Direct Observation of Thermal Relaxation in Artificial Spin Ice
A. Farhan,
1,2
P. M. Derlet,
3
A. Kleibert,
4
A. Balan,
4
R. V. Chopdekar,
1,4
M. Wyss,
1,5
J. Perron,
1,2,6
A. Scholl,
7
F. Nolting,
4
and L. J. Heyderman
1,2,
*
1
Laboratory for Micro- and Nanotechnology, Paul Scherrer Institute, 5232 Villigen PSI, Switzerland
2
Laboratory for Mesoscopic Systems, Department of Materials, ETH Zurich, 8093 Zurich, Switzerland
3
Condensed Matter Theory Group, NUM, Paul Scherrer Institute, 5232 Villigen PSI, Switzerland
4
Swiss Light Source, Paul Scherrer Institute, 5232 Villigen PSI, Switzerland
5
Swiss Nanoscience Institute, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland
6
Laboratoire de Chimie-Physique—Matie
`
re et Rayonnement (UMR 7614 UPMC/CNRS),
Universite
´
Pierre et Marie Curie, 75231 Paris Cedex 05, France
7
Lawrence Berkeley National Laboratory (LBNL), 1 Cyclotron Road, Berkeley, California 94720, USA
(Received 22 May 2013; revised manuscript received 2 July 2013; published 2 August 2013)
We study the thermal relaxation of artificial spin ice with photoemission electron microscopy, and are
able to directly observe how such a system finds its way from an energetically excited state to the ground
state. On plotting vertex-type populations as a function of time, we can characterize the relaxation, which
occurs in two stages, namely a string and a domain regime. Kinetic Monte Carlo simulations agree well
with the temporal evolution of the magnetic state when including disorder, and the experimental results
can be explained by considering the effective interaction energy associated with the separation of pairs of
vertex excitations.
DOI: 10.1103/PhysRevLett.111.057204 PACS numbers: 75.10.Hk, 75.70.i, 75.78.n
Relaxation phenomena in various condensed matter sys-
tems, including dielectrics [1], glassy systems [2], and pro-
teins [3], have long caught the attention of the research
community. While studies of such relaxation processes
involved measurements of macroscopic quantities, the recent
creation of artificial spin systems has allowed the dynamics
to be inspected microscopically [4,5]. A prominent example
of such systems is artificial spin ice, which consists of
dipolar-coupled nanoscale ferromagnets arranged in two-
dimensional frustrated geometries [6] and is considered to
be a two-dimensional analogue to the naturally occurring
pyrochlore spin ice [7]. Each nanomagnet is monodomain
and elongated so that the magnetic moments point in one of
two directions parallel to the island long axis, thus mimick-
ing a single Ising spin. The main advantage of artificial spin
ice systems is that their geometry can be tailored and their
magnetic configurations can be directly visualized and inves-
tigated using appropriate imaging techniques [6,8–12].
However, due to the high blocking temperatures of the
patterned nanomagnets, it has not been possible to observe
thermal fluctuations in most of these systems and low-energy
states could only partially be accessed using demagnetiza-
tion protocols [6,9,10]. In terms of thermally active systems,
in the past, investigations on superparamagnetic nanomag-
nets have been carried out [13–16],buttemporallyresolved
real-space observations of thermally active artificial spin
ice structures have been performed only recently [5].
In this Letter we now study thermally active extended
arrays of artificial square ice [6] [see Fig. 1(a)], demonstrat-
ing at temperatures close to room temperature how such a
system relaxes from a well-defined higher energy state to
the lowest energy state, a behavior that compares well with
our corresponding lattice kinetic Monte Carlo model.
Specifically, we are able to observe the creation, separation,
and annihilation of vertex excitations, which through their
migration trigger the investigated relaxation processes. The
results can be explained by an effective interaction energy
(or string tension) that causes vertex excitations to separate
and propagate, or converge and annihilate, depending on the
background configuration through which they migrate.
For this work, a Permalloy (Ni83%Fe17%) wedge film
on a silicon (100) substrate was patterned using electron
beam lithography. Artificial square ice arrays consisting of
nanomagnets, with length L ¼ 470 nm, width W ¼
170 nm, and center-to-center distance of nearest-neighbor
nanomagnets a ¼ 425 nm, were fabricated, ranging in
thickness from 0 to 20 nm over a distance of 3 mm, with
a 3 nm-thick aluminum capping layer. Magnetic images
[see Fig. 1(b)] were taken in a photoemission electron
microscope (PEEM) [17], employing x-ray magnetic cir-
cular dichroism (XMCD) at the Fe L
3
edge [18]. The
XMCD images were obtained with a total exposure time
of six seconds (three seconds for each x-ray helicity). The
dark and bright contrast in the obtained images is a measure
of the orientation of the nanomagnet’s magnetic moment
relative to the x-ray polarization vector. Nanomagnets with
magnetic moments pointing toward the x-ray polarization
vector will appear dark, while nanomagnets with magnetic
moments opposing the x-ray polarization vector will appear
bright [see Figs. 1(b) and 1(c)].
In order to investigate the relaxation processes, obser-
vations were performed at a film thickness of 3 0:3nm,
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which allowed the details of evolution of the magnetic state
to be recorded over a reasonable time scale. To understand
the mechanisms behind the observed relaxation, it is
important to first characterize the investigated system en-
ergetically. For artificial square ice, this is best done by
considering the local vertex configurations [6], where each
vertex is located at the center point of a cross consisting of
four neighboring nanomagnets. The vertex type is given by
the orientation of the moments of the nanomagnets and
may be classified into four different types with increasing
dipolar energy (type I to type IV) [see Fig. 1(c)]. The lowest
energy configuration in an artificial square ice consists of an
array of type I vertices, resulting in alternating clockwise
and anticlockwise plaquettes. This ground state has a
degeneracy of two, which can give rise to domains of
opposite chirality [8]. While both type I and type II
vertices have two moments pointing toward the vertex
and two pointing away, obeying the so-called ice rule
analogue to that found in the bulk systems [7], type II
vertices have a higher energy since the interactions between
the nanomagnets at a vertex in these two-dimensional sys-
tems are not equivalent [6,19]. Under an appropriate mag-
netic field, a magnetic configuration containing only type II
vertices becomes the lowest energy configuration. Type III
vertices can be considered as mobile thermal excitations
and, as we will show, mediate configurational changes
through their migration. Type IV vertices have all moments
pointing in or all moments pointing out, and represent the
most energetic configuration, which is never observed in
our experiments.
An array of type II vertices was created in the artificial
square ice after application of a saturating magnetic field
(35 mT) along the negative [11] direction, and image
acquisition was begun immediately after the magnetic field
was switched off. Initially all moments point toward one
direction (see bright XMCD contrast in Fig. 2(d) and the
Supplemental Material [20]), and then the investigated
system was found to pass through two distinct regimes
while undergoing relaxation from the excited type II mag-
netic configuration to the lowest energy type I magnetic
configuration (see Fig. 2). During the first regime we see
isolated chains of type I vertices emerge within a type II
vertex background via the creation of type III vertex pairs
and their separation [see Figs. 2(a) and 2(d)]. Since this
involves neighboring reversed island moments appearing
as black lines, we refer to this regime as the string regime
[see Figs. 2(a) and 2(d)]. During the string propagation, the
reversal of nearest neighbor moments (in orthogonal nano-
magnets) creates a chain of type I vertices within the
initially saturated type II background. However, the less
frequent reversal of next nearest neighbor moments (par-
allel islands at a vertex) results in trapped type II vertices,
which have a different configuration to those forming the
initial saturated state [see Fig. 2(d)] and become part of the
boundaries in the domain regime [see Figs. 2(b) and 2(e),
and the Supplemental Material [20]]. With an increasing
number of strings, the ends of the strings meet (type III
vertices annihilate) in either the same or across adjacent
rows, and eventually form areas of (lowest energy) type I
vertices. As there are two possible orientations for the
type I vertex [8], such type I areas can be divided into
two domain types of opposite chirality [see Fig. 1(b)].
This stage marks the beginning of the domain regime
[see Fig. 2(b) and [20]], and subsequently these lowest
energy type I domains of opposite chirality, separated by
type II boundaries, evolve via type III vertex creation and
propagation along the domain boundaries [see Fig. 2(e),
and the Supplemental Material [20]]. Bigger domains
expand at the cost of smaller domains through domain
boundary movement until a uniform single domain
[Fig. 2(c) and the Supplemental Material [20]] is achieved.
Taking a closer look at the vertex statistics with time [see
Fig. 3(a)], we see that the string regime is characterized by
a rapid increase (decrease) of type I (type II) vertices. In
addition, the increasing number of strings in the early
FIG. 1 (color). Artificial square ice. (a) Scanning electron
microscope image (a ¼ 425 nm) and (b) associated XMCD
image resolving vertex-type configurations detailed in (c).
Highlighted in orange and yellow are type I vertices, and green
and blue are a type II and a type III vertex, respectively. Present
in (b) are two type I ground state domains of opposite chirality
separated by a domain boundary (pink line) consisting of type II
and type III vertices.
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stages is reflected by an initial sharp increase in the number
of type III vertices, which subsequently decrease in num-
ber as the string ends meet. In contrast, the domain regime
is characterized by a general slow change in the vertex
population, interspersed with small jumps in the number of
type I (type II) vertices, associated with the creation and
fast movement of type III vertices along a domain bound-
ary. While this movement often occurs beyond the tempo-
ral limits of the XMCD imaging, several observations
indicate that the movement of the type III vertices occurs
via a sequential reversal of neighboring nanomagnets.
We now compare our experimental observations with
kinetic Monte Carlo simulations. Here a simplified dipolar
Hamiltonian is used, defining each nanomagnet to have
either one of the two possible magnetic states with the
moments aligned along the long axis of the nanomagnets.
These mesoscopic magnetic moments interact via the mag-
netic dipolar interaction,
Vðr
ij
; m
i
; m
j
Þ¼
0
4r
3
ij
½3ðm
i
^
r
ij
Þðm
j
^
r
ij
Þm
i
m
j
;
(1)
where r
ij
is the distance vector separating the ith and jth
nanomagnets with magnetic moments m
i
and m
j
, giving
the final Hamiltonian
P
i<j
Vðr
ij
; m
i
; m
j
Þ. Our calculations
are performed for up to six nearest neighbors.
In order to realize the temporal evolution of the magne-
tization dynamics, the magnetic moment of each nano-
magnet is assumed to have a reorientation rate given
by the Arrhenius form
0
expðE=k
B
TÞ. Thus the mag-
netic moments are reoriented via thermal activation where
0
is an intrinsic reorientation prefactor, and E is a reor-
ientation barrier energy equal to the sum of the intrinsic
energy barrier of a nanomagnet, E
0
, and half the dipolar
energy gain associated with the particular moment reor-
ientation [5]. For the calculation of the dipolar energy, we
treat each nanomagnet as a point source, with each moment
equal to the product of the nanomagnet’s magnetization,
FIG. 3 (color). Temporal evolution of vertex-type population.
(a) Experimental data obtained at a constant temperature of
350 K, showing a 100% ground state ordering within eight
hours. (b) Kinetic Monte Carlo simulation data for a Gaussian
disorder in the intrinsic nanomagnet energy barrier with a
standard deviation ¼ 0:05 and also for the case of no disorder,
¼ 0.
FIG. 2 (color). XMCD images of thermal relaxation in artifi-
cial spin ice: (a) string regime, (b) domain regime, and (c) the
ground state (field of view 20 m; see also the Supplemental
Material [20]). Migration of type III vertices (green and yellow
dots) in detail: (d) in the string regime, pairs of type III vertices
are created and separate, and (e) in the domain regime, a type III
vertex travels along various routes within a type II domain
boundary (pink line) so that the bigger type I domain expands
at the cost of the smaller domain. The path taken by the type III
vertex is indicated with colored arrows and a ¼ 425 nm.
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M, and the experimental nanomagnet volume. The standard
kinetic Monte Carlo technique [21] is then used to stochas-
tically evolve the configuration and its physical time.
For reasonable agreement to experiment, values of E
0
¼
1:05 eV, M ¼ 350 kA=m and
0
¼ 0:5 10
12
s
1
were
used in the kinetic Monte Carlo simulations. These are
realistic values as explained in Ref. [5]. It was also found
that a certain degree of disorder (variation in the nano-
magnet anisotropy energy) was required in the kinetic
Monte Carlo model to better reproduce the experimental
observations of the increase in type I vertex population at
the start of the string regime (see Fig. 3). Using a similar
approach to that taken by Budrikis et al. [22], this was
achieved by randomly varying the intrinsic energy barrier
of each nanomagnet using a Gaussian distribution centered
on E
0
with a standard deviation of ¼ 0:05 eV. This is a
lower value for than that found in the literature for
artificial spin ice [12] and other magnetic structures [23],
which is likely to be a result of a more homogeneous
microstructure at lower film thickness [24]. Disorder
allows the system to explore many more pathways to the
lowest energy state [22,25], so requiring more time to
develop from the perfect type II configuration into a multi-
domain low-energy configuration. Indeed, when we
neglect disorder ( ¼ 0), we find an instantaneous initial
increase in the type I vertex population, with the string
regime being completed in only a few minutes (see
Fig. 3(b), dashed black curve) rather than taking several
hours to develop as seen in experiment. In addition, the
inclusion of disorder causes the propagating type III verti-
ces to periodically shift to neighboring rows, better match-
ing the experimental observations [see Fig. 2(d)].
In general, the kinetic Monte Carlo simulations compare
well with the experiment over the full relaxation process
including both the string and domain regimes. This
agreement was achieved by including dipolar interactions
between first and second nearest neighbors only. The
effect of adding the contributions from further nanomag-
nets up to six nearest neighbors (6a) was also investigated.
However, this did not modify the temporal evolution in
any significant way. In particular, there is good quantitative
agreement in the temporal evolution of the vertex popula-
tions [Figs. 3(a) and 3(b)], where the initial 100% type II
vertex population rapidly decays as time evolves with a
corresponding increase of the type I vertices. Inspection of
the simulated domain regime occurring at later times
reveals similar activity as that seen by Budrikis et al.
[22], where the motion of domain boundaries (of type II
vertices) is mediated by type III vertex creation, propaga-
tion, and annihilation.
For both experiment and simulation, the early stages of
the string regime are characterized by the creation of
isolated pairs of neighboring type III vertices [see
Fig. 2(d)]. This is followed by further moment reorienta-
tions resulting in a rapid separation of the type III vertex
pairs associated with the expansion of a chain of type I
vertices between them along the [11] direction. This can be
explained by considering the energy change as a function
of the string length connecting the type III vertex pair in an
otherwise perfect type II background, which may be read-
ily calculated by determining the dipolar interaction
energy between all nanomagnets, Eq. (1). The energy as
a function of string length is plotted in Fig. 4, revealing that
while the system initially increases its energy for the
creation of the type III vertex pair, subsequent separation
reduces the energy with a roughly linear decrease in energy
as a function of their separation (see Fig. 4, orange line and
bottom schematic). Thus, upon their creation, the type III
vertex pair separate and rapidly move toward the edges of
the array. This is in contrast to excitations above a type I
background, where the separating type III vertex pair is
now connected by a chain of type II vertices. Again there is
a nearly linear change in energy, but now with a positive
gradient (see Fig. 4, dashed blue line and upper schematic).
Indeed, in the experiment we see that as soon the system
falls into a single ground state domain consisting only of
type I vertices, no further configurational changes are
observed. This is an example of the confining potential
associated with the type III vertices occurring in two
dimensional artificial square ice systems that dominates
over the negligible Coulomb interaction between charged
vertex pairs [26,27].
In conclusion, the realization of thermally active artifi-
cial spin ice at room temperature means that we now have
access to model systems that are closer in nature to their
FIG. 4 (color). Interaction energy as a function of string length
connecting two isolated type III vertices. Type I and II vertices
are indicated by orange and blue dots, respectively, and the
numbers in the schematics indicate the string length as the
right-hand type III vertex follows a zigzag path.
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three dimensional counterparts [7], and allow the magnetic
configurations during relaxation to be detected in micro-
scopic detail. This next generation of thermally active
artificial spin ice systems offers the exciting prospect of
direct observations of various relaxation phenomena.
Under controllable thermal equilibrium conditions, this
will also allow the long sought-after observation of phase
transitions [28–30]. In terms of applications, the control of
vertex excitations [11,12,31] will bring novel spintronic
devices that make specific use of their thermally active
magnetic properties.
The authors would like to thank Juri Honegger for
technical support. This work was supported by the Swiss
National Science Foundation and the Swiss Nanoscience
Institute, Basel, Switzerland. Part of this work was per-
formed at the Swiss Light Source, Paul Scherrer Institute,
Villigen, Switzerland and the Advanced Light Source,
Lawrence Berkeley National Laboratory (LBNL),
Berkeley, California.
*laura.heyderman@psi.ch
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![FIG. 2 (color). XMCD images of thermal relaxation in artificial spin ice: (a) string regime, (b) domain regime, and (c) the ground state (field of view 20 m; see also the Supplemental Material [20]). Migration of type III vertices (green and yellow dots) in detail: (d) in the string regime, pairs of type III vertices are created and separate, and (e) in the domain regime, a type III vertex travels along various routes within a type II domain boundary (pink line) so that the bigger type I domain expands at the cost of the smaller domain. The path taken by the type III vertex is indicated with colored arrows and a ¼ 425 nm.](/figures/fig-2-color-xmcd-images-of-thermal-relaxation-in-artificial-s2txw6uu.webp)
