Carsten Timm: Theory of superconductivity

The fact that the gap closes at some k points implies that the quasiparticle density of states does not have a gap.
At low energies we can estimate it from our expansion of the quasiparticle energy,
D s (E) = 1 ∑
δ(E − E k )
N
k
∼= 4 ∑
(
)
δ E −
√(v F · q)
N
2 + (v qp · q) 2
q
∫
d
∼= 2 (
)
q
4a uc
(2π) 2 δ E −
√(v F · q) 2 + (v qp · q) 2
= 4a ∫
uc
v F v qp
d 2 u
(
(2π) 2 δ E −
0
√ )
u 2 x + u 2 y
= 2a ∫ ∞
uc
du u δ(E − u) =
2a uc
E, (12.16)
πv F v qp πv F v qp
where a uc is the area of the two-dimensional unit cell. We see that the density of states starts linearly at small
energies. The full dependence is sketched here:
D s (E )
D n ( E F )
2
0
∆
max
E
An additional nice feature of the d-wave gap is the following: The interaction considered above is presumably
not of BCS (Coulomb + phonons) type. However, there should also be a strong short-range Coulomb repulsion,
which is not overscreened by phonon exchange. This repulsion can again be modeled by a constant V 0 > 0 in
k-space. This additional interaction adds the term
− 1 ∑ 1 − n F (E k ′)
V 0 ∆ k ′ (12.17)
N 2E k
k ′ ′
to the gap equation. But since E k ′ does not change sign under rotation of k ′ by π/2 (i.e., 90 ◦ ), while ∆ k ′ does
change sign under this rotation, the sum over k ′ vanishes. d-wave pairing is thus robust against on-site Coulomb
repulsion.
12.2 Cuprates
Estimates of T c based on phonon-exchange and using experimentally known values of the Debye frequency,
the electron-phonon coupling, and the normal-state density of states are much lower than the observed critical
temperatures. Also, as we have seen, such an interaction is flat in k-space, which favors an s-wave gap. An s-wave
gap is inconsistant with nearly all experiments on the cuprates that are sensitive to the gap. The last section has
shown that d x2 −y2-wave pairing in the cuprates is plausible if there is an attractive interaction for momentum
transfers q ≈ (π/a, π/a). We will now discuss where this attraction could be coming from.
123