138 3 Results for High–Tc Cuprates: Doping Dependence

sensitive to the details of the band structure and the anisotropy of the spin fluctuation interaction. Comparison of the low–frequency B1g and B2g responses in the normal state could provide an indication of the strength and anisotropy of the interaction. The pair–breaking peak in the B1g response carries information about quasiparticle scattering. However, no microscopic theory exists which is able to describe simultaneously

–the di erent peak positions in the relevant scattering geometries,

–the particular lineshape of the pair-breaking spectrum,

–the power laws for small Raman shifts, and

–the relative scattering intensities.

In this section, we present results for electronic Raman scattering from hole–doped cuprates and the quasiparticle scattering rate for optimally doped and overdoped cuprates, based on the FLEX approximation for the spin and quasiparticle excitations which solve the generalized Eliashberg equations for the two–dimensional one–band Hubbard model. We find for T > Tc and intermediate coupling strengths a flat background in the Raman intensity; this can be understood in terms of the quasiparticle scattering rate, which turns out to be strongly anisotropic in momentum space but less temperature dependent for large ω. For ω > 4T , a linear behavior of the quasiparticle damping (as suggested within a nested–Fermi liquid picture) is obtained. Below Tc, the feedback e ect of the one-particle properties on the spin fluctuation spectrum is taken into account self–consistently and has important consequences in the superconducting state. For example, the quasiparticle damping which varies linearly with frequency in the normal state in accordance with the MFL [119] and NFL [129] theories, is strongly suppressed at lower frequencies in the superconducting state. These properties of the quasiparticle damping determine to a large extent the Raman spectra and also the results of INS, ARPES, tunneling, and optical–conductivity experiments in the normal and superconducting states, as discussed earlier. Thus, below Tc, we find a large pair–breaking peak in the Raman intensity and a gap developing in the B1g spectrum, while the e ect of superconductivity on the B2g spectrum is found to be much smaller.

3.3.1 Raman Response and its Relation to the Anisotropy and Temperature Dependence of the Scattering Rate

First we present our results for the Raman response function given in (2.141) in the absence of vertex corrections to the bare Raman vertices in (2.142) (J = 0 in (2.154)). For this purpose, we have solved the generalized Eliashberg equations within the FLEX approximation for the 2D tight–binding band mentioned above with B = 0.45 and µ = −1.1. This describes approximately the Fermi surfaces of the Bi2212 and YBCO compounds. Furthermore, we have taken an e ective Coulomb repulsion U (q) which has a maximum value U = 3.6 at q = Q and decreases monotonically with decreasing q to a

Fig. 3.24. Raman spectra Im χγ (q = 0, ω) for B1g polarization in the normal state at T = 0.1t and 0.023t (solid lines with increasing slopes), and in the superconducting state (Tc = 0.022t) at T = 0.021t and 0.017t, or T /Tc = 0.77 (dashed lines with increasing peaks).

value U (0) = 0.62 at q = 0. This functional form provides good agreement with INS data and also approximates the calculated vertex corrections to the spin susceptibility χs0. With these parameters, we obtain a superconducting transition at Tc = 0.022t. We remark that the vertex corrections for the irreducible spin susceptibility χs0(q, ω) are similar to those in (2.155) apart from an opposite sign and the dependence on q. It turns out that the frequency and temperature dependences are rather weak and that the dispersion with respect to q around Q can be well approximated by the phenomenological spin–spin coupling which has been used to describe the NMR data for YBCO compounds [130].

One can see from Figs. 3.24 and 3.25, for B1g and B2g symmetry, respectively, that in the normal state (solid curves) both spectra start linearly in the frequency ω and become flat at high frequencies. The slope at ω = 0 increases with decreasing temperature T , while the spectrum at high frequencies decreases with decreasing T . In the B2g spectrum, a low–frequency peak develops with decreasing T . These results are similar to normal–state results obtained from the theory of nearly antiferromagnetic Fermi liquids in the z = 1 pseudo-scaling and z = 2 mean–field–scaling regimes [132]. The experimental data available at present do not show a peak in the normal–state B2g response. It has been pointed out that observation of this structure in the B2g response and its absence in the B1g response would lend support to the current models of the Fermi topology and of the strength and anisotropy of the interaction [133].

In Fig. 3.24, for the B1g response, we recognize that a gap at lower ω and a pair-breaking peak at a threshold energy of ω = 0.15t (3/2)∆0 develop as T decreases below Tc (dashed curves). Here, ∆0 is the amplitude of the

140 3 Results for High–Tc Cuprates: Doping Dependence

Fig. 3.25. Raman response function for the B2g channel for the same temperatures as in Fig. 3.24. In the superconducting state, the slope at ω = 0 decreases with decreasing T .

dx2−y2 –wave gap, which can be estimated from the binding energy at the midpoint of the leading edge in the calculated photoemission spectrum near the antinode of the gap [134]. This gap amplitude ∆0 rises much more rapidly below Tc than does the BCS d–wave gap and reaches a value of about ∆0 = 0.1t at our lowest temperature T = 0.017t (T /Tc = 0.77). Comparison with the weak-coupling theory shows that the singularity at the pair–breaking threshold [127] is removed here by strong quasiparticle damping, while according to the weak–coupling theory of [76] this singularity is removed by a screening term arising from vertex corrections due to the pairing interaction. Electron– electron scattering due to short–range Coulomb interaction can describe the observed broadening above the pair–breaking peak in the B1g Raman spectrum of YBCO [76]. Our results for the B1g response in the superconducting state (see Fig. 3.24) agree qualitatively with the results of non–self–consistent calculations which include the e ect of inelastic scattering [133].

The Raman response function for B2g symmetry shown in Fig. 3.25 does not exhibit such dramatic e ects below Tc as those for B1g symmetry shown in Fig. 3.24. One notices that the spectrum is linear in ω for small ω, and that the slope at ω = 0 decreases and the normal–state peak broadens and shifts to somewhat higher frequency as T decreases below Tc (dashed curves in Fig. 3.25). The spectrum above this peak is somewhat enhanced up to frequencies near the pair-breaking threshold. In contrast to our results shown in Fig. 3.25, the non–self–consistent calculation yields a distinct pair–breaking peak below Tc in the B2g response which occurs much closer to the B1g pair– breaking peak [133]. We do not show the calculated Raman spectrum for A1g symmetry because it is quite similar to that for the B1g symmetry. In order to obtain the measured Ax x spectrum, we have to add the B2g spectrum to the A1g spectrum. The resulting Ax x response starts linearly in ω because,

Fig. 3.26. Results for optimum doping concentration (x = 0.15): Raman scattering intensity I Im χγ (q = 0, ω) for γ = t[cos kx − cos ky ] (B1g polarization) as a function of the transferred energy ω/t. The solid curve corresponds to T ≈ 2Tc and the dashed curve corresponds to T ≈ 1.05Tc . For comparison, the result for the superconducting state (T ≈ 0.75Tc , dash–dotted curve) is also displayed. Inset: e ective–mass ratio Re Z(k, ω) for k = ka = (0.15, 1)π as a function of ω/t.

at low frequencies, it is dominated by the B2g spectrum up to a shoulder corresponding to the small peak in the B2g spectrum. At higher frequencies, the Ax x spectrum is dominated by the A1g component, which exhibits a the large pair–breaking peak.

We come now to the discussion of the e ect of vertex corrections on the Raman response functions derived in Sect. 2.3.3. From (2.154), one sees that the general trend of the vertex correction J is to suppress the response in the B1g channel and to enhance the response in the B2g channel, while we have a mixed e ect on the A1g channel because the component of γA1g proportional to t is suppressed and the component proportional to t = −Bt is enhanced.

In Figs. 3.26 and 3.27a, we use the canonical parameters U = 4t and t = 0 and focus on optimum doping (x = 0.15). Below Tc, a pair–breaking peak in the Raman scattering intensity develops. For T ≤ 0.75Tc, the threshold position 2∆0 is approximately the peak–to–peak value calculated for the superconducting density of states, which has been discussed in Sect. 3.2.1. For temperatures T > Tc, we find a structureless (incoherent) background for large Raman shifts and only a small temperature dependence, which is in good agreement with experiment. Such a behavior is also expected in nested–Fermi liquid theory [120], where Im χ tanh(ω/4T ). We indeed find the linear behavior of Γ for ω > 4T predicted in NFL theory. However, to calculate the scattering rate τ −1, one has to take into account Re Z also (see inset). For example, we can clearly see that m /m = Re Z(ω = 0) depends strongly on temperature.

In Fig. 3.27b, we show our results for Γ and Re Z for the overdoped case x = 0.22. Again, the quasiparticle damping Γ is anisotropic. Furthermore,

142 3 Results for High–Tc Cuprates: Doping Dependence

Fig. 3.27. Comparison of the quasiparticle damping Γ = ω Im Z(k, ω) for (a) optimum doping (x = 0.15) and (b) the overdoped case (x = 0.22): we use the same notation as in Fig. 3.26. The upper curves correspond again to k = ka = (0.15, 1)π, whereas the lower curves correspond to k = kb = 0.41π(1, 1) on the Fermi line. Inset: e ective–mass ratio Re Z(k, ω) for x = 0.22.

we find that the e ective–mass ratio and Γ have decreased. Such a behavior is expected far away from the antiferromagnetic phase; this behavior is responsible for the pairing and for the lifetime e ects via spin fluctuations. Note that for small ω we do not even find an anisotropy for di erent k vectors at the same temperature. From these pictures we can conclude that the scattering rate in optimally and overdoped cuprates is strongly anisotropic. Furthermore, for large frequencies we find a linear behavior in ω, in agreement with NFL theory. This provides a possible explanation for the structureless background in the Raman scattering intensity of high–Tc superconductors in the normal state. Below Tc, a gap opens in the quasiparticle scattering at approximately ω = 3∆ − ωsf , where ωsf denotes the spin fluctuation energy (the position of the peak in the Ornstein–Zernicke–type spin susceptibility),

as discussed in Sect. 3.2.1. This leads to the observed pair–breaking Raman peak and thus reflects mainly the density of states in the superconducting state.

Finally, in order to discuss the influence of vertex corrections in the A1g channel (which are more complicated), it is intructive to go back to the weak-coupling limit and (2.156). In Fig. 3.28, we present our results for an expansion of the bare Raman vertex in Fermi surface harmonics (FSH), i.e.

L,µ

and for the integration over the Fermi circle. The unscreened response (i.e. the first term on the right-hand side) in the A1g and B1g scattering geometries diverges logarithmically at the threshold ω = 2∆0, whereas the B2g scattering intensity is small. The screening term in the B1g channel cancels the divergence without changing its position. A similar situation is realized for the A1g polarization, where a cusp remains at the threshold energy ω = 2∆0. However, existence of the same threshold energy for the A1g and B1g scattering geometries is in clear contradiction to experiment. Therefore, one might introduce higher harmonics in the expansion,

This leads to the results presented in the inset of Fig. 3.28, where only the A1g response is shown (γ0 = 0). Although the low–frequency Raman response remains less a ected, one can clearly see that the peak position is extremely sensitive to the admixture of higher harmonics. Because neither the expansion coe cients nor the convergence behavior of such an expansion is known, we may safely conclude that this approach is an unsatisfactory way to obtain a detailed description of the Raman response in the A1g polarization. The problem concerning the position of the A1g pair–breaking peak also remains in the e ective–mass approximation, which has the advantage that no additional parameters are introduced. It has been shown in [35] that the e ective mass approximation depends strongly on the details of the tight–binding band structure used in the calculation. Recently, several groups have shown that if magnon states at higher energies are taken into account, the peak position in the A1g polarization becomes more stable and resonable agreement with experiment can be achieved.

To briefly summarize this section, we have calculated the electronic Raman response function within the framework of the generalized Eliashberg equations using the 2D Hubbard model. The FLEX approximation is capable of describing the most important properties of the high–Tc cuprates, namely, their unusual normal–state behavior arising from strong electronic correlations, and the unconventional superconducting state, which is widely believed to have dx2−y2 wave pairing. These properties are reflected in the

144 3 Results for High–Tc Cuprates: Doping Dependence

Fig. 3.28. Raman scattering intensity versus Raman shift for a clean superconductor with a d–wave gap ∆(φ) = ∆0cos(2φ) calculated using an expansion in FSHs. The unscreened A1g (solid line) and B1g (medium–dashed line) responses diverge logarithmically at the threshold energy ω = 2∆0. The screened B1g intensity (short–dashed line) and the B2g channel (long–dashed line) have approximately the same (small) scattering intensity; for A1g a cusp at the threshold energy remains. Inset: unscreened (solid lines) and screened (dashed lines) A1g Raman response due to the admixture of higher harmonics α = 0, −0.15, −0.30, and −0.45 (peak positions of the screened response from right to left in this sequence). For α ≥ 0, no screening occurs.

calculated Raman response functions for the A1g or Ax x , B1g , and B2g polarizations. In the normal state these spectra start linearly in the frequency ω with a slope that increases with decreasing temperature T , and at high frequencies these spectra become almost constant. The latter property is a consequence of the linear frequency variation of the quasiparticle damping. In the superconducting state one obtains a gap and a pair–breaking peak in the B1g channel because this polarization probes the region in momentum space around the antinode of the gap. The e ect of superconductivity on the B2g spectrum is much smaller, which is not surprising, because the B2g channel probes the region around the node of the gap. Thus our results for the Raman spectra agree qualitatively with experiments on optimally doped cuprates.

3.4 Collective Modes in Hole–Doped Cuprates

In general, the internal structure of a Cooper pair can be investigated through its dynamics, i.e. the ω (and k) dependence of the condensate. In particular, in unconventional superconductors, where at least one an additional symmetry is broken, many low–frequency collective modes are present. For example, a wide variety of collective modes has been observed in the three phases of