Transcript sclecture5

The Pippard coherence length
In 1953 Sir Brian Pippard considered
1.
N/S boundaries have positive
surface energy
2.
In zero magnetic field
superconducting transitions in
pure superconductors can be as
little as 10-5K wide
Thus all electrons in the sample
must participate in superconductivity
and there must be long range order
or coherence between the electrons
3.
Small particles of superconductors
have penetration depths greater
than those of bulk samples
Therefore superconducting electron
densities must change at a relatively
slow rate through the sample
Lecture 5
He concluded:
The superconducting electron
density ns cannot change rapidly
with position...
….it can only change appreciably
of a distance of ~10-4cm,
The boundary between normal and
superconducting regions therefore
cannot be sharp….
…..ns has to rise from zero at the
boundary to a maximum value
over a distance 
 is the Pippard coherence
length
Superconductivity and Superfluidity
The Pippard coherence length
The superconducting electron
density ns cannot change rapidly
with position...
The boundary between normal and
superconducting regions therefore
cannot be sharp….
…..ns has to rise from zero at the
boundary to a maximum value
over a distance 
ns
surface
….it can only change appreciably
of a distance of ~10-4cm,

superconductor
x
 is the Pippard coherence
length
Lecture 5
Superconductivity and Superfluidity
Surface energy considerations
We now have two fundamental length scales of the superconducting state:
The penetration depth, , is the length scale over
which magnetic flux can penetrate a superconductor
The coherence length, , is the length scale over
which the superelectron density can change
We also know that the superconducting region is “more ordered” than the
normal region so that
which changes on
1
2
1
gn  gs  oHc
the
length
scale
of

2
Whilst in a magnetic field the superconductor acquires a magnetisation to
cancel the internal flux density, hence deep in the material
1
which changes on
2
gs  gn  oHc2
the
length
scale
of

2
Deep inside the superconductor these free energy terms (1 and 2) cancel
exactly, but what happens closer to the surface?
Lecture 5
Superconductivity and Superfluidity
Positive and negative surface energy
For  > 
Surface energy is positive:
Type I superconductivity
Lecture 5
For  < 
Surface energy is negative:
Type II superconductivity
Superconductivity and Superfluidity
Conditions for Type II Superconductivity
If the surface energy is negative we expect
Type II superconductivity
Ns
Normal “cores” ,“flux lines” or “vortices” will
appear and arrange themselves into an
hexagonal lattice due to the repulsion of the
associated magnetic dipoles
A normal core increases the free energy per
unit length of core by an amount
1
2 2 oHc2
2
d
 = radius over which
B
superconductivity is
destroyed
…but over a length scale  the material is not
fully diamagnetic so in a field Ha there is a
local decrease in magnetic energy of
2 2 oHa2
1
So for a net reduction of energy
2 2 oHc2  2 2 oHa2
1
Lecture 5
1
2
d
 = radius of vortex
Superconductivity and Superfluidity
The Lower and Upper Critical Fields
2 2 oHc2  2 2 oHa2
1
1
0
Therefore magnetic cores or flux lines
will spontaneously form for
2Ha2

Hc
Hc2
Ha
Areas
approximately
equal
2Hc2

Ha  Hc1  Hc


  1
providing

Hc1
ie, if
Mv
where  is the Ginzburg-Landau parameter
Hc1 is known as the lower critical field
The Mixed State
(A more rigorous G-L treatment
shows  must be greater than 2
-see later lectures)
As some magnetic flux has entered the sample it has lower free energy than if it
was perfectly diamagnetic, therefore a field greater than Hc is required to drive it
fully normal
Note: for Nb, ~1
This field, Hc2, is the upper critical field.
Lecture 5
Superconductivity and Superfluidity
Ginzburg-Landau Theory
Everything we have considered so far has treated superconductivity semiclassically
However we know that superconductivity must be a deeply quantum
phenomenon
In the early 1950s Ginzburg and Landau developed a theory that put
superconductivity on a much stronger quantum footing
Their theory, which actually predicts the existence of Type II superconductivity,
is based upon the general Landau theory of “second order” or “continuous”
phase transitions
In particular they were able to incorporate the concept of a spatially dependent
superconducting electron density ns, and allowed ns to vary with external
parameters
Note: in the London theory   m onse but ns does not
depend upon distance as the Pippard model demands. The
concept of coherence length is entirely absent.
2
Lecture 5
Superconductivity and Superfluidity
Landau Theory of Phase Transitions
As a reminder of Landau theory, take the example of a ferromagnetic to
paramagnetic transition where the free energy is expressed as
F(T,M)  F(T,0)  a(T  TCM )M2  bM4  c M
2
M is the magnetisation - the so-called order parameter of the magnetised
ferromagnetic state and M  gradM is associated with variations in
magnetisation (or applied field)
F(T,M)
The stable state is found at the
minimum of the free energy, ie when
T>TCM
F(T,M)
0
M
We find M=0 for T>TCM
M0 for T<TCM
Any second order transition can be
described in the same way, replacing
M with an order parameter that goes
to zero as T approaches TC
Lecture 5
T=TCM
M
T<TCM
Superconductivity and Superfluidity
The Superconducting Order Parameter
We have already suggested that superconductivity is carried by superelectrons
of density ns
ns could thus be the “order parameter” as it goes to zero at Tc
However, Ginzburg and Landau chose a quantum mechanical approach, using a
wave function to describe the superelectrons, ie
(r )  (r ) ei(r )
This complex scalar is the Ginzburg-Landau order parameter
(i) its modulus  *  is roughly interpreted as the
number density of superelectrons at point r
(ii) The phase factor (r ) is related to the supercurrent
that flows through the material below Tc
(iii)   0 in the superconducting state, but   0
above Tc
Lecture 5
Superconductivity and Superfluidity
Free energy of a superconductor
The free energy of a superconductor in the absence
of a magnetic field and spatial variations of ns can be
 4
2
written as
Fs  Fn     
2
 and  are parameters to be determined,and
it is assumed that  is positive irrespective of
T and that  = a(T-Tc) as in Landau theory
Fs-Fn
>0

2
Assuming that ns   the equilibrium value
of the order parameter is obtained from
(Fs  Fn )
2
 0    
ns
Fs-Fn
<0
we find:
2
for  >0 minimum must be when   0

2
 
for  <0 minimum is when  

where  is defined as  in the interior of the
sample, far from any gradients in 
2
Lecture 5



Superconductivity and Superfluidity