Ginzburg Landau phenomenological Theory

 
Ginzburg Landau phenomenological Theory
 
The superconducting state and the normal metallic
state are separate thermodynamic phases of matter
in just the same way as gas, liquid and solid are
different phases
.
Similarly, the normal Bose gas and BEC, or normal
liquid He
4
 and super-fluid He II are separated by a
thermodynamic phase transitions
.
Each such phase transition can be characterized by
the nature of the singularities in specific heat and
other thermodynamic variables at the transition Tc.
The problems of superfluidity and superconductivity
were examined from the point of view of the
thermodynamics of phase transitions.
 
Ginzburg Landau phenomenological Theory
 
The theory of superconductivity introduced by
Ginzburg and Landau in 1950 describes the
superconducting phase transition from this
thermodynamic point of view.
It was originally introduced as a
phenomenological theory, but later Gor’kov
showed that it can be derived from full the mi-
croscopic BCS theory in a suitable limit.
Condensation Energy
 
Ginzburg Landau phenomenological Theory
 
The quantity 
μ
0
H
2
c
 
/2 is the condensation energy in SI unit.
It is a measure of the gain in free energy per unit volume in
the superconducting state compared to the normal state at
the same temperature
.
As an example lets consider niobium. Here T
c
 = 9K, and H
c 
=
160kA/m (B
c
 = 
μ
0
Hc = 0.2T). The condensation energy 
μ
0
H
2
c
 
/2
= 16.5kJ/m3.
Given that Nb has a bcc crystal structure with a 0.33nm
lattice constant
we can work out the volume per atom and find that the
condensation energy is only around 2μeV/atom!
Such tiny energies were a mystery until the BCS theory, which
shows that the condensation energy is of order (k
B
T
c
)2g(E
F
 ),
where g(E
F
 ) is the density of states at the Fermi level.
The energy is so small because k
B
T
c
 is many orders of
magnitude smaller than the Fermi energy, 
ε
F
 
.
 
Ginzburg Landau phenomenological Theory
 
The Ginzburg-Landau theory of superconductivity is built upon a
general approach to the theory of second order phase
transitions which Landau had developed in the 1930’s.
Landau had noticed that typically second order phase
transitions, such as the Curie temperature in a ferromagnet,
involve some change in symmetry of the system. For example a
magnet above the Curie temperature, T
C
, has no magnetic
moment.
But below T
C
 a spontaneous magnetic moment develops. In
principle could point in any one of a number of different
directions, each with an equal energy, but the system
spontaneously chooses one particular direction
.
In Landau’s theory such phase transitions are characterized by
an order parameter which is zero in the disordered state above
Tc, but becomes non-zero below T
C
.
In the case of a magnet the magnetization, M(r), is a suitable
order parameter.
 
Ginzburg Landau phenomenological Theory
 
For superconductivity Ginzburg and Landau (GL)
postulated the existence of an order parameter
denoted by 
ψ
 . This characterizes the supercon-
ducting state, in the same way as the
magnetization does in a ferromagnet.
The order parameter is assumed to be some
(unspecified) physical quantity which
characterizes the state of the system.
In the normal metallic state above the critical
temperature Tc of the superconductor it is zero.
While in the superconducting state below Tc it is
non-zero.
 
Ginzburg Landau phenomenological Theory
 
Therefore it is assumed to obey:
 
 
 
Ginzburg and Landau postulated that the order parameter   should
be a complex number, thinking of it as a macroscopic wave
function for the superconductor in analogy with superfluid 
4
He.
At the time of their original work the physical significance of this
complex in superconductors was not at all clear.
But, in the microscopic BCS theory of superconductivity there
appears a parameter, 
Δ
, which is also complex.
Gor’kov was able to derive the Ginzburg-Landau theory from BCS
theory, and show that 
ψ 
is essentially the same as 
Δ
 , except for
some constant.
numerical factors. In fact, we can even identify |
ψ
|
2
 as the density
of BCS “Cooper pairs” present in the sample.
 
Ginzburg Landau phenomenological Theory
 
Ginzburg and Landau assumed that the free
energy of the superconductor must depend
smoothly on the parameter 
ψ
 .
Since  
ψ
 is complex and the free energy must be
real, the free energy can only depend on |
ψ
|.
Furthermore, 
ψ
 since   goes to zero at the critical
temperature, T
C
, we can Tailor expand the free
energy in powers of |
ψ
|. For temperatures close
to T
C
 only the first two terms in the expansion
should be necessary, and so the free energy
density (F) must be of the form:
 
Ginzburg Landau phenomenological Theory
 
 
Here fs(T) and fn(T) are the superconducting state and
 
 
Clearly  above Eq. is the only possible function which is
real for any complex 
ψ
 
near 
ψ
 
 = 0 and which is a
differentiable function of 
ψ 
and 
ψ
*
  near to 
ψ
 = 0.
The parameters 
(T) and 
(T) are, in general,
temperature dependent pheonomenological parameters
of the theory.
However it is assumed that they must be smooth
functions of temperature. We must also assume that 
(T)
is positive, since otherwise the free energy density would
have no minimum, which would be unphysical (or we
would have to extend the expansion to include higher
powers such as |
ψ
|
6
).
 
Ginzburg Landau phenomenological Theory
 
Plotting F
s
 − F
n
 as a function of  
ψ
 is easy to see that there are two
possible curves, depending on the sign of the parameter 
(T), as
shown in Fig.   In the case 
(T) > 0, the curve has one minimum at
ψ
 = 0.  On the other hand, for 
(T) < 0 there are minima wherever
|
 ψ
 |
2
 = −
(T)/2
(T).
Landau and Ginzburg assumed that at high temperatures, above
T
C
, we have 
(T) positive, and hence the minimum free energy
solution is one with 
ψ 
= 0, i.e. the normal state.
But if 
(T) gradually decreases as the temperature T is reduced,
then the state of the system will change suddenly when we reach
the point 
(T) = 0. Below this temperature the minimum free
energy solution changes to one with 
ψ ≠
 0. Therefore we can
identify temperature ,where , 
(T) becomes zero as the critical
Temperature  T
C
.
 
Ginzburg Landau phenomenological Theory
 
Near to this critical temperature, Tc, assuming
that the coefficients 
(T) and 
(T) change
smoothly with temperature, we can make a Taylor
expansion,

(T) ≈ 
’ × (T − Tc) + . . .
              
(T) ≈  
 + . . . ,
 
Ginzburg Landau phenomenological Theory
 
 
 
 
 
                                                                                   (1)
 
F is now the total free energy. Eq. (1) in its present form does not
model the increase in energy  associated with a spatial distortion of
the order parameter, i.e., effects associated with a coherence  length,
ξ
.  To account for such effects Ginzburg and Landau added a 'gradient
energy' term to (1)  0f the form
 
Ginzburg Landau phenomenological Theory
 
 
 
Ginzburg Landau phenomenological Theory
 
 
 
 
 
Hence
 
 
  
Finally we must add the contribution of the
magnetic field to the energy density
 
Ginzburg Landau phenomenological Theory
 
 
 
 
Combining the above
 
 
 
                                                                             (2)
 
                                                                                
(2)
 
variation with respect to 
ψ
, which is an independent
Variable, yields the complex conjugate of (2). To minimize ,
we set the integrand of the of above  to zero; this yields the
first  G-L equation.
 
 
Ginzburg Landau Equation
 
 
 
 
Variation of Eq. (2) w.r.t    
A 
where
 
 
 
 
Provided we identify 
J(r) 
as
 
 
 
 
Or equivalently
 
 
 
 
This is the 2
nd
 Ginzburg landau equation which is
same as the current density in quantum Mechanics
 
Boundaries and boundary conditions
 
We first examine a simple case involving an
inhomogeneous order parameter generated
by the  presence of a boundary, in the absence
of a magnetic field.
Assume we have a superconducting half space
occupying the region x > 0.
We further assume that the order parameter is
driven to zero at this interface.
Experimentally this can be accomplished by
coating the surface of the superconductor with
a film of ferromagnetic material.
 
continue
 
G L equation
 
 
 
 
In one dimension becomes
 
 
 
 
continue
 
 is negative in the superconducting state
 
 
 
Substituting
 
The above diff. equation  becomes
 
continue
 
Multiplying by 
ƒ
’ the diff. Eqn. can be written as
 
 
 
 
This means that the quantity in bracket is constant.
 
                                                          =  C
 
continue
 
 Far from the boundary the 
ψ
 
and hence
 f  
is
constant.  This employs that 
f’=0 and
 
 
Then the above diff. equation becomes
 
 
Which has the solution
 
 
continue
 
 
 
or
 
Thus we see that the 
ξ
 is the measure of
distance  over which it responds to
perturbation.
Since
 
continue
 
  we have
 
 
 
 We see that the G L coherence length diverges as
 
this divergence is a general property of the coherence
length at all second order phase transitions (although
the exponent differs in general from this 'mean field'
value of 1/2 close to T
C
).
 
 
continue
 
  we have
 
 
 
 We see that the G L coherence length diverges as
 
this divergence is a general property of the coherence
length at all second order phase transitions (although
the exponent differs in general from this 'mean field'
value of 1/2 close to T
C
).
 
 
London equation from GL equation
 
2
nd
 GL equation
 
 
In the limit when the two first term is negligible
then the equation is nothing but London equation
 
        J(r)   
=
 
Provided
 
London equation from GL equation
 
 
Or,
 
Thus we see that in G-L theory 
L
 and   
ξ
 
both
diverges as
The ratio, called G-L parameter
 
 
is independent of temperature and constant.
 
 
Surface energy at phase boundary
 
At phase boundary
By definition
 
 
 
 
Or,
 
 
London equation from GL equation
 
Inserting for  F in G
 
 
 
And
  and using definition of
 
                 G
 
Surface energy at phase boundary
 
We obtain for   ,
 
The vanishing of the cross term          in both in
above equation  and the first G-L equation
allows us to  choose 
ψ
 real; it then follows
from 2
nd
 G-L  that
 
Surface energy at phase boundary continue
 
And
 
To compute 

we must simultaneously
solve the first and second G-L equations,
for  
(x)
 and  
B(x) = -(dA(x)/dx)), 
subject to
boundary  conditions  and substitute in the
equation for 

 
Surface energy at phase boundary
 
To simplify the calculation we rewrite in scaled var-
iable
 
 
The GL eqn becomes
 
 
and
 
Surface energy at phase boundary
 
The solution for arbitrary 

must be obtained
numerically. The appropriate boundary condition
 
And
 
 
To obtain the first integral of GL 1
st
 , multiply by 
 
Surface energy at phase boundary
 
Multiplying 2
nd
 by A’
 
 
 
Combining the above two eqn.
 
 
The value of constant was fixed by boundary
condition
 
Surface energy at phase boundary
 
 
Using                                             and scaled variables
 
 
 
Combining this eqn. with the earlier eqn
 
Surface energy at phase boundary
 
In general the soln of
 
 
 
Can be computed if  A’(x) and y’(x) are given.
For  A=0
 
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The Ginzburg-Landau phenomenological theory explains superconductivity and superfluidity as distinct thermodynamic phases. It focuses on phase transitions characterized by singularities in specific heat at the transition temperature. Derived from BCS theory, it quantifies condensation energy, emphasizing the small energy scale relative to Fermi energy. Landau's general approach to second-order phase transitions underlies this theory, emphasizing order parameters that distinguish between states.


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  1. Ginzburg Landau phenomenological Theory The superconducting state and the normal metallic state are separate thermodynamic phases of matter in just the same way as gas, liquid and solid are different phases. Similarly, the normal Bose gas and BEC, or normal liquid He4and super-fluid He II are separated by a thermodynamic phase transitions. Each such phase transition can be characterized by the nature of the singularities in specific heat and other thermodynamic variables at the transition Tc. The problems of superfluidity and superconductivity were examined from the point of view of the thermodynamics of phase transitions.

  2. Ginzburg Landau phenomenological Theory The theory of superconductivity introduced by Ginzburg and Landau in 1950 describes the superconducting phase thermodynamic point of view. It was originally phenomenological theory, but later Gor kov showed that it can be derived from full the mi- croscopic BCS theory in a suitable limit. Condensation Energy transition from this introduced as a

  3. Ginzburg Landau phenomenological Theory The quantity 0H2c/2 is the condensation energy in SI unit. It is a measure of the gain in free energy per unit volume in the superconducting state compared to the normal state at the same temperature. As an example lets consider niobium. Here Tc= 9K, and Hc= 160kA/m (Bc= 0Hc = 0.2T). The condensation energy 0H2c/2 = 16.5kJ/m3. Given that Nb has a bcc crystal structure with a 0.33nm lattice constant we can work out the volume per atom and find that the condensation energy is only around 2 eV/atom! Such tiny energies were a mystery until the BCS theory, which shows that the condensation energy is of order (kBTc)2g(EF), where g(EF) is the density of states at the Fermi level. The energy is so small because kBTcis many orders of magnitude smaller than the Fermi energy, F.

  4. Ginzburg Landau phenomenological Theory The Ginzburg-Landau theory of superconductivity is built upon a general approach to the theory of second order phase transitions which Landau had developed in the 1930 s. Landau had noticed that typically second order phase transitions, such as the Curie temperature in a ferromagnet, involve some change in symmetry of the system. For example a magnet above the Curie temperature, TC, has no magnetic moment. But below TCa spontaneous magnetic moment develops. In principle could point in any one of a number of different directions, each with an equal energy, but the system spontaneously chooses one particular direction. In Landau s theory such phase transitions are characterized by an order parameter which is zero in the disordered state above Tc, but becomes non-zero below TC. In the case of a magnet the magnetization, M(r), is a suitable order parameter.

  5. Ginzburg Landau phenomenological Theory For superconductivity Ginzburg and Landau (GL) postulated the existence of an order parameter denoted by . This characterizes the supercon- ducting state, in the magnetization does in a ferromagnet. The order parameter is assumed to be some (unspecified) physical characterizes the state of the system. In the normal metallic state above the critical temperature Tc of the superconductor it is zero. While in the superconducting state below Tc it is non-zero. same way as the quantity which

  6. Ginzburg Landau phenomenological Theory Therefore it is assumed to obey: Ginzburg and Landau postulated that the order parameter should be a complex number, thinking of it as a macroscopic wave function for the superconductor in analogy with superfluid4He. At the time of their original work the physical significance of this complex in superconductors was not at all clear. But, in the microscopic BCS theory of superconductivity there appears a parameter, , which is also complex. Gor kov was able to derive the Ginzburg-Landau theory from BCS theory, and show that is essentially the same as , except for some constant. numerical factors. In fact, we can even identify | |2as the density

  7. Ginzburg Landau phenomenological Theory Ginzburg and Landau assumed that the free energy of the superconductor must depend smoothly on the parameter . Since is complex and the free energy must be real, the free energy can only depend on | |. Furthermore, since goes to zero at the critical temperature, TC, we can Tailor expand the free energy in powers of | |. For temperatures close to TConly the first two terms in the expansion should be necessary, and so the free energy density (F) must be of the form:

  8. Ginzburg Landau phenomenological Theory Here fs(T) and fn(T) are the superconducting state and Clearly above Eq. is the only possible function which is real for any complex near differentiable function of and *near to = 0. The parameters a(T) and temperature dependent pheonomenological parameters of the theory. However it is assumed that they must be smooth functions of temperature. We must also assume that b(T) is positive, since otherwise the free energy density would have no minimum, which would be unphysical (or we would have to extend the expansion to include higher powers such as | |6). = 0 and which is a b(T) are, in general,

  9. Ginzburg Landau phenomenological Theory Plotting Fs Fnas a function of is easy to see that there are two possible curves, depending on the sign of the parameter a(T), as shown in Fig. In the case a(T) > 0, the curve has one minimum at = 0. On the other hand, for a(T) < 0 there are minima wherever | |2= a(T)/2b(T). Landau and Ginzburg assumed that at high temperatures, above TC, we have a(T) positive, and hence the minimum free energy solution is one with = 0, i.e. the normal state. But if a(T) gradually decreases as the temperature T is reduced, then the state of the system will change suddenly when we reach the point a(T) = 0. Below this temperature the minimum free energy solution changes to one with 0. Therefore we can identify temperature ,where , a(T) becomes zero as the critical Temperature TC.

  10. Ginzburg Landau phenomenological Theory Near to this critical temperature, Tc, assuming that the coefficients a(T) and b(T) change smoothly with temperature, we can make a Taylor expansion, a(T) a (T Tc) + . . . b(T) b + . . . ,

  11. Ginzburg Landau phenomenological Theory (1) F is now the total free energy. Eq. (1) in its present form does not model the increase in energy associated with a spatial distortion of the order parameter, i.e., effects associated with a coherence length, . To account for such effects Ginzburg and Landau added a 'gradient energy' term to (1) 0f the form

  12. Ginzburg Landau phenomenological Theory

  13. Ginzburg Landau phenomenological Theory Hence Finally we must add the contribution of the magnetic field to the energy density

  14. Ginzburg Landau phenomenological Theory Combining the above (2) (2)

  15. variation with respect to , which is an independent Variable, yields the complex conjugate of (2). To minimize , we set the integrand of the of above to zero; this yields the first G-L equation.

  16. Ginzburg Landau Equation Variation of Eq. (2) w.r.t A where

  17. Provided we identify J(r) as Or equivalently This is the 2nd Ginzburg landau equation which is same as the current density in quantum Mechanics

  18. Boundaries and boundary conditions We first examine a simple case involving an inhomogeneous order parameter generated by the presence of a boundary, in the absence of a magnetic field. Assume we have a superconducting half space occupying the region x > 0. We further assume that the order parameter is driven to zero at this interface. Experimentally this can be accomplished by coating the surface of the superconductor with a film of ferromagnetic material.

  19. continue G L equation In one dimension becomes

  20. continue a is negative in the superconducting state Substituting The above diff. equation becomes

  21. continue Multiplying by the diff. Eqn. can be written as This means that the quantity in bracket is constant. = C

  22. continue Far from the boundary the and hence f is constant. This employs that f =0 and Then the above diff. equation becomes Which has the solution

  23. continue or Thus we see that the is the measure of distance over which it responds to perturbation. Since

  24. continue we have We see that the G L coherence length diverges as this divergence is a general property of the coherence length at all second order phase transitions (although the exponent differs in general from this 'mean field' value of 1/2 close to TC).

  25. continue we have We see that the G L coherence length diverges as this divergence is a general property of the coherence length at all second order phase transitions (although the exponent differs in general from this 'mean field' value of 1/2 close to TC).

  26. London equation from GL equation 2nd GL equation In the limit when the two first term is negligible then the equation is nothing but London equation J(r) = Provided

  27. London equation from GL equation Or, Thus we see that in G-L theory L and both diverges as The ratio, called G-L parameter is independent of temperature and constant.

  28. Surface energy at phase boundary At phase boundary By definition Or,

  29. London equation from GL equation Inserting for F in G And and using definition of G

  30. Surface energy at phase boundary We obtain for , The vanishing of the cross term in both in above equation and the first G-L equation allows us to choose real; it then follows from 2nd G-L that

  31. Surface energy at phase boundary continue And To compute g we must simultaneously solve the first and second G-L equations, for y(x) and B(x) = -(dA(x)/dx)), subject to boundary conditions and substitute in the equation for g.

  32. Surface energy at phase boundary To simplify the calculation we rewrite in scaled var- iable The GL eqn becomes and

  33. Surface energy at phase boundary The solution for arbitrary k must be obtained numerically. The appropriate boundary condition And To obtain the first integral of GL 1st , multiply by y

  34. Surface energy at phase boundary Multiplying 2nd by A Combining the above two eqn. The value of constant was fixed by boundary condition

  35. Surface energy at phase boundary Using and scaled variables Combining this eqn. with the earlier eqn

  36. Surface energy at phase boundary In general the soln of Can be computed if A (x) and y (x) are given. For A=0

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