This document summarizes a phase field model used to study nucleation and avalanches in films with labyrinthine magnetic domains. The model uses a phase field approach with a power expansion energy functional to simulate the system. It produces two different limit behaviors depending on film thickness and disorder strength: multiple nucleation and coalescence, or expansion by a single branching domain. The model is used to analyze characteristic lengths, domain shapes, and avalanche statistics under variations of parameters like thickness, disorder strength, and temperature. Adding random field and anisotropy terms introduces additional parameters for random field strength and temperature noise.

1. Nuclea'on
and
avalanches
in
films
with
labyrinthine
magne'c
domains
Andrea
Benassi
&
Stefano
Zapperi

2. Outline
Experiments
on
labyrinthine
domains
Our
phase
field
model
Characteris>cs
Lengths
and
avalanche
sta>s>cs
A
new
version
of
the
phase
field
model
Ironing
stripe
domains
Memory
effects
In-­‐plane
magne>za>on:
very
preliminary
results
(yesterday)

3. Labyrinthine
domains
Phys.Rev.Le*.
92,
077206
(2004)
Phys.Rev.B
71,
104431
(2005)
Deconvolving
nucleaAon
they
find
a
0.5
exponent

4. Labyrinthine
domains
Avalanche
staAsAcs
taken
over
different
intervals
of
the
hysteresis
loop
show
different
criAcal
exponents
Appl.Phys.Le*.
95,
182504
(2009)

5. A
phase
field
model
V =
Ku
4
m2
2
−
m4
4
m =
M(r)
Ms
= m(x, y)

6. A
phase
field
model
V =
Ku
4
m2
2
−
m4
4
m =
M(r)
Ms
= m(x, y)
∂M(r, t)
∂t
= −Γ
δH[M(r, t)]
δM(r, t)
Energy
funcAonal
power
expansion
+
linear
relaAon
between
Ame
and
energy
fluctuaAons
Small
Ame
fluctuaAons
hypothesis

7. A
phase
field
model
hr(r) = 0
hr(r)hr(r
) = Dδ(r − r
)
V =
Ku
4
m2
2
−
m4
4
m =
M(r)
Ms
= m(x, y)
∂M(r, t)
∂t
= −Γ
δH[M(r, t)]
δM(r, t)
Energy
funcAonal
power
expansion
+
linear
relaAon
between
Ame
and
energy
fluctuaAons
Small
Ame
fluctuaAons
hypothesis
2
dimensionless
parameters
= 2
A/Ku
α = Ku/4µ0M2
s
γ = d/4π

8. A
phase
field
model
hr(r) = 0
hr(r)hr(r
) = Dδ(r − r
)
V =
Ku
4
m2
2
−
m4
4
m =
M(r)
Ms
= m(x, y)
∂M(r, t)
∂t
= −Γ
δH[M(r, t)]
δM(r, t)
Energy
funcAonal
power
expansion
+
linear
relaAon
between
Ame
and
energy
fluctuaAons
Small
Ame
fluctuaAons
hypothesis
2
dimensionless
parameters
= 2
A/Ku
α = Ku/4µ0M2
s
γ = d/4π

9. Two
different
limit
behaviors
Depending
on
the
film
thickness
and
on
the
disorder
strength
we
can
have
two
limit
behaviors
-4 -2 0
-0.5
0
0.5
42
b
c
d
f a
γ = 0.5
γ = 0.6
γ = 0.7
h
e
g
b c da
f g he

10. Two
different
limit
behavors
MulAple
nucleaAon
and
coalescence
by
bridging
Expansion
by
branching
of
a
single
domain
and
lateral
fa*ening

11. Characteris'c
lengths
m(x, y, d) = sin
πx
d
m(x, y, w) = tanh
x
w
d = α/γ domain width
w =
√
2 domain wall width
n nucleation diameter
MinimizaAon
of
the
energy
with
respect
to
a
fixed
magneAzaAon
configuraAon
with
one
parameter:
NucleaAon
depends
strongly
on
disorder,
any
analyAcal
theory
is
useless!!!

12. Characteris'c
lengths
m(x, y, d) = sin
πx
d
m(x, y, w) = tanh
x
w
d = α/γ domain width
w =
√
2 domain wall width
n nucleation diameter
MinimizaAon
of
the
energy
with
respect
to
a
fixed
magneAzaAon
configuraAon
with
one
parameter:
NucleaAon
depends
strongly
on
disorder,
any
analyAcal
theory
is
useless!!!
Avalanches

13. Triggering
of
minor
avalanches
The
difference
between
consecuAve
magneAzaAon
maps
allows
a
direct
imaging
of
avalanches

14. Avalanche
sta's'cs
Analysis
of
different
loop
regions:
• The
maximum
avalanche
size
decreases
as
the
domain
density
reaches
its
maximum
• NucleaAon
and
bridging,
with
their
characterisAc
size,
affect
the
size
distribuAon
NucleaAon
and
annihilaAon:
• For
nucleaAon
to
take
place
a
barrier
must
be
overcame,
its
value
goes
as
1/γ
• AnnihilaAon
is
almost
independent
of
the
dipolar
field
strength
• At
zero
temperature
the
gaussian
distribuAon
is
due
to
the
spaAal
disorder

15. Avalanche
sta's'cs
Different
film
thickness:
• The
avalanche
cutoff
increases
when
γ
is
decreased,
following
the
corresponding
increase
of
the
domain
width
and
confirming
that
α/γ
is
the
relevant
parameter
controlling
the
size
of
the
scaling
regime
Different
Disorder
strength:
• The
Larger
D
the
larger
the
external
field
at
which
walls
depin,
the
larger
their
jumps.
• Increasing
D
the
domains
shape
is
slightly
affected
by
the
disorder
strength
but
is
almost
independent
of
D,
• NucleaAon
diameter
decreases
with
increasing
D
d = α
n nucleation diame

16. Phase
field
model
reloaded
V = [1 − λ(r)]
¯Ku
4
m2
2
−
m4
4
˙m = α
dV
dm
+ ∇2
m
− γ
dr m(r
)
|r − r|3
+ hr(r) + he(t) + R(t)
hr(r) = 0 hr(r)hr(r
) = Dδ(r − r
)
Two
new
randomness
sources
means
two
new
physical
parameters
to
be
introduced…
λ(r) = 0 λ(r)λ(r
) = Aδ(r − r
)
R(r)R(r
) = 2KBTδ(r − r
)δ(t − t
)R(r) = 0
Random
field
Temperature
noise
Anisotropy
disorder
Random
field
and
random
anisotropy
has
the
same
effect
on
the
domains
topography,
except
that
the
type
of
domain
dynamics
(nucleaAon/coalescence
or
branching)
seems
to
be
a
bit
more
sensiAve
to
A
than
D.

17. Phase
field
model
reloaded
V = [1 − λ(r)]
¯Ku
4
m2
2
−
m4
4
˙m = α
dV
dm
+ ∇2
m
− γ
dr m(r
)
|r − r|3
+ hr(r) + he(t) + R(t)
hr(r) = 0 hr(r)hr(r
) = Dδ(r − r
)
Two
new
randomness
sources
means
two
new
physical
parameters
to
be
introduced…
λ(r) = 0 λ(r)λ(r
) = Aδ(r − r
)
R(r)R(r
) = 2KBTδ(r − r
)δ(t − t
)R(r) = 0
Random
field
Temperature
noise
Anisotropy
disorder
Random
field
and
random
anisotropy
has
the
same
effect
on
the
domains
topography,
except
that
the
type
of
domain
dynamics
(nucleaAon/coalescence
or
branching)
seems
to
be
a
bit
more
sensiAve
to
A
than
D.

18. Ironing
stripe
domains
No
disorder
(realizaAon
1)
No
disorder
(realizaAon
2)
Gaussian
disorder
• The
final
orientaAon
of
the
parallel
stripes
depends
on
the
iniAal
random
configuraAon
• The
presence
of
disorder
inhibits
the
complete
reorientaAon
OscillaAng
external
field
perpendicular
to
the
film
surface:
he(r) = h0 sin(ωt)
ω = 0.0126 Γµ0 ≡ 1 h0 = 2 hsat 4

19. Memory
effects
Hysteresis
loop
unrolled:
Ame
m

20. Memory
effects
Hysteresis
loop
unrolled:
Ame
Φ

21. ˙m = α
dV
dm
+ ∇2
m
− γ
dr m(r
)
|r − r|3
+ hr(r) + h
In-­‐plane
Magne'za'on
Just
modifying
the
dipolar
(stray)
field,
our
scalar
model
seems
to
be
able
to
reproduce
the
domain
dynamics
of
in-­‐plane
films.
Now
the
magneAzaAon
is
assumed
to
be
oriented
only
along
the
x-­‐axis
ranging
in
[-­‐1,+1]
an
External
field
is
applied
along
the
same
axis
to
record
hysteresis
loops.
+γ
dr 2(x − x
)2
− (y − y
)2
|r − r|
m(r
)

22. Open
Issues:
Which
quanAAes
can
be
used
to
characterize
the
memory
effects
and
the
stripes
domains?
One
Hysteresis
loop
takes
24
hours:
• Do
we
really
need
to
be
so
slow
in
increasing
the
field?
• How
many
loops
to
test
memory
effects?
• (Easy)
ParallelizaAon
will
speed
up
our
calculaAons
by
a
factor
of
4
Up
to
now
we
used
only
white
noise,
does
it
make
sense
to
define
a
characterisAc
length
for
the
noise
correlaAon?
Working
in
reciprocal
space
enable
us
to
deal
with
large
systems
but
we
are
forced
to
use
periodic
boundary
condiAons.
Edge
effects
cannot
be
taken
into
account
in
the
simulaAons
In
the
case
of
a
bubbles
lamce,
can
we
play
with
an
external
oscillaAng
field
in
the
same
way
we
do
for
stripe
domains,
to
try
to
order
the
lamce?