Structure in the mixed phase

1. Structure in the mixed phase Gautam I. Menon IMSc, Chennai, India

2. The Problem • Describe structure in a compactmanner • Correlation functions • Distinguish ordered and disordered states. Also unusual orderings: hexatic Information: Flux-line coordinates as functions of time

3. Vortex Structures • Lines/tilted lines • Pancake vortices in layered systems in fields applied normal to the layers • Josephson vortices in layered systems for fields applied parallel to the planes • Vortex chains and crossing lattices for layered systems in general tilted fields

4. Address via correlation functions Probability of finding a “pancake” vortex a specified distance away from another one

5. Correlation Functions Defines average density at r: Sum over all particles A correlation function Related to the probability of finding a particle at r1, given a distinct particle at r2 The two-point correlation function in a fluid depends only on the relative distance between two points, by rotational and translational invariance.

6. Correlation Functions II Brackets denote a thermodynamic average Defines a structure factor From the previous definition of (r) In terms of Fourier components of the density

7. Correlation Functions in a Solid This sum is over lattice sites. It is non-zero only if q=G (a reciprocal lattice vector), in which case it has value N, i.e. (q) = Nq,G Implies

8. Correlation Functions III Inserting the definition In terms of n2

9. Correlations IV Defines g(r) From g(r), S(q) Just removes an uninteresting q=0 delta-function

10. Why are correlation functions interesting? Experiments measure them! Theorists like them ……

11. The generic scattering experiment measures precisely a correlation function and from there g(r)

12. Physical Picture of g(r) Area under first peak measures number of neighbours in first coordination shell

13. Scattering

14. Intensities as functions of q

15. Melting from Neutron Scattering Bragg spots go to rings: Evidence for a melting transition Ling and collaborators

16. The Disordered Superconductor • Larkin/Imry/Ma: No translational long-range order in a crystal with a quenched disordered background. • Natterman/Giamarchi/Le Doussal: This doesn’t preclude a more exotic order, power-law translational correlations The Bragg Glass

17. Different types of Ordering What does long range order mean? What does quasi-long range order mean? What does short-range order mean?

18. The Bragg Glass proposal Precise consequence for small angle neutron scattering experiments: S(q) decay about (quasi-) Bragg spots

19. More exotic forms of ordering

20. Hexatics • In 2-d systems, thermal fluctuations destroy crystalline LRO except at T=0. Positional order decays as a power law at low T • But, orientationallong-range order can exist at finite but low temperatures

21. Hexatics • In the liquid, short range order in positional and orientational correlations • How do power-law translational order and the orientational long-range order go away as T is increased? • Must be a transition – one or more?

22. Hexatics: Nelson/Halperin • Two transitions out of the low T phase • Intermediate hexatic phase, power-law decay of orientational correlations, short-ranged translational order. • Topological defects: transitions driven by dislocation and disclination unbinding

23. Orientational Correlations Hexatic

24. Hexatic vs Fluid Structure

28. Muon-Spin Rotation

29. The -SR Method I Positively charged muons from an accelerator Muons polarized transverse to applied magnetic field. Implanted within the sample

30. What the muons see

31. Muon Spin Rotation II Muons precess in magnetic field due to vortex lines Muons are unstable particles. Decay into positrons, anti-neutrinos and gamma rays

32. Muon Spin Rotation III Muon lifetime » 10-6 s. Muon decay ! positron emitted preferentially with respect to muon polarization. Emitted positron polarization recorded

33. Muon Spin Rotation IV The Principle: Can reconstruct the local magnetic field from knowledge of the polarization state of the muon when it decays Need to average over a large number of muons for good statistics Muons are local probes

34. Muon Spin Rotation V The magnetic field distribution function Moments of the field distribution function Moments contain important information, obtain l

35. Muon-Spin Rotation Field at point r Density of vortex lines In Fourier space. A is the area of the system

36. Muon Spin Rotation II Flux quantum

37. Muon Spin Rotation VI Assuming a perfect lattice The sum is over reciprocal lattice vectors of a triangular lattice

38. _ <ΔB>1 λ2 Sonier, Brewer and Kiefl, Rev. Mod. Phys. 72, 769 (2000). Muon-Spin Rotation Spectra

39. The rate of muon depolarisation in zero-field µSR (ZF-µSR) is a sensitive probe for spontaneous internal magnetic fields. 0.1G 0.05G MgCNi3 This experiment: •no spontaneous fields present greater than ~0.03G above 2.5K

40. MgCNi3  ns/m*-2 Results: •Tc=7K • Functional form implies s-wave gap Important information about the superconducting gap

41. Results from m-Spin Rotation Underdoped LSCO, Divakar et al.

42. Muon Spin Rotation LSCO Why do line-widths increase with field? Strong disorder in-plane, almost rigid rods The “true” vortex glass U.K. Divakar et al. PRL (2004)

43. Phase Behavior from mSR Probing the glassy state and its local correlations

44. Lee and collaborators

45. Lee and collaborators

46. Lee and collaborators

47. Menon, Drew, Lee, Forgan, Mesot, Dewhurst ++…..

48. Three body correlations in the flux-line glass phase

49. Nontrivial Information about the Nature of superconductivity: Uemura Plot

50. NMR and the Mixed Phase