1. Superconductivity and Superfluidity Flux line motion In principlea Type II superconductor should be “better” than a Type I for most applications - it remains superconducting to much higher fields ……..but is it? Flux lines tend to move easily so as to reach equilibrium. But if a current flows in a superconductor above H c1 there will be a Lorentz force acting between the current (ie charge carriers) and the flux. Flux lines J FLFL FLFL FLFL FLFL FLFL So the flux lines move perpendicular to the current and induce an electric field v = velocity of flux lines Now, E is parallel to J, so acts like a resistive voltage, and power is dissipated 1 watt of dissipated power at 4.2K requires 300MW of refrigerator power! The solution is to introduce “pins” by creating defects within the superconductor Lecture 8

2. Superconductivity and Superfluidity The Bean Model A Type II superconductor without any pinning is said to be reversible - flux enters abruptly at H c1 and produces a uniform flux density throughout. If there are pinning centres within the Type II superconductor they hold the flux lines back near the surface creating a gradient of flux lines - such superconductors are said to be irreversible surface Lecture 8

3. Superconductivity and Superfluidity D superconductor The Bean Model The pins can be thought of as introducing “friction” inhibiting the movement of flux lines into the supercoductor In this respect the superconductor is a little like a sandpile with the flux lines behaving like grains of sand However big we make the sandpile the sloping sides always have the same gradient In an analogous fashion, however large the magnetic field the gradient of flux lines remains constant: This is the basis of the Bean Model B1B1 B1B1 B* B2B2 B2B2 Lecture 8

4. Superconductivity and Superfluidity The Bean Model Remember that curl B=  o J So a field gradient implies that a current is flowing perpendicular to B The Bean Model assumes that the effect of pinning is to: D B1B1 B1B1 B* B2B2 B2B2 x y zIf B is in the z-direction, and the gradient is in the x- direction, then in the y-direction (b) produce a maximum current density J and therefore to (a) produce a maximum gradient From another viewpoint the Bean model assumes (a) there exists a maximum current density J c (b) any emf, however small induces this current to flow The Bean model is therefore also known as the critical state model : only three current states are allowed - zero current for regions that have not felt B and ±J c (ie the critical current density) depending upon the sense of the emf generated by the last field change Lecture 8

5. Superconductivity and Superfluidity Critical state model - increasing field D superconductor D current density 0 +J c -J c B*/2 B* 2B* and Lecture 8

6. Superconductivity and Superfluidity D current density 0 +J c -J c Critical state model - decreasing field D superconductor then reduce B to zero again Because the flux density gradient must remain constant, flux is trapped inside the superconducting sample, even at B=0 B=0 BoBo BoBo BoBo First increase B to a value of, say B o a bced a bced Lecture 8

7. Superconductivity and Superfluidity Predictions of the Bean model The magnetisation of a sample can be calculated using the Bean model from diagrams such as the previous ones, with B* as the only free parameter Lecture 8

8. Superconductivity and Superfluidity Calculating the critical current density From the Bean model the critical current density is easily calculated from the hysteretic magnetisation loop in SI units, where J c is measured in A.m -2, in A.m -1 and D, the diameter of the sample in m 2M2M The strength of the pinning force, F, can also be determined: F usually shows a peak, at a field corresponding to that at which the pins “break” F B Lecture 8

9. Superconductivity and Superfluidity Magnetic superconductors Notice that in the 2nd quadrant of the hysteresis loop the magnetisation of the sample is positive, ie strongly (or even very strongly) paramagnetic. So - depending upon the magnetic history of the sample - the superconductor can be attracted to a magnet! So this….....can become this 1 23 4 As the magnetic falls away the field decreases from a-b, so the magnetisation increases. The magnet therefore moves closer to the superconductor (b-c) and the field increases, but the magnetisation decreases and the magnet falls away….etc a b c Lecture 8

10. Superconductivity and Superfluidity Critical current densities Lecture 8

11. Superconductivity and Superfluidity Critical current densities Lecture 8

12. Superconductivity and Superfluidity Small Angle Neutron Scattering R scattering angle 2  Superconducting sample multi-detector 64x64cm 2 incident neutron beam B L Bragg’s Law: =2dsin  neutron wavelength = 10Å flux lattice spacing, d = 1000Å sin  ~R/L = 1/100 Flux distribution determined from SANS Lecture 9

13. Superconductivity and Superfluidity Flux lattice melting R scattering angle 2  Superconducting sample multi-detector 64x64cm 2 incident neutron beam B L Lecture 9