Download Superconductivity and Superfluidity PHYS3430 Professor Bob

Direct currents
Zero resistance implies no voltage drop across a superconductor
Therefore irrespective of length no power is generated!
This is really true only if the current is dc
A superconductor can be considered as a mixture of two “fluids”
superelectrons
normal electrons
Temperature
At T=0 all electrons are superelectrons, for T>Tc all electrons are normal,
with superelectrons converting to normal electrons as Tc is approached
The dc current must be carried by the superelectrons, and there must be no
electric field, otherwise the superelectrons would continue to accelerate and
the current would increase
The normal electrons are effectively “shorted out”
Lecture 2
Superconductivity and Superfluidity
Alternating currents
If an ac voltage is applied across a superconductor there will be a time
varying electric field
Superelectrons, like normal electrons, have mass and hence inertia
So, the supercurrent lags the electric field and therefore produces an
inductive impedence
An inductive impedence in turn implies that there is an electric field
present, so the normal electrons also carry some current
The superconductor is therefore resistive, and appears as a perfect
inductance in parallel with a resistance
The inductive component is small (~10-12 that of normal resistance) at
100kHz and only 10-6 of the total current is carried by normal electrons
But…...
At higher (optical) frequencies (~1011Hz) the superconductor appears
entirely normal
…..to be discussed later!!
Lecture 2
Superconductivity and Superfluidity
Some definitions
In free space:
 H  dl  I
 B  dl  oI
H is “magnetic field” in A/m
B is magnetic flux density measured in Tesla
N turns/unit length
BA
I
I
Flux density in empty infinitely long solenoid, by
Ampere’s law, is |B| = o NI
(B = oH)
Flux density in solenoid containing infinitely long sample
with a magnetisation per unit volume of Mv is
B = o(H + Mv)
Lecture 2
(Mv has units of A/m)
Superconductivity and Superfluidity
Susceptibility
For most materials (except ferromagnets, and paramagnets in very high
magnetic fields and low temperatures)
Mv
Mv  H with Mv = H
paramagnet
where  is the (dimensionless) susceptibitity
so:
B = o H(1 + )
For most paramagnetic materials  ~10-3,
for diamagnets  ~ -10-5
diamagnet
H
If a superconductor always maintains
B=0 within its interior, then  = -1
Perfect diamagnet/superconductor
A superconductor can therefore be described as
(a) a “perfect diamagnet”
or
(b) having screening currents flowing at the surface producing
a field of magnitude MV equal and opposite to H
Note that B=0 but H  0 within the superconductor
Lecture 2
Superconductivity and Superfluidity
Demagnetisation
N turns/unit length, carrying current i
F
A
B
C
E
D
With a superconducting sample in the solenoid
Around ABCDEF
 H  dl  Ni
and
 H  dl  AB Hi  dl  BCDEFHe  dl
1
But with the sample removed from the solenoid
Around ABCDEF
 H  dl  Ni
Ha = field applied to sample,
H’e = external field without sample
Lecture 2
and
 H  dl  AB Ha  dl  BCDEFHe  dl
2
Hi = internal field within sample,
He= external field with sample
Superconductivity and Superfluidity
Demagnetisation
N turns/unit length
F
A
E
B X
So this term...
Equating 2 and 1
Y
...is always greater than
or equal to this term
C
D
AB Ha  dl  BCDEFHe  dl  AB Hi  dl  BCDEFHe  dl
At X, screening currents cause He to be less than He’
But at Y, the effects of the screening currents are negligible, and He = He’
Therefore and Hi  Ha
Lecture 2
and the field inside the superconductor
can exceed the applied field!
Superconductivity and Superfluidity
Demagnetisation corrections
In general we write
 to axis
Hi = Ha - HD
n
For the special case of an ellipsoid, the
field is uniform throughout the body
and
Hi = Ha - nMv
where n is the demagnetising factor
For a superconductor Mv < 0, so Hi>Ha
or
Mv = Hi = -Hi
so
Hi (1-n) = Ha
and
Ha
Hi 
(1  n)
 to axis
1.0
 to axis
0.8
nx+ ny+ nz=1
 to axis
0.6
0.5
0.4
0.2
sphere
0
0
1
This will be needed later!
Lecture 2
2
3
4
5
6
Length/diameter
ratio
Superconductivity and Superfluidity
The London Model
An important consequence of flux exclusion in superconductors is that
If magnetic flux density must remain zero in the bulk of a
superconductor, then any currents flowing through the
superconductor can flow only at the surface
However a current cannot flow entirely at the surface or the current density
would be infinite
The concept of “penetration depth” must be introduced
In 1934 F and H London proposed a macroscopic
phenomenological model of superconductivity based
upon the two-fluid model
The London model introduced the concept of the
(London) penetration depth and described the
Meissner effect by considering superconducting
electrodynamics
Lecture 2
Superconductivity and Superfluidity
Some electrodynamics
Consider a perfect conductor in which the current is carried by n electrons
current density
J  nev
1
and in an electric field
mv  eE
2
so the rate of increase of current density is
Maxwell’s equations give
J  ne2 E m

curlH  J  D
4
  curlE
B
  0 and (1  )  1
Assuming the displacement current D
and
3
5
curlB  o J
6
Then 5 and 3 give
   m curl J ne2
B
7
and 6 and 7 give
m

B
curl curlB
2
one
8
equation 4 gives
Lecture 2
Superconductivity and Superfluidity
Some more electrodynamics
m
B  
curl curlB can be simplified using the standard identity
2
one
(div B  0, div B  0)
curl curlB  grad div B  2 B
So equation 8 becomes
B

 B

2
with  
m
one2
In one dimension this is simply
 2 B B

2

x
To which the solution is
B A
B A exp  x

B ( x )  B A exp  x  
So, for x   ~10-6cm, ie inside a
perfect conductor B does not change
( B ( x )  0 ) when BA changes
x
B decreases exponentially when we move into the material
Lecture 2
Superconductivity and Superfluidity
The London penetration depth
  0 but also B  0 within a superconductor
Experiment had shown that not only B
B

F and H London suggested that not only  B 

B
2
but also  B 

2
To which the solution is
B( x )  BA exp  x L 
where L    m onse2
L is
BA
BA exp  x  
known as the London penetration
depth
x
It is a fundamental length scale of the
superconducting state
Lecture 2
Superconductivity and Superfluidity
Surface currents
Working backwards from the London equation  2 B 
gives
B   m curl Js ns e2
B

to equation 7
So, for a uniform field parallel to the surface
(z-direction) the “new” equation 6 becomes
B
  o Jy
x
B
B
and as
  A exp(  x L )
x
L
B
Jy   A exp(  x L )
 o L

or
Jy  JA exp  x L 
BA
Jy  JA exp  x L 

x
So current flows not just at the surface, but
within a penetration depth L
Lecture 2
Superconductivity and Superfluidity