Self-organization of topological defects for a triangular-lattice magnetic dots array subject to a perpendicular magnetic field
The regular array of magnetic particles (magnetic dots) of the form of a two-dimensional triangular lattice in the presence of external magnetic field demonstrates complicated magnetic structures. The magnetic symmetry of the ground state for such a system is lower than that for the underlying latti...
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irk-123456789-1534952019-06-15T01:27:22Z Self-organization of topological defects for a triangular-lattice magnetic dots array subject to a perpendicular magnetic field Khymyn, R.S. Kireev, V.E. Ivanov, B.O. The regular array of magnetic particles (magnetic dots) of the form of a two-dimensional triangular lattice in the presence of external magnetic field demonstrates complicated magnetic structures. The magnetic symmetry of the ground state for such a system is lower than that for the underlying lattice. Long range dipole-dipole interaction leads to a specific antiferromagnetic order in small fields, whereas a set of linear topological defects appears with the growth of the magnetic field. Self-organization of such defects determines the magnetization process for a system within a wide range of external magnetic fields. Перiодичний масив магнiтних частинок (магнiтних точок), сформований в трикутну двовимiрну гратку, утворює складну струтуру магнiтного впорядкування. Магнiтна симетрiя основного стану для такої системи виявляється нижчою нiж симетрiя гратки. Далекодiюча диполь-дипольна взаємодiя призводить до специфiчного антиферомагнiтного впорядкування при малих полях, в той час як при збiльшеннi магнiтного поля виникає серiя лiнiйних топологiчних дефектiв. Самоорганiзацiя таких дефектiв визначає процес намагнiчування системи в широкому дiапазонi зовнiшнього магнiтного поля. 2014 Article Self-organization of topological defects for a triangular-lattice magnetic dots array subject to a perpendicular magnetic field / R.S. Khymyn, V.E. Kireev, B.O. Ivanov // Condensed Matter Physics. — 2014. — Т. 17, № 3. — С. 33701:1-10. — Бібліогр.: 57 назв. — англ. 1607-324X PACS: 75.10.Hk, 75.50.Tt, 75.30.Kz DOI:10.5488/CMP.17.33701 arXiv:1411.4897 http://dspace.nbuv.gov.ua/handle/123456789/153495 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The regular array of magnetic particles (magnetic dots) of the form of a two-dimensional triangular lattice in the presence of external magnetic field demonstrates complicated magnetic structures. The magnetic symmetry of the ground state for such a system is lower than that for the underlying lattice. Long range dipole-dipole interaction leads to a specific antiferromagnetic order in small fields, whereas a set of linear topological defects appears with the growth of the magnetic field. Self-organization of such defects determines the magnetization process for a system within a wide range of external magnetic fields. |
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Khymyn, R.S. Kireev, V.E. Ivanov, B.O. |
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Khymyn, R.S. Kireev, V.E. Ivanov, B.O. Self-organization of topological defects for a triangular-lattice magnetic dots array subject to a perpendicular magnetic field Condensed Matter Physics |
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Khymyn, R.S. Kireev, V.E. Ivanov, B.O. |
| author_sort |
Khymyn, R.S. |
| title |
Self-organization of topological defects for a triangular-lattice magnetic dots array subject to a perpendicular magnetic field |
| title_short |
Self-organization of topological defects for a triangular-lattice magnetic dots array subject to a perpendicular magnetic field |
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Self-organization of topological defects for a triangular-lattice magnetic dots array subject to a perpendicular magnetic field |
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Self-organization of topological defects for a triangular-lattice magnetic dots array subject to a perpendicular magnetic field |
| title_full_unstemmed |
Self-organization of topological defects for a triangular-lattice magnetic dots array subject to a perpendicular magnetic field |
| title_sort |
self-organization of topological defects for a triangular-lattice magnetic dots array subject to a perpendicular magnetic field |
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Інститут фізики конденсованих систем НАН України |
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2014 |
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Self-organization of topological defects for a triangular-lattice magnetic dots array subject to a perpendicular magnetic field / R.S. Khymyn, V.E. Kireev, B.O. Ivanov // Condensed Matter Physics. — 2014. — Т. 17, № 3. — С. 33701:1-10. — Бібліогр.: 57 назв. — англ. |
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Condensed Matter Physics |
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AT khymynrs selforganizationoftopologicaldefectsforatriangularlatticemagneticdotsarraysubjecttoaperpendicularmagneticfield AT kireevve selforganizationoftopologicaldefectsforatriangularlatticemagneticdotsarraysubjecttoaperpendicularmagneticfield AT ivanovbo selforganizationoftopologicaldefectsforatriangularlatticemagneticdotsarraysubjecttoaperpendicularmagneticfield |
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2025-07-14T04:38:02Z |
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| fulltext |
Condensed Matter Physics, 2014, Vol. 17, No 3, 33701: 1–10
DOI: 10.5488/CMP.17.33701
http://www.icmp.lviv.ua/journal
Self-organization of topological defects
for a triangular-lattice magnetic dots array
subject to a perpendicular magnetic field
R.S. Khymyn1, V.E. Kireev2, B.O. Ivanov2
1 Oakland University, 48309 Rochester Hills, MI, USA
2 Institute of Magnetism, 03142 Kiev, Ukraine
Received April 18, 2014, in final form May 15, 2014
The regular array of magnetic particles (magnetic dots) of the form of a two-dimensional triangular lattice in the
presence of external magnetic field demonstrates complicated magnetic structures. The magnetic symmetry
of the ground state for such a system is lower than that for the underlying lattice. Long range dipole-dipole
interaction leads to a specific antiferromagnetic order in small fields, whereas a set of linear topological defects
appears with the growth of the magnetic field. Self-organization of such defects determines the magnetization
process for a system within a wide range of external magnetic fields.
Key words: magnetic dot, topological defect
PACS: 75.10.Hk, 75.50.Tt, 75.30.Kz
1. Introduction and motivation
Self-organization phenomena are usually associated with complex systems far from equilibrium [1–
3]. In that case, the standard approach, depicting condensed matter as a gas of weakly interacting quasi-
particles (phonons, magnons, etc.) obtained within the framework of linearized theory, may become in-
appropriate, and the main role is played by soliton-type excitations [3, 4]. One can point out numerous
examples of such a behavior in very different physical systems. In particular, the Berezinskii-Kosterlitz-
Thouless phase transition in quasi-two-dimensional magnets is driven by the appearance of a finite den-
sity of free magnetic vortices [5–7]. Plastic deformation of solids may lead to the emergent complexity of
a defect structure, resulting in a hierarchy of super-defects such as dislocation and disclination systems
[8]. A fast quench across the phase transition line may produce a finite density of frozen topological de-
fects of various types; such a behavior is known for a wide range of systems from all fields of physics,
either for classical [9] or for quantum [10] systems. Note the defect line scenario of phase transitions
for two-dimensional systems with discrete symmetry breaking, see, for example, [11]. The common fea-
ture of all the above examples is a considerable role of stochastic factors, connected primarily to thermal
fluctuations, which leads to an irregular distribution of defects [9, 12].
Magnetic ordering is usually attributed to the exchange interaction of atomic spins, leading to rather
simplemagnetically ordered states [13]. Long-rangemagnetic dipole interaction usually produces smooth
non-uniformity (domain structures of different kinds) above this simple exchange structure [14–16].
However, the theoretical investigation of the systems of magnetic moments with pure dipolar interac-
tion, the so-called dipolar magnets, shows that many physical properties, lacking in the spin-exchanged
systems, are present for those systems. We should first note the presence of a non-unique ground state
with nontrivial continuous degeneracy for quite simple bipartite lattices, such as three-dimensional cu-
bic lattice, [17, 18] and for a two-dimensional square lattice, [19–21] as well as specific phase transitions
induced by an external magnetic field [22–24].
© R. Khymyn, V. Kireev, B. Ivanov, 2014 33701-1
http://dx.doi.org/10.5488/CMP.17.33701
http://www.icmp.lviv.ua/journal
R. Khymyn, V. Kireev, B. Ivanov
The models of dipolar magnets were originally discussed in regard to real crystalline spin systems.
A renewed interest to such models has been caused by investigation of two-dimensional lattices of sub-
micron magnetic particles (the so-called magnetic dots), see [25–27] for a recent review. The ability to
create the dots with practically exactly equal sizes and precisely controlled distance between them, leads
to the long-range order phenomena and can be treated as the creation of artificial crystals.
A direct exchange interaction between the dots is negligible, and the dipolar interaction is the sole
source of coupling between the dots. For small enough dots of a size smaller than 100 nm, the magneti-
zation inside a dot is almost uniform, producing the total magnetic moment m0 ≫ µB, where µB is the
Bohr magneton, i.e., the typical value for an atomic magnetic moment. For rather small magnetic dots
of volume 103 ÷105 nm3, the value of m0 exceeds 104
µB, and for dense arrays the characteristic energy
is higher than the energy of thermal motion at room temperature [22, 23, 28]. Moreover, the Mermin-
Wagner theorem is not valid for two-dimensional magnets with a dipolar coupling of spins having con-
tinuous degeneracy, and a true long range order can exist even for a purely two-dimensional case at finite
temperatures, either for ferromagnets [29, 30] and antiferromagnets [31]. Thus, one can expect that ther-
mal effects are less important for magnetic dot arrays up to high temperatures compared with the Curie
temperature of the magnetic material.
Such systems represent dipolar magnets and fill their theoretical investigation with a new physi-
cal content. Owing to the absence of exchange, magnetic dot arrays constitute a promising material for
high-density magnetic storage media. For this purpose, the dense arrays of small enough magnetic dots
with magnetic moments perpendicular to the array plane are optimal, see Refs. [32, 33, 35]. Currently,
the ordered arrays of magnetic sub-micron elements have been discussed as materials for the so-called
magnonics. In this new field in the applied physics of magnetism, magnonmodeswith a discrete spectrum
present for magnetic nanoelements are used for processing the microwave signals [37]. Novel prospects
are opened by the observation of the excitation of collective spin oscillations by femtosecond laser pulses
[34, 36]. Magnon spectra for dot arrays demonstrate a non-analytic behavior either for small wavevectors,
[38–41] or at some symmetrical points within the Brillouin zone [40]. Non-reciprocal effects for magnetic
dot lattices [42, 43] and non-trivial properties of the magnon modes localized on the defects of dot arrays
have been recently found [44]. The presence of phase transitions opens novel opportunities for design
of magnonic devices with the band structure operated by external parameters, e.g., magnetic field, see
for the recent review [45]. Thus, magnetic dot arrays are interesting as radically new objects for both the
fundamental and applied physics of magnetism.
If the magnetic moment of an individual magnetic particle is perpendicular to the array plane (x y -
plane), m = ±m0ez , the system can be described on the basis of the Ising model. The energy of dipolar
interaction of Ising moments perpendicular to the system’s plane is minimal for antiparallel orientation
of magnetic moments. Within the nearest-neighbors approximation, such interactions lead to antiferro-
magnetic (AFM) structures, e.g., simple chessboard AFM ordering is known for a two-dimensional Ising
square lattice with dipolar interaction [22, 27]. However, the close-packed triangular lattices of the mag-
netic dots are also frequently used in experiments. In particular, these lattices of cylindrical particles
considerably extended in the direction normal to the array plane are naturally obtained when the array
is prepared by controlled self-organization [46]. However, the triangular lattice is not bipartite. Such a
lattice with AFM interaction of the moments is a typical example of frustrated antiferromagnets [47]. It
is worth mentioning here that the behavior of a dot array with in-plane anisotropy is essentially differ-
ent from our case. In this case, the dipolar interaction is not clearly AFM; and for a model of infinite
unbounded array, the ferromagnetic state is stable [21], whereas for finite array, there appears a meso-
scopic non-uniform state of a form of either domain wall [48] or magnetic vortex [48, 49].
For a nearest-neighbor Ising triangular lattice with AFM interaction, thermodynamic properties are
quite unusual [47]. It is enough to mention that in this model there is no magnetic ordering at any finite
temperature T , 0; the ordering appears as a result of accounting for the next-nearest-neighbor interac-
tions only [50–52]. This counterintuitive feature can be explained within the concept of creation of linear
topological defects with zero energy [53]. For the case of magnetic dipole interaction, (common to what is
observed for many next-nearest-neighbor interaction models), our analysis shows the presence of AFM
ordering for small enough magnetic fields.
In the present work, a cascade of phases with different patterns of dot magnetization has been found
for a triangular lattice of mesoscopic magnetic dots with perpendicular magnetization and in an external
33701-2
Self-organization of topological defects for a magnetic dots array
magnetic field also perpendicular to the plane of the dot lattice. The transition between these states is
governed by a novel mechanism involving the creation of an ordered system of linear topological defects
with non-zero magnetization. For those transitions, thermal fluctuations are insignificant and one can
expect quite regular spatial distribution of such defects.
2. Model description
Consider a system of magnetic moments of magnetic particles mn placed in the sites of a triangular
lattice n, parallel to the x y -plane,
n = akex +
al
2
(
ex +
p
3ey
)
, (2.1)
where a is a lattice constant, k, l are integers, and ex and ey are unit vectors parallel to x and y axes,
respectively. We assume that any particle has easy-axial (Ising-like) anisotropy of the form of wn,a =
Ha(m2
n,x +m2
n,y )/2m0, and that the corresponding internal anisotropy field Ha is substantially higher
than the characteristicfield of the dot dipolar interaction, H∗ = m0/a3. Such systems ofmagnetic particles
made of soft ferromagnets, with the Ha originating from the shape anisotropy Ha ∼ 2πMs , where Ms
is the saturation magnetization of the material, are used for magnetic memory applications [35]. The
interaction field H∗ is proportional to a small geometric factor, H∗ = Ms(v0/a3), where v0 is the dot
volume. This interaction field can be much less than Ha even for dense enough lattices [22]; and it is not
capable of deflecting the dot magnetic moment from the z-axis. Thus, we can assume that all magnetic
moments are mn = m0ezσn, where σn =±1. Then, the Hamiltonian of this system of magnetic moments
can be written as
W = m2
0
∑
n,n′
σnσn′
|n−n′|3
−m0H
∑
n
σn , (2.2)
where an external magnetic field H = Hez is applied perpendicularly to the plane of the array. The first
term describes dipolar interaction, with the summation performed over all of the pairs of the lattice sites.
Below, for the sake of simplicity, we present the energy (per one magnetic particle) in the units of m2
0/a3
and we use the dimensionless magnetic field, h = H/H∗, where the characteristic value H∗ = m0/a3.
The present model applies directly to any triangular lattice of identical dipoles that are restricted to the
two directions of normal orientation [21]. It is interesting that the model formulated in this paper can be
used to describe a system of vortex state magnetic dots [22] accounting for the interaction of a magnetic
moment of vortex cores. The direction of the core moments is connected with the topological invariant,
and the vortices with upward or downward directions of the moments survive until non-small values of
magnetic field [54] are reached.
As has been mentioned above, two sources of degeneracy are present in our system, and it is not a
priori obvious which structure will constitute the ground state. In this situation it is natural to start with
the numerical analysis of the problem.
3. Ground states for infinite system: Numerical analysis
To find the global minimum of the energy of a system, there was used a Monte-Carlo (MC) approach
combined with a simulated annealing (SA) method, see the textbook [55] for details. The standard MC
method is based on the attempts to reverse the moment on the random site, and the only reversals favor-
able to energy are allowed. By contrast, for MC-SA, the probability of the reversal is non-zero even if the
energy grows after the reversal; otherwise, the system with a high probability will be “frozen” in some
local minimum. The probability depends not only on the energy gain∆E but also on the global parameter
T called the temperature. If the reversal is favorable in the energy, the moment is always reversed, irre-
spective of T . But even if the reversal is unfavorable, the non-zero probability of a reversal is chosen as
follows: flip-over takes place if ∆E < T | log p|, where T is the current value of temperature, p is a random
value 0 < p É 1. Here, the parameter temperature determines the strategy of minimization and the mean-
ing of the temperature is the same as for annealing in metallurgy involving initial heating and controlled
cooling of a material, thereby avoiding the formation of defects. T changes according to the quantity of
33701-3
R. Khymyn, V. Kireev, B. Ivanov
full steps of MC-SA such that the initial temperature is high enough compared with the interaction energy,
and then the temperature decreases. For a concrete analysis, we took the rhombus-shaped samples with
consecutively increasing periods up to 16×16 and used periodic boundary conditions.
3.1. Simplest ground states of the system: zero field and saturation
For a triangular Ising lattice in a several-neighbor approximation, a simple AFM order with two sub-
lattices can be implemented [56]. We found the same configuration for a long-range dipolar interaction
in the absence of the field and for a small enough magnetic field. For these states, the magnetic elemen-
tary cell is rectangular having lower symmetry than for the underlying triangular lattice, and this state
possesses a much higher discrete degeneracy than a simple chessboard structure for a square lattice dis-
cussed before. Several AFM states can occur in a system, which are different but fully equivalent by their
energies. Figure 1 presents three of these states, while the other three states are obtained from them by
changing the magnetic moment sign σn at all of the particles. Note the deviation from the standard AFM
case with the antiparallel orientation of all the nearest neighbors, which is a typical manifestation of
frustration in a system.
Figure 1. Uniform antiferromagnetic states giving the energy minimum at a small magnetic field. Here
and below, in all figures, the open and closed circles denote the particles with the upward and downward
moments, respectively.
3.2. Monte-Carlo analysis for intermediate field values
MC-SA analysis shows that for some small but finite values of the magnetic field, at least up to h = 0.7,
the mean value of the magnetic moment 〈m〉 equals zero, which indicates a simple AFM structure. For
higher fields, numerous more complex structures with 0 < 〈m〉 < m0 occur in the intermediate region
between AFM state and saturated state. The mean value of the magnetic moment (per one particle) 〈m〉
corresponding to these configurations, is present in figure 2.
Note the specific regions of this dependence present at different field intervals; first, the region with
small values of 〈m〉 É 0.2m0 having a rather non-regular dependence of 〈m〉 on h; second, the regions
with constant values of 〈m〉 independent of the magnetic field (shelves); and third, the saturation region.
The characteristic magnetic structures found in these regions are depicted in figure 3.
Monte-Carlo data are not too clear in the region of small fields such as 0.7÷ 0.9, and the magnetic
structures are far from the simple AFM structures, see figures 3(a)–3(d). Within this region, the afore-
mentioned structure with a set of topological defects appears. A detailed discussion of this region is the
main issue of our article.
Within the shelf regions, almost all initial Monte-Carlo configurations lead to the samemagnetic struc-
tures, which correspond to the formation of triangular superlattices for the minority dots (antiparallel to
the magnetic field) with different lattice spacings. As an example, note the ideal triangular superlattices
with 〈m〉 = m0/3 and with the period asl/a =
p
3, see figure 3(f) present at the values 2.4 . h . 6.4.
For higher fields, the superlattices with asl/a = 2 [figure 3(h)], asl/a =
p
7, [figure 3(i)] and asl/a = 3 [in
figure 3(j)] correspond to such shelves. For high magnetic fields near the saturation region, the magneti-
zation process is going through creating a superlattice of flipped dots of small density.
In the region of low magnetic fields, as well as in the regions of a magnetic field where the transitions
between the superlattices occur, resettability of Monte-Carlo result lowers, and the results become un-
reliable. The observed magnetic structures in these transition regions are characterized by much lower
symmetry than for the shelf regions. For example, at the values 0.9. h . 1.5, where the finite (but small)
33701-4
Self-organization of topological defects for a magnetic dots array
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.5 1.0
0.0
0.1
h
m
Figure 2. The mean value of magnetization 〈m0〉 (in units of m0, per one dot) of the array as a function
of magnetic field (in units of H∗ = M0/a3) found by Monte-Carlo simulations. Magnetization function
at low fields found by an exhaustive search of the states of rhombus-shaped samples (full line) together
with the Monte-Carlo data (symbols) are presented in the insert.
magnetic moment 〈m0〉 is formed, the translational symmetry for a set of flipped dots cannot be at-
tributed to a simple superlattice structure, see figure 3(a). However, in this figure one can clearly see
a novel element, an additional zigzag line of the sites oriented parallel to one of the translation vectors
of the lattice. The resulting magnetic structure can be interpreted as an antiferromagnetic domain struc-
ture in a system with a zigzag line as a domain wall. Such topological linear defects were described for
a two-sublattice antiferromagnetic state with an interaction of a few neighboring moments [57]. For the
region of small fields, an increase of magnetic field leads to an increase of the density of topological lin-
ear defects, see figures 3(a)–3(c). A minimal field for the start of this process corresponds to a low density
of such defects, and to find the critical field one needs to consider larger and larger systems. Namely, to
present a stripe of width n we need a system of at least (2n+1)×(2n+1) size. Below, in section 4 we find
the starting field for the creation of a set of topological defects by an analytical calculation.
To refine the Monte-Carlo data and to clarify the magnetic states at the fields of interest, 0 É h É 3, we
(a) h = 1.0 (b) h = 1.1 (c) h = 1.5 (d) h = 2.1
(e) h = 2.3 (f) h = 4.0 (g) h = 6.77 (h) h = 7.5
(i) h = 9.5 (j) h = 10.1
Figure 3. Ground states for characteristic values of magnetic field found by Monte-Carlo simulations.
33701-5
R. Khymyn, V. Kireev, B. Ivanov
perform a direct exhaustive search based on the picture of stripe AFM domain structures for rhombus-
shaped space regions with various (not necessarily equidistant) geometries of domain lines, up to the
size 60×60. It appears that for all fields the only equidistant structure corresponds to the minimal con-
figurations, with linear system of stripes at h < 1.5 or triangular superlattice at h > 2.0. Then, the only
equidistant structures with the size up to 300×300 were examined. The magnetization curve based on
these calculations is represented above in the insert in figure 2 and is compared with the Monte-Carlo
data.
The common “topological” scenarios are present for other transition regions, both below and above
the shelf regions with the magnetic structure of a form of ideal triangular superlattices of minority dots.
For example, the structure present at 2.0. h . 2.4 can be described as a “compression” of the domains of
the superlattice of period a
p
3 by the lines of dots with downmagnetic moments, see figures 3(d) and 3(e),
whereas the state at the opposite end of this shelf can be seen as a “rarefication” of the a
p
3 superlattice,
see figure 3(g). The transition structures corresponding to the “higher” shelves have C6 symmetry, higher
than for low field structures.
4. Magnetic ground states: Analytic description
For a non-frustrated square lattice of Ising magnetic moments, the destruction of both types of states
is started through the creation of a point defect in the state, the single magnetic dot with the magnetic
moment being reversed with respect to the regular structure of a given state [22]. First, in this section we
check the validity of such a scenario for saturated and AFM states of a triangular lattice array of magnetic
dots. Then, there follows a theoretical description of the novel topological mechanisms.
4.1. Point defect scenarios
In order to determine the values of the magnetic fields which correspond to “point defect instability”
we have calculated the change in dipolar interaction energy that occurs when the magnetic moment of
a single dot is reversed with respect to the ferromagnetic and two-sublattice AFM structures [22]. This
energy change is determined by the energy per dot in the initial states, which can be expressed by simple
lattice sums calculated with high precision. These sums here and below were calculated using a standard
software packageMathematica.
The point defect scenario well describes the instability of the saturated state. It is easy to see that the
change of the energy of the saturated state with the flip of a single magnetic moment can be presented as
W1 = 2m0(H −Hsat), where
Hsat = m0
∑
ni,0
1
|ni |3
≡ hsat
m0
a3
, hsat ≈ 11.034176 . (4.1)
The quantity Hsat determines the saturation field for a triangular lattice of Ising magnetic moments.
If the magnetic field H < Hsat , the value of W1 is negative and the flipping of a dot becomes favorable.
In principle, the common calculations can be performed for AFM state, as well as for any state with
the superlattice of flipped dots of the same symmetry as for underlying dot lattice. This approach well
describes the instability point for the AFM state for a square lattice of magnetic dots [22]. Its application
for the triangular dot lattice shows that the reversal of the magnetic moment of one dot in the AFM state
becomes favorable at H Ê HAFM, with the value hAFM = 1.8377, is much higher than the instability field
h ≃ 0.7 found numerically in the previous section. Thus, the reversal of a single magnetic moment cannot
describe the instability of the AFM state observed for a triangular lattice at h É 1.
4.2. Instability of AFM state through the creation of topological lines
As we found byMonte-Carlo analysis, the AFM state looses its stability as the field increases due to the
creation of topological defect lines (domain walls). This defect line corresponds to an additional zigzag
line of the particles withmagnetic moments, which are oriented as their neighbors, and should be normal
to one of the elementary translation vectors, see figure 4.
33701-6
Self-organization of topological defects for a magnetic dots array
Figure 4. (Color online). (a)–(c) The structure of topological defect with the different sets of characteristic
lines used for calculation of the defect energy, see details in the text. (d) The ideal AFM structure and
the characteristic lines used for energy calculation. As usually, the open and closed circles denote the
particles with the upward and downward magnetic moments, respectively, but the dots with upward
magnetic moments within the defect line are mentioned by grey.
The importance of defect lines in thermodynamics is a well-known property of two-dimensional sys-
tems with a discrete symmetry breaking. Due to the creation of a finite density of such lines, the long
range order is destroyed at finite temperature determined by the energy of the defect, see, for example,
[11]. However, for frustrated AFM states, the behavior can be very unusual. In particular, there is nomag-
netic order for AFM Ising system with the nearest-neighbor interaction at any finite temperature T > 0
[47, 50–52]. In general, this behavior can be explained using the defect line picture of the phase transition,
with the vanishing of energy (more exactly, the free energy) of a certain linear topological defect [53].
For our system of mesoscopic magnetic particles with a dipole interaction, the effect of a magnetic
field, instead of thermal effects, should be significant. The topological defect line with non-zero magne-
tization and having zero energy in the nearest-neighbor approximation was recently found [57]. This
defect coincides with that observed in our numerical simulations, compare figure 4 and figures 3(a)–3(d)
above. A physical consequence of a nonzero magnetic moment is that for such a defect, an additional
energy gain m0H per defect particle appears in the magnetic field H . Then, the defect energy decreases
as the field increases and becomes zero at H = HDW ≡ EDW/m0. For H Ê HDW , the finite density of such
defects will be present in the ground state. This is exactly the scenario observed in our numerical simu-
lations at magnetic field at the range 0.7÷1.5.
In order to find the critical value of the field HDW , let us calculate the energy of the domain wall
EDW. It is convenient to divide the full lattice into lines of dots parallel to the defect line, as it is shown in
figure 4. The energy of the system with domain wall (per one dot in the defect line) can be presented as a
sum over these lines as follows:
E = EGS +EDW =−
1
2
∑
n
m0σn Hn , (4.2)
where EGS is the energy of the ground state, the integer n describes the distance an of the given line from
the defect line, an = an/2, σn =±1 gives the sign of the moment for the n-th line, and Hn is the magnetic
field created on the dot in the n-th line by other dots in the system. To find the field Hn , it is convenient
to group all other dots to pairs of lines equidistant from the n-th line, as it is shown for n = 0,1,2 in
figure 4(a), (b) and (c), respectively. Let us enumerate these pairs by an integer k so that the distance
between the n-th line and one component of the k-th pair is equal to ak/2, the pairs with k = 1,2,3 are
presented in figure 4. Then, the energy of the magnetic state with a domain wall can be presented by a
double sum, over n > 0 and k > 0.
It is easy to see that for any finite n, the only pairs with limited k < n contribute to the energy of
the state with the domain wall. For example, for the lines directly entering the defect line [n = 0, see
figure 4(a)], the contributions of two lines composing any pair cancel each other. For this line, the non-
zero contribution to the energy is given by the dots from the same line, and we denote this contribution
as ε0. Then, for the line with n = 1, only one pair gives non-zero contribution, see figure 4(b), and the
energy can be written as ε0 −2ε1. Similarly, for n = 2, the energy is ε0 −2ε1 +2ε2, see figure 4(c), and so
on. Finally, the energy of the state with a domain wall is presented through ε0 and the particular finite
33701-7
R. Khymyn, V. Kireev, B. Ivanov
sums of the positive quantities εn ,
ε2n+1 =
∞
∑
k=1
4
[(n+1/2)2 +3(k −1/2)2]3/2
, ε2n =
∞
∑
k=1
4
(n2 +3k2)3/2
+
2
n3
.
The energy of the domain wall equals the difference of the energy of the state with the domain wall,
equation (4.2) and the ground state energy EGS. To find the ground state energy, it is convenient to use
the same presentation by a parallel lines, see figure 4(d), and to present it by the same sums εn . It is clear
that the energy per one dot in any line in the ground state is proportional to an infinite sum of the form
εGS = ε0+2
∑∞
n=1(−1)n
εn . Then, the domain wall energy can be found by term-by-term summation of the
corresponding contributions of the form [(ε0 −εGS)+ (ε0 −2ε1 −εGS)+ . . .] ≡ hDW = 2ε1 −4ε2 +6ε3 + . . . .
The corresponding infinite series hDW =−2
∑∞
n=1(−1)nnεn are sign-alternating and converge quite well.
Finally, the domain wall energy per one dot EDW can be presented as follows:
EDW = m0HDW , HDW = hDW
m0
a3
, hDW = 0.70858944. (4.3)
Here, we also present the characteristic value of the magnetic field, and HDW = EDW/m0 determining
the border of stability of the simple AFM state; for H > HDW, AFM state becomes unstable against the
creation of domain walls. Note, that the calculated value (4.3) is in good agreement with that found by
numerical simulations, but it is much lower than the field of point defect instability for AFM state, HAFM =
1.8377m0/a3.
4.3. Plateau description
Monte-Carlo simulations show some peculiarities (see figure 2) in the dependence of the magneti-
zation on the applied magnetic field in the form of plateaus, where the value of the function does not
change over a wide range of the argument. These peculiarities have a simple explanation. The magneti-
zation of the array increases at a small external field due to the formation of parallel topological defects
in the form of domain walls. At some critical concentration of such walls, the resulting state is nothing
but the superlattice of flipped dots which has a triangular structure that coincides with the array symme-
try; see figure 3(f), (h), (i), (j). Such a superlattice transforms into a self-similar structure but with another
step (lattice constant) as the applied field increases. Since the superlattice constant has discrete values
asl/a =
p
3, 2,
p
7, 3, 2
p
3,
p
13, 4,
p
19,
p
21, 5. . . , such a superstructure has good stability against the
alteration of the external field, and magnetization can be changed only stepwise. One can see that such
a structure consists of two inversely magnetized states with the lattice constants a and asl and with the
magnetization 〈m〉 = m0−2m0(a/asl)
2. The value of the field of the stability loss of such a superstructure
relative to the transition to another lattice constant can be easily calculated on the same principle as the
field of the saturated state stability which was done above,
Hsl = hsat
m0
a3
[
1−2
(
a
asl
)3]
, (4.4)
where themultiplier 2 in the numerator responds to the change ofmagnetization of the dot in comparison
with the saturated state and hsat = 11.034176 is the field of the transition to the saturated state. Though
this effect occurs in a narrow range of the field, it leads to instability of the superlattice at a little bit
smaller value of the field than it is predicted in equation (4.4).
5. Conclusions
The triangular regular array of magnetic particles demonstrates quite non-trivial scenario of phase
transitions in the presence of an external magnetic field. The combination of two kinds of the origin of
frustration, the first, present for non-bipartite triangular lattice with any kind antiferromagnetic interac-
tion, and the second, connected with the long-range character of magnetic dipole interaction, leads to the
creation of a set of linear topological defects with the growth of the magnetic field. Self-organization of
33701-8
Self-organization of topological defects for a magnetic dots array
such defects determines the magnetization process for a systemwithin a wide range of external magnetic
fields. It is worth noting an essential difference from the standard topological transitions known for an
atomic spin system of low-dimensional magnets. For a system of microparticles (atomic spins) of standard
low-dimensional magnets, the topological defects (vortices, domain walls, etc.) appear as a consequence
of thermal fluctuations, the resulting structure being quite irregular. By contrast, in our case of the sys-
tem of mesoscopic elements, the characteristic energy of the interaction of two particles is much higher
than the Curie temperature. The creation/annihilation of the defects takes place at the critical value of the
magnetic field, where the energy of the defect vanishes, whereas other characteristics (such as the typical
interaction energy of the defects) are still finite and high, up to the values of 105 Kelvin [22]. In our case,
a quite regular pattern of these defects is realized in the system even at finite temperatures below the
Curie temperature.
Acknowledgements
This work was partly supported by the State Foundation of Fundamental Research of Ukraine via
grant No. F53.2/045
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Самоорганiзацiя топологiчних дефектiв в трикутнiй гратцi
магнiтних точок пiд впливом перпендикулярного
магнiтного поля
Р.С. Химин1, В.Є. Кiрєєв2, Б.О. Iванов2
1 Iнститут магнетизму НАН України та МОН України, бульв. Вернадського, 36-б, 03142 Київ, Україна
2 Унiверситет Окленд, Рочестер Хiллс, 48309 Мiчиган, США
Перiодичний масив магнiтних частинок (магнiтних точок), сформований в трикутну двовимiрну гратку,
утворює складну струтуру магнiтного впорядкування. Магнiтна симетрiя основного стану для такої си-
стеми виявляється нижчою нiж симетрiя гратки. Далекодiюча диполь-дипольна взаємодiя призводить
до специфiчного антиферомагнiтного впорядкування при малих полях, в той час як при збiльшеннi ма-
гнiтного поля виникає серiя лiнiйних топологiчних дефектiв. Самоорганiзацiя таких дефектiв визначає
процес намагнiчування системи в широкому дiапазонi зовнiшнього магнiтного поля.
Ключовi слова: магнiтна точка, топологiчний дефект
33701-10
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http://dx.doi.org/10.1103/PhysRevB.87.094402
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http://dx.doi.org/10.1109/LMAG.2013.2277995
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http://dx.doi.org/10.1016/0031-8914(50)90130-3
http://dx.doi.org/10.1103/PhysRevB.7.5017
http://dx.doi.org/10.1103/PhysRevB.72.144417
http://dx.doi.org/10.1103/PhysRevB.65.134434
http://dx.doi.org/10.1088/0022-3719/17/26/014
http://dx.doi.org/10.1134/S0021364009240035
Introduction and motivation
Model description
Ground states for infinite system: Numerical analysis
Simplest ground states of the system: zero field and saturation
Monte-Carlo analysis for intermediate field values
Magnetic ground states: Analytic description
Point defect scenarios
Instability of AFM state through the creation of topological lines
Plateau description
Conclusions
|