4. Landau Theory of Phase
Transitions
Let Ψ be a complex order parameter.
Consider a normal phase n, and an ordered phase A. Choose
the density of ordered particles to be
For a superconducting system, the ordered phase is the
superconducting phase s, and the density of superconducting
particles (the density of normal particles is n*).
We expand the Gibbs Free Energy G about the order
parameter Ψ:
(we omit odd powers since is real as is G)
2
Sn
...
2
1 42
nA GG
2
9. Work done on SC in bringing it into non-zero B is
-∫M∙dBA
10. For the ordered state of a type I superconductor we can evaluate
the inside magnetic field. The magnetization
M is given by (SI), and B= inside field
= applied field
Consider a Type I SC again:
At and
=> energy/vol. required to
suppress SC is:
area =
0,0
Applied magnetic field
MBB a 0
aB
M0
CH
0, BHC CBM 0
2
0
2
1
))((
2
1
C
CC
H
HB
11.
12. This intuition is clearer if one considers that the
gradient term is just the kinetic energy term in the
presence of a magnetic field
½ m l(-ih/(2π) -q*A) ψ(r)l
2
40. Conclusions
• Theory of second order transitions and expansion
in terms of order parameter is powerful tool for
many different applications
– Limited to regions close to transition
– Macroscopic physics – no microscopic
• GLT makes key predictions capturing fundamental
physics of superconductivity – especially type II
(Hc2)
– Same limitations as 2nd order phase transitions
– Cannot predict transport properties
