1

The Peak Effect

Gautam I. Menon

IMSc, Chennai, India

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 propertyof dynamics in the mixed

phase

3

How do vortex lines move under the action of an external force?

How are forces exerted on vortex lines?

4

Lorentz Force on Flux Lines

Magnetic pressureTension along lines of force

Force/unit volume

Local supercurrent density Local induction

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Dissipation from Line MotionViscous 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

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?

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)

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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

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The Peak Effect

The Peak Effect refers to thenon-monotonic behavior of

the critical force/currentdensity as H or T are varied

Critical force to set the fluxline system into motion

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How is this critical force computed?

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Larkin Lengths

• At large scales, disorder induced relative displacements of the lattice increase

• Define Larkin lengths

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.

13T.G. Berlincourt, R.D. Hake and D.H. Leslie

No peak effect

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Surface Plot of jc

The peak effect in

superconducting response

Rise in critical currents implies

a drop in measured resistivity

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Why does the peak effect occur?

Many explanations …

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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

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?

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Peak Effects in ac susceptibility measurements

Dips in the real part of ac susceptibility translate to peaks in the critical current

Sarkar et al.

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Will concentrate principally on transport measurements

G. Ravikumar’s lecture: Magnetization, susceptibility

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Peak Effect in Transport: 2H-NbSe2

Fixed H, varying T; Fixed T varying H

Peak effectprobed in resistivity

measurements

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Nonlinearity, Location

A highly non-linear phenomenon

Transition in relation to Hc2

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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

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

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IV curves and their evolution

Differential resistivityEvolution of dynamics

IV curves are convex upwards in the peak region

Peak in differential resistivity in the peak

region

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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

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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)

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 Lv

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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

29

Numerical Simulations

• Brandt, Jensen, Berlinsky, Shi, Brass..

• Koshelev, Vinokur• Faleski, Marchetti,

Middleton• Nori, Reichhardt,

Olson• Scalettar, Zimanyi,

Chandran ..

And a whole lot more …

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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

31

Numerical Simulations

Depinning as a function of pinning strengths. Differential resistivity

Faleski, Marchetti, Middleton: PRE (1996)

Bimodal structure of velocity distributions: Plastic flow

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FMM: Velocity Distributions

Velocity distributions appear to have two components

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Chandran, Zimanyi, Scalettar (CZS)

More realistic models for interactions

Defect densities Hysteresis

Dynamic transition in T=0 flow

34

CSZ: Flow behaviour

Large regime of Disordered flow

All roughly consistent with the physical ideas of

the dominance of plasticity at depinning

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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

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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

37

Return to the experiments

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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

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Peak Effect vs Peak Effect Anomalies

The Peak Regime

Tpl Tp

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?

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

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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

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Corbino geometry:Zeldov and collaborators

Surface effects can be

eliminated by working in a

Corbino geometry. Peak effect sharpens, associated with

Hp

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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

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

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

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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

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Memory and Reorganization I

Andrei group

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Memory and Reorganization II

Response resumes

where it left off

Andrei group

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Generalized Dynamic Phase Diagram

More complex intermediate “Phases” in a

disordered system under

flow

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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

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Phase Behaviour: Reentrant Melting

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The phase diagram angle

and a personal angle ….

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Phase Behaviour of Disordered Type-II superconductors

The ordered phase

The disordered phase

The conventionalview

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Phase Behaviour in the Mixed Phase

The conventional picture An alternative view

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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

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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?