Models of exchange-bias in thin films have been able to describe various aspects of this technologically relevant effect. Through appropriate choices of free parameters the modelled hysteresis loops adequately match experiment, and typical domain structures can be simulated. However, the use of these parameters, notably the coupling strength between the systems' ferromagnetic (F) and antiferromagnetic (AF) layers, obscures conclusions about their influence on the magnetization reversal processes. Here we develop a 2D phase-field model of the magnetization process in exchange-biased CoO/(Co/Pt)×n that incorporates the 10 nm-resolved measured local biasing characteristics of the antiferromagnet. Just three interrelated parameters set to measured physical quantities of the ferromagnet and the measured density of uncompensated spins thus suffice to match the experiment in microscopic and macroscopic detail. We use the model to study changes in bias and coercivity caused by different distributions of pinned uncompensated spins of the antiferromagnet, in application-relevant situations where domain wall motion dominates the ferromagnetic reversal. We show the excess coercivity can arise solely from inhomogeneity in the density of biasing- and anti-biasing pinned uncompensated spins in the antiferromagnet. Counter to conventional wisdom, irreversible processes in the latter are not essential.
Cultivation of KODO MILLET . made by Ghanshyam pptx
the role of exchange bias in domain dynamics control
1. The
Role
of
Exchange
Bias
in
Domain
Dynamics
Control
A.
Benassi1,
M.A.
Marioni1,
D.
Passerone1
and
H.J.
Hug1,2
1-‐
EMPA
Swiss
Federal
InsHtute
for
Materials
Science
and
Technology,
Dübendorf
(Switzerland)
2-‐
Department
of
Physics,
Universität
Basel,
Basel
(Switzerland)
2. Sample
and
measurements
A
perpendicular
anisotropy
ferromagneHc
film
(FF)
is
anH-‐coupled
with
a
thinner
anHferromagneHc
film
(AF)
grown
on
top
Upon
cooling
below
the
Neel
temperature,
the
AF
becomes
ordered
except
for
few
atomic
layer
at
the
interface,
here
the
defects
at
the
interface
give
rise
to
a
distribuHon
of
uncompensated
spins
(UCS)
Being
the
Neel
temperature
of
the
AF
smaller
than
the
Curie
temperature
of
the
FF,
the
presence
of
the
ferromagneHc
domains
can
orient
the
uncompensated
spin
at
the
interface
during
the
cooling.
This
allow
us
to
fix
stably
the
FF
domain
structure
on
the
cooled
AF.
the
domains
image
is
taken
through
the
AF
layer
because
the
FF
field
is
orders
of
magnitude
stronger
the
UCS
image
is
taken
saturaHng
the
FF
domains
with
an
external
field
Schmid
e
al.
PRL
105
197201
(2010)
Joshi
et
al.
Appl.Phys.LeY.
98
082502
(2011)
Hext
uncompensated
frustrated
AF
FF
3. The
model
system
The
Landau-‐Lifshitz-‐Gilbert
(LLG)
equaHon
rules
the
precession
of
a
magneHc
dipole
in
an
external
field:
∂m
∂t
= −
γ
1 + ξ2
m ×
B + ξ
m × B
B = −
1
Ms
δH[m]
δm
+ Q(R, t)
Q(R, t) = 0
Q(R, t)Q(R
, t
) = δ(t − t
)δ(R − R
)2KBTξ/Msγ
Bm
precession
term
Bm
damping
term
dissipaHon
by
microscopic
degrees
of
freedom
Bm
stochasHc
term
thermal
fluctuaHons
Under
the
following
approximaHons:
• scalar
magneHzaHon
uniform
along
the
FF
thickness
d
• domain
walls
smaller
than
the
domain
size
• small
FF
thickness
d
the
magneHzaHon
in
the
FF
can
be
described
by
the
following
hamiltonian
power
expansion:
Gilbert
IEEE
Trans.
On
MagneHcs
40
3434
(2004)
Brown
Phys.Rev.
130
1677
(1963)
Usadel
PRB
73
212405
(2006)
m(r, t) = m(x, y, t)ˆz
H =
d3
R
− Ku(R)
m2
2
+
A
2
(∇Rm)2
+
µ0M2
s d
8π
d2
R m(R
)m(R)
|R − R|3
− µ0Msm(Hext − HUCS(R))
4. The
model
system
∂m
∂t
= −
γ
1 + ξ2
m ×
B + ξ
m × B
B = −
1
Ms
δH[m]
δm
+ Q(R, t)
Q(R, t) = 0
Q(R, t)Q(R
, t
) = δ(t − t
)δ(R − R
)2KBTξ/Msγ
Bm Bm
Under
the
following
approximaHons:
• scalar
magneHzaHon
uniform
along
the
FF
thickness
d
• domain
walls
smaller
than
the
domain
size
• small
FF
thickness
d
the
magneHzaHon
in
the
FF
can
be
described
by
the
following
hamiltonian
power
expansion:
Jagla
PRB
72
094406
(2005)
Jagla
PRB
70
046204
(2004)
m(r, t) = m(x, y, t)ˆz
The
Landau-‐Lifshitz-‐Gilbert
(LLG)
equaHon
rules
the
precession
of
a
magneHc
dipole
in
an
external
field:
precession
term
Bm
damping
term
dissipaHon
by
microscopic
degrees
of
freedom
stochasHc
term
thermal
fluctuaHons
H =
d3
R
− Ku(R)
m2
2
+
A
2
(∇Rm)2
+
µ0M2
s d
8π
d2
R m(R
)m(R)
|R − R|3
− µ0Msm(Hext − HUCS(R))
5. The
model
system
Under
the
following
approximaHons:
• scalar
magneHzaHon
uniform
along
the
FF
thickness
d
• domain
walls
smaller
than
the
domain
size
• small
FF
thickness
d
the
magneHzaHon
in
the
FF
can
be
described
by
the
following
hamiltonian
power
expansion:
m(r, t) = m(x, y, t)ˆz
Jagla
PRB
72
094406
(2005)
Jagla
PRB
70
046204
(2004)
anisotropy
term:
its
fluctuaHons
around
an
average
value
provides
strong
pinning
points
for
the
domain
walls
Ku(R) = Ku(1 − P(x, y))
P(R) = 0
P(R)P(R
) = θδ(R − R
)
H =
d3
R
− Ku(R)
m2
2
+
A
2
(∇Rm)2
+
µ0M2
s d
8π
d2
R m(R
)m(R)
|R − R|3
− µ0Msm(Hext − HUCS(R))
6. H =
d3
R
− Ku(R)
m2
2
+
A
2
(∇Rm)2
+
µ0M2
s d
8π
d2
R m(R
)m(R)
|R − R|3
− µ0Msm(Hext − HUCS(R))
The
model
system
anisotropy
term:
its
fluctuaHons
around
an
average
value
provides
strong
pinning
points
for
the
domain
walls
Under
the
following
approximaHons:
• scalar
magneHzaHon
uniform
along
the
FF
thickness
d
• domain
walls
smaller
than
the
domain
size
• small
FF
thickness
d
the
magneHzaHon
in
the
FF
can
be
described
by
the
following
hamiltonian
power
expansion:
m(r, t) = m(x, y, t)ˆz
Ku(R) = Ku(1 − P(x, y))
P(R) = 0
P(R)P(R
) = θδ(R − R
)
Jagla
PRB
72
094406
(2005)
Jagla
PRB
70
046204
(2004)
anisotropy
term:
it
represents
the
energy
cost
for
the
domain
walls.
we
do
not
have
real
Block
or
Neel
walls,
just
their
projecHon
along
z
7. H =
d3
R
− Ku(R)
m2
2
+
A
2
(∇Rm)2
+
µ0M2
s d
8π
d2
R m(R
)m(R)
|R − R|3
− µ0Msm(Hext − HUCS(R))
The
model
system
anisotropy
term:
its
fluctuaHons
around
an
average
value
provides
strong
pinning
points
for
the
domain
walls
Under
the
following
approximaHons:
• scalar
magneHzaHon
uniform
along
the
FF
thickness
d
• domain
walls
smaller
than
the
domain
size
• small
FF
thickness
d
the
magneHzaHon
in
the
FF
can
be
described
by
the
following
hamiltonian
power
expansion:
m(r, t) = m(x, y, t)ˆz
Ku(R) = Ku(1 − P(x, y))
P(R) = 0
P(R)P(R
) = θδ(R − R
)
Jagla
PRB
72
094406
(2005)
Jagla
PRB
70
046204
(2004)
anisotropy
term:
it
represents
the
energy
cost
for
the
domain
walls.
we
do
not
have
real
Block
or
Neel
walls,
just
their
projecHon
along
z
stray
field
energy:
responsible
for
the
domain
formaHon
non
local
term
to
be
treated
in
reciprocal
space
8. H =
d3
R
− Ku(R)
m2
2
+
A
2
(∇Rm)2
+
µ0M2
s d
8π
d2
R m(R
)m(R)
|R − R|3
− µ0Msm(Hext − HUCS(R))
The
model
system
anisotropy
term:
its
fluctuaHons
around
an
average
value
provides
strong
pinning
points
for
the
domain
walls
Under
the
following
approximaHons:
• scalar
magneHzaHon
uniform
along
the
FF
thickness
d
• domain
walls
smaller
than
the
domain
size
• small
FF
thickness
d
the
magneHzaHon
in
the
FF
can
be
described
by
the
following
hamiltonian
power
expansion:
m(r, t) = m(x, y, t)ˆz
Ku(R) = Ku(1 − P(x, y))
P(R) = 0
P(R)P(R
) = θδ(R − R
)
Jagla
PRB
72
094406
(2005)
Jagla
PRB
70
046204
(2004)
anisotropy
term:
it
represents
the
energy
cost
for
the
domain
walls.
we
do
not
have
real
Block
or
Neel
walls,
just
their
projecHon
along
z
stray
field
energy:
responsible
for
the
domain
formaHon
non
local
term
to
be
treated
in
reciprocal
space
UCS
field:
as
measured
in
the
experiment
9. H =
d3
R
− Ku(R)
m2
2
+
A
2
(∇Rm)2
+
µ0M2
s d
8π
d2
R m(R
)m(R)
|R − R|3
− µ0Msm(Hext − HUCS(R))
The
model
system
anisotropy
term:
its
fluctuaHons
around
an
average
value
provides
strong
pinning
points
for
the
domain
walls
Under
the
following
approximaHons:
• scalar
magneHzaHon
uniform
along
the
FF
thickness
d
• domain
walls
smaller
than
the
domain
size
• small
FF
thickness
d
the
magneHzaHon
in
the
FF
can
be
described
by
the
following
hamiltonian
power
expansion:
m(r, t) = m(x, y, t)ˆz
Ku(R) = Ku(1 − P(x, y))
P(R) = 0
P(R)P(R
) = θδ(R − R
)
Jagla
PRB
72
094406
(2005)
Jagla
PRB
70
046204
(2004)
anisotropy
term:
it
represents
the
energy
cost
for
the
domain
walls.
we
do
not
have
real
Block
or
Neel
walls,
just
their
projecHon
along
z
stray
field
energy:
responsible
for
the
domain
formaHon
non
local
term
to
be
treated
in
reciprocal
space
UCS
field:
as
measured
in
the
experiment
External
field:
uniform
but
Hme
dependent
10. Pueng
this
approximate
hamiltonian
inside
the
LLG
equaHon
we
obtain
the
following
equaHon
of
moHon:
where
everything
is
now
in
dimensionless
units:
the
model
contains
only
three
non-‐independent
dimensionless
parameters
related
to
the
material
proper.es:
∂m
∂τ
= (1 − m2
)
α(1 − p(r)) m −
1
4π
d2
r m(r
)
|r − r|3
+ hext(t) − hUCS(r) + q(r, τ)
+ β∇2
rm
The
model
system
Jagla
PRB
72
094406
(2005)
Jagla
PRB
70
046204
(2004)
Zhirnov
Zh.Eksp.Teor.Fiz.
35
1175
(1958)
r = R/d
τ = tγµ0Ms/ξ
hext = Hext/Ms
hUCS = HUCS/Ms
q(r, τ) = Q(R, t)/µ0Ms
KBT = KBT/µ0M2
s d3
dimensionless
posiHon
dimensionless
Hme
dimensionless
fields
η = θ/d3
β = A/µ0M2
s d2
α = Ku/µ0M2
s
dimensionless
temperature
dimensionless
thermal
noise
p(r)p(r
) = ηδ(r − r
) dimensionless
anisotropy
noise
uniaxial
anisotropy
exchange
sHffness
anisotropy
fluctuaHons
(strength
on
the
pinning
disorder)
11. Pueng
this
approximate
hamiltonian
inside
the
LLG
equaHon
we
obtain
the
following
equaHon
of
moHon:
where
everything
is
now
in
dimensionless
units:
the
model
contains
only
three
non-‐independent
dimensionless
parameters
related
to
the
material
proper.es:
The
model
system
Jagla
PRB
72
094406
(2005)
Jagla
PRB
70
046204
(2004)
Zhirnov
Zh.Eksp.Teor.Fiz.
35
1175
(1958)
r = R/d
τ = tγµ0Ms/ξ
hext = Hext/Ms
hUCS = HUCS/Ms
q(r, τ) = Q(R, t)/µ0Ms
KBT = KBT/µ0M2
s d3
dimensionless
posiHon
dimensionless
Hme
dimensionless
fields
η = θ/d3
β = A/µ0M2
s d2
α = Ku/µ0M2
s
dimensionless
temperature
dimensionless
thermal
noise
p(r)p(r
) = ηδ(r − r
) dimensionless
anisotropy
noise
uniaxial
anisotropy
exchange
sHffness
anisotropy
fluctuaHons
(strength
on
the
pinning
disorder)
∂m
∂τ
= (1 − m2
)
α(1 − p(r)) m −
1
4π
d2
r m(r
)
|r − r|3
+ hext(t) − hUCS(r) + q(r, τ)
+ β∇2
rm
12. Pueng
this
approximate
hamiltonian
inside
the
LLG
equaHon
we
obtain
the
following
equaHon
of
moHon:
where
everything
is
now
in
dimensionless
units:
the
model
contains
only
three
non-‐independent
dimensionless
parameters
related
to
the
material
proper.es:
The
model
system
Jagla
PRB
72
094406
(2005)
Jagla
PRB
70
046204
(2004)
Zhirnov
Zh.Eksp.Teor.Fiz.
35
1175
(1958)
r = R/d
τ = tγµ0Ms/ξ
hext = Hext/Ms
hUCS = HUCS/Ms
q(r, τ) = Q(R, t)/µ0Ms
KBT = KBT/µ0M2
s d3
dimensionless
posiHon
dimensionless
Hme
dimensionless
fields
η = θ/d3
β = A/µ0M2
s d2
α = Ku/µ0M2
s
dimensionless
temperature
dimensionless
thermal
noise
p(r)p(r
) = ηδ(r − r
) dimensionless
anisotropy
noise
uniaxial
anisotropy
exchange
sHffness
anisotropy
fluctuaHons
(strength
on
the
pinning
disorder)
domain
wall
energy
domain
wall
width
domain
size
domain
morphology
∝
β/α
∝
αβ
∂m
∂τ
= (1 − m2
)
α(1 − p(r)) m −
1
4π
d2
r m(r
)
|r − r|3
+ hext(t) − hUCS(r) + q(r, τ)
+ β∇2
rm
13. Model
validaHon:
from
micro
to
macro
The
procedure
for
the
determinaHon
of
the
three
material
parameters
is
made
in
such
a
way
to
fit
both
macroscopic
and
microscopic
properHes
of
the
sample:
A
good
iniHal
guess
for
α,β
and
η
makes
the
measured
domain
image
at
0
mT
a
steady
state
of
our
equaHon
of
moHon.
A
good
choice
of
α,β
and
η
is
such
that,
if
we
use
the
measured
image
as
the
iniHal
condiHon
of
our
equaHon
of
moHon
and
we
let
it
evolve
in
Hme,
it
will
not
change.
Benassi
et
al.
(waiHng
for
PRL
rejecHon)
14. Model
validaHon:
from
micro
to
macro
The
procedure
for
the
determinaHon
of
the
three
material
parameters
is
made
in
such
a
way
to
fit
both
macroscopic
and
microscopic
properHes
of
the
sample:
A
good
iniHal
guess
for
α,β
and
η
makes
the
measured
domain
image
at
0
mT
a
steady
state
of
our
equaHon
of
moHon.
A
good
choice
of
α,β
and
η
is
such
that,
if
we
use
the
measured
image
as
the
iniHal
condiHon
of
our
equaHon
of
moHon
and
we
let
it
evolve
in
Hme,
it
will
not
change.
The
measured
field
from
the
UCS
distribuHon
is
also
included
in
the
equaHon
of
moHon
and
it
helps
in
stabilizing
the
domain
configuraHon.
Benassi
et
al.
(waiHng
for
PRL
rejecHon)
15. Model
validaHon:
from
micro
to
macro
The
procedure
for
the
determinaHon
of
the
three
material
parameters
is
made
in
such
a
way
to
fit
both
macroscopic
and
microscopic
properHes
of
the
sample:
Than
we
ramp
up
the
external
uniform
field
and
we
trim
the
parameters
in
such
a
way
to
reproduce
the
correct
path
to
saturaHon
looking
also
at
the
MFM
taken
at
100
mT,
200
mT
and
300
mT.
Benassi
et
al.
(waiHng
for
PRL
rejecHon)
16. Model
validaHon:
from
micro
to
macro
The
procedure
for
the
determinaHon
of
the
three
material
parameters
is
made
in
such
a
way
to
fit
both
macroscopic
and
microscopic
properHes
of
the
sample:
Once
that
the
microscopic
properHes
are
well
reproduced
we
can
check
the
macroscopic
ones
(hysteresis
loops).
We
can
sHll
trim
a
bit
the
model
parameters
to
adjust
the
fine
detail.
Eventually
we
have
to
go
back
and
control
again
the
microscopic
behavior.
The
loops
were
measured
with
a
sample
cooled
in
a
saturaHng
field
so
the
printed
UCS
distribuHon
is
different
from
the
previous
one.
Benassi
et
al.
(waiHng
for
PRL
rejecHon)
17. Model
validaHon:
from
micro
to
macro
The
procedure
for
the
determinaHon
of
the
three
material
parameters
is
made
in
such
a
way
to
fit
both
macroscopic
and
microscopic
properHes
of
the
sample:
Benassi
et
al.
(waiHng
for
PRL
rejecHon)
αinit = 6.25
βinit = 0.85
ηinit = 1.5 × 10−4
α = Ku/µ0M2
s = 6.6 → Ku = 3.1 × 106
J/m3
β = A/µ0M2
s d2
= 0.88 → A = 2.2 × 10−10
J/m
η = 1.88 × 10−4
perfectly
in
the
expected
range
too
big
but
our
1D
walls
are
less
expensive
than
a
real
block
wall
and
A
must
compensate!
18. Path
to
saturaHon:
the
full
dynamics
Now
that
the
parameters
are
fixed,
the
theoreHcal
model
allows
us
to
access
the
full
dynamics
in
Hme,
we
can
thus
invesHgate
the
domain
behavior
with
more
than
few
MFM
images
Benassi
et
al.
(waiHng
for
PRL
rejecHon)
As
demonstrated
by
the
experiments
the
domains,
retracHng
with
increasing
external
field,
will
try
to
avoid
the
frustrated
F/AF
coupling
regions.
19. UnmounHng
the
machinery
Benassi
et
al.
(waiHng
for
PRL
rejecHon)
Now
we
can
switch
off
separately
the
different
hamiltonian
terms
and
try
to
understand
and
quanHfy
their
contribuHon:
The
UCS
distribuHon
alone
has
not
enough
strength
to
pin
the
domains
and
even
at
0
mT
the
shape
of
the
steady
configuraHon
is
quite
different
from
the
original
one.
The
saturaHon
occurs
too
early!
The
anisotropy
fluctuaHons
have
enough
strength
to
keep
the
iniHal
configuraHon
pinned,
however,
without
the
help
of
the
UCS
local
field,
the
pinning
sHll
occurs
too
early.
20. Now
we
can
also
try
to
predict
which
is
the
effect
of
a
different
UCS
distribuHon
on
the
exchange-‐bias
(EB)
effect
and
on
the
coercivity
of
the
FF.
• As
expected,
switching
off
the
UCS
field
the
EB
effect
goes
to
zero
• Doubling
the
average
value
of
the
UCS
field,
without
changing
the
fluctuaHons
strength,
increases
the
EB
effect
without
affecHng
the
coercivity
• Doubling
the
fluctuaHons
of
the
UCS
field,
without
shiking
its
average,
increase
strongly
the
coercivity
with
minor
changings
in
the
EB
loop
shik.
Some
predicHons
on
the
macroscopic
properHes
calc. 10 K
-0.5
0.0
0.5
1.0
-1.0
-0.4 -0.2 -0.1 0.0 0.1 0.2 0.4-0.3 0.3
no UCS
double UCS
average
double UCS
fluctuations
Benassi
et
al.
(waiHng
for
PRL
rejecHon)
21. Up
to
now
the
fluctuaHons
of
the
uniaxial
anisotropy
have
been
considered
to
be
uncorrelated
(white
noise),
however
they
have
something
to
do
with
the
granularity
of
the
sample.
Something
more
about
the
strength
and
the
correlaHon
of
these
fluctuaHon
can
be
inferred
from
Barkhausen
noise
measurements.
The
presence
of
Chromium
atoms
in
the
AF
decouples
the
magneHc
moment
of
neighboring
grains,
increasing
the
UCS
field
by
the
40%.
The
model
will
be
used
to
study
this
new
sample
in
which
the
role
of
the
UCS
map
as
been
enhanced.
ParHcular
aYenHon
will
be
given
to
the
return
point
memory
effects.
The
code
is
easily
parallelizable
allowing
for
the
descripHon
of
lager
system
or
for
the
coupling
of
two
interacHng
magneHzed
films
Further
developments
Benassi
et
al.
PRB
84
214441
(2011)