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