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Inversion of ARPES measurements in high T cuprates c S. Verga, A. Knigavko,and F. Marsiglio Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2J1 Recent energy dispersion measurements in several families of the hole-doped copper oxides have 3 revealed a kink in the energy vs. momentum relation. These have tentatively been identified as 0 due to electron phonon coupling. We invert this data directly to determine the bosonic spectral 0 function; the kinkgives rise to a singular function in thephonon energy region. 2 n a J Thedeterminationofmechanismforsuperconductivity Inthispaperwe outlineaninversionprocedure,deter- 0 3 in the high temperature oxides has occupied researchers mine the ‘bosonic spectrum’ to which the electrons are for the past fifteen years. The most definitive signature coupled, and assess the possibility that these bosons are ] fordeterminingmechanisminconventionalsuperconduc- the phonons. The data for the dispersion for three dif- n tors traditionally has been the measurement of the sin- ferent dopant concentrations in LSCO is reproduced in o gle particle tunneling I-V characteristic1, and the con- Fig. 17. Particularly for the underdoped sample there c - comitant inversion procedure2. Researchers3 have re- is a well-defined kink which occurs at approximately 70 pr ported some success with this procedure for the high meV. Lanzara et al.7 attributed this kink to an electron u temperature superconductors; nonetheless the applica- selfenergyeffectduetocouplingtophonons. Wewishto s bility of such a procedure is unknown outside a weak investigatethisclaimbasedonsomemicroscopicmodels. . t coupling electron phonon framework, and the process We first examine the result obtained from ‘standard’ a is complicated in the superconducting state due to the phonon models, namely the Einstein and Debye mod- m non-isotropic nature of the order parameter. Other pro- els. Each in turn is used to model the electron phonon - cedures have been suggested, such as the inversion of spectralfunction,α2F(ν),andthen,withinthestandard d n the normal state optical conductivity4, the conductivity framework8,9, the electron self energy Σ(ω) ≡ Σ1(ω)+ o in the superconducting state in conjunction with neu- iΣ (ω) is obtained (at T =0): 2 c tronscatteringdata5,andtheinversionofphotoemission ∞ [ data6 in the superconducting state. Recent very high Σ1(ω+iδ)= dν α2F(ν)log ω−ν (1) ω+ν 2 resolution photoemission measurements on a variety of Z0 (cid:12) (cid:12) (cid:12) (cid:12) v cupratematerialshavesuggestedthatasizeableelectron from which the electron dispersion Ek c(cid:12)an be(cid:12)obtained: 5 phonon coupling exists7, and the possibility of inverting (cid:12) (cid:12) 4 this data (in the normal state) has been re-opened. Ek =ǫk+Σ1(Ek), (2) 1 where ǫ is the bare (with respect to electron phonon 7 k ARPES quasiparticle dispersion versus momentum interactions) quasiparticle energy. The model spectrum, 0 2 0 along with the real part of the self energy and the dis- δ = 0.07 0 δ = 0.15 persion is plotted in Fig. 2(3) for the Einstein (Debye) / δ = 0.22 models, respectively. Note that for some values of ǫ the t k a dispersionas shownis multivalued. This is because mul- m tiple poles exist in these regions. In fact, the spectral -100 - eV) function generally evolves in the following manner8 (we d m n E (k usethe EinsteinmodeldepictedinFig. 2forsimplicity): o ε at low energies a pole exists just above the real axis (i.e. F v:c -200εk εk ~ 2hm2k2 w2iothr 3inifisntitheastmtahlewwidetighh).t oWf thhaitsipsonleot(id.ee.pitchteedreinsidFuieg). goes to zero as |ǫ | → ∞. So, beyond about |ǫ | ≈ 140 i k k X kF k meV, the weight of this pole becomes very small. The r 0 0.025 0.05 0.075 actualenergy(Ek)atthis pointbecomes verynearlythe a k’ = |k-k | F Einsteinfrequency. Inasensethe electronsandphonons FIG. 1. Dispersion of occupied states vs. momentum for have‘hybridized’andthisbranch,whichstartedoutvery LSCO with threedifferent dopingconcentrations. The insert ‘electron-like’nearE ≈0isnowvery’phonon-like’. The shows a schematic which puts this data in the context of a k parabolic band. A modification due to electron phonon cou- branch just below -80 meV in Fig. 2c is more or less ir- plingoccursneartheFermisurface. Theregioncorresponding relevant,since the selfenergy has a very largeimaginary to the data of the main figure is highlighted in bold on the part (see Fig. 2b). Finally, as |ǫk| continues to increase left side of theparabola. the lowest branch begins to dominate. In this limit this branch becomes very ‘electron-like’ albeit with a finite width. 1 ∞ Einstein phonon spectrum (T=0K) Self energy 1 1.2 Σ(ω+iδ)= dν α2F(ν) −2πi n(ν)+ 2αωF() 00..48 (a) ωλEE == 08.05 meV ω) (meV)15000 (b) -RIme[[ΣΣ((ωω))]] +ψ(cid:18)Z210+iν2−πTω(cid:19)−(cid:20)ψ(cid:18)12 −(cid:18)iν2+πTω(cid:19)2(cid:19)(cid:21) (3) Σ( whereψ(x)isthedigammafunctionandn(ν)istheBose 0.0 0 0 100 200 0 -100 -200 distribution function. Similarly, for the Debye model ω (meV) ω (meV) there is a well pronounced shoulder, but still no clear Quasiparticle dispersion kink. To determine whatsortofspectrum doesleadto a 0 Ek (c) kink we utilize the procedure described below. ε k Real part of the self energy - Einstein Quasiparticle dispersion E (meV)k-100 ωλEE == 08.05 meV 100 (a) TTT === 13200000 K KK 0 (b) TTT === 13200000 K KK -200 Ek=εk+Re[Σ(Ek)] Σω()] (meV) 50 ωλEE == 08.05 meV E (meV)k-100 0 100 200 e[ |εk| (meV) R ω = 80 meV FIG. 2. The (a) boson spectrum vs. frequency, (b) real E λ = 0.5 and imaginary parts of the electron self energy vs. energy E below the Fermi level, and (c) the resulting dispersion vs. 0 -200 0 -100 -200 0 100 200 bare quasiparticle energy. These figures are for the Einstein ω (meV) |εk| (meV) model for phonons, with ωE = 80 meV and λE = 0.5. See FIG.4. Temperaturedependenceof(a)therealpartofthe text for a discussion of themultivalued portion. self energy, and (b) the dispersion relation for the Einstein model shown in Fig. 2. Note that the singularity is washed out by finitetemperature. Debye phonon spectrum (T=0K) Self energy As a preliminary analysis we note that the following 1.2 80 (a) ωD = 80 meV (b) Re[Σ(ω)] function fits the data in the underdoped regime (where 2αωF() 00..48 λD = 0.8 Σω() (meV)40 -Im[Σ(ω)] the kink is particula−rlλyEpkrominenitf)|Ereka|s<onωabDly well: Σ1(Ek)= −λωD2 if |E |>ω . (4) 0.0 0 ( Ek k D 0 100 200 0 -100 -200 ω (meV) ω (meV) This parametrization is motivated by the following con- Quasiparticle dispersion 0 siderations: we wanted an analytical form that would Ek (c) contain an explicit ‘kink’, we wanted as simple a form ε k as possible, and we wanted a form with which we could analytically perform the Kramers-Kronig integration to V) obtain the imaginary part of the self energy. We have E (mek-100 ωλDD == 08.08 meV had to use an additional fit for the high energy region, to relate ǫ to k −k . For simplicity, we have used a k F ′ ′ linearfit, i.e. ǫ =xk , withk =|k−k |. Inclusionof a k F Ek=εk+Re[Σ(Ek)] quadratic term (see Ref.10 for a discussion of quadratic -200 corrections) results in a correction which is negligible. 0 1|ε0k0| (meV) 200 The proportionality constant x has units of eV ˚A and is FIG. 3. Same as in Fig. 2, except for the Debye model, relatedtothebareFermivelocity(x=h¯v ). Fortheun- F with ωD =80 meV and λD =0.8. derdoped sample, the proportionality constant has been determined by the requirement that, at high energy, the full dispersion becomes the bare one (the model self en- In addition,in the caseofthe Einsteinspectrum there ergy decreases to zero). Because the kink is washed out is an unphysical singularity. While this is easily washed by increased doping, using the same function to fit the outby temperature,thermaleffects still do notreconcile data in the optimally doped and especially in the over- theorywithexperiment. Examplecalculationsareshown doped regime is difficult. To overcome this problem we in Fig. 4. We use the full finite temperature expression renormalize the momentum such that the three experi- for the self energy9: mentalcurvesoverlapinthehighenergyrange. We then 2 use the fit for the underdoped data to calculate the pro- The imaginary part of the self-energy can be deter- portionality constant x for the high energy fit for the mined from the real part through a Kramers-Kronig in- optimally doped and overdoped sample. tegral: Theresultofusingthreefittingparameters,λandωD, 1 ∞ Σ (ω′+iδ) ′ 1 and the background slope x is shown in Fig. 5; it is Σ (ω+iδ)−Σ (∞+iδ)= dω . (5) clear that the fit is very good. The important feature, 2 2 π Z−∞ ω−ω′ captured by the fit, is the initial linear dependence of Substituting Eq. (4) results in energy on wavevector, followed by an abrupt change at 2 ω some characteristic frequency. In Fig. 5b we show the Σ (ω+iδ)−Σ (∞+iδ)= λ ω f , (6) 2 2 D π ω extractedrealandimaginarypartsoftheselfenergyfrom (cid:18) D(cid:19) thedata. Weshowthemodelfit,theactualdataoncethe where f(x) = 1 + (x2−1)ln|1−x|. With the standard highenergypartis extracted,andthe smoothenedcurve 2 4x 1+x approximations9,11, one can relate the imaginary part of used to carry out the Kramers-Kroniganalysis. We find the self energy to the underlying electron phonon spec- parameter values of λ = 0.89, ω = 72 meV, and v = D F trum. The required result is: 4.1eV-˚A/¯h(=6.2X107cm/s). Varyingtheseparameters ‘by hand’ results in a less than 10 % change in λ, for α2F(Ω)=−1 d Σ (Ω+iδ). (7) 2 example, before the fit becomes visually poor; we thus πdΩ regard this as a rough measure of the uncertainty. The result from the fit shown in Fig. 5 is plotted in Fig. 6. Note that f(x) is the same function which ap- Quasiparticle dispersion versus momentum pearsintheHartree-Fockcalculationforthefreeelectron 0 gas12, and as is well known, its derivative has a logarth- experiment, δ = 0.07 micsingularityatthe characteristicfrequency,asshown. model fit (a) bare ε Aside from the logarithmic singularity, the spectrum is k peculiar (as an electron phonon spectrum) in that it is linear at low frequencies, and has a long tail at high fre- E (meV)k-100 ωEk= =εk 7+1R.8e [mΣ(eEVk)] qthueenficty.givAenmiondEelq-i.n4d)epfoerndeeancthcoaflcthueladtioopnin(gi.ec.onwcietnhtoruat- λx D== 40..08991 eV A° (|εk| = x k’) tisiocnlseaprrloyvaideroduinndFinigg.o1f tihsealssiongsuhloawrintyi1n3,Faiglt.ho6u.gThhtehree overall strength of the interaction, as indicated by the area under the spectrum, is of the same magnitude. -200 0 0.025 0.05 Bosonic spectra k’ = |k-kF| 0.8 Self energy experiment, δ = 0.07 (b) experiment: δ = 0.15 δ = 0.22 raw data model, δ = 0.07 (see Eq. 3) smoothened data spin fluctuations (Ref. 14) 50 ω) 0.4 eV) model fit 2αF( m Σω()] ( Im[Σ(ω)] e[ R 50 0.0 experiment model fit 0 100 200 0 ω (meV) 0 -100 -200 FIG. 6. The bosonic spectral functions which result from 0 an inversion of the data for underdoped, optimally doped, 0 -100 -200 ω (meV) andoverdopedsamples(solid,short-dashed,andlong-dashed FIG. 5. (a)The fit (solid curve) to the measured disper- curves, respectively). Parameters for the fits are given in siondata(shownwithsymbols)fortheunderdopedcase. We Table. 1. The model based on Eq. 3, designed specifically have used Eq. (3) with ωD = 72 meV and λ = 0.89. The for theunderdoped case, δ=0.07, is also shown (dot-dashed dotted line shows the linear fit that results using x=4.1 eV curve), along with that expected for spin fluctuations14. In ˚A. (b) Real part of the self energy vs. energy, as extracted the spin fluctuation model of14 there is considerable spectral from the data(symbols), themodel fit (solid curve),and the weight at high frequencies, including weight up to 400 meV smoothened fit to thedata (dashed curve). The insert shows (not shown). Note that in the inverted experimental results the imaginary part obtained through Kramers-Kronig analy- there have slight negative portions, which are set to zero in sis. thesubsequentanalysis in thetext. 3 SeveralquestionsarisefromtheresultsofFig. 6. First, Another interesting question is whether these spectral whatsortofcouplingstrengthdothesespectrarepresent, functions represent phonons or not. As pointed out in as measured by a superconducting critical temperature, Ref.7, the most compelling evidence favouring phonons assuming that these are utilized in an Eliashberg-type is that the frequency domain is consistent with that ob- analysis ? To answer this question, one would ideally served in neutron scattering. Fig. 6 does show high like to perform a calculation with an order parameter frequency spectral weight, however, and one can ask with d-wave symmetry. However, for this to be mean- whether these high frequency tails (clearly beyond the ingful one would require a series of results as shown in phonon energies in these materials) rule out phonons as Fig. 1 for various directions in the Brillouin zone. Then a possibility. To address this question we cut off the the same inversion scheme would result in a momentum spectrumat100meV, andthenreadjustedotherparam- dependent spectrum, which would then be used in a d- eters in the fit (by hand) to recover an improved fit to waveEliashbergequation. Sincethisinformationislack- the data originally presented in Fig. 1. While the fit ing, we simply use an s-wavecalculation,with the direct is never as good as the original one, we find that it is Coulomb interaction set to zero. This would apply if sufficiently good to be a plausible possibility. Thus, un- the repulsionwasprimarilyshort-range;thenthed-wave fortunately, we are unable to say anything very definite symmetrywouldbeunaffectedbyit. Ontheotherhand, on this issue. It is true that spin fluctuations (one of there is no guarantee that the momentum dependence the competing alternatives to phonons) are expected to of the data acquired in this way would necessarily lead have significant spectral weight at higher frequency. An to d-wave superconductivity. We proceed in this sim- exampleisshowninFig. 6,takenfromRef.14,whichhas plistic way nonetheless, and obtain the results for the considerable spectral weight extending up to 400 meV. three doping concentrations summarized in Table 1. As It is clear that significant spectral weight exists at fre- judged both by the value of T and the parameter λ, quencies much higher than indicated by the experimen- c the coupling strength decreases as the doping increases. tal data; furthermore, this particular spectrum appears Probably data from more dopant concentrations is re- to be considerably softer below the 50 meV region than quired before one can assess how significant this trend the data indicates. Once again more definite statements is, and what additional physics may causing T to de- couldbepossibleonceadetailedmomentumdependence c crease in the underdoped regime. We should emphasize of the spectral functions is available. that the main part of the analysis is done in the normal Finally, we have focussed on the real part of the dis- state. Since momentum dependent data is lacking, we persion. One might well ask why we didn’t examine the have tacitly assumed that the coupling strength is inde- imaginarypartdirectly. The partialanswerto this ques- pendent of momentum; an important refinement will be tion is that there are other (i.e. non-pairing) scattering to eventually try to extract this momentum dependence processes which affect the imaginary part of the self en- and see how it correlates with d-wave pairing. ergyandnotthe realpart(e.g. impurities—seebelow). Even more critically, as was attempted in the original angle-resolved photoemission spectroscopy (ARPES) in- dopant level δ=0.07 δ=0.15 δ=0.22 versionwork6,onemighttrytoinvertthe entirespectral λ 0.94 0.82 0.51 function. The difficulty is exemplified in Fig. 7, where ωln(meV) 42 40 38 we show an energy distribution curve taken at kF for Tc (K) 58 46 19 the underdoped sample. We have also plotted the energy distribution calculated with our model fit for the TABLE I. Spectral function parameters as a function of self energy, Eqs. (4) and (6), additionally including an dopant concentration. The parameters λ and ωln are deter- energy resolution function15 and elastic scattering from minedbynumericalintegrals9forthespectralfunctionsshown impurities (which affects the imaginary part of the self inFig. 6. Allindicatorsshowasomewhatenhancedcoupling energy,butnotthe realpart). The fitisexcellentatlow at optimal doping. energies, but there is a clear and very large discrepancy athighenergies. The originofthis discrepancyissimply not understood at present. It may or may not represent newphysics(theremaybealackofunderstandinginthe analysisofARPES),butitclearlywillnotbeunderstood with the approach adopted in this paper, which focuses onthe quasiparticlepeak as anessentialfeature. Mainly for this reason we felt that an examination of the real part of the self energy (i.e. the dispersion) was the best procedure for extracting a potential pairing interaction. 4 Energy Distribution Curves (EDC) for δ = 0.07 Conduction in Oxides, 2nd Edition, (Springer, New York, 300 2000), and references therein. experiment, model fit 4F. Marsiglio, T. Startseva and J.P. Carbotte, Phys. Lett. F A245172(1998); F.Marsiglio, Molecular PhysicsReports k at 24 73 (1999). nits) 200 5J. P. Carbotte, E. Schachinger, D. N. Basov, Nature 401 y u 354 (1999). arbitrar 6GLe.Btt..A6r7no2l5d6,9F.(M19.9M1)u;eJll.eCr,. aSnwdihJa.Crt.,SWwi.hHa.rtB,uPthleyrs,.FR.eMv.. ntesity (100 7AM.uLelalenrz,aaran,dPG..VB..BAorgndoaldn,oPv,hyXs..JR.eZvh.oBu4,6S.58A6.1K(1el9l9a2r,).D. I L. Feng, E. D. Lu, T. Yoshida, H. Eisaki, A. Fujimori, K. Kishio, J.-I. Shimoyama, T. Noda, S. Uchida, Z. Hussain, 0 and Z.-X.Shen, Nature412 510 (2001). -0.2 -0.1 0 ω (eV) 8S. Englesberg and J.R. Schrieffer, Phys. Rev. 131 993 FIG.7. Theenergydistributionfunction(spectralfunction (1963). timesthefermifunctionconvolutedwithanenergyresolution 9P.B. Allen and B. Mitrović, in Solid State Physics, edited function15 vs. energy,fortheunderdopedsample,atkF. The by H. Ehrenreich, F. Seitz, and D. Turnbull (Academic, impurityscatteringrateusedwas1/τ =80meV,andtheen- New York,1982) Vol. 37, p.1. ergybroadeningwas8meV.Thefitisexcellentatlowenergy, 10S.LaShell,E.Jensen,andT.Balasubramanian,Phys.Rev. but clearly fails at high energy. B61 2371 (2000). 11F. Marsiglio and J.P. Carbotte, in ‘The Physics of Con- ventional and Unconventional Superconductors’ edited by In summary we have used the electron dispersion, as K.H. Bennemann and J.B. Ketterson (Springer-Verlag, measuredbyARPES,toextractanelectronicselfenergy 2003)p. 233; cond-mat/0106143 (2001). (real part). Through Kramers-Kronigwe are able to ex- 12N.W. Ashcroft and N.D. Mermin Solid State Physics tract the imaginary part of the self energy, from which (SaundersCollege Publishing, New York 1976)p. 334. an inelastic scattering spectral function is extracted in 13This is due to two reasons: first, the data does not, after a straightforward manner. The result is summarized in all,displayaperfectkink,andsecond,wehavehadtoem- Fig. 6 for a variety of doping concentrations in LSCO. ploysome smoothing tothedata toperform theKramers- The result is clearly compatible with phonons; the ex- Kronig and subsequent analysis. tracted coupling strength would then be able to account 14E.Schachinger,J.P. Carbotte, andD.N.Basov,Europhys. for the superconductivity. However, based on the avail- Lett. 54, 380 (2001) able data we are unable to rule out, for example, spin 15R.Fehrenbacher,Phys. Rev.B54, 6632 (1996). fluctuations as a possibility. Measurements over more doping concentrations and in different directions in the Brillouinzonewouldaidconsiderablyinnarrowingdown the possibilities. We would like to acknowledge Z.-X. Shen for initially encouragingus to perform this study. We thank him, A. Lanzara and Xingjiang Zhou for the communication of their datain electronicform,andforfurther discussions. We also thank an anonymous referee for comments and questions which have improved this paper. This work was supported by the Natural Sciences and Engineering ResearchCouncil(NSERC)ofCanadaandtheCanadian Institute for Advanced Research. 1J.M.Rowell,P.W.Anderson,andD.E.Thomas,Phys.Rev. Lett. 10 334 (1963). 2W.L. McMillan and J.M. Rowell, Phys. Rev. Lett. 14 108 (1965). 3N.Tsuda,K.Nasu,A.Fujimori,andK.Siratori,Electronic 5