Superconductivity
Generic forms of superconductivity
A generic form of superconducting Hamiltonian,
^ = ∑ ϵ→ c† c→ − 1 ∑ Δ→ c †→ c † → + Δ ∗k,σ → 1 c −k,σ
→ 2
H →
k k,σ k,σ k,σ 1 ,σ 2 k,σ 1 −k,σ 2 → 1 ,σ 2 c k,σ
→
2 →
k,σ k
can be characterized by a superconducting matrix,
→
Δ k,↑↑ →
Δ k,↑↓
Δ k→ = [ ]
→
Δ k,↓↑ →
Δ k,↓↓
The symmetry of the SC order determines the nature of the SC order
A generic type of superconductor is characterized by the order parameter,
Real space
r, r→′ ) ∼< c r→↑ c r→′ ↓ >
Δ ↑↓ (→
Momentum space
→ ∼< c → c → >
Δ ↑↓ (k) k↑ −k↓
→
The superconducting state can be characterized by the symmetry of Δ ↑↓ (k)
In order to classify the superconductivity, the superconducting order is expanded in terms of
harmonics (l=1, 2, 3 etc,.) in the momentum space. If the gap matrix is l=0 then we have s-wave
superconductor, l=2, then we have d-wave superconductor and so on.
SPIN-SINGLET(EVEN) :
→ = Δ ↑↓ (−k)
Δ ↑↓ (k) →
SPIN-TRIPLET(ODD) :
→ = −Δ ↑↑ (−k)
Δ ↑↑ (k) →
Gapped and gapless superconductors
Fully gapped : |Δ ↑↓ (k)| → 2 > 0 (NbSe2)
Gapless : |Δ ↑↓ (k→α )| 2 = 0 (Twisted trilayer graphene)
Source : 31st Jyväskylä Summer School:
