Lecture 17: Type II SuperconductorsFluxoid Quantization and Type II SuperconductorsThe Vortex StateQuantized VorticesNormal Core of theVortexCoherence Length xTemperature DependenceVortex in a CylinderVortex in a cylinderVortex in a cylinder k >> 1Energy of a single VortexModified London Equation k >> l/xGeneral Thermodynamic ConceptsW: Electromagnetic EnergyThermodynamic FieldsEntropy and the Second LawConcept of Reservoir and SubsystemGibbs Free EnergyGibbs Free Energy and Co-energyGibbs Free Energy and EquilibruimPhase Diagram and Critical FieldCritical Field for Type ICritical Fields for Type IIMassachusetts Institute of Technology 6.763 2003 Lecture 17Lecture 17: Type II SuperconductorsOutline1. A Superconducting Vortex2. Vortex Fields and Currents3. General Thermodynamic Concepts• First and Second Law• Entropy• Gibbs Free Energy (and co-energy)4. Equilibrium Phase diagrams5. Critical FieldsOctober 30, 2003Massachusetts Institute of Technology 6.763 2003 Lecture 17Fluxoid Quantization and Type II SuperconductorsMassachusetts Institute of Technology 6.763 2003 Lecture 17The Vortex StatenVis the areal density of vortices, the number per unit area.Top view of Bitter decoration experiment on YBCOMassachusetts Institute of Technology 6.763 2003 Lecture 17Quantized Vortices Fluxoid Quantization along C1 But along the hexagonal path C1B is a mininum, so that J vanishes along this path.Therefore,And experiments give n = 1, so each vortex has one flux quantum associated with it.Along path C2, For small C2,Massachusetts Institute of Technology 6.763 2003 Lecture 17Normal Core of theVortexThe current densitydiverges near the vortex center,Which would mean that the kinetic energy of the superelectrons would also diverge. So to prevent this, below some core radius ξ the electrons become normal. This happens when the increase in kinetic energy is of the order of the gap energy. The maximum current density is thenIn the absence of any current flux, the superelectrons have zero net velocity but have a speed of the fermi velocity, vF. Hence the kinetic energy with currents isMassachusetts Institute of Technology 6.763 2003 Lecture 17Coherence Length ⌧The energy of a superelectron at the core is The difference in energy, is to first order in the change in velocity,With this givesThe full BCS theory gives the coherence length asTherefore the maximum current density, known as the depairing current density, isMassachusetts Institute of Technology 6.763 2003 Lecture 17Temperature DependenceBoth the coherence length and the penetration depth diverge at TCBut there ratio, the Ginzburg-Landau parameter is independent of temperature near TCAl, NbType I superconductorType II superconductorNb, Most magnet materialsMassachusetts Institute of Technology 6.763 2003 Lecture 17Vortex in a CylinderyξφzLondon’s Equations hold in the superconductorxWith Ampere’s Law givesLzBecause B is in the z-direction, this becomes a scalar Helmholtz EquationMassachusetts Institute of Technology 6.763 2003 Lecture 17Vortex in a cylinderWhich as a solution for an azimuthally symmetric fieldC0is found from flux quantization around the core,Which forMassachusetts Institute of Technology 6.763 2003 Lecture 17Vortex in a cylinder κ >> 1ln r1/rMassachusetts Institute of Technology 6.763 2003 Lecture 17Energy of a single VortexThe Electromagnetic energy in the superconducting region for a vortex isThis gives the energy per unit length of the vortex asIn the high κ limit this isMassachusetts Institute of Technology 6.763 2003 Lecture 17Modified London Equation κ >> λ/ξGiven that one is most concerned with the high κ limit, one approximates the core of the vortex ξ as a delta function which satisfies the fluxoidquantization condition. This is known as the Modified London Equation:The vorticity is given by delta function along the direction of the core of the vortex and the strength of the vortex is Φ0For a single vortex along the z-axis: For multiple vorticesMassachusetts Institute of Technology 6.763 2003 Lecture 17General Thermodynamic ConceptsFirst Law of Thermodynamics: conservation of energyInternal energyHeat in E&M energy storedwork done by the systemMassachusetts Institute of Technology 6.763 2003 Lecture 17W: Electromagnetic EnergySuperconducting region of Volume VsNormal region of Volume VnIn the absence of applied currents, in Method II, we have found that Moreover, for the simple geometries H is a constant, proportional to the applied field. For a H along a cylinder or for a slab, H is just the applied field. Therefore,Massachusetts Institute of Technology 6.763 2003 Lecture 17Thermodynamic Fieldsthermodynamic magnetic field thermodynamic flux densitythermodynamic magnetization densityTherefore, the thermodynamic energy stored can be written simply asMassachusetts Institute of Technology 6.763 2003 Lecture 17Entropy and the Second LawThe entropy S is defined in terms of the heat delivered to a system at a temperature T Second Law of Thermodynamics:For an isolated system in equilibrium ∆S = 0The first law for thermodynamics for a system in equilibrium can be written asThen the internal energy is a function of S, B, and ηConjugate variablesMassachusetts Institute of Technology 6.763 2003 Lecture 17Concept of Reservoir and SubsystemBecause we have more control over the conjugate variables we seek a rewrite the thermodynamics in terms of these controllable variables., Isolated system = Subsystem + Reservoir∆QAThe change in entropy of the reservoir isTherefore,Massachusetts Institute of Technology 6.763 2003 Lecture 17Gibbs Free EnergyThe change total entropy is then where the Gibbs Free Energy is defined by At equilibrium, the available work is just ∆G (the energy that can be freed up to do work) and the force isFree Energy of subsystem decreasesMassachusetts Institute of Technology 6.763 2003 Lecture 17Gibbs Free Energy and Co-energyThe Gibbs free energy isThe differential of G isand with the use of the first lawTherefore, the Gibbs free energy is a function ofthe co-energyAt constant temperature and no work, then Note minus sign!Massachusetts Institute of Technology 6.763 2003 Lecture 17Gibbs Free Energy and EquilibruimIn Equilibrium ∆G = 0Consider the system made up of two phases 1 and 2Phase 1, G = G1Phase 2, G= G2Therefore,Mixed phaseis minimized when Two phases in equilibrium with each other have the same Gibbs Free EnergyMassachusetts Institute of