ELECTRON SELF-ENERGY OF HIGH TEMPERATURE SUPERCONDUCTORSAS REVEALED BY ANGLE-RESOLVED PHOTOEMISSION

M. R. NORMANa, H. DINGa,b, M. RANDERIAc and J. C. CAMPUZANOa,b

aMaterials Sciences Division, Argonne National Laboratory, Argonne, IL 60439, USAbDepartment of Physics, University of Illinois at Chicago, Chicago, IL 60607, USA

cTata Institute of Fundamental Research, Mumbai 400005, India

Abstract—In this paper, we review some of the work our group has done in the past few years to obtain theelectron self-energy of high temperature superconductors by analysis of angle-resolved photoemission data. Wefocus on three examples which have revealed: (1) a d-wave superconducting gap, (2) a collective mode in thesuperconducting state, and (3) pairing correlations in the pseudogap phase. In each case, although a novel result isobtained which captures the essence of the data, the conventional physics used leads to an incomplete picture. Thisindicates that new physics needs to be developed to obtain a proper understanding of these materials.q 1998Elsevier Science Ltd. All rights reserved

Keywords:self energy, photoemission, D. superconductivity

Eleven years after their discovery, the physics of hightemperature superconductors is still not well understoodbecause of their complex nature. One of the key toolsused to obtain information on these materials has beenangle-resolved photoemission spectroscopy (ARPES).Although a surface sensitive probe, ARPES has theadvantage of being resolved both in energy and momen-tum space, thus providing information difficult to obtainfrom other methods. Given the fact that ARPES measuresthe single particle spectral function [1], then, in principle,one should be able to obtain the electron self-energy fromthe data. In some sense, this would ‘solve’ the hightemperature superconductor problem, assuming one hadsome microscopic theory which produced the same self-energy.

In this spirit, our group has worked several years nowanalyzing ARPES data in an attempt to extract usefulinformation about the electron self-energy for high tem-perature cuprate superconductors. The amount of workdone is too extensive to review in this short paper, so wewill confine ourselves to three examples. In each case wefind a non-trivial result which captures the essence of thedata. But in each case, we find that our ‘conventional’explanation is in some sense incomplete. We will use thisto show that any ‘mean field’ explanation of the data willalways lead to inconsistencies and relate this to the longstanding ‘x’ versus ‘1þ x’ debate on the doping depen-dence of physical quantities. The conclusion is that newphysics will need to be developed to obtain a completepicture of the data.

Our first example concerns the determination of thelow temperature superconducting gap. Traditionally,workers in ARPES [2] have defined the gap by themidpoint of the leading edge of the spectrum. Although

this midpoint is related to the superconducting gap, it isnot the same [3]. At low temperatures, and ignoringlinewidth broadening and momentum resolution, themidpoint of spectra at the Fermi momentum is the super-conducting gap minus the HWHM of the energy resolu-tion, if the gap is large enough so that the Fermi functionplays no role. Even with these restrictions, this statementassumes one can equate the photocurrent to the spectralfunction, and that one knows the Fermi momentum, eachof which involves a number of assumptions. We havetaken the first step beyond this midpoint criterium in anattempt to give a well-defined meaning to the measure-ment of the gap by ARPES [4].

We first assert that the measured photocurrent isproportional to the spectral function times the Fermifunction, the proportionality constant being the dipolematrix element connecting the initial and final states (thesignal above the Fermi energy, due to higher harmonicsof the photon beam, is obviously subtracted beforemaking this identification). This assumes (1) the suddenapproximation is valid, (2) contributions due to thegradient of the photon vector potential can be ignored,and (3) ‘secondaries’ (due to inelastic scattering of thephotoelectron) are either small or have also been sub-tracted. Although this seems a lot to stomach at once,there are ways to test this. For instance, if valid, then afrequency integral of the ARPES spectra should beproportional to the momentum distribution function,nk.Our studies [1] indeed indicate that the frequencyintegrated ARPES data are consistent with such anidentification. Exploiting this, a rigorous method can besuggested to determine the Fermi momentum, that pointwhere the gradient of the integrated data (i.e.l=nkl) has amaximum [5]. Doing this, we find a large hole-like Fermi

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q 1998 Elsevier Science Ltd. All rights reservedPII: S0022-3697(98)00144-9Pergamon

surface centered about the (p,p) points of the squarelattice Brillouin zone with a volume consistent with 1þ x[6].

We next need some model spectral function by whichto fit the data. Since we wish to determine the BCS gap,then it is natural to use a BCS spectral function. At lowtemperatures we are fortunate, in that the leading edge ofthe spectrum is resolution limited. This implies that theimaginary part of the electron self-energy is small atfrequencies of order the superconducting gap, and thusthere is some justification for using a BCS-like ansatz(limitations of this picture will be discussed in our nextexample). For frequencies beyond the gap, however, self-energy effects cannot be ignored. One sees this in the dataas a large non-quasiparticle peak contribution to thespectrum. Whether all of this weight is part of the truespectral function or not (i.e. secondaries) is still a matterof debate. In our case, we simply subtracted this inco-herent part by assuming it could be treated as ‘second-aries’ (using the standard Shirley procedure). In practice,for extracting the gap, this makes little difference sinceone is fitting only the leading edge of the spectrum, whichis not much affected by this subtraction (though thissubtraction becomes more of an issue as the temperatureis raised). But this does illustrate the point made at thebeginning. Although by doing this subtraction we are ableto fit the data to a BCS spectral function, and thus obtain areliable estimate of the BCS gap, we have in essence onlyobtained a single number. Although it is a very usefulnumber, as we will see, it encapsulates only one aspect ofthe very interesting ARPES spectra in the superconduct-ing state.

Our first attempt at this procedure revealed a gap inBi2212 which was a strong function of the Fermimomentum [4], in support of earlier work by the groupof Shen [2]. Unlike this earlier work, evidence for twozeros of the gap as a function of momentum (per zonequadrant), rather than the single zero expected for ad-wave order parameter, was found. At that time, wesuggested two possibilities for interpreting this: (1) ananisotropic s-wave gap, or (2) a d-wave gap which was

either being measured on the true Fermi surface or oneof the ghost images of the Fermi surface (the ghosts dueto diffraction of the outgoing photoelectrons by theincommensurate BiO superlattice) depending on whatparticular value of momentum one was measuring.Subsequently, by exploiting the photon polarizationdependence of the dipole matrix elements, we wereable to show that explanation (2) was actually the correctone [7, 8]. As extensively discussed in these papers, thesuperlattice complications can be avoided by measuringthe gap in theYquadrant of the Brillouin zone. Doing soreveals a gap which beautifully follows the behaviorpredicted for an order parameter with dx2 ¹ y2 symmetry[8] (see Fig. 1). In fact, one learns more than this. Sincethe data follow the form cos(kxa) ¹ cos(kya) quiteclosely, this indicates that the pairing interaction isfairly local in real space. Data taken on another optimallydoped sample where the large gap region was sampledmore closely [8] actually indicate the presence of a weakmaximum in the gap at locations on the Fermi surfaceconnected by (p,p) wavevectors. Similar effects havebeen seen in calculations where spin fluctuations areconsidered as the pairing mechanism.

As said above, even though a lot of useful informationis obtained from knowing the value ofDk, it is only asmall part of the overall story. This can be seen in Fig. 2,where the spectrum at the (p,0) point of the zone for aslightly overdoped sample at low temperatures is shown.Rather interestingly, this spectrum agrees with that of thenormal state for energies beyond about 90 meV, which isequivalent to stating that the self-energies agree beyondthis energy. For lower energies, though, one sees adramatic departure of the superconducting state spectrafrom the normal state one, as first noted by Dessau et al.[9]. The superconducting (SC) state spectrum first drops(thus leading to a dip/hump structure) then rises to form asharp, essentially resolution limited, quasiparticle peak.Since this change in behavior is all occuring on the scaleof the energy resolution, this indicates that the imaginarypart of the self-energy (ImS) must drop from its large

1903Electron self-energy of superconductors

Fig. 1.Yquadrant gap (in meV) versus angle on the Fermi surface(filled circles) compared to a dx2 ¹ y2 gap (———————).

Fig. 2. Comparison of SC data at (p,0) to a model fit based onelectrons interacting with a collective mode. - - -, an assumed

background contribution.

normal state value to a small value over a narrow energyrange. Fits we have done using model self-energies revealthat the drop in ImS must be rather abrupt, essentially astep edge (the standard d-wave prediction of crossingover fromq to q3 is too weak to give a dip). In fact, theobserved dip is so deep, it is best fit by a peak in ImS

followed by a rapid drop.There are a number of consequences of such behavior.

By Kramers–Kronig transformation, a step in ImS

implies a peak in ReS. Such a peak will lead to anadditional mass renormalization relative to the normalstate which acts to suppress the quasiparticle dispersion.This explains the rather puzzling observation that thequasiparticle peak does not appear to disperse much whenmoving away from the (p,0) point. In fact, data on anumber of our samples indicate that a sharp low energypeak is still present when moving towards the (0,0) pointeven when the higher binding energy feature (the hump)has begun to disperse (the hump dispersion rapidlyapproaches the dispersion of the single broad peak seenin the normal state). This result was so puzzling that thedata were not published for several years. The peak inReS, however, naturally explains this, since it can lead toa low energy quasiparticle pole even when the normalstate binding energy moves well away from the Fermienergy. We note that a step edge in ImS is equivalent tothe problem of an electron interacting with a dispersion-less mode, as previously treated by Engelsberg andSchrieffer [10]. Such an interaction leads to the predic-tion of a spectral function composed of two features (peakand hump) whose dispersion is remarkably similar to thatextracted from the data [11]. The crucial difference hereis that this behavior appears only in the superconductingstate, and thus the mode is not a phonon. Rather, it mustbe of collective origin. Detailed calculations we havedone of a superconducting electron interacting with adispersionless mode give a good description of the data[12], with the peak in ImS due to the peak in the SCdensity of states. They also give a good fit to the observeddispersion of the two features.

Microscopically, these findings imply that the domi-nant contribution to the electron self-energy is fromelectron–electron scattering processes. The low fre-quency reduction in ImS is a consequence of the gappingof the spectrum causing the scattering (i.e.a2F) by thesuperconducting gap (the 2D effect first discussed byKuroda and Varma [13]). To get a step edge, though, onemust assume that the gapped weight shows up as a sharpmode inside of this ‘2D’ gap. Recent spin fluctuationtheories have indeed predicted such behavior [14]. Fits tothe data indicate a mode with an energy of 41 meV,equivalent to that of the collective mode seen by neutronscattering in YBCO [15]. Whether this is a coincidence ornot remains to be seen. The YBCO neutron scatteringdata indicate that the mode is associated with (p,p)

scattering events, whereas in the ARPES data the stepedge in ImS implies dipsersionless behavior. On the otherhand, the low energy ARPES peak exists over about thesame momentum range along (p,0)–(p,p) as it doesalong (p,0)–(0,0). As these directions are related by a(p,p) translation, the ARPES data also indicate that (p,p)scattering is indeed playing an important role. This ismost obviously seen in the fact that the dip/hump struc-ture is most pronounced in spectra at the (p,0) points, asrecently emphasized by Shen and Schrieffer [16].

It is important to remark that the above description isincomplete. In Fig. 2, we show that our model gives avery good fit to the spectra, but at a price. The price is thata large ‘background’ contribution has to be subtracted offthe data. This background is modeled by a step, and isessentially equivalent to the total ARPES spectra forunoccupied states (with the step edge at the Fermienergy in the normal state, but pushed back byD in thesuperconducting state). In reality, there are indicationsthat most if not all of this background is part of the truespectral function. This has led to a recent speculation thatthis large background actually represents the instabilityof the photohole to decay into spinons and holons [17]. Insuch a model, the quasiparticle peak represents a boundstate split from this continuum. To look into these mattersin more detail, we have recently attempted to extract theactual experimental self-energy by direct inversion of theARPES data. This inversion reveals the predicted peakand step edge in ImS of our model, as well as the peak inReS [18].

We now ask ourselves the question of how the abovepicture changes as the doping is reduced, moving towardsthe Mott insulating phase. What is found is that the lowtemperature gap again has the expected d-wave form[19]. On the other hand, the spectrum does changesignificantly as the doping is reduced. The quasiparticlepeak becomes smaller and the hump becomes morepronounced, moving to higher binding energy. Perhapsthe most significant change is seen upon heating thesample. In overdoped Bi2212, the superconducting gapis observed to close at or nearTc. Unpublished fits wehave done similar to those of Fig. 1 (which are obviouslymore suspect as the temperature is raised) reveal a gapwhoseT dependence is much flatter than the BCS pre-diction, which then closes rapidly nearTc. This isaccompanied by a strong increase in the low frequencybroadening back to its large normal state value (thebroadening is found to drop approximately likeT6

below Tc, similar to what is seen in conductivitymeasurements, and again a strong indication of theelectron–electron scattering origin ofS). In the under-doped case, however, something quite different occurs.The spectral gap is seen to exist well aboveTc, onlydisappearing at a higher temperature (denotedT*)[19–21] and has a similar anisotropy as that seen below

1904 M. R. NORMANet al.

Tc. The gap as measured by the midpoint of the leadingedge goes smoothly throughTc [19] indicating that thegap aboveTc has the same origin as the gap belowTc, aspredicted by theories with pairing correlations aboveTc

[22].This so-called pseudogap has been seen in a variety of

other measurements, most of them predating the ARPESones. Its origin is a matter of intense debate and encap-sulates one of the most fundamental issues of hightemperature superconductivity: how the unusual super-conducting state seen in the cuprates evolves into theequally unusual Mott insulating state. Again, the advan-tage of ARPES is that it provides both momentum andfrequency resolved information. What does it find? First,the quasiparticle peaks appear only belowTc (i.e. notbelow T*), and again one finds that the low frequencybroadening drops off roughly asT6 belowTc. AboveTc,the spectra near (p,0) are quite unusual, being rather flat,but with a sharp leading edge with a large gap (note, thisleading edge gap discussed in Refs. [19, 21] is not thesame as the hump position discussed in Ref. [20], a pointof confusion in the literature). Second, this leading edgegap, as characterized by its midpoint, smoothly evolvesthroughTc [19, 23] implying that it is of the same originas the superconducting gap. Third, the low temperaturegap actually increases as the doping decreases, reflectingthe increase ofT* with underdoping [23–25], againshowing the strong connection between the supercon-ducting gap and pseudogap. Fourth, the pseudogap has asimilar anisotropy aboveTc to that below [19, 21]. Fifth,the pseudogap is tied to the normal state Fermi surface[26], as expected if the gap were of pairing origin, ratherthan of CDW or SDW origin.

Recently, we have looked again into the anisotropyissue by taking temperature sweeps at differentk pointson the Fermi surface. We found something quite unusual,in that the pseudogap disappears at different temperaturesfor different k [23]. This means that the d-wave nodebelowTc becomes a gapless arc aboveTc which expandsin length with temperature, until the entire Fermi surfaceis recovered atT*. The evolution is smooth, as predictedby theories with d-wave pairing correlations [27], ratherthan the abrupt change one would expect if the gap aboveTc were of different origin than the one belowTc.

Closer inspection, though, reveals that more is goingon. This is most clearly seen by employing a recentmethod we have suggested for removing the effects ofthe Fermi function from ARPES data [23]. If one assumesthe spectral function is particle-hole symmetric, a mildassumption for spectra atkF over a sufficiently narrowenergy range about the Fermi energy, then one canformally eliminate the Fermi function from the data bysumming the ARPES intensity at positive and negativeenergies (with respect to the chemical potential). Bydoing so, one acquires a dramatic visual picture of what

is going on. Such symmetrized data are shown in Fig. 3 atthe (p,0)–(p,p) Fermi crossing for a moderately under-doped (Tc ¼ 83 K) sample. One clearly sees from this thatthe spectral gap fills in rather than closes. In contrast,halfway along the Fermi surface between (p,0) to (p,p)the behavior is quite different, as the gap is seen toactually close (at a temperature betweenTc and T*)rather than fill in. The momentum dependence of theelectron self-energy is highly non-trivial.

These results have motivated us to find a model self-energy which captures the unusual behavior seen near the(p,0) point in the pseudogap phase. One which repro-duces the low energy data quite well (see Fig. 3) is of theform ¹ iG1 þ D2=(q þ iG0) [28]. BCS theory is recoveredby settingG0 to zero. The surprising finding is thatG0 isproportional toT ¹ Tc (and thus zero belowTc), with D

essentiallyT independent (the latter having been inferredearlier from specific heat data in YBCO [29]).T* thencorresponds to whereD ¼ G0(T), i.e. although the spectralgap closes,D is still non-zero. The interesting point is thatthere is only one known quantity which is proportional toT¹ Tc, the inverse Cooper pair lifetime [30]. This almostcertainly means that the pseudogap is due to pairingcorrelations [28]. In CDW or SDW type theories, thequantity T ¹ Tc would not naturally arise. In fact, theabove form for the self-energy can be motivated by at-matrix calculation of the self-energy due to pairingfluctuations [28]. The derivation is especially transparentin the limit where the bands are dispersionless, whichthus motivates why the behavior is seen only near the(p,0) points of the zone.

This ‘zero dimensional’ behavior again emphasizes theunconventional nature of the cuprates. A model havingthese characteristics has been recently proposed byGeshkenbein et al. [31], who emphasized that suchbehavior can explain the existence of large pairingcorrelations without the large expected signatures offluctuational conductivity or diamagnetism. In essence,the states near (p,0) have no measurable Fermi velocityand not even a remnant of a quasiparticle peak in the

1905Electron self-energy of superconductors

Fig. 3. Symmetrized data on aTc ¼ 83 K sample at the (p,0)–(p,p) Fermi crossing at three temperatures (14 K, 90 K, and170 K) compared to model fits based on pairing fluctuations.

pseudogap phase, and consequently they do not contri-bute to the supercurrent response. Any type of mean fieldtreatment would therefore be totally inadequate.Although these electrons near (p,0) have a largeD, theyhave noW in the Ginzburg–Landau sense. As the dopingdecreases, the anomalous region expands, eventuallytaking over the entire zone, giving rise to the non-super-conducting, insulating state (data taken on lowTc Bi2212samples show this type of behavior [19, 26]).

Any mean field description of the above would forceone again into the two gap picture, which is inconsistentwith the data, in that the gap for eachk smoothly evolvesthroughTc, and at low temperatures has an anisotropycompletely consistent with a simple d-wave order para-meter. This same kind of inconsistency is also seen if oneattempts to force a Fermi liquid picture in the underdopedsuperconducting state [32], and is related to the ‘x’ versus‘1 þ x’ debate which has been prevalent in the cuprateliterature over the past decade, a debate whichencapsulates the unconventional nature of the cuprates.Is it x, or 1þ x? In some sense, the ARPES data say it isboth, analogous to quantum mechanics whose objectsbehave like waves or particles depending on whatquestion one is asking. Is there a large Fermi surfaceenclosing a volume 1þ x, even for reduced doping? Theanswer is yes, but [26]. The ‘but’ is due to the fact that onecan quite reasonably define a Fermi crossing along (p,0)–(p,p) due to the intensity drop in the spectra (thel=nklmaximal argument discussed earlier), but this hides thefact that there is no true dispersion in the conventionalband theory sense. In essence, states near the (p,0) pointsdo not behave like a normal Fermi liquid, and the restform a liquid whose effective volume is increasinglyreduced as the doping is reduced, leading tox-like scaling(note, this is not the same as the ‘mean field’x picturewith small hole pockets, something we find no evidencefor). This behavior is perhaps most exhaustivelydiscussed in the recent book of Anderson [33].

In conclusion, much useful information concerning theelectron self-energy can be extracted from an analysis ofthe ARPES data. In each example studied above, aconventional, but exotic, explanation is found (d-wavesuperconducting gap, collective mode in the supercon-ducting state, pairing correlations aboveTc). But, in each

case, the conventional explanation, although capturingthe essence of the data, is incomplete. This indicates thatnew physics needs to be developed before a true under-standing of the cuprates is achieved.

Acknowledgements—This work was supported by the USDepartment of Energy, Basic Energy Sciences, under contractW-31-109-ENG-38, the National Science Foundation DMR9624048, and DMR 91-20000 through the Science and Technol-ogy Center for Superconductivity.

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