Superconductivity and Superfluidity Type I and Type II superconductivity Using the first Ginzburg Landau equation, and limiting the analysis to first order

Superconductivity and Superfluidity

Type I and Type II superconductivityType 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

2A*ei*m2

1

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

c21

oE

c is the cyclotron frequency*m*e

Bc

so*m2B*e

Eo

Recognising that we have a non zero solution for , hence superconductivity when B<Bc2, we obtain from the Schroedinger equation:

*m2

B*e 2c

Lecture 7


Type I and Type II superconductivityType I and Type II superconductivity

*m2

B*e 2cWe have and, from earlier 2

2

*m2

Combining these two equations, and recognising the temperature dependence of Bc2 and explicitly, gives

)T(*e)T(B

22c

However we have also just shown that the thermodynamic critical flux density, Bc, is given by

*e21

)T(*)T()T(Bc

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

So, we obtain )T(B2)T(B c2c where is theGinzburg-Landau parameter

)T()T(

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 I Superconductivity

TYPE II Superconductivity

If Bc2=3 Tesla

then = 10nm

Lecture 7


The Quantisation of FluxThe 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

)r(ie)r()r( )r.p(ie)r(

where p=m*v is the momentum of the superelectrons

In one dimension (x) this wave function can be written in the standard form

vtx

2sinp

where =h/p

Note that here is the wavelength of the wavefunction not the penetration depth

Lecture 7



vtx

2sinp

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 dl.

x̂2

Y

X

YXXY

Now the supercurrent density is *ss nv*eJ so

s

*s

J*mn*he

and dl.J*ehn

*m2Y

X

S*s

XY

this is the phase difference arising just from the flow of current

Lecture 7



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

A*ev*mp

The additional phase difference between X and Y on applying a magnetic field is therefore

dl.Ah

*e2Y

X

And the total phase difference is

dl.Ah

*e2dl.J

*ehn

*m2Y

X

Y

X

S*s

XY

Phase difference due to current

Phase difference due to change of flux density

Lecture 7



We know that in the centre of a superconductor the current density is zero

dl.Ah

*e2dl.J

*ehn

*m2Y

X

Y

X

S*s

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

dl.Ah

*e22N

Y

X

XY Hence

XY

Lecture 7



XYdl.A

Y

X

The integral

is simply the flux threading the loop, ,

and *e

hN

So the flux threading a superconducting loop must be quantised in units of

*eh

o

oN

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

2N

h*e2

dl.Ah

*e2Y

X

so

Lecture 7


mag

netis

atio

n

time

Quantisation of fluxQuantisation of flux

*eh

o 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

Superconductivity is therefore a manifestation of macroscopic quantum mechanics, and is the basis of many quantum devices such as SQUIDS

Lecture 7


A single vortexA 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

2o

o1cH

Also the energy per unit length associated with the creation of the flux line is so E o

2

2o2

11c

H

However if a single flux line contained n flux quanta the associated energy wouldbe E n2o

2

It is clearly energetically more favourable to create n flux lines each of one flux quanta, for which E no

2

the flux lines on the micrograph above therefore represent single flux quanta!

Lecture 7


The mixed state in Type II superconductorsThe mixed state in Type II superconductors

B Hc1< H <Hc2

The bulk is diamagnetic but it is threaded with normal cores

The flux within each core is generated by a vortex of supercurrent

Hc2Hc1H

0

-M

Lecture 7


The flux line latticeThe flux line lattice

Lower density of flux lines

Lower density of flux lines

Defects and disorder

Defects and disorder

Hexagonal lattice

Hexagonal lattice

flux linecurvatureflux line

curvature

Lecture 7

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Superconductivity and Superfluidity Type I and Type II superconductivity Using the first Ginzburg Landau equation, and limiting the analysis to first order