Lecture 23: Superconductivity II Theory (Kittel Ch. 10) 23: Superconductivity II Theory (Kittel Ch...

Physics 460 F 2000 Lect 23 1

Lecture 23: Superconductivity II Theory (Kittel Ch. 10)

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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 )


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


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



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



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)


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)


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

E

D(E)

EF

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∆E


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


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


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


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


Superconductor

λL

Normal state or vacuum


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


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


Superconductor

λL

Normal state or vacuum


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


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


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


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


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!


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


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


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


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!


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


BCS theory - IV • Result

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D(E)

EF

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E

D(E)

EF

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Gap ∆E


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


Superconducting elements• Elements that have large electron-phonon coupling

NOT the “best” metals, NOT the magnetic elements

SuperconductingSuper

conducting


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)


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)


Next time• Magnetism

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Lecture 23: Superconductivity II Theory (Kittel Ch. 10) 23: Superconductivity II Theory (Kittel Ch...

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