11

Last lecture:Last lecture:

We found that the low-lying elementary excitations of a weakly interacting BEC are density fluctuations with a linear energy spectrum Ek = hck at low k. This leads to a finite Landau critical velocity and superfluidity. We also made contact between microscopic theory and the phenomenological Ginzburg-Landau model. The order parameter is a coherent state (similar to thatfor a laser) involving a superposition of many-particle wavefunctions with different numbers of particles. When g = 0 the order parameter reduces to the single-particle wavefunctionthat is macroscopically populated.

We now turn to the BCS theory of superconductivity. Some of the mathematical techniques will look similar to that of the last lecture, but there are very important technical and conceptual differences.

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Lecture 10: The BCS TheoryLecture 10: The BCS Theory

BEC-BCS Crossover

Cooper Pairs

The BCS Wavefunction

Mean-Field Approximation and Bogoliubov Transformation

The Energy Gap Equation

The Bogoliubov Quasiparticles

Experimental Support for the BCS Model

Order Parameter and the Ginzburg-Landau Coherence Length

Literature: Annett ch. 6, Waldram ch. 7 & Schmidt ch. 6

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Consider a BEC where each boson is a molecule of two tightly bound fermions, and then slowly reduce the attractive interaction between the two fermions. What we get is a gradual crossover from the BEC to the BCS state.

BECBEC--BCS CrossoverBCS Crossover

Superfluidity (or superconductivity if the particles are charged) can survive this crossover, but the microscopic nature of the BCS state is very different from that of the BEC state.

In the BEC limit the binding of the fermions occurs at tempera-tures above the BEC condensation temperature. In the BCS limit the opposite is the case. Also, the fermion pairs, known as Cooper pairs, are not true bosons in that the creation and annihilation operators for Cooper pairs do not obey the Bose commutation rules.

Picture credits: M. Greiner

(real space) (wavevector space)

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The Cooper-pair wavefunction extends over distances much larger than the inter-particle spacing, so that our usual notion of molecules undergoing a Bose-Einstein condensation is an oversimplification. We need a new approach starting with fermions occupying states inside the Fermi surface. A full treatment of this problem, particularly that of superconductivity in real materials, is extremely difficult.

Surprisingly, important insights can be gained by considering a very simple model: a homogeneous gas of fermions with a weak attractive interaction g = -|g| for low-energy transitions, |δε| < εc , around the Fermi surface at low temperatures. Sums over k will thus be effectively restricted in this lecture to a shell in k-space kF-kc<k<kF+kc where kF is the Fermi wavevector and kc = εc/(hvF).

We show that the Fermi liquid state is unstable to a formation of a coherent state of Cooper pairs. This is the BCS state that is characterized by an energy gap in the spectrum of elementary excitations. The origin of the attractive interaction will be discussed in the next lecture.

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We know from elementary quantum mechanics that two particles in 3D cannot form a bound state unless the attractive potential is strong enough. However, Cooper pairs can bind for any finite |g| if g is negative. It turns out that the bound state is stabilized by the presence of the filled Fermi sea (examples).

Since only states near the Fermi surface will be affected by g it is useful to consider a wavevector representation of the Cooper state. The transitions which are produced by g are as shown in the figure for the case where the total momentum of the Cooper state is zero. g produces repeated scattering from a state (k,-k) to a state (k’,-k’) in the shaded shell near the Fermi surface. Since the scattering is s-wave the spin state must be a singlet.

Cooper PairsCooper Pairs

FilledFermi sea–k

k–k’

k’-k↓

k↑

-k’↓

k’↑

Region of Attraction

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Cooper’s finding that a bound state of pairs lowers the energy of the Fermi system suggests that the true ground will involve somecoherent state of Cooper pairs. Of special interest now is the pair creation operator

which creates a pair of fermions at (k + q,-k) with total momentum hq and total spin zero. This is the analogue of the operator , which we introduced in the last lecture. In this lecture a+ and a satisfy the commutation rules for fermions.

BCS conjectured that the ground state should be a coherent statebuilt up from q=0 pair creation operators

This can be normalized by replacing γk by vk and the first term by uk, where

The BCS The BCS WavefunctionWavefunction

+↓−

+↑+ kqkaa

kqk aa++

)0)(( ...1]exp[ 2 =++= ++↓−

+↑

+↓−

+↑

aaaaakkkkkk γγ

22 ||1

||1

1

k

kk

kk vu

γ

γ

γ +=

+=

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Including all k we arrive at the BCS wave-function

∏ =+>+=> +↓−

+↑

kkkkkkkBCS vuvacaavu 1||||,|)(| 22Ψ

This is a trial wavefunction defined by the variational parameter uk and vk. These are to be determined by minimizing the energy

It is sufficient to work with the reduced Hamiltonian

Instead of minimizing EBCS, we apply a mean-field approximation and the Bogoliubov transformation similar to that used in the last lecture.

-k↓

k↑

-k’↓

k’↑

><= BCSBCSBCS HE ΨΨ |ˆ|

∑ ∑ ↑↓−+

↓−+

↑+ −=

k kkkkkkkkk aaaa

Vg

aaH',

''

||ˆ ε

g

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The mean-field approximation in this context replaces an operatorby . This assumes that

can be ignored, where and similarly for .*

Applying this approximation to Ĥ yields, to a constant term,

where ∆ is in general a complex quantity defined by

Ĥ can be diagonalized by a Bogoliubov transformation from bare fermion operators to new fermion operators .

*More generally use Wick’s theorem which gives both Hartree and Fock terms when averaging

products of creation and annihilation operators in (see also footnote in lecture 9).

MeanMean--Field Approximation andField Approximation andBogoliubovBogoliubov TransformationTransformation

BAˆˆ ><><−><+>< BABABA ˆˆˆˆˆˆ >< BA ˆˆδδ><−= AAA ˆˆˆδ B̂δ

∑ ∑ ↑↓−+

↓−+

↑+ +−=

σσσ ∆∆ε

k kkkkkkkk aaaaaaH )(ˆ *

∑ ><−= ↑↓−k

kk Vaag /||∆

),( +σσ kk aa ),( +

σσ αα kk

V̂

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The required linear transformation can be expressed in the normalized form

As in the last lecture, we can show by substitution that, to a constant term, Ĥ has the diagonal form

if sk and Ek satisfy

where µ is the chemical potential. |∆| is the BCS energy gap.

2

2

||1/][

||1/][

kkkkk

kkkkk

ssa

ssa

++=

++=

+↑−↓↓

+↓−↑↑

αα

αα

µε

∆ε

∆ε

αασ

σσ

−=

+=

−=

= ∑ +

mk

E

Es

EH

k

kk

kkk

kkkk

2

||

||/)(||

ˆ

22

222

h

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The Energy Gap EquationThe Energy Gap Equation

The energy gap is obtained by replacing in the expression for ∆the original fermion operators by the new fermion operators and setting <αk

+αk> equal to the Fermi function for excitations

This yields the self-consistent equation

A simple numerical analysis gives |∆(T)| as shown in the figure (next page), where

N(εF) is the single-particle density of states per spin per volume at the Fermi level εF. ∆ is taken to be real in the following except where otherwise indicated.

)1/(exp1)( +

=

TkE

EfB

kk

∑−

−=k k

kVE

Efg

2

))(21(|||||| ∆∆

||)(

1exp13.1 1.76 |0)(|

−===

gNTkTkT

FccBcB ε

ε∆

(The relevant chemical potential for excitations is zero as in the case of photons and phonons)

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From f(Ek), one can also calculate the entropy and thus the heat capacity. The numerical result is shown in the figure above. The ratios ∆/(kBTc) and ∆Ce/Ce,n as well as other properties are in striking agreement with experiment in conventional supercon-ductors, namely, s-wave spin-singlet superconductors in the weak coupling limit.

T

∆

Tc

∆ = 1.76 kBTc

T

Ce/T

Tc

∆C = 1.43 Ce,n

at Tc

C ∼∼∼∼ e-∆/kBT

at low T

Verified by experiment!

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The The BogoliubovBogoliubov QuasiparticlesQuasiparticles

The new operator creates a fermion with wavevector and spin (k,σ). Well above the Fermi surface reduces to and so adds a bare fermion with (k,σ). Well below the Fermi surface

reduces to a-k-σ and so removes a bare fermion with (-k,-σ). This is as expected.

At intermediate energies, however, is quite strange. It creates a fermion with well-defined wavevector and spin, but indeterminate particle number. In particular, the new fermion has zero charge at µ even if the bare fermions carry a charge. This is a remarkable example of a fermionic excitation in which spin and charge are separated. Recall that the corresponding boson operator in the last lecture was interpreted as the creation operator for a density fluctuation in a BEC.

+σαk

+σαk

+σka

+σαk

+σαk

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εk/∆

|uk|2|vk|

2

hkvF/∆

E/∆Ek

εk (electrons)

−εk(holes)

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Isotope effect: for a given N(εF)|g|, Tc is expected to scale as εc, which is proportional to the Debye frequency ωD for the electron-phonon interaction (next lecture). ωD can be varied by changing the isotopic mass. Lighter isotopes have higher ωD and hence have higher Tc, assuming that other factors are equal (figure left).

Experimental Support for the BCS ModelExperimental Support for the BCS Model

Picture credits: Hyperphysics@GSU/ I. Giaever

Tunneling conductivity provides a direct measure of the density of states of elementary excitations and thus the BCS energy gap (figures below and right).

k

E E

g(E)

eV

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The energy gap ∆ can also be determined from the heat capacity and thermal conductivity, which contain contributions from the Bogoliubov quasiparticles at low T < Tc as well as from phonons.

Tc/T ∼∼∼∼ ∆/kBT

Ce(log-scale!)

Picture credits: C. A. Bryant / E. Schuberth

The sharp change in the thermal conductivity of a pure type-I supercon-ductor on crossing the critical field Bc can be used to produce high-performance heat switches used in laboratory- and space-based applications.

(Tc (Sn) = 3.7 K)

phonons

electrons

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Order Parameter andOrder Parameter andthe the GinzburgGinzburg--Landau Coherence LengthLandau Coherence Length

The Ginzburg-Landau model was derived from the BCS model by Gorkov. The derivation is more subtle than for a BEC. In the uniform state the order parameter turns out to be proportional to ∆(T) or equivalently from the definition of ∆,

Note that in general ∆ = |∆|eiθ, i.e., ∆ has an amplitude and a phase as expected for the Ginzburg-Landau order parameter.

The superconducting coherence length ξ may be interpreted as the characteristic size of the Cooper-pair state. This may be estimated via the uncertainty principle δkδx ~ 1. If ∆ ~ hvFδk and δx ~ ξ, we find ξ ~ hvF/∆. The BCS model gives a similar result

ξ ≅ hvF/(π∆) ≅ 0.18 hvF/(kBTc)

space real in )()(or space- in ><>< ↑↓↑↓−∑ rrkaak

kk ψψ