1. Don McKenzie Paul University of Warwick Don McKenzie Paul University of Warwick The Vortex Lattice in Superconductors as seen by neutron diffraction An Introduction and some examples. Charles Dewhurst, Bob Cubitt Institut Laüe Langevin Charles Dewhurst, Bob Cubitt Institut Laüe Langevin Mohana Yethiraj Oak Ridge National Lab. Mohana Yethiraj Oak Ridge National Lab. Simon Levett, Nicola Bancroft Sonya Crowe, Geetha Balakrishnan Simon Levett, Nicola Bancroft Sonya Crowe, Geetha Balakrishnan Ted Forgan & his group

2. Type II Superconductivity is characterised by the mixed state or vortex state. Predicted by A. A. Abrikosov in 1957 First observed by Cribier et. al. in 1967 by neutron diffraction Later imaged directly by Essman & Traüble in 1968 by Bitter decoration Type II Superconductivity is characterised by the mixed state or vortex state. Predicted by A. A. Abrikosov in 1957 First observed by Cribier et. al. in 1967 by neutron diffraction Later imaged directly by Essman & Traüble in 1968 by Bitter decoration Hexagonal VL in an PbIn alloy at 1.1 K and 40 mT (Traüble and Essmann, 1968) Hexagonal VL in an PbIn alloy at 1.1 K and 40 mT (Traüble and Essmann, 1968) Vortex penetration into single crystal ErNi 2 B 2 C (N. Saha et al., 2000) Vortex penetration into single crystal ErNi 2 B 2 C (N. Saha et al., 2000) Type II Superconductivity : the mixed state and vortex lattice also can use µSR to determine density of states for the magnetic field B B n(B) Basics of Vortices in Superconductors

3. Type II Superconductivity : the mixed state and vortex lattice Length scales: ξ – coherence length (distance over which the SC order parameter can be suppressed) λ – penetration depth (screening length of magnetic fields) Ratio:κ = λ / ξ defines Type II behaviour Length scales: ξ – coherence length (distance over which the SC order parameter can be suppressed) λ – penetration depth (screening length of magnetic fields) Ratio:κ = λ / ξ defines Type II behaviour Determines ‘neutron contrast’ and controls intensity as 1 /λ 4 Abrikosov initially predicted a Square VL … he changed his mind later! A Hexagonal VL is more energetically favorable but the energy difference is very small. Abrikosov initially predicted a Square VL … he changed his mind later! A Hexagonal VL is more energetically favorable but the energy difference is very small. Basics of Vortices in Superconductors These individual vortices interact through their currents and the state of minimal energy is a lattice configuration

4. Choice of field Orientation? Horizontal field parallel to neutron beam Horizontal field parallel to neutron beam Vertical field perpendicular to neutron beam Vertical field perpendicular to neutron beam which is the better orientation for experiments? OK, I’ll state my preference in most cases really depends on many things!

5. Type II Superconductivity : the mixed state and vortex lattice Images from an Experiment

6. Type II Superconductivity : the mixed state and vortex lattice Rocking Curve Diffraction from a Vortex Lattice High Intensity - Investigate the VL close to the temperature or field where superconductivity is destroyed. - Investigate materials with long penetration depth e.g. High T c ’s, Organic SC’s. - Time resolved studies High Resolution - Determine complex vortex morphologies. - Phase transitions in the vortex lattice. - Spatially resolved studies. High Intensity - Investigate the VL close to the temperature or field where superconductivity is destroyed. - Investigate materials with long penetration depth e.g. High T c ’s, Organic SC’s. - Time resolved studies High Resolution - Determine complex vortex morphologies. - Phase transitions in the vortex lattice. - Spatially resolved studies. D22

7. Wiggle (10%, field x sine) We ‘shake’ the VL to try to induce better perfection of the lattice, closer to the equilibrium vortex distribution(magnetisation). Procedure is analogous to de-magnetising a ferromagnet. Before (field cooled) YNi 2 B 2 C, 2.5K, 100mT After (field cooled + wiggle) YNi 2 B 2 C, 2.5K, 100mT + 10% Small perturbation or ‘shaking’ of the disordered field-cooled vortex lattice introduces better orientational order. Optimum ~ 10% amplitude. Type II Superconductivity : the mixed state and vortex lattice Shake that Vortex Lattice

8. Miniature magnetic Hall sensors allow the ‘local-induction’ to be measured and therefore monitor static and dynamic properties of the vortex lattice. This local induction tool is quite a different probe to the approach using Neutrons! Schematic of a crystal mounted on a miniature Hall sensor array. Optical microscope image of the Hall sensor array. ‘Optically smooth’ YNi 2 B 2 C single crystal ~200x90x70μm. Single crystal HoNi 2 B 2 C polished into a prism to reduce the geometrical barrier. ~1000x150x50μm. Type II Superconductivity : the mixed state and vortex lattice Source of the Disorder in the Vortex Lattice

9. Local magnetic probe arrays work like conventional magnetic sensors, but with spatial resolution. Only type- II superconductors exhibit a spatially varying magnetic induction on the macroscopic scale. Detailed field profiles show the asymmetric penetration of vortices and the workings of vortex pinning, surface and geometrical barrier effects. Non-magnetic species dominated by surface barriers, bulk pinning controls the Vortex lattice in the magnetic states. ‘Local’ magnetisation curves. Collaborations with: Weizmann Institute, Israel University of Cambridge University of Warwick University of Leiden Type II Superconductivity : the mixed state and vortex lattice Source of the Disorder in the Vortex Lattice

10. Advantages & Disadvantages of using Neutron Diffraction to study the Vortex Lattice Advantages & Disadvantages of using Neutron Diffraction to study the Vortex Lattice Investigation of the VL in the bulk of the sample not surface dependent Investigation of the VL in the bulk of the sample not surface dependent Almost any combination of temperature & field possible T (50mK - 100K) H (20 Oe - 100,000 Oe) Almost any combination of temperature & field possible T (50mK - 100K) H (20 Oe - 100,000 Oe) Neutrons go through walls pressure experiments should be possible! Neutrons go through walls pressure experiments should be possible! Shame that neutron diffraction is flux limited and extremely sensitive to the contrast large samples required large penetration depth is hard 1000 Å is relatively easy 10,000 Å is ~10,000 times more difficult Shame that neutron diffraction is flux limited and extremely sensitive to the contrast large samples required large penetration depth is hard 1000 Å is relatively easy 10,000 Å is ~10,000 times more difficult

11. What do neutrons see well about the vortex lattice ? Average Morphology and in particular changes in symmetry of the VL. Average Morphology and in particular changes in symmetry of the VL. Should be able to extract the form-factor and hence the distribution of magnetic induction around the vortex core, but it’s hard. TmNi2B2C Phase transition in a magnetically ordered superconducting state TmNi2B2C Phase transition in a magnetically ordered superconducting state

12. Type II Superconductivity : the mixed state and vortex lattice What has been done? YBCO Is there really a lattice? Role of twin planes Zig-Zag vortices BSSCO Melting and Decomposition Sr2RuO4 Supporting evidence for p-wave superconductivity UPt3 Changes in superconducting order parameter YNi2B2C Changes in Vortex Lattice structure with magnetic field and temperature Magnetic Superconductors Changes in morphology with magnetic order Changes in core size with susceptibility YBCO Is there really a lattice? Role of twin planes Zig-Zag vortices BSSCO Melting and Decomposition Sr2RuO4 Supporting evidence for p-wave superconductivity UPt3 Changes in superconducting order parameter YNi2B2C Changes in Vortex Lattice structure with magnetic field and temperature Magnetic Superconductors Changes in morphology with magnetic order Changes in core size with susceptibility

13. all figures from S.T. Johnson et. al. PRL 82, 2792, 1999 Square Lattice in YBCO? Early work on YBCO showed an “apparent” square lattice? E.M.Forgan et. al. Nature 343, 735,1990 Eventually, we got “de-twinned” samples good enough YBCO 0,51 T H || c, a axis vertical YBCO 0,51 T H || c, a axis vertical YBCO 0,20 T H ~1º off c, a axis vertical YBCO 0,20 T H ~1º off c, a axis vertical

14. Square Lattices do exist however Strong supporting evidence for p-wave superconductivity in Sr2RuO4

15. Flux-Line Decomposition & Melting BSSCO Loss of coherent lattice structure with the application of a magnetic field, pancake vortices and in-plane pinning Loss of coherent lattice structure with increasing temperature vortex lattice melting Loss of coherent lattice structure with increasing temperature vortex lattice melting

16. Increasing field Type II Superconductivity : the mixed state and vortex lattice Field dependent Transition in the Vortex Lattice H2 H1

17. At high enough fields a stable square configuration is reached. There is no evidence for the presence of any residual low-field hexagonal VL Type II Superconductivity : the mixed state and vortex lattice Field dependent Transition in the Vortex Lattice H > H2

18. As H → H 2, β smoothly opens up and approaches 90 ° β β Type II Superconductivity : the mixed state and vortex lattice Field dependent Transition in the Vortex Lattice H1 < H < H2

19. Decrease in β ⇒ [110] becomes the nearest neighbour direction β β Type II Superconductivity : the mixed state and vortex lattice Field dependent Transition in the Vortex Lattice H < H1

20. Type II Superconductivity : the mixed state and vortex lattice Diffraction from a Vortex Lattice β β Coexistence of all possible domains. No way we can go smoothly from one configuration to the other. First order transition?. β β

21. Type II Superconductivity : the mixed state and vortex lattice Fermi Surface Anisotropy and non-local Electrodynamics  Deviations from the Abrikosov (hexagonal) lattice have been reported in many conventional superconductors, showing strong correlation to the symmetry of the underlying electronic structure perpendicular to H.  The physical argument in many cases is that nonlocality introduces a distortion of the distribution of supercurrent flowing around the normal core of a vortex, resulting in an anisotropic contribution to the intervortex interaction.  The magnitude of the distortion of the distribution of supercurrents is proportional to the degree of anisotropy of the Fermi surface and cleanness of the electronic system.  With increasing applied field the density of the mutually repulsive vortices increases, forming a close-packed structure under the influence of the distortion of the distribution of supercurrents. Fermi surface anisotropy and high-field flux line arrangement in YNi 2 B 2 C

22. Type II Superconductivity : the mixed state and vortex lattice Fermi Surface Anisotropy and non-local Electrodynamics Kogan et al. have developed a model which incorporates nonlocal corrections to the London theory of superconductivity to describe the morphology of the VL. The model adds extra terms to describe the distribution of supercurrents within each vortex, depending on the Fermi velocities averaged over the Fermi surface (j(r) is determined by A within a domain ~ ξ 0 around r). VL free energy density is then calculated using knowledge of the magnetic field distribution about each vortex. V. G. Kogan et al., “The Superconducting State in Magnetic Fields”, ed. Sa de Melo (World Scientific, Singapore, 1998) 127 FOR MORE INFO...

23. Type II Superconductivity : the mixed state and vortex lattice Fermi Surface Anisotropy and non-local Electrodynamics II

24. Izawa et al. Angular resolved thermal conductivity, Izawa et al. PRL 86, 2653 (2002). 4-fold symmetric H c2 (θ) in TmNi 2 B 2 C, Warwick data Unpublished. Type II Superconductivity : the mixed state and vortex lattice Other Influences : Nodes and Gap Anisotropy Substantial evidence has been presented to show that the Superconducting Gap in the Borocarbides is NOT isotropic s-wave. Recent thermal conductivity work by Izawa et al. show Δ has point nodes along [100] and [010]. 4-fold symmetric in-plane H c2 (θ) 4-fold symmetric c-axis thermal conductivity vs. in-plane field. 4-fold symmetric in-plane H c2 (θ) 4-fold symmetric c-axis thermal conductivity vs. in-plane field.

25. Point Node Gap, Izawa et al. Anisotropic Fermi-Surface Nakai et al. (Pre-print) Type II Superconductivity : the mixed state and vortex lattice Other Influences : Nodes and Gap Anisotropy More complicated picture predicts additional phase transitions as a function of field and temperature

26. Nonlocal effects should weaken with increasing temperature and in fact may disappear close to T c2 (H) due to thermal fluctuations. Similar behaviour with impurity doping (Eskildsen et al.). Apex angle, β, vs. field @ temperature Type II Superconductivity : the mixed state and vortex lattice Other Influences : Nodes and Gap Anisotropy No evidence for anything but similar effects as seen at low temperature

27. The order parameter symmetry should remain the same over the entire phase diagram. What is the overall VL phase diagram in the presence of both Fermi surface anisotropy + non-locality and an anisotropic order parameter? LuNi 2 B 2 C B//c Eskildsen What is ‘clear’ is that the VL in the borocarbides does not appear to be a simple Hexagonal lattice even with weak non-locality. Need to consider both effects, the underlying anisotropies e.g. Fermi surface and gap anisotropy. Type II Superconductivity : the mixed state and vortex lattice Other Influences : Nodes and Gap Anisotropy

28. UPt3, Phase Transition by a change in superconducting order parameter

29. Type II Superconductivity : the mixed state and vortex lattice V3Si : another non-local superconductor

30. ErNi 2 B 2 C shows examples of reorientation and square phase transitions T = 2.0 K H = 450 mT T = 5.8 K H = 200 mT T = 4.0 K H = 20 mT Similar to YNi 2 B 2 C

32. ErNi 2 B 2 C T N = 6 K T C = 11 K Fairly simple modulated structure in zero field but with “squaring up” at low T and the development of even harmonics. Below 1.6 K a “ferromagnetic” component develops. Becomes even more complicated in a field. Vortex distortion responds to the changes at T N. Vortex lattice is square at lower fields near to T N. Small change in penetration depth but slope is different. Certain change in superconducting state through T N again probably due to changes in core size and coherence length “Ferromagnetic” component appears to be a series of randomly oriented “domain wall planes” with moments aligned parallel to each other, hence the rods of scattering. These objects cut the vortices and can act as pinning centres

33. TmNi 2 B 2 C : the Role of Paramagnetic Moments Tc = 10.5 K Tn = 1.5 K Tc = 10.5 K Tn = 1.5 K Low Tn and high Tc large paramagnetic susceptibility with field along the c-axis Low Tn and high Tc large paramagnetic susceptibility with field along the c-axis

34. Type II Superconductivity : the mixed state and vortex lattice Some Final Thoughts A powerful technique to look at an interesting phase of matter A subtle, soft solid of wobbly lines Crystal Growth & Annealing Novel and unusual phase transitions Disorder, Dimensionality and Anisotropy are of interest bending, pinning, twisting,melting, decomposition etc Need better theoretical models even for the ‘simplest’ materials A powerful technique to look at an interesting phase of matter A subtle, soft solid of wobbly lines Crystal Growth & Annealing Novel and unusual phase transitions Disorder, Dimensionality and Anisotropy are of interest bending, pinning, twisting,melting, decomposition etc Need better theoretical models even for the ‘simplest’ materials