Although the low frequency electronic Raman response in the superconducting
state of the cuprates can be largely understood in terms of a d-wave energy
gap, a long standing problem has been an explanation for the spectra observed
in the $A_{1g}$ polarization orientations. We present calculations which
suggest that the peak position of the observed $A_{1g}$ spectra is due to a
collective spin fluctuation mode.Comment: 4 pages, 5 eps figure

A. P. Kampf

A.P. Kampf

A.V. Chubukov

D. Manske

F. Venturini

F. Wenger

H.A. Mook

H.F. Fong

H.J. Vidberg

J. Annett

N. Bulut

O.K. Andersen

T. P. Devereaux

T. Strohm

T.P. Devereaux

U. Michelucci

English

arXiv

Venturini, F.

Michelucci, Umberto

Devereaux, T. P.

Kampf, Arno P. (Prof.)

OPUS - Augsburg University Publication Server

Collective spin fluctuation mode and Raman scattering in superconducting cuprates

Although the low frequency electronic Raman response in the superconducting state of the cuprates can be largely understood in terms of a d-wave energy gap, a long standing problem has been an explanation for the spectra observed in the A{1g} polarization orientations. We present calculations which suggest that the peak position of the observed A{1g} spectra is due to a collective spin fluctuation mode

Venturini, Francesca

Michelucci, U.

Kampf, A. P.

ZHAW digitalcollection (Zurich Univ.  of Applied Sciences)

Collective spin fluctuation mode and raman scattering in superconducting  cuprates

Crossref

Online-Publikationserver Augsburg

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PHYSICAL REVIEW B 1 DECEMBER 2000-IIVOLUME 62, NUMBER 22Collective spin fluctuation mode and Raman scattering in superconducting cupratesF. Venturini,1 U. Michelucci,2 T. P. Devereaux,3 and A. P. Kampf21Walther Meissner Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany2Theoretische Physik III, Elektronische Korrelationen und Magnetismus, Institut für Physik, Universität Augsburg,86135 Augsburg, Germany3Department of Physics, University of Waterloo, Waterloo, ON, Canada N2L 3G1~Received 10 April 2000!Although the low-frequency electronic Raman response in the superconducting state of the cuprates can belargely understood in terms of a d-wave energy gap, a long-standing problem has been an explanation for thespectra observed in the A1g polarization orientations. We present calculations which suggest that the peakposition of the observed A1g spectra is due to a collective spin fluctuation mode.Electronic Raman scattering has proven to be a useful toolin exploring the superconducting state in the cuprate materi-als. The possibility of probing selectively electronic excita-tions in different regions of the Brillouin zone ~BZ! by thechoice of polarization geometries has allowed us to explorethe superconducting gap anisotropy. The successful explana-tion of the Raman data in B1g and B2g scatteringgeometries1,2 has provided one piece of evidence for the bynow widely accepted dx22y2 pairing symmetry in hole-dopedcuprate superconductors.3 In the context of impurity effectsas a testing ground for unconventional superconductivity, theobserved v3 to v crossover in the low-frequency B1g Ramanresponse fits consistently with the power-law crossovers atlow temperatures in the NMR relaxation rate and in the mag-netic penetration depth. An even quantitatively consistentpicture of electronic Raman scattering and infrared conduc-tivity was achieved when the T-matrix approximation in the‘‘dirty’’ d-wave scenario is extended to include a spatial ex-tension of the impurity potential.4However, up to now the discrepancy between Raman datain A1g and B1g ,B2g geometries has remained unresolved.5,6Previous results for the A1g scattering geometry were foundto be very sensitive to changes in the Raman vertex functiong(k)5 making a comprehensive explanation difficult for theexperimental data in different cuprate materials.In this paper we calculate the Raman response of a dx22y2superconductor including the contribution from a collectivespin-fluctuation ~SF! mode which is identified with the(p ,p ,p)-resonance ~in short p-resonance! near vR'41 meV observed by inelastic neutron scattering ~INS! onbilayer cuprates.7,8 Our results suggest that the A1g peak po-sition is largely controlled by the strength and frequency ofthe p-resonance mode which on the other hand does notaffect the Raman response in the B1g and B2g channels. Theinclusion of the collective SF mode allows for a simulta-neous fit of the Raman data in all channels in optimallydoped materials. Furthermore, we find that the inclusion ofthe SF term significantly reduces the sensitivity to the specialchoice of the underlying tight-binding band structure, i.e.,the sensitivity to the choice of the Raman vertex in A1gsymmetry, and thus offers a possible route for resolving thepreviously encountered problems in the symmetry analysisof the light-scattering amplitude.2,5PRB 620163-1829/2000/62~22!/15204~4!/$15.00On the basis of the observation of the collective SF modein Y-123 and Bi-2212, we consider a bilayer model repre-sented by a tight-binding band structureek522t@cos~kx!1cos~ky!#14t8cos~kx!cos~ky!2t'~k!,with an interplane hopping given by9t'~k!52t'cos~kz!@cos~kx!2cos~ky!#2, ~1!where kz is 0 or p for the bonding or antibonding bands ofthe bilayer, respectively. Although there is no experimentalevidence of the INS peak in the monolayer cuprates,10 theresults of this model are not qualitatively changed when amonolayer structure is considered.The spin susceptibility (xs) is modeled by extending theweak-coupling form of a BCS superconductor in a dx22y2pairing state to include antiferromagnetic spin fluctuations bya random-phase-approximation ~RPA! form with an effectiveinteraction Ū; i.e., xs(q,iv)5x0(q,iv)/@12Ūx0(q,iv)#where9x0~q,iv !51bTr (k,iv8Ĝ~k,iv8!Ĝ~k1q,iv81iv !. ~2!Tr denotes the trace and b5T21. Ĝ(k,iv) is the BCSGreen’s function in Nambu spaceĜ~k,iv !5ivt̂01jkt̂31Dkt̂1~ iv !22jk22Dk2 ~3!with t̂ i(i51,2,3) being the Pauli matrices, t̂0 the 232 unitmatrix, jk5ek2m , and Dk5D0@cos(kx)2cos(ky)#/2. Thisform of the spin susceptibility contains a strong magneticresonance peak at q5Q[(p ,p ,p) which was proposed9 toexplain the INS resonance at energies near 41 meV in Y-123~Ref. 8! and Bi-2212.7 Other forms for the spin susceptibilitycan be straightforwardly used within our model. However,the results are mainly determined by the collective mode atQ. Therefore, we take the bilayer susceptibility for a repre-sentative calculation.The intensity of scattered light I(v) in Raman experi-ments is proportional to the imaginary part of the responsefunction for the effective density operator15 204 ©2000 The American Physical SocietyPRB 62 15 205COLLECTIVE SPIN FLUCTUATION MODE AND RAMAN . . .r̃q5(k,sg~k!cs ,k1q† cs ,k ~4!in the long-wavelength limit q→0. SpecificallyI~V !}@11n~V !#Imx~V1i01!,x~ iV !5E01/Tdt e2iVt ^Tt@ r̃~t !, r̃~0 !#&, ~5!with the Bose function n(v) and the time ordering operatorTt .The bare Raman vertex ĝ(k)5 t̂3g(k) in different scat-tering geometries are classified according to the elements ofthe D4h point group. For the limiting case of vanishinglysmall scattered (vS) and incident (v I) photon energies, itcan be represented in the effective-mass approximation~EMA!g~k!5(a ,beaI ]2ek]ka]kbebS , ~6!where eI and eS are the unit vectors for in-plane polarizations~i.e., a ,bP$x ,y%) of the incoming and the scattered light,respectively. Using Eq. ~6! and the bilayer tight-binding dis-persion Eq. ~1! we obtaingkB1g52tgkdS 11 4t't cos~k z!@cos~kx!1cos~ky!# D ~7!gkA1g52tgks22t'cos~k z!@cos~2kx!1cos~2ky!#24 cos~kx!cos~ky!@ t812t'cos~k z!# ~8!where gkd ,s5@cos(kx)7cos(ky)#/2. However, the EMA has aquestionable region of validity for all Raman measurements5on the cuprates since the incoming photons have energy;2 eV, which is of the order of the bandwidth and of theinterband excitations according to local densitycalculations.11 EMA-based arguments in previous worksabout relative Raman intensities for different channels aretherefore questionable.6 We hence consider other forms forthe vertices as well which obey the proper symmetry trans-formations. For the A1g geometry some symmetry compat-ible choices are g(k)5cos(kx)1cos(ky) and g(k)5cos(kx)cos(ky). These basis functions assign weight to dif-ferent regions of the BZ and this is the reason why previousresults for the A1g response were particularly sensitive to thespecific choice of the bare Raman coupling vertex.The spin fluctuations lead to an additional contribution forthe Raman response via a two-magnon-like process as showndiagrammatically in Fig. 1.12 Here, the SF propagator is in-corporated in its RPA form for the bilayer ~as describedabove! by the ladder diagram series with an effective on-siteHubbard interaction Ū .We, therefore, write the Raman response function at finitetemperature as the sum of a pair-breaking ~PB! and a SFcontributionxgg~q,iv !5xggPB~q,iv !1xggSF~q,iv ! ~9!with the Raman vertex specifying the scattering geometry. Inthe limit q→0 the diagram for the SF contribution translatesintoxggSF~ iV !51b (q8,ivVg~q8,iV ,iv !xs~2q8,2iv !3xs~q8,iv1iV !Vg~q8,2iV ,2iv !. ~10!The vertex function Vg(q8,iV ,iv) includes the bare Ramanvertex and is evaluated asVg~q8,iV ,iv !5TrH 1b (k,iv8 ĝ~k!Ĝ~k,iv81iV !3 t̂0ŪĜ~k1q8,iv81iV1iv !3 t̂0ŪĜ~k,iv8!J . ~11!iV ,iv denote bosonic and iv8 fermionic Matsubara fre-quencies. Similarly, xggPB is evaluated asxggPB~ iV !5Trb (k,iv8ĝ~k!Ĝ~k,iv8!ĝ~k!Ĝ~k,iv81iV !.~12!The total Raman response is calculated in the gauge-invariant form which results from taking into account thelong-wavelength fluctuations of the order parameter to guar-antee local charge conservation.2,13 The total Raman suscep-tibility thus follows asx~ iV !5xgg~ iV !2xg12 ~ iV !x11~ iV !, ~13!where x1g and x11 are obtained by the replacement g(k)→1 in one or both bare Raman vertices in the vertex func-tion Eq. ~11!. The analytical continuation to the real axis isperformed using Padé approximants.14FIG. 1. ~a! Feynman diagrams for the Raman response functionincluding pair-breaking quasiparticle excitations and ‘‘two-magnonprocesses.’’ Dashed, wiggly, and solid lines represent photons, spinfluctuations, and fermionic propagators. g denotes the bare Ramanvertex in a selected scattering geometry. ~b! Ladder series for thespin-fluctuation propagator. ~c! Explicit diagram for the spin fluc-tuation contribution.15 206 PRB 62VENTURINI, MICHELUCCI, DEVEREAUX, AND KAMPFThe band-structure parameters are chosen for all the nu-merical calculations to be applicable to optimally doped sys-tems: ^n &50.85, t8/ t50.45, t' / t50.1,5 while the gap hasbeen chosen as D0 / t50.25. The effective interaction Ū andthe hopping integral t are used as the only two fit parameters,as explained below. The Raman response is evaluated at thefixed temperature T / t50.08.Let us first consider xSF alone. In Fig. 2 we plot xSF9 forthe A 1g channel versus frequency. xSF is the convolution oftwo spin susceptibilities @see Eq. ~10!# and therefore has amaximum near twice the magnetic resonance frequencies inImx s(q,v) at q5Q and q5Q85(p ,p ,0). For the chosenparameters Imx s has a strong peak for q5Q near vR'0.4and a weaker peak for q5Q8 at a slightly higher frequency;we note that vR and the peak frequency in ImxSF scale lin-early with D0 . An important point is that in the B 1g and B 2ggeometries the SF term introduces vanishingly small correc-tions to the total response, rendering the presence of the SFterm important only in the A 1g geometry. This is due to thesharpness in momentum space of the resonance peak at Q inthe SF propagator. In fact, if the transfer is taken only at Q,both the B 1g and B 2g contributions to xSF vanish identically,as can be seen from Eq. ~11!. Therefore, for these channelsthe response is given by the PB term alone.In Fig. 3 we illustrate the frequency dependence of x9(V)in the A 1g geometry for three different values of the effectiveinteraction with both the PB and SF contributions included.The shape of the Raman response is modified varying Ū / tand in particular the position of the resonance is shifted to-wards higher energies for increasing Ū . With the inclusionof the SF term we now obtain a peak slightly above D0 asobserved experimentally in Y-123 and Bi-2212.A comment is in order on the relative magnitude of the SFand the PB term. Comparing Figs. 2 and 3 it is clear that theSF term is much smaller than the PB term, but the effect ofthis new term is nevertheless visible since the backflow @thesecond term in Eq. ~13!# mixes in a nontrivial way the twocontributions. Since the SF term varies as Ū 4 in our model itstarts to dominate for larger Ū , leading to a shift in spectralweight out towards 2vR , which also changes with Ū .The experimental position of the peak is not sample de-pendent in the A 1g geometry as already mentioned, and isFIG. 2. ‘‘Two-magnon process’’ contribution xggSF in the A 1ggeometry. The parameters used are defined in the text.almost the same in different cuprates. On the other hand, thetheoretical description with the PB term alone is very sensi-tive to Raman vertex changes, which can produce variationof its position between D0 and 2D0 ,5 not allowing for acomprehensive modeling of different cuprates.In Fig. 4 we address the problem of the sensitivity of theresult to changes in the bare Raman vertex g(k). To inves-tigate the effect of changes of the vertex function, we havecalculated the final response using the three different formsfor the vertex g(k)5cos(kx)1cos(ky), g(k)5cos(kx)cos(ky)FIG. 3. The total response in the A 1g channel. Solid, dashed, anddotted lines correspond to Ū50, Ū / t51.3, and Ū / t52.0 respec-tively.FIG. 4. The total A 1g response for different vertices: g obtainedfrom effective mass approximation ~solid line!, cos(kx)1cos(ky)~dashed line!, cos(kx)cos(ky) ~dotted line!. In the upper panel theRaman response is plotted for Ū50, in the lower panel for Ū / t51.3.PRB 62 15 207COLLECTIVE SPIN FLUCTUATION MODE AND RAMAN . . .and in the EMA which posses the correct transformationproperties required by symmetry. All curves are renormal-ized to their peak height to allow for an easier comparison.Clearly the strong sensitivity to changes of the bare Ramanvertex ~first panel! is much reduced when the SF term isadded ~second panel!. As we have verified in detail, thisreduced sensitivity is not altered by different model param-eter choices and different broadening of the response.In Fig. 5 we compare the theoretical results with experi-mental data on optimally doped Bi-2212.5 Adding the SFFIG. 5. Comparison of the A1g and B1g response with Bi-2212data. The parameters used are t8/t50.45,t5130 meV, D0 /t50.25,Ū /t51.3, and ^n&50.85.contribution leads to a shift of the peak position from near;D0 for Ū50 to higher frequencies, and thus to a betteragreement with the experimental relative peak positions inA1g and B1g geometries. For the fit we have adjusted t toachieve a good agreement with the B1g channel, and thenadjusted Ū to match the A1g peak position.The value of t obtained from the fit is t5130 meV. Thisvalue has to be compared with t'105 meV, which resultsfrom the condition vR'40 meV. This slight discrepancy ismost probably related to our simple modeling of the propa-gators which neglects strong renormalizations from interac-tions as well as impurities. We note that the excess intensityin the A1g channel ~see Fig. 5! above the theory line is some-what sample dependent and is possibly related to disorder-induced effects of phonons.15,16From this work we conclude that including the SF contri-bution in the Raman response solves the previously unex-plained sensitivity of the A1g response to small changes inthe Raman vertex. Also, within our model it is now possibleto obtain the correct relative peak positions of the A1g andthe B1g scattering geometry. Whereas the SF ~two-magnon!contribution controls the A1g peak, the B1g and B2g scatter-ing geometries are essentially unaffected and determined bypair-breaking processes alone.We would like to thank R. Hackl for numerous discus-sions. F.V. would like to thank the Gottlieb Daimler and KarlBenz Foundation for financial support. This work was par-tially supported by the Deutsche Forschungsgemeinschaftthrough SFB 484.1T.P. Devereaux, D. Einzel, B. Stadlober, R. Hackl, D.H. Leach,and J.J. Neumeier, Phys. Rev. Lett. 72, 396 ~1994!.2T.P. Devereaux and D. Einzel, Phys. Rev. B 51, 16 336 ~1995!;54, 15 547 ~1996!.3J. Annett, N. Goldenfeld, and A.J. Leggett, in Physical Propertiesof High Temperature Superconductors, edited by D. M. Gins-berg ~World Scientific, Singapore, 1996!, Vol. 5.4T.P. Devereaux and A.P. Kampf, Phys. Rev. B 59, 6411 ~1999!.5T.P. Devereaux, A. Virosztek, and A. Zawadowski, Phys. Rev. B54, 12 523 ~1996!.6F. Wenger and M. Käll, Phys. Rev. B 55, 97 ~1997!; T. Strohmand M. Cardona, ibid. 55, 12 725 ~1997!; 58, 8839 ~1998!; D.Manske, C. Rieck, R. Das Sharma, A. Bock, and D. Fay, ibid.56, R2940 ~1997!; 58, 8841 ~1998!; A.V. Chubukov, D.K.Moor, and G. Blumberg, Solid State Commun. 112, 183 ~1999!.7H.F. Fong, B. Keimer, P.W. Anderson, D. Reznik, F. Dogan, andI.A. Aksay, Phys. Rev. Lett. 75, 316 ~1995!; H.F. Fong, P.Bourges, Y. Siddis, L.P. Regnault, A. Ivanov, G.D. Gu, N. Ko-shizuka, and B. Keimer, Nature ~London! 398, 588 ~1999!.8H.A. Mook, M. Yethiraj, G. Aeppli, T.E. Mason, and T. Arm-strong, Phys. Rev. Lett. 70, 3490 ~1993!.9N. Bulut and D.J. Scalapino, Phys. Rev. B 53, 5149 ~1996!.10T.E. Mason, Handbook on the Physics and Chemistry of RareEarths, edited by K.A. Gschneider Jr., L. Eyring and M.B.Maple ~Elsevier, Amsterdam, 1997!, Vol. 24.11O.K. Andersen, O. Jepsen, A.I. Liechtenstein, and I.I. Mazin,Phys. Rev. B 49, 4145 ~1994!.12A.P. Kampf and W. Brenig, Z. Phys. B: Condens. Matter 89, 313~1992!.13Note that the term ‘‘screening’’ has been used to describe thebackflow yielding the second term in Eq. ~13!. However, thebackflow is independent of the charge of the carriers and comesinto play solely from the pair interactions responsible for super-conductivity. It yields the well-known Anderson-Bogoliubovmode which restores gauge invariance. Details can be found inRef. 2.14H.J. Vidberg and J.W. Serene, J. Low Temp. Phys. 29, 179~1977!.15R. Hackl ~private communication!.16The A1g symmetry is not directly observable in tetragonal sys-tems but is obtained by subtraction of two measured symmetries~e.g., xx2x8x8). However, each of the two measured spectracontains phonons of the corresponding symmetry, which have tobe subtracted, too, to extract the electronic contribution. It is,therefore, possible that disorder-induced effects of phonons arenot completely subtractable and produce the additional spectralweight.

Collective Spin Fluctuation Mode and Raman Scattering in Superconducting
  Cuprates

Collective Spin Fluctuation Mode and Raman Scattering in Superconducting Cuprates

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