This document summarizes key concepts in phenomenological theories of superconductivity. It discusses the Meissner effect where magnetic fields are expelled from the interior of a superconductor below its critical temperature. The London equations are introduced, which describe electromagnetic properties in superconductors. The magnetic field penetration depth is derived from the London equations. Fluxoids are also summarized, where the magnetic flux through any surface entirely within the superconductor must be quantized.

1. Lecture 2.
Phenomenological Theories
of Superconductivity

2. Superfluids and their properties
• Electrodynamics and the magnetic
penetration depth
• The London Equations and magnetic effects
• Fluxoids

3. 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

9. 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.

15. 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

16. 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
dB
Ex 
 (leave off vector signs, we’ll
ultimately solve a 1-dimensional
case)

17. dt
dB
m
qn
xE
m
qn
dt
dJ
x sss
*
2
*
2


*
2
m
Bqn
xJ s
s or
B
m
qn
or
xJxBxJxB
s
ss
*
2
0
00
................................ 


Now
B
m
qn
B
BBxBx
s
*
2
02
2
)(


and

18. 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

22. 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
1
4
)/(1)0()(

 TcTT 

29. 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)
Ej
dt
d
Bjx
s
s


)(
)(


OB
D
t
B
xE
t
D
JxH










30. Additionally,
And,
Take
Now differentiate (7) with respect to t, and use (2)
Take of each side =>
Whose solution is:









BB
B
xHx
andusex
continuity
t
J
EJJJJ SNS




)6(),4(),1(),3(
)8.......(..........
)7.......(

)exp()exp(
0.
1
21 tBtA 






31. Where and are roots of:
One can show that:
One can estimate
1 2
2
1
4
,
0
1
2
2
21
2
e

















112
2
119
1
sec10~
sec10~




21,

32. 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 00
02



Superconductor
Current density
Normal current
density
Displacement
current density

35. 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
L
SSC djdaB 0.. 