R. Zeyher and A. Greco Electronic Raman scattering in
superconduc-tor probes charge excitations across the su-perconducting gap providing thus information on the magnitude and the anisotropy of the gap. Commonly used weak-coupling theo-ries fail to explain the spectra at optimal dop-ing and in the underdoped regime, mainly be-cause of the appearance of two energy scales in the experiments. The larger one is
typ-measured by ARPES (Angle-Resolved Photo-emission Spectroscopy). It increases mono-tonically with decreasing doping in contrast to the parabolic-like doping dependence of Tc. The doping dependence of the smaller energy scale, typical for A1g and to some extent for B2g spectra, resembles that of Tc. The pres-ence of two energy scales is especially con-spicuous in optimally doped YBa2Cu3O7: The
600 cm1 whereas superconductivity-induced phonon renormalizations indicate a gap of about 310 cm1.
A possible key for an understanding of the data in the optimal and underdoped region is provided by the t-J - model. Its phase dia-gram in the large N limit (N is the number of spin components) is largely determined by the onset of a charge-density wave (CDW) at a doping δ=δ0 with a wavevector Q near (ππ). Its order parameter has the form Φ(k) =Φ γ(k) withγ(k) =coskxcosky2, i.e., exhibits d-wave symmetry. Furthermore, there exists for all dopings an instability to-wards d-wave superconductivity with an order parameter ∆(k) =∆γ(k). In general, it is nec-essary to include also a Coulomb potential VC in order to stabilize the d-CDW with respect to phase separation. The resulting superconduct-ing transition temperature Tcdecreases with in-creasing doping forδδ0. Forδδ0 the two order parameters compete with each other lead-ing to a strongly decreaslead-ing Tc with decreasing doping. As a result optimal doping is essentially determined byδ0.
Figure 19: Order parameters Φ (d-CDW) and ∆ (superconductivity) as a function of doping at T = 0.
Figure 19 shows the calculated doping depen-dence of Φ and ∆ at zero temperature, using tt = –0.35, Jt = 0.3, and VCt = 0.06, where
t and t are nearest and next-nearest neighbor hopping integrals, and J and VC Heisenberg and Coulomb coupling constants between near-est neighbors. In the overdoped region δδ0, Φ is zero and∆ increases monotonically with decreasing δ. After the onset of the d-CDW at δ0, Φ is first suppressed by superconductiv-ity and then, with decreasing doping, increases steeply and suppresses now the superconduct-ing order parameter. In our calculation we as-sumed the value (ππ) for the wavevector Q.
Figure 19 shows that superconductivity coexists in the underdoped region with the d-CDW and that the competition between the two order pa-rameters leads to the rapid decay of Tc towards small dopings. The gap in the one-particle spec-trum consists in general of a CDW and a super-conducting part. The former increases mono-tonically with decreasing δ whereas the latter shows a maximum atδ0and decreases towards the overdoped and the underdoped regions.
Figure 20: Electronic Raman spectra of B-symmetry for three different dopingsδ, calculated for tt = –0.35, Jt = 0.3, and T = 0. The B2g spec-trum has been multiplied by 10 in the two upper and by 100 in the lower diagrams.
Figure 20 shows theoretical B1gand B2gspectra.
The upper, middle and lower panels are typi-cal for the overdoped, optimally doped and un-derdoped regions, respectively. The B2g spec-tra contains no collective effects and is mainly determined by the superconducting part of the gap. Moreover, its bare vertices weight heavily transitions near the diagonals in k-space where the gap vanishes. As a result, the B2gspectrum in the upper and middle panels exhibit a broad peak well below the total gap which lies near the maxima of the curves B1g0. The peak shifts to-wards lower frequencies at low dopings (lower panel) and looses intensity.
The B1g spectrum without vertex corrections, denoted by B1g0 in Fig. 20, is determined by free particle-hole excitations mainly at the X-and Y-points of the Brillouin zone. This leads to a well-pronounced peak at the maximum of the gap as shown by the dashed, blue curves in Fig. 20. This maximal gap is caused both by the CDW and superconductivity and increases monotonically with decreasing δ. Including also vertex corrections we obtain the curves B1g. Practically the total spectral weight of the B1g0 curve is shifted down in energy into one peak which still increases slightly in en-ergy with decreasingδ. On the overdoped side this peak describes an exciton state of the pure superconductor, on the underdoped side it de-scribes amplitude fluctuations of the d-CDW order parameter. The results in Fig. 20 agree at least qualitatively with experiments in the cuprates, in particular, the increase of the fre-quency of the B1gpeak with decreasing doping, the non-monotonic behavior of the B2gpeak as a function of doping similar to that of Tc, and the fact that the (two-particle) Raman gap is al-ways smaller than two times the (one-particle) ARPES gap.
Quantitatively, the predicted final state effects seem to be too large. For instance, two times the ARPES gap in slightly overdoped Bi2212 amounts to 70–75 meV whereas the B1gpeak
lies near 59–63 meV for the same doping. The analogous values in untwinned YBa2CuO7 are 58 and 88 meV at the X- and Y-points from ARPES, whereas the corresponding peaks in Raman scattering are at 50 and 60 meV.
Figure 21: ˜A1g0and ˜A1g: Unperturbed and full A1g spectra calculated with the indirect coupling; A1gs: Screened, full A1gspectrum multiplied by 50 (upper two diagrams) or 10 (lower diagram).
The red lines in Fig. 21, called A1gs, show screened A1gspectra which include correlation effects as given by the t-J - model in the large N limit. Their absolute intensities are about 2 orders in magnitude too small to be able to account for the experimental spectra. Density fluctuations induced by the the A1gRaman ver-tex may, however, also couple to the super-conducting order parameter via the modulation of the density of states at the Fermi energy.
Figure 21 shows calculated A1g spectra using this coupling which is unaffected by Coulomb screening. The dashed and solid lines, denoted by ˜A1g0 and ˜A1g, respectively, have been calcu-lated without and with vertex corrections, re-spectively.
A˜1g0 exhibits a rather broad peak at the maxi-mum gap similar as B1g0. Including also ver-tex corrections the spectral weight shifts down into a pronounced collective peak describing amplitude fluctuations of the superconducting order parameter. Going from the overdoped to the underdoped regime this peak first increases, then passes through a maximum around optimal doping, and then decreases in frequency and in-tensity, becoming at the same time rather broad.
The decrease in its frequency can be understood from the fact that the frequency of collective amplitude fluctuations is proportional to∆and thus would vanish with vanishing ∆ if damp-ing effects could be neglected. Figure 21 is in agreement with experiments in Bi2212 where the A1g peak passes through a maximum and then decreases on the underdoped side.
In conclusion, we have shown that the observed different behavior of the three symmetry com-ponents of the electronic Raman spectrum in high-Tc superconductors as a function of dop-ing can be explained within a t-J - model in the large N limit. Basic ingredients of this approach are the strong competition of the superconduct-ing and the d-CDW order parameters in the un-derdoped regime and the importance of collec-tive effects. The peak in the B1g spectrum in the superconducting state is explained by am-plitude fluctuations of the d-CDW order param-eter which, in the optimal and overdoped re-gion, can also be viewed as excitonic states. We also found that the indirect coupling of light to the superconducting order parameter is impor-tant leading to the conclusion that the A1gpeak is mainly due to amplitude fluctuations of the superconducting order parameter, at least, if the interlayer hopping is sufficiently small.