Emergent ice rule and magnetic charge screening from vertex frustration in artificial spin ice

Abstract

Artificial spin ice comprises a class of frustrated arrays of interacting single-domain ferromagnetic nanostructures. Previous studies of artificial spin ice have focused on simple lattices based on natural frustrated materials. Here we experimentally examine artificial spin ice created on the shakti lattice, a structure that does not directly correspond to any known natural magnetic material. On the shakti lattice, none of the near-neighbour interactions is locally frustrated, but instead the lattice topology frustrates the interactions leading to a high degree of degeneracy. We demonstrate that the shakti system achieves a physical realization of the classic six-vertex model ground state. Furthermore, we observe that the mixed coordination of the shakti lattice leads to crystallization of effective magnetic charges and the screening of magnetic excitations, underscoring the importance of magnetic charge as the relevant degree of freedom in artificial spin ice and opening new possibilities for studie

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PUBLISHED ONLINE: 3 AUGUST 2014 | DOI: 10.1038/NPHYS3037
Emergent ice rule and magnetic charge screening
from ver tex frustration in artificial spin ice
Ian Gilbert
1
, Gia-Wei Chern
2
, Sheng Zhang
3
, Liam O’Brien
4,5
, Bryce Fore
1
, Cristiano Nisoli
2
and Peter Schier
1
*
Artificial spin ice comprises a class of frustrated arrays of interacting single-domain ferromagnetic nanostructures. Previous
studies of artificial spin ice have focused on simple lattices based on natural frustrated materials. Here we experimentally
examine artificial spin ice created on the shakti lattice, a structure that does not directly correspond to any known natural
magnetic material. On the shakti lattice, none of the near-neighbour interactions is locally frustrated, but instead the lattice
topology frustrates the interactions leading to a high degree of degeneracy. We demonstrate that the shakti system achieves
a physical realization of the classic six-vertex model ground state. Furthermore, we observe that the mixed coordination of the
shakti lattice leads to crystallization of eective magnetic charges and the screening of magnetic excitations, underscoring
the importance of magnetic charge as the relevant degree of freedom in artificial spin ice and opening new possibilities for
studies of its dynamics.
A
rtificial spin ices
1–3
are artificial materials-by-design that
place nanometre-scale ferromagnetic nanostructures in
close proximity so that the moment interactions lead to
complex correlated behaviour. Although artificial spin-ice systems
were initially introduced to mimic the rare earth titanate spin-ice
materials
4–6
, which are interesting systems in their own r ight ow-
ing to their zero-point entropy
7,8
and magnetic monopole exci-
tations
9–12
, artificial spin-ice systems have emerged as prototype
materials-by-design that can be engineered to study frustrated spin
systems
13–18
, monopole-like excitations
19–21
and out-of-equilibrium
thermodynamics
22,23
. The utility of artificial spin ice as a model
system has recently been enhanced by the development of new
thermalization techniques
24–29
.
Artificial spin ice has primarily been studied so far in t wo simple
lattice structures: the square lattice, with four-moment coordination
at each vertex and a twofold degenerate ground state
1,24,30
, and the
kagome lattice, with three-moment coordination at each vertex and
a richer phase diagram associated with effective magnet ic charges
intrinsic to the three-moment vertices
13,14,19,20,29
(Fig. 1a,b).
The ultimate promise of artificial spin ice, however, has been
the ability to design and study a non-trivial frustrated geometry
inaccessible in any natural system, and the exotic behaviour that
would be observable as a result. Such a geometry has been recently
proposed in the form of a ‘shakti’ lattice
31,32
as shown in Fig. 1c,d.
This lattice has mixed coordination (that is, the moments can meet
at vertices of two, three or four), which distinguishes it from any
other artificial spin-ice material studied so far, and its ground state
is determined by a new type of frustration associated with the
lattice topology.
We have fabricated and studied shakti artificial spin-ice
arrays, using a recently developed t hermalization technique
29
to experimentally confirm the predicted ground state of the
shakti lattice and study the excitations above that manifold. We
find evidence of screening dynamics between magnetic charge
a
d
b
c
Figure 1 | Sketches of several dierent artificial spin-ice lattices. a,b, The
square (a) and kagome (b) lattices that have been extensively investigated
in previous studies. c,d, The short-island shakti (c) and long-island shakti
(d) lattices considered in this work. The darkened islands in c,d define a
plaquette of the lattice, which is the basis for understanding its frustration.
excitations at the four-moment vertices and the ground-state
magnetic charges embedded on three-moment vertices, and we
demonstrate through simulations that this screening effect can be
explained quantitatively by the inclusion of magnetic charge–charge
interactions. In the context of future applications of delocalized
1
Department of Physics and Frederick Seitz Materials Research Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA,
2
Theoretical Division, and Center for Nonlinear Studies MS B258, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA,
3
Department of
Physics and Materials Research Institute, Pennsylvania State University, University Park, Pennsylvania 16802, USA,
4
Department of Chemical Engineering
and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455, USA,
5
Thin Film Magnetism Group, Department of Physics, Cavendish
Laboratory, University of Cambridge, Cambridge CB3 0HE, UK.
*
e-mail: pschie@illinois.edu
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NATURE PHYSICS DOI: 10.1038/NPHYS3037
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1 µm
abc
de f
gh i
Figure 2 | Experimental realization of the shakti lattice. a–c, Scanning electron micrographs of artificial square spin ice (a), the short-island shakti lattice
(b) and the long-island shakti lattice (c). The island separation is 360 nm for all of these images, and the scale bar in a also applies to all of the images.
d–f, Magnetic force microscope images of the corresponding lattices in a–c. The black and white contrast indicates the magnetic poles of the nanomagnets.
g–i, Representation of the magnetic moments extracted from the corresponding images in d–f as arrows. Note that the ordered moments of the square
lattice demonstrate the eectiveness of our thermalization protocol.
monopolar degrees of freedom in magnetic materials, our results
suggest the likelihood of polaronic bound states with highly
non-trivial dynamics.
We fabricated samples of shakti artificial spin ice with lattice
spacings ranging from 320 to 880 nm on a silicon nitride
substrate using electron-beam lithography followed by electron-
beam evaporation of permalloy. The details of our fabrication
process can be found elsewhere
1,29
. We fabricated b oth the originally
proposed lattice with long islands crossing each plaquette of the
lattice
31,32
, and a modified version that includes only a single island
size and is simply a decimation of the artificial square spin-ice lattice,
with a quarter of the islands removed (Fig. 1). We refer to these
as the short-island and long-island shakti lattices, noting that they
are effectively equivalent because both arrangements have the same
ground-state frustration descr ibed below.
Images of our structures are shown in Fig. 2, with the short islands
220 nm long, 80 nm wide and 25 nm high. The length of the long
islands varied with the lattice constant (between 540 and 1,100 nm).
Magnetic force microscope (MFM) images demonstrated that all
the islands were single domain with the magnetization constrained
to point along t he islands’ long axes. We thermally annealed the
arrays using the process described in ref. 29 to place the arrays
in a thermalized, low-energy state, as evinced by the simultaneous
annealing of square ice samples into large ground state domains
(Fig. 2d). MFM images of the annealed arrays allowed us to
extract the moment configuration from each image for analysis,
and two different samples treated with the same thermal annealing
protocol y ielde d qualitatively equivalent results, thus demonstrating
reproducibility.
Ground-state degeneracy in the shakti lattice
The most striking qualitative difference between the shakti lattice
and previously studied artificial spin-ice lattices is the multiple
coordination of islands at different vertices on the shakti lattice.
Each plaquette of the shakti lattice, as defined in Fig. 1, includes
both four-moment and three-moment vertices (four of each). In
Fig. 3a we enumerate the possible magnetic moment configurat ions
for these vertices. The shakti-lattice ground state can be obtained
by direct inspection of the vertices on a given plaquette of the
lattice (Fig. 2e,h): all of the four-moment vertices have a Type I
4
moment configuration (the lowest-energy moment configuration
of those four moments), and the three-moment vertices are evenly
divided between the lowest-energy Type I
3
moment configuration
and the ‘defect’ Type II
3
moment configurations. In other words, to
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NATURE PHYSICS DOI: 10.1038/NPHYS3037
bc
Type I
4
(2) Type II
4
(4) Type III
4
(8) Type IV
4
(2)
Type I
3
(2) Type II
3
(4) Type III
3
(2) Type II
2
(2)Type I
2
(2)
a
Figure 3 | Mapping the shakti lattice to the six-vertex model. a, The vertex types found in the shakti lattice, showing only the short-island shakti lattice.
Consistent with the standard literature nomenclature for the vertices in artificial square spin ice, we number the vertices with Roman numerals in order of
increasing energy. We distinguish vertices with dierent numbers of islands by a subscript indicating the number of islands. b, A map of the three-island
vertex configurations in the magnetic force microscope image in Fig. 2e. The solid lines indicate the boundaries of the plaquettes, the dashed lines indicate
the two-island vertices in the middle of each plaquette, and the directions of the island moments are reproduced as red arrows in the upper-leftmost
plaquette for illustrative purposes. The circles indicate the location of the three-moment vertices; the Type I
3
vertices are denoted by open circles, and the
higher energy Type II
3
vertices are denoted by filled circles. c, The six degenerate ground-state configurations for arrangement of the three-moment vertex
types on a plaquette.
minimize the global energy of the system, the lattice topology re-
quires that half the three-moment vertices are in the local magneto-
static ground state and half are in a higher-energy defect state
31,32
.
As a result of the mixture of states on the three-moment vertices,
the shakti ground state has intrinsic degeneracy arising from the
freedom in allocating the two higher-energy vertices among the
four possibilities on each plaquette; t he two defect vertices on each
plaquette can be equivalently located at four different sites for a
total of six possible energetically equivalent configurations (Fig. 3c).
Therefore, each plaquette can be mapped precisely onto a vertex of
a classic two-dimensional six-vertex model obeying the ice rule
31,32
.
This correspondence is illustrated in Fig. 3b, where we take t he
moment configuration from Fig. 2—the first physical realization
of the six-vertex model ground state. The correspondence between
individual plaquettes and vertices of the six-vertex model is shown
in Fig. 3c. Note that the observed emergent frustration and high
degeneracy that arises from the topology of the shakti lattice is a
direct consequence of its design and has no obvious analogue in any
naturally occurring lattice.
By plotting the vertex population as a function of lattice spacing
(Fig. 4), we demonstrate explicitly that we are obtaining the ground
state of the shakti lattice by tuning the strength of the moment
interactions. Although the vertex distribution is close to random
at large lattice spacing, when the spacing is small, we obtain the
precisely expected vertex distribution for the short-island lattice.
The long-island shakti lattice does not quite achieve this degenerate
six-vertex state within the range of our experiments, presumably
owing to the more-constrained dynamics of the longer islands.
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NATURE PHYSICS DOI: 10.1038/NPHYS3037
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1.0
0.8
0.6
Vertex fraction
0.4
0.2
300 400 500 600
Lattice spacing (nm)
700 800 900
300 400 500 600
Lattice spacing (nm)
700 800 900
300 400 500 600
Lattice spacing (nm)
700 800 900 300 400 500 600
Lattice spacing (nm)
700 800 900
300 400 500 600
Lattice spacing (nm)
700 800 900
300 400 500 600
Lattice spacing (nm)
700 800 900
0.0
1.0
0.8
0.6
Vertex fraction
0.4
0.2
0.0
1.0
0.8
0.6
Vertex fraction
0.4
0.2
0.0
Short-island shakti
Type II
4
Type III
4
Type IV
4
Type I
4
Long-island shakti
Type II
4
Type III
4
Type IV
4
Type I
4
a
1.0
0.8
0.6
Vertex fraction
0.4
0.2
0.0
1.0
0.8
0.6
Vertex fraction
0.4
0.2
0.0
1.0
0.8
0.6
Vertex fraction
0.4
0.2
0.0
d
be
cf
Short-island shakti
Type II
3
Type III
3
Type I
3
Long-island shakti
Type II
3
Type III
3
Type I
3
Short-island shakti
Type II
2
Type I
2
Square lattice
Type II
Type III
Type IV
Type I
Figure 4 | Vertex population fractions for the shakti lattice. a–f, Depiction of the vertex fractions for the short-island shakti lattice (a–c), the long-island
shakti lattice (d,e) and the square lattice (f), all as a function of lattice spacing. These plots show that the system converged to the ground state at small
lattice spacing as expected. In the ground states of all three lattices, the four-island vertices will all be in the Type I
4
configuration. In the ground state of
both the shakti lattices, half of the three-island vertices will be in the (minimum energy) Type I
3
configuration and half will be in the (defect) Type II
3
configuration. Finally, all of the two-island vertices in the short-island shakti lattice will be Type I
2
in the ground state. Error bars correspond to the inverse
square root of the number of vertices observed for each type.
Magnetic charge ordering and screening
As we have demonstrated the existence of topologically induced
frustration in the shakti lattice, we now can consider the dynamics
of the effective magnetic charges at the vertices. The three-moment
vertices necessarily possess an effective charge, even in their lowest-
energy state
13,14,29
, whereas four-moment vertices have an effective
charge only when excited out of the vertex ground state (that is, in
a Typ e III
4
vertex configuration). The dynamics of such effective
magnetic charges has been studied recently in a range of artificial
spin-ice systems
19,20,24,28
, and ordering of the effective magnetic
charges was seen in the kagome ice I manifold of artificial kagome
spin ice after thermalizat ion
29
. Analogous to the kagome case, the
ice-state manifold of the shakti lattice allows for the ordering of the
charges on alternating three-moment vert ices, and such ordering
is favoured by longer-range interactions among the ver tices
31,32
.
Indeed, as shown in Fig. 5 we observe incipient crystallization
in precisely such arrangement, t hus demonstrating the collective
nature of the dynamics in this system and the importance of
considering magnetic charges as the relevant degrees of freedom.
To better understand the nature of charge dynamics in the shakti
lattice, we fit our vertex data using Monte Carlo simulations based
on a vertex model and interacting magnetic charges
33,34
. We consider
a simplified model including primarily interactions within vertices
(that is, only between nearest neighbours). To account for further-
neighbour interactions, we add a magnetic charge term of the form
K
P
hmni
q
m
q
n
, where q
m
denotes the mag net ic charge of vertex
m, as the main effect of interactions beyond the first-neighbour
pairs is to produce the non-trivial charge–charge correlations. The
Monte Carlo simulations are able to creditably reproduce the data
in Fig. 4 (Supplementary Information), validating the inclusion of
charge–charge interactions. As suggested in previous studies of as-
grown artificial spin ice, the effect of weak
35
disorder can be folded
into a redefinition of the effect ive temperature
36
and should not
qualitatively alter the behaviour of this system (for example, the
vertex populations) in the regime accessed experimentally here. The
experimental vertex populations for all of the lattices (including
the well-understood square lattice), are fitted well by Monte Carlo
simulations that neglec t disorder altogether, indicating that the
samples are in the weak-disorder regime. Further simulations
incorporating disorder in the ver tex energies at the 20% level show
negligible differences in vertex populations (Supplementary Fig. 3)
and indicate that the statistical mechanical model of ref. 32
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NATURE PHYSICS DOI: 10.1038/NPHYS3037
2 µm
ab
cd
2 µm
Figure 5 | Charge ordering in the shakti lattices. a,b, Magnetic force
microscope (MFM) images of the 320 nm short-island (a) and long-island
(b) shakti lattices. c,d, Schematics of the charge-ordered three-moment
vertices, where the red and blue dots correspond to three-island vertices
belonging to the two degenerate antiferromagnetic charge orderings. Note
that the charge order of the three-moment vertices coexists in a
background of disordered island moments (as evidenced by the absence of
order in the MFM images).
for the shakti lattice is robust to the effects of disorder. We
conclude that although the formation of crystallites of magnetic
charge shows the theoretically predicted features, our effectively
thermalized ensemble naturally still has finite entropy t hat inhibits
full crystallization. This remaining entropy (and the finite size
of the MFM images) also explains the absence of the power-law
correlations predicted for the shakti lattice
32
.
In addition to simply fitting the vertex populations, we also
used the Monte Carlo simulations (using an effective temperature
that scales with the cube of the lattice constant, Supplementary
Information) to investigate the relevance of charge–charge
interactions. In particular, because of its mixed coordination
structure, the shakti lattice raises the possibilities of interactions
between the intrinsic charges on the three-moment vertices and
charge excitations on the four-moment vertices (Type III
4
vertex
configurations), including the intriguing possibility of formation
of bound states, or magnetic polarons
37
. We characterized such
interactions by measuring the charge of the nearest neighbours
of Type III
4
excitations using the quantity
h
Q
nn
i
=
1
N
III
P
i
δ
i
P
j
q
i,j
,
where i runs over all of the Type III
4
vertices in the lattice, j
runs over the nearest neighbours of Type III
4
vertices, N
III
is the
number of Type III
4
excitations, δ
i
is the sign of the Type III
4
vertex i’s
charge, and q
i,j
is the charge of the nearest-neighbour three-moment
vertex j.
We compared the experimentally measured
h
Q
nn
i
with two
distinct Monte Carlo simulations of the same quantity: one
simulation included only nearest-neighbour-ver tex interactions—
whose weak screening comes simply from the constraints of the
underlying spin manifold—and the other also included long-range
interactions between magnetically charged ver tices (taking K to
be 10% of the interaction energy between aligned moments on
a vertex). As shown in Fig. 6a, the experimental data compare
favourably only with the simulation that incorporates long-range
interactions between the magnetic charges.
a
b
300
−3.5
0.2
0.0
0.4
0.6
0.8
1.0
−3.0
−2.5
−2
−3
0.0 0.2 0.4
Excitation fraction
Long-island shakti
Kagome
−2.0
−1.5
〈Q
nn
〉
〈Qnn〉
P (Q
nn
= −4)
−1.0
400 500 600 700 800
Lattice constant (nm)
900
300 400 500 600 700 800
Lattice constant (nm)
900
Long-island shakti
Experiment
MC 1
MC 2
Long-island shakti
Experiment
MC 1
MC 2
Figure 6 | Evidence of magnetic charge screening in the long-island shakti
lattice. a, Average of the relative net charge (
h
Q
nn
i
) of an excited Type III
4
vertex’s nearest neighbours. The blue line (MC 1) represents the Monte
Carlo simulation that included only nearest-neighbour vertex interactions,
and the red line (MC 2) represents the simulation that also included
long-range charge–charge interactions between the magnetically charged
vertices; inset:
h
Q
nn
i
plotted as a function of the fraction of charge
excitations for the shakti long-island lattices and for the kagome lattice,
showing opposite behaviour; as temperature decreases, the screening in
kagome also decreases. The lines are Monte Carlo simulations. b, The
probability of Q
nn
= δ
i
P
j
q
i,j
= −4. Again, the blue line (MC 1) represents
the Monte Carlo simulation that included only nearest-neighbour vertex
interactions, and the red line (MC 2) represents the simulation that also
included long-range charge–charge interactions between the magnetically
charged vertices. In both a and b, only simulations that include long-range
charge–charge interactions can replicate the behaviour of the experimental
data. Error bars for each lattice spacing are calculated from the standard
deviation of Q
nn
over all of the excitations for those arrays.
As a f urther test of the role of long-range magnetic charge
interactions in the screening of the Type III
4
excitations, we
compared the probabilities for various values of the quantity
Q
nn
= δ
i
P
j
q
i,j
. Figure 6b compares the experimentally determine d
fraction of nearest neighbours with a net relative charge of
−4q (physically, this corresponds to a positive Type III
4
monopole
surrounded by four negatively charged three-island vertices, or vice
versa) to two different Monte Carlo simulations of the fract ion of
excitations for which Q
nn
takes this value. Again, one simulation
neglected magnetic charge interactions and one included them, and
again the data agree with simulations only when the magnetic charge
interactions are included.
Finally, we contrast the screening of monopole-like charge
excitations in the shakti lattice with the s creening of ±3 charged
excitations in the pseudo-ice manifold of artificial kagome spin
ice (using data from ref. 29). The inset of Fig. 6a shows two
opposite temperature behaviours for the two geometries: screening
in the kagome lattice approaches saturation at large lattice constants,
where other excitations also contribute and entropic contributions
dominate. At low entropy (small lattice constant), screening is
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NATURE PHYSICS DOI: 10.1038/NPHYS3037
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reduced, as diluted vertex excitations in the kagome lattice place
a strong constraint on the three neighbouring vertices, locally
reducing the entropy. In the case of the shakti lattice, however,
screening increases at low entropy, when Type III
4
excitations
become sparse, pointing to the formation of genuine bound states.
This trend is robust in the long-island shakti case as shown in
the figure, and also in the short-island shakti case (Supplementary
Information and Supplementary Fig. 4).
Our data show that the shakti lattice, owing to its mixed
coordination, offers a new spectrum of phenomena beyond those
already observed in other spin-ice materials. In particu lar, our
data on charge ordering and screening fur ther underscore the
importance of magnetic charge as a paradigm for understanding
the behaviour of these systems. The results open the door to
experiments on a wide variety of other artif icial spin-ice lattices,
predicted to host a variety of interesting phenomena, including
sliding phases, smectic phases and emerging chirality associated
with the explicit design of exotic lattice structures where the
topology governs the physics
31
.
Methods
The samples studied in this work were fabricated in a process simil ar to t hat used
in previous experiments
1,29
. A bilayer resist stack (bottom layer,
polymethylglutarimide; top layer, polymethyl methacrylate, molecular mass
950,000 daltons, 2% in anisole) was spin-coated on a si licon wafer with a 200 nm
layer of low-pressure chemical vapour deposition silicon nitride, which acted as a
diffusion barrier between film and substrate during annealing. The square,
short-island shakti, and long-island shakti lattices were then defined w ith a Vistec
5200 electron-beam lithography writer. A 25 nm permalloy (Ni
81
Fe
19
) film with a
3 nm Al capping layer was deposited by electron-beam evaporation. The patterns
were lifted off in Microchem’s Remover PG liftoff solvent at 75
◦
C. The samples
were finally sonicated in acetone.
The samples were polarized in the (1,1) direction and then thermally
annealed in vacuum at 545
◦
C for 15 min and cooled at a rate of 1
◦
C min
−1
.
Magnetic force microscopy verified t hat all of the islands were single domain, and
the square lattices were no longer magnetized but had been annealed into a
low-energy state with large ground-state domains of Type I vertices. Two
identical samples were fabricated and annealed in this fashion, and both yielded
similar results. The magnetic moment configurations were extracted from the
MFM images using automatic image recognition routines.
Received 23 December 2013; accepted 17 June 2014;
published online 3 August 2014
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Acknowledgements
This project was funded by the US Department of Energy, Office of Basic Energy
Sciences, Materials Sciences and Engineering Division under Grant No. DE-SC0010778.
Lithography was performed in part with the support of the National Nanotechnology
Infrastructure Network. The work of G-W.C. and C.N. was carried out under the auspices
of the US Department of Energy at LANL under contract no. DE-AC52-06NA253962.
Work performed at the University of Minnesota was supported by EU Marie Curie IOF
project no. 299376.
Author contributions
C.N. and P.S. designed this study and supervised the experiments, simulations and data
analysis. Artificial spin-ice samples were fabricated by I.G. and S.Z. Thin-film deposition
and thermal annealing was done by L.O’B. Magnetic force microscopy and data analysis
was performed by I.G. with assistance from B.F. Simulations and theoretical
interpretation was given by G-W.C. and C.N. The paper was written by I.G., G-W.C.,
C.N. and P.S. with input from all of the co-authors.
Additional information
Supplementary information is available in the online version of the paper. Reprints and
permissions information is available online at www.nature.com/reprints.
Correspondence and requests for materials should be addressed to P.S.
Competing financial interests
The authors declare no competing financial interests.
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