50 years of bcs school lecture 1: basics of pairing - lsu · 50 years of bcs school basics of...
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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
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
=
Tkm B3v2 =
2/3v Bnkc =
2
2v
3v
nemc
T=
σκ Classical gas
3v v
2 τκ c=
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.
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
Metals: 1925 -- 1930• W. Pauli: exclusion principle• Fermi-Dirac statistics:• Sommerfeld theory of conductivity
CV << nkB
Tkm B3v2 =
2/3v Bnkc =
2
2v
3v
nemc
T=
σκ
)/(v FBB ETknkc ≈
)/2(v2F TkETkm BFB=
July 2007 Cargese
Metals: 1925 -- 1930• W. Pauli: exclusion principle• Fermi-Dirac statistics:• Sommerfeld theory of conductivity
CV << nkB
2822
/1044.23
KWe
kT
B Ω×≈⎟⎠⎞
⎜⎝⎛= −π
σκ
measurable at low T
July 2007 Cargese
Metals: 1925 -- 1930• W. Pauli: exclusion principle• Fermi-Dirac statistics:• Sommerfeld theory of conductivity
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
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
Superconductivity: 1930-1937
• 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
Superconductivity: 1939-1949
• 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
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
β/ax ≈ mM /=βTypical atomic displacement
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
β/ax ≈ mM /=βTypical atomic displacement
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
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],[
, σσσσ δδNot the usual bosonicoperators!
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],[
, σσσσ δδNot the usual bosonicoperators!
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