Transcript Document 7462971

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
Free energy of a superconductor
In the superconducting state we have
 4
2
Fs  Fn     
2
with
2
 

2
 

Fs-Fn
<0
 changes sign at Tc and  is always
positive for a second order transition
also at equilibrium
 2
Fs  Fn 
2



1
2

H
o
c
2
But we have already shown that
1
Fs  Fn   oHc2
2
2
2
so oHc 
We will use this later

Lecture 6
Superconductivity and Superfluidity
The full G-L free energy
If we now take the full expression for the Ginzburg-Landau free energy at a
point r in the presence of magnetic fields and spatial gradients we have:
 4
Fs  Fn     
2
1
 oH2 (r )
2
2

1
 i  e * A  2
2m *
the term we have already discussed
the magnetic energy associated
with the magnetisation in a local
field H(r)
A kinetic energy term associated
with the fact that  is not uniform
in space, but has a gradient
e* and m* are the charge and mass of the superelectrons and A is the
vector potential
We should look at the origin of the kinetic energy term in more detail.
Lecture 6
Superconductivity and Superfluidity
A charged particle in a field
Consider a particle of charge e* and mass m* moving in a field free region with
velocity v1 when a magnetic field is switched on at time t=0
The field can only increase at a finite rate, and while it builds up there is an
induced electric field which satisfies Maxwell’s equations, ie
curlE  B
If A is the magnetic vector potential (B=curl A) then
d
(curlA )
dt
Integration with respect to spatial coordinates gives
curlE  
E
dA
dt
So the momentum at time t is

t
m * v 2  m * v1  e * E dt  m * v1  e *
0
and m * v 2  m * v1  e * A
Lecture 6
or

t
dA dt
0 dt
m * v1  m * v 2  e * A
Superconductivity and Superfluidity
A charged particle in a field
If
m * v 2  m * v1  e * A
and m * v1  m * v 2  e * A
the vector p  m * v  e * A must be conserved during the
application of a magnetic field
The kinetic energy, , depends only upon m*v so if  = f(m*v) before
the field is applied we must write  = f(p-e*A) after the field is applied
Quantum mechanically we can replace p by the momentum
operator -iħ
So the final energy in the presence of a field is:
1
 i  e * A  2
2m *
Lecture 6
Superconductivity and Superfluidity
Back to G-L Free Energy - 1st GL Equation
Remember that the total free energy is
2
Fs  Fn    
 4
1
1
 i  e * A  2
  oH2 (r ) 
2
2
2m *
This free energy, Fs((r), A(r)), must now be minimised with respect to the
order parameter, (r) , and also with respect to the vector potential A(r)
To do this we must use the Euler-Lagrange equations:
1
F



j

F
0
x j ( j )
2
1 Is easy to evaluate - we only need
2
     
Lecture 6
F

A

j

F
0
x j (A j x j )
F
 0 ie

1
i  e * A 2   0
2m *
This is the First
G-L equation
Superconductivity and Superfluidity
The second G-L equation
Evaluation of the second derivative in 2
F

A

j

F
0
x j (A j x j )
1
curl curl A
gives
o
Remember that B=curl A, and that curl B = oJ
Therefore 2
where J is the current density
gives
J


e* 2
 (i  e * A )
m*
This is the Second
G-L equation
This is the same quantum mechanical expression for a current of particles
described by a wavefunction 
Lecture 6
Superconductivity and Superfluidity
Magnetic penetration within G-L Theory


e* 2
Taking the second GL equation: J  
 (i  e * A )
m*
spatial variations of :
and neglecting
e *2
2
J
A 
m*
So, if curl B = oJ and using ns  
2
o e * 2 n * s
curl(o J)  curlcurlB  
curlA
m*
and as curlcurlB  grad divB  2B  2B
This gives
and finally
with
Lecture 6
o e * 2 n * s
 B  
B
m*
2
B   *2 2B  0
m*
 *2 
oe * 2 n * s
Compare these equations
directly with the London
equations
Superconductivity and Superfluidity
A comparison of GL and London theory
We will now pre-empt a result we shall derive later in the course
and recognise that superconductivity is related to the pairing of
electrons.
(This was not known at the time of Ginzburg and Landau’s theory)
If electrons are paired in the superconducting state then:
m* = 2me
e* = 2e
n*s = ns/2
and hence
Lecture 6
 *2 
2m
m
2



L
o 4e2ns / 2 oe2ns
Superconductivity and Superfluidity
The coherence length
We shall now look at how the concept of the coherence length arises in the G-L
Theory
Taking the 1st G-L equation in 1d without a magnetic field, ie:
1
2
i  e * A 2   0
     
2m *
 2 d2
2
Eq 1
     
0
becomes
2
2m * dx
Earlier we showed that the square of the order parameter can be written
2
 


with <0
However we believe that the order parameter can vary slowly with distance, so
we shall now change variables and use instead a normalised order parameter

f

1
2
where f varies with distance
Lecture 6
Superconductivity and Superfluidity
The coherence length
Substituting the normalised order parameter f into equation 1 on the previous
slide, and noting that  2     , we obtain a “non linear Schrodinger
equation”
1
1
1
2
2
2
2
2

  df
3 
“-” disappears as <0

 f
 f
0
2
2m *  dx

 
and we introduce | |
 2 d2f
3



f


f
0
2
2m * dx

 2 d2f
3
hence

f

f
0
Canceling gives
2
2m *  dx
2
2
Making the substitutions f=1+ f´ where f´ is small and negative, and  
2m * 
2
d f'
we have
2 2  1  f '(1  3f '.....)  0
dx
d2 f ' 2f '
 2
hence
2
dx

the solution of which is
f ' ( x )  exp(  x 2 )
 is therefore the coherence length, characteristic distance over which the order
parameter  varies
Lecture 6
Superconductivity and Superfluidity
Relationship between Bc, * and 
To summarise we have
2

2m * 2
m*

  ns 



o e * 2  * 2
2
1
Solving for  and 
Finally, using
1
o2Hc2e *2  *2

m*
from
3oHc2e * 4  * 4

m *2
by substituting in 2
2 and
2


3
3
we have
 H 
o c
2
1  
  *     cons tan t
2e*
2 2
or
2 and
oHc2
2
Bc22 *2  cons tan t
So, although Bc, * and  are all temperature dependent, their product is not
although experimentally it is found to be not quite independent of T
Lecture 6
Superconductivity and Superfluidity