Vortex avalanches in superconductors:
Size distribution and Mechanism
Daniel Shantsev
Tom Johansen
and
Yuri Galperin
AMCS group
Department of Physics,
University of Oslo
A. V. Bobyl
A. F. Ioffe Institute, St. Petersburg, Russia
Vortices in Superconductors
Vortex lattice
A. A. Abrikosov
2003
(published 1957)
Hc2
Normal state
Mixed state
(vortex matter)
Hc1
Type II
Meissner state
Temperature
Tc
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
Avalanches ???
Motivation
to study vortex avalanches
The slope of the vortex pile - the critical current density Jc –
is the key parameter for many applications of superconductors
Trapped field magnets
High-current cables
Jc
Record trapped field:
17 Tesla
~100 times better than Cu wire
Measuring avalanches
H
Hall probe
YBCO
Size distribution
SOC
or
Statistics of vortex avalanches
Reference
Geometry
Material
Sensor
Avalanche
type
Avalanche
distribution
Heiden &
Rochlin
PRL (1968)
Hollow
cylinder
Pb-In
Coil
Off the
edge
Exponential
Field et al
PRL (1995)
Hollow
cylinder
Nb-Ti
Coil
Off the
edge
Power law
(slow ramps)
Zieve et al
PRB (1996)
Planar
YBCO
crystal
1 Hall
probe
Internal
Peaked
2 Hall
probes
Off the
edge &
internal
Peaked or
Ring
Nb
film
Planar
BSCCO
crystal
SQUID
Off the
edge
Planar
Nb
film
Hall probe
arrang.
Nowak et al
PRB (1997)
Aegerter
PRE (1998)
Behnia et al
PRB (2000)
Why peaked?
Power law
(dep. on T)
Exp or
Power law
(dep. on T & t)
Peaked or
Internal
Power Law
(dep. on H & T)
Thermal effects
1) Flux motion releases heat
2) T rise weakens flux pinning
T0  Jc    Q  T > T0
Can it also affect
the statistics of small
avalanches?
and in what way?
The thermal instability can lead to catastrophic avalanches with thermal runaways (flux jumps)
and sometimes remarkable flux patterns
1 mm
Magneto-optical movie of
flux penetration in MgB2 film
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
Down to small scales...
Flux penetration
on small scales :
5 mm
in space:
MgB2 film
• highly
non-uniform
Ba
rise
100 mm
in time:
• gradual
or
abrupt ???
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
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
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
Adiabatic approach
Heat stays where it has been released
OK if thermal diffusion is much slower than flux diffusion DT<<DM
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
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
Conclusions
Trivial conclusions:
• Flux avalanches are observed in superconducting films
using magneto-optical imaging
• They have a charactristic size (~1000 0) that grows with Ba
• Adiabatic model for the size of thermal flux avalanche
in a thin film is developed
• Agreement with experiment
(the thershold Ba, threshold T, size(Ba)-dependence)
Deep conclusions:
Thermal mechanism can be responsible for
microscopic avalanches (not only catastrophic jumps)
and leads to a peaked size distribution
Thermal effects contribute to formation of the critical state
(and modify Jc ) without destroying it
Phys. Rev. B 72, 024541 (2005)
http://www.fys.uio.no/super/
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)
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%