Superconductivity and BCS Theory

Introduction Electron-phonon interaction, Cooper pairsBCS wave function, energy gap and quasiparticle statesPredictions of the BCS theoryLimits of the BCS gap equation: strong coupling effects

I. Eremin, Max-Planck Institut für Physik komplexer Systeme, Dresden, Germany

Institut für Mathematische/Theoretische Physik, TU-Braunschweig, Germany

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IntroductionIntroduction: : conductivity in a metalconductivity in a metal

• firstly observed in mercury (Hg) below 4K by Kamerlingh Onnes in 1911

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• Drude theory: metals are good electrical conductors because electrons can move nearly freely between the atoms in solids

mne τσ

2

= n - density of conduction electrons, m is an effective mass of conduction electrons, and τ - average lifetime for the free motion of electrons between collision with impurities, other electrons, etc

1111 −−

−−

−− ++= phelelelimp ττττ

2~ T 5~ T)(T

...520 +++= bTaTρρ

12

1 −− == τσρnem

⇒⇒

• common feature of many elements

IntroductionIntroduction: : Superconducting MaterialsSuperconducting Materials

23Nb3Ge

18Nb3Sn

4.2Hg

0.0019Pt

9.25Nb

Tc (K)Element

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IntroductionIntroduction: : Superconducting MaterialsSuperconducting Materials

5Na0.3CoO2-1.3H2O

0.65CeCu2Si2

0.5UPt3

38MgB2

17YNi2B2C

1.5Sr2RuO4

39La1.85Sr0.15CuO4

92YBa2Cu3O7

138HgBa2Ca2Cu3O8+δ

Tc (K)Material

{High-Tc layered cuprates (1986) Bednorz Müller

{Borocarbide superconductors

{discovered in 2001

{Heavy-fermion superconductors

Possible triplet superconductors {

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IntroductionIntroduction: : Basic Facts Basic Facts ⇒⇒ MeissnerMeissner--OchsenfeldOchsenfeld EffectEffect

What does it mean to have ρ =0 ?

jΕ ρ= 1) E = 0, j is finite inside all points of superconductors2) From the Maxwell equation: 0=

∂∂

−=×∇tBE

• Below Tc the magnetic field does not penetrate into superconductor(if we start from B=0 in the normal state)

• if above Tc there some magnetic field B ≠ 0, then below Tc it is expelled out of the system ⇒ Meissner effect

Superconductors are perfect diamagnets!

IntroductionIntroduction: : Basic Facts Basic Facts ⇒⇒ Type I and type II superconductivityType I and type II superconductivity

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What happens for the large magnetic field ?

• in type I superconductor the B field remains zero until suddenly the superconductivity is destroyed, Hc

• in type II superconductor there are two critical fields, Hc1 and Hc2

Magnetic field enters in the form of vortices(Hc1 < H < Hc2)

Abrikosov

Microscipic BCS Theory of SuperconductivityMicroscipic BCS Theory of Superconductivity

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First truly microscopic theory of superconductivity! (Bardeen-Cooper-Schrieffer 1957)

(i) The effective forces between electrons can sometimes be attractive in a solid rather than repulsive

(ii) „Cooper problem“ ⇒ two electrons outside of an occupied Fermi surface form a stable pair bound state, and this is true however weak the attractive force

(iii) Schrieffer constructed a many-particle wave functionwhich all the electrons near the Fermi surface are paired up

Three major insights:

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BCS theory: the electronBCS theory: the electron--electron interaction electron interaction (i)(i)

Bare electrons repel each other with electrostatic potential:

|'|4)'(

0

2

rrrr

−=−

πεeV

In a metal we are dealing with quasiparticles (electron with surroundingexchange-correlated hole)

TFreeV /'||

0

2

|'|4)'( rr

rrrr −−

−=−

πε

Thomas-Fermi screening rTF reduces substantially the repulsive force

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BCS theory: the electronBCS theory: the electron--phonon interaction (Frphonon interaction (Fröölich 1950) lich 1950) (i)(i)

Such a displacement means a creation of phonons ⇒ a set of quantum harmonic oscillators

due to vibrations the ion positions at Ri will be displaced by δRi

Electrons move in a solid in the field of positively charged ions

Modulation of the charge density and the effective potential V1(r) for the electrons

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BCS theory: the electronBCS theory: the electron--phonon interaction (Frphonon interaction (Fröölich 1950) lich 1950) (i)(i)

ii i

VV RR

rr δδ ∑ ∂∂

=)()( 1

1 with wavelength 2π/q

a scattering of 1 electron )()( ' rr qkk −Ψ⇒Ψ nn with emission of phonon

a scattering of 2 electron )()( ' rr qkk +Ψ⇒Ψ mm with absorption of phonon

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BCS theory: the electronBCS theory: the electron--phonon interaction (Frphonon interaction (Fröölich 1950) lich 1950) (i)(i)

Effective interaction of electrons due to exchange of phonons

22

2 1),(λ

λ ωωω

qqq

−= gVeff

gqλ is a constant of electron-phonon interaction ⇒Mm~ small number

ωqλ is a phonon frequency

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BCS theory: the electronBCS theory: the electron--phonon interaction (Frphonon interaction (Fröölich 1950) lich 1950) (i)(i)

Effective interaction of electrons due to exchange of phonons

• geff is an effective constant of electron-phonon interaction

22

2 1)(D

effeff gVωω

ω−

=

• ωD is a typical Debye phonon frequency

after averaging

For ω < ω D and low temperaturesDeffeff gV ωωω <−= ,)(

2

Attraction!

DFiωεε h<−k

ξ coherence length

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BCS theory: Cooper problem BCS theory: Cooper problem (ii)(ii)What is the effect of the attraction just for a single pair of electrons outside of Fermi sea:

Trial two-electron wave functionspini cmcme

21 ,212211 )(),,,( σσχϕσσ rrrr Rk −=Ψ

1) kcm =0, Cooper pair without center of mass motion2) Spin wave function

( )↑↓−↑↓=2

121,

spinσσχSpin singlet (S=0)

Spin triplet (S=1) ( )⎪⎪

⎩

⎪⎪

⎨

⎧

↓↓

↑↓+↑↓

↑↑

=2

121,

spinσσχ

),,,(),,,( 11222211 σσσσ rrrr Ψ−=Ψ

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BCS theory: Cooper problem BCS theory: Cooper problem (ii)(ii)

3) Orbital part of the wave function

Spin singlet )()( 1221 rrrr −+=− ϕϕ even function

Spin triplet )()( 1221 rrrr −−=− ϕϕ odd function

BCS: For the spin singlet state and s-wave symmetry a subsititon of the trial wave function in Schrödinger equation

)(,22/1

FeffD NgeE ελω λ ==− −h

),(|)(|)( 2121 φθϕ lmf Υ−=− rrrr

λ - electron-phonon coupling parameter

Bound state exists independent of the value of λ !!!

Ψ=Ψ EH

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BCS wave function BCS wave function (iii)(iii)

Whole Fermi surface would be unstable to the creation of such pairs

Pair creation operator +↓−

+↑

+ = kkk ccP̂

[ ] 1ˆ,ˆ ≠+kk PP [ ] 0ˆ,ˆ =++

kk PP ( ) 0ˆ 2=+

kP

Cooper pairs are not bosons!

( ) 0**∏ ++=Ψk

kkk PvuBCS

uk and vk are the normalizing coefficients (parameters)

1=ΨΨ BCSBCS122 =+ kk vu

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BCS wave function: variational apporach at T=0 BCS wave function: variational apporach at T=0 (iii)(iii)

BCSBCS HE ΨΨ= ˆminimize with constN =ˆ

∑∑ ↑↓−+

↓−+

↑+ −=

',''

2

,

ˆkk

kkkkk

kkk ccccgccH effσ

σσε

( ) ∑∑ −+−=',

*'

*'

222 1kk

kkkkk

kkk uuvvguvE effε

( )∑ +−=k

kk 122 uvN

122 =+ kk vu

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BCS wave function: variational approach at T=0 BCS wave function: variational approach at T=0 (iii)(iii)

Minimization with Lagrange multipliers μ and Ek

kkkk

kkkk

vEvN

vE

uEuN

uE

+∂∂

−∂∂

=

+∂∂

−∂∂

=

**

**

0

0

μ

μ ( )( ) kkkkk

kkkkk

vuvE

vuuE

με

με

−−Δ=

Δ+−=*

∑=Δk

kk*2

vugeff BCS gap parameter

( ) 22 Δ+−±=± μεkkE

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−=

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+=

k

kk

k

kk

Ev

Eu

με

με

121

121

2

2

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BCS energy gap and quasiparticle states BCS energy gap and quasiparticle states (iii)(iii)

∑ Δ=Δ

k kEgeff 2

2λω /12 −=Δ eDh

equals to the binding energy of a single Cooper pair at T=0!

What about the excited states and finite temperatures?Consider and small excitations relative to this state BCSΨ

↑↓−+

↓−+

↑↑↓−+

↓−+

↑↑↓−+

↓−+

↑ +≈ '''''' kkkkkkkkkkkk cccccccccccc

( ) ( )∑∑ +↓−

+↑↑↓−

+ Δ+Δ−−=k

kkkkk

kkk ccccccH *

,

ˆσ

σσμε

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BCS energy gap and quasiparticle states BCS energy gap and quasiparticle states

Unitary transformations: diagHUHU ˆˆ =+

⎟⎟⎠

⎞⎜⎜⎝

⎛

−= *

*

kk

kk

uvvu

U ( ) 22 Δ+−±=± μεkkE

new pair of operators:

+↓−↑

+↓−

+↓−↑↑

+=

−=

kkkkk

kkkkk

cucvb

cvcub **

( )∑ ↓−+

↓−↑+↑ +=

kkkkkk bbbbEHdiag

ˆ

What is the physical meaning of these operators?

u and v are Bogolyubov-Valatin coeffiecents:

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BCS quasiparticle statesBCS quasiparticle states1) b operators are fermionic:

{ } { } { } 0,,0,,, '''''''''' === +++σσσσσσσσ δδ kkkkkkkk bbbbbb

2) b „particles“ are not present in the ground state: 0=Ψ↑ BCSbk

3) The excited state corresponds to adding 1,2, ... of the new quasiparticles to this state

4) b –quasiparticle is superposition of an electron and a hole

+↓−↑↑ −= kkkkk cvcub **

5) the energy gap is 2Δ

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BCS gap equation at finite temperature: TBCS gap equation at finite temperature: Tcc

( ) ( )kkkkkk EfbbEfbb −== +↓−↓−↑

+↑ 1,

allows to determine temperature dependence of the gap

⎟⎟⎠

⎞⎜⎜⎝

⎛Δ=Δ ∑ Tk

EE

gB

eff 2tanh

22 k

k k

( ) cBTkT 52.302 ==Δ

( )λω /1exp13.1 −= DcBTk h

21

−∝ MTc

isotope effect !

∑ ↑↓−=Δk

kk ccgeff

2

[ ] ( ) 2

21

Δ−=−≈ ∑ Fcond NEE εεk

kkCondensation energy

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Some thermodynamic propertiesSome thermodynamic properties

1) Specific heat discontinuity at T=Tc

2nd order phase transiton ⇒ discontinuity of specific heat

43.1=−

=Δ

== cc TTn

n

TTn CCC

CC

2) Key quantity: density of states ( ) ( )∑ −=k

kEEN ωδ2

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Predictions of the BCS theoryPredictions of the BCS theory

1) Paramagnetic susceptibility of the conduction electrons HM χ=

( ) ( ) ↓↑ −→∝→→ nnN Fεωχ 0,0q

spin of Cooper Pairs S=0

( )[ ]2

1

1FN

TTε∝ effect of coherence

factors

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Predictions of the BCS theoryPredictions of the BCS theory

2) Andreev scattering: electron in a metal Δ<− Fεεk

a) an electron will be reflected at the interface

b) An electron will combine another electron and form Cooper pair

σσ −⇒−⇒

⇒−kk

ee

e and h are exactly time reversed

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Time inversion symmetry consequenciesTime inversion symmetry consequencies

1) Even if k is not a good qunatum number

( ) ( )rr *, ↓↑ ΨΨ ii

a) To reformulate the BCS theory in terms of field operators

works for alloys

Non-magnetic impurities do not influence s-wave superconductivity

Anderson (1959)

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Paramagnetic limit of superconductivityParamagnetic limit of superconductivity

Lack of inversion symmetry

qrq

ie0Δ=Δ

1) Condensation energy vs polarization energy

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Predictions of the BCS theory: Josephson effectPredictions of the BCS theory: Josephson effect

ϕ2iBCSBCS eΨ⇒Ψ

ϕiecc ↑↑ ⇒ kk1) Violation of U(1) gauge symmetry

2) Coherent tunneling of a Cooper-pairs

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Interplay of Coulomb repulsion and attraction: AndersonInterplay of Coulomb repulsion and attraction: Anderson--Morel modelMorel model

Coulomb repulsion is larger than attractive

⎥⎦

⎤⎢⎣

⎡−

−= *

1exp14.1μλ

ωDcBTk h

( )DW ωμμμ

h/ln1*

+= μλ <≠ evenTc 0

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Strong coupling effects: Eliashberg theoryStrong coupling effects: Eliashberg theory

Assumption of BCS: λ << 1

for λ =0.2÷0.5 the effect of electrons on phonons also has to be taken intoaccount

• Phonon frequencies renormalize due to electrons

• Self-consistent inclusion of screened Couolomb repulsion between the electrons ⇒ μ*

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+−+

=λμλ

λω62.01)1(04.1exp

45.1 *D

cBTk h

390.35MgB2

0.620.2Os0.90.33Mo7.20.49Pb0.90.45Zn

Tc (K)αCompound

α−∝ MTc

Further reading:Further reading:

1. J.-R. Schrieffer, Theory of Superconductivity (1964)2. M. Tinkham, Introduction to Superconductivity, 2nd Ed.

(1995)3. J.B. Ketterson and S.N. Song, Superconductivity (1999) 4. G. Rickayzen, Green‘s Functions and Condensed Matter

(1980).