Lecture 3 Phenomenology:

Ginsburg-Landau Theory

•Landau Theory of Phase Transitions

•Ginsburg-Landau Expansion

•Coherence Length

•The Ginsburg-Landau Equations

•Abrikosov Lattice and Flux Pinning

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

Next we introduce the superconductor into a magnetic field B= xA

Work done on SC in bringing it into non-zero B is -∫M∙dBA

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

02

1

))((2

1

C

CC

H

HB

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

Great success

(London 1950)

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