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Physics 460 F 2000 Lect 23 1
Lecture 23: Superconductivity II Theory (Kittel Ch. 10)
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Physics 460 F 2000 Lect 23 2
Outline• Superconductivity - Concepts and Theory• Key points
Exclusion of magnetic fields can be used toderive energy of the superconducting state
Heat Capacity shows there is a gap Isotope effect
• How does a superconductor exclude B field?London penetration depth (1930’s)
• Flux QuantizationHow we know currents are persistent!
• Cooper instability - electron pairsBardeen, Cooper, Schrieffer theory (1957)(Nobel Prize for work done in UIUC Physics)
• (Kittel Ch 10 )
Physics 460 F 2000 Lect 23 3
Meisner Effect• Magnetic field B is excluded
B = H + µ0M • For type I superconductors, µ0M = - H for T < Tc
• Perfect Diamagnetism !
Hc H
B
NormalSuper-
conducting
Hc H
- µ0M
NormalSuper-
conducting
From previous lecture
Physics 460 F 2000 Lect 23 4
Meisner Effect (1934)• A superconductor can actively push out a magnetic
field - Meisner effect• (For H < Hc in type I superconductors
and H < Hc1 in type II superconductors)
H
0
T > Tc T < Tc
Zero Field Cooled
H
0
T > Tc T < Tc
Field CooledExcludes Magnetic Field
From previous lecture
Physics 460 F 2000 Lect 23 5
Effect of a Magnetic Field• Magnetic fields tend to destroy superconductivity
Tc
T
H
Hc Normal
Super-conducting
Note: H = external applied fieldB = internal field
B = H + µ0MM = Magnetization
Phase TransitionSUPERCONDUCTING
STATE ISA NEW PHASE OF MATTER
From previous lecture
Physics 460 F 2000 Lect 23 6
Energy : normal vs. superconducting• The free energy F of the superconductor plus
magnetic field is increased because magnetic field B is excluded
• The normal state energy is nearly independent of field• Transition at Hc
Hc H
F
Normal
Superconducting FS(H) = FS(0) + H2/2µ0
FN
FS(0)
Physics 460 F 2000 Lect 23 7
Energy : normal vs. superconducting• Therefore FS(Hc) = FS(0) + Hc
2/2µ0 = FN(0)
or ∆F = FN(0) - FS(0) = Hc2/2µ0
• Typical Values: ∆F ~ 10-7 eV/electron ! SMALL !
Hc H
F
Normal
Superconduc. FS(H) =
FS(0) + H2/2m0
FN
FS(0)
Physics 460 F 2000 Lect 23 8
Energy : normal vs. superconducting• How do we understand the small values
∆F ~ 10-7 eV/electron ?• Similar to the description of thermal energy
∆F ~ D(EF) ∆E2 ~ ∆k2
where ∆E is the region affected - as shown by the gap in the heat capacity - agrees with experiment
kF
∆k
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∆E
Physics 460 F 2000 Lect 23 9
Coherence Length• The typical length associated with the mechanism
of superconductivity is the feature associated with the Fermi surface is ξ = 1/∆k = hvF/2 ∆E where ∆E is the region affected
(Understood from the BCS theory – see later)
kF
∆k
Typical valuesAl Tc = 1.19K ξ = 1,600 nmPb Tc = 7.18K ξ = 83 nm
Physics 460 F 2000 Lect 23 10
How is a field excluded?• What makes B = 0 inside superonductor?• Supercurrents flowing on the boundary!• Easiest geometry - long thin rod
H
Current around boundary causesfield inside that
cancels the external field
-A supercurrentthat flows with
no decay
B = 0 inside
Physics 460 F 2000 Lect 23 11
Thickness of region where current flows• Supercurrents J flowing on the boundary!
H H
Both B fieldand J decay intosuperconductor
Superconductor Normal state or vacuum
Supercurrent
Physics 460 F 2000 Lect 23 12
Thickness of region where current flows• London Penetration
Depth λL
• Maxwell’s Eq.: ∇ × B = µ0 j∇ × ∇ × B = - ∇2B = µ0 ∇ × j
• Also B = ∇ × A (A not unique)
• London PROPOSEDthat in the gauge ∇A = 0, Anormal = 0,
j = - A/(µ0 λL2 )
so∇ × j = - B /(µ0 λL
2 )
H
Both B fieldand J decay intosuperconductor
Superconductor
λL
Normal state or vacuum
Physics 460 F 2000 Lect 23 13
London Equations • Here we give a derivation of the London equations that gives
physical insight and the expression for the penetration depth λL
• The free energy for the system with a supercurrent and the penetrating B field is
F = F0 + Ekin + EmagwhereEmag = ∫ dr B2/8π and Ekin = ∫ dr ½ mv2 ns with j(r) = ns q v(r)
• Using ∇ × B = µ0 j we find F = F0 + (1/8π)∫ dr [B(r)2 + λL
2(∇ × B(r))2], where λL2 = ε0 mc2 /nsq 2
• Varying the form of B(r) by adding δB(r) the change δF isδF = (1/4π)∫ dr [B(r) δB(r) + λL
2 (∇ × B(r)) (∇ × δB(r)) ]= (1/4π)∫ dr [B(r) - λL
2 ∇ × ∇ × B(r)] δB(r)• At the minimum, δF = 0 for all possible δB(r) which requires that
B(r) - λL2 ∇ × ∇ × B(r) = B(r) + λL
2 ∇2B(r) = 0 • Which leads to the London Equation
ns = superfluid densityv(r) = velocity
Integrationby parts
Physics 460 F 2000 Lect 23 14
Thickness of region where current flows• Therefore
∇2B = B/ λL2
Solution: B decays into superconductor with theform
B(x) = B(0) exp(-x/λL)
• Explains Meisner effectB vanishes inside thesuperconductor
H
Both B fieldand J decay intosuperconductor
Superconductor
λL
Normal state or vacuum
Physics 460 F 2000 Lect 23 15
The superconducting state is a quantum state
• Landau and Ginsburg (before the BCS theory)proposed all the electrons act together to form a new state Ψ, with | Ψ |2 = ns where ns is the superfluid density
• Ground state: ΨG = ns1/2 - No current flowing
• Consider now Ψ = ( ns 1/2 ) exp( iθ(r)) - the phase in
a wavefunction corresponds to a current • The velocity operator is
v = p/m = (1/m)( - i h ∇ - (q/c)A)Thus
j = q Ψ∗ v Ψ = (ns q/m) (h ∇θ - (q/c)A) and
curl j = - (ns q 2 /mc) B
Physics 460 F 2000 Lect 23 16
The superconducting state is a quantum state - II
• This quantum state leads to a theory of the London penetration depth
• The equation curl j = - (ns q 2 /mc) B
and the London proposal curl j = - B /(µ0 λL
2 ) leads to
λL2 = ε0 mc2 / ns q 2
• Agrees with experiment!
BUT what is m? What is q? How do we really know it is quantum in nature?
See earlier slidefor alternative
derivation
Physics 460 F 2000 Lect 23 17
Quantized Flux• The flux enclosed in a ring is quantized!• Consider a line inside the superconductor
The current j = 0 inside• h ∇θ - (q/c)A = 0 inside the superconductor
H
Magnetic field threading ring
Current onlynear surface
Physics 460 F 2000 Lect 23 18
Quantized Flux -II• The line integral of ∇θ is the change in θ around
the loop = 2π x integer • The line integral of A is the surface integral of B
(See Kittel p 281) = total flux Φ enclosed in the ring• Result: Φ = (2π hc/q) x integer -- quantized!
• Result: Charge q = 2e - pairs !
Line integral ona closed contour
inside the superconductor
Physics 460 F 2000 Lect 23 19
Persistent Currents• How can the current stop flowing? • Only if some of the flux Φ leaks out of the ring• But the flux can only decrease by quanta!
• There is an energy barrier for the flux to go through the superconductor to escape - time for current to decrease can be ~ age of universe!
Physics 460 F 2000 Lect 23 20
Two length scales in superconductivity• London Penetration depth
λL2 = ε0mc2/nq2 (particles of mass m, charge q)
• (Understood from the BCS theory that m and q are for an electron pair – see later)
Typical valuesAl Tc = 1.19K ξ = 1,600 nm λL = 160 nm ξ/λL = 0.01Pb Tc = 7.18K ξ = 83 nm λL = 370 nm ξ/λL = 0.45
The ratio determines type I (ξ/λL <<1) and type II (ξ/λL > ~1) superconductors
see later
Other examples are given in Kittel
Physics 460 F 2000 Lect 23 21
Type II• Type II superconductors are ones where it is
favorable to break up the field into quanta - the smallest posible unit of flux in each “vortex”shown - for Hc1 < H < Hc2
• Lattice of quantized flux units
HappliedMagnetic flux penetrates through the superconductor by creating
small regions normal metal
Physics 460 F 2000 Lect 23 22
BCS theory • Hints: Must involve phonons, small energy scale • First: Cooper instability• If for some reason there were an attractive
interaction between two electrons above the Fermi energy in a metal, they would form a bound pair below the Fermi energy no matter how weak the interaction!
• Two electrons of opposite k and opposite spin form a bound state
• Fermi surface is unstable!
kF
Physics 460 F 2000 Lect 23 23
BCS theory - II • What could cause the attraction? - phonons! • The Coulomb interaction is repulsive• But phonons can cause the “Mattress effect” - one
electron causes the lattice to distort - the second electron is attracted the the distortion even after the first electron has left!
• Two electrons of opposite k and opposite spin form a bound state!
Physics 460 F 2000 Lect 23 24
BCS theory - III • The Cooper idea shows there is a problem for two
electrons - but what do all the electrons do?• This is the key advance of BCS - to construct a
new quantum wavefunction for all the electrons • Fundamental change only for electrons within a
energy range ∆E near the Fermi surface
• Opens an energy gap -explains the specific heat
• Forms single quantum state Ψ separated by a gap from other states
Physics 460 F 2000 Lect 23 25
BCS theory - IV • Result
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D(E)
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Gap ∆E
Physics 460 F 2000 Lect 23 26
Superconducting transition Tc• BCS prediction: Tc = 1.14 ΘD exp(-1/UD(EF))
where is the Debye temperature (measure of phonon energy), D(EF) is then density of states at Fermi energy, and U = typical electron-phonon coupling energy
• Fits experiments for ratio of energy gap to TcHard to actually predict Tc !
• Experiment:Al 1.2 K Hg 4.6 K Pb 7.2 KAu < 0.001 K - not found to be superconducting! Na3C60 40 K (1990)YBa2Cu3O7 93 K (1987)
Record today 140 K
Physics 460 F 2000 Lect 23 27
Superconducting elements• Elements that have large electron-phonon coupling
NOT the “best” metals, NOT the magnetic elements
SuperconductingSuper
conducting
Physics 460 F 2000 Lect 23 28
What is the “Order Parameter”?• If superconductivity is a new state of matter and there
is a phase transition between the normal and superconducting states:What is the order parameter?(Analogous to magnetization vector M in a magnet)
Tc T
H
Hc Normal
Super-conducting
• The wavefunctionΨ = ( ns
1/2 ) exp( iθ(r))• Two components:
magnitude ns 1/2, phase θ
• The ground state is forθ = constant
• Variations in θ(r) describehigher energy currentcarrying states (analogous magnons in a magnet)
Physics 460 F 2000 Lect 23 29
Summary • Superconductivity - Concepts and Theory • Exclusion of magnetic fields can be used to derive
energy of the superconducting state• Shows very small energy ∆F ~ D(EF) ∆E2 ~ ∆k2
where the gap is consistent with heat capacity • How does a superconductor exclude B field?
London penetration depth (1930’s)• Superconductor forms a quantum state• Flux Quantization
How we know currents are persistent!• Cooper instability - electron pairs• Bardeen, Cooper, Schrieffer theory (1957)
(Nobel Prize for work done in UIUC Physics)
Physics 460 F 2000 Lect 23 30
Next time• Magnetism