1. Ginzburg-Landau theory
Brian Chapler
PHYS 211A
12/10/07

2. Outline Landau theory of 2nd order phase transitions
Structural phase transition examples
Order parameter and T dependence
Physical quantities which can be measured
Application to magnetic transitions
Ginzburg Landau theory of superconductors
The rise and fall of the hallmarks of superconductivity
the universal characteristic of SC – the order parameter
Fundamental lengths
Application to type II superconductivity
Conclusions

3. Change in symmetry – order (2nd order)
Order: if number of lattice sites = number of atoms  completely ordered
Probability of finding atom in neighborhood of lattice site = 1.
consider atoms having “own” sites in completely ordered crystal – and “other” sites which can be occupied by some atoms as crystal becomes disordered
“own” and “other” sites differ only by probabilities of finding atom in question
If probabilities become equal  crystal is disordered
state of the body changes continuously
Symmetry changes discontinuously

4. 2nd order transitions
1st order – change in symmetry subject to no restrictions (symmetries of two phases can be unrelated)
2nd order – symmetry of one phase is always higher than that of the other

5. The order parameter Phase with higher symmetry – “symmetrical”
lower symmetry – “unsymmetrical”
Ex:
h = displacement
h = magnetic moment per unit volume (ferromagnet)
h = magnetic moment of sub-lattice (antiferromagnet)

6. near the transition point…
Lets consider the case where,
thus only one condition at transition point
Limiting factors:
only valid near transition point
macroscopic rather than microscopic

7. Temperature dependence
We can expand near the transition point
(a > 0)
unsymmetrical
symmetrical

8. What do we measure In a second order transition
Quantities which depend on 1st derivative of potential are continuous (U, S, V, …)
Discontinuities appear in quantities which depend on the 2nd derivatives (C, thermal expansion, compressibility, …)

9. Magnetic transitions
The beauty of the theory is how it can be applied to so many different systems
However, application might not be easy
To actually calculate something you need to figure out all these functions

10. Superconductivity Zero resistivity (Onnes 1911)
Perfect diamagnetism (1933 Meissner and Ochensfeld)
Energy gap (1950’s)

11. What is superconductivity
Perfect diamagnetism? – No. magnetic vortices
Energy gap? – No. Pseudogap
Perfect conductivity? – No. Vortex lines experience Lorentz force:
the motion of flux lines creates E-field, which creates dissipation
There can exist a mixed state – Type II superconductors

12. Universal Characteristic of Superconductivity
The existence of an order parameter having a non-zero value.
Analogous to wavefunction for superconducting electrons

13. Basics of GLT Free energy (no magnetic field)
Free energy with magnetic field

14. GL equations
Minimizing the free energy and using a two fluid model interpretation
Ginzburg-Landau equations

15. Fundamental lengths Penetration depth Coherence length
Ginzburg Landau parameter

16. Determining k
Only the difference of k from one material to another prevents the equations from scaling to a single law valid for all superconductors

17. Application of GLT Calculation of critical fields
Calculation of critical currents
Type II superconductivity
NOTE: transport properties cannot be handled by this approach
GLE can be used in microscopic theory to aid in solving coupled equations to get at transport properties

18. Critical fields in type II
Linearizing GLE
Details of this mixed state described by Abrikosov

19. Great success
(London 1950)

20. 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

22. Structural phase changes of the 2nd order
Discontinuous change in symmetry - displacement
Discontinuous change is symmetry - order

23. Change is symmetry - displacement (2nd order)
Consider BaTiO3
As temperature is lowered below a certain value, O and Ti atoms begin to move relative to Ba
As soon as this motion begins – symmetry is affected
Motion is continuous – however, arbitrarily small displacement of atoms form original symmetric positions is sufficient to change symmetry.
sudden rearrangement of lattice (discontinuous)  1st order
(continuous) displacement of atoms  2nd order
state of the body changes continuously
Symmetry changes discontinuously

24. London Equation London 1935 l Phenomenological parameter (length)
Density of superconducting electrons
Penetration depth
Great success!

25. Ginzburg Landau 1950 – points out London theory is unsatisfactory
Cannot account for destruction of superconductivity by current
Cannot determine surface tension at boundary between normal and superconducting states
Introduced order parameter, y, to describe superconductivity

26. BCS theory 1957 Bardeen, Cooper and Schrieffer
Pairing theory of superconductivity (cooper pairs)
Energy gap predicted

27. Microscopic Ginzburg Landau theory
Gor’kov 1959
Showed GL was limiting form of the microscopic theory valid near Tc
Firm basis given to existence of order parameter