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
I. Eremin, MPIPKS
Introduction: Superconducting Materials
Element
Tc (K)
Nb
9.25
Pt
• common
feature of many elements
I. Eremin, MPIPKS
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
I. Eremin, MPIPKS
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!
I. Eremin, MPIPKS
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 I. Eremin, MPIPKS
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 I. Eremin, MPIPKS
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
I. Eremin, MPIPKS
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 I. Eremin, MPIPKS
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
I. Eremin, MPIPKS
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
I. Eremin, MPIPKS
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 I. Eremin, MPIPKS
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
I. Eremin, MPIPKS
(
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
I. Eremin, MPIPKS
!!!
BCS wave function (iii) Whole Fermi surface would be unstable to the creation of such pairs Pair creation operator
[Pˆ ,Pˆ ] ≠ 1 k
+ k
Pˆk+ = 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
I. Eremin, MPIPKS
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
I. Eremin, MPIPKS
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 I. Eremin, MPIPKS
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
I. Eremin, MPIPKS
)
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? I. Eremin, MPIPKS
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
I. Eremin, MPIPKS
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
I. Eremin, MPIPKS
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
I. Eremin, MPIPKS
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
I. Eremin, MPIPKS
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
I. Eremin, MPIPKS
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)
I. Eremin, MPIPKS
Paramagnetic limit of superconductivity Lack of inversion symmetry
1) Condensation energy vs polarization energy
Δ q = Δ 0 e iqr
I. Eremin, MPIPKS
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
I. Eremin, MPIPKS
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
I. Eremin, MPIPKS
Strong coupling effects: Eliashberg theory Assumption of BCS: λ
