fermi surface of an optimally doped Bi Sr CaCu O2 2 2 8qd

system, measured at room temperature using the angle scanningmethod. The Fermi surface map measured in the ‘even’ geometry.The intensity distribution is given by gray scale and increasesfrom light to dark colors and the maximum corresponds to theblack colour. The wavevector of the electrons in the CuO plane2

˚Ž .is given in units of p rd d is Cu–O–Cu distances3.8 A . WeŽ . Ž .have used the following notations: G s 0,0 , M s p ,0 , Ms1

Ž . Ž . Ž .0,p , X s p ,p , Y s yp ,p , where G – M is along the Cu–O–Cu bond direction. The circular arcs represent the expectedFermi surface.

identified two well-defined wavevectors on the FSconnecting the points where a suppression of thespectral weight occurs. These points in the first

Ž . Ž .Brillouin zone are P 0.2p ,0.8p , P 0.8p ,0.2p ,1 2Ž . Ž .P y0.8p ,y0.2p , P y0.2p ,y0.8p . They3 4

Ž .are connected by the vectors G p ,p andŽ .q y0.4p ,0.4p as shown in Fig. 2. These vectors

are the wavevectors of collective excitations thatsuppress the spectral weight of the quasiparticles at

Ž .P . We notice that the wavevector q y0.4p ,0.4pn

is diagonal and it is the second harmonic of theincommensurate and anharmonic lattice fluctuationtherefore it can be associated with diagonal chargedensity waves CDW. On the other hand, the vectorŽ .G p ,p is the antiferromagnetic wavevector and it is

the wavevector of spin density waves that have beenw xobserved in the superconducting phase 13 .

It should be stressed that the suppression has anintrinsic nature and is not a result of other effects

( )A. Bianconi et al.rPhysica C 317–318 1999 304–311308

such as crossing of the ‘umklapp’ satellite bandsŽdisplaced by wavevector "Q due to diffraction

.from the superstructure in the Bi–O plane andsatellite ‘shadow’ bands displaced by wavevector Gwith the main Fermi surface. To further highlightthis fact, we have shown in Fig. 3, the spectralphotointensity higher than 50% of the maximum, andcompared with the calculated tight binding curvesw x26–28 for the main, ‘umklapp’ and ‘shadow’ bands.It is evident that the points P , where the largen

suppression of spectral weight occur, lie at the fourcrossing points of the main and the shadow bands.Moreover these points can be located at the crossingpoint of the main Fermi surface and the lines con-necting the M–M points.1

We have assigned the suppression of photointen-sity on the Fermi surface map to charge fluctuationsassociated with lattice stripe ordering in the CuO2

Ž .plane along the diagonal direction, i.e., yp ,p .The present results identify the wavevector ofthe collective excitation coupled with electrons atthe Fermi level. This diagonal wavevectorŽ .q y0.4p ,0.4p indicates that this mode is an in-

commensurate CDW mode, the second harmonic of

Fig. 3. Plot of the experimental photointensity around the Fermisurface after cutting off the low intensity features due to satellitesin the first Brillouin zone. The points P of highest suppression ofn

spectral weight are indicated.

the 1D lattice fluctuations. Moreover the joint inter-action of electrons with a SDW of wavevectorŽ .G p ,p is clearly indicated by the fact the points Pn

are connected also by the G vector.The presence of strong scattering of electrons by

SDW has been predicted for a doped antiferromag-w xnetic system 32–34 . In the presence of strong

scattering with SDW we expect the suppression ofthe spectral weight at the eight points in the firstBrillouin zone at the crossing of the main Fermisurface with a shadow band shifted in the k-space byŽ .G 0.5p ,0.5p from the main one. We observe that

the suppression of spectral weight occurs only atŽ .four points see Fig. 2 instead of eight, indicating a

preferential coupling of electrons at the Fermi levelalso with a 1D SDW, however, only in the lattice

Ž . Žstripe direction p ,p i.e., in the direction of lattice.modulation .

Moreover, the presence of strong scattering ofelectrons with spin fluctuations is expected to giveadditional spectral weight in k-space, out of the mainFermi surface, the so-called shadow band. Theseshadow features were first seen by Osterwalder et al.w x w x18 and Aebi et al. 19 in a photoemission studymade in the angle scanning mode. However thepresence of these features have been always an issueof controversy because of their weak intensity. Thepresence of the spectral weight on the shadow bandshas been confirmed by other groups investigating

w xsamples at optimum doping 26–28 and in the over-w xdoped regime 35 , however, the identification of

these features is difficult since they overlap in manypoints with the so called umklapp bands along theŽ .yp ,p direction. We have recently shown theshadow features with significant intensity aroundŽ . w x"1.5p ,"1.5p in constant energy contour 30measured in the ‘even’ geometry. However, the tran-

Ž .sition in the p ,p direction in the first Brillouinzone is dipole forbidden for an initial state of perfect3d 2 2 symmetry. Actually we observe that thex yy

dipole selection rule completely suppresses the exci-tation from initial states with wavevectors in the kdirection perpendicular to the stripe directionŽ .yp ,p , but it is not completely active for initial

Ž .states with k direction in the stripe direction p ,p .In fact for this direction we have observed a non-zerointensity also in the ‘even’ symmetry and an associ-ated weak intensity on the shadow band.

( )A. Bianconi et al.rPhysica C 317–318 1999 304–311 309

Ž .The initial states in the "p ,"p directions canbe selected and studied by using the experimental‘odd’ geometry. The constant energy contour mea-sured in the ‘odd’ geometry is shown in Fig. 4. Itcould be seen that the photointensity along the G–Mand G–M directions is absent due to matrix element1

effects in this experimental geometry. In fact in thisexperimental geometry the transitions are allowedfrom initial states having odd symmetry with respectto the mirror plane in the G–X and G–Y directions.The contribution from the shadow bands could be

Ž .well differentiated along the p ,p direction that isfree from any overlapping ‘umklapp’ band while itappears to be overlapping with ‘umklapp’ in theŽ .yp ,p direction.

To further investigate k-space topology of theshadow bands we have measured k-dependent pho-

Ž .tointensity along X p ,p directions, shown in Fig.5. The main Fermi surface appears as a broad peakstructure around 0.35p while the shadow featureappears with a peak around 0.65p . The direct mea-surement of the photointensity, plotted in the con-stant energy scans in the two orthorhombic direc-tions show that the contribution of shadow band issignificant. It is evident that the photointensity due to

Fig. 4. Fermi surface contour measured at room temperature usingthe angle scanning method in the ‘odd’ geometry. The notationsare similar to the one used for the map measured in the ‘even’

Ž .geometry Fig. 2 .

Fig. 5. Photointensity measured by polar scan along the G – Xdirection using ‘odd’ geometry.

main band at the Fermi surface is different in the twoŽ .directions, being higher in the X p ,p direction.

This may be due to interfering ‘umklapp’ bands atthe Fermi surface in this direction. We have foundexact location of the Fermi surface crossing of these

w xfeatures by gradient method 36 . The location of themain band at the Fermi surface is found to be around0.40p while the shadow band appears to be at 0.6p .

Ž .The small pocket centered at y0.5p ,0.5p ,formed by the main band and the shadow band has

Ž .nesting vector N y0.2p ,0.2p , going from the pointŽ .at y0.4p ,0.4p on the main Fermi surface to the

Ž .point y0.6p ,0.6p on the shadow band. It is rele-vant to mention that the wavevector of the lattice

Ž .stripe modulation Q ; 0.21 " 0.02 p ,p ;Ž .N y0.2p ,0.2p . This is the reason for overlapping

Ž .of ‘umklapp’ and shadow band in the yp ,pdirection.

ŽThe ratio of the photointensity within a window.of 50 meV of the shadow band along the X direc-

tion to the main band is found to be 0.24"0.2. Weassign the shadow features to coupling of electronswith SDW. The amplitude of the spin fluctuationshas been found to decrease in the overdoped regime,however a large coupling of electrons with SDW isobserved in the present work in the optimum dopingregime. There are other experimental evidenceswhich support the presence of antiferromagnetic cor-

w xrelations even at overdoping 37 . It has also beenargued that these shadow features may appear due to

( )A. Bianconi et al.rPhysica C 317–318 1999 304–311310

orthorhombic symmetry of the Bi2212 system whichw xgives folding of the Brillouin zone 38–40 , how-

ever, this possibility may be rare as argued on thebasis of energy dependence of the photointensity

w xratio of shadow band to the main band 35 .We have shown that in the lattice stripe scenario

w x41 , the optimum doping is at the crossover from a1D to a 2D behavior in the CuO plane, that taking2

into account the c-axis transport corresponds to atransition from 2D to 3D. This dimensional crossoveris due to the variation of the chemical potential withcharge density moving from a partial gap in thetransversal direction of the superlattice of quantumstripes in underdoped regime to an energy regionwhere the transversal hopping is allowed in theoverdoped regime. Therefore at the optimum andoverdoped regime the more delocalized electrons are

Ž .allowed to move in the stripes of localized polaroniccharges where spin fluctuations are expected to belocated. In the broken Fermi surface with largeanisotropy in the DOS found in the present work,momentum decoupling associated with small q-scattering is expected to dominate the pairing inter-

w xaction 42 .In summary, a complete k-space mapping of the

Fermi surface features of Bi2212 superconductor atthe optimum doping is done by angle scanning pho-toemission in the even and odd symmetry of thetransition, allowing us to identify the wavevector ofthe 1D CDW and the SDW interacting with theelectron gas at the Fermi level. The line shapes showa strong analogy with 1D conductors. In conclusion,the Fermi surface reported in the present work indi-cates that electrons moving in a superlattice of quan-tum stripes are strongly interacting with spin andcharge collective excitations.

Acknowledgements

This work was partially funded by IstitutoŽ .Nazionale di Fisica della Materia INFM and Con-

Ž .siglio Nazionale delle Ricerche CNR of Italy. Theexperiment was done at the Spanish–French beamline of LURE with the support of the large scaleInstallation Program and the Spanish agency DGI-CYT under grant PB-94-0022-C02-01.

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