1. 1 Most of the type II superconductors are anisotropic. In extreme cases of layered high Tc materials like BSCCO the anisotropy is so large that the material can be considered two dimensional. It is important to distinguish the anisotropy in directions parallel and perpendicular to the magnetic field direction. V. ANISOTROPIC and LAYERED SUPERCONDUCTORS We start with the simplest case of anisotropic GL theory neglecting layered structure. A. Some phenomenology

2. 2 Various types (old and new) of the “conventional” or the “BCS” superconductors 7.2830370800-0.4 1 7.477690160015911 2026240013050901 15 1400 26302 25 2000 9015070

3. 3 93151440500260100105 6520255030015011012130.1 34292800705013013150.1 120252500150 10018300.5 2578020002.125-0.9 Various types of “unconventional” (or “non BCS” SC) 1986 High T c Superconductors Alex Muller, Georg Bednorz

4. 4 for YBCO First we assume that the material is rotationally symmetric in the plane perpendicular to magnetic field. While the potential and magnetic terms are always symmetric, the gradient term generally is not: The asymmetry factor is defined by for BSCCO 1. Anisotropic GL model B. Anisotropic GL and Lawrence- Doniach models

5. 5 One repeats the calculations in the anisotropic (and even in the “tilted” geometry when magnetic field is not oriented parallel to one of the symmetry axes) using scaling transformations. Coherence length in the c direction is typically much smaller while the corresponding penetration depth is larger: Blatter et al RMP (1994)

6. 6 It is much easier to create vortices to be oriented in the ab plane. We don’t have to solve again the GL equations: they do not change. A A ~2,000  ~10 Type II:  ~ 200 >>  c

7. 7 However when the material consists of well separated superconducting layers, the continuum field theory might not approximate the situation well enough: one should use the LD tunneling model: Interlayer distance d CuO plane (layer or bilayer) Layer width s 2. The Lawrence - Doniach model Bi 2 Sr 2 Ca 1 Cu 2 O 8+ 

8. 8 Lawrence-Doniach model Hamiltonian of LD model Order parameter in nth layer γ t :Tunneling factor d: interlayer spacing (2)

9. 9 Criterion of applicability of GL for layered material is when coherence length in the c direction is not smaller than the interlayer spacing: finite differences can be replaced by derivatives and sums by an integral

10. 10 YBCO GL still OK BSCCO Anisotropic GL invalid The condition is obeyed in most low Tc materials and barely in optimal doping YBCO at temperatures not very far from Tc, but generally not obeyed in BSCCO and other high Tc superconductors

11. 11 Until now we have assumed that the system is in plane O(2) rotationally symmetric: Real materials are usually not symmetric. However if the material is “just” fourfold ( ) symmetric In YBCO there is sizable explicit O(2) ( in plane ) breaking due to the d-wave character of pairing. However asymmetry is not always related to the non s – wave nature of pairing. 3. Fourfold anisotropy

12. 12 to include effects of O(2) breaking, one has to use “small” or “irrelevant” four derivative terms. There is no quadratic in covariant derivative terms that break O(2) but preserve There are three such terms but only the last breaks the O(2) and is thereby a “dangerous irrelevant”. One therefore adds the following gradient term: With dimensionless constant characterizing the strength of the rotational asymmetry

13. 13 Most remarkable phenomenon structural phase transition. Body centered rectangular lattice becomes square This term leads to anisotropic shape of the vortex and an angle dependent vortex – vortex interaction leading to emergence of lattices other than hexagonal: the symmetric rhombic lattices. Structural phase transitions in vortex lattices

14. 14 Pearl’s solution for thin film C. Vortices in thin films and layered SC 1. Pearl’s vortices in a thin film Anti-monopole field Magnetic monopole field

15. 15 2D London’s equation inside the film, z=0 Where is the polar angle (see the derivation of the vortex Londons’ eq. in part I). Now I drop curl using Londons’ gauge Where the vector field is defined by

16. 16 For, and almost do not vary inside the film as function of z. The 2D supercurrent density consequently is: Where the effective 2D penetration depth is defined by

17. 17 Since the current flows only inside the film, the Maxwell equation in the whole space is: Two different Fourier transforms

18. 18 Integrating over k, one gets: The 3D equation takes a form:

19. 19 Substituting back into eq.(*) and performing the k and the angle integrations one obtains the vector potential: The magnetic field z component in the film is: The effective penetration depth indeed describes magnetic field scale in thin fielm

20. 20 For example, the flux crossing the film within radius r is: for For this gives monopole field: Performing the k and angle integration in the inverse 3D Fourier transform one obtains:

21. 21 Supercurrent for

22. 22 The potential energy therefore is: Force that a vortex at exerts on a vortex at is: where the standard unit of the line energy is used

23. 23 Energy to create a Pearl vortex is The film therefore behaves as a superconductor with The two features, logarithmic interaction and finite creation energy make statistical mechanics of Pearl’s vortices subject to thermal fluctuations a very nontrivial 2D system. How to make a good type II superconductor from a type I material?

24. 24 “Pancake” vortex Pearl’s region 2. “Pancake” vortices in layered superconductors Two magnetic field scales

25. 25 Fourier transform for one pancake vortex in the layer n=0 London’s eqs. for a pancake vortex centered at

26. 26

27. 27 Magnetic field extends beyond the Pearl’s region: Total flux through cillinder of height z and radius r is: Flux through the central layler where core is located

28. 28 Current in the central layer Due to squeezing of magnetic field the cutoff disappeared for all distances ! In higher layers: Interaction in the same plane

29. 29 Energy of a single pancake vortex is logarithmically infinite in infrared: cannot be isolated. Pancake vortices in different layers also interact: for

30. 30 The Ginzburg-Landau string tension is recovered in the case of straight vortices with and replacing and. Pancake vortices in neighbour planes attract each other. Abrikosov flux line in layered superconductors

31. 31 … for

32. 32 Summary 1. In thin films the field “leaks” out and the vortex envelop (effective penetration depth l) becomes large. The material becomes therefore more type II and interaction acquire longer range. 2. Layered SC (a superlattice) causes interaction between vortices (which become “pancakes”) to be truly long range logarithmic. 3. While moderately anisotropic layered SC still can be described by the anisotropic GL theory for strongly anisotropic ones Lawrence-Doniach tunneling theory should be used.