obtains no valid solution of the equation Re g(ω0) = 0 because the solution

ω¯0 = 3/2 violates the condition that ω¯0 ≤ 1. However, for su ently large values of Γ (γ ≥ ω/¯ 2), one obtains a solution of the equation Re g(ω0) = 0 which satisfies the condition ω¯0 ≤ 1.

We have also calculated the resonance frequency of the exciton-like s-wave mode of the order parameter which is caused by an additional s-wave pairing

component |g | smaller than the main d-wave pairing component |g¯ | (see

0 2

[87]). The method of Refs. [88] and [86] yields the following contribution χexc from this order parameter fluctuation mode to the charge susceptibility χc0

where

From (2.167) we obtain the following approximate result:

where V0−1 is given by (2.162) and ω¯ and γ by (2.163). We have carried out the summation over k in (2.167) numerically and find in agreement with Ref. [87], that a solution of the equation Re gexc(ω0) = 0 for given ∆0 and Γ exists

only for very small values of the parameter (g¯ /g ) − 1 (≤ 0.1). This means

2 0

that the s-wave pairing coupling has to be almost as strong as the d-wave pairing component, which is quite unrealistic. However, with increasing Γ the resonance frequency ω0 decreases and becomes much smaller than the pairbreaking threshold 2∆0 for reasonably large scattering rates (Γ/2∆0 1/2). This means that the contribution Im χexc(ω) of the exciton-like mode to the Raman scattering intensity with the B1g polarization shows up as a small peak below the pair-breaking threshold. Since the damping Γ in the direction of the momentum of the antinode of the order parameter rises rapidly with ω, it may be that this peak becomes observable for smaller values of the ratio

g /g¯ of the s-wave and d-wave pairing couplings than those obtained from

0 2

weak-coupling theory [87]. This will be discussed in the next chapter.

2.3.4 Optical Conductivity

Similarly to the Raman response, the optical conductivity σ(ω) is calculated here with the help of the current–current correlation function using the spectral densities which solve the generalized Eliashberg equations. For underdoped cuprates, in the presence of a pseudogap, the in-plane conductivity σab(ω), neglecting vertex corrections, is given by

72 2 Theory of Cooper Pairing Due to Exchange of Spin Fluctuations

where vk,i = ∂ k/∂ki are the band velocities within the ab plane and are calculated for the corresponding tight-binding energy dispersion of the quasiparticles. Again, N is the normal spectral function, and A1 and Ag are the anomalous spectral functions with respect to the superconducting gap and the pseudogap, respectively. These spectral functions have been taken from a self-consistent solution of the generalized Eliashberg equations in the presence of the pseudogap, as already descibed in (2.63)–(2.65):

We again want to emphasize that it is necessary to include the bubble contribution due to Ag in the conductivities and susceptibilities. Although Ward’s identities are satisfied, neglect of this term leads to disagreement with experimental data. In the optimally and overdoped cases where no pseudogap is present, we take Ag ≡ 0.

It is interesting to remark that for high-Tc cuprates the transport properties perpendicular to the CuO2 planes are also of significant interest. For example, measurements of the c-axis conductivity suggest that the conductance in c direction is coherent in the overdoped regime [90] and successively becomes incoherent in the underdoped regime [91, 92]. In this work we shall therefore study the two limits of coherent and incoherent c-axis conductivity. The coherent conductivity along the interplane c direction is given, to lowest

where c0 and a0 are the c-axis and ab-plane lattice constants taken from experiment. On the other hand, incoherent conductivity corresponds to di use c-axis transmission and amounts to taking the averages of the spectral functions N (k, ω), A1(k, ω), and Ag (k, ω) over all momenta (see the discussion in [94]). This means that N (k, ω) is replaced by the density of states