londons equations
TRANSCRIPT
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Massachusetts Institute of Technology 6.763 2003 Lecture 4
Lecture 4: London’s Equations
Outline
1. Drude Model of Conductivity2. Superelectron model of perfect conductivity
• First London Equation• Perfect Conductor vs “Perfect Conducting Regime
3. Superconductor: more than a perfect conductor4. Second London Equation5. Classical Model of a Superconductor
September 15, 2003
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Massachusetts Institute of Technology 6.763 2003 Lecture 4
Drude Model of ConductivityFirst microscopic explanation of Ohm’s Law (1900)
1. The conduction electrons are modeled as a gas of particles with no coulomb repulsion (screening)
2. Independent Electron Approximation• The response to applied fields is calculated for
each electron separately. • The total response is the sum of the individual
responses.3. Electrons undergo collisions which randomize
their velocities.4. Electrons are in thermal equilibrium with the
lattice.
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Massachusetts Institute of Technology 6.763 2003 Lecture 4
Response of individual electronsConsider an electron of mass m and velocity v in an applied electric E and magnetic B.
Ohm’s Law Hall EffectTransport scattering time
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Massachusetts Institute of Technology 6.763 2003 Lecture 4
Response of a single electron
Consider a sinusoidal drive and response of a single electron
Then,
and
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Massachusetts Institute of Technology 6.763 2003 Lecture 4
Total Response of conduction electronsThe density of conduction electrons, the number per unit volume, is n. The current density is
ω
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Massachusetts Institute of Technology 6.763 2003 Lecture 4
Scattering timeTo estimate the scattering time
Hence for frequencies even as large at 1 THz,
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Massachusetts Institute of Technology 6.763 2003 Lecture 4
Equivalent Circuit for a Metal
v
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Massachusetts Institute of Technology 6.763 2003 Lecture 4
Perfect Conductor vs. Perfectly Conducting Regime
Perfect conductor:
A perfect inductorPurely reactiveLossless
Perfectly conducting regime:A perfect resistorPurely resistiveLossy
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Massachusetts Institute of Technology 6.763 2003 Lecture 4
Ordering of time constants
Lossless & dispersivequasistaticNondispersive σ = σ0
Cannot be quasistatic and losses
1/τem 1/τtr
Lossless & dispersivquasistatic
nondispersiveCan be Quasistatic and losses
1/τtr 1/τem
1/τem
quasistatic Lossless & dispersiveCan be Quasistatic and losses for all frequencies: Perfect conductor
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Massachusetts Institute of Technology 6.763 2003 Lecture 4
First London Equation
S
SuperelectronOr Cooper Pair:
and
Therefore,
And we have the First London Equation
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Massachusetts Institute of Technology 6.763 2003 Lecture 4
MQS and First Londonand describe the perfect conductor.
So that
0
using the first London EQN
Therefore,
governs a perfect conductor.
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Massachusetts Institute of Technology 6.763 2003 Lecture 4
The Penetration Depth
So that
is independent of frequency.The penetration depth
And is of the order of about 0.1 microns for Nb
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Massachusetts Institute of Technology 6.763 2003 Lecture 4
Perfectly Conducting Infinite Slab
Let
Therefore,
and2a
x
Boundary Conditions demand
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Massachusetts Institute of Technology 6.763 2003 Lecture 4
Fields and Currents for |y|< a
Thin film limit
λ λ
Bulk limit
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Massachusetts Institute of Technology 6.763 2003 Lecture 4
Ohmic vs. perfect conductorOhmic conductor
Complex k mean a damped wave: lossy
Perfect conductor
Real λ means an evanescent wave: Lossless
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Massachusetts Institute of Technology 6.763 2003 Lecture 4
Modeling a perfect conductor
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Massachusetts Institute of Technology 6.763 2003 Lecture 4
Perfectly Conducting Infinite Slab: General Solution
Let
Therefore,
2a
x
Boundary Conditions demand
Integrating over time gives:
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Massachusetts Institute of Technology 6.763 2003 Lecture 4
Perfectly Conducting Infinite Slab: General Solution
2a
x
Bulk limit near surface y = a
Deep in the perfect conductor
For a thin film, H(y,t) = H(a,t) for all time.
The perfect conductor preserves the original flux distribution, in the bulk limit.
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Massachusetts Institute of Technology 6.763 2003 Lecture 4
A perfect conductor is a flux conserving medium; a superconductor is a flux expelling medium.
Perfect Conductor Superconductor
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Massachusetts Institute of Technology 6.763 2003 Lecture 4
Towards a superconductorPerfectly conducting regime Perfect conductor Superconductor
τm > τc
Flux “expulsion” in the bulk limit, not for ω = 0.
Flux conserving in the bulk limit
Flux expulsion in the bulk limit, even for ω = 0.
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Massachusetts Institute of Technology 6.763 2003 Lecture 4
Second London EquationFor a superconductor we want to have
Working backwards
Therefore, the second London Equation
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Massachusetts Institute of Technology 6.763 2003 Lecture 4
Superconductor: Classical Model
first London Equation
second London Equation
penetration depth
When combined with Maxwell’s equation in the MQS limit