Carsten Timm: Theory of superconductivity

k y
0
Q
k x
This type of gap is called a d x2 −y 2-wave (or just d-wave) gap since it has the symmetry of a d x 2 −y 2-orbital
(though in k-space, not in real space). The simplest gap function with this symmetry and consistent with the
lattice structure is
∆ k = ∆ 0 (cos k x a − cos k y a). (12.8)
Recall that the gap function away from the Fermi surface is of limited importance.
The d-wave gap ∆ k is distinct from the conventional, approximately constant s-wave gap in that it has zeroes
on the Fermi surface. These zeroes are called gap nodes. In the present case they appear in the (11) and equivalent
directions. The quasiparticle dispersion in the vicinity of such a node is
√
E k =
√ξk 2 + |∆ k| 2 ∼ = (ϵ k − µ) 2 + ∆ 2 0 (cos k xa − cos k y a) 2 . (12.9)
One node is at
k 0 = k ( )
F 1
√ . (12.10)
2 1
Writing k = k 0 + q and expanding for small q, we obtain
E ∼ √
k0+q = (v F · q) 2 + ∆ 2 0 [−(sin k0 xa)q x a + (sin kya)q 0 y a] 2 , (12.11)
where
v F := ∂ϵ k
∂k
∣
∣
k=k0
= v ( )
F 1
√
2 1
(12.12)
is the normal-state Fermi velocity at the node. Thus
√
[(
E ∼ k0 +q = (v F · q) 2 + ∆ 2 0 −a sin k F a
√ , a sin k ) 2 √
F a
√ · q]
= (v F · q) 2 + (v qp · q) 2 , (12.13)
2 2
where
v qp := ∆ 0 a sin k ( )
F a −1
√ ⊥ v 2 1
F . (12.14)
Thus the quasiparticle dispersion close to the node is a cone like for massless relativistic particles, but with
different velocities in the directions normal and tangential to the Fermi surface. Usually one finds
The sketch shows equipotential lines of E k .
k y
v F > v qp . (12.15)
k x
122