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Lecture XXI 61

Lecture XXI: Cooper instability

In the final section of the course, we will explore a pairing instability of the electron gas

which leads to condensate formation and the phenomenon of superconductivity.

History:

1911 discovery of superconductivity (Onnes) 1951 isotope effect clue to (conventional) mechanism 1956 Development of (correct) phenomenology (Ginzburg-Landau) 1957 BCS theory of conventional superconductivity (Bardeen-Cooper-Schrieffer) 1976 Discovery of unconventional superconductivity in heavy fermions (Steglich) 1986 Discovery of high temperature superconductivity in cuprates (Bednorz-Muller) ???? awaiting theory?

(Conventional) mechanism: exchange of phonons can induce (space-nonlocal)attractive pairwise interaction between electrons

H = H0 |M|2kkq

q

22q (k kq)2ckqc

k+qckck

Physically electrons can lower their energy by sharing lattice polarisation of another

By exploiting interaction, electron pairs can condense into macroscopic phase

coherent state with energy gap to quasi-particle excitations

To understand why, let us consider the argument marshalled by Cooper which lead to

the development of a consistent many-body theory.

Cooper instability

Consider two electrons propagating above a filled Fermi sea:Is a weak pairwise interaction V(r1 r2) sufficient to create a bound state?

Consider variational state

(r1, r2) =

spin singlet 1

2(| 1 | 2 | 2 | 1)

spatial symm. gk = gk |k|kF

gkeik(r1r2)

Applied to Schrodinger equation: H = Ek

gk [2k + V(r1 r2)] eik(r1r2) = Ek

gkeik(r1r2)

Lecture Notes October 2005

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Lecture XXI 62

Fourier transforming equation: Ld d(r1 r2)eik(r1r2)k

Vkkgk = (E 2k)gk, Vkk = 1Ld

dr V(r)ei(kk

)r

If we assume Vkk = VLd {|k F|, |k F|} < D

0 otherwise

VLd

k

gk = (E 2k)gk VLd

k

1

E 2kk

gk =k

gk VLd

k

1

E 2k = 1

Using1

Ld

k

=

ddk

(2)d=

() d (F)

d, where () =

1

|kk| is DoS

V

Ld k1

2k

E

(F)V F+D

F

d

2

E=

(F)V

2ln

2F + 2D E2

F E = 1

In limit of weak coupling, i.e. (F)V 1

E 2F 2De2

(F)V

i.e. pair forms a bound state (no matter how small interaction!) energy of bound state is non-perturbative in (F)V

Radius of pair wavefunction: g(r) = k gkeikr, gk = 12kE const., k =

kk

r2 =

ddr r2|g(r)|2ddr|g(r)|2 =

k |kgk|2k |gk|2

v2FF+DF

4d(2E)4F+D

F

d(2E)2

=4

3

v2F(2F E)2

if binding energy 2F E kBTc, Tc 10K, vF 108cm/s, 0 = r21/2 104A,i.e. other electrons must be important

BCS wavefunction

Two electrons in a paired state has wavefunction

(r1 r2) = (| 1 | 2 | 1 | 2)g(r1 r2)with zero centre of mass momentum

Drawing analogy with Bose condensate, let us examine variational state

(r1 r2N) = NNn=1

(r2n1 r2n)

Lecture Notes October 2005

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Lecture XXI 63

Is state compatible with Pauli principle? Using g(r1 r2) =

k gkeik(r1r2)

or, in Fourier representation

ddr1

Ldeik1r1

ddr2

Ldeik2r2g(r1 r2) =

kgkk1,kk2,k

or in second quantised form,

FT

g(r1 r2)c(r1)c(r2)|

=k

gk ckc

k|

Then, of the terms in the expansion of

| =Nn=1

kn

gknckn

ckn

|

those with all kns different survive

Generally, more convenient to work in grand canonical ensemblewhere one allows for (small) fluctuations in the total particle density

| =k

(uk + vkckc

k)|

cf. coherent state exp

k

gkckc

k

|

i.e. statistical independence of pair occupation

In non-interacting electron gas vk = 1 |k| < kF

0 |k| > kFIn interacting system, to determine the variational parameters vk,

one can use a variational principle, i.e. to minimise

|H FN|

BCS Hamiltonian

However, since we are interested in both the ground state energy, and the spectrum of

quasi-particle excitations, we will follow a different route and explore a simplified modelHamiltonian

H =k

kckck V

kk

ckc

kckck

Lecture Notes October 2005