where Dν denotes the corresponding di usion constant. Only pairs with size r ≤ rω contribute to the screening. Unfortunately, Dν is not easy to calculate. In the absence of pinning, the theory of Bardeen and Stephen yields [82]

where c is the speed of light, ξab r0/2 corresponds to the coherence length, ρn denotes the normal-state resistivity, φ0 = hc/2e is the elementary super-

˜

conducting flux quantum, and d corresponds to an e ective layer thickness. However, in cuprate superconductors, pinning becomes important. Thus we have assumed a simple Arrhenius law

where Ep denotes the corresponding pinning energy barrier. We insert (2.137) and (2.136) into (2.135), yielding a new length scale l in (2.131) and (2.132), namely a new upper limit l = ln(rω /r0). Then, with the help of (2.135)– (2.137) and (2.131) and (2.132), we calculate the dynamical conductivity σ(ω) via [20]

where e is the elementary charge, and Im σ(ω) has been obtained from the current–current correlation function and the Kubo formula using the FLEX approximation (see (2.128)).

2.3.3 Raman Scattering Intensity Including Vertex Corrections

In general, the di erential cross section in a Raman scattering experiment is proportional to the imaginary part of the Raman response function χΓ γ , which is given by the analytic continuation of

ˆ

where G is the Green’s function in Nambu space and τˆi are the Pauli matrices. Since the momentum transfer from the scattered photon to the electronic system q is much smaller than the extension of the Brillouin zone, we put q = 0. γk is the bare vertex that describes the coupling of light to e ective density fluctuations, and Γ denotes the dressed vertex, which includes renormalization e ects due to the pairing interaction and elastic electron–electron

66 2 Theory of Cooper Pairing Due to Exchange of Spin Fluctuations

scattering [83]. γ(k) can be parameterized by the so-called e ective–mass approximation, i.e.

Here, eˆI and eˆS are the polarization vectors of the incoming and scattered light, respectively, and k is the normal-state dispersion for which a twodimensional tight-binding band structure

k = −2t [cos(kx) + cos(ky ) − 2B cos(kx) cos(ky ) + µ/2]

introduced earlier in (2.4), is assumed. This approach is often used and is believed to be valid in the nonresonant limit (i.e. neglecting interband transitions). Analytic continuation from imaginary to real frequencies leads to the following expression for the Raman response function:

Here, N (k, ω) = A0(k, ω) + A3(k, ω) and A1(k, ω) are the spectral densities of the Green’s functions G and F . The bare Raman vertices for the di erent polarization symmetries B1g , B2g, and A1g are the following [83]:

Here, t is the nearest–neighbor hopping energy and t = −Bt (with B = 0.45) is the next–nearest–neighbor hopping energy in the tight-binding band [84]. It should be pointed out that we have subtracted from the vertex for Ax x symmetry given in [83] the vertex for B2g symmetry in order to obtain an A1g component which is fully symmetric with respect to the D4h point group.

In order to derive the vertex corrections and thus the equation Γ (γ), let us first show that Ward’s identity for the electromagnetic kernel holds also for the FLEX approximation. The general expression for the current–charge correlation function in the 2 × 2 Nambu matrix formalism is given by [19]

Again, here Γν is the dressed vertex function, and γµ is the bare current– charge vertex (µ = 1, 2, 3):

Note that P00(q) = e2χc0(q), where χc0 is the irreducible charge susceptibility. The Dyson equation yields the dressed 2 × 2 matrix Green’s function G in terms of the bare Green’s function G0 and the self-energy Σ:

G−1(k) = G−0 1(k) − Σ(k)iωnZ(k)τ0 − [ (k) + ξ(k)] τ3 − φ(k)τ1 . (2.145)

As discussed earlier in this chapter, the self-energy Σ in the FLEX approximation for the Hubbard Hamiltonian is determined by the following generalized Eliashberg equations:

where Ps(q) and Pc(q) are defined in (2.31) and (2.32), respectively. The ladder approximation to the vertex function Γµ corresponding to the FLEX approximation to Σ then yields the following linear equation:

Γµ(k + q, k) = γµ(k + q, k)

+ [τ0G(k + q)Γµ(k + q, k )G(k )τ0Ps(k − k )

k

+ τ3G(k + q)Γµ(k + q, k )G(k )τ3Pc(k − k )] . (2.146)

Gauge invariance of the electromagnetic kernel requires that Γµ satisfy Ward’s identity [19]:

µ

One can derive Ward’s identity (2.147) from (2.146) by inserting (2.147) on the right–hand side in (2.145) and then making use of the self-energy equations. For q = 0, the important relationship P00(q = 0, iνm) = e2χc0(q = 0, iνm) = 0 follows from (2.147). Furthermore, we obtain from Ward’s identity in (2.147) for q = 0 the following expression for the vertex Γ0:

Γ0(k, ω + ν, ω) = Z(k, ω + ν)τ3 + ω [Z(k, ω + ν) − Z(k, ω)] ν−1τ3

The last term, proportional to τ3τ1 = iτ2, in (2.148) diverges for ν → 0 and corresponds to the collective gauge mode [19, 85]. This is renormalized by the Coulomb interaction to the 2D plasmon.

The functions f and b in (2.151) and (2.155) are again the Fermi and Bose functions.

In the weak-coupling limit, where the gap function ∆(k) is independent of ω, Manske et al. have shown that the Raman response functions for the three relevant polarization geometries including vertex corrections, are approximately given by the following (assuming particle–hole symmetry and T = 0) [83]:

Here, the averages f (k) are defined in terms of the Tsuneto function (see (10) in [83]). Let us emphasize that the second term in (2.157) is present only for a d-wave gap, and vanishes for an s-wave symmetry of ∆.

We come now to the theoretical description of order parameter collective modes in Raman scattering experiments. These modes can be calculated analogously to those in p-wave–pairing superconductors [86]. In general, it can be said that the dx2−y2 -wave pairing component in weak-coupling theory gives rise to a phase fluctuation mode which is renormalized into a 2D plasmon [87], and to an amplitude fluctuation mode of the d-wave gap. For each additional (weaker) pairing component, such as an extended s-wave component, one obtains an amplitude (real) and a phase (imaginary) fluctuation mode. Let us consider first the amplitude fluctuation mode of the dx2−y2 -wave gap. We have calculated the mode frequency ω0 from the weak-coupling expression

The e ect of quasiparticle damping is taken into account by carrying out an analytical continuation of this result from iωm to ω + iΓ . For a gap ∆(k) = (∆0/2)(cos kx − cos ky ) and a band (k) with t = 0 and chemical potential µ, the summation over k in the square Brillouin zone has been carried out numerically for the following expression for T = 0:

Here, Ek2 = 2(k)+∆2(k). We obtain a peak in the function of ω, Re χs0(Q, ω) at the kinematical gap ω = 2|µ| [89], whose height decreases with increasing Γ . The approximate analytic result for T = 0 is given by

70 2 Theory of Cooper Pairing Due to Exchange of Spin Fluctuations

where NF is the density of states at the Fermi energy,

z = 4µ2 − (ω + iΓ )2 / (2∆0)2 and , V0−1 = NF log (2W/∆0) . (2.162)

Here, W = 4t is the half bandwidth. The function Re χs0(Q, ω) in (2.162) first rises with ω2 and then exhibits a peak at the kinematical gap 2|µ|, whose height is about V0−1 − (π/2)NF (Γ/2∆0). A low-frequency mode, i.e. a zero of the equation Re χs0(Q, ω) = 1/U , is obtained only for a finite range of U values which decreases with increasing Γ . For t = −0.45t, a kinematical gap no longer exists and the e ective |µ| is nearly zero. The approximate analytical result for the expression in (2.161) becomes then equal to

Here, K is the first elliptic integral. By summing numerically over k in the

square Brillouin zone, we obtain, for t = 0 in (k) and T = 0, two solutions

√

of (2.159) with frequencies ω0 3∆0 provided that the damping Γ is su ciently large, namely, ω0 < 3.5Γ . For t = −0.45t and T = 0, we obtain two solutions whose frequencies are somewhat larger, ω0 2∆0, where again the condition ω0 < 3.5Γ has to be satisfied. For a mode frequency ω0 = 2∆0 0.2t at T /Tc = 0.77, we find a damping Γ (ω0) 0.1t at the anti-node ka

which means that the condition ω < 3.5Γ is satisfied. In [87], Wu and Gri n

0√

have obtained a frequency ω0 = 3∆0 for the amplitude collective mode; however, the coupling of this mode to the charge fluctuations was neglected. We find that the coupling of this fluctuation in the particle–particle channel to the charge fluctuation in the particle-hole channel yields approximately the following contribution χf l to the charge susceptibility χc0 at T = 0 (see [88] and [86]):

the Fermi energy ω = 0. One notices from (2.165) that in the limit γ → 0 one