Magnetic Frustration in Circular Arrays of Dipoles

Abstract

Magnetostatically interacting structures consisting of 1.5 μm diameter rings of radially oriented nanomagnets were investigated using magnetic force microscopy. An even number of nanomagnets, 16 in this case, formed a ground state in which the magnetizations of neighboring nanomagnets were oriented antiparallel. However, an odd number of nanomagnets, 15, led to a frustrated state in which at least one pair of neighboring nanomagnets was magnetized parallel, giving a net radial magnetic charge. The “defect” in the structure was moved around the ring by a modest external field.

Full text

IEEE MAGNETICS LETTERS, Volume 3 (2012) 4000104
Nanomagnetics
Magnetic Frustration in Circular Arrays of Dipoles
Seungha Yoon1,2, Youngman Jang1, Chunghee Nam1,JeanAnneCurrivan
1, B. K. Cho2,
and C. A. Ross1∗
1Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
2Department of Nanobio Materials and Electronics and School of Material Science and Engineering, Gwangju Institute of Science and
Technology, Gwangju 500-712, Korea
∗Senior Member, IEEE
Received 16 November 2011, revised 13 December 2011, accepted 13 December 2011, published 10 February 2012.
Abstract—Magnetostatically interacting structures consisting of 1.5 µm diameter rings of radially oriented nanomagnets
were investigated using magnetic force microscopy. An even number of nanomagnets, 16 in this case, formed a ground
state in which the magnetizations of neighboring nanomagnets were oriented antiparallel. However, an odd number of
nanomagnets, 15, led to a frustrated state in which at least one pair of neighboring nanomagnets was magnetized parallel,
giving a net radial magnetic charge. The “defect” in the structure was moved around the ring by a modest external field.
Index Terms—Nanomagnetics, magnetostatic interactions, magnetic frustration.
I. INTRODUCTION
New architectures for binary computation using nanomag-
nets have been developed over the past 15 years, with the
promise of low power, high speed, and nonvolatile operation
compared to standard semiconductor technology. One logic
architecture is based on magnetic quantum cellular automata
(MQCA) [Cowburn 2000, Imre 2006, Carlton 2008, Orlov 2008,
Varga 2010, Lambson 2011] in which magnetostatic interac-
tions between adjacent lithographically defined nanomagnets
are used to transfer data, and information is read back by de-
tecting the magnetization state of an output nanomagnet. A row
of closely spaced nanomagnets arranged with their easy axes
perpendicular to the row, in which magnetostatic interactions
favor antiparallel magnetization of neighboring nanomagnets,
has been proposed as a conduit for transmission of data in
MQCA logic [Imre 2006]. Arrays of magnetostatically interacting
nanomagnets have also been explored as a model for spin-ice
behavior [Castelnovo 2008], in which the geometrical arrange-
ment of the nanomagnets leads to frustration of the interactions
and the existence of a number of energetically equivalent states
[Wang 2006, Moessner 2009, Lammert 2010, Mellado 2010,
Mengotti 2011]. These two-dimensional (2-D) structures contain
many magnetically charged defects, and enable macroscopic
studies of the thermodynamics of, and correlations within, frus-
trated systems.
In both spin ice and MQCA, the formation, propagation, and
stability of magnetically charged defects is, therefore, of cen-
tral importance, but it is difficult to isolate and manipulate a
single defect in a large or complex nanomagnet arrangement.
In this letter, a small circular array of nanomagnets was used
as a model system to generate an individual magnetic defect,
and the response of the defect to a modest external field was
examined.
Corresponding author: C. A. Ross (caross@mit.edu).
Digital Object Identifier: 10.1109/LMAG.2011.2180704
II. EXPERIMENTAL METHODS AND RESULTS
The samples consisted of a circular flower-like arrangement
of elliptical nanomagnets, in which the long axis of the particles
was oriented radially and the particles were close enough to in-
teract magnetostatically. For an even number of nanomagnets,
magnetostatic energy is minimized by an alternating orientation
of the magnetization directions, and the system has two pos-
sible ground states in which the magnetization directions were
IOIOIO... or OIOIOI... where I =radially inward and O =ra-
dially outward orientations of magnetization. However, for an
odd number of nanomagnets, there necessarily exists at least
one defect in the arrangement corresponding to a net magnetic
charge at the perimeter of the ring denoted +Qor−Q.
The flower structures consisted of 150 nm ×300 nm ellipti-
cal thin film nanomagnets arranged in a ring with outer diameter
1.5 µm. Two types of structures with 15 or 16 petals were made,
as in Fig. 1(a) and (b), using electron-beam lithography and con-
ventional lift-off processing of a film of Ti (1.5 nm)/Co (7 nm)/Ti
(2 nm) which was deposited by triode sputtering with an Ar pres-
sure of 1 mTorr in a system with a base pressure of ∼10−8Torr.
The structures were ac-demagnetized by rotating the sample
in a decreasing in-plane magnetic field, as shown in Fig. 1(c)
[Wang 2006]. The external magnetic field was reduced slowly
at 5 Oe/s from 1000 to 0 Oe, while rotating the sample holder at
180◦/s. A dc-demagnetized state was achieved by applying then
removing an in-plane field without sample rotation. The samples
were imaged by scanning electron microscope (SEM), atomic
force microscopy, and magnetic force microscopy (MFM). MFM
images were taken in a Digital Instruments Nanoscope IV with
commercial low-moment magnetic probes.
Fig. 2(a)–(d) shows initialized states of the flower structures
after the ac-demagnetizing procedure. The nanomagnets were
single-domain particles and showed only two possible states, I
or O, at remanence. For the array composed of 16 nanomag-
nets, there were two possible defect-free ground states shown
in Fig. 2(a) and (b), which represent opposite magnetization
states. On the contrary, many different states were observed
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4000104 IEEE MAGNETICS LETTERS, Volume 3 (2012)
Fig. 1. (a) and (b) SEM images of flower-shaped devices, consisting of
(a) 16 and (b) 15 nanomagnets. (c) Schematic of the ac-demagnetizing
process, using a decreasing magnetic field with sample rotation. (d)
Nearest-neighbor interaction field as a function of the outer diameter
of the ring array for a 15-magnet ring consisting of 150 nm ×300 nm
elliptical nanomagnets of Co, 7 nm thick, with bulk Ms=1400 kA/m,
using a dipole approximation.
for the array with 15 nanomagnets, each containing exactly one
defect, such as II in Fig 2(c) with a net inward spin of −Q and OO
in Fig 2(d) with a net spin of +Q. States with more defects were
also observed for both odd and even numbers of petals (not
shown). Fig. 2(e) and (f) shows configurations after dc demag-
netization. The magnets on one side of the ring were oriented
inward while the others were oriented outward. The magnetic
orientation of the nanomagnets could be detected even in a low-
resolution magnetic image, so the MFM scan height was cho-
sen to be high deliberately to minimize interactions between the
tip and sample during acquisition of the magnetic contrast. The
strongest interactions occur when the tip contacts the sample to
measure the topographic contrast. The tip-induced switching of
nanomagnets during imaging was uncommon but one example
is indicated by ∗in Fig. 2(f).
The configurations obtained by ac demagnetization, in which
neighboring magnets had antiparallel magnetization, indicate
that magnetostatic interactions between the nanomagnets dom-
inated the behavior of the arrays. As the sample rotated in
a decreasing field, the highest coercivity nanomagnets would
be the first to stop reversing, and their stray field would pro-
mote antiparallel orientation of their lower coercivity neighbors
as the field decreased further. A simple parallel-dipole approxi-
mation yields an interaction field of Hi=29 Oe between nearest-
neighbor elliptical particles in the 15-magnet ring, and depend-
ing on their magnetization direction the two nearest neighbors
could, therefore, produce a field of 2Hi,0,or−2Hion a given
nanomagnet. The switching field Hsw was estimated by assum-
ing that the particles follow the Stoner–Wohlfarth model, which
predicts that switching occurs at Hsw/2 when the field is at 45◦
to the particle axis, in the absence of nearest-neighbor inter-
action fields. MFM measurements of rings at different fields
showed that nanomagnets oriented at 45◦to the field switched
Fig. 2. (a) and (b) MFM images of the two possible ground states of
the 16-magnet flower. The black and white dotted lines in some images
indicate inward-oriented (I) and outward-oriented (O) magnetizations.
(c) and (d) Two possible ground states of the 15-magnet flower. A +Q
defect is present in (c) and −Qin(d).(e)and(f)Remanentstatesof
the samples after saturation in a field oriented along the arrow. The ∗
indicates a nanomagnet that was reversed by the MFM tip field. The
contrast corresponds to a 3◦phase change in tip oscillation.
at Hsw/2 =∼200 Oe for a 16-magnet ring and at 200–250 Oe
for a 15-magnet ring. This gave Hsw =∼400 Oe compared to
a magnetostatic interaction field of 2Hi=58 Oe.
The dominant effects of magnetostatic interactions on the
magnetic configuration of the rings are somewhat surprising in
view of the considerable shape variation between nanomag-
nets evident from the SEM images in Fig. 1, which is expected
to cause a spread in the switching field of the nanomagnets. The
data suggest that the majority of the nanomagnets differed in
their switching fields by less than ∼2Hiin the 1.5-µm diameter
rings. A different situation arose for arrays with diameters of 2 or
2.5 µm, for which MFM showed no significant tendency for pre-
ferred antiparallel alignment of the magnetization of neighboring
nanomagnets. The calculated nearest-neighbor field decreased
as the ring diameter increased, as shown in Fig. 1(d), and the
lack of antiparallel ground states in 2–2.5 µmdiameterrings
after ac demagnetization suggested that the nearest-neighbor
field was unable to promote an antiparallel ground state for more
widely spaced nanomagnets.

IEEE MAGNETICS LETTERS, Volume 3 (2012) 4000104
Fig. 3. Defect rotation process. (a) State 1: the defect was initially
located at magnets 9 and 10. (b) State 2: after applying a reverse field
at 45◦to magnet 10, magnets 10 and 12 switched. (c) State 3: a field at
45◦to magnet 12 corrected its orientation leading to a ground state with
an opposite defect at magnets 10 and 11. (d) State 4: a field at 45◦to
magnet 10 returned the defect to its original position, but also reversed
magnet 5. A further field reversed magnet 5, restoring State 1.
III. MANIPULATION OF DEFECTS
One of the initialized states of the 15-magnet 1.5-µm diameter
array was selected to demonstrate how the magnetic defect
can be manipulated. The nanomagnets are numbered from 1
to 15 in Fig. 3(a). The defect position is at nanomagnets 9
and 10 with a +Q charge. Fig. 3(b) and (c) shows the process
for clockwise defect rotation in an external magnetic field. A
reverse clocking field (H=250 Oe) was applied at 45◦to the
axis of nanomagnet 10 [see Fig. 4(a)]. The nearest-neighbor
fields acting on nanomagnet 10 destabilized it with respect to
other nanomagnets and the external field reversed it, which
would ideally move the defect by one position to nanomagnets
10 and 11 and change its sign to −Q. However, there was a
failure in that nanomagnet 12 was also reversed leading to the
configuration in Fig. 3(b). This was corrected by applying a field
H=250 Oe at 45◦to the axis of nanomagnet 12 [see Fig. 4(b)],
which induced the state shown in Fig. 3(c) with the −Q defect
at nanomagnets 10 and 11.
In a similar manner, the defect was translated in a counter-
clockwise manner by switching nanomagnet 10 using a 250-Oe
clocking field as shown in Fig. 3(d). This erroneously also
switched nanomagnet 5, which was corrected by a 250-Oe field
to restore the initial state in Fig. 3(a) with the +Qdefectatnano-
magnets 9 and 10. This shows that the defect could be reversed
in sign and translated around the ring by an in-plane field.
The movement of the defect can be understood by consider-
ing that a nanomagnet surrounded by antiparallel neighbors is
stabilized by an interaction field of 2Hi, but there is no magne-
tostatic stabilization of a nanomagnet that is part of a magnetic
defect because its two neighbors have oppositely aligned mag-
netization, so their stray fields cancel midway between them.
The nanomagnet at the defect can, therefore, be irreversibly
Fig. 4. Fields required to transition between states (a) 1 to 2, (b) 2
to 3, (c) 3 to 4, and (d) 4 to 5. The red (in a and c) and blue (in b
and d) arrows indicate the direction for clocking and correction fields,
respectively.
switched by a field of Hsw/2 applied at 45◦to its axis, while a
nanomagnet outside the defect or one oriented at a different
angle to the applied field would be undisturbed. It is, therefore,
in principle possible to reverse just a nanomagnet at the defect
site without affecting its neighbors. The inadvertent switching
of nanomagnets 5 and 12 in Fig. 3 indicates that their switch-
ing field is relatively low, arising from the variability in size and
shape evident in the SEM image.
Because this array geometry allows for the controlled move-
ment of a magnetic defect, it could be used to provide input
to an MQCA system. For example, if nanomagnet 10 in Fig. 3
were an input magnet to a logic gate or data conduit, it could be
switched between its two states at a low field. A ring of nano-
magnets could also provide a shift register in which bits are
stored as magnetic defects, which can be translated around the
ring by a field. The structure also allows the dynamics of indi-
vidual defects to be examined in a simple system, analogous
to the study of 2-D artificial spin ice [Wang 2006], as well as
studies of the effects of different demagnetizing protocols [Ke
2008].
IV. CONCLUSION
In summary, the demagnetized states of a ring of radially
oriented magnetic dipoles were investigated using MFM. For a
ring with an even number of dipoles, a ground state is possible
in which the magnetizations of nearest neighbors are all ori-
ented antiparallel. However, in an odd-numbered ring, at least
one defect is necessarily present consisting of a pair of parallel-
oriented dipoles, with a ‘charge’ of +Qor−Qdependingonthe
direction of net magnetization, outward or inward. By application
of an external field at 45◦to the axis of the dipole in the defect,

4000104 IEEE MAGNETICS LETTERS, Volume 3 (2012)
the defect can be rotated around the ring, changing its sign on
each move. For a system with a small switching field distribu-
tion, defect motion may be possible without disturbing the other
dipoles; but in practice, the reversal of dipoles away from the
defect with lower switching field was corrected by application of
an additional field.
ACKNOWLEDGMENT
This work was supported by the National Science Founda-
tion, the Singapore-MIT Alliance, NRI INDEX Program, and by
the Basic Science Research Program through the National Re-
search Foundation of Korea funded by the Ministry of Education,
Science and Technology (grant 2009-0078928).
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