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Lecture XXII 64

Lecture XXII: BCS Superconductivity

⊲ Recall: We have seen that phonon exchange a pairing interaction which rendersa single pair of electrons unstable towards the formation of a bound state (Cooper)

Motivated by this consideration, we have proposed a many-body generalisationof the pair state in the form of the variational BCS state

|ψ =k

(uk + vkc†k↑c

†−k↓)|Ω

and within which one may show that the “anomalous average” bk = ψ|c†k↑c

†−k↓|ψ

acquires a non-zero expectation

In principle, we could now proceed with the Ansatz for |ψ and employ a variational

analysis. However, instead, we will make use of this Ansatz to develop an approximation scheme to expand the second quantised BCS Hamiltonian. Indeed, such an approach will

lead to the same phenomenology.

Since we expect quantum fluctuations in bk to be small, we may set

c†k↑c

†−k↓ = bk+

small c†k↑c

†−k↓ − bk

(cf. our approach to BEC where a†0 was replaced by a C-number) so that

H − µN =kσ

ζ kc†kσckσ − V

kk′

c†k′↑c

†−k′↓c−k↓ck↑, ζ k = ǫk − µ

≃ k

ζ kc†kσckσ − V kk′

bkc−k′↓ck′↑ + bk′c†k↑c†−k↓ − bkbk′

Then, if we set V

kbk ≡ ∆, we obtain Bogoliubov-de Gennes or Gor’kov Hamiltonian

H − µN =k

ζ kc†kσc

kσ −k

∆c−k′↓c

k′↑ + ∆c†k↑c

†−k↓

+|∆|2

V

=k

c†k↑ c−k↓

ζ k −∆−∆ −ζ k

ck↑

c†−k↓

+|∆|2

V

For simplicity, let us for now assume that ∆ is real

Bilinear in fermion operators, H − µN is diagonalised by canonical transformation

ck↑

c†−k↓

=

OT uk −vkvk uk

γ k↑

γ †−k↓

where anticommutation relation requires OT O = 1,i.e. u2

k+ v2

k= 1 (orthogonal transformations)

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Lecture XXII 65

Substituting, one finds that the Hamiltonian diagonalised if

2ζ kukvk + ∆(v2k − u2k

) = 0

i.e. setting uk = sin θk and vk = cos θk,

tan2θk = −∆ζ

, sin2θk = ∆ ζ 2k

+ ∆2, cos2θk = − ζ k

ζ 2k

+ ∆2

(N.B. for complex ∆ = |∆|eiφ, vk = eiφ cos θk)

As a result

H − µN =k

(ζ k − (ζ 2k

+ ∆2)1/2) −∆2

V +kσ

(ζ 2k

+ ∆2)1/2 γ †kσγ kσ

Quasi-particle excitations, created by γ †kσ, have minimum energy ∆

Energy gap

rigidity of ground state

Ground state wavefunction identified as vacuum state of algebra γ kσ, γ †kσ,

i.e state which is annihilated by all the quasi-particle operators γ kσ.

Condition met uniquely by the state

|ψ ≡k

γ −k↓γ k↑|Ω =

k

(ukc−k↓ − vkc†k↑)(ukck↑ + vkc

†−k↓)|Ω

=k

(vk)(ukc−k↓c†k↓ − vkc

†k↑c

†−k↓)|Ω = const. ×

k

uk + vkc

†k↑c

†−k↓

|Ω

cf. variational analysis in fact, const. = 1

Note that phase of ∆ is arbitrary,i.e. ground state is continuously degenerate (cf. BEC)

⊲ Self-consistency condition: BCS gap equation

∆ = V k

bk = V k

ψ|c†k↑c

†−k↓|ψ = V

k

ukvk =V

2

k

sin2θk =V

2

k

∆ ζ 2k

+ ∆2

i.e. 1 =V

2k

1 ζ 2k

+ ∆2 =V Ldν (ǫ

F )

2 ωD−ωD

dζ 1

ζ 2 + ∆2

if ωD ≫ ∆, ∆ ≃ 2ωDe− 1ν(ǫF )V L

d

Ground state: In limit ∆ → 0, v2k→ θ(ǫF − ǫk), and the ground state collapses

to the filled Fermi sea with chemical potential ǫF

As ∆ becomes non-zero, states in the vicinity of the Fermi surface rearrange into acondensate of paired states

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Lecture XXII 66

Excitations: Spectrum of quasi-particle excitations

ζ 2k

+ ∆2 shows rigid energy gap ∆.An excitation can be either the creation of a quasi-particle at positive energy or theelimination of a quasi-particle (the creation of a quasi-hole) at negative energy. Inthe ground state, all negative-energy quasi-particle states are filled.

Density of quasi-particle excitations near Fermi surface

ρ(ǫ) =1

Ld

kσ

δ (ǫ −

ζ k + ∆2) =

dζ

1

Ld

kσ

δ (ζ − ζ k)

ν (ζ )

δ (ǫ −

ζ + ∆2)

≈ ν (ǫF )s=±1

∞0

dζ δ

ζ − s[ǫ2 − ∆2]1/2

∂ [ζ 2+∆2]1/2

∂ζ

= 2ν (ǫF )Θ(ǫ − ∆)ǫ

(ǫ2 − ∆2)1/2,

Spectral weight transferred from Fermi surface to interval [∆,∞]

Lecture Notes October 2005