Lecture 2.

Phenomenological Theories of Superconductivity

Superfluids and their properties

• Electrodynamics and the magnetic penetration depth

• The London Equations and magnetic effects

• Fluxoids

WHAT IS SUPERCONDUCTIVITY??

For some materials, the resistivity vanishes at some low temperature: they become superconducting.

Superconductivity is the ability of certain materials to conduct electrical current with no resistance. Thus, superconductors can carry large amounts of current with little or no loss of energy.

Type I superconductors: pure metals, have low critical field Type II superconductors: primarily of alloys or intermetallic compounds.

MEISSNER EFFECT

B

T >Tc T < Tc

B

When you place a superconductor in a magnetic field, the field is expelled below TC.

Magnet

Superconductor

Currents i appear, to cancel B. i x B on the superconductor

produces repulsion.

In a normal conductor, consider a particle of mass ‘m*’ and charge ‘q’ in motion:

v

m

qE

dt

dv

*

Normal relaxation term due to scattering

Here ‘v’ is the average velocity =

In a superconductor, there is no scattering

Now ,

nq

J

J

m

Enq

dt

dJ

*

2

*

2

m

Eqn

dt

dJ ss

dt

dBEx (leave off vector signs, we’ll

ultimately solve a 1-dimensional case)

dt

dB

m

qnxE

m

qn

dt

dJx sss

*

2

*

2

*

2

m

BqnxJ s

s or

Bm

qnor

xJxBxJxB

s

ss

*

2

0

00

................................

Now

Bm

qnB

BBxBx

s

*

2

02

2)(

and

2

1

2

0

*

qn

m

s

In 1-D, this has solution

x

eBxB

)0()(

If the dimensions of SC >> λ B=0 in the interior (Meissner Effect) If the dimensions are comparable to λ, get exponentially decreasing flux penetration.

where

2

1

*

2

01

m

qns

, a magnetic field penetration depth

We will not cover the 2-fluid model , but it can be shown* that in the 2-fluid model of a superconductor,

* from Gibb’s free energy considerations

and

4

1

c

s

T

T

n

n

2

14)/(1)0()(

TcTT

London Theory In 1935, Fritz and Heintz London postulated 2 equations:

I (1)

II (2)

These are the 2 London Equations

Additionally, we’ll write the Maxwell Equations as:

(3)

(4)

(5) (D=ϵE)

(6)

Ejdt

d

Bjx

s

s

)(

)(

OB

D

t

BxE

t

DJxH

Additionally,

And,

Take

Now differentiate (7) with respect to t, and use (2)

Take of each side =>

Whose solution is:

BBB

xHx

andusex

continuityt

J

EJJJJ SNS

)6(),4(),1(),3(

)8.......(..........

)7.......(

)exp()exp(

0.1

21 tBtA

Where and are roots of:

One can show that:

One can estimate

1 2

2

14

,

01

2

2

21

2

e

112

2

119

1

sec10~

sec10~

21,

Rate of change in SC is controlled by slower relaxation

Hence for use only

Frequencies Supercurrents

It can be shown also that

0,10

sec10~1

12

12

2

Hz

JJJj 0002

Superconductor Current density

Normal current density

Displacement current density

If path contains no hole, use Stoke’s Theorem for

Deep in SC,

and flux is excluded (part of Meissner effect)

i.e. fluxoid vanishes for any surface entirely in the SC (assuming

there is no hole).

Sj

L

S S

SSS daBdajxdj ...

0Sj

S

LSSC djdaB 0..