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

σ 2 ne τ n - density of conduction electrons, m is an effective mass of = conduction electrons, and τ - average lifetime for the free motion m of electrons between collision with impurities, other electrons, etc ρ σ τ m τ(T) ~ T 2 ~ T 5 = −1 = τ −1 τ ne2 ⇒⇒ −1 −1 −1 −1 = imp + el−el +τ el− ph

ρ 2 5 = ρ0 + aT + bT +...

• firstly observed in mercury (Hg) below4Kby Kamerlingh Onnes in 1911

I. Eremin, MPIPKS Introduction: Superconducting Materials

Element Tc (K)

Nb 9.25

Pt 0.0019 Hg 4.2

Nb3Sn 18

Nb3Ge 23

• common feature of many elements

I. Eremin, MPIPKS Introduction: Superconducting Materials

Material Tc (K)

HgBa2Ca2Cu3O8+δ 138 YBa Cu O 92 High-Tc layered cuprates 2 3 7 (1986) Bednorz Müller { La Sr CuO 39 1.85 0.15 4

Borocarbide superconductors YNi B C 17 { 2 2 MgB 38 discovered in 2001 { 2

CeCu2Si2 0.65 Heavy-fermion superconductors { UPt 0.5 3

Na0.3CoO2-1.3H2O 5 Possible triplet superconductors { Sr2RuO4 1.5

I. Eremin, MPIPKS Introduction: Basic Facts ⇒ Meissner-Ochsenfeld Effect

What does it mean to have ρ =0 ? 1) E = 0, j is finite inside all points of superconductors Ε = ρ j ∂B 2) From the Maxwell equation: ∇×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') = 4πε0 | r − r'|

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

e2 V (r − r') = e−|r−r'|/ rTF 4πε0 | 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 δBCS theory: the electron-phonon interaction (Frölich 1950) (i) ∂V (r) 1 with wavelength 2π/q V1(r) = ∑ δ Ri i ∂Ri

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 electronsω due to exchange of phonons λ 2 1 Veff (q, ) = gq ω 2 2 −ωqλ m g is a constant of electron-phonon interaction ~ ⇒ small number qλ M

ωqλ is a phonon frequency

I. Eremin, MPIPKS BCS theory: the electron-phonon interaction (Frölich 1950) (i) Effective interaction of electronsω due to exchange of phonons

2 1 Veff ( ) = geff ω 2 2 after averaging −ωD

• geff is an effective constant of electron-phonon Attraction! interaction ω • ωD is a typical Debye phonon frequency

For ω < ω and low temperatures 2 ω D V ( ) = − g , < ω effε eff D ε − < ω k i F h D

ξ coherence length

I. Eremin, MPIPKS BCS theory: Cooper problem (ii) What is the effect of the attraction just for a singleσ pair of electrons outside of Fermi sea: σ Trial two-electron wave functionϕ ik R spin Ψ(r , ,r , ) = e cm cm (r − r )χσ 1 1 2 2 1 2 1 ,σ 2

Ψ(r1,σ1,r2 ,σ 2 ) = −Ψ(r2 ,σ 2 ,r1,σ1)

1) kcm =0, Cooper pair without center of mass motion 2) Spin wave function

spin 1 χσ = (↑↓ − ↑↓ ) Spin singlet (S=0) 1 ,σ 2 2 ⎧ ↑↑ ⎪ spin ⎪ 1 Spin triplet (S=1) χσ = ()↑↓ + ↑↓ 1 ,σ 2 ⎨ ⎪ 2 ⎪ ⎩ ↓↓ I. Eremin, MPIPKS 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(r − r ) = f (| r − r |)Υ (θ,φ) 1ω 2 1 2 lm BCS: For the spin singlet state and s-wave symmetry a subsititon of the trial wave function in Schrödinger equation HΨ = EΨ λ −1/ λ 2 − E = 2h De , = geff 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 ˆ + + + Pk = ck↑c−k↓ [Pˆ ,Pˆ + ]≠ 1 [Pˆ + ,Pˆ + ]= 0 ˆ + 2 k k k k (Pk ) = 0

Cooper pairs are not bosons!

* * + ΨBCS = ∏(uk + vk Pk )0 k

uk and vk are the normalizing coefficients (parameters)

u 2 + v 2 =1 ΨBCS ΨBCS =1 k k

I. Eremin, MPIPKS BCS wave function: variational apporach at T=0 (iii) ˆ ˆ minimize E = ΨBCS H ΨBCS with N = const

ˆ + 2 + + H = ∑ε kck ck − geff ∑ck↑c−k↓c−k'↓ck'↑ k,σ σ k,k' σ

2 2 2 * * E = ∑ε k (vk − uk +1)− geff ∑vk vk'uk'uk k k,k'

2 2 N = ∑(vk − uk +1) k 2 2 uk + vk =1

I. Eremin, MPIPKS BCS wave function: variational approach at T=0 (iii)

Minimization with Lagrange multipliers μ and Ek ∂E ∂N 0 = μ− + E u * * k k E u = (ε − μ)u + Δ v ∂uk ∂uk k k k k k * ()ε ∂E ∂N Ekvk = Δ uk − k − μ vk 0 = * − μ * + Ekvk ∂vk ∂vk

2 * BCS gap parameter Δ = geff ∑ukvk k ε μ 2 1 ⎛ − ⎞ ⎜ k ⎟ uk = ⎜1+ ⎟ ε 2 2 E ± E = ± ()− μ 2 + Δ ⎝ k ⎠ k k ε 2 1 ⎛ − μ ⎞ ⎜ k ⎟ vk = ⎜1− ⎟ 2 ⎝ Ek ⎠

I. Eremin, MPIPKS BCS energy gap and quasiparticle states (iii) 2 Δ Δ = g −1/ λ eff ∑ Δ = 2hωDe k 2Ek

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

What about the excited states and finite temperatures?

Consider Ψ BCS 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'↑

ε ˆ ( ) + ( * + + ) H = ∑ k − μ ck ck − ∑ Δ c−k↓ck↑ + Δck↑c−k↓ k,σ k σ σ

I. Eremin, MPIPKS BCS energy gap and quasiparticle states + Unitary transformations: ˆ ˆ U HU = H diag

* ⎛ uk vk ⎞ ε 2 2 U = ⎜ ⎟ ± Ek = ± ()k − μ + Δ ⎜ * ⎟ − vk uk ⎝ ⎠ u and v are Bogolyubov- Valatin coeffiecents: new pair of operators: * * + bk↑ = uk ck↑ − vkc−k↓ + + b−k↓ = vk ck↑ + uk c−k↓

ˆ ( + + ) H diag = ∑ Ek bk↑bk↑ + b−k↓b−k↓ k

What is the physical meaning of these operators?

I. Eremin, MPIPKS σ σ

BCS quasiparticleσσ states σ 1) b operators are fermionic: σ

+ δ +σ + {bk ,bk' '}= kk'δ ' , {bk ,bk' '}= 0, {bk' ' ,bk'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↑ = ukck↑ − 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

2 2 Δ ⎛ E ⎞ Δ = geff c c k ∑ −k↓ k↑ Δ = geff tanh⎜ ⎟ k ∑ ⎜ ⎟ k 2Ek ⎝ 2kBT ⎠

2Δ()T = 0 = 3.52kBTc

kBTc =1.13hωD exp(−1/ λ)

1 − 2 isotope effect ! Tεc ∝ M

1 2 Condensation energy Econd ≈ ∑[]k − Ek = − N()ε F Δ k 2

I. Eremin, MPIPKS Some thermodynamic properties

1) Specific heat discontinuity at T=Tc 2nd order phase transiton ⇒ discontinuity of specific heat ΔC C − C = n =1.43 Cn Cn T =Tc T =Tc

2) Key quantity: density of states δ N(E)= 2∑ (ω − Ek ) k

I. Eremin, MPIPKS Predictions of the BCS theory

1) Paramagnetic susceptibility of the conduction electrons M = χH

χ(q → 0,ω → 0)∝ N(ε F )→ n↑ − n↓

spin of Cooper Pairs S=0

1 ∝ []N()ε 2 effect of coherence F factors T1T

I. Eremin, MPIPKS Predictions of the BCS theory

2) Andreev scattering: electron in a metal ε k −ε F < Δ

e and h are exactly time reversed

− e ⇒ e σk ⇒ − k ⇒ −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 s- wave superconductivity

Anderson (1959)

I. Eremin, MPIPKS Paramagnetic limit of superconductivity

Lack of inversion symmetry

1) Condensation energy vs polarization energy

iqr Δq = Δ0e

I. Eremin, MPIPKS Predictions of the BCS theory: Josephson effect

iϕ 1) Violation of U(1) gauge symmetry ck↑ ⇒ ck↑e

i2ϕ ΨBCS ⇒ ΨBCS e 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 ω ⎡ 1 ⎤ kBTc =1.14h D exp⎢− λ * ⎥ ⎣ − μ ⎦

* μ μ T ≠ 0 even λ < μ μ = c 1+ ln()W / hωD

I. Eremin, MPIPKS 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

• Self-consistent inclusion of • Phononω frequencies renormalize due to electrons screened Couolomb repulsion between the electrons ⇒ μ* λ λ h D ⎛ 1μ.04(1+ ) ⎞ k T = exp⎜ ⎟ Compound α Tc (K) B c ⎜ * ⎟ 1.45 ⎝ − ()1+ 0.62λ ⎠ Zn 0.45 0.9 Pb 0.49 7.2 Mo 0.33 0.9 −α Tc ∝ M Os 0.2 0.62

MgB2 0.35 39

I. Eremin, MPIPKS 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).