1. 1 In a vortex system a force applied on a vortex by all the others can be written within London appr. as a sum: IV. FLUX LINE LATTICE The total energy of the vortex system is: A. General overview of phase diagram and magnetic properties

2. 2 minimal magntic field H at which vortices appear To minimize this energy distribution is likely to be as homogeneous as possible : triangular or (sometimes) square lattice (FLL). maximal magnetic field H at which vortices disappear Critical fields H T Mixed Meissner H c2 H c1 TcTc Normal

3. 3 Steep second order phase transition at Hc1. Magnetization curves

4. 4 Magnetization of BSCCO B - H [G] H [G] B m (75K) =92G B m (70K) =140G B m (60K) =250G T = 60, 70, 75, 80 K

5. 5 For the “strongly bound” lattice (London) For the “sparse” lattice (London) For “weakly superconducting” lattice (LLL)

6. 6 B. Relatively sparse lattice. London appr. 1. Lower critical field.

7. 7 Due to flux quantization, the fluxon density is When first vortices appear at, the distances are large and interactions exponentially negligible for triangular square The transition field therefore will be determined by the energy magnetic field has to overcome to push vortices through the material.

8. 8 - Surface area -line energy The condition of the phase transition is:

9. 9 One therefore obtains an expression for the lower critical field

10. 10 Since for a very sparse lattice, (near Hc1) there is just exponentially small repulsion, only nearest neighbors contribute significantly 2. FLL energy and magnetization just above Hc1 Number of vortices z- coordination number ( # of nearest neighbors)

11. 11 Since this is exponential in c and just linear in z, the hexagonal (triangular) FLL is preferable energetically, at least for sufficiently small B. We will see that at larger fields B this feature still holds for rotation invariant interactions (Kepler’s theorem). Using the asymptotic expression for large separations and the expression for the lattice spacing a one obtains:

12. 12 For a given external field H, the induction B should minimize G[B] Since for, B=0 is the minimum. However for H>Hc 1, the minimum is nontrivial and is defined by: Magnetization

13. 13

14. 14 3. Intermediate vortex densities. London eqs. And energy. The “logarithmic repulsion” dominates FLL for Magnetic field becomes quite homogeneous

15. 15 In high Tc SC this region is very wide. Example: BSCCO at T=70K B [G] H [G] to B c2 (70K) Normal material: B center = B ext H c1 (70K) Meissner B=0

16. 16 Unit cell For the square array In this case one still can use London approximation neglecting cores, but not only nearest neighbors now interact. Summations over whole lattice can be performed using Fourier transform. London equations

17. 17 For the hexagonal FLL array two reciprocal lattice vectors perpendicular to hexagonal translation symmetry vectors are:

18. 18 London’s equations are: Solution is a sum over reciprocal lattice (*) Where is the density of vortices

19. 19 Using (*) for one obtains: London interaction energy

20. 20 It logarithmically diverges at due to neglect of the core cut off Minimization of the Gibbs energy with respect to B gives 4. A problem with London approximation and its solution

21. 21 Since the field is rather uniform, especially for the term dominates and better to be separated. “Rounding” the Brillouin zone gives a good approx.: Approximation of the “round Brillouin zone”

22. 22 Performing the integral one obtains Where the expression for the upper critical field was used (see next section)

23. 23 C. Dense lattice. The LLL appr. and Hc2. Order parameter is greatly reduced and only small “islands” between core centers remain superconducting. Still superconductivity dominates electromagnetic properties of the material

24. 24 Some history 1967 Bitter Decoration U. Essmann H. Trauble 1989 STM H.F. Hess Bell labs A.A. Abrikosov 1957 2003

25. 25 Supercurrent in turn is very small since it is proportional to Since magnetization is small we replace the field inside superconductor B by external field H which is essentially homogeneous. magnetization is relatively small (smaller than field) We will use the Landau gauge

26. 26 Since order parameter is small everywhere Naively we can drop nonlinear higher powers of 1. Linearization of the GL eqs. Near Hc2. This is the usual linear Schrodinger equation of quantum mechanics.

27. 27 In the Landau gauge it still has the manifest translation symmetries in both z and y directions, while the x translation invariance is “masked” by the choice of gauge. Therefore one can disentangle variables: getting the shifted harmonic oscillator equation eigenvalue

28. 28 solved by the Landau harmonics Eigenvalues of harmonic oscillator are (for any shift): The corresponding magnetic fields are The highest such value of H is for

29. 29 For yet higher fields the only solution of GL equations is normal: This is Hc2 – the highest field for which a nontrivial solution exists obtained for Hc2 and the LLL appr. normal lattice

30. 30 The two pair breaking phenomena, temperature and magnetic field appear symmetrically, the second order transition line being a straight line For N=0 and one has and the Larmor orbital center X still can be anywhere inside the sample. Eigenvalues of the linearized GL equation however are still highly degenerate (the Landau degeneracy): How should we chose the correct one and its normalization?

31. 31 Below the nonlinear term will lift the degeneracy. It is reasonable to try an Ansatz: most general state on the lowest Landau level (LLL): However the number of the variational parameters is unmanageble. To narrow possible choices of coefficients one has to utilize all the symmetries of the lattice solution. 2. The Abrikosov lattice solution. Heuristic approach

32. 32 Periodicity in the y direction with lattice constant a means that X is quantized: Periodicity in the x direction is possible when absolute values of coefficients C are the same and in addition phases are periodic in n.

33. 33 For the square FLL: For the hexagonal (also called sometimes triangular) FLL: Geometry + flux quantization gives us, as before in units of lattice spacing a instead of customary x : Square lattice

34. 34 The x direction translation takes a form This is “regauging” which generally accompany a symmetry transformation.

35. 35 The symmetry is not “manifest”, for example in our Landau gauge the symmetry along the x direction is “hidden”. The gauge invariant quantities like modulus is invariant and, in particular, zeros are repeated. Hexagonal lattice

36. 36 The y direction translation is the same with different a as a unit of length: The translation in diagonal direction is

37. 37 Shifting n by 1 switches even and odds:

38. 38 Again regauging. Exercise 3: construct the Abrikosov lattices for the rhombic symmetry with an arbitrary angle [0,1] [1,0] Q

39. 39 First let us write the GL for constant magnetic field using dimensionless variables with as the unit of length, as unit of magnetic field as the unit of energy and 3. Abrikosov lattice solution: a systematic expansion approach

40. 40 Here is the (shifted) Schroedinger operator of an electron in magnetic field with nonnegative Landau spectrum: With the hexagonal symmetric eigenfunctions that we will normalize as

41. 41 For the positive and small the energy of a LLL lattice becomes negative provided it is proportional to the Abrikosov solution. normal lattice

42. 42 One therefore develops perturbation theory in small around the Hc2 line: The leading ( ) order equation gives the lowest LLL restriction already motivated in the heuristic approach: order equation is: With normalization undetermined. The next to leading

43. 43 Multiplying it with and integrating one obtains Exercise 4: Fix the normalization of  and calculate the Abrikosov ratio for the square lattice.

44. 44 Free energy of the leading order solution is indeed negative To find the most favorable lattice symmetry (the minimum of free energy) one should minimize Free energy

45. 45 Therefore since the hexagonal lattice energy is slightly (by about 2%) lower than that of the square lattice. This sound small, but for comparison typical latent heat at melting (difference between lattice and homogeneous liquid) is of the same order of magnitude. One can show that energies of other lattices are also higher than that of the hexagonal. Autler et al (67) One also observes that while the first derivative with respect to T is smooth, while the second jumps:

46. 46 To calculate magnetization we return to standard units for energy ensity and add the Maxwell term Minimizing G with respect to b one obtains Magnetization

47. 47

48. 48 Multiplying the GL equation with and integrating over unit cell, then using orthonormality relations one obtains: Higher order correction will include higher Landau level contributions Corrections

49. 49 The LLL component is found from the order etc. All the corrections are very small numerically partially due to factor 1/n with multiples of 6 contributing due to symmetry LLL leading LLL next to leadings 6th LL

50. 50 Therefore the perturbation theory in LLL is by far the leading contribution above this line. converges well up to surprisingly low fields and temperatures, roughly above the line

51. 51 Summary 1. Vortex lattice is a strongly bound elastic medium. Not too close to Hc1 the field becomes homogeneous due to the vortex overlaps. 2. The diamagnetism in the mixed is far from ideal but still very strong compared with the normal state one. 3. One can rely on standard solid state methods far from Hc2. 4. One must use a delicate bifurcation perturbation theory in an interesting region not far from Hc2.