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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
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