50 years of BCS School Lecture 1: Basics of pairing - LSU · 50 years of BCS School Basics of pairing Ilya Vekhter Louisiana State University, USA July 2007 Cargese. ... • London

50 years of BCS School

Basics of pairing

Ilya Vekhter

Louisiana State University, USA

July 2007 Cargese

July 2007 Cargese

Timescale

H. Kamerlingh-Onnes1911, Nobel Prize 1913

G. Holst“carefully carried out

all the measurements”

J. Bardeen, L. Cooper and R. Schrieffer1957, Nobel Prize 1972

Why so long?

July 2007 Cargese

Solids: 1900 -- 1925• 1897 J.J. Thomson: discovery of an electron• 1900 P. Drude: transport theory

– classical electrons scatter on atoms. Wiedemann-Franz ratio.

mne τσ

2

=

2

2v

3v

nemc

T=

σκ

3v v

2 τκ c=

July 2007 Cargese


– classical electrons scatter on atoms. Wiedemann-Franz ratio.

mne τσ

2

=

Tkm B3v2 =

2/3v Bnkc =

2

2v

3v

nemc

T=

σκ Classical gas

3v v

2 τκ c=

July 2007 Cargese


classical electrons scatter on atoms. Wiedemann-Franz.

mne τσ

2

=28

2

2

2v /1011.1

23

3v KW

ek

nemc

TB Ω×≈⎟⎠⎞

⎜⎝⎛== −

σκ

3v v

2 τκ c=

•1904 H. Lorentz: Maxwell-Boltzmann to treat electron-atom collisions. Wrong temperature dependence, wrong WF•No understanding of metal physics•1911: superconductivity… what is it about?

July 2007 Cargese

Missing specific heat1912: P. Debye: specific heat of lattice vibrations (phonons),

http://hyperphysics.phy-astr.gsu.edu/hbase/solids/phonon.html

No trace of classical electronic specific heat

July 2007 Cargese

Metals: 1925 -- 1930

• W. Pauli exclusion principle, beginning of solid state theory

W. Pauli

Then he said, "Who are you?" "I am Weisskopf, you asked me to be your assistant." "Yes," he said, "first I wanted to take Bethe, but he works on solid state theory,which I don't like, although I started it."

V. Weisskopf

July 2007 Cargese

Metals: 1925 -- 1930• W. Pauli: exclusion principle• Fermi-Dirac statistics

filled states

empty states• Fermi energy EF ~ 10000K

EF

July 2007 Cargese

Metals: 1925 -- 1930• W. Pauli: exclusion principle• Fermi-Dirac statistics

filled states

empty states• Fermi energy EF ~ 10000K

• Absorbed energy ∆E~n(kBT)2/EF~(kBT)2N0

• Specific heat CV ~ nkB (kBT/EF)<< nkB

kBT

July 2007 Cargese

Metals: 1925 -- 1930• W. Pauli: exclusion principle• Fermi-Dirac statistics:• Sommerfeld theory of conductivity

CV ~kB2N0kT<< nkB

Tkm B3v2 =

2/3v Bnkc =

2

2v

3v

nemc

T=

σκ

July 2007 Cargese


CV << nkB

Tkm B3v2 =

2/3v Bnkc =

2

2v

3v

nemc

T=

σκ

)/(v FBB ETknkc ≈

)/2(v2F TkETkm BFB=

July 2007 Cargese


CV << nkB

2822

/1044.23

KWe

kT

B Ω×≈⎟⎠⎞

⎜⎝⎛= −π

σκ

measurable at low T

July 2007 Cargese


CV << nkB

2822

/1044.23

KWe

kT

B Ω×≈⎟⎠⎞

⎜⎝⎛= −π

σκ

• Problem: Mean free path too long

• Bloch’s theorem for a crystal

al F >>= τv

)()( rr kkr

k uei=ψ

• Infinite mean free path in a periodic lattice

measurable at low T

MM

Me a

July 2007 Cargese

Metals: summary

2

22

maae h

≈=ε2

2

meaB

h=MM

Baa 54 −≈

Mm

D εω ≈h2

222

221

maaM D

h≈⎟

⎠⎞

⎜⎝⎛ωDebye frequency:

cmaF

210v −≈≈h

aNaN

LnkF

122 max =≈≈ππ

Electron momenta:

300≈=mMβK

maEF

42

2

10≈≈h

Energy scales KD 300≈ωh

DFc EKT ω,10 <<≤ Very small scale

July 2007 Cargese

Bloch, Pauli, and superconductivity

F. Bloch 1980

July 2007 Cargese

Second Bloch’s theorem

July 2007 Cargese

Second Bloch’s theorem

Flaw: thinking of a superconductor as

a state of a metal with finite current

July 2007 Cargese

Superconductivity: 1930-1937

J. C. McLennan 1934

Sn

• 1932 Keesom, Kok: specific heat anomaly in tin

July 2007 Cargese

Superconductivity: 1930-1937• 1932 Keesom, Kok: specific heat anomaly in tin

Superconductivity is a stable thermodynamic

phase different from the usual metallic state

J. C. McLennan 1934

Sn He

superfluid

July 2007 Cargese


• 1933 Meissner-Ochsenfeld: Meissner effect

Superconductors are not just perfect conductors

July 2007 Cargese

Superconductivity: 1930-1937• Gorter: B=0 inside a superconductor • London equations: zero resistivity and Meissner effect

Ev edtdm =

tc

∂∂

−=×∇ − HE 1vj ens=

Hj 12

−−=×∇ cen

m

sEj =⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

2enm

t s

• Penetration depth

jHcπ4

=×∇

022 =−∇ − HH Lλ H0

)/exp(0 LxHB λ−=

2

22

4 enmc

sL πλ =

July 2007 Cargese

Rigidity of wave function and gap

Ej =⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

2enm

t s tc ∂∂−=

×∇=− /1 AE

AHHj 12

−−=×∇ cen

m

s

00=⋅∇=⋅∇

Aj

Ajmc

ens2

−=gauge invariance?but

F.London 1937

July 2007 Cargese

Rigidity of wave function and gap

Ej =⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

2enm

t s tc ∂∂−=

×∇=− /1 AE

AHHj 1

2−−=×∇ c

enm

s

00=⋅∇=⋅∇

Aj

Ajmc

ens2

−=gauge invariance?but

•Rigidity: wave function does not respond to A (phase coherence over large distances)

[ ]∗Ψ∇Ψ−Ψ∇Ψ−∝ *

2me

phj Aj 2

2

Ψ−∝mce

d

•Energy gap between continuum and superconducting state

July 2007 Cargese


• Pippard: there is another scale besides λL

• Non-local electrodynamics: current averaged over ξ0

• Ginzburg-Landau: effective theory for order parameter

• Discussed in other lectures

July 2007 Cargese

Isotope effect: 1950

Reynolds, Serin, Nesbitt 1950Maxwell 1950

July 2007 Cargese



H. Frohlich 1980

July 2007 Cargese



Ionic motion is important for superconductivity

July 2007 Cargese

Phonon attraction: formalismAmplitudes of processes:

1p 2p

kp h−1 kp h+2

kh

kh−

)()()(||

11

2

kkpEpEV

ωhh −−−+

)()()(||

22

2

kkpEpEV

ωhh −+−

2211

2

)]([)]()([)(||

kkpEpEkV

ωω

hh

h

−−−)()()()( 2121 kpEkpEpEpE hh ++−=+)()()()( 2211 pEkpEkpEpE −+=−− hh

Phonon-mediated attractive e-e interaction

for electrons within ωD of the Fermi surfaceH. Fröhlich 1952

July 2007 Cargese

Phonon attraction: simple pictureFa v/≈τTime electron spends near an atoma

MMe ττ )/( 22 aeFp ≈≈Typical momentum transfer

MpxM D /222 ≈ωTypical atomic displacement

βωωa

aeMmM

ae

maa

Maea

Mpx

DFD≈≈≈≈

)/(/

)/(1

v 22

2

2

2 h

h

July 2007 Cargese

Phonon attraction: simple pictureFa v/≈τTime electron spends near an atoma

MM e

aττ )/( 22 aeFp ≈≈Typical momentum transfer

β/ax ≈ mM /=βTypical atomic displacement

Dae

ax

aeU ω

βh−≈−≈−≈∆

22Change in potential

July 2007 Cargese

Phonon attraction: simple picture

Fa v/≈τTime electron spends near an atom

MM e

MM

ττ )/( 22 aeFp ≈≈Typical momentum transfer


DU ωh−≈∆Change in potentialM M1−≈ DM ωτTime before atoms relax back

M Mavl MF βτ ≈≈Displacement trail length

July 2007 Cargese

Phonon attraction: simple picture

Fa v/≈τTime electron spends near an atom

ττ )/( 22 aeFp ≈≈Typical momentum transfer


DU ωh−≈∆Change in potential

l βωτ )v/(1FDM a≈≈ −

Time before atoms relax back

avl MF βτ ≈≈Displacement trail length

“Tube” of attractive potential: most advantageous for electrons moving in opposite directions

V. Weisskopf 1979

July 2007 Cargese

and still

July 2007 Cargese

and still

R. Feynman 1956

July 2007 Cargese

and still

R. Feynman 1956

July 2007 Cargese

Cooper phenomenon

Fpair EE 2<Bound states of two electrons near the Fermi surface:

( ) ( ) ),(2),(),(2 222

22

21

2

rrrrrr 111 Ψ+=Ψ⎥⎦

⎤⎢⎣

⎡+∇+∇− FEEV

mh

[ ])(exp)(),( 22 rrkkrr 1k

1 −⋅=Ψ ∑ iψOur “interaction tube”: CM at rest

( ) )(2)()(22

kkkk

kk ψψψ FEEVmk

+=′+∑ ′h

[ ] rkkrrkk diVLV ∫ ′−⋅= −′ )(exp)(3What is the matrix element?

July 2007 Cargese

Matrix element

al β≈

Both electrons near the Fermi surface: rapidly oscillating phases

[ ]drirLUVl

∫ ′−⋅−≈ −′

0

30 ||exp kkkk

DmaakkEE ω

ββh

hhh ≈≈≤′−≈′− 2

2F

Fv||v||

July 2007 Cargese

Bound State( ) )(2)()()(2 kkkk

kkk ψψψ FEEVE +=′+∑ ′

e

e

Fkk <= for0)(kψKey: filled Fermi sea

))((2

)()(

FEEE

V

−−

′=

∑ ′

k

kk k

kk ψψSolution:

1

00 )2(

))((211 −−≈

−−= ∫∑ EdVN

EEEV

D

Fξξ

ωh

k kSelf-consistency:

Bound state of two electrons with opposite momenta: Cooper pair ]/2exp[2 00 VNE D −−= ωh

L. N. Cooper 1956

00 ∆≈ENB:

July 2007 Cargese

Importance of the Fermi surface• Same problem without FS:

)(2)(2

)()(

kk

kk k

kk

EEC

EE

V

−=

−

′=∑ ′ψ

ψ

• Self-consistency on C is equation for energy E

∑+

=k k ||/

1122 EmV h

Is there a bound state for a small attractive potential?

For small E integral finite: need critical potential value∫ +

=||/

4122

2

Emkdkk

V h

π3D

)/1exp( 0VNWE −−=Fermi surface makes it “2D”

||ln

||/21

022 EWN

Emkkdk

V≈

+= ∫

h

π2D

July 2007 Cargese

Comments

• Non-perturbative result: no expansion V~0• Cooper pairs: bosonic objects

• Bose condensation of pairs leads to superconductivity.M. Schafroth, S. Butler, J. Blatt 1957

4He is also a collection of bound fermions (6, not 2).

What is special about gas of Cooper pairs?

July 2007 Cargese

Cooper pair gas

3−≈ an

pF∆≈∆ v0

)/(/v 000 ∆≈∆≈ FF EahξFF Ep

p 0∆≈∆

Spread of momentum

Cooper pair size

Electron density

Cooper pair density

03/1

03/1 )/( ξ<<∆≈≈ −

Fc EandDistance between pairs

10

20

3 )/( −−− ≈∆≈ ξaEan Fc

Strongly overlapping gas of Cooper pairs

July 2007 Cargese

Dilute Bose gas

13 <<ξnSingle-particle states Ensemble states

k≠0 excited

k=0 ground state

ξ condensatecondensate

Gapless spectrum: when one particle is excited (starts moving) it has little chance of scattering off the particles in the condensate

July 2007 Cargese

Dense neutral Bose gas

1~3ξnSingle-particle states Ensemble states

k≠0 excited

k=0 gap ∆

ground statecondensate

Low energy excitation: rotation of atoms relative to one another: typical energy

22 / ξMh≈∆ Rotons

July 2007 Cargese

Cooper pair gas

Ensemble states Overlapping charged pairs: roton energy high

excited

Energy gap=energy to dissociate a pair

gap ∆

ground state

How to describe such a state? What is the “right” combination?

July 2007 Cargese

BCS wave function

kkkk

kkkk

k bbVnH +′

′′∑∑ += σ

σεReduced Hamiltonian:

electron pairing

( )( ) 0

02

2

=

=+

k

k

b

b+↓−

+↑

+

+

=

=

kkk

kkk

ccb

ccn σσσ

↓−↑+

′′′′+

−−=

=

kkkk

kkkk

nnbb

cc

1],[

, σσσσ δδNot the usual bosonicoperators!

July 2007 Cargese

BCS wave function

kkkk

kkkk

kk bbVbbH +′

′′

+ ∑∑ +=σ

ε2Reduced Hamiltonian: electron pairing

( )( ) 0

02

2

=

=+

k

k

b

b+↓−

+↑

+

+

=

=

kkk

kkk

ccb

ccn σσσ

↓−↑+

′′′′+

−−=

=

kkkk

kkkk

nnbb

cc

1],[


Variational approach: unoccupied / doubly occupied states, no single occupancy

Inspiration: polaron

July 2007 Cargese

BCS wave function

kkkk

kkkk

kk bbVbbH +′

′′

+ ∑∑ +=σ

ε2Reduced Hamiltonian: electron pairing

( )( ) 0

02

2

=

=+

k

k

b

b+↓−

+↑

+

+

=

=

kkk

kkk

ccb

ccn σσσ

↓−↑+

′′′′+

−−=

=

kkkk

kkkk

nnbb

cc

1],[


Variational approach: unoccupied / doubly occupied states, no single occupancy

Inspiration: polaron Static distortion of the lattice around an electron: emission of multiple phonons into the same state

[ ] vacaagk

kkk∏ −+ +∝Ψ )(exp

e

T.D. Lee, F. Low, D. Pines 1953

July 2007 Cargese

BCS wave function IIvacbgvace kk

kk

bg kk )1( +∏∏ +≡∝Ψ+

vacccvu kkkk

kBCS )( +↓−

+↑∏ +=ΨNormalization:

122 =+ kk vuProbability empty

Probability occupied

July 2007 Cargese

Ground state energykkkk

kkkk

kkkBCSBCSg vuvuVvNHE ′′

′′∑∑ +=Ψ−Ψ=

σεµ 22

Choice of coefficients: variational method

P. Hirschfeld talk

kk

kk

vu

θθ

cossin

== 22 ∆+= kkE ε

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

k

kk

k

kk

Ev

Eu

ε

ε

121

121

2

2 excitation energy

kkk

kkk vuV ′′′

′∑−=∆

( ) 00/1exp2 EVND ≈−≈∆ ωh

energy gap

0=∂∂θ

gE

2/)0()( 20∆−==∆−∆ NEE ggCondensation energy

July 2007 Cargese

Occupation numbers

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

k

kk E

v ε1212Probability that the states with

momenta k and –k are occupied

2kv

FE

∆~ At T=0 looks like thermally smeared Fermi distribution

No discontinuity in occupation numbers: no gap in momentum space

Gap in excitations that are not coherent with the pairs: different phase

kε

July 2007 Cargese

Particle number

vacccvu kkkk

kBCS )( +↓−

+↑∏ +=Ψ particle number is not

conserved!

∑∑ =ΨΨ= ↑+↑

kk

kBCSkkBCS vccN 222

NNN 1

∝δ

( ) ∑=−=k

kk vuNNN 2222 4δ

Small relative fluctuations of the particle number

July 2007 Cargese

Particle number II

∑ Ψ=Ψ NNBCS λProject on a state with a definite # of electrons

State with N electrons has phase 2/ϕiN vacccevu kk

ik

kk )( +

↓−+↑∏ +=Ψ ϕ

ϕ

ϕϕ

πϕϕ

π

ϕϕ Ψ=+=Ψ −+↓−

+↑

− ∫∏∫ 2/2

0

2/2

0)( iN

kki

kk

kiN

N edvacccevued

All phases needed for definite particle number

NiiN∂∂⇔∂∂−⇔

//

ϕϕNumber and phase are

conjugate variables1≈∆∆ ϕN

July 2007 Cargese

Summary I

• Reduction of the problem to essential physics, and then trial for new ground states/excitations

• Solve problems even seemingly difficult ones: electrons form pairs despite Coulomb repulsion

July 2007 Cargese

Summary I

• Reduction of the problem to essential physics, and then trial for new ground states/excitations

• Solve problems even seemingly difficult ones: electrons form pairs despite Coulomb repulsion

• Works for other complicated problems: Laughlin wave function for fractional quantum Hall.

• Does the reduction method always work?

iyxzezzzz z

ijjin +=−=Ψ −∏ ,)(),...,(

2||31

July 2007 Cargese

Extension of BCS: AFM, RVB and all that

• Doped Hubbard systems (High-Tc, cobaltates)

ji

ijij

i JctcH SS∑∑ += +σσ

tU >>double occupancy expensive

C. Gros 2007

• Superexchange: AFM interaction

• If not 1 electron per site: vacancies that move

↓↑+ ∑∑ += i

iij

iji nUnctcH σσ

July 2007 Cargese

Resonating valence bonds

21

=• Form singlets on neighboring sites

• Look at 1D chain, compare energy

valence bond wins in 1D

energies may be close in 2D, with frustration…4

3J−

43J

−4J

−4J

−4J

−

• Make a linear superposition of bonds: allow them to resonate to lower the kinetic energy: mobile singletsAnderson, Anderson and Fazekas, 1973

July 2007 Cargese

RVB trial wave function

• Two electrons on a mobilevalence bond of length r

• Vary bond length/ no double occupancy

• Bonds condense/resonate

• Project out doubly occupied (Gutzwiller)

ikrk

kkri

iir eccccb +

↓−+↑

+↓+

+↑

+ ∑∑ ==

0)( =∑k

ka+↓−

+↑

+ ∑= kk

k cckab )(

( ) vacbN 2/+ double occupancy

( )( ) vacbnnN

iiiRVB

2/1 +

↓↑∏ −=Ψ

Anderson, 1987

July 2007 Cargese

BCS and RVB+↓−

+↑

+ = kkk cckab )(• Consider an unusual pairing

vacbgvace kkk

bg kk )exp(∑=∝Ψ +∏+

• Construct a BCS wave function

• Project a fixed number vacbvN

kkkNBCS

2/

, ⎥⎦⎤

⎢⎣⎡=Ψ ∑ +

• Prohibit double occupancy ( ) NBCSi

iiRVB nn ,1 Ψ−=Ψ ∏ ↓↑

Resonating valence bonds and Cooper pairs look similar: basis for theories of exotic SC

Anderson, 1987-2007,Gros et al. 2007,Paramekanti et al. 2004…

July 2007 Cargese

Summary and challenge

• (Correctly) guessing a wave function remains the best way to solve a strongly correlated problem

• Challenge: wave function describing an electronic system with most exotic properties.

• Bonus points: Hamiltonian for which this wave function is a good variational ground state

• Prize: TBD

50 years of BCS School Lecture 1: Basics of pairing - LSU · 50 years of BCS School Basics of pairing Ilya Vekhter Louisiana State University, USA July 2007 Cargese. ... • London

Documents