lecture 2 phenomenology
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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..