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
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?
1. Solsby Rule
H
Magnetic field
created by current,
H = I / 2p R < Hc
R
should not exceed
the critical magnetic field
Jc(1) = 2Hc / R
I
2. Depairing current density
Ginsburg-Landau equations
have a solution only if
R
I
J < Jc(2)  Hc / l
For J>Jc the kinetic energy of
Cooper pairs exceeds
the superconducting energy gap
Meissner effect
normal core
x~10 Å
Vortex lattice
J
B(r)
l
 B dA
= h/2e = 0
Flux quantum:
Vortices are driven by Lorentz force and
their motion creates electric field E ~ dB/dt
Lorentz
force
F = j 0
Vortices get pinned by tiny defects
and start moving only if
Lorentz force > Pinning force
Ba
Lorentz
force
current
J
pinning
force
3. Depinning current density
U(r)
Superconductor remains
in the non-resistive state only if
Lorentz force < Pinning force, i.e. if
J < Jc(3) = U / 0
Ideal pinning center is
a non-SC column of radius ~ x
so that U ~ Hc2x2 and
Jc(3) ~ Hc / l
similar to the depairing Jc
E ~ dB/dt
Vortex motion
dissipates energy,
J*E
Local Temperature
Increases
velocity
positive
feedback
current
It is easier for vortices
to overcome pinning barriers
+kT
Vortices move
faster
Thermal instability criterion
~ Swartz &Bean, JAP 1968
dQM = Jc(T) d = H2/2Jc dJc/dT dT
j
dQT = C(T) dT
H
dQM > dQT - instability starts
0
d
x
H > Hfj = (2C Jc [dJc/dT]-1)1/2
Jc(4) = (2C Jc(3) [d Jc(3) /dT]-1)1/2/2w
d<<w
j
H
Hfj  Hfjslab (d/w)1/2
d
2w
x
D. S. et al. PRB 2005
List of current-limiting mechanisms
1.
2.
3.
4.
Solsby, Jc ~ Hc/R
Depairing current Jc ~ Hc / l
Depinning current, Jc (U)
Thermal instability current, Jc(C,..)
We need to know which Jc is the most important i.e. the smallest!
Jc(3) < Jc(4)
<
Jc(1) < Jc(2)
Achieved
How to distinguish between Jc’s
 J >Jc(3)
 J >Jc(4)
a small finite resistance appears
a catastrophic flux jump occurs
(T rises to ~Tc or higher)
Gaevski et al, APL 1997
Brull et al, Annalen der Physik 1992, v.1, p.243
Global flux jumps
M(H) loop
 DM ~ M
 Critical state is destroyed
Muller & Andrikidis, PRB-94
Dendritic flux jumps
 DM ~ 0.01 M
 Critical state is destroyed locally
MgB2 film
Magneto-optical imaging
Zhao et al, PRB 2002
Europhys. Lett. 59, 599-605 (2002)
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
Analyzing difference images
7.15 mT
=
—
MO image (7.150mT)
local increase of flux density -
linear
ramp
of Ba
15 MO
images
MO image (7.165mT)
flux jump
23000
T=3.6K
11000
7.40 mT
2500
The problem with microscopic jumps
before jump
after jump
Flux density B (mT)
50
40
30
7,5000
20
Ba=11.6mT
31,0000
10
Ba=5.6mT
edge
0
-100
0
100
200
distance (mm)
 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
edge
Flux profiles before and after a flux
jump have similar shapes
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
before jump
after jump
Flux density B (mT)
50
power-law
4mT
Number of Jumps
10
40mT
1
10
10
2
10
3
10
4
10
5
10
6
10
7
Jump size (0)
10
Ba=5.6mT
edge
actions – improve C, heat removal conditions etc,
 if not, then Jc=Jc(3), determined by pinning;
actions – create better pinning centers
100
200
before jump
after jump
1.0
0.8
0.6
Ba = 2Bc
0.4
0.2
Ba = Bc
0.0
-1.5
 If they fit, then Jc=Jc(4) , determined by instability;
0
1.2
(thermal
mechanism)
2
Ba=11.6mT
distance (mm)
peak
10mT
20
-100
B / m0 jcd
3
10
30
0
Altshuler et al.
PRB 2005
Jump size (0)
40
-1.0
-0.5
x/w
0.0
Two Jc’s in one sample
300 mm
70 mm
Jcleft  2 Jcright
Jc(3)
Jc(4)
Dendritic instability can be suppressed by a contact with normal metal
Baziljevich et al 2002
Two Jc’s in one sample
9 mm
3 mm
w
MgB2
Au
300 mm
70 mm
Au suppresses jumps,
Jc is determined by pinning
Jc(3)
Jc is determined by jumps
Jc(4)
A graphical way to determine Jc’s: d-lines
J
H
3 mm
MgB2
Au
Jc1
?
Jc2
α
α
d1 jc 2
=
= cos  = cosp  2  =  cos2  = 2 cos2   1
d2
jc1
α ≈ π/3
! jc1 ≈ 2jc2 !
d2 
1
β
jc 2
α

 1
jc 2  

 arccos
1 



2
j

c1 



 =
 1
jc 2  


p  arccos  2 1  j  
c1 




α
d1 
1
jc1
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
http://www.fys.uio.no/super/
Evolution of local flux density
5x5 mm2
local B (m T)
30
Local B grows by
small and repeated steps
20
10
7.4mT
7.9mT
7mT
0
6.8
7.2
7.6
8.0
8.4
B a (mT)
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
Jc is determined by
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
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
Flux jump size
0.1Tc
0.1
0.2Tc
0.01
Bfj
1E-3
0.3Tc
Flux jump size (0)
Jump Size,
 / m0 jc0 dw
10
6
T=0.1Tc
10
5
10
4
10
3
10
2
1E-4
1
4
Applied field, Ba / Bc(T0)
8
12
Ba (mT)
We fit
• Bfj ~ 2 mT
• Tth ~ 13 K
• (Ba) dependence
using only
one parameter:
Thermal origin
of avalanches
Irreproducibility
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
Magneto-optical Imaging
image
q F(H)
Faraday-active crystal
A
small
Linearly
polarized
light
large
Faraday
rotation
polarizer P
H
Magnetic field
light source
MO indicator
mirror
Square YBaCuO film
N
S
small