1. Ginzburg-Landau-Abrikosov Theory of Type II Superconductors - Phase diagram of vortex matter 李定平 北京大学物理学院理论物理研究所 , 台湾,台北
在Hc2附近, 可假設 B是constant magnetic field.因為在此區域很多vortices overlap,1/k^2<<1,所以B=H

2. Main Collaborator: Prof
Main Collaborator: Prof. Rosenstein, National Chiao Tung University, Hsinchu, Taiwan
Dingping Li and Baruch Rosenstein, Phys. Rev. Lett. 86, (2001).
Dingping Li and Baruch Rosenstein, Phys. Rev. B 65, (2002).
Dingping Li, Rosenstein, Phys.Rev.Lett . 90, (2003).
Dingping Li, Rosenstein, Phys. Rev. B 65, (2002), Rapid Communication
Dingping Li,Rosenstein,Phy. Rev. B70, (2004)
Dingping Li, Rosenstein, Vinokur, Journal of Superconductivity: Incorporating Novel Magnetism, vol 19, 369 (2006).
H. Beibdenkopf, Myasoedov, Shtrikman, Zeldov, Rosenstein, Dingping Li, Tamegai, PRL 98, (2007)
Tinh, Li, Rosenstein, PRB81,224521(2010) 2

3. By Rosenstein and Dingping Li,
Review of Modern Physics, “The Ginzburg-Landau Theory of Type II superconductors in magnetic field”,
By Rosenstein and Dingping Li,
VOLUME 82, JANUARY–MARCH 2010, Page 109. 3

4. Properties of a superconductor:
1. Zero resistivity Perfect diamagnetism H
Kamerlingh Onnes, (Mercury, 1911) H Hc
Normal
Meissner
Persistent current Tc T
Meissner and Ochsenfeld, (1933)

5. Magnetic (Abrikosov) vortices in a type II superconductorH x l
Most new materials
Including high Tc are
Type II superconductors,
Most part of phase diagram
Is the mixed state.
Normal
Mixed
Meissner H
Hc2
Hc1 Tc T
The perfect diamagnetism is lost

6. STM (scanning tunneling microscope)Topographic Image of vortex lattice of a superconductor NbSe2
Temperature: 400mK, Field: 0.5T
Magnetic Materials Laboratory, RIKEN, Dr. Hanaguri

7. Vortex line repel each other forming highly ordered structures like flux line lattice ( STM and neutron scattering)
Pan et al (2002)
Normal
Abrikosov
lattice
Meissner H
Hc2
Hc1 Tc T
Park et al (2000)

8. Troyanovsky et al (04) Flux flow
Fluxons can move. Electric current “induces” the flux flow if there
is no pinning force, and the flux flow will cause voltage. B J F
zero resistivity, is also lost in magnetic field above Hc1 if there is no pinning
Field driven flux motion probed by STM on NbSe2
Troyanovsky et al (04)

9. Intrinsic random disorder Artificial
Pinning of vortices
Intrinsic random disorder
Artificial
STM of both the pinning centers (top) and the vortices (bottom)
Pan et al,
PRL 85, 1536 (2000)
Schuler et al, PRL79, 1930 (1996)
For the materials to be superconducting, we must pin vortices (vortices can not move)

10. Due to huge thermal fluctuation in high Tc,
Large part of Vortex lattice can melt to vortex liquid
Vortex lattice
Vortex liquid

11. Phase Behaviour in the Mixed Phase in high Tc
An alternative view
The conventional picture 11

12. Melting of the flux line lattice
local magnetisation
measurements on BSCCO
E. Zeldov et al.
Nature 1995
1st – order melting transition:-
revealed by jump
in magnetisation at Bm
E.Zeldov 12

13. Melting of the flux line lattice
B = 0 T
B = 9 T
Crabtree
heat capacity
measurements on YBCO
A. Schilling et al.
PRL 1997
Entropy jumps at 1st order flux lattice melting 13

14. Nernst effect The electric field is induced in a metal under
magnetic field by the temperature gradient perpendicular to the magnetic field ,
phenomenon known as Nernst effect.
The Nernst signal is defined as
In the vortex liquid phase of superconductor the Nernst effect is large due to vortex motion, while in the normal state and in the vortex lattice or glass states it is typically smaller.
The vortex-Nernst effect in a type-II
superconductor. Concentric circles
represent vorties

15. N.P.Ong ZA, Xu, nature 406, 486(2000) N. P. Ong’s group (YBCO)
PRL88, (2002)
N.P.Ong
Yu Yang et al
(BSCCO)
PRB 73, (2006)

16. Two complementary theoretical approaches to the mixed state
For vortex cores almost overlap. Instead of lines one just sees array of superconducting “islands”
Normal
Mixed
Meissner H
Hc2
Hc1 Tc T
The Landau level description for constant B
London appr. for infinitely thin lines
For vortices are well separated and have very thin cores

17. Outline The Ginzburg – Landau Theory, GL model of superconductivity.
Type I and Type II superconductor
Thermal Fluctutation
Phase diagram
Dynamics of vortex liquids
conclusion 17

18. Laudau Theory For Phase Transition
Example:Ising model can be mapped to a model with order parameter m by using hubbard transformation
Near mean Tc, we can expand in as it is small and we can expand as field
Magnetization density field
As coherence is long, we can omit higher derivative term.

19. Above theory can correctly describe the physics
Near Tc, can be considered as constant,
Is linear dpendent on T and change sign at mean-field transition temperature
Above theory can correctly describe the physics
near Tc (including thermal fluctuation) for Ising
Model.

20. Second Order Phase Transitions
Landau Theory of
Second Order Phase Transitions
L.D. Landau ( 1937 )
Order parameter field and spontaneous symmetry breaking
Any second order phase transition is described by:
a) Order parameter field
b) Symmetry group G.
The Laudau theory is a lower energy effective theory
In contrast to original microscopic theory

21. GL Free energy for superconductors.
Without external magnetic
field the free energy near transition is:
V.Ginzburg
Ginzburg and Landau (1950) generalized this to the case of arbitrary magnetic field
using gauge invariance of electrodynamics.

22. The GL free energy with magnetic field is:
In equilibrium under fixed external magnetic field the relevant thermodynamic quantity is the Gibbs free energy:

23. The theory is generally believed to be a correct theory describing the physics near Tc, even the microscopic origin of order parameters are unknown!
Historically GL theory of superconductor was formulated at 1950 without knowing any paring mechanism. Gorkov, later on based on BCS, derived GL theory for superconductor.
Generally we believe GL theory is valid also for high Tc even though its microscopic theory is unknown.

24. Mean field equation (ignoring the thermal fluctuation)-Ginzburg – Landau equations
Minimizing the free energy with covariant derivatives one arrives at GL equations. The nonlinear Schrodinger equation (variation of ):
And the super-current equation (variation of A):

25. For finite sample, the equations should be supplemented by the boundary conditions. The covariant gradient is perpendicular to the surface
While magnetization is parallel to it:

26. Time dependence and disorder
The GL energy for inhomogeneos superconductor
The friction dominated dynamics is described by the TDGL
the thermal noise, the normal state inverse diffusion constant.
The electric field is also minimally coupled to the order parameter:
Disorder can be introduced by adding a white noise term to Tc 26

27. GL equations possess two scales. Coherence length
Two characteristic scales
GL equations possess two scales. Coherence length
characterizes variations of , while the penetration depth characterizes variations of
Both diverge at T=Tc.

28. Ginzburg – Landau parameter
The only dimensionless parameter one can construct from the two lengths is temperature independent:
Properties of solutions crucially depend on GL parameter. If
(so called type II superconductivity ) there exist topological solution –
the Abrikosov vortices.
Abrikosov (1957)

29. Meissner Phase, where no magnetic field in the bulk
Type-I
Meissner Phase, where no magnetic field in the bulk
Type-II
In addition of Meissner phase, there is also a mix state where magnetic field can penetrate the superconductor

30. Near Hc2, it is convenient to do follow scaling: length
in unit of , magnetic field in unit of Hc in
Unit of Boltzmann factor is
With following parameters:

31. Gi is the so called Ginzburg parameters which characterize the thermal fluctuation strength.
In high Tc, Gi is so large and thermal fluctuation is very significant! Mean field the solution (Ginzburg-Landau eq.) is not enough. Contrast to low Tc superconductor, one must include the thermal fluctuation effect in high Tc!

32. Increased role of thermal fluctuations in high Tc materials
1 Ginzburg number is much larger
Metals::
High Tc:
Remark: The thermal fluctuation (TF) is so small in low Tc sup., melting and other TF effect can not be observed.
2 Magnetic field effectively reduces dimensionality of fluctuations from D to D-2, and it will enhance the fluct..

33. Thermal fluctuations of vortices in high Tc superconductors will lead to significant effects, melt the Abrikosov vortex lattice into “vortex liquid”.
The thermal fluctuation changes the second order phase transition of mean field solution into a first order melting transition.
The disorder will change part of phase diagram to glass phase (therefore becoming superconducting states), including vortex glass and Bragg glass (Giamarchi,Le Doussal

34. Thermal fluctuation To calculate the thermal effect
How to calculate this very complicated functional integral.
Various method-Perturbative and Non Perturbative.

35. Phenomenological approach based on elastic theory
Elastic theory of vortices (lower energy approximation of the GL model) in the vortex lattice, E.H. Brandt, 1977.
Based on the elastic theory, and using Lindermann criterion of theory of tradition melting (solid to liquid) , when the dislocation exceeds to some extent, the vortex lattice melts to vortex liquid (normal state).
The Lindermann criterion is a pure phenomenological criterion without a sound theoretical basis. Also it can not predict the latent heat et. al. along the melting line.
Furthermore they are qualitative, not quantitative.

36. Key theoretical developments
Second order transition between the vortex lattice and vortex liquid based on the mean field solution of GL theory of type-II superconductor (Abrikosov,1957).
GL model in the lowest Landau level modes with thermal fluctuation included, studied G.J. Ruggeri and D.J. Thouless, The phase transition was speculated as the second order phase transition.
The model studied by G.J. Ruggeri et. al. was reinvestigated using RG. The phase transition was speculated as the first order phase transition (melting). E. Brezin, D.R. Nelson and A. Thiaville, 1985.

37. Larkin-Ovchinnikov collective-pinning theory (1979)
Brag glass theory by Giamarchi and Le Doussal (1995)
Review by Blatter etal.,RMP 1994; Brandt, Rep. Prog. Phys (1995);Nattermann and Scheidl, Adv.Phys.,200; Larkin, Varlamov, book by Oxford (2005)

38. Our approach: Various method (perturbative and nonperturbative) of field theory are used in calculating the thermal fluctuation from the GL model . The main approximation is the LLL mode approximation which is valid near Hc2 line. Using the theory, the melting line location can be determined and magnetization and specific heat jumps can be calculated , which agree well with experimental results on YBCO and Monte Carlo simulations. Considering disorder effects, we also locate the glass transition line or superconducting transition line

39. Phase diagram of vortex matter -Spontaneous symmetry breaking pattern
RS (replica symmetric)
RSB (replica symmetry broken)
TS (translation invariant)
liquid
Vortex glass
TSB (translational symmetry broken)
solid
Bragg glass
Different translation symmetry breaking patterns leads
square or hexagonal lattice-structure phase transition . 39

40. Generic Phase diagrams in high Tc40

41. 41

42. 42

43. 43

44. 44

45. Dynamics of vortex liquid
The friction dominated dynamics is described by the TDGL
for a constant electric field
If E=0, the state in the long time limit will go to Equilibrium state described by free energy F.
Large portion of mix state in phase diagram is liquid due to very
strong thermal fluctuations in high Tc. Vortex Lattice melts to liquid 45

46. Vortex liquid flow (response to electric field)
Tinh, Li, Rosenstein, PRB81,224521(2010)
Gaussian approximation in TDGL is approximating the nonlinear term
The quadratic TDGL can be then solved exactly, and conductivity or Nernst coefficient can be calculated. In particular, Lawrence-Doniach model can be solved too considering all Landau level.

47. As layered superconductor free energy
Lawrence Doniach model

48. In rescaled unit, TDGL can be written as
Gaussian variation approximation, the nonlinear
Term is approximated by

49. The factor 2 is due to two contractions between
and The nonlinear equation is replaced
By the linear one
can be solved consistently.

50. Then we can solve the gap equation completely
And the final theory is cut off independent contrast
previous theory!
Nonlinear conductivity
Can be obtained and
compared to experiments
Nernst effect also can be
Calculated.

51. Noticing in actually is the mean
field
Actually Tc is related to (details can be found in paper)
The energy cutoff is Fermi energy as in high Tc,
the coherence length is of order atomic distance
Comment: 2d limit, d is very large, the above eq. shows
there is no superconductivity which is consistent with
Mermin Wagner theorem-in 2d, there is no U(1)
Symmetry breaking, therefore no superconductivity

52. After using the renormalized Tc (experimental observed Tc
After using the renormalized Tc (experimental observed Tc!)in the gap eq., one can show that the gap eq. becomes Cutoff independent, as it should be for a renormalizable theory.
For a sample YBCO, real Tc is 87K, mean
field Tc is 101K. Between two Tc, there are
Local orderings, but lost global phase coherence,
This might be the pesudogap region.
One can also use the theory to explain the Nernst
Effect.

53. Conclusion
The GL model is systematically studied and used to explain the experimental phase diagrams and calculate thermodynamic quantities and dynamical quantities. . 53