Lecture 10: Supercurrent EquationMacroscopic Quantum ModelWave functionSupercurrent Equation with n* constantLondon’s EquationsFirst London revisitedFlux QuantizationFluxoid QuantizationSimply and Multiply Connected Regions in a SuperconductorsFlux QuantizationFlux Quantization ExperimentsInduced CurrentsEnergy of Hollow CylinderThe Normal StateFluxoid Quantization and Type II SuperconductorsThe Vortex StateQuantized VorticesNormal Core of theVortexCoherence Length xMassachusetts Institute of Technology 6.763 2003 Lecture 10Lecture 10: Supercurrent EquationOutline1. Macroscopic Quantum Model2. Supercurrent Equation and the London Equations3. Fluxoid Quantization4. The Normal State5. Quantized VorticesOctober 7, 2003Massachusetts Institute of Technology 6.763 2003 Lecture 10Macroscopic Quantum Model1. The wavefunction describes the whole ensemble of superelectrons such thatdensityand Total number2. The flow of probability becomes the flow of particles, with the physical current density given by3. This macroscopic quantum wavefunction followsMassachusetts Institute of Technology 6.763 2003 Lecture 10Wave functionWriting , we findThe real part of the S-Eqn givesThe imaginary part of the S-Eqn gives the supercurrent equation:Massachusetts Institute of Technology 6.763 2003 Lecture 10Supercurrent Equation with n* constantLet be a constant, so that we findwithandEnergy of a superelectronMassachusetts Institute of Technology 6.763 2003 Lecture 10London’s Equations1.Take the curl of the supercurrent equationgives the Second London Equation:2. Take the time derivative of the supercurrent equation:with givesSomething more thanFirst London Equation?Massachusetts Institute of Technology 6.763 2003 Lecture 10First London revisitedFull First LondonWith a number of vector identities, this can be shown to be equivalent to full Lorentz forceHence, the above is the full first London Equation. However, for MQS problems we never used the first London Equation!! So all our previous results are valid.Our “short” form of the first London equation is valid in the limit where we ignored the magnetic field, that is ignored the Hall effect. One can show that this is true as long as orMassachusetts Institute of Technology 6.763 2003 Lecture 10Flux Quantization3. Take the line integral of the supercurrent equation around a closed contour within a superconductor:The line integral of each of the parts:fluxTherefore,integern = - n’Massachusetts Institute of Technology 6.763 2003 Lecture 10Fluxoid QuantizationThe flux quantum is defined as And the Fluxoid Quantization condition becomesFluxiodExperiments testing fluxiod quantization will determine q*= -2e, so thatMassachusetts Institute of Technology 6.763 2003 Lecture 10Simply and Multiply Connected Regions in a SuperconductorsCCsimply connected region multiply connected regionFor the simply connected region, fluxoid quantization holds for every contour C, no matter how small. As the contour shrinks to zero, both integrals vanishes and n=0.For the mutiply connected region, the contour can only be shrink to the contour outlining the normal region. Hence, the integrals need not vanish, so n can be any integer.Massachusetts Institute of Technology 6.763 2003 Lecture 10Flux QuantizationConsider a hollow cylinder, in bulk limit so that the thickness of the walls is less than the penetration depth.Let an applied field Happbe trapped in the hole as the cylinder is cooled down.Let the contour be deep within the superconductor where J=0. ThenFlux is quantized in the bulk limit.Massachusetts Institute of Technology 6.763 2003 Lecture 10Flux Quantization ExperimentsFlux trapped in hollow cylinderDeaver and Fairbank, 1961, measure and show that q*=2e;Cooper Pairs.Massachusetts Institute of Technology 6.763 2003 Lecture 10Induced CurrentsTo have flux quantization, currents must be induced in the cylinder to add to or oppose the applied magnetic field.Induced flux= L iMassachusetts Institute of Technology 6.763 2003 Lecture 10Energy of Hollow CylinderIf the walls of the cylinder are thin compared to the total area of the hole but thickerthan the penetration depth, the difference in electromagnetic energy in the superconducting is n=0 n=2n=1n=-1WMassachusetts Institute of Technology 6.763 2003 Lecture 10The Normal StateIf the electrons in both the superconducting and the normal state are described by quantum mechanics, what is the wavefunction for the normal state and how does it differ from the superconducting state?Quantum Mechanics describes both states as a wavefunction that depends on the coordinates of all the electrons. The MQM wavefunction for the superconductor is the spatial average of this phase coherent wavefunction and is preserved with an applied field. The coherence persists over the macroscopic scale of the superconductor.In the normal state, the applied field causes dissipation; this energy loss causes the phase of the wavefunction is randomized, on a length scale which is usually much smaller than the scale of the material. The electrons is the normal state have a background speed, the fermi velocity vF. The electrons undergo collisions on a time scale τtr. The average distance between collisions is the mean free path,Massachusetts Institute of Technology 6.763 2003 Lecture 10Fluxoid Quantization and Type II SuperconductorsMassachusetts Institute of Technology 6.763 2003 Lecture 10The 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 10Quantized 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 10Normal 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