Lecture 10: Supercurrent Equation Outline 1. Macroscopic Quantum Model 2. Supercurrent Equation and the London Equations 3. Fluxoid Quantization 4. The Normal State 5. Quantized Vortices October 13, 2005 Massachusetts Institute of Technology 6.763 2005 Lecture 10 Macroscopic Quantum Model and density Massachusetts Institute of Technology 6.763 2005 Lecture 10 Total number 1. The wavefunction describes the whole ensemble of superelectrons such that 2. The flow of probability becomes the flow of particles, with the physical current density given by 3. This macroscopic quantum wavefunction follows 1Wave function , we find Massachusetts Institute of Technology 6.763 2005 Lecture 10 Writing The real part of the S-Eqn gives The imaginary part of the S-Eqn gives the supercurrent equation: Let and Massachusetts Institute of Technology 6.763 2005 Lecture 10 Supercurrent Equation with n* constant we find be a constant, so that with Energy of a superelectron 2London’s Equations with gives Massachusetts Institute of Technology 6.763 2005 Lecture 10 1.Take the curl of the supercurrent equation gives the Second London Equation: 2. Take the time derivative of the supercurrent equation: Something more than First London Equation? First London revisited full long as or Massachusetts Institute of Technology 6.763 2005 Lecture 10 With a number of vector identities, this can be shown to be equivalent to Lorentz force Hence, 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 Full First London 3Flux Quantization integer flux Massachusetts Institute of Technology 6.763 2005 Lecture 10 3. Take the line integral of the supercurrent equation around a closed contour within a superconductor: The line integral of each of the parts: Therefore, n = - n’ The And the Fluxiod Massachusetts Institute of Technology 6.763 2005 Lecture 10 Fluxoid Quantization flux quantum is defined as Fluxoid Quantization condition becomes Experiments testing fluxiod quantization will determine q*= -2e, so that 4Simply and Multiply Connected Regions in a Superconductors C C simply connected region multiply connected region For 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 2005 Lecture 10 Flux Quantization Consider a hollow cylinder, in bulk limit so that the thickness of the walls is less Image removed for copyright reasons. than the penetration depth. Please see: Figure 5.2, page 247, from Orlando, T., and K. Delin. Foundations of Applied Superconductivity. Reading, MA: Let an applied field Happ beAddison-Wesley, 1991. ISBN: 0201183234. trapped in the hole as the cylinder is cooled down. Let the contour be deep within the superconductor where J=0. Then Flux is quantized in the bulk limit. Massachusetts Institute of Technology 6.763 2005 Lecture 10 5Flux Quantization Experiments Image removed for copyright reasons. Please see: Figure 5.3, page 249, from Orlando, T., and K. Delin. Foundations of Applied Superconductivity. Reading, MA: Flux trapped in Addison-Wesley, 1991. ISBN: 0201183234. hollow cylinder Deaver and Fairbank, 1961, measure and show that q*=2e; Cooper Pairs. Massachusetts Institute of Technology 6.763 2005 Lecture 10 Induced Currents To have flux quantization, currents must be induced in the cylinder to add to or oppose the applied magnetic field. Induced flux= L i Image removed for copyright reasons. Please see: Figure 5.4, page 250, 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 10 6Energy of Hollow Cylinder If the walls of the cylinder are thin compared to the total area of the hole but thicker than the penetration depth, the difference in electromagnetic energy in the superconducting is n=0 n=2n=1n=-1 W Massachusetts Institute of Technology 6.763 2005 Lecture 10 The Normal State If 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 2005 Lecture 10 7Fluxoid Quantization and Type II Superconductors Image removed for copyright reasons. Please see: Figure 6.1, page 259, 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 10 The Vortex State nV is the areal density of vortices, the number per unit area. Image removed for copyright reasons. Please see: Figure 6.2a, page 262, from Orlando, T., and K. Delin. Foundations of Applied Superconductivity. Reading, MA: Addison-Wesley, 1991. ISBN: 0201183234. Image removed for copyright reasons. Please see: "A current-carrying type II superconductor in the mixed state" from http://phys.kent.edu/pages/cep.htm Top view of Bitter decoration experiment on YBCO Massachusetts Institute of Technology 6.763 2005 Lecture 10 8Quantized Vortices 1 But along 1 B mininum, so that J Along path C2, 2, Image removed for copyright reasons. Please see: Figure 6.2b, page 262, from