Lecture 4: London’s Equations Outline 1. Drude Model of Conductivity 2. Superelectron model of perfect conductivity • First London Equation • Perfect Conductor vs “Perfect Conducting Regime 3. Superconductor: more than a perfect conductor 4. Second London Equation 5. Classical Model of a Superconductor September 20, 2005 Massachusetts Institute of Technology 6.763 2005 Lecture 4 Drude Model of Conductivity First 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. Massachusetts Institute of Technology 6.763 2005 Lecture 4 1Response of individual electrons m and velocity v electric E B. Massachusetts Institute of Technology 6.763 2005 Lecture 4 Consider an electron of mass in an applied and magnetic Ohm’s Law Hall Effect Transport scattering time Response of a single electron Then, and Massachusetts Institute of Technology 6.763 2005 Lecture 4 Consider a sinusoidal drive and response of a single electron 2Total Response of conduction electrons n. The current density is ω Scattering time Massachusetts Institute of Technology 6.763 2005 Lecture 4 The density of conduction electrons, the number per unit volume, is Massachusetts Institute of Technology 6.763 2005 Lecture 4 To estimate the scattering time Hence for frequencies even as large at 1 THz, 3Equivalent Circuit for a Metal v Massachusetts Institute of Technology 6.763 2005 Lecture 4 Perfect Conductor vs. Perfectly Conducting Regime Perfect conductor: Perfectly conducting regime: A perfect inductor Purely reactive Lossless A perfect resistor Purely resistive Lossy Massachusetts Institute of Technology 6.763 2005 Lecture 4 4Ordering of time constants quasistatic Nondispersive σ = σ0 Lossless & dispersive Cannot be quasistatic and losses 1/τem 1/τtr Lossless & dispersiv quasistaticnondispersive Can be Quasistatic and losses 1/τtr 1/τem Can be Quasistatic and losses for all frequencies: Perfect conductor quasistatic Lossless & dispersive 1/τem Massachusetts Institute of Technology 6.763 2005 Lecture 4 First London Equation Or Cooper Pair: S and Therefore, And Massachusetts Institute of Technology 6.763 2005 Lecture 4 Superelectron we have the First London Equation 5MQS and First London 0 and So that Therefore, governs a perfect conductor. Massachusetts Institute of Technology 6.763 2005 Lecture 4 describe the perfect conductor. using the first London EQN The Penetration Depth So that is independent of frequency.The penetration depth And is of the order of about 0.1 microns for Nb Massachusetts Institute of Technology 6.763 2005 Lecture 4 6Perfectly Conducting Infinite Slab Let Therefore, and Image removed for copyright reasons. Please see: Figure 2.13, page 43, from Orlando, T., and K. Delin Foundations of Applied Superconductivity. Reading, MA: Addison-Wesley, 1991. ISBN: 0201183234. Massachusetts Institute of Technology 6.763 2005 Lecture 4 Boundary Conditions demand λ λ Massachusetts Institute of Technology 6.763 2005 Lecture 4 Fields and Currents for |y|< a Thin film limit Bulk limit 7k Real λ Massachusetts Institute of Technology 6.763 2005 Lecture 4 Ohmic vs. perfect conductor Ohmic conductor Complex mean a damped wave: lossy Perfect conductor means an evanescent wave: Lossless Modeling a perfect conductor Massachusetts Institute of Technology 6.763 2005 Lecture 4 8Let Therefore, Image removed for copyright reasons. Please see: Figure 2.13, page 43, from Orlando, T., and K. Delin Foundations of Applied Superconductivity. Reading, MA: Addison-Wesley, 1991. ISBN: 0201183234. Massachusetts Institute of Technology 6.763 2005 Lecture 4 Perfectly Conducting Infinite Slab: General Solution Boundary Conditions demand Integrating over time gives: y = a . Image removed for copyright reasons. Please see: Figure 2.13, page 43, from Orlando, T., and K. Delin Foundations of Applied Superconductivity. Reading, MA: Addison-Wesley, 1991. ISBN: 0201183234. Massachusetts Institute of Technology 6.763 2005 Lecture 4 Perfectly Conducting Infinite Slab: General Solution Bulk limit near surface Deep in the perfect conductor The perfect conductor preserves the original flux distribution, in the bulk limitFor a thin film, H(y,t) = H(a,t) for all time. 9A perfect conductor is a flux conserving medium; a superconductor is a flux expelling medium. Images removed for copyright reasons. Massachusetts Institute of Technology 6.763 2005 Lecture 4 Towards a superconductor Perfectly conducting regime Perfect conductor τm > τc ω = 0.ω = 0. Massachusetts Institute of Technology 6.763 2005 Lecture 4 Superconductor Flux conserving in the bulk limit Flux expulsion in the bulk limit, even for Flux “expulsion” in the bulk limit, not for 10Second London Equation For a superconductor we want to have Working backwards Therefore, the second London Equation Massachusetts Institute of Technology 6.763 2005 Lecture 4 Superconductor: Classical Model second London Equation first London Equation penetration depth Massachusetts Institute of Technology 6.763 2005 Lecture 4 When combined with Maxwell’s equation in the MQS limit