1. 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

2. 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

3. 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

4. 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

5. 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

6. 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

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

8. 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

9. 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

10. 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

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