Lecture 23: Superconductivity II Theory (Kittel Ch. 10) D(E) D(E) - - PowerPoint PPT Presentation

Physics 460 F 2000 Lect 23 1

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

E D(E) EF Filled Empty E D(E) EF Filled Empty

Physics 460 F 2000 Lect 23 2

Outline

• Superconductivity - Concepts and Theory
• Key points

Exclusion of magnetic fields can be used to derive 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 Quantization

How we know currents are persistent!

• Cooper instability - electron pairs

Bardeen, 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 Normal Super- conducting Hc H

• µ0M

Normal Super- 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 T > Tc T < Tc Zero Field Cooled H T > Tc T < Tc Field Cooled Excludes 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 field B = internal field B = H + µ0M M = Magnetization

Phase Transition SUPERCONDUCTING STATE IS A 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)

• r

∆F = FN(0) - FS(0) = Hc

2/2µ0

• 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 E D(E) EF Filled Empty ∆E

Physics 460 F 2000 Lect 23 9

Coherence Length

• The typical length associated with the mechanism
• f 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 values Al Tc = 1.19K

ξ = 1,600 nm

Pb 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 causes field inside that cancels the external field

• A supercurrent

that 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 field and J decay into superconductor Superconductor Normal state

• r 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 PROPOSED

that in the gauge ∇A = 0, Anormal = 0, j = - A/(µ0 λL

2 )

so ∇ × j = - B /(µ0 λL

2 )

H Both B field and J decay into superconductor Superconductor

λL

Normal state

• r 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 + Emag where Emag = ∫ 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 λL 2 = ε0 mc2 /nsq 2

• Varying the form of B(r) by adding δB(r) the change δF is

δF = (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) - λL

2 ∇ × ∇ × B(r) = B(r) + λL 2 ∇2B(r) = 0

• Which leads to the London Equation

ns = superfluid density v(r) = velocity Integration by parts

Physics 460 F 2000 Lect 23 14

Thickness of region where current flows

• Therefore

∇2B = B/ λL

2

Solution: B decays into superconductor with the form B(x) = B(0) exp(-x/λL)

• Explains Meisner effect

B vanishes inside the superconductor

H Both B field and J decay into superconductor Superconductor

λL

Normal state

• r 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 = ns

1/2 - 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 λL

2 = 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 slide for 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 only near 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 on a 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

λL

2 = 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 values Al Tc = 1.19K ξ = 1,600 nm

λL = 160 nm ξ/λL = 0.01

Pb 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

Happlied Magnetic 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
• pposite k and
• pposite 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

E D(E) EF Filled Empty E D(E) EF Filled Empty 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 Tc

Hard to actually predict Tc !

• Experiment:

Al 1.2 K Hg 4.6 K Pb 7.2 K Au < 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

Superconducting Super 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 wavefunction

Ψ = ( ns

1/2 ) exp( iθ(r))

• Two components:

magnitude ns

1/2, phase θ

• The ground state is for

θ = constant

• Variations in θ(r) describe

higher energy current carrying 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