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BCS TheoryHuge effect of a small gap:the BCS theory of
supercon-ductivity
GSI PhD Seminar, January 31, 2001
Thomas Neff
1
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Nobel Prizes
1913 Heike Kamerlingh-Onnes for his investigations on the
prop-erties of matter at low temperatures which led, inter alia to
the pro-duction of liquid helium
1962 Lev Davidovich Landau for his pioneering theories for
con-densed matter, especially liquid helium
1972 John Bardeen, Leon N. Cooper, J. RobertSchrieffer for their
jointly developed theoryof superconductivity, usually called the
BCS-theory
1973 Ivar Giaever for their experimental discoveries re-garding
tunneling phenomena in superconductors1973 Brian D. Josephson for
his theoretical predictions ofthe properties of a supercurrent
through a tunnel barrier, inparticular those phenomena which are
generally known asthe Josephson effects
1978 Pyotr Leonidovich Kapitza for his basic inventions and
dis-coveries in the area of low-temperature physics
1987 J. Georg Bednorz, K. Alexander Muller for their im-portant
breakthrough in the discovery of superconductivityin ceramic
materials
1996 David M. Lee, Douglas D. Osheroff, RobertC. Richardson for
their discovery of superfluid-ity in helium-3
2
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Superconductor Phenomenology
Infinite Conductivity Meissner effect
Crtitical Field Persistent currents and fluxquantization
Specific Heat Isotope effect
The transition temperature Tcvaries with the ionic mass M
Tc M1/2
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Phenomenological Theories
London Theory
two-fluid model: normal and superconducting electrons
London equation permits no magnetic field in the interior
j = nse2
mcB
allow calculation of penetration depth
Ginzburg-Landau Theory
superconductor is characterized by complex order parameter (r)
whichvanishes above Tc
Free Energy density of the superconductor
Fs = Fn + a||2 +12
b||4 +1
2m?
(~
i +
e?Ac)
2+
h2
8
allows description of superconductors with nonuniform fields and
surfaceeffects, flux quantization and gives description of Type
II
superconductors
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Microscopic BCS Theory
Wanted
A microscopic theory should describe thephenomenology based on
first principles (electron and
crystal structure of the metal, Hamiltonian of the system)
The physical nature of the order-parameter should beidentified,
the special non-classical features of the
superconducting phase should be outlined
tungsten fermi surface
Simplifications
Superconductivity is observed in many metals,therefore a
principle understanding should not
depend on the band-structure of the metal and thedetailed form
of electron-electron and
electron-lattice-interaction
The BCS theory will be presented for the T = 0case. A
temperature dependent description shouldmake use of thermal Greens
functions (see Fetter,
Walecka, Quantum Theory of Many-ParticleSystems)
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Fermions in a box
Fermi Gas / Fermi Liquid
Ground-state of fermi-liquid (in which the electrons are
described asindependent particles) has no correlations between
electrons of oppo-site spin and only statistical, by
pauli-principle, correlations for elec-trons of same spin.
Single particle excitations are described by moving an electron
withmomentum ki from the the fermi sphere to a momentum k j above
thefermi sphere.
Two Electrons above the fermi-sphere
The real electron-electron interaction is weak and slowly
varying overthe fermi surface.
The energy involved in the transition to the superconducting
state issmall
4 only excitations near the fermi-surface play a role
small attractive interaction between electrons at the fermi
surface
4 electrons try to minimize their energy, using the attractive
potential,have to pay with kinetic energy
4 widespread relative wavefunction of the order of 104cm , if
the in-teraction is also attractive at short distances the
electrons will form aspin-singulett
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Cooper Pairs
Solution of Bethe-Goldstone-equation is possible to solve
numerically.Using a simplified interaction, an analytical solution
is possible.
The Hamilton-Operator of the two electrons
H=
12m
k
21 +
12m
k
22 + V
Define binding-energy relative to fermi-energy
H
= E
= (EB + 2F)
Total momentum is conserved. only states above the fermi surface
canbe occupied. Therefore pair wavefunction has one electron
spin-up,one spin-down (spin-singulett)
=
k1>kF ,k2>kF
k1, k2
=
|12 P+k|>kF ,|
12 Pk|>kF
P, k
(1)
Lowest energy for P = 0, therefore use
k
=
P = 0, k
k
H
= (EB + 2F)
k
2k
k
+
k
k
V
k
k
= (EB + 2F)
k
This is the simplified interaction, attractive in small region
at the fermisurface
k
V
k
= V(k, k) = V(D |k F |)(D |k F |)
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Cooper Pairs, continued
Going from the momentum to the energy basis, using the density
ofstates N()
(EB + 2( F))
= V F+D
F
dN()
With the further assumptions D F and EB D we can assumeN()
N(F)
EB = 2D exp(
2
N(F)V
)
4 Theres a bound-state for an arbitrarily small potential. This
is a many-body effect.
4 We have an instability of the normal fermi-liquid against the
forma-tion of cooper-pairs. The task is now to formulate a
self-consistenttreatment of all electrons, the electrons are
indistinguishable !
4 At higher Temperature kT D the surface is fuzzy and the
instabilitygoes away.
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BCS many-particle state
We have to find a many-particle state, which has two-body
correla-tions in the form of cooper-pairs and respects the fermi
character ofthe electrons.
To get the correlations in the many-body case we use the
variationalprinciple
.
We could use in principle an ansatz of the form
BCS
=
(
k
g(k)a
ka
k
)(N/2)
0
8 This ansatz has technical problems. Therefore we use something
else.
BCS
= exp(
k
g(k)a
ka
k
)
0
8 This state has no definite particle number !
Using the anticommutation relations of the fermions we get
BCS
=
k
(
1 + g(k) a
ka
k
)
0
Usual one uses a slightly different notation
BCS
=
k
(uk + vk a
ka
k)
0
The normal ground state (filled fermi sphere) is given in this
ansatz by
|vk| = 1, uk = 0; k < Fvk = 0, |uk| = 1; k > F
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Reduced Interaction
We are using a reduced Hamiltonian, here only contributions
whichcontain two cooper-pairs are considered. it is assumed that
the missingparts of the Hamiltonian dont destroy the structure of
the solution -they should give more or less the same contributions
in the normal andin the superconducting state
H red =
k
k(a
kak + a
kak) +
k,kVk,k a
ka
kakak
We have no fixed particle number. Introduce chemical potential
which fixes the expectation value of the particle number
BCS
N
BCS
= N
For the Variation we need the matrix elements of the reduced
Hamilto-nian. After some tedious algebra we get
BCS
BCS
=
k
(
|uk|2 + |vk|2)
BCS
N
BCS
= 2
k
|vk|2
k,k
(
|uk |2 + |vk |2)
BCS
H red
BCS
=2
k
k|vk|2
k,k
(
|uk |2 + |vk |2)
+
k,kVk,ku?kv
?k ukvk
p,k,p,k
(
|up|2 + |vp|2)
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Gap Equation
To fulfill the normalization
|uk|2 + |vk|2 = 1
we make the ansatz
uk = sin k, vk = eik cos k
The variation procedure now tells us to minimize
BCS
H red N
BCS
The result is
k = , tan 2k = eikk
withk =
kVk,ku?kvk
We finally arrive at the gap equation
|k | = 12
kVk,k
|k |
(k )2 + |k|2
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Solution of the Gap Equation
8 The gap equation is for a general potential an integral
equation. Asimple solution is possible for the electron-electron
interaction as dis-cussed for the Cooper Pairs
Using this simple form of the interaction we get
k =
; |k | < D0 ; elsewhere
|| = 2D exp(
1
N(F)V
)
and change of the groundstate energy of
E= 12 N(F)||
2
Bogoliubov Transformation
To describe the excitations of the BCS ground-state the
BogoliubovTransformation to quasi-particle-operators
kand
k are used which
live on the BCS-vacuum and satisfy the usual anticommutation
rela-tions
k = ukak vka
kk = ukak + vka
k
excitation spectrum has a gap of 2
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Outlook
Applications
High-Tc superconductors Liquid 3He neutron stars
Order Parameter
Bose condensation Off-diagonal long-range order Meissner effect
and Flux Quantization Josephson effects
Order Parameter
Temperature Dependence Crystal structure, Fermi surface
Electron-electron interaction
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Title PageNobel PrizesSuperconductor
PhenomenologyPhenomenological TheoriesMicroscopic BCS
TheoryFermions in a boxCooper PairsCooper Pairs, continuedBCS
many-particle stateReduced InteractionGap EquationSolution of the
Gap EquationOutlook
