Size distribution of flux avalanches in MgB2 films
visualized by magneto-optical imaging
Daniel V. Shantsev
Department of Physics, University of Oslo, Norway
Collaboration:
A. L. Rakhmanov, Inst. Th&Appl. Electrodyn., Moscow, Russia
D. Denisov, Y. M. Galperin, T. H. Johansen, University of Oslo, Norway
A. V. Bobyl, A. F. Ioffe Institute, St. Petersburg, Russia
Support:
FunMat@UiO, Nanomat (Norway)
Picture: Folgefonna Glacier , Norway
Vortex Avalanches
Magneto-Optical imaging
MgB2 films
Small: 50 - 50,000 vortices
Round shape
Big: ~5,000,000 vortices
Dendritic shape
1 mm
20 mm
“Shape” Model
“uniform” shape
dendritic shape
“Size” Model
Thermal origin of avalanches
SOC
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
N
S
small
Flux pattern produced by instability
Sample: MgB2 film
[Sung-Ik Lee, POSTECH, Korea]
Ba
T=3.6K
68 mT
Magneto-optical
movie
t
250 sec
50 sec
1 mm
Dendritic patterns in other MgB2 films
Pulse Laser Deposition on 1102 Al2O3 substrate
400nm, Tc=39K
S.I. Lee, Pohang Univ., Korea
Screen printing, Al2O3 substrate
3000 nm, Tc=35K
G. Gritzner, Univ. of Linz, Austria
PLD, SrTiO3 substrate,
250nm, Tc=28K
S.X. Dou, Wollongong, Australia
Dendritic patterns in other materials
Nb:
Nb3Sn:
Pb:
C.A. Duran et al. PRB 52, 75 (1995)
Oslo, Cryogenics 2003
Menghini et al, cond-mat/0409391
What is the role of
sample inhomogeneities
?
NbN
YNi2B2C
YBaCuO, induced by laser
Oslo, cond-mat/0411489
Wimbush et al. JAP 2004
P. Leiderer et al. PRL (1993)
Irreproducibility
3 identical experiments: field ramp from 0 to 13.6 mT for 10 sec
the nucleation place:
the exact flux pattern:
well reproduced
never reproduced
Irreproducibility at high fields
Ba = 51 mT
T=8.0 K
The exact pattern is
never reproduced
Sample Inhomogeneities
OR
Instability-driven
D=1.05
D=1.35
D=1.65
D=1.75
Tthresh ~ 10K
“Shape” Model
Phys. Rev. B 70, 224502 (2004)
Conventional
flux jump mechanism:
1) Flux motion releases heat
2) T rise weakens flux pinning
T0  Jc    Q  T > T0
Thermal runaway
We look for spatially-nonuniform solutions
Slab
Thermal diffusion + electrodynamics
z
j,E
y
H
0
x
l
penetrated
by flux
no flux
Linear Analysis
unstable if Re  > 0
non-uniform if
ky0
Thermal diffusion
Nb disk,
Goodman et al.,
Phys. Lett. 18, 236 (1965)
Favors uniform jumps, ky=0
Why narrow fingers ?
J
the local J remains high
J
maximal jE
fastest possible growth
The instability increment
Fastest
growth:
0.0
Re

0.5
-0.5
Ba
-1.0
0
3
6
ky
Only when
9
Numerical Solution
Temperature
Electric field
y
T(t=0)=0.0001 * ``white noise”
to introduce all ky
sample
edge
Finger pattern with some characteristic ky is formed in a self-organized way
Numerical Solution, finger propagation
Increasing applied magnetic field
B
linearized j(E)
full non-linear j(E)
E
T
Beyond the linear regime
• a few strongest fingers survive
• and propagate into the flux free area
. . . . . in agreement with experiment
H(E) stability diagram
H
Fingering is not sensitive to
• initial T(x,y), E(x,y)
• boundary conditions
• Jc(B) dependence
Uniform
jumps
dynamic
criterion
Fingering
adiabatic criterion
Hadiab
S
t
Ec/n
a
b
l
Ec
e
E ~dH/dt
Slab
Thin strip
z
d<<w
j
Ba
B
y
x
Non-local electrodynamics:
2w
Heat removal into the substrate:
H > Hfj (E, h0)
Heat release by vortex motion
is slower than
Heat removal to the substrate
Uniform
jumps
H
Hadiab
Fingering
dynamic adiabatic criterion
criterion
S
t
a
b
l
e
slab
Stable
strip
E ~ dH/dt
Thin films are
* more unstable (avalanches occur at smaller H, dH/dt)
* stronger tendency for dendritic pattern (it forms at smaller dH/dT)
In agreement
with experiment
2 mT
more unstable
100 mT
10-4V/m
more dendritic
0.1 V/m
Dendrite core
width
In agreement with Leiderer’s
ultrafast experiments
We predict increase in dendrite width
close to the threshold T
Experimentally, dendrites become
more branching
Bulk
Experiment
Linear
theory
Film
…
Fingering
Fingering +
Branching
Fingering (easier)
Branching
No branching
Numerics &
Similations
Aranson et al.
Comparison with Aranson et al. PRL 2005
Uniform
jumps
H
Hadiab
Fingering
dynamic adiabatic criterion
criterion
S
t
a
b
l
e
Stable
Uniform/Fingering boundary
is not predicted
by Aranson et al.
slab
strip
E
Agrees with Aranson et al.
Compatible, but not overlapping with
Leiderer et al.
who finds the propagation velocity
• Big dendritic avalanches
1 mm
• “Shape” model
explaining dendritic pattern
We are here
• Small round avalanches
• “Size” model
20 mm
Down to small scales...
Ba
movie
17mT
t
200 sec
T=3.6K
5 mm
MgB2
film
Analyzing difference images
7.15 mT
=
MO image (7.165mT)
—
MO image (7.150mT)
Ba= 0.015mT, t=2.5 sec
local increase of flux density -
linear
ramp
of Ba
15 MO
images
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
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
Irreproducibility: small jumps
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
Avalanche size
10
6
10
5
10
4
2.500.0000
4mT
3
10
10
200
10
3
Number of Jumps
Flux jump size (0)
1. Typical size exists
2. It grows with Ba
2
4
8
12
16
10mT
2
10
40mT
1
10
20
10
Ba (mT)
2
10
3
10
4
10
5
Jump size (0)
10
6
10
7
Adiabatic approach
Heat stays where it has been released
OK if thermal diffusion is much slower than flux diffusion DT<<DM,
which is true since dendritic avalanches occur
Originally used by Swartz &Bean in 1968
Adiabatic critical state for a thin strip
is given by a set of equations:
Adiabatic :
All energy released by
flux motion is absorbed
Critical state
Flux that has passed through
“x” during avalanche
Biot-Savart
Intermediate result:
the adiabatic instability field for a thin strip
Consistent with our
“shape” model
(the limiting case no thermal
diffusion)
Demonstrates existence
of a threshold T
(above which jumps do not
occur no matter how large field
is applied)
Tth
temperature
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
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
Applied field, Ba / Bc(T0)
4
8
12
Ba (mT)
We fit
• Bfj ~ 2 mT
• Tth ~ 10 K
• (Ba) dependence
using only
one parameter:
Thermal origin
of avalanches
16
20
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
The model assumes uniformity along y,
and also applicable for “round” avalanches
Experiment:
Model:
xy
x
y
Thermal effects:
no difference
Magnetic effects: a factor ~ p/2
For dendrites:
the model is not applicable
The model assumesyuniformity along y,
but also applicable for “round” avalanches
Experiment:
Model:
xy
x
y
Thermal effects:
no difference
Magnetic effects: a factor ~ p/2
x
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%
10
6
10
5
10
4
10
3
10
2
Jump size
Flux jump size (0)
Imaging on global
scale to collect
global statistics
4
8
12
16
10
7
10
6
10
5
10
4
20
Ba (mT)
0
20
Ba (mT)
40
60
C ~ T3
Stabilization
at small dT, small scale
QT
[Swartz &Bean, JAP 1968]
heat diffusion k
[Wipf, Phys.Rev. 1967]
[Mints, Rakhmanov, J.Phys.D 1983]
Nonuniforn B(r)
Jump acceleration
at T~Tc, large scale
MO image
QM
Jc
0 at Tc
Thin films:
field enhancement due to current bending
J
“propagating flux jump”
Conservation of magnetic flux
flux jump results in a
local decrease of flux
density near the film edge
big
dendritic
jump
black regions
on difference images
small
jump
Large electric field needed,
Ramping magnetic field:
E ~ w*dH/dt
one needs:
E > 0.1 V/m
dH/dt ~ 100 T/s
our experiment:
0.001 T/s
Flux jumping in thin films
R.G.Mints and E.H. Brandt, PRB 1996
d=1mm, dH/dt=100T/s
Can local time-dependent fields be larger?
H=1T
E(r,t)  <E>
High resoltuion magneto-optical movie
25 mm
increasing applied field
real-time
superconductor
non-thermal
vortex avalanches
edge
NbSe crystal from P. Gammel
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
Counting vortices
B (r)
B (r)
1500
5 vortices has moved
11 vortices has entered from the edge
dH/dt
local E ~ 400 <E>
perhaps much more: 1ms << t << 1/24s
:
<E>=2.4 10-10 V/m
local B: local E ~ 10-7 V/m
Metal film suppresses thermomagnetic instability
MO indicator
Al-foil (10 micron)
MgB2 film
Heat removal (thermal contact)?
Electrodynamics?
dB/dt => E => j
50% dendrites back
Physica C 369, 93 (2002)
MO indicator
Al-foil (10 micron)
MO indicator
Polyethylene
micron)
Al-foil (10 (40
micron)
Polyethylene (40 micron)
MgB2 film
Poor thermal contact, but the instability is still suppressed
Conventional flux penetration
Flux density  Brightness
Magneto-optical
movie
Ba
1 mm
t
YBCO film