Superconductivity and BCS Theory
I. Eremin,
Max-Planck Institut für Physik komplexer Systeme, Dresden, Germany
Institut für Mathematische/Theoretische Physik, TU-Braunschweig, Germany
Introduction
Electron-phonon interaction, Cooper pairs
BCS wave function, energy gap and quasiparticle states
Predictions of the BCS theory
Limits of the BCS gap equation: strong coupling effects
I. Eremin, MPIPKS
Introduction: conductivity in a metal
• Drude theory: metals are good electrical conductors because electrons can move nearly
freely between the atoms in solids
ne 2τ
σ=
m
ρ =σ
−1
m −1
= 2τ
ne
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
⇒
(T ) ~ T 2 ~ T 5
−1
τ −1 = τ imp
+ τ el−1−el + τ el−1− ph
ρ = ρ 0 + aT 2 + bT 5 + ...
• firstly observed in mercury (Hg) below 4K by
Kamerlingh Onnes in 1911
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Introduction: Superconducting Materials
Element
Tc (K)
Nb
9.25
Pt
• common
feature of many elements
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0.0019
Hg
4.2
Nb3Sn
18
Nb3Ge
23
Introduction: Superconducting Materials
{
High-Tc layered cuprates
(1986) Bednorz Müller
Borocarbide superconductors
discovered in 2001
Heavy-fermion superconductors
Possible triplet superconductors
{
{
{
{
Material
Tc (K)
HgBa2Ca2Cu3O8+δ
138
YBa2Cu3O7
92
La1.85Sr0.15CuO4
39
YNi2B2C
17
MgB2
38
CeCu2Si2
0.65
UPt3
0.5
Na0.3CoO2-1.3H2O
5
Sr2RuO4
1.5
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Introduction: Basic Facts ⇒ Meissner-Ochsenfeld Effect
What does it mean to have ρ =0 ?
Ε=ρ j
•
•
1) E = 0, j is finite inside all points of superconductors
2) From the Maxwell equation:
∂B
∇×E = −
=0
∂t
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!
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Introduction: Basic Facts ⇒ Type I and type II superconductivity
What happens for the large magnetic field ?
Magnetic field enters
in the form of vortices
(Hc1 < H < Hc2)
Abrikosov
•
•
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
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Microscipic BCS Theory of Superconductivity
First truly microscopic theory of superconductivity!
(Bardeen-Cooper-Schrieffer 1957)
Three major insights:
(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 function
which all the electrons near the Fermi surface are paired
up
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BCS theory: the electron-electron interaction (i)
Bare electrons repel each other with electrostatic potential:
e2
V (r − r ' ) =
4πε 0 | r − r ' |
In a metal we are dealing with quasiparticles (electron with surrounding
exchange-correlated hole)
e2
V (r − r ' ) =
e −|r −r '|/ rTF
4πε 0 | r − r ' |
Thomas-Fermi screening rTF reduces substantially the repulsive force
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BCS theory: the electron-phonon interaction (Frölich 1950) (i)
Electrons move in a solid in the field of positively charged ions
due to vibrations the ion positions at Ri will be displaced by δRi
Such a displacement means a creation of phonons ⇒ a set of quantum harmonic oscillators
Modulation of the charge density and the effective potential V1(r) for the electrons
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BCS theory: the electron-phonon interaction (Frölich 1950) (i)
∂V1 (r )
δ V1 (r ) = ∑
δ Ri
∂R i
i
with wavelength 2π/q
a scattering of 1 electron
Ψnk (r ) ⇒ Ψn 'k −q (r ) with emission of phonon
a scattering of 2 electron
Ψmk (r ) ⇒ Ψm 'k +q (r ) with absorption of phonon
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BCS theory: the electron-phonon interaction (Frölich 1950) (i)
Effective interaction of electrons due to exchange of phonons
Veff (q, ω ) = g qλ
2
1
ω 2 − ωq2λ
gqλ is a constant of electron-phonon interaction
ωqλ is a phonon frequency
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m
~
⇒
M
small number
BCS theory: the electron-phonon interaction (Frölich 1950) (i)
Effective interaction of electrons due to exchange of phonons
Veff (ω ) = g eff
2
1
ω 2 − ω D2
after averaging
• geff is an effective constant of electron-phonon
interaction
Attraction!
• ωD is a typical Debye phonon frequency
For ω
< ω D and low temperatures
2
Veff (ω ) = − g eff ,
ω < ωD
ε k − ε F < hω D
i
ξ
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coherence length
BCS theory: Cooper problem (ii)
What is the effect of the attraction just for a single pair of electrons outside of Fermi sea:
Trial two-electron wave function
Ψ (r1 , σ 1 , r2 , σ 2 ) = e ik cm R cm ϕ (r1 − r2 ) χσspin
1 ,σ 2
Ψ (r1 , σ 1 , r2 , σ 2 ) = − Ψ (r2 , σ 2 , r1 , σ 1 )
1) kcm =0, Cooper pair without center of mass motion
2) Spin wave function
Spin singlet (S=0)
Spin triplet (S=1)
χσspin,σ =
1
χσspin,σ
1
2
2
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(
1
↑↓ − ↑↓
2
)
⎧ ↑↑
⎪
⎪ 1
=⎨
↑↓ + ↑↓
⎪ 2
⎪ ↓↓
⎩
(
)
BCS theory: Cooper problem (ii)
3) Orbital part of the wave function
Spin singlet
ϕ (r1 − r2 ) = +ϕ (r2 − r1 ) even function
Spin triplet
ϕ (r1 − r2 ) = −ϕ (r2 − r1 ) odd function
ϕ (r1 − r2 ) = f (| r1 − r2 |)Υlm (θ , φ )
BCS: For the spin singlet state and s-wave symmetry a subsititon of the trial wave function
in Schrödinger equation HΨ = EΨ
− E = 2hω D e
−1 / λ
2
, λ = g eff N (ε F )
λ - electron-phonon coupling parameter
Bound state exists independent of the value of λ
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!!!
BCS wave function (iii)
Whole Fermi surface would be unstable to the creation of such pairs
Pair creation operator
[Pˆ ,Pˆ ] ≠ 1
k
+
k
P̂k+ = ck+↑ c−+k ↓
[Pˆ ,Pˆ ] = 0
+
k
( )
+
k
Pˆk+
2
=0
Cooper pairs are not bosons!
(
)
ΨBCS = ∏ uk* + vk* Pk+ 0
k
uk and vk are the normalizing coefficients (parameters)
uk + vk = 1
2
ΨBCS ΨBCS = 1
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2
BCS wave function: variational apporach at T=0 (iii)
E = ΨBCS Hˆ ΨBCS
minimize
Hˆ = ∑ ε k ck+σ ckσ − g eff
2
k ,σ
(
k ,k '
)
2
2
k
(
*
*
v
v
u
u
∑ k k' k' k
k ,k '
)
N = ∑ vk − uk + 1
2
2
k
uk + vk = 1
2
Nˆ = const
+
+
c
c
∑ k ↑ −k ↓c−k '↓ck '↑
E = ∑ ε k vk − uk + 1 − g eff
2
with
2
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BCS wave function: variational approach at T=0 (iii)
Minimization with Lagrange multipliers μ and Ek
∂E
∂N
0 = * − μ * + Ek uk
∂uk
∂uk
Ek uk = (ε k − μ )uk + Δ vk
Ek vk = Δ*uk − (ε k − μ )vk
∂E
∂N
0 = * − μ * + Ek vk
∂vk
∂vk
Δ = g eff
2
*
u
v
∑ kk
BCS gap parameter
k
± Ek = ± (ε k − μ ) + Δ
2
uk
2
2
vk
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2
1 ⎛ εk − μ ⎞
⎟⎟
= ⎜⎜1 +
2⎝
Ek ⎠
1 ⎛ εk − μ ⎞
⎟⎟
= ⎜⎜1 −
2⎝
Ek ⎠
BCS energy gap and quasiparticle states (iii)
Δ = g eff
2
Δ
∑k 2 E
k
Δ = 2hω D e −1/ λ
equals to the binding energy of a single Cooper pair at T=0!
What about the excited states and finite temperatures?
Consider
ΨBCS
and small excitations relative to this state
ck+↑ c−+k ↓ c−k '↓ ck '↑ ≈ ck+↑ c−+k ↓ c−k '↓ ck '↑ + ck+↑ c−+k ↓ c−k '↓ ck '↑
(
Hˆ = ∑ (ε k − μ )ck+σ ckσ − ∑ Δ*c−k ↓ ck ↑ + Δck+↑ c−+k ↓
k ,σ
k
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)
BCS energy gap and quasiparticle states
Unitary transformations:
⎛ uk
U = ⎜⎜
⎝ − vk
U + Hˆ U = Hˆ diag
v ⎞
⎟
u ⎟⎠
*
k
*
k
new pair of operators:
± Ek = ± (ε k − μ ) + Δ
2
2
u and v are BogolyubovValatin coeffiecents:
bk ↑ = uk* ck ↑ − vk* c−+k ↓
b−+k ↓ = vk ck ↑ + uk c−+k ↓
(
Hˆ diag = ∑ Ek bk+↑bk ↑ + b−+k ↓b−k ↓
)
k
What is the physical meaning of these operators?
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BCS quasiparticle states
1) b operators are fermionic:
{b
}
{
}
+
+
+
{
}
,
b
=
δ
δ
,
b
,
b
=
0
,
b
,
b
kσ
k 'σ '
kk ' σσ '
kσ
k 'σ '
k 'σ ' k 'σ ' = 0
2) b „particles“ are not present in the ground state:
bk ↑ ΨBCS = 0
3) The excited state corresponds to adding 1,2, ... of the new quasiparticles to this state
bk ↑ = uk* ck ↑ − vk* c−+k ↓
4) b –quasiparticle is superposition of
an electron and a hole
5) the energy gap is 2Δ
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BCS gap equation at finite temperature: Tc
bk+↑bk ↑ = f (Ek ), b−k ↓b−+k ↓ = 1 − f (Ek )
allows to determine temperature dependence of the gap
Δ = g eff
2
∑
k
c−k ↓ ck ↑
Δ = g eff
2
⎛ Ek ⎞
Δ
∑k 2 E tanh⎜⎜ 2k T ⎟⎟
k
⎝ B ⎠
2Δ(T = 0 ) = 3.52k BTc
k BTc = 1.13hω D exp(− 1 / λ )
Tc ∝ M
Condensation energy
Econd
−
1
2
isotope effect !
1
2
≈ ∑ [ε k − Ek ] = − N (ε F ) Δ
2
k
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Some thermodynamic properties
1) Specific heat discontinuity at T=Tc
2nd order phase transiton ⇒ discontinuity of
specific heat
C − Cn
ΔC
=
Cn T =T
Cn
c
= 1.43
T =Tc
2) Key quantity: density of states
N (E ) = 2∑ δ (ω − Ek )
k
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Predictions of the BCS theory
M = χH
1) Paramagnetic susceptibility of the conduction electrons
χ (q → 0, ω → 0) ∝ N (ε F ) → n↑ − n↓
spin of Cooper Pairs S=0
1
2
∝ [N (ε F )]
T1T
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effect of coherence
factors
Predictions of the BCS theory
2) Andreev scattering: electron in a metal
εk − ε F < Δ
e and h are exactly time reversed
−e⇒ e
k ⇒ −k
σ ⇒ −σ
a) an electron will be reflected at the interface
b) An electron will combine another electron and form Cooper pair
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Time inversion symmetry consequencies
1) Even if k is not a good qunatum number
a) To reformulate the BCS theory in terms of
field operators
Ψi↑ (r ), Ψi*↓ (r )
works for alloys
Non-magnetic impurities do not influence swave superconductivity
Anderson (1959)
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Paramagnetic limit of superconductivity
Lack of inversion symmetry
1) Condensation energy vs polarization energy
Δ q = Δ 0 e iqr
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Predictions of the BCS theory: Josephson effect
1) Violation of U(1) gauge symmetry
ck ↑ ⇒ ck ↑ eiϕ
ΨBCS ⇒ ΨBCS e i 2ϕ
2) Coherent tunneling of a Cooper-pairs
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Interplay of Coulomb repulsion and attraction: Anderson-Morel model
Coulomb repulsion is larger than attractive
⎡
1 ⎤
k BTc = 1.14hω D exp ⎢−
*⎥
−
λ
μ
⎣
⎦
μ =
*
μ
1 + μ ln (W / hω D )
Tc ≠ 0 even λ < μ
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Strong 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 into
account
•
Phonon frequencies
renormalize due to electrons
•
Self-consistent inclusion of
screened Couolomb repulsion
between the electrons ⇒ μ*
⎛ 1.04(1 + λ ) ⎞
hω D
⎟⎟
k BTc =
exp⎜⎜
*
1.45
⎝ λ − μ (1 + 0.62λ ) ⎠
Tc ∝ M
−α
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Compound
α
Tc (K)
Zn
0.45
0.9
Pb
0.49
7.2
Mo
0.33
0.9
Os
0.2
0.62
MgB2
0.35
39
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).