1. Critical state controlled by microscopic flux jumps in superconductors
Daniel Shantsev
Physics Department, University of Oslo, Norway
in collaboration with
Vitali Yurchenko, Alexander Bobyl, Yuri Galperin, Tom Johansen
Eun-Mi Choi, Sung-Ik Lee
Pohang University of Science and Technology, Korea

2. a superconductor can carry?
What determines
the maximal current
a superconductor can carry?

3. the critical magnetic field
1. Solsby Rule H
Magnetic field
created by current,
should not exceed
the critical magnetic field
H = I / 2p R < Hc R
Jc(1) = 2Hc / R I

4. I J < Jc(2)  Hc / l 2. Depairing current density
Ginsburg-Landau equations
have a solution only if R
J < Jc(2)  Hc / l
For J>Jc the kinetic energy of
Cooper pairs exceeds
the superconducting energy gap I

5. x~10 Å J l  B dA = h/2e = 0 Meissner effect normal core
Flux quantum:
x~10 Å l J
B(r)
normal core
Vortex lattice

6. Lorentz force F = j F0 current Ba J
Vortices are driven by Lorentz force and
their motion creates electric field E ~ dB/dt
Lorentz force
F = j F0 Ba J
pinning force
Lorentz force
Vortices get pinned by tiny defects
and start moving only if Lorentz force > Pinning force
current

7. J < Jc(3) = U / F0 Jc(3) ~ Hc / l U(r)
3. Depinning current density
Superconductor remains in the non-resistive state only if
Lorentz force < Pinning force, i.e. if
U(r)
J < Jc(3) = U / F0
Ideal pinning center is a non-SC column of radius ~ x so that U ~ Hc2x2 and
similar to the depairing Jc
Jc(3) ~ Hc / l

8. positive feedback J*E velocity current +kT E ~ dB/dt Vortex motion
dissipates energy,
J*E
Local Temperature
Increases
velocity
+kT
It is easier for vortices
to overcome pinning barriers
positive
feedback
current
Vortices move
faster

9. x Hfj  Hfjslab (d/w)1/2 x H Jc(4) = (2C Jc(3) [d Jc(3) /dT]-1)1/2/2w
Thermal instability criterion
~ Swartz &Bean, JAP 1968
dQM > dQT - instability starts
dQT = C(T) dT
dQM = Jc(T) dF = H2/2Jc dJc/dT dT
H > Hfj = (2C Jc [dJc/dT]-1)1/2
Jc(4) = (2C Jc(3) [d Jc(3) /dT]-1)1/2/2w j H dF x
Hfj  Hfjslab (d/w)1/2 x H j 2w
d<<w dF
D. S. et al. PRB 2005

10. List of current-limiting mechanisms
Solsby, Jc ~ Hc/R
Depairing current Jc ~ Hc / l
Depinning current, Jc (U)
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

11. J >Jc(3) a small finite resistance appears
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.243
Gaevski et al, APL 1997

12. Critical state is destroyed
Global flux jumps
M(H) loop
DM ~ M
Critical state is destroyed
Muller & Andrikidis, PRB-94

13. Critical state is destroyed locally
Dendritic flux jumps
MgB2 film
DM ~ 0.01 M
Critical state is destroyed locally
Europhys. Lett. 59, (2002)
Magneto-optical imaging
Zhao et al, PRB 2002

14. Microscopic flux jumps
5 mm
MgB2 film
fabricated by
S.I. Lee (Pohang, Korea)
MgB2 film
100 mm
Magneto-optical movie shows
that flux penetration proceeds
via small jumps

15. 2300F0 1100F0 250F0 Analyzing difference images flux jump
7.15 mT
7.40 mT
linear
ramp
of Ba
15 MO images
T=3.6K
= MO image (7.165mT) — MO image (7.150mT)
local increase of flux density -
flux jump
2300F0
1100F0
250F0

16. Jc(3) OR Jc(4) ? The problem with microscopic jumps
31,000F0
7,500F0
Too small, DM ~ 10-5 M : invisible in M(H)
Critical state is not destroyed B-distribution looks as usual x
From the standard measurements one can not tell what limits Jc:
vortex pinning OR thermal instabilities
Jc(3) OR Jc(4) ?
edge
Flux profiles before and after a flux jump have similar shapes

17. 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
Jump size (F0)
Number of jumps
power-law
Altshuler et al. PRB 2005
peak (thermal mechanism)

18. Two Jc’s in one sample
300 mm
70 mm
Jcleft  2 Jcright
Jc(3) Jc(4)

19. Dendritic instability can be suppressed by a contact with normal metal
Baziljevich et al 2002

20. Jc(3) Jc(4) MgB2 300 mm 70 mm Two Jc’s in one sample Au 9 mm w 3 mm
Au suppresses jumps,
Jc is determined by pinning
Jc is determined by jumps
Jc(3) Jc(4)

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

22. MgB2
3 mm Au
Jc1
Jc2 ?

23. α

24. α ≈ π/3 β α α
! jc1 ≈ 2jc2 !

25. Conclusions
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

27. Evolution of local flux density
5x5 mm2
Local B grows by small and repeated steps
7.9mT
7.4mT
7mT
linear ramp 6 mT/s
local flux density calculated from local intensity of MO image;
each point on the curve corresponds to one MO image

28. Jc is determined by Jc depends on
stability with respect to thermal avalanches
Jc depends on
thermal coupling to environment,
specific heat, sample dimensions
But we need to prove that the observed microscopic avalanches are indeed of thermal origin

29. Adiabatic critical state for a thin strip
In the spirit of Swartz &Bean in 1968
Adiabatic :
All energy released by flux motion is absorbed
Critical state
Biot-Savart for thin film
Flux that has passed through “x” during avalanche

30. Flux jump size Thermal origin of avalanches T=0.1Tc We fit Bfj ~ 2 mT
Tth ~ 13 K
F(Ba) dependence
using only
one parameter:
Thermal origin
of avalanches

31. Irreproducibility B(r) DB(r) The final pattern is the same
T=3.6K
Ba = 13.6 mT
B(r)
the flux pattern almost repeats itself
MOI(8.7mT) - MOI(8.5mT)
DB(r)
DB(r) is irreproducible!
The final pattern is the same
but
the sequences of avalanches are different

32. Magneto-optical Imaging
polarizer P A
mirror
MO indicator
image
large
small
Faraday
rotation S N
light source
Linearly
polarized
light
Faraday-active crystal
Magnetic field H q
(H) F
Square YBaCuO film