214 A Solution Method for the Generalized Eliashberg Equations for Cuprates

and can be calculated self-consistently, including all scattering and damping e ects of the quasiparticles by the procedure described above. This strong feedback e ect has important consequences: for example, the dynamical spin susceptibility Im χs(Q, ω) (for simplicity we restrict our discussion to the antiferromagnetic nesting vector q = Q = (π, π)) shows a rearrangement of its spectral weight for small frequencies ω < 2φ(Q, ω), followed by a peak at approximately ω = 2φ. As discussed in the Introduction, this behavior provides a possible explanation for the observed ”resonance peak” in inelastic neutron scattering for many cuprate superconductors. Therefore, the strong feedback e ect of superconductivity on the elementary excitations and thus on Im χs(Q, ω) can be viewed as a general phenomenon of a strongly interacting system and, in particular, as an important fingerprint of spin-fluctuation- mediated pairing in cuprate high-Tc superconductors. To demonstrate this, the self-consistent procedure shown in Fig. A.1 is required.

References

1.N. E. Bickers, D. J. Scalapino, and S. R. White, Phys. Rev. Lett. 62, 961 (1989); N. E. Bickers and D. J. Scalapino, Ann. Phys. (N.Y.) 193, 206 (1989). 211

2.C.-H. Pao and N. E. Bickers, Phys. Rev. Lett. 72, 1870 (1994); Phys. Rev. B 51, 16310 (1995). 211

3.D. J. Scalapino, Phys. Rep. 250, 329 (1995). 211

4.P. Monthoux and D. J. Scalapino, Phys. Rev. Lett. 72, 1874 (1994). 211

5.M. Langer, J. Schmalian, S. Grabowski, and K. H. Bennemann, Phys. Rev. Lett. 75, 4508 (1995). 211

6.J. Schmalian, S. Grabowski, and K. H. Bennemann, Phys. Rev. B 56, R509 (1997). 211

7.J. Schmalian, M. Langer, S. Grabowski, and K. H. Bennemann, Comput. Phys. Comunn. 93, 141 (1996). 211

8.J. W. Serene and D. W. Hess, Phys. Rev. B 44, 3391 (1991). 212

B Derivation of the Self-Energy

(Weak-Coupling Case)

The self-energy in the one-loop approximation is given by

For spin fluctuations, we have τˆi = τˆ0. The Coulomb interaction U (p) is taken to be a constant and the full Green’s function is approximated by its noninteracting counterpart

and the coherence factors

218 B Derivation of the Self-Energy (Weak-Coupling Case)

+ [1 − f (Ek−p) + n(Eq − Ep+q )]

Now the analytic continuation iωn → ω + iη can be performed and the components of the self-energy can be obtained. The imaginary part of the self-energy which corresponds to the τˆ0 component then reads

Im Σ0(k, ω) = πU 2 [f (Ek−p) + n(Ep+q + ω)] 8

p

×c+(p + q, q) [f (Eq ) − f (Ep+q )]

q

× [δ(ω + Ek−p − Eq + Ep+q ) − δ(ω + Ek−p + Eq − Ep+q )] + c−(p + q, q) [1 − f (Eq ) − f (Ep+q )]

× [δ(ω + Ek−p + Eq + Ep+q ) − δ(ω + Ek−p − Eq − Ep+q )]

×[f (Ek−p) + n(Eq − Ep+q )]

×ω + iη − Ek−p + Eq − Ep+q − ω + iη + Ek−p − Eq + Ep+q

+ [1 − f (Ek−p) − n(Eq + Ep+q )]

×−ω + iη + Ek−p + Eq − Ep+q − ω + iη − Ek−p − Eq + Ep+q

+ c−(p + q, q) [f (Eq ) + f (Ep+q ) − 1]

×[f (Ek−p) + n(Eq − Ep+q )]

×−ω + iη + Ek−p − Eq − Ep+q + ω + iη − Ek−p + Eq + Ep+q

(B.14)

and thus, for the corresponding imaginary part,

×c+(p + q, q) [f (Eq ) − f (Ep+q )]

q

× [δ(ω + Ek−p + Eq − Ep+q ) − δ(ω + Ek−p − Eq + Ep+q )]

220B Derivation of the Self-Energy (Weak-Coupling Case)

+c−(p + q, q) [1 − f (Eq ) − f (Ep+q )]

× [δ(ω + Ek−p − Eq − Ep+q ) − δ(ω + Ek−p + Eq + Ep+q )]

+ k−p [f (Ek−p) + n(Ek−p − ω)]

p Ek−p

×c+(p + q, q) [f (Eq ) − f (Ep+q )]

q

× [δ(ω − Ek−p + Eq − Ep+q ) − δ(ω − Ek−p − Eq + Ep+q )] + c−(p + q, q) [1 − f (Eq ) − f (Ep+q )]

× [δ(ω − Ek−p − Eq − Ep+q ) − δ(ω − Ek−p + Eq + Ep+q )] .

(B.15)

The self-energy which corresponds to the τˆ1 component can be evaluated as

Σ1(k, ω)

×[f (Ek−p) + n(Eq − Ep+q )]

×ω + iη − Ek−p + Eq − Ep+q − ω + iη + Ek−p − Eq + Ep+q

×−ω + iη + Ek−p + Eq − Ep+q − ω + iη − Ek−p − Eq + Ep+q

+ c−(p + q, q) [f (Eq ) + f (Ep+q ) − 1]

In the other cases of the τˆ1 and τˆ3 components we obtain similar results, which can be written in a unified way using the nesting approximation. In this approximation one assumes that the main contribution to the momentum sum comes from Im χ at the (transferred) nesting vector Q = (π, π). In this case, the self-energy no longer depends on k and one can write (ν = 0, 1, 3)

This completes the derivation of (3.8).

C dx2−y2-Wave Superconductivity

Due to Phonons?

In this appendix we analyze how the magnetic mode which is usually peaked at q = Q = (π, π) leads to a kink in the ARPES results and to a dx2−y2 -wave order parameter that is maximal around (π, 0). In particular, we consider to what extend phonons contribute to this result. This is an extension of the general remarks made in sect. 1.4.3.

In general, the generalized Eliashberg equations read, after the inclusion of attractive phonons (branch i) via their spectral function α2Fi(q, Ω),

For α2Fi(q, Ω), we employ a Lorentzian in the frequency Ω around Ω0 ≈ ωD (Debye frequency), and a normalized form factor Fi(q) peaked at q = qpair as indicated in Fig. C.1. The spin fluctuations that dominate Ve (q, ωsf ) are peaked at q = Qpair.

It is instructive to write down the weak-coupling limit of the τˆ component

1

where again Ek = ∆2(k) + 2k is the dispersion of the quasiparticles in the superconducting state. Note that the contribution to the pairing potential is repulsive for spin fluctuations and attractive for phonons. In the case where no phonons contribute to the Cooper pairing (α2Fi(q) = 0), Ve (q) bridges parts of the Fermi surface where the superconducting order parameter has opposite signs. This momentum dependence of the pairing interaction is required for solving (C.1) and is typical of unconventional superconductivity. Note that for a repulsive and momentum-independent pairing potential, Ve (q) = const, no solution of (C.1) can be obtained (see sect. 1.4.3).

226 C dx2−y2 -Wave Superconductivity Due to Phonons?

Fig. C.1. Illustration of dx2−y2 -wave Cooper pairing for a fixed frequency Ω = Ω0 ≈ ωsf ≈ ωD due to spin fluctuations peaked at momentum k − k = q = Qpair and phonons peaked at q = qpair. The solid lines denote the Fermi surface and the dashed lines the nodes of the dx2−y2 -wave order parameter. The corresponding sign of the order parameter is also displayed.

How is the kink related to the pairing mechanism? Physically speaking, the interdependence of the elementary excitations that dominate Ve (q) leads to dx2−y2 -wave Cooper pairing, as well as to the kink structure observed by ARPES experiments. In other words, the quasiparticles around the hot spots

couple strongly to spin fluctuations. This coupling leads to (a) a dx2−y2 -wave order parameter, and (b) a kink in the nodal direction that occurs close to

the Fermi level where Qpair = (π, π), as indicated in Fig. 3.19.

It follows also from (C.1) that attractive phonons with a corresponding spectral function α2F (q) peaked at q = qpair contribute constructively to

dx2−y2 -wave pairing, as long as the main pairing interaction is provided by spin fluctuations. However, the kink close to the antinodal points occurs only below Tc and is a result of the fact that ∆(ω) is maximal around (0, π). Therefore, the kink structure in the antinodal direction is connected mainly to spin excitations peaked at Qpair = (π, π) and not to the phonon branch peaked at qpair.

Note that in the case where no spin fluctuations were present, i.e. Ve (q) = 0, the attractive phonon contribution would cancel the minus sign on the right-hand-side of (C.1), yielding an order parameter with s-wave symmetry. Thus we can safely conclude that dx2−y2 -wave Cooper pairing due to phonons and the anisotropy of the kink feature in the elementary excitations are hard to reconcile within the same physical picture.