1. Flux pinning in general Adrian Crisan School of Metallurgy and Materials, University of Birmingham, UK and National Institute of Materials Physics, Bucharest, Romania

3. CONTENTS Introduction: type I vs. type II Vortices Pinning Bulk Pinning Force Density Pinning Potential

4. INTRODUCTION: Type I vs. Type II Type I superconductors -They cannot be penetrated by magnetic flux lines (complete Meissner effect) -They have only a single critical field at which the material ceases to superconduct, becoming resistive -They are usually elementary metals, such as aluminium, mercury, lead

5. Type II superconductors -Gradual transition from superconducting to normal with an increasing magnetic field -Typically they superconduct at higher temperatures and fields than Type I -Between Meissner and normal state there is a large “mixed” or “vortex” state -They have two critical fields (upper and lower) -They are ussually metal alloys, intermetallic compounds, complex oxides (e.g., Cu-based HTSC) and, recently discovered, pnictides and chalcogenides

6. Phase diagram of “classical” superconductors

7. Penetration depth ( ) -Diamagnetic material (no internal flux) -Currents to repel external flux confined to surface -Surface currents must flow in finite thickness (penetration depth  )

8. Coherence length (  ) -characterises the distance over which the superconducting wave function  (r) can vary without undue energy increase -the distance over which the superconducting carriers concentration decreases by Euler’s number e -GL parameter  = /  if   then Type I;   then Type II

11. II. VORTICES Vortex (mixed) state -Normal regions thread through superconductor - Ratio between surface and volume of the normal phase is maximised -Cylinders of normal material parallel to the applied field (normal cores) -Cores arranged in regular pattern to minimize repulsion between cores (close-packed hexagonal lattice) – flux lattice

13. Flux quanta - vortex

14. Phase diagram of High-Tc superconductors and Vortex Melting Lines The vortex lattice undergoes a first-order melting transition transforming the vortex solid into a vortex liquid [Fisher et al, PRB 43,130, 1991]. For high anisotropy, at low magnetic fields (approx 1 Oe in BSCCO [A.C. et al, SuST 24, 115001, 2011), there is a reentrance of the melting line [Blatter et al, PRB 54, 72, 1996]. The flux lines in the vortex -liquid are entangled resulting in an ohmic longitudinal response, hence the vortex liquid and normal metallic phases are separated by a crossover at H c2. For low enough currents -VL- linear dissipation: E ≈ J -VS (VGlass)- strongly nonlinear dissipation: E ≈ exp[-(J T /J)  ]

15. III. PINNING Lorenz force (F L ) and pinning force (F p ) In the presence of a magnetic field perpendicular to the current direction, a Lorentz force F L = j ×  0, where j is the current and  0 is the magnetic flux quantum, acts on the vortices If F L is smaller than the pinning force F p, vortices do not move. Defect-free sample Point defects Columnar defects

16. Dimensionality and strength of PCs

17. IV. BULK PINNING FORCE DENSITY F P determined from magnetization loops M(H a ) F p =BxJ c J c =Ct.  M (thin films; m=  M/2; d-thickness; a,b-rectangle dim.)

18. Dew-Hughes model F = F p /F pmax = h p (1-h q ) ;h = B/B irr p and q depend on the types of pinning centres. -Classified by the number of dimensions that are large compared with the inter-flux-line spacing; and -by the type of the core: “  pinning” and “normal pinning” Ususlly there are several types of pinning centres. F = Ah p1 (1-h q1 )+Bh p2 (1-h q2 )+Ch p3 (1-h q3 )+....... D. Dew-Hughes, Philosophical Magazine 30 (1974) 293

19. Geometry of pin Type of centre Pinning function p, q Position of maximum Max. Const. VolumeNormal A(1-h) 2 p=0; q=2 - A=1 Δκ Bh(1-h)p=1; q=1 h=0.5 B=4 SurfaceNormal Ch 1/2 (1-h) 2 p=1/2; q=2 h=0.2 C=3.5 Δκ Dh 3/2 (1-h)p=3/2; q=1 h=0.6 D=5.37 PointNormal Eh(1-h) 2 p=1; q=2 h=0.33 E=6.76 ΔκFh 2 (1-h)p=2; q=1h=0.67F=6.76

20. Volume normal; (1-h) 2, no max; F=1 Volume  k; h(1-h), max at 0.5, Fm=0.25, A=4 Surface n; h 1/2 (1-h) 2 ; max at 0.2, Fm=0.286, B= 3.5 Surface  k; h 3/2 (1-h) ; max at 0.6, Fmax=0.186, C=5.37

21. Point n; h(1-h) 2 ; max at 0.33, Fmax=0.148, D=6.76 Point  k; h 2 (1-h) ; max at 0.67, Fmax=0.148; E=6.76

22. V. PINNING POTENTIAL Energy needed by the flux line to escape from the potential well crated by the pinning centre Shape and influence on superconducting properties modelled in several ways, depending on material, strength and distribution of pinning centres In 1962 Anderson predicted that movement of vortices with a drift velocity v will create dissipation (electric field) E=Bxv

23. Dissipation occurs through two mechanisms: 1. Dipolar currents which surround each moving flux line (eddy currents) and which have to pass through “normal conducting vortex core” 2. Retarded relaxation of the order parameter when vortex core moves Anderson and Kim predicted that thermal depinning of flux lines can occur at finite temperatures T (“flux creep”).

24. Anderson-Kim model Model assumes that flux creep occurs due to thermally-activated jump of isolated bundles of flux lines between two adjacent pinning centres. The jump is correlated for a bundle of vortices of volume (correlated volume), V c due to the interactions between them In the absence of transport current (i.e., no Lorentz force) the bundle is placed in a rectangular potential well of height U o. Due to thermal energy, there are jumps over the barier with a frequency = o exp(-U o /kT)

25. (1) (3) (2) (3) (1): I=0 (2): 0<I<I o (3): I=I o Second term is ussualy neglected, since current densities of interest are smaller than J 0 Critical current density is defined arbitrarily at a certain electric field, e.g., 10 -6 V/cm. It follows:

26. Logarithmic decay, magnetic relaxation J; M (a.u.) ln (t) K-A model: -pinning potential decreases linearly with current -remnant magnetization and persistent current (or critical current density) decay logarithmically with time

27. Modified Anderson-Kim model Tilted-washboard cosine potential, which leads to U=U 0 [1-(J/J c )] 3/2 The two forms can be generalized as U=U 0 [1-(J/J c )]  Such forms focus on the detailed behavior near J c, which is appropriate for the classic superconductors where fluctuation effects cause only slight degradation of J c

28. Larkin-Ovchinnikov collective pinning model Cooperative aspects of vortex dynamics Formation of vortex lattice will be a result of a competition between: -vortex-vortex interaction, which tends to place a vortex on a lattice point of a periodic hexagonal/triangular lattice; and - vortex-pin interaction, which tends to place a vortex on the local minimum of the pinning potential v-v interaction promotes global translational invariant order v-p interaction tend to suppress such long-range order, if pinning potential varies randomly

29. Long-range order of an Abrikosov lattice is destroyed by a random pinning potential, no matter how weak it is. Periodic arrangement is preserved only in a small corellated volume v c which depends on the strength of the pinning potential and the elasticity of the vortex lines Correlated volume v c increase strongly with decreasing current density J, which leads to a power-law dependence of effective pinning potential on the current density

30. The above dependence leads to a non-ohmic current-voltage characteristic of the form: In an inductive circuit, V is proportional to dJ/dt T 0  10 -6 s

31. Zeldov effective pinning Magneto-resistivity and I-V curves of YBCO films Potential well having a cone-like structure exhibiting a cusp at its minimum and a broad logarithmic decay with the distance E. Zeldov et al, PRL 62, (1989) 3093, PRL. 56, (1990) 680 A.C. et al, SuST 22, 045014, 2009