Critical state controlled by microscopic flux jumps in superconductors

Daniel ShantsevPhysics Department, University of Oslo, Norway

in collaboration with

Vitali Yurchenko, Alexander Bobyl, Yuri Galperin, Tom Johansen

Physics Department, University of Oslo, Norway

Eun-Mi Choi, Sung-Ik Lee

Pohang University of Science and Technology, Korea

What determines the maximal current

a superconductor can carry?

R

II

Magnetic field created by current,

should not exceed the critical magnetic field

H = I / 2 R < Hc

H

1. Solsby Rule

Jc(1) = 2Hc / R

2. Depairing current density

R

II

Ginsburg-Landau equations

have a solution only if

J < Jc(2) Hc /

For J>Jc the kinetic energy of Cooper pairs exceeds

the superconducting energy gap

Vortex lattice

Meissner effect

B dA = h/2e = 0 Flux

quantum:

Å

J

B(r)

normal core

current

Lorentz forceF = j

Vortices are driven by Lorentz force andtheir motion creates electric field E ~ dB/dt

Ba

J

pinningforce

Lorentzforce

Vortices get pinned by tiny defects and start moving only if

Lorentz force > Pinning force

U(r)

3. Depinning current density

J < Jc(3) = U / 0

Superconductor remains in the non-resistive state only if

Lorentz force < Pinning force, i.e. if

Ideal pinning center is

a non-SC column of radius ~ so that U ~ Hc

22 and

similar to the depairing Jc

Jc(3) ~ Hc /

current

velocity

E ~ dB/dt Vortex motiondissipates energy,

J*E

Local TemperatureIncreases

+kT

It is easier for vortices to overcome pinning barriers

Vortices movefaster

positivefeedback

Thermal instability criterion

~ Swartz &Bean, JAP 1968

H

0 x

j

QM > QT - instability starts

QT = C(T) T

QM = Jc(T) JcdJc/dT

H > Hfj = (2C Jc [dJc/dT]-1)1/2

Jc(4) = (2C Jc

(3) [d Jc(3) /dT]-1)1/2/2w

Hfj Hfjslab (d/w)1/2

x

Hj

2w

d<<w

D. S. et al. PRB 2005

List of current-limiting mechanisms

1. Solsby, Jc ~ Hc/R2. Depairing current Jc ~ Hc / 3. Depinning current, Jc (U)4. Thermal instability current, Jc(C,..)

Jc(3) < Jc

(4) < Jc(1) < Jc

(2)

We need to know which Jc is the most important i.e. the smallest!

Achieved

How to distinguish between Jc’s

J >Jc(3) a small finite resistance appears

J >Jc(4) a catastrophic flux jump occurs

(T rises to ~Tc or higher)

Brull et al, Annalen der Physik 1992, v.1, p.243Gaevski et al, APL 1997

Global flux jumps

Muller & Andrikidis, PRB-94

M(H) loop

M ~ M Critical state is destroyed

Dendritic flux jumps

Zhao et al, PRB 2002

M ~ 0.01 M Critical state is destroyed locally

Europhys. Lett. 59, 599-605 (2002)

Magneto-optical imaging

MgB2 film

Microscopic flux jumps

5 mm

MgB2 film

100 m

MgB2 filmfabricated byS.I. Lee (Pohang, Korea)

Magneto-optical movie showsthat flux penetration proceeds

via small jumps

Analyzing difference images

7.15 mT

7.40 mT

linearrampof Ba

15 MO images

T=3.6K

= MO image (7.165mT) — MO image (7.150mT)

local increase of flux density -

flux jump

23000

11000

2500

-100 0 100 2000

10

20

30

40

50 before jump after jump

Ba=5.6mTFlu

x de

nsity

B (

mT

)

distance (m)

Ba=11.6mT

edge

x

edge

31,0000

7,5000

Too small, M ~ 10-5 M : invisible in M(H) Critical state is not destroyed B-distribution looks as usual

The problem with microscopic jumps

Flux profiles before and after a flux jump have similar shapes

From the standard measurementsone can not tell what limits Jc:

vortex pinning OR thermal instabilities

Jc(3) OR Jc

(4) ?

What can be done

One should measure dynamics of flux penetration and look for jumps If any, compare their statistics, B-profiles etc with thermal instability theories

If they fit, then Jc=Jc(4) , determined by instability;

actions – improve C, heat removal conditions etc, if not, then Jc=Jc

(3), determined by pinning;

actions – create better pinning centers

-1.5 -1.0 -0.5 0.00.0

0.2

0.4

0.6

0.8

1.0

1.2

Ba = 2Bc

Ba = Bc

before jump after jump

B / 0

j cd

x / w

-100 0 100 2000

10

20

30

40

50 before jump after jump

Ba=5.6mTFlu

x d

en

sity

B (

mT

)

distance (m)

Ba=11.6mT

edge

Jump size (0)

Nu

mb

er

of

jum

ps

power-law

peak(thermalmechanism)

Altshuler et al. PRB 2005

300 m70 m

Two Jc’s in one sample

Jcleft 2 Jc

right

Jc(3) Jc

(4)

Dendritic instability can be suppressed by a contact with normal metal

Baziljevich et al 2002

Au suppresses jumps,Jc is determined by pinning

300 m70 m

Two Jc’s in one sample

Jc(3) Jc

(4)

3 m

m

9 mm

w

Au

MgB2

Jc is determined by jumps

H

J

A graphical way to determine Jc’s: d-lines

3 mm

Au

MgB2

Jc1

Jc2?

αα

ααβ

11

1

cjd

22

1

cjd

1cos22cos2coscos 2

1

2

2

1 c

c

j

j

d

d

1

2

1

2

12

1arccos

12

1arccos

c

c

c

c

j

j

j

j

α ≈ π/3

! jc1 ≈ 2jc2 !

Thermal avalanches can be truly microscopic as observed by MOI and described by a proposed adiabatic model

These avalanches can not be detected either in M(H) loops or in static MO images =>

“What determines Jc?” - is an open question

MO images of MgB2 films partly covered with Au show two distinct Jc’s: - Jc determined by stability with respect to thermal avalanches - a higher Jc determined by pinning

http://www.fys.uio.no/super/

Conclusions

6.8 7.2 7.6 8.0 8.4

0

10

20

30

loc

al

B (

mT

)

Ba (mT)

local flux density calculated from local intensity of MO image;each point on the curve corresponds to one MO image

5x5 m2

linear ramp 6 T/s

Evolution of local flux density

7mT7.4mT

7.9mT

Local B grows bysmall and repeated steps

Jc is determined by

stability with respect to thermal avalanches

But we need to prove that the observed microscopic avalanches are indeed of thermal origin

Jc depends on

thermal coupling to environment, specific heat, sample dimensions

Adiabatic :All energy released by flux motion is absorbed

Flux that has passed through “x” during avalanche

Biot-Savart for thin film

Adiabatic critical state for a thin strip

Critical state

In the spirit of Swartz &Bean in 1968

We fit• Bfj ~ 2 mT• Tth ~ 13 K• (Ba) dependence

using only one parameter:

4 8 12 16 20

102

103

104

105

106

Ba (mT)

Flux

jum

p si

ze (

0) T=0.1Tc

0.3Tc

Thermal originof avalanches

Flux jump size

Ba = 13.6 mT

the flux pattern almost repeats itself

Irreproducibility

B(r)

B(r) is irreproducible!

The final pattern is the same but

the sequences of avalanches are different

MOI(8.7mT) - MOI(8.5mT)

B(r)

T=3.6K

polarizer P

A

mirrorMO indicator

image

largesmall

Faraday rotation

small

SN

light source

Linearlypolarized light

Faraday-active crystal

Magnetic fieldH

(H)F

Magneto-optical Imaging

Square YBaCuO film