2. 1 The Peak Effect Gautam I. Menon IMSc, Chennai, India

3. 2 Type-II Superconductivity Structure of a vortex line The mixed (Abrikosov) phase of vortex lines in a type-II superconductor The peak effect is a property of dynamics in the mixed phase

4. 3 How do vortex lines move under the action of an external force? How are forces exerted on vortex lines?

5. 4 Lorentz Force on Flux Lines Magnetic pressure Tension along lines of force Force/unit volume Local supercurrent densityLocal induction

6. 5 Dissipation from Line Motion Viscous forces oppose motion, damping coefficient  lines move with velocity v Competition of applied and viscous forces yields a steady state, motion of vortices produces an electric field Power dissipation from EJ, thus nonzero resistivity from flux flow

7. 6 Random Pinning Forces To prevent dissipation, pin lines by quenched random disorder Line feels sum of many random forces Summation problem: Adding effects of these random forces. How does quenched randomness affect the crystal?

8. 7 Elasticity and Pinning compete In the experimental situation, a random potential from pinning sites The lattice deforms to accommodate to the pinning, but pays elastic energy Pinning always wins at the largest length scales: no translational long-range order (Larkin)

9. 8 Depinning From random pinning: critical force to set flux lines into motion Transition from pinned to depinned state at a critical current density Competition of elasticity, randomness and external drive

10. 9 The Peak Effect The Peak Effect refers to the non-monotonic behavior of the critical force/current density as H or T are varied Critical force to set the flux line system into motion

11. 10 How is this critical force computed?

12. 11 Larkin Lengths At large scales, disorder induced relative displacements of the lattice increase Define Larkin lengths

13. 12 Estimating Jc Larkin and Ovchinnikov J. Low Temp. Phys 34 409 (1979) Role of the Larkin lengths/Larkin Volume collective pinning theory Pinning induces Larkin domains. External drive balances gain from domain formation.

14. 13 T.G. Berlincourt, R.D. Hake and D.H. Leslie No peak effect

15. 14 Surface Plot of j c The peak effect in superconducting response Rise in critical currents implies a drop in measured resistivity

16. 15 Why does the peak effect occur? Many explanations …

17. 16 The Pippard Mechanism Pippard: Softer lattices are better pinned [Phil. Mag. 34 409 (1974) Close to Hc2, shear modulus is vastly reduced (vanishes at Hc2), so lines adjust better to pinning sites Critical current increases sharply

18. 17 PE as Phase Transition? Shear moduli also collapse at a melting transition Could the PE be signalling a melting transition? (In some systems …) Disorder is crucial for the peak effect. What does disorder do to the transition?

19. 18 Peak Effects in ac susceptibility measurements Dips in the real part of ac susceptibility translate to peaks in the critical current Sarkar et al.

20. 19 Will concentrate principally on transport measurements G. Ravikumar’s lecture: Magnetization, susceptibility

21. 20 Peak Effect in Transport: 2H-NbSe 2 Fixed H, varying T; Fixed T varying H Peak effect probed in resistivity measurements

22. 21 Nonlinearity, Location A highly non-linear phenomenon Transition in relation to H c2

23. 22 In-phase and out-of-phase response Apply ac drive, measure in phase and out-of- phase response Dip in in-phase response, peak in out-of-phase response: superconductor becomes more superconducting Similar response probed in ac susceptibility measurements

24. 23 Systematics of I-V Curves I-V curves away from the peak behave conventionally. Concave upwards. Such curves are non- trivially different in the peak regime

25. 24 IV curves and their evolution Differential resistivity Evolution of dynamics IV curves are convex upwards in the peak region Peak in differential resistivity in the peak region

26. 25 Fingerprint effect Differential resistivity in peak regime shows jagged structure Reproducible: increase and lower field Such structure absent outside the peak regime Power-laws in IV curves outside; monotonic differential resistivity

27. 26 Interpretation of Fingerprints? A “Fingerprint” of the structure of disorder? Depinning of the flux-line lattice proceeds via a series of specific and reproducible near-jumps in I-V curves This type of finger print is the generic outcome of the breaking up of the flux- line lattice due to plastic flow in a regime intermediate between elastic and fluid flow (Higgins and Bhattacharya)

28. 27 Noise If plastic flow is key, flow should be noisy Measure frequency dependence of differential resistivity in the peak region Yes: Anomalously slow dynamics is associated with plastic flow. Occurs at small velocities and heals at large velocities where the lattice becomes more correlated. A velocity correlation length L v

29. 28 Dynamic Phase Diagram Force on y-axis, thermodynamic parameter on x-axis (non-equilibrium) Close to the peak, a regime of plastic flow Peak onset marks onset of plastic flow Peak maximum is solid-fluid transition

30. 29 Numerical Simulations Brandt, Jensen, Berlinsky, Shi, Brass.. Koshelev, Vinokur Faleski, Marchetti, Middleton Nori, Reichhardt, Olson Scalettar, Zimanyi, Chandran.. And a whole lot more …

31. 30 Simulations: The General Idea Interaction – soft (numerically easy) or realistic Disorder, typically large number of weak pinning sites, but also correlated disorder Apply forces, overdamped eqn of motion, measure response Depinning thresholds, top defects, diff resistivity, healing defects through motion, Equilibrium aspects: the phase diagram

32. 31 Numerical Simulations Depinning as a function of pinning strengths. Differential resistivity Faleski, Marchetti, Middleton: PRE (1996) Bimodal structure of velocity distributions: Plastic flow

33. 32 FMM: Velocity Distributions Velocity distributions appear to have two components

34. 33 Chandran, Zimanyi, Scalettar (CZS) More realistic models for interactions Defect densities Hysteresis Dynamic transition in T=0 flow

35. 34 CSZ: Flow behaviour Large regime of Disordered flow All roughly consistent with the physical ideas of the dominance of plasticity at depinning

36. 35 Dynamic Phase Diagram Predict a dynamic phase transition at a characteristic current Phase at high drives is a crystal The crystallization current diverges as the temperature approaches the melting temperature Fluctuating component of the pinning force acts like a “shaking temperature” Koshelev and Vinokur, PRL(94).. Lots of later work

37. 36 Simulations: Summary We now know a lot more about the depinning behaviour of two-dimensional solids in a quenched disorder background. Variety of new characterizations from the simulations of plastic flow phenomena Dynamic phase transitions in disordered systems Yet.. May not have told us much about the peak effect phenomenon itself

38. 37 Return to the experiments

39. 38 History Dependence in PE region Critical Currents differ between FC and ZFC routes Henderson, Andrei, Higgins, Bhattacharya Two distinct states of the flux-line lattice, one relatively ordered one highly disordered. Can anneal the disordered state into the ordered one

40. 39 Peak Effect vs Peak Effect Anomalies The Peak Regime T pl TpTp Let us assume that the PE is a consequence of an order-disorder transition in the flux line system Given just this, how do we understand the anomalies in the peak regime?

41. 40 Zeldov and collaborators: Peak Effect anomalies as a consequence of the injection of a meta-stable phase at the sample boundaries and annealing within the bulk Boundaries may play a significant role in PE physics

42. 41 The Effects of Sample Edges Role of barriers to flux entry and departure at sample surfaces Bean-Livingston barrier Currents flow near surface to ensure entry and departure of lines Significant dissipation from surfaces

43. 42 Corbino geometry:Zeldov and collaborators Surface effects can be eliminated by working in a Corbino geometry. Peak effect sharpens, associated with Hp

44. 43 Relevance of Edges Both dc and ac drives Hall probe measurements Measure critical currents for both ac and dc through lock-in techniques Intermediate regime of coexistence from edge contamination

45. 44 Direct access to currents Map current flow using Maxwells equations and measured magnetic induction using the Hall probe method Most of the current flows at the edges, little at the bulk Dissipation mostly edge driven?

46. 45 Zeldov et al : Edge contamination At the end of a half cycle, for an ac drive, current flow pattern interchanges Metastability from the injection of disordered phase at the edges and subsequent annealing in the bulk

47. 46 Andrei and collaborators Start with ZFC state, ramp current up and then down Different critical current.. “Jumpy” behavior on first ramp Lower threshold on subsequent ramps

48. 47 Plastic motion/Alternating Currents Steady state response to bi- directional pulses vs unidirectional pulses Motion if bi-directional current even if amplitude is below the dc critical current No response to unidirectional pulses Henderson, Andrei, Higgins

49. 48 Memory and Reorganization I Andrei group

50. 49 Memory and Reorganization II Response resumes where it left off Andrei group

51. 50 Generalized Dynamic Phase Diagram More complex intermediate “Phases” in a disordered system under flow

52. 51 Reentrant Peak Effect Reentrant nature of the peak effect boundary at very low fields Connection to reentrant melting? See in both field and temperature scans Later work by Zeldov and collaborators TIFR/BARC/WARWICK/NEC COLLABORATION

53. 52 Phase Behaviour: Reentrant Melting

54. 53 PE Reentrance (Corbino) Zeldov and collaborators Corbino geometry Smeared out in the strip geometry Nature of ordered and disordered phase?

55. 54 The phase diagram angle and a personal angle ….

56. 55 Phase Behaviour of Disordered Type-II superconductors The ordered phase The disordered phase The conventional view

57. 56 Peak Effect as Phase Transition One refinement: Ling and collaborators

58. 57 Phase Behaviour in the Mixed Phase The conventional picture An alternative view

59. 58 Properties of the Phase Diagram Peak effect associated with the sliver of glassy phase which is the continuation of the high field glassy state to low fields Domain-like structure in the intermediate (multi- domain) state Domains can be very large for weak disorder and high temperature A generic two-step transition Lots of very suggestive data from TIFR/BARC etc

60. 59 The Last Word.. Alternative approaches: Critical currents may be dominated by surface pinning, effects of surface treatment (Simon/Mathieu). PE seems to survive, though How to compute the transport properties of the multi-domain glass? If the Zeldov et al. disordered phase injection at surfaces scenario is correct, what about the simulations? More theory which is experiment directed Other peak effects without transitions?