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# Lecture 3 gl theory

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### Lecture 3 gl theory

1. 1. 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
2. 2. 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 
3. 3. Next we introduce the superconductor into a magnetic field B= xA
4. 4. Work done on SC in bringing it into non-zero B is -∫M∙dBA
5. 5. 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 
6. 6. 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
7. 7. Great success (London 1950)
8. 8. 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