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PHYSICAL REVIEW B 87, 094402 (2013)
Multipole magnetostatic interactions and collective vortex excitations in dot pairs,
chains, and two-dimensional arrays
O. V. Sukhostavets,1J. Gonz´
alez,1and K. Y. Guslienko1,2,*
1Departamento de F´
ısica de Materiales, Universidad del Pa´
ıs Vasco, UPV/EHU, 20018 San Sebasti´
an, Spain
2IKERBASQUE, The Basque Foundation for Science, 48011 Bilbao, Spain
(Received 13 August 2012; revised manuscript received 14 January 2013; published 4 March 2013)
The multipole expansion of magnetostatic interaction energy between nonuniformly magnetized ferromagnetic
particles is developed and applied for calculation of the gyrotropic frequencies of the coupled vortex state dots.
A pair of coupled dots in a vortex state is considered as a model system, and then the calculations are extended to
lateral arrays of the interacting vortex state dots [one- and two-dimensional (2D) square and hexagonal lattices].
The odd rank multipole moments (dipolar, octupolar, etc.) play the main role in the interdot magnetostatic
interaction. The vortex collective frequencies and group velocities are calculated within the pole-free and rigid
vortex models and compared with the experimental observations of the dynamics by broadband ferromagnetic
resonance and x-ray imaging. The cases of different vortex core polarizations are considered, and their strong
influence on the excitation spectra is shown. Frequency bandwidths of the vortex gyrotropic modes for dense
2D square and hexagonal arrays can exceed 1/3 of the eigenfrequency of an isolated dot. The group velocity for
transferring the total microwave magnetic energy from one dot to another is found to be proportional to the dot
thickness and inversely proportional to the squared lattice period.
DOI: 10.1103/PhysRevB.87.094402 PACS number(s): 75.75.Jn, 75.30.Ds, 75.50.Tt
I. INTRODUCTION
Electromagnetic interactions play an important role for
closely located charged particles. These interactions are
usually accounted in dipolar approximation introducing the
particle dipole moments. If charge density distributions within
the particles are complicated (e.g. in molecules),1,2the dipolar
approach is not sufficient, and higher multipolar moments
have to be taken into account. The multipole expansions are
well known for calculation of the electrostatic (Coulomb)
interaction energy between molecules, whereas they are not
so widely used in magnetism. It is also well known that the
magnetic charges (magnetic monopoles) do not exist. Never-
theless, if we assume that each particle possesses a fictitious
magnetic charge that can be calculated as a divergence of its
magnetization, then the interaction between magnetic particles
can be written as Coulomb type. Therefore, the formalism of
electrostatic multipole expansion can be applied to describe the
magnetostatic interaction energy replacing the electric charges
by magnetic ones. It can be useful for arrays of small magnetic
particles that attract attention nowadays due to their possible
applications as patterned magnetic recording media, logic
operation devices, artificial magnonic crystals, and different
biomagnetic applications (hyperthermia, drug delivery, etc.).3
The presence of other magnetic particles nearby leads to the in-
creasing influence of magnetostatic and exchange interparticle
interactions. The long-range magnetostatic interaction plays
a leading role for the nontouching particles or for particles
with nonmagnetic (dielectric) shells, which exclude the direct
exchange of the particle spins.
If the magnetic particles of arbitrary shape are considered
as pointlike objects having multipole moments located in their
centers, then the magnetostatic interaction between them can
be expanded via spherical tensors.4Such decomposition has
the advantage that a coupling between the particle multipole
magnetic moments of any rank can be written within a unified
analytical approach, and the geometry of spatial arrangement
of the particles can be accounted apart from single particle
physical properties described by their multipole moments. The
multipole decomposition is expressed as an asymptotic series
on inverse center-to-center particle distance, and the symmetry
properties of the in-particle magnetization distributions are
accounted explicitly. The technique of the multipole expansion
allows calculating the magnetostatic interaction energy with
any desirable accuracy, and contributions of the different
multipole moments can be analyzed separately. So, the
problem of the calculating of magnetostatic interaction is
reduced to accounting the particle multipole moments and
relative spatial locations of the particles that can be done
within an appropriate approximation using concrete models
of the particle magnetization distributions.
The multipole expansion has already been applied to
describe the interaction between single-domain in-plane mag-
netized nanoparticles.5Another typical example of magnetic
charge distributions is a vortex structure of the magnetization
being often a ground state of small soft magnetic particles
for certain lateral parameters.6Such vortices in flat quasi-two-
dimensional magnetic particles (dots) are characterized by the
in-plane curling magnetization and vortex core region with out-
of-plane magnetization. After shifting this vortex core from
a static equilibrium position by applying external magnetic
field or spin-polarized electric current, it oscillates with the
frequency in the 100 MHz range, providing a narrow frequency
linewidth and quite high microwave power output.7,8A pair of
dots with oscillating magnetic vortices attracts attention due to
behavior of the vortices as coupled oscillators9–15 and ability to
transfer energy periodically from one dot to another.13 It also
serves as a model system for better understanding the vortex
dynamics in one- (1D) and two-dimensional (2D) dot arrays.
In order to be able to use the coupled dots, for example, in
magnonic crystals, where the interaction leads to appearance of
frequency bands of the coupled single-dot spin excitations, it is
important to get better theoretical insight into the dependences
of this coupling on different parameters.
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1098-0121/2013/87(9)/094402(14) ©2013 American Physical Society
SUKHOSTAVETS, GONZ ´
ALEZ, AND GUSLIENKO PHYSICAL REVIEW B 87, 094402 (2013)
Usually, the interaction between two dots is calculated
numerically, and the dependence of this interaction on the
separation between the dots is fitted using simple analytic
functions. In this paper, we show that the multipole expansion
serves as an excellent tool to describe interaction between
flat ferromagnetic particles in a vortex state with a desirable
accuracy without any numerical fitting and to analyze the con-
tribution of different multipole moments to the magnetostatic
interaction. The dipole-dipole is known to be the dominant
interaction for large separation between dots, but contribution
of the high-order multipole moments with decreasing the
interdot distance dis not clearly understood. In this paper, we
show that the high-order multipole terms play a significant role
with decreasing the interdot separation. It turns out that there is
no reason to characterize the interaction by single-power law
behavior like 1/d6(Refs. 9and 16)or1/d3.9(Refs. 13 and 14)
because the energy is decomposed over odd powers of 1/d.
The multipole interactions are responsible for nonmonotonous
dependence of the spin eigenfrequencies on the vortex dot
lattice period measured in Ref. 17. The quadrupolar interaction
leads, in particular, to appearance of the fourfold magnetic
anisotropy in the square arrays of single-domain circular
dots.18–21 It was also shown that the high-order multipole
interaction changes the dipolar ordering and hysteresis loops
in 2D dot arrays.5,22
A series of the experimental16,17,23–26 and theoretical
papers27–29 on the coupled magnetic vortex dynamics in
lateral nanostructures have been published recently. Two-
dimensional arrays,16,17,23,24 chains,25,26 and clusters23 of the
circular and square dots were investigated by different experi-
mental techniques (broadband ferromagnetic resonance, time-
resolved magneto-optic Kerr effect, and x-ray imaging). These
coupled vortex dynamics can be used for information-signal
processing.11,13 We calculate the low-frequency excitation
spectra of different arrays of the vortex state dots (1D and
2D arrays) on the basis of the multipole decompositions
of the interdot coupling integrals assuming a zero in-plane
magnetic field. The existing experiments on broadband ferro-
magnetic resonance in the arrays of circular and square FeNi
dots16,17,23,26 are analyzed in this paper within the developed
approach.
The paper is organized as the following. We consider a
general approach to the interdot multipole interactions in
Sec. II. We calculate the multipole moments of a vortex state
magnetic dot in Sec. III. The interaction energy and dynamics
of a coupled vortex dot pair are considered in Sec. IV.The
collective vortex excitations of array of the laterally situated
vortex dots (1D chains and 2D dot lattices) are considered in
Secs. V–VI. Discussion of the obtained results and comparison
with experiments are conducted in Sec. VII. A summary is
presented in Sec. VIII.
II. MULTIPOLE DECOMPOSITION OF MAGNETOSTATIC
INTERACTION ENERGY
Two magnetic particles with nonuniform magnetization
distributions M(r) are assumed to interact similar to elec-
trically charged particles via the Coulomb law. Instead of
the electric charges, we use the fictitious magnetic charges
that can be expressed as a divergence of magnetization
FIG. 1. (Color online) Schematic picture of (a) a magnetic dot
pair and (b) 2D square lattice with the coordinate system used. Here,
is the center-to-center interdot distance and rand rare the local
dot coordinates.
ρ(r)=−Msdivm(r) due to symmetry of the Maxwell’s equa-
tions. Here, m=M/Msis the magnetization unit vector, and
Msis the saturation magnetization. Thus, the magnetic inter-
action energy is written in the following Coulomb-like form:
Uint =dVdV ρ(r1)ρ(r2)
|r1−r2|,(1)
where r1∈Vand r2∈Vbelong to the first and second
particle [see Fig. 1(a)], respectively. All the equations in this
paper are written in the CGS system of units. In the SI system,
one has to multiply Eq. (1) by μ0/4π.
The interaction energy Eq. (1) can be rewritten via a
common set of multipole moments of ferromagnetic particles.
For obtaining a multipolar energy decomposition of Eq. (1),we
introduce a vector
connecting the particle centers [Fig. 1(a)].
We use the relation |r1−r2|=|
−(r−r)|, where rand r
are the local coordinates within the first and second particles
[Fig. 1(a)]. The Coulomb kernel decomposition is given by
1
|
−
|
=1
lm
4π
(2l+1)
l
¯
Ym
l(ˆ
)Ym
l(ˆ
),<(2)
via the spherical harmonics Ym
l(ˆ
r)—irreducible tensors of the
rank lhaving (2l+1) independent components numbered
by the index m=−l,−l+1, ...l.4A vector with the hat
symbol ∧indicates the corresponding unit vector in the given
vector direction. A bar over a variable means the complex
conjugation. Then, introducing
=r−r, we separate the
in-particle variables rand rby using the summation theorem4
for the solid spherical harmonics rlYm
l(ˆ
r):
LYM
L(ˆ
)
=4π(2L+1)!
llmm
(−1)l
√(2l+1)!(2l+1)!
×CLM
lmlmrlrlYm
l(ˆ
r)Ym
l(ˆ
r),(3)
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MULTIPOLE MAGNETOSTATIC INTERACTIONS AND ... PHYSICAL REVIEW B 87, 094402 (2013)
where CLM
lmlmare the Clebsch–Gordan coefficients,4and
L=l+l,M=m+m, and l,l=0,1,...,L.
The interaction energy in Eq. (1) can be rewritten as
decomposition via the particle multipole moments:1,2
Uint =
l,l,m,m
Tllmm(
)Qm
l(1)Qm
l(2),(4)
where Qm
l(1) and Qm
l(2) are the 2lmultipole moments
of the nonoverlapping magnetic charge distributions
ρ(r)→−divM(r) and ρ(r)→−divM(r) calculated
in the local frames of the 1st and 2nd particles, and
Tllmm(
) is a purely geometrical interaction tensor
that depends only on the relative positions of the
particle centers
. The multipole moments of the rank
l(l=0,1,2,..., |m|l) for the charge distribution
ρ(r) can be written as functionals of the in-particle
magnetizations Qm
l=(4π/(2l+1))1/2d3rρ(r)rlYm
l(ˆ
r),
where ris the radius vector in the local frame [Fig. 1(a)],
Ym
l(θ,ϕ)=√(2l+1)/4π√(l−m)!/(l+m)!Pm
l(cos θ)eimϕ
are the normalized-to-unit spherical harmonics,4Pm
l(x)are
the Legendre polynomials, and θand ϕare the spherical
angles. This is a common language to write a coupling of
the particle multipole moments of any rank within a unified
analytical approach. Equation (4) allows one to separate
geometrical properties of the particle arrays from physical
properties of a single particle described by the moments Qm
l
specific for given M(r). Such multipole expansion has been
used for electrostatic energy between unperturbed molecules
(e.g. Refs. 1and 2). The interaction tensor in Eq. (4) is given
by the following expression:
Tllmm(
)
=(−1)l(−1)ML+M
l+m1/2L−M
l−m1/2
I−M
L(
),
(5)
where ( n
k)=n!
k!(n−k)! for 0 knare the binomial coeffi-
cients, and L=l+l,M=m+m, and
I−M
L(
)=4π
2L+11/2Y−M
L(ˆ
)
L+1(6)
are the irregular solid spherical harmonics.
Spatial dependence of the interaction of two multipole
moments of the ranks land lseparated by distance obeys the
power law 1/l+l+1. Each multipole behaves as if it is located
in the particle center, and the magnetostatic potential (and mag-
netic field) created by the multipole depends on the value of the
multipole moment, distance to the given point and orienta-
tion of the vector
. The components of the interaction tensor
Tllmm(
) for given
can be easily found from Eqs. (5)–(6).
They are also tabulated in Ref. 1. So, the energy is defined by
the single particle multipole moments Qm
l(1) and Qm
l(2) that
are functionals of the particular magnetization distributions.
Let us apply Eqs. (4)–(6) to describe a system of flat
ferromagnetic particles arranged in the xy plane [Fig. 1(b)].
For such a case, we put θ=π/2 in the definition of the
spherical harmonic YM
L(ˆ
)=YM
L(θ,ϕ), where θand ϕ
are the angles between the center-to-center radius vector
and zand xaxes, respectively. The relatively simple function
YM
L(π/2,ϕ) can be found in Chap. 5 of Ref. 4. This allows
simplifying Eq. (5) for the interdot coupling tensor:
Tllmm(
)=(−1)le−iMϕ
L+1SLM L+M
l+m1/2L−M
l−m1/2
,
(7)
SLM =(−1)(L+M)/2(L+M−1)!!(L−M−1)!!
(L+M)!!(L−M)!! .
Thus, the multipole expansion can be written as the explicit
series in 1/|
| using the terms with the fixed total index L=
l+l:
Uint =
L
UL(ϕ)
L+1,
UL(ϕ)=
lmM
(−1)lSLM L+M
l+m1/2
×L−M
l−m1/2
e−iMϕQm
lQM−m
L−l.(8)
From Eqs. (7)–(8), we can calculate the interparticle
magnetostatic interaction energy in Eq. (1) as a result of
interaction of the extended magnetic charges within the
particles having the charge density proportional to divm(r).
Equation (8) is valid for arbitrary particle shape. It can be
directly applicable for spherical particles. For magnetic disks,
it is natural to use the cylindrical coordinates (ρ,ϕ,z), so we
substitute cos θ=z/z2+ρ2(zvaries from −h/2toh/2,
where his the dot thickness). Convergence of the series in
Eq. (8) is determined by dependence of the particle multipole
moments Qm
lon the index l.
Equation (8) can be generalized for a 2D regular array of
the particles located in the positions
nin the xy plane:
Uint =
Lnn
UL(ϕnn)
L+1
nn
,(9)
where
nnis the vector connecting the centers of the nth and
nth particles.
Further simplification of Eq. (9) can be achieved by using
symmetries of the particular particle arrays (
n) and in-particle
magnetization distributions described by the moments Qm
l(r).
A typical example of the vortex state dots is considered in the
next section.
III. MULTIPOLE MOMENTS OF A VORTEX
STATE SINGLE DOT
In this section, we calculate the multipole moments that
a circular vortex state dot with a shifted vortex possesses.
The magnetization distributions M(r) of the vortex state dots
are essentially inhomogeneous. We show below that, apart
from the dominant dipole moment, the shifted vortices exhibit
the additional multipole moments Qm
lof a rank l>1 that
influence significantly on the dynamics of interacting vortices
(especially octupole moment with l=3).
We consider a cylindrical dot of the thickness hand
radius Rin the vortex state. This state is characterized by
magnetization circulating in plane of the dot around the vortex
094402-3
SUKHOSTAVETS, GONZ ´
ALEZ, AND GUSLIENKO PHYSICAL REVIEW B 87, 094402 (2013)
core.6As this vortex core is magnetized perpendicular to the
dot plane, there are charges produced by the out-of-plane
magnetization. The radius of vortex core Rcdepends on the
material parameters (like the saturation magnetization Msand
exchange stiffness A). For typically used material (Permalloy),
Rcis about 10 nm; that is quite small compared to the dot
radius (several hundreds of nanometers). For this reason,
we can neglect the small vortex core and assume that there
are no magnetic charges in such centered flux closure state.
The charges are generated by off-centered vortices applying
external magnetic field or exciting the vortex core motion.
The vortex core magnetic moment is perpendicular to both the
dynamical (in-plane) dot magnetic moment and to the radius
vector connecting the dot centers. Therefore, this interaction
is equal to zero in the main approximation. The vortex core
displacement from the equilibrium position in the dot center
can be described by a complex parameter s=sx+isy, where
sxand syare the vortex core coordinates in the units of dot
radius R. The simplest model of the displaced vortex (rigid
vortex model) assumes the presence of side surface charges
on the dot circumference and no volume charges.6This model
overestimates the interaction energy between magnetic dots as
well as the vortex eigenfrequencies. Alternatively, a two-vortex
or pole-free model can be used to calculate the magnetic
charge distribution outside the vortex core.6In this model,
one vortex with coordinate sis located within the dot, while
the second image vortex is located outside the dot. We assume
that the dot is thin enough, and any dependence of magne-
tization on the coordinate zalong the dot thickness can be
neglected. The components of the magnetization M(x,y)are
given by
Mx+iMy=2Msw/(1 +w¯w),
Mz=pMs(1 −w¯w)/(1 +w¯w).(10)
We describe the magnetic charges ρ(r)=−Msdivmout-
side the vortex core, using the function w(ζ,¯
ζ)=f(ζ)/|f(ζ)|6
with f(ζ)=−iC(ζ−s)(¯
sζ −1)/c(1 +|s|2) for the pole-
free model or f(ζ)=iC(ζ−s)/c for the rigid vortex
model, where Cis the vortex chirality (C=±1 for coun-
terclockwise/clockwise rotation of in-plane magnetization),
p=Mz(0)/Msis the vortex core polarization defined as
a sign of the magnetization component along the zaxis
in the vortex center,6ζ=(x+iy)/R, and xand yare
Cartesian coordinates. The real parameter c=Rc/R is the
reduced vortex core radius. So, in the region outside of
the vortex core, Mz=0 and reduced magnetization can be
written as mx+imy=w(ζ,¯
ζ). In the expression for divm=
2Re{∂w(ζ,¯
ζ)/∂ ζ }, we keep only terms proportional to the
small parameter sas they correspond to quadratic approxima-
tion on the scomponents in the interdot coupling energy in
Eq. (1). Such approximation is sufficient for consideration
of the linear vortex dynamics. Then the magnetic charge
density can be written as divm=Cχ(ρ)(sxsin ϕ−sycos ϕ),
where ϕis the polar angle of ρ=(x,y) and χ(ρ)isa
model dependent function (χ(ρ)=2forthetwo-vortex
model and χ(ρ)=−δ(ρ−1) for the rigid vortex model, see
Appendix A).
Within the two-vortex model, we can write the multipole
moment of the rank lfor the dot in an off-centered vortex state
TABLE I. Calculated multipole moments of the rank lfor
magnetic cylindrical dots with shifted vortices [the vortex core has
the coordinates (X,Y), s=(X+iY)/R ]. The function is fm(s)=
−iπC(sδm,1+¯
sδm,−1)MsR3,whereCis a vortex chirality, Msis the
saturation magnetization, and Ris a dot radius.
Rank of the moment Qm
lExplicit expression for Qm
l
Dipole moment l=1Qm
1=(√2/3)βfm(s)
Octupole moment l=3Qm
3≈−(√3/10)R2βfm(s)
Triacontadipole moment l=5Qm
5≈(√30/56)R4βfm(s)
Octacosahectapole moment l=7Qm
7≈−(5√14/288)R6βfm(s)
as
Qm
l=2CMs
R(l−m)!
(l+m)! ρdρdϕdz(ρ2+z2)l/2
×Pm
lz
z2+ρ2eimϕ(sxsin ϕ−sycos ϕ),(11)
where −lmlfor a given l, and the integration over
ρand zruns within the intervals 0 ρRand −h/2
zh/2. The monopole term (l=0) is equal to zero due
to the symmetry of in-dot magnetization distribution. Other
multipole moments [dipolar (l=1), quadrupolar (l=2),
octupolar (l=3), hexadecapolar (l=4), triacontadipolar
(l=5), etc.] are not equal, in general, to zero. So, the index
lis l1 for the vortex dots, and consequently, the index
L2inEq.(8). The integration over the angle ϕyields
that only the indices m=±1 contribute to the summation in
Eq. (8). By using the explicit expression for the associated
Legendre polynomials Pm
l(x),4we calculate the multipole
moments in Eq. (11) of the vortex state dots (see Table I).
As for thin magnetic dots, the ratio β=h/R 1; so for the
dipole, octupole, and higher-order moments, we can neglect
the terms proportional to β3and higher-order terms in β.In
Table I, we present the moments Qm
lwith odd values of l
that are linearly proportional to β. To calculate the moments
Q±1
l, we used the properties of the Pm
l(x) polynomials30
shown in Appendix B,Eq.(B1). The moments with even l
(quadrupolar, hexadecapolar, etc.) are equal to zero for thin
dots (β1) within the linear approximation in β. Due to the
relation ¯
Ym
l=(−1)mY−m
l,wehave ¯
Qm
l=(−1)mQ−m
l,soit
is sufficient to calculate only the moments with m0. The
index m=±1 within both the models of displaced vortex;
consequently, only the components Q+1
lare independent.
We can write an analytical expression for the moments Qm
l
(m=+1/−1) with odd values of l:
Qm
l=2β(−1)(l−1)/2l!!
(l+2)(l−1)!!√l(l+1) Rl−1fm(s)+O(β3),
fm(s)=−iπC(sδm,1+¯
sδm,−1)MsR3(12)
within the pole-free vortex model. The additional multiplier
−(l+2)/2 appears in Eq. (12) if we use the rigid vortex model.
It can be shown using the moments Qm
lin Eq. (12) that the L
terms in Eq. (8) are decreasing as 1/(L2dL)or1/(LdL) with L
increasing (L1) for the pole-free and rigid vortex models,
respectively (d=/R).
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MULTIPOLE MAGNETOSTATIC INTERACTIONS AND ... PHYSICAL REVIEW B 87, 094402 (2013)
In the next sections, calculating the vortex collective
excitations, we assume that there is no in-plane bias magnetic
field applied to the dots (static vortex core displacement sis
equal to zero). Therefore, the multipole moments in Eq. (12)
and corresponding interdot magnetostatic interaction in
Eq. (9) have pure dynamical origin.
IV. INTERACTING VORTEX DOT PAIR
We consider the magnetostatic interaction energy and
excitation frequencies of the vortex state pair of cylindrical
dots. The dots are situated in the xy plane [Fig. 1(a)] and
have identical thicknesses and radii (the dot aspect ratio
β=h/R < 0.1). Their chiralities are C1and C2, and the
vortex core polarizations are p1and p2.
A. Interaction energy
The expression for the magnetization divergence allows us
to rewrite the interaction energy between two dots in Eq. (1) as
Uint =αβ ηαβ sα1sβ2, where the indices (1 and 2) correspond
to the first and second dot, respectively. The coefficients ηαβ =
C1C2M2
sR3Iαβ reflect the value of interaction along xand y
directions (α,β =x,y). If we write |r1−r2|2=(z−z)2+
(ρ−ρ−
)2, then the dimensionless coupling integrals Iαβ
are given by the following expression:
Iαβ =1
0
ρρdρdρ 2π
0
dϕdϕβ
0
dzdz
×χ(ρ)χ(ρ)
(z−z)2+(ρ−ρ−
)2
ˆϕαˆϕ
β,(13)
where d=/R,ˆϕ=−ˆ
xsin ϕ+ˆ
ycos ϕ, and ρand ρare the
vectors lying in the xy plane. If we direct the xcoordinate
axis along the vector
=ˆ
xconnecting the centers of two
dots, only diagonal components of Imn are nonzero in Eq. (13).
We denote them as Ixx =Ixand Iyy =Iy. So, the interaction
between two dots can be found by numerical integration of
Eq. (13). The integration over z,zin Eq. (13) can be performed
analytically [see Appendix B,Eq.(B2)].
Following Sec. II, the interaction energy between magnetic
dots is proportional to the products of the dot multipole
moments. For vortex state dots, the products of the moments
Q±1
lQ±1
lwith odd lthat are proportional to β2give the
main contribution to the energy neglecting the higher powers
of β. For instance, as the quadrupole-quadrupole interaction
∝Qm
2Qm
2∝β4is equal to zero in the quadratic approximation
on s, it means that it plays a negligibly small role in the
interaction between ideally shaped thin magnetic dots with m
being independent on z, but dot shape imperfections, defects,
etc. may lead to inducing the dot quadrupolar moments. So,
the interaction energy presented by Eq. (8) with the multipole
terms ∝1/l+l+1can be written as decomposition in odd
powers of the inverse dot center-to-center distance 1/only.
The expansion by the multipole moments in Eq. (4) assumes
well-separated dots, i.e. this is a kind of perturbation theory
using the small parameter ε=R/|
| = 1/d. The condition
of the applicability of the multipole decomposition is that
the center-to-center particle distance is larger than all other
linear dimensions of the particles (diameter, height, etc.). This
decomposition is valid even for touching dots as ε≈1/2 still
remains a small parameter in this case.
After some algebra, we get the multipole coefficients ULof
the vortex dots in the form
UL=−8π2C1C2β2[(L−1)A+(L+1)B]PLRL+4M2
s,
PL=(L−1)!!(L−3)!!
l
glgL−l
(l+1)!(L−l+1)! ,(14)
where gl=l!!/[(l−1)!!(l+2)] for the two-vortex model
or gl=l!!/[2(l−1)!!] for the rigid vortex model,
and A=(s1·s2) and B=(s1ys2y−s1xs2x) cos 2ϕ−
(s1xs2y+s1ys2x)sin2ϕare functions of the vortex core
coordinates.
The expression ULfor the energy contribution of the
different multipole moments to the magnetostatic interaction
energy of a dot pair can be calculated by using Eq. (14).For
a pair of thin magnetic dots (or a linear dot chain) situated
along the xaxis, the angle ϕis equal to zero, and the total
interaction energy is
Uint =M2
sC1C2R3(λxs1xs2x+λys1ys2y),(15)
where the coefficients λx,y are the multipole approxima-
tions of the coupling interaction integrals Ix,y in Eq. (13).
Using the expressions for the different multipole terms,
written in Table I, we can find the explicit multipole ex-
pansions for the interaction coefficients within the pole-free
model:
λx=π2β24
9d3+1
5d5+113
560d7+5·197
8·16 ·27d9+O(d−11)
+O(β4),(16)
λy=−π2β28
9d3+4
5d5+6·113
560d7+5·197
16 ·27d9+O(d−11)
+O(β4).(16)
It is shown in Fig. 2how the coefficients λx,y approach the
numerically calculated coupling integrals Ix,y if we increase
the accounted number of terms of different powers of 1/d.
With decreasing distance between dots, the dipole-dipole and
dipole-octupole interactions are not sufficient for representing
the total interdot magnetostatic interaction. The pure dipolar
approximation (L=2) yields error about 15% for Ix(d) and
about 40% for Iy(d) in the case of closely spaced dots d→2.
The higher-order interactions such as dipole-octupole (L=4),
octupole-octupole (L=6), dipole-triacontadipole (L=6),
and so on are important. The multipole decomposition up to
L=8(1/d 9term) is necessary if d<2.5. Note that the
dipole-quadrupole and quadrupole-quadrupole interactions are
negligibly small.
B. Dynamics of coupled magnetic vortices
Recent experiments demonstrate that, in a pair of closely
situated magnetic dots, the excitation of the vortex core
gyrotropic motion in one dot induces the vortex core movement
in the second dot due to the presence of magnetostatic
interaction.10–12 Moreover, in a pair of magnetic dots with
oscillating vortices, the energy can be transferred periodically
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SUKHOSTAVETS, GONZ ´
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∼
∼
∼
∼
∼
∼
∼
∼
FIG. 2. (Color online) The contributions of the different multipole
terms in the expansion of λx,y in Eq. (16) to the interdot interaction
energy of two vortex state dots are compared with the numerically
found values of the coupling integrals in Eq. (13) Ix,y (violet solid
lines) along (a) xand (b) ydirections (β=0.1). Here, uL=U2/
(M2
sR6), where U2is defined in Eq. (8) at L=2. dis in the units of
dot radius R.
from one dot to another.13 As it was shown in Refs. 9and
14 by solving the Thiele equations of motion31 iR2Gzn ˙
sn=
2∂U/∂ ¯
sn, the coupled system of two magnetic dots in a vortex
state has two gyrotropic eigenfrequencies ωp
1and ωp
2. Here,
Gzn =−pn|G|and |G|=2πhMs/γ are the zprojection and
absolute value of the gyrovector of the nth dot (for dot pair n=
1,2), ∂/∂¯
s=(∂/∂sx+i∂/∂sy)/2, U=nκnsn¯
sn/2+Uint is
the total magnetic energy of the vortex pair, and pnand κnare
the vortex core polarization and stiffness coefficient32 of the
nth dot. These eigenfrequencies can be written as (ωp
1,2)2=
ω2
0(1 ±a)(1 ±pb), where ω0is the eigenfrequency of isolated
dot, and the dimensionless interdot coupling parameters are
a=C1C2ηx/(|G|ω0) and b=C1C2ηy/(|G|ω0). Here, ωp
1,2
depends on the interaction between the dots ηx,y (defined in
Sec. IV A) and on the product of the vortex core polarizations
p=p1p2=±1 (the cores almost do not contribute to the
dynamic energy; they only determine signs of the gyrovectors).
The degeneracy of two eigenfrequencies is removed due
to the interaction, and a finite difference between these
eigenfrequencies (the frequency splitting) appears. The value
of the splitting |fp
1−fp
2|that is measured experimentally11–13
can be calculated via the interdot coupling integrals
FIG. 3. (Color online) Calculated (solid and dot-dashed lines)
dependences of the frequencies of the vortex gyrotropic modes of a
dot pair on the interdot distance d=/R for the dot aspect ratio
β=0.013. The triangles and circles correspond to the positive
and negative product of polarizations of the dots p=+1/ −1,
respectively, and are taken from the experiment by Arai et al.12
Ix,y as
fp
1−fp
2=ωp
1−ωp
2
2π=γM
s
4π2β(|Iy|−pIx),(17)
where Ms=700 −800 G (kA/m in SI units) and γ/2π=
2.95 MHz/Oe (29.5 GHz/T in SI units) for Permalloy. The
values of the integrals depend on the particular model within
which they are calculated. We can see that the frequency
splitting depends only on the geometrical parameters of the
dots (thickness hand radius R) and the distance between the
dot centers d=/R. The eigenfrequencies fp
n=ωp
n/2πvs
dare depicted in Fig. 3as solid and dot-dashed lines. The
splitting is larger for p=−1. In the limit of small β1,
the integral Ix,y is approximately proportional to β2,14 so
the splitting is |fp
1−fp
2|∝βF(d). An explicit form of the
function F(d) can be found by the multipolar decompositions
in Eq. (16). The dependence of the frequency splitting in
Eq. (17) on the geometrical parameters of the system is then
given by the explicit multipole decompositions (d2) for the
pole-free model
fp
1−fp
2=βγM
s
44(2 −p)
9d3+4−p
5d5+O(d−7)
and for the rigid vortex model
fp
1−fp
2=βγM
s
42−p
d3+3(4 −p)
4d5+O(d−7).
The splitting calculated within the rigid vortex model is
approximately two times larger than the one calculated within
the pole-free model. The frequency splitting is linearly propor-
tional to the interdot coupling integral and as a consequence
to the dot aspect ratio β.
The calculated frequency splitting in Eq. (17) is higher for
the pair of magnetic dots with different polarizations (p=−1)
like observed in the experiments.11–13 The agreement between
the theoretical and experimental results11 is reasonable for
p=+1, but it is not good for p=−1 for the measurements
conducted in Refs. 11 and 13.InFig.3, we compare the
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calculated eigenfrequencies for FeNi dots with parameters
of Ref. 12 and the frequencies measured in Ref. 12.The
agreement is quite good. Only two experimental points for
p=+1/−1 (for the distances d≈2.06/2.27) for low
eigenfrequencies12 differ about 5% from the calculated
dependencies.
V. TWO-DIMENSIONAL ARRAYS OF VORTEX DOTS
(SQUARE AND HEXAGONAL LATTICES)
For 2D regular dot arrays produced by modern experimental
technology, the typical number of the dots in an array is N=
106, and the coordinate approach used for a dot pair is not
adequate to describe the collective vortex dot dynamics. For
description of the spin excitations in the periodical arrays (2D
dot lattices and 1D chains), we use the Born–von Karman
periodic boundary conditions and introduce a finite in-plane
Bloch wave vector k= 0 and the corresponding Brillouin
zones. For any dot array (number of the dots can vary from
three to several millions), we assume that the magnetostatic
interactions between the dots are of pair type, and Eq. (9)
can be applied for calculation of this interaction. The set of
vectors
ndescribe the dot array [Fig. 1(b)], and ϕnis the polar
angle of
n. We can apply the calculation scheme described
above for the case of 2D regular dot arrays (typically square,
rectangular, or hexagonal lattices), 1D dot chains, or clusters.
Rewriting the functions Aand Bin the form Ann=(sn·sn)
and Bnn=(sn·sn)−2(sn·
nn)(sn·
nn)/|
nn|2, we get
the magnetostatic coupling energy of a dot array in the
following form:
Uint =−16π2M2
sR3β2
Lnn
PL
RL+1
L+1
nn
CnCn
×Lsn·sn−(L+1)
2
nn
(sn·
nn)(sn·
nn).(18)
For consideration of magnetization dynamics in the coupled
vortex dot arrays with the anisotropic interaction energy in
Eq. (18), it is convenient to write the Thiele equations of
vortex motion31 via the complex variables sn=sxn +isyn:
iGzn˙
sn=2
R2
∂U
∂¯
sn
,−iGzn˙
¯
sn=2
R2
∂U
∂sn
,(19)
where U=nκnsn¯
sn/2+Uint is the total magnetic energy
of the vortex dot array, Gzn =−pn|G|is the zprojection of
the gyrovector, pnis the vortex core polarization, and κnis
the stiffness coefficient32 of the nth dot in the array. We are
interested in finding eigenfrequencies of the coupled vortex
dots; therefore, the damping term calculated in Ref. 33 is not
included to Eq. (19).
We account in Eq. (18) all the dot multipole moments in
2D vortex dot arrays and are not restricted by the nearest
neighbors’ approximation as in Ref. 24 or by dipole-dipole
approximation as in Ref. 25. The selection rule for M=m+
m=0,±4(±6),...in Eq. (9) leads to only one allowed value
of M=0 for the square or hexagonal arrays of identical dots
having the same values of Cnsn,pn, and k=0 (these dense
arrays are considered below). For such dots, the lattice sums
of the coefficients Bnnare equal to zero due to symmetry of
the arrays. This leads to the important consequence that the
interaction energy for such dot arrays can be written in the
Heisenberg-like isotropic form:
Uint =
nn(nn)CnCn(sn·sn),
(nn)=M2
sR3β2
L
fL
RL+1
L+1
nn
,(20)
where snis the reduced vortex core shift in the nth dot
(proportional to the dot averaged in-plane magnetization),
(nn) is the magnetostatic coupling integral calculated
from the multipole decomposition in Eq. (9), and fL=
−8π2(L−1)PL. All the multipole contributions to the integral
(nn) are negative; therefore, (nn)<0.
The system of the coupled Thiele equations of motions
in Eq. (19) after conducting the Fourier transform sn(t)=
snexp(iωt) is reduced to the pairs of conjugated equations
(pnω−ω0)sn=2
|G|R2
∂Uint
∂¯
sn
,
(21)
(pnω−ω0)¯
sn=2
|G|R2
∂Uint
∂sn
,
where ω0=κ/|G|.
So, the main in-phase (k=0, all the vortex core positions
snare oscillating in-phase assuming identical chiralities of all
the dots Cn) vortex gyrotropic mode of the coupled dot arrays
has eigenfrequency ωG:
ωG(d)=ω0+γM
s
2πβ
L
fLσL(0)
dL+1,
(22)
σL(0) =
n1,n2
1
n2
1+n2
2+2n1n2cos νL+1
2
,
where the angle νis equal to π/2 for square or π/3for
hexagonal (triangular) dot lattices.
The frequencies of the collective vortex modes with k=0
[following Eq. (22)] are plotted in Fig. 4as functions of the
FIG. 4. (Color online) The main gyrotropic frequency (in-phase
mode, k=0) of the square and hexagonal 2D lattices of identical dots
vs the lattice period d(β=0.1). The solid lines represent calculations
by using Eq. (22) accounting L10 for square (1) and hexagonal
(2) lattice. The dashed (dot-dashed) lines correspond to the dipolar
approximation for square (hexagonal) dot lattice.
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SUKHOSTAVETS, GONZ ´
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dot lattice periods dfor 2D square and hexagonal lattices. The
dipolar approximation fails for dense arrays with d<2.5.
The relative error of the dipolar approximation is not large
(7% for square and 10% for hexagonal lattice). These fre-
quencies and additional frequency shift due to the high-order
multipole interactions can be easily detected by ferromagnetic
resonance.16,17
Equation (22) for the in-phase mode (k=0) can be
generalized for the case of identical vortex core polarizations
of the dots, but a finite in-plane Bloch wave vector k= 0. The
symmetry (allowed M=0, ±4(±6), ...)21 is broken for
k= 0(M=±2 terms appear) and can be restored for some
symmetrical values of kat the edges of the first Brillouin
zone. The high-symmetry situations with nonzero kat the
edges of the Brillouin zone will be considered elsewhere. The
aligned state of the parallel polarizations can be reached by
applying perpendicular bias field to the dot array and then
releasing the field. The vortex core polarizations essentially
influence the collective excitation frequencies. The effect of the
vortex polarizations on the interdot magnetostatic interactions
is neglected, accounting small vortex core size (Rc=10 nm)
in comparison to the dot radius (typical R=200–500 nm).
The eigenfrequencies of the coupled vortices depend on the
vortex polarizations only via the dot gyrovectors.
To find the eigenfrequencies for k= 0, we insert Eq. (18)
for the interdot interaction energy Uint to the equations of
motion in Eq. (19). Using the Fourier transform sn(t)=
kskexp[i(k
n−ωt)] and the formalism developed in
Ref. 28, the system of the Thiele equations of motion in
Eq. (19) can be solved yielding the following dispersion
relations for the eigenmodes of the vortex collective precession
in the periodical dot arrays:
ω2(k)=[ωG+ωMβF−(k)][ωG+ωMβF+(k)],(23)
where ωM=γ4πMs,F±(k)=
1(0) −
1(k)±|
2(k)|, and
the multipole decompositions of the coupling integrals can be
presented in the form
1(k)=
L
(L−1)PL
σL(k)
dL+1,
(24)
2(k)=
L
(L+1)PL
σc
L(k)
dL+1.
Here, we use the 2D lattice sums (L=2,4,6,...is even)
σL(k)=
n1,n2
eik·
n
n2
1+n2
2+2n1n2cos νL+1
2
,
(25)
σc
L(k)=
n1,n2n1+n2eiν2
n2
1+n2
2+2n1n2cos νL+3
2
eik·
n,
over the square
n=D(n1,n2) or hexagonal
n=
D(n1+n2/2,√3n2/2) dot 2D Bravais lattices, n1and n2are
integers. σL(σc
L)inEq.(25) are derived by summation of the
terms Ann(Bnn) over the dot lattice. We can rewrite Eq. (23)
as ω2(k)/ω2
0=[1 +(9π/5)F1
−(k)][1 +(9π/5)F1
+(k)], where
F1
±(k)=−
1(k)±|
2(k)|, and frequency of the isolated dot
ω0is calculated within the pole free model. The lattice
sums in Eq. (25) converge quite slowly with increasing the
number of the dots. To improve accuracy of the calculations
and accelerate them, we use the series conversion for the
square lattices (see Appendix C). The lattice sums in Eq. (25)
and corresponding eigenfrequencies can also be calculated
for other dense dot arrays, i.e. for the hexagonal lattices.
The eigenfrequencies for symmetrical directions in the first
Brillouin zone are plotted in Fig. 5.
The group velocity of the collective excitations can be
calculated by using the dispersion relation in Eq. (23) as
vg(k)=∂ω(k)/∂k. The dispersion ω(k)=ω(0) +vg·kis
linear for kD 1; therefore, the velocity can be calculated
analytically near the point k=0. The behavior of the sums
in Eq. (25) near this point strongly depends on the index
L. It was shown that, for L=2, σ2(k)=σ2(0) −2π|k|
and σc
2(k)=−2π|k|exp(2iϕk)/3 for the square lattice34 and
ω ω
ω ω
(a) (b)
FIG. 5. (Color online) Frequencies of the vortex collective excitations for (a) square and (b) hexagonal lattices of ferromagnetic dots for
different lattice periods d(in units of the dot radius R). (a) The wave vector kis plotted along the square side 10or along the diagonal 11
in the first Brillouin zone. (b) The wave vector kis plotted perpendicular to the hexagonal unit cell side (ky) or along the diagonal (kx)ofthe
first Brillouin zone. The values of L10 are used.
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σ2(k)=σ2(0) −4π|k|/√3 and σc
2(k)=−4π|k|exp(2iϕk)/
3√3 for the hexagonal lattice.35 For L>2, σL(k)=σL(0) −
O(|k|2) and σc
L(k)∝k2
α. The group velocity then is determined
by the dipolar (L=2) terms
vg(k)=2πP2
d2ωMLk
|k|,(26)
where P2=1/36(1/16) for the pole-free (rigid) vortex model.
This velocity reaches the values of the order of 400 m/s
for the closely packed square Permalloy dot arrays (d=2)
and is maximal (0.125 ω0D) for the wave vector kdirection
11at kD ≈0.07. The additional multiplier (2/√3) ≈1.155
appears in Eq. (26) in the case of the hexagonal dot arrays,
reflecting larger density of the 2D hexagonal lattice in
comparison to the square lattice with the same period D.
Another important case is the chessboard state (CB)
of the vortex core polarizations {pn}. This state is the
ground state of the square array of the vortex dots, al-
though the energy difference between the aligned pn=
const and CB polarization states is not so large as for
the single-domain perpendicularly magnetized dots36 due
to small sizes of the vortex cores. The polarization of
the dot in the position
n=D(n1,n2) can be written as
pn=(−1)n1+n2. We represent the CB square dot array as a
superposition of two square sublattices with the period √2D.
The uniform (k=0) mode frequency then can be written
as ω2
CB =ωG(ωG+ω1), ω1=βωMn1(n)(1 −pn)/2,
and 1(n)=L(L−1)PL(R/n)L+1. Here, ωCB ωG
because the frequency ω1is positive. The analysis of the
collective excitations of the CB dot arrays with k= 0 will
be conducted elsewhere.
In the papers by Vogel et al.,16,23,26 the frequency splitting
was measured for different sizes of the Permalloy circular
and square dot arrays by broadband ferromagnetic resonance
and x-ray imaging. The nonuniform vortex resonance line
broadening vs the period of square lattice of NiFe circular
dots16 accounting the large number of the interacting dots
(N1200) can be interpreted as a collective frequency
bandwidth and described by Eqs. (23)–(25) (see Fig. 6).
FIG. 6. (Color online) The frequency bandwidth for square array
of the vortex state dots vs the dot lattice period d. Points are taken from
the broadband ferromagnetic resonance experiment by Vogel et al.16
The error bars are plotted using the experimental details presented
in Ref. 16.
The agreement with experiment is good, accounting different
directions of the wave vector k. According to Ref. 16,
the error of the linewidth measurements is ±0.05ω0/2πor
±18 MHz in absolute units, accounting the experimental value
of ω0/2π=355 MHz. The experimental points practically
coincide with the theoretical lines in Fig. 6within such
accuracy.
Recalling that the average dot magnetization
within the main approximation is μ=mV=
−ξCˆ
z×s, we can conduct further simplifications
of Eq. (18). The relation Cnsn=ˆ
z×μn/ξ leads to
the identities CnCn(sn·sn)=(μn·μn) and ξ2CnCn
Bnn=−(μn·μn)+2(μn·
nn)(μn·
nn)/|
nn|2and it
is equivalent to the counterclockwise rotation of all the
vectors Cnsnin the dot array xy plane by the angle π/2. The
interaction energy in new variables is
Uint =16π2M2
sR3β2
ξ2
Lnn
RL+1
L+1
nn
PL
×μn·μn−(L+1)
2
nn
(μn·
nn)(μn·
nn),(27)
where ξ=2/3 or 1 for the pole-free or rigid vortex model,
respectively.
Equation (27) generalizes the dipole-dipole coupling
(L=2) for the case of coupling of the dot multipole moments
Qm
land Qm
lof arbitrary order (L=l+l). The coupling
of any order Lis expressed via the average magnetizations
μn=mnVof the vortex dots. The Hamiltonian in Eq. (27) is
reduced to the standard Heisenberg Hamiltonian in the case of
oscillating identical (oscillating in phase and equal pn) dots or
in the static case, when finite μnappear due to applied in-plane
magnetic field:
Uint =1
ξ2
nn(nn)(μn·μn),(28)
The negative values (nn)<0 of the interdot coupling
integrals [see Eq. (20)] correspond to ferromagnetic interaction
of the dot magnetic moments in the square and hexagonal ar-
rays regardless the magnetostatic character of this interaction.
This is a counterintuitive conclusion because the magnetostatic
interaction in 2D square arrays of single-domain dots37 or
point dipoles35 results in an antiferromagnetic coupling or
antiferromagnetic ground state. The ground state of an infinite
hexagonal lattice of the point dipoles is ferromagnetic,34,35
but for any finite hexagonal lattice, it is a macrovortex with
zero net magnetization.38 In the static case of response of the
coupled dot arrays to the applied in-plane field H, the analysis
can be conducted on the basis of the total magnetic energy
U=nκn|μn|2/2ξ2+Uint +UZ, where the Zeeman energy
is UZ=−VdMsH·nμnand Vd=πR2his the dot volume.
VI. ONE-DIMENSIONAL ARRAYS OF VORTEX DOTS
The collective vortex dynamics in linear chains of circular
dots with the same dimensions is considered in this section.
Equation (25) derived for the 2D lattice sums is also applicable
to 1D dot chains. Assuming that the xaxis is directed along
the chain and making n2=0, kx=k, and ky=0inEq.(25),
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FIG. 7. (Color online) Frequencies of the vortex collective ex-
citations in a linear chain of ferromagnetic dots for different chain
periods d=D/R calculated by Eqs. (30) and (31). The wave vector
kis parallel to the line connecting the dot centers. The red and pink
dashed lines correspond to the equal vortex core polarizations. The
green solid and blue dot-dashed lines are for the alternating vortex
core polarizations, pn=(−1)n.
we get
σL(k)=σc
L(k)=2∞
n=1
cos(nk)
nL+1.(29)
The dipolar sum σ2(k)=ζ(3) +k2(ln k−3/2)/2−k4/288
can be calculated analytically, where ζ(3) ≈1.202 is the
Riemann’s ζfunction. The dispersion ω(k) for the dot chain
with pn=const is weaker than in the 2D arrays case due to
reduced number of the neighboring dots. For touching dots
(d=2), we obtain ω(0) =0.902ω0and ω(π)=1.07ω0.The
group velocity vg(k)=dω(k)/dk reaches the maximum value
of 0.07ω0Dat kD ≈1.1 (150 m/s for the typical parameters
D=1μm, ω0/2π=300 MHz) for the dot chains. For
the chain with alternating core polarizations pn=(−1)n,
the excitation spectrum is essentially different. To describe
the eigenfrequencies of such dot chains, we introduce the
functions σe
L(k) and σo
L(k), which correspond to even or odd
values of the summation index nin Eq. (29). Then, the
frequencies can be expressed as the single band for the aligned
vortex core polarizations
ω2
P(k)/ω2
0=[1 +(18π/5)u+(k)][1 +(9π/5)w−(k)] (30)
and two bands (due to two dot sublattices) for the alternating
polarizations
ω2
±(k)/ω2
0=[1 +(18π/5)u±(k)][1 +(9π/5)w±(k)],(31)
where the multipole coupling coefficients are
u±(k)=
L
PL
dL+1σe
L(k)±σo
L(k),
w±(k)=
L
LPL
dL+1−σe
L(k)±σo
L(k).
The eigenfrequencies in Eqs. (30) and (31) are plotted in
Fig. 7, and the group velocity for the case pn=(−1)nis
depicted in Fig. 8. The eigenfrequency of the aligned chain
FIG. 8. (Color online) Group velocity (in the units of ω0D)for
the chains of vortex state dots having the alternative polarizations
pn=+/−1 vs the chain period d=D/R.
ωP(k) is always in between the branches ω−(k) and ω+(k),
except the point kD =π/2, where all three frequencies are
degenerated because the intersublattice coupling is equal to
zero. The group velocity is approximately twice as high for the
alternating core polarizations, reaching the value 0.142ω0D
for kD =π/2 compared to the case of identical polarizations.
We can deduce from the solution of Eq. (19) the symmetry
relations between the components of the vortex core positions:
s2
kx =±s1
kx and s2
ky =∓s1
ky for the eigenmodes ω±(k)inthe
case pn=(−1)n. That is the mode ω+(k) corresponds to
in-phase (out-of-phase) oscillations of x(y) components of
the vortex core position vectors s1,2
k, while the mode ω−(k)
corresponds to in-phase (out-of-phase) oscillations of y(x)
components of the vectors s1,2
k. We note that the phase relations
are the same, and the vortex trajectory ellipticities are similar
to ones calculated in Ref. 14 for the dot pair with opposite
vortex core polarizations. The uniform oscillating magnetic
field directed along the chain (hx) or perpendicularly to the
chain (hy) can excite only ω−(0) or ω+(0) modes if Cn=
const. If the vortex chiralities of the dot sublattices are different
(C1,C2), then hxexcites only the mode ω±(0) for C1C2=∓1,
and hyexcites the mode ω±(0) for C1C2=±1.
VII. DISCUSSION
We use the obtained multipole expansion to calculate the
vortex gyrotropic eigenfrequencies in coupled 2D dot arrays
(square and hexagonal) and 1D dot chains assuming pair-type
interactions.16,17,23–26 We consider that the frequency splitting
of the degenerated vortex gyrotropic frequencies of the
isolated dots composing a dot pair and the formation of the
finite bandwidth of the vortex collective excitations in the
case of large dot arrays are caused by presence of the interdot
magnetostatic coupling. Therefore, measuring the frequency
splitting or bandwidth (or ferromagnetic resonance frequency
shift) can supply information about the magnetostatic
coupling integrals Ix,y. In the case of the dot pair, it was
shown experimentally11–13 that this frequency splitting
increases with decreasing the distance between the dots
because of strong increasing of the magnetostatic interaction.
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Following the theory presented above, the magnetostatic
interaction energy is decomposed over odd powers of 1/d
in Eq. (16), while other attempts to characterize the interdot
interaction energy and corresponding splitting of degenerated
vortex gyrotropic modes by a single-power law led to
1/d6,11,16 1/d 3.9,13,39 or 1/d 3.6(Ref. 14) dependences. Using
the expression for the frequency splitting in Eq. (17) written
above for the pole-free model, we compared our results with
the experiments.11–13 The overall agreement is not good. We
think that it might be due to two reasons. Firstly, the pole-free
model describes quite well the frequencies of isolated dots,
but for dot pairs and dot arrays, it gives the results different
from experimental ones. It may happen because some surface
magnetic charges can be generated (although the in-plane
bias field is absent) that are not accounted in this analytical
model. Secondly, the differences between the calculated
and measured frequencies of the dot pairs can come from
difficulties in accurate measurements of the eigenfrequencies
because of experimental setup limited resolution and not
exact control of the dot sizes and interdot separations.
The in-phase mode eigenfrequency in Eq. (22) is the
lowest in the excitation spectra of 2D array of vortex state
dots because any set of {pn}different from the aligned
polarizations pn=const leads to increasing the interdot
dynamic interaction energy in Eq. (18). This is explicitly
shown for the CB state (ωCB ωG). The eigenfrequency ωG
decreases monotonically with decreasing the dot lattice period
d=D/R [the explicit expressions for σL(0) are given in
Appendix C], whereas a nonmonotonous dependence ωG(d)
was observed in Ref. 17 by broadband ferromagnetic reso-
nance. Such dependence of the main vortex eigenfrequency
(having maximal in-plane dynamical magnetization) on the
period allows one to conclude that the dots in Ref. 17 are not
in identical vortex states and differ in the vortex polarizations
and/or chiralities. We can conclude from Eq. (22) that the term
∝1/5used in Ref. 17 for interpretation of the results has
a dipole-octupole origin because the quadrupole-quadrupole
term is negligibly (at least in one order of magnitude) small.
We note that, for our calculation of the dispersion relations
in Eqs. (23),(30), and (31), we need neither the approximation
of the nearest neighbors27 nor the dipole-dipole coupling
approximation.28 The long-range magnetostatic interactions
of all the dot multipole moments Qm
lare explicitly accounted
in Eq. (23). The function F±(k=0) =0 due to 2(0) =0
for square lattice and, therefore, the frequency of in-phase
oscillations ωGin Eq. (22) corresponds to the bottom of the
vortex excitation band. The shift of ωGdue to the interdot
coupling is of the order of 100 MHz for a closely packed array
of the typical dots made of Permalloy. The excitation band-
width ω[k=(π/d,0)] −ω(0) also is about 100 MHz. In the
case of the square dot array, the minimal excitation frequency
ωG=ω(0) =0.766ω0for d=2 and ωG=ω(0) =0.835ω0
for d=2.2 is essentially lower than ω(0) =0.982ω0of the
pure dipolar approximation by Galkin et al.28 for d=2.2. The
analytically calculated value for d=2.15, ω(0) =0.821ω0,is
in good agreement with the value of ω(0) =0.815ω0simulated
by the dynamical matrix method29 for the square dot arrays
(we used the value of ω0/2π=908 MHz calculated following
Ref. 32 for the given dot parameters h=15 nm and R
=52 nm). The uniform mode frequency ω(0) =0.708ω0is
lower for the denser hexagonal dot arrays [ω(0) =0.783ω0
in the pure dipolar approximation] for touching dots. The
excitation bandwidths in 10and 11directions in the
reciprocal square lattice are strong functions of the square
dot lattice period d(see Fig. 6). The value of frequency
bandwidth ω[k=(π/d,π/d)] −ω(0) ≈0.35ω0at d→2+
(this notation means that the distance dapproaches the point
d=2 from above) is quite considerable. Although most of the
patterned films are prepared in the form of 2D square arrays,
preparation of wide-area hexagonal dot arrays was recently
demonstrated.40 The frequency bandwidth of the hexagonal
lattice in K direction [Fig. 5(b)] can reach the considerable
value of ω[k=(4π/3d,0)] −ω(0) ≈0.37ω0at d→2+.
Usually the excitation field is applied locally to one dot in
the chain, and the signal propagation from the excited dot to
the next dots in the chain is detected. More efficient dot-to-dot
microwave energy transfer in the chains of square NiFe dots
for alternating core polarizations pn=(−1)n(Ref. 26) can
be explained by wider excitation bandwidth ω−(this band is
excited applying a linearly polarized along-the-chain-direction
variable magnetic field) in comparison to the bandwidth of the
aligned polarizations case pn=const. The dot-to-dot energy
transfer time13 is smaller, τ−=π/ω−τp=π/ωp, and
efficiency of the transfer is higher, exp(−τ−)exp(−τp)
for alternating pn=(−1)n, where =αeff ω0is the relax-
ation frequency. The estimated maximal signal velocity in
the chain of circular dots 0.144ω0D≈330 m/susingthe
parameters of experiment26 (f0=167 MHz, D=2.2μm) is
a bit smaller than the experimental value 370 m/s. That can be
explained by the square shape of the dots and limited number of
the dots in the measured chain.26 The simulations by Barman
et al.25 showed that the propagation of the microwave signal in
a chain of circular vortex dots is more effective for the aligned
polarizations pn=const. This effect is due to excitation of
the first dot in the chain by circularly polarized magnetic field
with frequency close to ω0. The simulated signal propagation
velocities for d=2.25 and L=40 nm (160–200 m/s) are
close to the analytical calculations on the base of Eq. (30).
The phase relations for the ω±(0)-modes in the dot chain
can be used to design/interpret experiments on broadband
ferromagnetic resonance in such patterned films.
The group velocity of the collective vortex excitations in the
vortex dot arrays is essentially higher than the maximal Walker
velocity for the vortex/transverse domain walls propagation in
magnetic nanostripes41 with the similar thickness hand width
w=2R. The vortices in each dot rotate with the linear velocity
of about υ0≈ω0R|s|or typically ≈100 m/s. Accounting
ω0=(5/9π)ωMβand comparing υ0to the group velocity
υg=2πP2ωML/d2in Eq. (26) of the square dot lattices,
we see that the ratio υ0/υg≈d2|s|<1for|s|≈0.1 and
d≈2. Even for the maximal vortex core velocity42 defined
as υm=ωMLe/4π(Le=√2A/Msis the exchange length,
≈18 nm for Permalloy), the ratio υg/υm=(2π2/9d2)L/Le≈
0.5L/Lecan be comparable to and exceed 1 for the closely
packed dots, i.e. the signal propagation velocity in the dot
array can be higher than the maximal velocity of the vortex
motion in each dot. This is also correct for the hexagonal dot
lattices and the linear dot chains with alternating vortex core
polarizations, which have larger maximal group velocity in
comparison to the square arrays.
094402-11
SUKHOSTAVETS, GONZ ´
ALEZ, AND GUSLIENKO PHYSICAL REVIEW B 87, 094402 (2013)
The calculated group velocity is isotropic in Xand Y
direction for the square dot arrays in the case of kD 1.
However, as noted in Ref. 29, applying a finite in-plane
magnetic bias field leads to changing of the dot ground state
(a finite static vortex core displacement) and to removing
this isotropy. The velocity in direction of the bias field
can be essentially larger than the velocity in the direction
perpendicular to the field. Such effect can be qualitatively
explained as redistribution of the interdot coupling energy
between the L-order multipole contributions and can be
calculated within the developed approach.
VIII. SUMMARY
The multipole expansion of the magnetostatic interaction
energy between small ferromagnetic particles is found and
applied to the vortex state thin circular dots. The expansion
simplifies calculation of the magnetostatic energy and allows
separating the in-particle variables such as an internal magne-
tization distribution and geometrical properties of the particle
arrays. From this expansion, one can conclude that, for closely
located thin magnetic dots, the pure dipolar approximation
fails, and the most important interactions are dipole-dipole,
dipole-octupole, and octupole-octupole that have to be taken
into account for interpreting experiments on the coupled vortex
dynamics. Meanwhile, the quadrupole-quadrupole interaction
is negligibly small for thin dots and can be neglected. The
pole-free and rigid vortex models of the moving vortices
give a lower and upper limit for the eigenfrequencies and
their splitting. The realistic dynamical magnetization m(r,t)
is somewhere in between these two limiting cases, being more
close to the pole-free model.
The vortex frequencies of isolated dots split or form
the collective excitation bands with decreasing the interdot
distances as a result of the interdot magnetostatic interactions.
The frequency splitting (excitation bandwidth) value can be
comparable to the gyrotropic frequency of the isolated dot for
dense dot arrays. The vortex eigenfrequencies in dot arrays do
not depend on the vortex chiralities, but strongly depend on the
vortex core polarizations. The group velocity for transferring
the total microwave energy from one dot to another is
proportional to the dot thickness and inversely proportional to
the squared lattice period. The velocity can reach 500 m/sfor
the typical dot sizes and closely packed (hexagonal) dot arrays
or dot chains with alternative polarizations. The presented
calculations can serve as a benchmark for interpretation of
experiments on the magnetic vortex dynamics in coupled dot
arrays and chains.
ACKNOWLEDGMENTS
The authors thank to O. Chubykalo-Fesenko, Y. Otani,
and Y. G. Pogorelov for valuable discussions. K.G. acknowl-
edges support by IKERBASQUE (the Basque Foundation
for Science). The work was partially supported by MEC
Grant Nos. PIB2010US-00153, FIS2010-20979-C02-01, and
by Joint Spanish-Portugal mobility Grant #PRI-AIBPT-2011-
1207. O.S. acknowledges financial support by University of
the Basque Country, UPV/EHU.
APPENDIX A
We calculate here the magnetization divergence within the
rigid vortex and two vortex models of the displaced vortex
core.
1. The rigid vortex model
The complex function describing the vortex magnetization
is f(ζ)=iC(ζ−s)/c. The divergence of magnetization is
due to surface magnetic charges σ=(m·n)Son the dot
circumference Sdescribed by the equation ρ=1. Here n=
(nx,ny) is the vector normal to the surface S, n=nx+iny=
cos ϕ+isin ϕ. The magnetization divergence is
divm=Re(w¯
n)δ(ρ−1)
=− C
|ζ−s|(sxsin ϕ−sycos ϕ)δ(ρ−1).(A1)
We get approximately from Eq. (A1) at small sdivm=
−C(sxy−syx)δ(ρ−1).
2. Two vortex model
The complex function describing the vortex magnetiza-
tion within the two vortex (pole-free model) is f(ζ)=
−i(C/c)(ζ−s)(¯
sζ −1)/(1 +|s|2). In the region outside of
the vortex core, we can write
∂w(ζ,¯
ζ)
∂ζ =∂
∂ζ f(ζ)
¯
f(¯
ζ)=1
2|f(ζ)|
∂f (ζ)
∂ζ ,
∂f (ζ)
∂ζ =iC
c1−2¯
sζ
(1 +|s|2).
The magnetization divergence is
divm=2Re ∂w(ζ,¯
ζ)
∂ζ =− 2C
|z−s||¯
sζ −1|Re(i¯
sζ).
(A2)
In the linear approximation on the small vortex core
shift s, we get from Eq. (A2) divm=2C(sxy−syx)/|ζ|.
In the polar coordinates x=ρcos ϕ, and y=ρsin ϕ, thus
the magnetization divergence within the model is divm=
2C(sxsin ϕ−sycos ϕ).
These expressions for divmserve as definition of the
auxiliary function χ(ρ) in Sec. III:χ(ρ)=2forthetwo-vortex
model and χ(ρ)=−δ(ρ−1) for the rigid vortex model.
APPENDIX B
The associated Legendre polynomials Pm
l(x) for the integer
indices land mare defined as30
Pm
l(x)=1
2ll!(1 −x2)m
2dl+m
dxl+m(x2−1)l,
−lml, l =0,1,2,...
The polynomials P1
l(x) have the following behavior near
the point x=0:30
P1
l(x)=(−1) l
2−1(l+1)!!
(l−2)!! x+O(x3)
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MULTIPOLE MAGNETOSTATIC INTERACTIONS AND ... PHYSICAL REVIEW B 87, 094402 (2013)
for even l, and
P1
l(x)=(−1) l−1
2(l)!!
(l−1)!! +O(x2)(B1)
for odd values of the index l,|x|1.
We can perform an integration over zand zin Eq. (13):
β
0
dzβ
0
dz1
(z−z)2+q2
=2lnβ
|q|+1+β2
q2+2|q|−2q2+β2,(B2)
where q2=ρ2+ρ2+d2−2ρρcos(ϕ−ϕ)−2d(ρcos ϕ−
ρcos ϕ), and ρand ρare in units of R. It simplifies the
further numerical integration of the coupling integrals Ix,y.
APPENDIX C
The lattice sums can be effectively calculated using the
rapidly converging series suggested by Benson et al.43 and
implemented for L>2inRef.44. The comprehensive review
on the problem can be found in Ref. 45 by Glasser and Zucker.
We consider below only square lattices. The sums for 2D
lattices of other symmetry (hexagonal, for instance) can be
found applying the methods described in Ref. 45.
We denote the square lattice vector n=(n1,n2) and |n|=
√n2
1+n2
2. The calculation consists of three steps. First, we
use the function46
1
|n|2z=1
(z)∞
0
dttz−1e−t|n|2.(C1)
Then, we use the Fourier series transform43,45
∞
l=−∞
e−tl2eikl =π/t ∞
μ=−∞
e−(πμ+k/2)2
t(C2)
and the integral representation of the second kind Bessel
function of imaginary argument:46
Kν(2βγ)=1
2γ
βν/2∞
0
dttν−1e−β/t−γt.(C3)
Using Eqs. (C1)–(C3) allows writing the lattice sums for
a 2D square lattice as a rapidly converging series due to
the asymptotic properties of the Bessel function Kν(z)≈
√π/2zexp(−z)atz1:46
σL(k)=
n1,n2
eik·n
n2
1+n2
2L+1
2=1
2[σ0L(kx,ky)+σ0L(ky,kx)],
(C4)
σc
L(k)=
n1,n2
(n1+in2)2
n2
1+n2
2L+3
2
eik·n=σ1L(kx,ky)−σ1L(ky,kx)
+2iSL(k),(C5)
where the contributions to the sums in Eqs. (C4) and
(C5) are
σ0L(kx,ky)=4√π
L+1
2
∞
n=1
∞
μ=−∞
cos(nky)|πμ +kx/2|L/2
nL/2KL/2(2n|πμ +kx/2|)+2∞
n=1
cos(nky)
nL+1,
σ1L(kx,ky)=
n1,n2
n2
1eik·n
n2
1+n2
2L+3
2=4√π
L+3
2
∞
n=1
∞
μ=−∞
cos(nkx)πμ +ky
2
L
2+1
nL
2−1KL
2+12n
πμ +ky
2,
SL(k)=
n1,n2
n1n2eik·n
n2
1+n2
2L+3
2=−4√π
L+3
2
∞
n=1
∞
μ=−∞
sin(nkx)(πμ +ky/2)|πμ +ky/2|L/2
nL
2−1KL
2(2n|πμ +ky/2|).
The lattice sums in Eqs. (C4) and (C5) include the term μ=0, which needs to be considered separately in the case kx
and/or ky→0 using the asymptotic relation zνKν(z)=2ν−1(ν)atz→0. For instance, for the point k=0, using the identity
[(L+1)/2] =(L−1)!!π1/2/2L
2for even L, we get46
σL(0) =2L
2+1L
2−1!
(L−1)!! ζ(L)+2ζ(L+1) +8(2π)L
2
(L−1)!!
∞
n=1
∞
μ=1μ
n
L
2
KL
2(2πnμ),(C6)
where ζ(z)=∞
n=1n−zis the zeta function.30
It is evident from Eq. (C5) that σc
L(k=0) =0 for all L. The lattice sums for other symmetries of 2D dot lattices (for instance,
hexagonal) can be calculated by using a similar approach. To reach accuracy of 0.1% in the lattice sums in Eqs. (C4)–(C6),it
is sufficient to truncate the series at the values of indices n,|μ|≈10 because of the exponential decrease of the functions Kν(z)
with n,|μ|increasing.
*Corresponding author: sckguslk@ehu.es
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