Nonequilibrium symmetry breaking
and pattern formation in magnetic films
Josh Deutsch
University of California
Santa Cruz
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 1/6
Collaborators
Michael S. Pierce, Conor R. Buechler, Larry B. Sorensen,
University of Washington
Eduardo A. Jagla,
The Abdus Salam International Centre for Theoretical Physics
Trieu Mai, Onuttom Narayan,
University of California, Santa Cruz
Joshua J. Turner, Steve D. Kevan,
University of Oregon, Eugene
Karine M. Chesnel, Jeff B. Kortright,
Lawrence Berkeley National Laboratory
Olav Hellwig, Eric E. Fullerton,
Hitachi Global Storage Technologies, San Jose,
Joseph E. Davies, Kai Liu,
University of California, Davis,
Jonathan Hunter Dunn,
MAX Laboratory, Lund
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 2/6
Outline
Introduction
Non-equilibrium statistical mechanics
Return point memory
Co/Pt multilayer films
Experiments
Correlations between states
Lack of configurational symmetry
Theoretical explanation
Simulations
Predictions
Conclusions
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 3/6
Equilibrium vs non-equilibrium
Simple example:water-vapor:
In equilibrium, we can calculate many things very precisely,
like velocity distribution function, specific heat.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 4/6
Equilibrium vs non-equilibrium
Simple example:water-vapor:
In equilibrium, we can calculate many things very precisely,
like velocity distribution function, specific heat.
If we cool down below saturation, we may start nucleating
water droplets. Their formation is an example of
non-equilibrium statistical mechanics.
Critical nucleus of a crystal of hard colloidal
spheres. (Daan Frenkel and Stefan Auer)
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 4/6
Harder examples
The equilibrium of a spin glass (e.g.
CuMn) below the spin glass transition
temperature. The ground state is hard to
find. This can be proved to be NP complete, or take exponentially long to compute.
(Michael Jünger)
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 5/6
Harder examples
The equilibrium of a spin glass (e.g.
CuMn) below the spin glass transition
temperature. The ground state is hard to
find. This can be proved to be NP complete, or take exponentially long to compute.
(Michael Jünger)
The nonequilibrium behavior of a
turbulent fluid. This is related to
computational complexity. Simplified lattice models are capable of showing universal computation. Many questions related to
such systems are provably undecidable.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 5/6
Irreversibility and Avalanches
Zoom in on part of a hysteresis loop:
M
H
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 6/6
Irreversibility and Avalanches
M
H
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 6/6
Irreversibility and Avalanches
M
H
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 6/6
Irreversibility and Avalanches
M
Irreversible
H
There is very fast motion during the avalanche, independent of the
speed at which the field is varied (within reason).
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 6/6
Irreversibility and Avalanches
M
Irreversible
H
There is very fast motion during the avalanche, independent of the
speed at which the field is varied (within reason).
This leads to irreversibility and hysteresis.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 6/6
Applications of Ferromagnetism
Doodle Pads
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 7/6
Applications of Ferromagnetism
Doodle Pads
Refrigerator Magnets
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 7/6
Return Point Memory
Magnetic systems containing only ferromagnetic couplings often
display "Return Point Memory": a system can return to the same
state after an excursion back to a previous magnetic field. (Perkovic,
Dahmen and Sethna PRL (1995).
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 8/6
Return Point Memory
Magnetic systems containing only ferromagnetic couplings often
display "Return Point Memory": a system can return to the same
state after an excursion back to a previous magnetic field. (Perkovic,
Dahmen and Sethna PRL (1995).
M
H
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 8/6
Return Point Memory
Magnetic systems containing only ferromagnetic couplings often
display "Return Point Memory": a system can return to the same
state after an excursion back to a previous magnetic field. (Perkovic,
Dahmen and Sethna PRL (1995).
M
H
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 8/6
Return Point Memory
Magnetic systems containing only ferromagnetic couplings often
display "Return Point Memory": a system can return to the same
state after an excursion back to a previous magnetic field. (Perkovic,
Dahmen and Sethna PRL (1995).
M
H
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 8/6
Return Point Memory
Magnetic systems containing only ferromagnetic couplings often
display "Return Point Memory": a system can return to the same
state after an excursion back to a previous magnetic field. (Perkovic,
Dahmen and Sethna PRL (1995).
M
H
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 8/6
Return Point Memory
Magnetic systems containing only ferromagnetic couplings often
display "Return Point Memory": a system can return to the same
state after an excursion back to a previous magnetic field. (Perkovic,
Dahmen and Sethna PRL (1995).
M
H
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 8/6
Return Point Memory
Magnetic systems containing only ferromagnetic couplings often
display "Return Point Memory": a system can return to the same
state after an excursion back to a previous magnetic field. (Perkovic,
Dahmen and Sethna PRL (1995).
M
H
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 8/6
Nested Branches
Nested sub-loops also return to the same state
A
M
C
(3)
(1)
Hmin
E
Hmax
D
H
(2)
B
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 9/6
RFIM
They proved this for the random field Ising Model (RFIM):
X
X
X
H=−
Ji,j Si Sj −
hi Si − H
Si .
hi,ji
i
i
Si = ±1.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 10/6
RFIM
They proved this for the random field Ising Model (RFIM):
X
X
X
H=−
Ji,j Si Sj −
hi Si − H
Si .
hi,ji
i
i
Si = ±1.
Arbitrary positive couplings Ji,j .
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 10/6
RFIM
They proved this for the random field Ising Model (RFIM):
X
X
X
H=−
Ji,j Si Sj −
hi Si − H
Si .
hi,ji
i
i
Si = ±1.
Arbitrary positive couplings Ji,j .
Arbitrary fields hi .
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 10/6
RFIM
They proved this for the random field Ising Model (RFIM):
X
X
X
H=−
Ji,j Si Sj −
hi Si − H
Si .
hi,ji
i
i
Si = ±1.
Arbitrary positive couplings Ji,j .
Arbitrary fields hi .
RPM does not hold in general when some of the couplings
become antiferromagnetic (negative). Also effects of finite
temperature destroy the exact nature of this result.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 10/6
Question:
To what extent does a real system return to its initial state after an
excursion away from it?
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 11/6
Question:
To what extent does a real system return to its initial state after an
excursion away from it?
A realistic Hamiltonian is expected to have a different form, for
example
H=−
X
i,j
Ji,j si · sj − B
X
i
s2z,i
−H
X
sz,i .
i
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 11/6
Question:
To what extent does a real system return to its initial state after an
excursion away from it?
A realistic Hamiltonian is expected to have a different form, for
example
H=−
X
i,j
Ji,j si · sj − B
X
i
s2z,i
−H
X
sz,i .
i
The couplings Ji,j can have a short range ferromagnetic component
but a long range antiferromagnetic component, and are not always
positive.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 11/6
Question:
To what extent does a real system return to its initial state after an
excursion away from it?
A realistic Hamiltonian is expected to have a different form, for
example
H=−
X
i,j
Ji,j si · sj − B
X
i
s2z,i
−H
X
sz,i .
i
The couplings Ji,j can have a short range ferromagnetic component
but a long range antiferromagnetic component, and are not always
positive.
There is no random field term because microscopically, the field is
produced by magnetic moments.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 11/6
Hamiltonian Symmetry
For a real system, we expect the Hamiltonian is invariant under
si → −si , H → −H
because all terms at bilinear in the variables s, H.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 12/6
Hamiltonian Symmetry
For a real system, we expect the Hamiltonian is invariant under
si → −si , H → −H
because all terms at bilinear in the variables s, H.
This symmetry is broken
P for the Random Field Ising Model
because of the term i hi Si
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 12/6
Finite Temperature
In addition a RPM is not expected to hold for finite temperature.
How do we characterize this case?
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 13/6
Co/Pt Multilayers
Experiments were performed at the Advanced Light Source (LBL).
The pinhole creates light coherent over a length of ≈ 40µm.
By exploiting resonant x-ray magnetic scattering, the scattering can
be made to have a strong magnetic component, producing a speckle
pattern.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 14/6
Speckle Pattern
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 15/6
3 mTorr Sample
[Nb(3.5 nm)/Si(3.0 nm)]40
XTEM image using pseudocolor. (Cross section transmission
electron microscopy). (E. E. Fullerton et al, PRB 48 17433 (1993))
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 16/6
15 mTorr Sample
(E. E. Fullerton et al)
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 17/6
Hysteresis loops
Pressures of 3, 7, 8.5, 10, and 12 mTorr
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 18/6
Correlations of states on loop
Look at correlations between states for different realizations, but the
same field.
M
H
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 19/6
Correlations of states on loop
Look at correlations between states for different realizations, but the
same field.
M
H
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 19/6
Correlations of states on loop
Look at correlations between states for different realizations, but the
same field.
M
H
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 19/6
Correlations of states on loop
Look at correlations between states for different realizations, but the
same field.
M
H
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 19/6
Correlations of states on loop
Look at correlations between states for different realizations, but the
same field.
M
H
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 19/6
Correlations of states on loop
Look at correlations between states for different realizations, but the
same field.
M
H
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 19/6
Correlations of states on loop
Look at correlations between states for different realizations, but the
same field.
M
H
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 19/6
Correlations of states on loop
Look at correlations between states for different realizations, but the
same field.
M
H
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 19/6
Definition of correlation
What we would like is to compare patterns in real space:
A state i with spins si (r) with
a state j with spins sj (r)
The un-normalized covariance between two spin configurations is
defined as:
cov(i, j) = hsi (r) · sj (r)ir − hsi (r)ir · hsi (r)ir
p
The normalized covariance is ρ = cov(i, j)/ cov(i, i)cov(j, j)
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 20/6
Definition of correlation
What we would like is to compare patterns in real space:
A state i with spins si (r) with
a state j with spins sj (r)
The un-normalized covariance between two spin configurations is
defined as:
cov(i, j) = hsi (r) · sj (r)ir − hsi (r)ir · hsi (r)ir
p
The normalized covariance is ρ = cov(i, j)/ cov(i, i)cov(j, j)
However experimentally, we have speckle data, which is in k-space.
It looks the same as above, substituting s for |ŝz (k)|2
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 20/6
Experimental RPM data
This correlation coefficient has been referred to as "RPM".
Data for the 8.5 mTorr sample.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 21/6
Compare complementary states
Look at correlations between complementary states: H → −H.
M
H
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 22/6
Compare complementary states
Look at correlations between complementary states: H → −H.
M
H
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 22/6
Compare complementary states
Look at correlations between complementary states: H → −H.
M
H
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 22/6
Compare complementary states
Look at correlations between complementary states: H → −H.
M
H
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 22/6
Compare complementary states
Look at correlations between complementary states: H → −H.
M
H
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 22/6
Compare complementary states
Look at correlations between complementary states: H → −H.
M
H
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 22/6
Compare complementary states
Look at correlations between complementary states: H → −H.
M
H
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 22/6
Compare complementary states
Look at correlations between complementary states: H → −H.
M
H
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 22/6
Definition of CPM
p
The normalized covariance is ρ = cov(i, j)/ cov(i, i)cov(j, j)
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 23/6
Definition of CPM
p
The normalized covariance is ρ = cov(i, j)/ cov(i, i)cov(j, j)
Consider spin configurations at field H on leg i of a hysteresis loop.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 23/6
Definition of CPM
p
The normalized covariance is ρ = cov(i, j)/ cov(i, i)cov(j, j)
Consider spin configurations at field H on leg i of a hysteresis loop.
The RPM normalized covariance is ρ(H, i; H, j), where i and j are
both legs going in the same direction.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 23/6
Definition of CPM
p
The normalized covariance is ρ = cov(i, j)/ cov(i, i)cov(j, j)
Consider spin configurations at field H on leg i of a hysteresis loop.
The RPM normalized covariance is ρ(H, i; H, j), where i and j are
both legs going in the same direction.
The "Complementary Point Memory" (CPM) normalized
covariance is ρ(H, i; −H, j) where i and j are legs going in opposite
directions.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 23/6
Experimental CPM data
Data for the 8.5 mTorr sample. Starting off at negative fields, going
to high fields.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 24/6
CPM < RPM
Why?
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 25/6
Unexpected Behavior
For a real physical system, one expects complete microscopic
symmetry because the Hamiltonian is invariant under
si → −si , H → −H
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 26/6
Unexpected Behavior
For a real physical system, one expects complete microscopic
symmetry because the Hamiltonian is invariant under
si → −si , H → −H
Not seen experimentally.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 26/6
RPM vs disorder
RPM and CPM both increase with disorder. That is disorder
stabilizes patterns.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 27/6
Possible explanation
There is a symmetry breaking term in the Hamiltonian, like
X
hi sz,i .
i
That is some elusive random fields.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 28/6
Possible explanation
There is a symmetry breaking term in the Hamiltonian, like
X
hi sz,i .
i
That is some elusive random fields.
But what is the source of these random fields?
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 28/6
Possible explanation
There is a symmetry breaking term in the Hamiltonian, like
X
hi sz,i .
i
That is some elusive random fields.
But what is the source of these random fields? Other magnetic
impurities with very high coercivity that aren’t completely
saturated.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 28/6
Possible explanation
But the loops look pretty saturated
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 29/6
More detailed analysis
The Landau-Lifshitz-Gilbert (LLG) equation describes
micromagnetic dynamics. It contains a reactive term and a
dissipative term:
ds
= −γ1 s × B − γ2 s × (s × B),
dt
s is a microscopic spin,
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 30/6
More detailed analysis
The Landau-Lifshitz-Gilbert (LLG) equation describes
micromagnetic dynamics. It contains a reactive term and a
dissipative term:
ds
= −γ1 s × B − γ2 s × (s × B),
dt
s is a microscopic spin,
B is the local effective field,
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 30/6
More detailed analysis
The Landau-Lifshitz-Gilbert (LLG) equation describes
micromagnetic dynamics. It contains a reactive term and a
dissipative term:
ds
= −γ1 s × B − γ2 s × (s × B),
dt
s is a microscopic spin,
B is the local effective field,
γ1 is a precession coefficient, and
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 30/6
More detailed analysis
The Landau-Lifshitz-Gilbert (LLG) equation describes
micromagnetic dynamics. It contains a reactive term and a
dissipative term:
ds
= −γ1 s × B − γ2 s × (s × B),
dt
s is a microscopic spin,
B is the local effective field,
γ1 is a precession coefficient, and
γ2 is a damping coefficient.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 30/6
More detailed analysis
The Landau-Lifshitz-Gilbert (LLG) equation describes
micromagnetic dynamics. It contains a reactive term and a
dissipative term:
ds
= −γ1 s × B − γ2 s × (s × B),
dt
s is a microscopic spin,
B is the local effective field,
γ1 is a precession coefficient, and
γ2 is a damping coefficient.
The effective field is B = −∂H/∂s + ζ, where H is the
Hamiltonian and ζ represents the effect of thermal noise.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 30/6
Dynamics Break Symmetry
If si → −si , H → −H then the effective field B → −B. Now
consider the LLG eqn:
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 31/6
Dynamics Break Symmetry
If si → −si , H → −H then the effective field B → −B. Now
consider the LLG eqn:
ds
= −γ1 s × B − γ2 s × (s × B),
dt
How does it change under inversion?
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 31/6
Dynamics Break Symmetry
If si → −si , H → −H then the effective field B → −B. Now
consider the LLG eqn:
ds
= −γ1 s × B − γ2 s × (s × B),
dt
-
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 31/6
Dynamics Break Symmetry
If si → −si , H → −H then the effective field B → −B. Now
consider the LLG eqn:
ds
= −γ1 s × B − γ2 s × (s × B),
dt
+
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 31/6
Dynamics Break Symmetry
If si → −si , H → −H then the effective field B → −B. Now
consider the LLG eqn:
ds
= −γ1 s × B − γ2 s × (s × B),
dt
-
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 31/6
Dynamics Break Symmetry
If si → −si , H → −H then the effective field B → −B. Now
consider the LLG eqn:
ds
= −γ1 s × B − γ2 s × (s × B),
dt
Therefore the dynamics do not preserve spin inversion symmetry.
More fundamentally, this can also be seen from the fact that
although the Hamiltonian has spin inversion symmetry, the spin
commutation relations (e.g. [Sx , Sy ] = i~Sz ), change sign under
spin inversion.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 31/6
What went wrong?
If instead of using precessional dynamics (LLG eqns), we left out
the precessional term (relaxational dynamics), the states would be
complementary.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 32/6
What went wrong?
If instead of using precessional dynamics (LLG eqns), we left out
the precessional term (relaxational dynamics), the states would be
complementary.
MISTAKE:
Leaving out the precessional nature of the dynamics.
Confusing Hamiltonian symmetry with configurational
symmetry.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 32/6
Counterargument
If the external field is varied adiabatically, the detailed fast time
scale precessional motion must be irrelevant.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 33/6
Counterargument
If the external field is varied adiabatically, the detailed fast time
scale precessional motion must be irrelevant.
Rebuttal:
It is the avalanches that give rise to hysteresis in the first place.
The motion during one is fast and the outcome is greatly
effected by the details of the precessional dynamics.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 33/6
Next Step
We describe a simulation using the LLG equations to find out how
well this agrees with experiment and make predictions.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 34/6
Simulation of Co/Pt films
We’ll simulate the LLG equations with the following ingredients in
the Hamiltonian:
Assume the films are disordered but strongly anisotropic. The
easy axis is randomly oriented but strongly biased
perpendicular to the film.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 35/6
Simulation of Co/Pt films
We’ll simulate the LLG equations with the following ingredients in
the Hamiltonian:
Assume the films are disordered but strongly anisotropic. The
easy axis is randomly oriented but strongly biased
perpendicular to the film.
Assume a long range dipolar interaction between points.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 35/6
Simulation of Co/Pt films
We’ll simulate the LLG equations with the following ingredients in
the Hamiltonian:
Assume the films are disordered but strongly anisotropic. The
easy axis is randomly oriented but strongly biased
perpendicular to the film.
Assume a long range dipolar interaction between points.
Assume a short range ferromagnetic coupling J + δi , where δi
is a random variable whose strength and statistics can be
adjusted.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 35/6
Simulation of Co/Pt films
We’ll simulate the LLG equations with the following ingredients in
the Hamiltonian:
Assume the films are disordered but strongly anisotropic. The
easy axis is randomly oriented but strongly biased
perpendicular to the film.
Assume a long range dipolar interaction between points.
Assume a short range ferromagnetic coupling J + δi , where δi
is a random variable whose strength and statistics can be
adjusted.
The usual interaction with an external field Hsz .
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 35/6
Simulation of Co/Pt films
We’ll simulate the LLG equations with the following ingredients in
the Hamiltonian:
Assume the films are disordered but strongly anisotropic. The
easy axis is randomly oriented but strongly biased
perpendicular to the film.
Assume a long range dipolar interaction between points.
Assume a short range ferromagnetic coupling J + δi , where δi
is a random variable whose strength and statistics can be
adjusted.
The usual interaction with an external field Hsz .
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 35/6
Precessional Term
α ≡ γ2 /γ1 measures how much damping there is compared to
precession .
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 36/6
Precessional Term
α ≡ γ2 /γ1 measures how much damping there is compared to
precession .
NeFe thin films, α ∼ .01.
CoCr/Pt multilayer films, α ∼ 1.
Co/Pt multilayer films, α ∼ .37.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 36/6
Precessional Term
α ≡ γ2 /γ1 measures how much damping there is compared to
precession .
NeFe thin films, α ∼ .01.
CoCr/Pt multilayer films, α ∼ 1.
Co/Pt multilayer films, α ∼ .37.
We conservatively set α to 1.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 36/6
Hysteresis Loops
Hysteresis loops for systems with different amounts of disorder; the
vertical axes are the magnetizations and the horizontal axes are the
external fields. The cliff in the hysteresis curve vanishes as disorder
is increased.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 37/6
Low disorder simulations
Here is a movie of the low disorder case.
Domain growth for a 2562 system with low disorder, λ = 1000 and
w = 0.15. The temperature is 10−4 .
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 38/6
Cliff Region
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 39/6
Low disorder images
O. Hellwig and E. E. Fullerton
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 40/6
Low disorder images
O. Hellwig and E. E. Fullerton
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 40/6
Low disorder images
O. Hellwig and E. E. Fullerton
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 40/6
Low disorder images
O. Hellwig and E. E. Fullerton
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 40/6
Low disorder images
O. Hellwig and E. E. Fullerton
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 40/6
Domains vs Disorder
Configurations near the coercive field. (T = 10−4 ).
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 41/6
Experimental Patterns
(O. Hellwig and E. E. Fullerton)
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 42/6
Corresponding Loops
Hysteresis loops for systems with different amounts of disorder; the
vertical axes are the magnetizations and the horizontal axes are the
external fields. The cliff in the hysteresis curve vanishes as disorder
is increased.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 43/6
Sensitivity To Noise
Look at domain evolution with different realizations of thermal
noise:
1
2
high disorder:
low disorder:
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 44/6
Sensitivity To Noise
Look at domain evolution with different realizations of thermal
noise:
1
2
high disorder:
low disorder:
The low disorder system is much more sensitive to thermal effects.
This implies that RPM will be much weaker in that case.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 44/6
RPM CPM plots
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-0.6 -0.4 -0.2
0
0.2
0.4
0.6
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-0.6 -0.4 -0.2
0
0.2
0.4
0.6
-0.6 -0.4 -0.2
0
0.2
0.4
0.6
-0.6 -0.4 -0.2
0
0.2
0.4
0.6
Blue RPM, Red CPM. As disorder increases, both RPM and CPM
increase. (T = 10−4 ).
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 45/6
RPM CPM vs Disorder
1
RPM, CPM
0.8
0.6
0.4
0.2
RPM
CPM
0
0
0.1
0.2
0.3
Disorder
0.4
0.5
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 46/6
Temperature dependence (predicted)
1
0.9
RPM
0.8
0.7
0.6
0.5
0.4
0.3
0.2
-0.4
T = 0.0
T = 0.00001
T = 0.0001
T = 0.001
-0.2
0
0.2
0.4
0.6
0.2
0.4
0.6
B
1
0.9
CPM
0.8
0.7
0.6
0.5
0.4
0.3
0.2
-0.4
T = 0.0
T = 0.00001
T = 0.0001
T = 0.001
-0.2
0
B
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 47/6
Low Disorder Prediction
1
0.8
0.6
0.4
0.2
0
-0.6 -0.4 -0.2
0
0.2
0.4
0.6
Initial growth sites should be the same at sufficiently low
temperatures.
(O. Hellwig and E. E. Fullerton)
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 48/6
Conclusions
Patterns in magnets are not symmetric between the upper and
lower branches of magnetic systems.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 49/6
Conclusions
Patterns in magnets are not symmetric between the upper and
lower branches of magnetic systems.
This can be explained by the presence of both damping and
precessional motion.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 49/6
Conclusions
Patterns in magnets are not symmetric between the upper and
lower branches of magnetic systems.
This can be explained by the presence of both damping and
precessional motion.
Simulations incorporating this, ferromagnetism, disorder, and
dipolar interactions explains the experiments fairly well.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 49/6
Conclusions
Patterns in magnets are not symmetric between the upper and
lower branches of magnetic systems.
This can be explained by the presence of both damping and
precessional motion.
Simulations incorporating this, ferromagnetism, disorder, and
dipolar interactions explains the experiments fairly well.
It also makes predictions about temperature dependence and
field dependence that should be testable by future experiments.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 49/6
Multicycle spin dynamics
Consider the 3d Edwards and spin glass Hamiltonian
X
X
H=−
Ji,j Si Sj − h
Si .
hi,ji
i
The couplings Ji,j are uniform random numbers between ±1.
The spins Si = ± 1.
Free boundary conditions.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 50/6
Multicycle spin dynamics
Consider the 3d Edwards and spin glass Hamiltonian
X
X
H=−
Ji,j Si Sj − h
Si .
hi,ji
i
The couplings Ji,j are uniform random numbers between ±1.
The spins Si = ± 1.
Free boundary conditions.
Use single spin-flip dynamics. At any step, we search for the next
value of h where a spin flip occurs. Once that happens we let any
subsequent avalanches occur before changing h again.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 50/6
Multicycle spin dynamics
Now we periodically cycle the field between hmin and hmax .
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 51/6
Multicycle spin dynamics
Now we periodically cycle the field between hmin and hmax .
In steady state, the hysteresis loop takes more than one cycle to
close on itself.
0.6
0.4
M
0.2
0
-0.2
-0.4
-0.6
-1.5
-1
-0.5
0
0.5
1
1.5
h
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 51/6
Power spectrum
The magnetization as a function of time in steady state shows
subharmonics. The power spectrum of M (t) has peaks at fractions
of the driving frequency. (Driving frequency = 1 below).
103 I
20
3
15
2
10
1
5
T=0
T=0.2
0
0.2 0.25 0.3 0.35
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
f
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 52/6
Nanomagnetic pillar arrays
Ni nanomagnets on silicon1 Magnetic Force Microscopy
[1] Courtesy of Holger Schmidt. Fabricated by T.Savas MIT
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 53/6
Nanomagnetic pillar arrays
Ni nanomagnets on silicon1 Magnetic Force Microscopy
[1] Courtesy of Holger Schmidt. Fabricated by T.Savas MIT
They can be fabricated to have a wide variety of shapes and sizes
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 53/6
Nanomagnetic pillar arrays
Ni nanomagnets on silicon1 Magnetic Force Microscopy
[1] Courtesy of Holger Schmidt. Fabricated by T.Savas MIT
They can be fabricated to have a wide variety of shapes and sizes
Can such a system show multicycles?
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 53/6
Modeling nanomagnetic pillars
These are single domain
nanomagnets where the crystalline orientation is random
in each pillar.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 54/6
Hamiltonian
The Hamiltonian is the addition of four pieces due to:
X
The external field: −
hsz,i
i
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 55/6
Hamiltonian
The Hamiltonian is the addition of four pieces due to:
X
The external field: −
hsz,i
i
Crystalline anisotropy:
X K1
2
2
2
4
4
4
[− (αx,i
]
αz,i
αy,i
) + K2 αx,i
+ αz,i
+ αy,i
2
i
α’s are direction cosines relative to the crystalline axes.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 55/6
Hamiltonian
The Hamiltonian is the addition of four pieces due to:
X
The external field: −
hsz,i
i
Crystalline anisotropy:
X K1
2
2
2
4
4
4
[− (αx,i
]
αz,i
αy,i
) + K2 αx,i
+ αz,i
+ αy,i
2
i
α’s are direction cosines relative to the crystalline axes.
X
Dipolar self energy, that is shape anisotropy: −
dz s2z,i
i
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 55/6
Hamiltonian
The Hamiltonian is the addition of four pieces due to:
X
The external field: −
hsz,i
i
Crystalline anisotropy:
X K1
2
2
2
4
4
4
[− (αx,i
]
αz,i
αy,i
) + K2 αx,i
+ αz,i
+ αy,i
2
i
α’s are direction cosines relative to the crystalline axes.
X
Dipolar self energy, that is shape anisotropy: −
dz s2z,i
i
Dipolar interactions between pillars:
X
si · A(rij ) · sj
j6=i
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 55/6
Results
We found that to get multi-cycles it was best to use a triangular
lattice. Here is a movie of a system showing two cycles.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 56/6
Results
We found that to get multi-cycles it was best to use a triangular
lattice. Here is a movie of a system showing two cycles.
Here part of the movie is in slow motion. showing details of
avalanches.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 56/6
Results
We found that to get multi-cycles it was best to use a triangular
lattice. Here is a movie of a system showing two cycles.
Here part of the movie is in slow motion. showing details of
avalanches.
We tried a range of pillar radii, heights and separations. The
probability of observing a multicycle is as high as ∼ .6 This
provides a viable system for observing multicycle behavior.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 56/6
Results
We found that to get multi-cycles it was best to use a triangular
lattice. Here is a movie of a system showing two cycles.
Here part of the movie is in slow motion. showing details of
avalanches.
We tried a range of pillar radii, heights and separations. The
probability of observing a multicycle is as high as ∼ .6 This
provides a viable system for observing multicycle behavior.
It would also be interesting to pursue the possibility of designing
these arrays to perform computation, by making cellular automata.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 56/6
Avalanches and Precession
with Andreas Berger, Hitachi Global Storage Technologies San Jose
In many magnetic materials spin dynamics are dominated for short
times by precessional motion as damping is relatively small.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 57/6
Avalanches and Precession
with Andreas Berger, Hitachi Global Storage Technologies San Jose
In many magnetic materials spin dynamics are dominated for short
times by precessional motion as damping is relatively small.
Values of α = γ2 /γ1 range from .01 to 1.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 57/6
Avalanches and Precession
with Andreas Berger, Hitachi Global Storage Technologies San Jose
In many magnetic materials spin dynamics are dominated for short
times by precessional motion as damping is relatively small.
Values of α = γ2 /γ1 range from .01 to 1.
What happens in the limit of α = 0 and (no thermal noise)?
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 57/6
Avalanches and Precession
with Andreas Berger, Hitachi Global Storage Technologies San Jose
In many magnetic materials spin dynamics are dominated for short
times by precessional motion as damping is relatively small.
Values of α = γ2 /γ1 range from .01 to 1.
What happens in the limit of α = 0 and (no thermal noise)?
In this limit, the dynamics conserve energy, but are highly
nonlinear.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 57/6
Dynamics with no damping
An avalanche can transition to an ergodic phase where the state is
equivalent to one at finite temperature.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 58/6
Dynamics with no damping
An avalanche can transition to an ergodic phase where the state is
equivalent to one at finite temperature.
The temperature is often above the ferromagnetic ordering
temperature. This is a movie of a 32 × 32 system. This is a
128 × 128 system.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 58/6
Dynamics with no damping
An avalanche can transition to an ergodic phase where the state is
equivalent to one at finite temperature.
The temperature is often above the ferromagnetic ordering
temperature. This is a movie of a 32 × 32 system. This is a
128 × 128 system.
However when the initial avalanche is below a critical size, it
usually dies out, with excess energy distributed in spin waves.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 58/6
Effect of finite damping
α = 0.9
final configuration
α = 0.8
during the avalanche
α=0
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 59/6
Effect of finite damping
α = 0.9
final configuration
α = 0.8
during the avalanche
α=0
Decreasing damping increases the size of avalanches
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 59/6
"Nucleation" with no damping
Model the system as being subdivided into metastable and ergodic
regions.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 60/6
"Nucleation" with no damping
Model the system as being subdivided into metastable and ergodic
regions.
ergodic
A growing ergodic
region
T
metastable
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 60/6
"Nucleation" with no damping
Model the system as being subdivided into metastable and ergodic
regions.
ergodic
Energy
released
when spin transitions
T
metastable
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 60/6
"Nucleation" with no damping
Model the system as being subdivided into metastable and ergodic
regions.
ergodic
This one
grow
T
doesn’t
metastable
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 60/6
"Nucleation" with no damping
Model the system as being subdivided into metastable and ergodic
regions.
ergodic
The
temperature
field diffuses and
decreases.
T
metastable
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 60/6
Model Results
35
30
w0
25
20
15
10
5
0
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
T0
1
Simulation results and analytical prediction of the critical line
separating growing and static avalanches. The initial avalanche of
size w0 to grow. T0 is the energy released by a site in transitioning
to an ergodic region.
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 61/6
Further Questions
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 62/6
Further Questions
If the total sz is only weakly broken, by dipolar and anisotropy
terms, then what effect does this have on the dynamics?
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 62/6
Further Questions
If the total sz is only weakly broken, by dipolar and anisotropy
terms, then what effect does this have on the dynamics?
How does precessional motion effect the size of the critical field
region of avalanche dynamics?
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 62/6
Further Questions
If the total sz is only weakly broken, by dipolar and anisotropy
terms, then what effect does this have on the dynamics?
How does precessional motion effect the size of the critical field
region of avalanche dynamics?
Can these considerations be extended to better understand
avalanches in granular media?
Nonequilibrium symmetry breaking and pattern formation in magnetic films. – p. 62/6