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

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

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

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

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)

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

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

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, …)

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

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

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

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

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

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

Fundamental lengths Penetration depth Coherence length Ginzburg Landau parameter

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

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

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

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

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

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

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

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

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

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