Global and local flux jumps in MgB2 films:
Magneto-optical imaging and theory
Daniel Shantsev, Yuri Galperin, Alexaner Bobyl, Tom Johansen
Physics Department, University of Oslo, Norway
Sung-Ik Lee
Pohang University of Science and Technology, Korea
What determines
the critical current density Jc ?
Thermal Vortex Avalanches
stable critical state
described by critical current Jc
Jc
due to
depinning of vortices
OR
AND
>
thermo-magnetic instability
(flux jumps)
Jc
due to
thermal vortex avalanches
at least for MgB2 films for T<15 K
Mechanism of Thermo-Magnetic Instability
 Flux motion releases heat
 Temperature rise weakens flux pinning
T0  Jc    Q  T > T0
Positive feedback loop
Catastrophic flux jumps
M(H) loop
 M ~ M
 Critical state is destroyed
 Temperature rises to ~Tc
Muller & Andrikidis, PRB-94
Dendritic flux jumps
 M ~ 0.01 M
 Critical state is destroyed locally
 Temperature rises to ~Tc 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 avalanches
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)
avalanche
23000
T=3.6K
11000
7.40 mT
2500
number of vortices
50 - 50000
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
Flux density profiles
before jump
after jump
Flux density B (mT)
50
Flux profiles
before and after a flux jump
have similar shapes
40
7,5000
30
20
31,0000
Ba=11.6mT
10
Ba=5.6mT
edge
0
-100
0
100
200
distance (mm)
x
Microscopic flux jumps
do not destroy
the critical state
film
edge
Catastrophic jumps
 M ~ M
 Critical state is destroyed
Dendritic jumps
 M ~ 0.01 M : noisy M(H)
 Critical state is destroyed locally
 Global Jc is suppressed
Microscopic jumps
 M ~ 10-5 M : invisible in M(H)
 Critical state is preserved
What determines Jc ?
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
Avalanche size distribution
hints to the thermal mechanism
4mT
3
Number of Jumps
10
10mT
2
10
40mT
1
10
10
2
10
3
10
4
10
5
10
6
10
7
Jump size (0)
The distribution has a peak at some typical size
(self-organized criticality suggests a power-law)
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
B, T - profiles
before jump
after jump
1.2
0.8
Ba = 2Bc
0.4
0.2
Ba = Bc
0.0
-1.5
-1.0
-0.5
0.0
x/w
0.3
40
7,5000
30
20
31,0000
Ba=11.6mT
10
Ba=5.6mT
edge
0
Ba = 2Bc
T / Tc
before jump
after jump
50
0.6
Flux density B (mT)
B / m0 jcd
1.0
-100
0
100
200
distance (mm)
Ba = Bc
0.2
0.1
-1.5
x
-1.0
-0.5
0.0
x/w
film
edge
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
Materials
Dendritic avalanches
seen by magneto-optics –
all kinds of MgB2 films (T<10K),
C-doped MgB2
Nb, NbN, Nb3Sn, YBaCuO, Pb, YB2C2O
Peaked size distribution of avalanches
measured by Hall probes
Nb, Pb
Results
• Small flux avalanches (~1000 0) are observed in MgB2 films
using magneto-optical imaging for T<15 K
• Adiabatic model for the size of flux avalanches in a thin film is developed
• Good agreement suggests the thermal mechanism of avalanches
Conclusions
 Thermal avalanches can be microscopic
 These avalanches can control formation of the critical state
without destroying it
 Jc is then determined by stability with respect to these thermal
avalanches rather than by pinning
 The avalanches are too small to be detected in M(H) loops
Phys. Rev. B 72, 024541 (2005)
http://www.fys.uio.no/super/
H-T
phase diagram
the instability field for a thin strip:
Dendritic
jumps
Microscopic
flux jumps
Tth
temperature
What determines
the critical current density Jc ?
Jc
<<
due to
depairing of Cooper pairs
Jc
due to
depinning of vortices
<
Jc
due to
thermal vortex avalanches
Breakdown of
critical state
thermal avalanches
(flux jumps)
a new type of critical state
with a new Jc
usually
Vortex Pinning
at least for MgB2 films for T<15 K
Critical state
Vortices :
• driven inside due to applied field
• get pinned by tiny inhomogeneities
=> Metastable critical state
Sandpile
Critical state in
a superconductor
Distribution of flux density
YBaCuO film, picture from R.Wijngarden
picture from E.Altshuler
Critical current
Critical angle
Pierre G. de Gennes
comments in his classic 1966 book
Superconductivity of Metals and Alloys:
‘‘We can get some physical feeling for this critical state by
thinking of a sand hill. If the slope of the sand hill exceeds
some critical value, the sand starts flowing downwards (avalanche).
The analogy is, in fact, rather good since it has been shown
(by careful experiments with pickup coils)
that, when the system becomes overcritical,
the lines do not move as single units, but rather
in the form of avalanches including typically 50 lines or more’’
Motivation
to study vortex avalanches
• To understand something about vortices
• To understand something about self-organization
(local interactions between vortices lead to long-scales correlations)
• To enhance Jc , i.e.
the slope of the vortex pile
(for various applications of superconductors)
Trapped field magnets
up to 17 Tesla
Jc
Statistics of vortex avalanches
From E. Altshuler and T. H. Johansen, Reviews of Modern Physics, 76, 471 (2004)
Using Magneto-optical Imaging to position the Hall probes
Magneto-optical imagin to measure avalanches
5 mm
MgB2 film
100 mm
Irreproducibility
T=3.6K
Ba = 13.6 mT
B(r)
the flux pattern almost repeats itself
MOI(8.7mT) - MOI(8.5mT)
B(r)
B(r) is irreproducible!
The final pattern is the same
but
the sequences of avalanches are different
Thermal effects
1) Flux motion releases heat
2) T rise weakens flux pinning
T0  Jc    Q  T > T0
The thermal instability is usually associated with catastrophic avalanches
Sometimes thermal avalanches are not complete,
but they are limited only by sample dimensions,
and obviously destroy the critical state
M ~ 0.2 M
Nb disk,
Goodman et al., Phys. Lett. 18, 236 (1965)
Are there small thermal avalanches ?
Are there thermal avalanches that do not destroy
the critical state?
Can thermal avalanches stop before reaching the
sample dimensions?
Can we calculate the size of a thermal avalanche?
Adiabatic energy balance
All energy released by
flux motion is absorbed
C dT = jE dt = jc d
the amount of flux that has
passed through the given point
during an avalanche
Adiabatic critical state for a thin strip
is given by a set of equations:
Biot-Savart
Critical state
Number of avalanches
dH/dt~1G/s
too small
large E
10
B (r)
10 mm
E > Ecdend
1000
1,000,000
20 mm
200 mm
“Size” model
avalanche size, 0
1 mm
“Shape” model
Conclusions
Small: 50 - 50,000 vortices
Round shape
20 mm
Vortex Avalanches
in MgB2 films
Big: ~5,000,000 vortices
Dendritic shape
1 mm
“Shape” Model
“uniform” shape
“Size” Model
Jump Size,
 / m0 jc0 dw
adiabatic critical state
Maxwell + Thermal diffusion
dendritic shape
• Criterion H(E,h0)
• Dendrite width
• Build-up time
0.1Tc
0.1
0.2Tc
0.01
Bfj
1E-3
1E-4
1
Applied field, Ba / Bc(T0)
Thermal effects control
• dendritic avalanches
• micro-avalanches down to 50 vortices
More info: http://www.fys.uio.no/super
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
normal core
x
The vortex core interacts with
tiny inhomogeneities
(x ~ nanometers)
J
=>
B(r)
l
Flux quantum:
 B dA
= h/2e = 0
vortices get pinned
(don’t want to move)
Detecting vortex jumps
B (r)
Ba=4G
Subtract subsequent images: B(r)
vortex
arrived
B (r)
vortex
left
10
90 %
no motion
40
10
We want to understand how the critical state is formed
because:
• it determines the critical current density Jc –
the key parameter for most applications of superconductors
(high-current cables, trapped-field magnets)
• to test models, e.g. self-organized criticality,
for applicability to vortices
(that move in a disordered landscape and don’t have inertia)
Evolution of local flux density
5x5 mm2
No long-range correlation
between the jumps
local B (m T)
30
20
Frequent jumps at the
same place
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
Number of Jumps
Why small and big jumps ?
40mT
1
10
10
3
10
4
10
5
10
6
10
7
Jump size (0)
Both types of jumps
have the same threshold T=10K
Nb films:
also 2 types of jumps,
big and small:
the same mechanism
James et al., Phys.C 2000
Nowak et al, PRB 1997
Distribution functions of jump sizes
10%
50%
4mT
10mT
10
10
resolution limit
Some flux penetrates into the
sample via very small jumps
or without jumps at all
Number of Jumps
10
3
2
1
10
Fraction of flux
arrived via jumps:
0
10
1
90%
40mT
10
2
10
3
4
10
10
Jump size (0)
Sall jumps i
final - initial
=
?
5
10
6
10
7
Dendritic
< 100%