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Influence of incommensurate dynamic charge-density wave scattering on the line

shape of superconducting high-Tccuprates

G. Seibold† and M. Grilli∗† Institut fur Physik, BTU Cottbus, PBox 101344, 03013 Cottbus, Germany

∗ Istituto Nazionale di Fisica della Materia e Dipartimento di Fisica, Universita di Roma “La Sapienza”,

Piazzale A. Moro 2, 00185 Roma, Italy

We show that the spectral lineshape of superconducting La2−xSrxCuO4 (LSCO) andBi2Sr2CaCu2O8+δ (Bi2212) can be well described by the coupling of the charge carriers to col-lective incommensurate charge-density wave (CDW) excitations. Our results imply that besidesantiferromagnetic (AF) fluctuations also low-energy CDW modes can contribute to the observeddip-hump structure in the Bi2212 photoemission spectra. In case of underdoped LSCO we proposea possible interpretation of ARPES data in terms of a grid pattern of fluctuating stripes where thecharge and spin scattering directions deviate by α = π/4. Within this scenario we find that thespectral intensity along (0, 0) → (π, π) is strongly suppressed consistent with recent photoemissionexperiments. In addition the incommensurate charge-density wave scattering leads to a significantbroadening of the quasiparticle-peak around (π, 0).

I. INTRODUCTION

There is an increasing experimental evidence [1,2] thatthe peculiar properties of the superconducting cuprates,both in the normal and the superconducting phase arerelated to the occurrence of a Quantum Critical Point(QCP) located near (slightly above) the optimal doping.These evidences corroborate the consensus that quan-tum criticality related to some “exotic” [3] or more tra-ditional [4–8] type of ordering is at work in these sys-tems. Accordingly the phase diagram of the cupratesis naturally partitioned in a (nearly) ordered, a quan-tum critical, and a quantum disordered region naturallycorresponding to the under-, optimally, and over-dopedregions of the phase diagram of the cuprates respectively.In this framework a key open issue is to determine thetype of ordering which is established (at least in a localand slowly dynamical sense) in the underdoped phase ofthe cuprates. While the direct detection of the exotic or-der is more difficult and elusive, it is quite natural thatthe spin and charge fluctuations associated to the moretraditional proposals [4–7] affect in a rather apparent waythe physical properties. Especially one would expect thatthe modification of the electronic structure due to dy-namical spin- or charge scattering should be observable inangle-resolved photoemission spectroscopy (ARPES). Infact, ARPES data of high-Tc Bi2Sr2CaCu2O8+δ (Bi2212)compounds show several specific features in the super-conducting state which can be interpreted in terms of acoupling of the electrons to a collective mode [9]. BelowTc, these spectra are characterized by an unsual line-shape around (π, 0) (M-point) which consists of a sharppeak at low energy followed by a hump at higher ener-gies. Both features are separated by a dip. In additionthe sharp peak persists at low energy over a large re-gion in k-space whereas the hump correlates well withthe underlying normal state dispersion [10]. Since the

energy difference between dip and peak position is sim-ilar to the energy of the (π, π) resonance observed ininelastic neutron scattering [11] is has been suggestedthat the mode responsible for the ARPES features corre-sponds to collective spin fluctuations [12]. Although thisproposal is quite appealing one needs to understand bywhich mechanism the substantial spin fluctuations canbe sustained in optimally and overdoped materials. Infact, since doping rapidly disrupts the antiferromagnetic(AF) order and with the occurrence of an AF-QCP inthe very underdoped region where the Neel temperaturevanishes, spin scattering should be a minor effect at largedopings. This difficulty can be overcome by consideringthe (traditional) ordering in the underdoped phase to bedriven by the charge. Then the occurrence of (a localand possibly slowly dynamical) charge ordering is natu-rally accompanied by substantial spin fluctuations sincethe spin coupling resurrects in the hole-depleted regions.This strong spin and charge coupling is also an effectivemechanism driving the weak charge modulation aroundthe second-order quantum transition to strongly anhar-monic charge (and spin) stripes in the deeply underdopedmaterials. The relevant role of charge ordering has a solidexperimental support at least in some classes of cuprates.First evidence for incommensurate stripe structures inthe high-Tc compounds was given by neutron scatteringexperiments [13] in Nd-doped lanthanum cuprate. Herethe Low-Temperature Tetragonal (LTT) lattice structureis suitable to effectively pin the charge stripes givingrise to static stripe ordering so that both magnetic andcharge-order peaks become detectable. However, also inNd-free materials including the YBCO compounds wherethe magnetic incommensurate peak has been observed[14,15] the simultaneous occurence of charge order hasbeen claimed in Ref. [16] and from NQR measurementsin Refs. [17–20]. Moreover, it has has been argued [1] thatthe QCP can be deduced from the incommensurate neu-

1

tron peak intensity which vansishes in the slightly over-doped regime x≈ 0.19. This coincides surprisingly wellwith experiments in La2−xSrxCuO4 (LSCO) compoundswhere superconductivity has been suppressed by pulsedmagnetic fields [2] showing an underlying metal-insulatortransition at about the same critical doping. Recent pho-toemission experiments [21–23] on superconducting un-derdoped La2−xSrxCuO4 (LSCO) have also revealed abroad quasiparticle peak (QP) around the (π, 0) point.On the other hand no QP can be identified along the(0, 0) → (π, π) direction in contrast to the Bi2212 com-pound where along the diagonals a clear Fermi surfacecrossing has been observed. Exact diagonalization stud-ies of the t-t’-t”-J model with an additional phenomeno-logical stripe potential [24] have demonstrated that thesefeatures may be due to the coupling of the holes to ver-tical charged (static) stripes.

In the present paper we want to focus on the dy-namic aspect of incommensurate charge-density wave(CDW) scattering within the framework of a QCP sce-nario. In this context we discuss the differences betweenthe photoemission spectra of Bi2212 and LSCO respec-tively. Especially we show that the different featurescan be explained by the assumption that in Bi2212 thecharge carriers couple to an incommensurate CDW ori-ented along the (1, 0)− and (0, 1)− directions whereas inLSCO the dynamic CDW scattering is along the diag-onals. The results presented below supplement consid-erations of Ref. [25] where we have shown that a two-dimensional ’eggbox-type’ charge-pattern can reproducethe essential features of the normal state Bi2212 Fermisurface, namely the reduction of spectral weight aroundthe M-points associated with the opening of a pseudo-gap in these regions of k-space. Here we extend ourconsiderations to the case of LSCO assuming that thek-dependence of the electron self-energy is an essentialingredient in the description of ARPES data.

We note that in a recent paper [26] Eschrig and Nor-man have analyzed ARPES data within a model of elec-trons interacting with a magnetic resonance using a simi-lar approach than in the present paper. Comparing theirresults with those reported below one can see that AFand CDW scattering have similar effects on the electronicstates around the M-points and the similarity betweenour results and those in Ref. [26] provides a support tothe idea that spin and charge fluctuations coexist andcooperate in determining the spectral properties.

After having introduced the formalism in Sec. II wewill present in Sec. III the analysis of ARPES spectrafor both LSCO and Bi2212 materials. We finally con-clude our discussion in Sec. IV.

II. FORMALISM

We consider a system of superconducting electrons ex-posed to an effective action

S = −λ2∑

q

∫ β

0

dτ1

∫ β

0

dτ2χq(τ1 − τ2)ρq(τ1)ρ−q(τ2) (1)

describing dynamical incommensurate CDW fluctua-tions. Using Nambu-Gorkov notation the unperturbedMatsubara Greens function matrix G is given by

G011(k, iω) =

u2k

iω − Ek

+v2

k

iω + Ek

(2)

G022(k, iω) =

v2k

iω − Ek

+u2

k

iω + Ek

(3)

G012(k, iω) = G0

21(k, iω) = −ukvk

[

1

iω − Ek

−1

iω + Ek

]

(4)

where the BCS coherence factors are defined as u2k =

12(1+ ǫk−µ

Ek) and v2

k = 12(1+ ǫk+µ

Ek) respectively. The lead-

ing order one-loop contribution to the self-energy readsas

Σ(k, iω) = −λ2

β

∑

q,ip

χq(ip)τzG0(k − q, iω − ip)τz (5)

which in turn allows for the calculation of G via

G = G0 + G0ΣG. (6)

It should be mentioned that the self-energy eq. (5) differsfrom the analogous expression for the coupling to spin-fluctuations by the τz Pauli matrices. Finally the spectral

function can be extracted from Ak(ω) = ImG11(k, ω).Note that our approach differs from the standard

Elisahberg treatment by the fact that already the unper-turbed system displays coherent superconducting order.Since the incommensurate CDW fluctuations have beenshown to be strongly attractive in the d-wave channel [7]the idea is to incorporate this feature already on the ze-roth order level by a frequency independent d-wave orderparameter. Thus the self-energy is gapped by G0, how-ever, for our present considerations this approximation issufficient since we are interested in line shape phenomenaoccuring at energies ω ≫ ∆SC .

III. RESULTS

In order to simplify the calculations we consider aKampf-Schrieffer-type model susceptibility [27] which isfactorized into an ω- and q-dependent part, i.e.

χq(iω) = W (iω)J(q) (7)

where W (ω) = −∫

dνg(ν)2ν/(ω2 + ν2) is some distribu-tion of dispersionless propagating bosons and

2

J(q) =N

4

∑

±qcx;±qc

y

Γ

Γ2 + 2 − cos(qx − qcx) − cos(qy − qc

y).

(8)

N is a suitable normalization factor introduced to keepthe total scattering strength constant while varying Γ.The above susceptibility contains the charge-charge cor-relations which are enhanced at the four equivalent crit-ical wave vectors (±qc

x,±qcy). As we will show below the

static limit g(ν) = δ(ν) together with an infinite charge-charge correlation length Γ → 0 allows one to relatethe coupling parameter λ to the parameter set of theHubbard-Holstein model with long-range Coulomb inter-action by following the approach in Ref. [25]. Concern-ing the normal state dispersion we use a tight-binding fitto normal state ARPES data [28,29] including the hop-ping between first, second and third nearest neighbors,i.e. ǫk = t(cos(kx) + cos(ky))/2 + t′ cos(kx) cos(ky) +t′′(cos(2kx) + cos(2ky))/2. For Bi2212 we take t =−0.59, t′ = 0.164, t′′ = −0.052 and for LSCO t =−0.35, t′ = 0.042, t′′ = −0.028. The BCS gap is assumedto be d-wave symmetry ∆SC(k) = ∆0(cos(kx)− cos(ky))where ∆0(Bi2212)= 32meV and ∆0(LSCO)= 10meV re-spectively.

A. Bi2Sr2CaCu2O8+δ

In Ref. [25] we have already demonstrated that a statictwo-dimensional eggbox-type charge modulation with qc

oriented along the vertical directions can account for thebasic Fermi surface (FS) features in the optimally and un-derdoped Bi2212 compounds. Here we supplement theseconsiderations by a detailed analysis of the photoemis-sion lineshape in the superconducting state. Concerningthe frequency dependent part of the susceptibility we re-strict ourselves to the simplest case of a single energyexcitation, i.e. we choose g(ν) = δ(ν − ω0).

The coupling of electrons with a dispersionless bosonicmode is a well known subject in solid state theory andhas been intensively investigated in [30]. In the presentcontext the new feature is a) the inclusion of supercon-ductivity and b) the role of a strongly q-dependent cou-pling between electrons and the incommensurate CDWmode.

−0.3 −0.2 −0.1 0.0 0.1 0.2energy (eV)

0

Spe

ctra

l fun

ctio

n (a

rb. u

nits

)−0.3 −0.2 −0.1 0.0 0.1

energy (eV)

0

10

20

Inte

nsity

(a.

u.)

−0.3 −0.2 −0.1 0.0 0.1 0.2energy (eV)

0

Spe

ctra

l fun

ctio

n (a

rb .

units

)

a)

b)

FIG. 1. Spectral functions for states from (top to bot-

tom) (a) (0.15π, 0.36π) → (0.5π, 0.36π) and (b) (0.5π, 0) →

(π, 0) → (π, 0.3π). The dashed line correpsonds to the M-

point. Parameters: ω0 = 40meV, λ = 10meV, Γ = 0.1, |qc| =

π/4. Temperature T = 40K. Inset: Spectral function at

(π, 0) in comparison with experimental data points for an

underdoped sample with Tc = 70K measured at 40K. A

step-like function has been added in order to incorporate the

background. Parameters: ω0 = 70meV, λ = 55meV, Γ =

0.5, |qc| = π/3.

In Figs. 1 (a) and (b) we show the spectral functionfor two “vertical” cuts, one through the order parameternode at the Fermi energy (0.2π, 0.36π) → (0.5π, 0.36π)and the other for a cut from (0.5π, 0) → (π, 0) →

(π, 0.3π). Parameters have been choosen in approximateanalogy to the ones used in [26].

The effect of the weight function J(q) is to stronglyreduce the scattering along the diagonals since the states(k, ω) and (k + qc, ω − ω0) which are coupled via theCDW fluctuations are very different in energy. As a con-sequence the quasiparticles are poorly scattered by thecharge collective modes. Therefore the hump feature issuppressed near the node and one observes a single dis-

3

persive quasiparticle peak at the Fermi energy. Scan-ning away from the node towards (0.2π, 0.36π) (Fig. 1a)the low energy peak rapidly looses weight in favor of thehump feature at higher binding energy. This latter broadpeak arises because moving away from the Fermi surfacethe quasiparticles have enough energy to excite chargefluctuations with q ∼ qc and are strongly damped. Inthis region of the momentum space the lower-energy nar-row peak progressively acquires the character of the col-lective mode while the hump becomes dispersive as a rem-nant of the underlying quasiparticle band. The behaviorof the spectral density along this first cut is rather sim-ilar to the case of electrons coupled to a dispersionlessphonon [30]. In this latter case, however, the phonons arecoupled to the quasiparticles for any momentum transferleading to broad incoherent features at high energy evenfor quasiparticles on the Fermi surface. On the contrary,in the present case, the scattering between the quasipar-ticles near the node is severely suppressed by the factorJ(q) and the incoherent hump is nearly absent when thequasiparticle is close to the Fermi surface (i.e. near thenode at (0.36π, 0.36π)). The switching of the underly-ing quasiparticle from the low-energy peak to the humpstructure leads to a break in the dispersion. This break ofthe dispersion is also observed experimentally [31] how-ever, only for momentum scans far from the node. Wewant to note that the dicrepancy of the break-locationis a consequence of our approximation of a factorizedsusceptiblity and should be less pronounced in a morerealistic model. An additional possible difficulty for thepresent scheme is that the quasiparticles near the nodeare only weakly scattered and therefore, according tothe standard Landau theory of Fermi liquids, they shoudhave a damping rate proportional to T 2 and to (bindingenergy)2. This Fermi-liquid behavior is violated in re-cent experiments carried above and below Tc [32]. How-ever, these experiments were not carried out at very lowtemperatures, where a fair comparison can be done withour T = 0 analysis, and where heat-transport experi-ments report a rather standard Fermi-liquid behavior ofthe quasiparticles [33] (thereby supporting our finding ofwell-defined quasiparticles near the nodes). This issueobviously deserves further experimental and theoreticalinvestigation.

Fig. 1b displays the spectral functions for states from(0.5π, 0) → (π, 0) → (π, 0.3π). The appearance of stateswith positive energy is due to coherence effects relatedto the finite superconducting gap in this region of mo-mentum space. Since the M-point defines the ’hot’ re-gion for incommensurate CDW scattering the supercon-ducting states couple strongly to the CDW mode and asa result the spectrum (dashed line) is characterized bythe peak-dip-hump feature as observed in ARPES. Fora suitable choice of parameters we can obtain a quitesatisfactory fit of our curve with experimental data fromphotoemission experiments (see inset of Fig. 1b).

Note that the structure of the hump feature is mainlydetermined by the charge-charge correlation functionJ(q). From our fit to the experimental data we can there-fore deduce that in Bi2212 the underlying CDW fluctua-tions are rather short-ranged i.e. the stripe correlationsextend over 2-3 unit cells only. The spectra for the scan(0.5π, 0) → (π, 0) show an interesting feature which isalso observed in ARPES (see e.g. [12], Fig. 3): Thesharp peak at low energies is essentially non-dispersivewhereas the hump exhibits a maximum binding energyωmax. For energies large than ωmax the hump disper-sion again corresponds to the underlying (normal state)band structure. The maximum in the hump dispersion isdue to the reduced scattering efficiency (determined byJ(q)) when moving away form the M-point. Since along(π, 0) → (0, 0) the underlying dispersion ǫk initially israther flat this leads to a shift of the hump to lower en-ergies. With kx becoming smaller the increasing banddispersion then starts to determine the dispersion of thehump which accordingly shifts to higher binding energies.

Scanning from M towards (π, π) the hump moves tolower energies and looses weight again in agreement withphotoemission experiments [12,31]. We want to pointout that our result for the spectral functions in Fig. 1are rather similar to analogous results of Ref. [26] wherethe coupling of electrons to an AF mode has been con-sidered. Indeed for both AF and incommensurate CDWscattering the states around (π, 0) are the ’hot points’,i.e. are most strongly affected by the scattering.

This indicates that in the real systems the spin and thecharge fluctuations may well coexist and cooperate in de-termining the spectral properties. Of course for a quan-titative comparison with experiments one should includeboth scattering channels. On the contrary, since we aimto explore and underline the role of charge fluctuationsonly, in the inset of Fig. 1(b) we fitted for an illustrativepurpose, the experimental data with the charge channelonly.

B. La2−xSrxCuO4

In underdoped La2−xSrxCuO4 the incommensuratemagnetic fluctuations display a four-fold pattern around(π, π). For the Nd-doped lanthanum cuprate it has beenargued [34] that this may be related to two types of twindomains, each with a single stripe orientation suitablypinned by the underlying LTT lattice structure. How-ever, this kind of argument no longer holds in case ofunderdoped LSCO where both the (tetragonal) a- andb- axes are at about 45◦ with respect to the CuO6 tiltdirection. Moreover it has been demonstrated in Refs.[35,36] that the positions of the elastic magnetic peaks inLa1.88Sr0.12CuO4 and excess-oxygen doped La2CuO4+y

are shifted off of the high-symmetry Cu-O-Cu directionsby a tilt angle of Θ ≈ 3◦. For a one-dimensional stripe

4

model this would correspond to one kink every ∼ 19 Cusites on the charge domain wall. However, as explicitlystated in Ref. [36] a grid pattern of orthogonal stripesoriented along the two orthorombic directions adequatelydescribes the data as well.

FIG. 2. Sketch of the two-dimensional charge- and spin

modulation which is consistent with the AF peak incommen-

surability of LSCO. The pattern can be constructed from

charge stripes running along the diagonal directions as in-

dicated by the dashed lines. The crossing of two stripes leads

to regions with high charge density (dark squares) whereas

the residual segments of the stripes have intermediate charge

(lightly shaded squares). The white squares illustrate the

charge depleted areas with maximal spin density while the

arrows indicate the sign of the AF order parameter.

In the following we therefore examine the consequencesof this kind of two-dimensional dynamic stripe fluctua-tions on the electronic structure of LSCO and comparewith photoemission experiments. Denoting the two or-thogonal charge scattering vectors (which for simplicitywe assume to have equal magnitude) by q±

c= qc(1,±1)

the charge modulation can be written as ρ(R) ∼

cos(qcx) cos(qcy). Fig. 2 shows a sketch of the corre-sponding charge pattern which can be thought of an arrayof stripes running along the (1, 1) and (1,−1) directionsrespectively. Although our calculations are restrictedon the charge channel we want to note that the corre-sponding pattern of the spin order can be constructedby assuming a sign change of the AF order parameterupon crossing the stripes perpendicular to their orienta-tion. This antiphase correlation between the spins acrossa charge stripe makes easier the transversal delocaliza-tion of the charges in the stripes and therefore it is ener-getically favorable. Hence these antiphase domain wallswhich are observed in Nd-doped LSCO around 1/8 of fill-ing [13] seem also to be a rather common feature of thetheoretically investigated stripe structures. As a conse-quence, in our charge-spin structure, the spin order fol-lows the relation ∆spin(R) ∼ cos[qc(x + y)/2] cos[qc(x −

y)/2] ∼ cos(q+sR) + cos(q−

sR) where the spin scattering

vectors are given by q+s

= qc(1, 0) and q−s

= qc(0, 1) re-spectively. Thus this kind of two-dimensional scatteringdisplays a deviation between charge- and spin directionof 45◦ where the latter is compatible with the results ofneutron scattering experiments [14,15]. Only in case of aLTT distortion both charge- and spin scattering collapseinto a single direction leading to a reduction of Tc [13].

−3.2 −2.4 −1.6 −0.8 0 0.8 1.6 2.4 3.2−3.2

−2.4

−1.6

−0.8

0

0.8

1.6

2.4

3.2

� MX

FIG. 3. Fermi surface for a system with two-dimensional

diagonal CDW modulation (qc = π/5, λ = 0.01). The plot

is for temperature T = 100K. The spectral weight has been

integrated over an energy window of 30meV around EF . In-

tensities: I > 50%: full points, 10% < I < 50%: circles,

1% < I < 10%: small dots.

As in the case of Bi2212 we restrict in the following onthe charge channel and as discussed above we considerscattering along the q±

c= (1,±1) directions. To gain

some qualitative insight we consider first the static non-superconducting case inspired by the approach describedin Ref. [25] where we have derived an effective interaction1

2N

∑

q Vqρqρ−q for charge carriers close to an incommen-surate CDW instability. Upon factorizing the interactionthe two-dimensional eggbox-type charge modulation canbe implemented by < ρq >egg=< ρq > [δq,q

+c

+ δq,q−

c]

which in the static limit identifies the coupling parameterλ with the order parameter of the CDW V (qc) < ρqc

>.The resulting one particle hamiltonian can now be diag-onalized and as a result we show in Fig. 3 the Fermisurface for an eggbox-type charge modulation with scat-tering vectors qc = π/5(1,±1).

Obviously diagonal CDW scattering strongly reducesthe spectral intensity of the Fermi surface along theΓ → X direction. This is connected with the fact thatalong the diagonals the scattering vector is parallel tothe contours of the bare bandstructure. As a conse-quence the CDW scattering is most efficient in these re-gions since qc can connect states with equal energies lead-ing to a redistribution of spectral weight near (π/2, π/2).

5

Note that since the band disperses rather rapidly alongΓ → X the vertical scattering we have adopted in caseof Bi2212 would leave the electronic structure in this di-rection nearly unchanged and would not reproduce theexperimental suppression of spectral weight along theΓ → X direction [21–23].

Quite obviously our simplified approach, where thespin degrees of freedom are neglected, cannot provide aquantitative description of the Fermi surface. However,we notice that a model based on spin degrees of freedomonly, constrained by the neutron experiments to have hotspots near the M points could hardly account for the sup-pression of spectral weight along the Γ → X direction.

−0.4 −0.2 0ω

0

2

4

6

8

10

Inte

nsity

(kx; ky)=� =

(0:6; 0)(0:8; 0)(1; 0)(1; 0:1)(1; 0:2)a) �!M ! X

−0.4 −0.2 0ω

0

2

4

6

8

10

Inte

nsity

kx=� =ky=� =

0:30:350:40:450:5b) �! XFIG. 4. Photoemission spectra for diagonal two-dimensional

CDW scattering. As in the case of Bi2212 we have added a

step-like function in order to model the background contri-

bution. Parameter: ω0 = 30meV, Γ = 0.01, qc = π/5(1,±1).

Let us now extend the photoemission lineshape analy-sis to the superconducting state including dynamic in-commensurate CDW scattering. We have found thatfor the LSCO compound a linear frequency distributiong(ν) = 2ν

ω20

Θ(ω0 − ν) up to a cutoff energy ω0 is more ap-

propriate consistent with the experimental observationof low energy magnetic fluctuations in this compound.Fig. 4 shows selected energy distribution curves alongΓ → M → X (Fig. 4a) and Γ → X (Fig 4b).

In the (0, 0) → (π, π) direction (Fig. 4b) one observes

only a broad and weak incoherent feature which disap-pears beyond kx = ky ∼ 0.45π corresponding to the FScrossing of the underlying bare bandstructure. In con-trast the spectra around (π, 0) (Fig. 4a) are composed ofan also broadened but still well pronounced quasiparticlepeak. Due to the small cutoff energy ω0 in comparisonwith Bi2212 and the additional linear frequency distri-bution the hump feature is no longer separated from thesharp peak but both collapse into the broadened peakshown in Fig. 4b. Note that we were forced to use arather large charge-charge correlation length (∼ 100 lat-tice constants) in order to effectively suppress the quasi-particle peak along (0, 0) → (π, π). This finding needsto be commented. On the one hand it is surely conceiv-able that the poor quality of the surface in 214 materialsis responsible for the width of the photoemission peaks.In this case this would be a spurious effect as far as thebulk electronic properties are concerned and our need ofa large charge-charge correlation length would be ficti-tious. On the other hand the authors of Refs. [21–23]claim that a careful attention was paid to the surfacetreatment. Moreover, recent photoemission experiments[37] on overdoped LSCO have revealed a sharp spectralfeature along the diagonal direction that is comparableto that of Bi2212. From this one would not expect strongdisorder effects in the underdoped samples which shouldimprove with decreasing Sr content. In this regard ourfinding of a large charge-charge correlation length couldbe taken seriously at least from a qualitative point ofview. The presence of rather long-ranged charge-orderingcorrelations would rather naturally correspond to themuch more visible presence of stripe correlations in 214materials. Indeed, but for some inelastic neutron scat-tering [16] and some NMR/NQR [18,19] experiments inYBCO, the evidence for stripes is rather clear in 214 ma-terials only, while it remains more elusive in the otherclasses of cuprates.

IV. CONCLUSION

We have demonstrated that the coupling of a supercon-ducting system to dynamical incommensurate CDW fluc-tuations can account for the observed ARPES photoe-mission lineshapes in both Bi2212 and LSCO materials.According to our analysis the vertically oriented CDWfluctuations in Bi2212 are rather fast and short rangedleading to the experimentally observed dip-hump struc-ture around the M-points whereas the lineshape alongthe diagonals is hardly affected by the scattering.

We emphasize again here that our analysis shares manysimilarities (both in the technical framework and in theresults) with the one carried out in Ref. [26] (the renor-malization function Z(ω) obtained from our approach isquasi identical to Fig. 1 in Ref. [26]). Therefore ourwork is not in contrast with Ref. [26] but rather comple-

6

mentar since the stripe fluctuations we consider here areanharmonic CDW’s with the strong correlation playinga relevant role in the hole-poor regions, where antiferro-magnetism is quite pronounced. For the sake of simplic-ity we only investigated the charge sector of the fluctu-ations. Nevertheless, the effects of spin [26] and charge(this work) fluctuations are similar indicating that thesefluctuations may easily coexist and that there is no com-petition but rather cooperation between the two sectors.Of course a long way is still to be followed in order to in-tegrate the two approaches by formalizing the interplaybetween charge and spin degrees of freedom. In particu-lar an open relevant issue is to establish how the criticalproperties related to the charge instability are mirroredin the spin criticality. This intriguing but difficult issueis definitely beyond the present scope.

In case of LSCO the photoemission data can be de-scribed by a two-dimensional, diagonally oriented CDWwhich is slowly fluctuating and characterized by a ratherlarge correlation length. This is consistent with the ideaof long-range AF order coexisting with superconductivityin the 214 systems and also with the more pronouncedtendency to form static charge textures in these mate-rials. We have shown that within this scenario the QPalong the (0, 0) → (π, π) direction can be effectively sup-pressed whereas around the M-points a broad quasipar-ticle peak persists. Finally we want to emphasize thatsince ∆SC enters the calculation as an input parame-ter, our model does not include the phase fluctuations ofthe particle-particle pairs. These fluctuations should playa major role in destroying the quasiparticle peak aboveTc [38]. However, the measurements in Refs. [21–23] onLSCO were done at rather low temperatures (T=15K)where such phase fluctuations are uneffective justifyingour simplified model for the photoemission spectra.

At the present stage of the research in the field ofsuperconducting cuprates our simplified analysis is par-ticularly urgent since it illustrates that charge fluctua-tions can generically be quite effective sources of scat-tering between the quasiparticles providing spectral fea-tures which are not in contrast with experiments bothin LSCO systems, where there seem to be little doubtthat (dynamical) stripes are present, and in Bi2212 com-pounds, where no stringent evidence for stripe fluctua-tions is available. In this regard we show that differentkinds of charge fluctuations can operate in different ma-terials. Moreover, in the present situation, where eventhe experimental situation is rapidly evolving, we believethat our results can be useful in keeping the commu-nity flexible enough to effectively determine the intricatephysical mechanisms of the superconducting cuprates.

ACKNOWLEDGMENTS

We would like to thank J. Mesot for providing the ex-perimental data in Fig. 1. We also greatfully acknowl-edge useful discussions with S. Caprara, C. Castellaniand C. Di Castro.

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