Type I and Type II superconductivity Using the first Ginzburg Landau equation, and limiting the analysis to first order in  (which is already small close to the transition) we have This is a well known quantum mechanical equation describing the motion of a charged particle, e*, in a magnetic field The lowest eigenvalue of this Schroedinger equation is c is the cyclotron frequency so Recognising that we have a non zero solution for , hence superconductivity when B<Bc2, we obtain from the Schroedinger equation: Lecture 7

Type I and Type II superconductivity and, from earlier We have Combining these two equations, and recognising the temperature dependence of Bc2 and  explicitly, gives If Bc2=3 Tesla then  = 10nm However we have also just shown that the thermodynamic critical flux density, Bc, is given by So, we obtain where is the Ginzburg-Landau parameter If <1/2 then Bc2 < Bc and as the magnetic field is decreased from a high value the superconducting state appears only at and below Bc TYPE I Superconductivity If >1/2 then Bc2 > Bc and as the magnetic field is decreased from a high value the superconducting state appears at and below Bc2 and flux exclusion is not complete TYPE II Superconductivity Lecture 7

The Quantisation of Flux We are used to the concept of magnetic flux density being able to take any value at all. However we shall see that in a Type II superconductor magnetic flux is quantised. To show this, we shall continue with the concept of the superconducting wavefunction introduced by Ginzburg and Landau where p=m*v is the momentum of the superelectrons In one dimension (x) this wave function can be written in the standard form where =h/p Note that here  is the wavelength of the wavefunction not the penetration depth Lecture 7

The Quantisation of Flux If no current flows, p=0, = and the phase of the wavefunction at X, at Y and all other points is constant X Y If current flows, p is small, =h/p and there is a phase difference between X and Y so Now the supercurrent density is and this is the phase difference arising just from the flow of current Lecture 7

The Quantisation of Flux An applied magnetic field can also affect the phase difference between X and Y by affecting the momentum of the superelectrons As before, the vector must be conserved during the application of a magnetic field The additional phase difference between X and Y on applying a magnetic field is therefore And the total phase difference is Phase difference due to change of flux density Phase difference due to current Lecture 7

The Quantisation of Flux We know that in the centre of a superconductor the current density is zero XY So if we now join the ends of the path XY to form a superconducting loop the line integral of the current density around a path through the centre of XY must be zero Also we know (eg from the Bohr-Sommerfeld model of the atom) that the phase at X and Y must now be the same…... …...so an integral number of wavelengths must be sustained around the loop as the field changes Hence Lecture 7

The Quantisation of Flux The integral is simply the flux threading the loop, , XY so and So the flux threading a superconducting loop must be quantised in units of where o is known as the flux quantum We shall see later that e* 2e, in which case o =2.07x10-15 Weber This is extremely small (10-6 of the earth’s magnetic field threading a 1cm2 loop) but it is a measurable quantity Lecture 7

Quantisation of flux Although the flux quantum is extremely small it is nevertheless measurable Quantisation of flux can be shown by repeatedly measuring the magnetisation of a superconducting loop repeatedly cooled to below Tc in a magnetic fields of varying strength …..analogous to Millikan’s oil drop experiment For all known superconductors it is found that e* 2e and o =2.07x10-15 Weber magnetisation time Superconductivity is therefore a manifestation of macroscopic quantum mechanics, and is the basis of many quantum devices such as SQUIDS Lecture 7

A single vortex We have seen that in a Type II superconductor small narrow tubes of flux start to enter the bulk of the superconductor at the lower critical field Hc1 We now know that these tubes must be quantised The very first flux line that enters at Hc1 must therefore contain a single flux quantum o Therefore Also the energy per unit length associated with the creation of the flux line is so E  o2 However if a single flux line contained n flux quanta the associated energy would be E  n2o2 It is clearly energetically more favourable to create n flux lines each of one flux quanta, for which E  no2 the flux lines on the micrograph above therefore represent single flux quanta! Lecture 7

The mixed state in Type II superconductors Hc1< H <Hc2 B The bulk is diamagnetic but it is threaded with normal cores The flux within each core is generated by a vortex of supercurrent Hc2 Hc1 H -M Lecture 7

Lower density of flux lines The flux line lattice flux line curvature Hexagonal lattice Defects and disorder Lower density of flux lines Lecture 7