Vortex polarity switching in magnets with surface anisotropy
Vortex core reversal in magnetic particle is essentially influenced by a surface anisotropy. Under the action of a perpendicular static magnetic field the vortex core undergoes a shape deformation of pillow- or barrel-shaped type, depending on the type of the surface anisotropy. This deformation pla...
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irk-123456789-1220712017-06-27T03:03:46Z Vortex polarity switching in magnets with surface anisotropy Pylypovskyi, O.V. Sheka, D.D. Kravchuk, V.P. Gaididei, Y. К 70-летию со дня рождения В. М. Локтева Vortex core reversal in magnetic particle is essentially influenced by a surface anisotropy. Under the action of a perpendicular static magnetic field the vortex core undergoes a shape deformation of pillow- or barrel-shaped type, depending on the type of the surface anisotropy. This deformation plays a key point in the switching mechanism: We predict that the vortex polarity switching is accompanied (i) by a linear singularity in case of Heisenberg magnet with bulk anisotropy only and (ii) by a point singularities in case of surface anisotropy or exchange anisotropy. We study in details the switching process using spin-lattice simulations and propose a simple analytical description using a wired core model, which provides an adequate description of the Bloch point statics, its dynamics and the Bloch point mediated switching process. Our analytical predictions are confirmed by spin-lattice simulations for Heisenberg magnet and micromagnetic simulations for nanomagnet with account of a dipolar interaction. 2015 Article Vortex polarity switching in magnets with surface anisotropy / O.V. Pylypovskyi, D.D. Sheka, V.P. Kravchuk, Y. Gaididei // Физика низких температур. — 2015. — Т. 41, № 5. — С. 466-481. — Бібліогр.: 74 назв. — англ. 0132-6414 PACS: 75.10.Hk, 05.45.–a, 75.70.Rf, 75.75.–c, 75.78.–n http://dspace.nbuv.gov.ua/handle/123456789/122071 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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К 70-летию со дня рождения В. М. Локтева К 70-летию со дня рождения В. М. Локтева |
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К 70-летию со дня рождения В. М. Локтева К 70-летию со дня рождения В. М. Локтева Pylypovskyi, O.V. Sheka, D.D. Kravchuk, V.P. Gaididei, Y. Vortex polarity switching in magnets with surface anisotropy Физика низких температур |
| description |
Vortex core reversal in magnetic particle is essentially influenced by a surface anisotropy. Under the action of a perpendicular static magnetic field the vortex core undergoes a shape deformation of pillow- or barrel-shaped type, depending on the type of the surface anisotropy. This deformation plays a key point in the switching mechanism: We predict that the vortex polarity switching is accompanied (i) by a linear singularity in case of Heisenberg magnet with bulk anisotropy only and (ii) by a point singularities in case of surface anisotropy or exchange anisotropy. We study in details the switching process using spin-lattice simulations and propose a simple analytical description using a wired core model, which provides an adequate description of the Bloch point statics, its dynamics and the Bloch point mediated switching process. Our analytical predictions are confirmed by spin-lattice simulations for Heisenberg magnet and micromagnetic simulations for nanomagnet with account of a dipolar interaction. |
| format |
Article |
| author |
Pylypovskyi, O.V. Sheka, D.D. Kravchuk, V.P. Gaididei, Y. |
| author_facet |
Pylypovskyi, O.V. Sheka, D.D. Kravchuk, V.P. Gaididei, Y. |
| author_sort |
Pylypovskyi, O.V. |
| title |
Vortex polarity switching in magnets with surface anisotropy |
| title_short |
Vortex polarity switching in magnets with surface anisotropy |
| title_full |
Vortex polarity switching in magnets with surface anisotropy |
| title_fullStr |
Vortex polarity switching in magnets with surface anisotropy |
| title_full_unstemmed |
Vortex polarity switching in magnets with surface anisotropy |
| title_sort |
vortex polarity switching in magnets with surface anisotropy |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2015 |
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К 70-летию со дня рождения В. М. Локтева |
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http://dspace.nbuv.gov.ua/handle/123456789/122071 |
| citation_txt |
Vortex polarity switching in magnets with surface anisotropy / O.V. Pylypovskyi, D.D. Sheka, V.P. Kravchuk, Y. Gaididei // Физика низких температур. — 2015. — Т. 41, № 5. — С. 466-481. — Бібліогр.: 74 назв. — англ. |
| series |
Физика низких температур |
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| first_indexed |
2025-07-08T21:05:30Z |
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2025-07-08T21:05:30Z |
| _version_ |
1837114308336549888 |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5, pp. 466–481
Vortex polarity switching in magnets with surface
anisotropy
Oleksandr V. Pylypovskyi and Denis D. Sheka
Taras Shevchenko National University of Kyiv, Kyiv 01601, Ukraine
E-mail: engraver@univ.net.ua
Volodymyr P. Kravchuk and Yuri Gaididei
Bogolyubov Institute for Theoretical Physics of NAS of Ukraine, Kyiv 03680, Ukraine
Received January 21, 2015, published online March 23, 2015
Vortex core reversal in magnetic particle is essentially influenced by a surface anisotropy. Under the ac-
tion of a perpendicular static magnetic field the vortex core undergoes a shape deformation of pillow- or bar-
rel-shaped type, depending on the type of the surface anisotropy. This deformation plays a key point in the
switching mechanism: We predict that the vortex polarity switching is accompanied (i) by a linear singularity
in case of Heisenberg magnet with bulk anisotropy only and (ii) by a point singularities in case of surface ani-
sotropy or exchange anisotropy. We study in details the switching process using spin-lattice simulations and
propose a simple analytical description using a wired core model, which provides an adequate description of
the Bloch point statics, its dynamics and the Bloch point mediated switching process. Our analytical predic-
tions are confirmed by spin-lattice simulations for Heisenberg magnet and micromagnetic simulations for
nanomagnet with account of a dipolar interaction.
PACS: 75.10.Hk Classical spin models;
05.45.–a Nonlinear dynamics and chaos;
75.70.Rf Surface magnetism;
75.75.–c Magnetic properties of nanostructures;
75.78.–n Magnetization dynamics.
Keywords: vortex core, surface anisotropy, magnetic field, Heisenberg magnet.
1. Introduction
Magnetization reversal in small magnetic particles is
one of the fundamental issues of the modern magnetism.
Different concepts of switching, including switching by
magnetic fields (quasistatic and precessional ones), switch-
ing by spin injection and all optical switching are inten-
sively studied and widely used for applications in magnetic
data storage [1,2]. Nowadays inhomogeneous switching
[3], i.e., the process of magnetization reversal for inhomo-
geneous magnetization configurations, attracts growing
interest. Particular attention is paid to topologically pro-
tected configuration like magnetic vortex. The vortex is
determined by a planar closed flux-free configuration in a
sample with a localized nonplanar magnetization, so called
vortex core. The sense of the core magnetization direction
is characterized by the vortex polarity (up, = 1p + or
down, = 1).p − Because of the topological stability of
vortices with different polarities, one can consider the vor-
tex polarity as a bit of information in nonvolatile magnetic
vortex random-access memories [4–6]. The corresponding
inhomogeneous magnetization reversal in the vortex-state
particles is called the vortex core reversal or the vortex
polarity switching.
There exist several concepts of the vortex polarity
switching, e.g. by magnetic fields, by spin injection, see
Ref. 7 and references therein. Independently of these con-
cepts one can separate two main scenarios: axially-
symmetric (or punch-through) scenario and axially-
asymmetric one. In the first scenario the vortex does not
make the macroscopic motion, the vortex core reversal is
caused by the direct excitation of radially symmetric
magnon modes. Such a switching occurs, e.g. under the
action of dc transversal magnetic field [8–11], or by ac
© Oleksandr V. Pylypovskyi, Denis D. Sheka, Volodymyr P. Kravchuk, and Yuri Gaididei, 2015
Vortex polarity switching in magnets with surface anisotropy
transversal fields [12–16]. The axially-asymmetric scenar-
io is caused by nonlinear resonance between coupled
magnon modes [17,18]; it is accompanied by the tempo-
rary creation of the vortex-antivortex pairs [19,20]. Such a
picture was observed in different setups [7].
Here we consider the axially-symmetrical switching of
the vortex polarity. During the reversal process the main
changes occur in a circular Bloch line [21], which is the
line passing trough the sample thickness and connecting
centers of all vortex cross-sections. The magnetization
direction in the Bloch line corresponds to the vortex polari-
ty: it has to reverse its direction during the switching pro-
cess. In case when a reversal occurs simultaneously for the
whole Bloch line ,γ there appears a line singularity during
switching: the magnetization vector m vanishes at the
switching moment for the whole line, ( ) = 0,m γ and the
temporary pure planar vortex (linear singularity) appears
during the switching. Such a two-dimensional (2D) switch-
ing process was considered in Ref. 11 to describe the re-
versal in very thin magnets. Another possibility was sug-
gested by Arrott et al. [22]. Arrott’s model is stray-field-
free, where the switching is made possible by propagating
of point singularities, Bloch points with coordinates BPr
[21–23]. In that case the reversal occurs nonsimul-
taneously: the magnetization vanishes only in one or a few
points, the so-called Bloch points, ( ) = 0,BPm r and the
switching is accompanied by the nucleation, motion and
final annihilation of the Bloch points. Such a process was
studied by Thiaville et al. [10] using micromagnetic simu-
lations: the switching process is typically accompanied by
the creation of two Bloch points, however, the single Bloch
point scenario was also mentioned [10].
The purpose of the current study is to describe process
of the axially-symmetrical switching, to find the mecha-
nism, which is responsible for simultaneous or non-
simultaneous switching, to understand how many Bloch
points are nucleated during the switching, and to describe
this process analytically.
Point singularities were introduced in magnetism by
[24] and [25]. For example, in the vicinity of the Bloch
point, situated at = 0,BPr the magnetization distribution
has a hedgehog-like configuration, = /Q r±m r with Q
being the constant orthogonal matrix [26]. They are ana-
logues of magnetic monopoles in elementary particle phys-
ics for spin waves [27]. Bloch points were observed as
twisting of the Bloch lines in garnet crystals [28,29] and as
singular point of the Bloch point domain wall in magnetic
nanowires [30]. According to simulations the Bloch points
appear as transient states in the vortex dynamics under the
action of dc [10] and ac [13,15] transversal fields; as tem-
poral formations (micromagnetic drops) during the mag-
netization reversal in soft magnetic cylinders [31]. The
Bloch points are also created during the vortex-antivortex
annihilation processes [32].
Theoretical treatment of the Bloch point is a complicat-
ed task. Even a “simple” problem of the Bloch point struc-
ture in a spherical particle caused a long-time discussion,
see Ref. 33 and references therein. The problem becomes
complicated due to the fact that the Bloch point distribu-
tion with ( ) = 0BPm r does not fulfil the condition of mag-
netization normalization, | |= 1.m There is no physical
singularity in reality. Due to the discreetness of the spin
lattice, the Bloch point (as well as linear singularity) is
situated in an interstitial site position, hence the singularity
appears only in continuum description. That is why the
modeling of Bloch points is also nontrivial task: there ap-
pear mesh dependent problems, e.g. mesh-friction effect
and a strong mesh dependence of the switching field [10].
Therefore specific simulators with atomistic resolution are
needed to overcome such difficulties [33,34]. The role of
discreetness of the magnetic lattice becomes of great im-
portance; in particular, namely the lattice creates an effec-
tive pinning field for the Bloch point [35]; it changes the
Bloch point behavior relatively to the main crystallograph-
ic directions when a depinning field for one direction can
be much lower than in other direction [34].
The problem of dynamics of Bloch points is not less
challenging. The spectrum of Bloch point oscillations
along the Bloch line was measured for yttrium garnet fer-
rite [36]. Dynamical properties of the Bloch point, its mo-
bility and mass were derived for the Bloch point inside the
Bloch line [37–40]. We are interesting in a specific dynam-
ics of the Bloch point, namely, the Bloch point mediated
switching of the vortex polarity. It is already known from
our recent studies [14,15] that the dominating contribution
to the switching mechanism is caused by the exchange
interaction inside the vortex core. Therefore one can expect
that the dipolar interaction, which is essential for the statics
and slow dynamics of the magnet, in particular, for the
vortex state nanodot, does not play the dominant role for
the switching process.
Argued by above mentioned reasons of the dominant
role of exchange interaction we do not take into account in
most cases the dipolar interaction. Instead we additionally
include into consideration the surface anisotropy effects. In
a disk-shaped particles the surface anisotropy favors mag-
netization curling states of onion (two half vortex [41] or
capacitor [42] configuration and a vortex one [41,42]. Be-
low we consider the vortex configuration. One has to note
that the vortex is the metastable state in model under con-
sideration [41], while it can form the ground state in mod-
els with finite skin depth of the surface anisotropy [43] or
in models with dipolar induced effective inhomogeneous
anisotropy [43]. In the last case there appear two effective
inhomogeneous anisotropy terms: one is effective easy-
plane anisotropy of face surface charges and another one is
effective anisotropy of edge surface charges [43]; namely
these two effective anisotropies are responsible for the
vortex state configuration.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5 467
Oleksandr V. Pylypovskyi, Denis D. Sheka, Volodymyr P. Kravchuk, and Yuri Gaididei
In the current study we consider the Heisenberg disk-
shaped magnet in a vortex configuration. Without the sur-
face anisotropy the vortex profile does not depend on a
thickness coordinate ,z it has a 2D shape. The presence of
the surface anisotropy breaks such a symmetry and results
in the deformation of the vortex core profile. Depending on
the type of surface anisotropy, the vortex profile becomes
barrel- or pillow-shaped deformed for the easy-surface
(ES) and easy-normal (EN) surface anisotropies, respec-
tively [41]. We will see below that inhomogeneity of the
Bloch line is crucial for understanding the Bloch point me-
diated switching mechanism: the switching process in Hei-
senberg three dimensional (3D) magnets with bulk anisot-
ropy only is mediated by the linear singularity. In contrast,
the switching in magnet with the surface anisotropy is ac-
companied by the simultaneous nucleation of two Bloch
points on the face surfaces of the sample for the EN case
and by the nucleation of two Bloch points inside the sam-
ple (at the center of the Bloch line) for the ES case. We
also analyze the case of the exchange anisotropy; qualita-
tively the mechanism is very similar to the EN surface ani-
sotropy: during the switching two Bloch points enter the
sample on face surfaces, move inside to each other and
finally annihilate at the center of the disk axis. We study in
details the process of the vortex core reversal, compute the
Bloch points trajectories, and find how the Bloch point
speed depends on the sample thickness. We propose a sim-
ple analytical picture which describes both statics of the
vortex in the cone phase with account of the surface anisot-
ropy effects and dynamics of the vortex polarity switching
process, which is in agreement with the full-scale spin-
lattice simulations. We also performed micromagnetic
OOMMF [45] simulations with account of surface anisot-
ropy and additional dipolar interaction, which confirm
main features of our vortex core deformation.
The paper is organized as follows. In the Sec. 2 we de-
scribe the model of the Heisenberg magnet with account of
both bulk and surface anisotropies. In the Sec. 3 we present
results of spin-lattice simulations of vortex polarity switching
for Heisenberg magnets with bulk easy-plane single-ion ani-
sotropy (Sec. 3.1), with additional easy-surface and easy-
normal surface anisotropies (Sec. 3.2) and with bulk exchange
anisotropy (Appendix B). We make analytical description of
the switching phenomenon in the framework of proposed
wired core model which demonstrates a good qualitative
agreement with simulations in Sec. 4. To validate our results
for nanoparticles we modeled a vortex state nanodisk using
OOMMF micromagnetic simulations with account of dipolar
interaction and surface anisotropy, which are discussed in Sec.
5. In Appendix A we consider how the homogeneous state is
deformed with account of the surface anisotropy. An influ-
ence of exchange anisotropy is considered in Appendix B.
Influence of magnetic field on the vortex state in a ring is con-
sidered in Appendix C. The basic equations for the wired
core model are derived in Appendix D.
2. The model and the vortex solution
Let us consider a classical lattice Heisenberg magnet
with a simple cubic lattice and Hamiltonian
2 an
( , )
ˆ= 2 ( ) ,BJ H z+− ⋅ − µ ⋅ +∑ ∑n n n
n n
m m m δ
δ
(1a)
where > 0J is the exchange integral, = 1/2 is the
length of a classical spin,
2 2= 1 cos , 1 sin ,m m m − ϕ − ϕ
n n n n n nm
is the normalized magnetic moment on a 3D site position
n, the 3D index δ runs over the nearest neighbors, Bµ is
the Bohr magneton, H is the intensity of external dc mag-
netic field, directed along ˆ-axis,z and an is the Hamil-
tonian of single-ion anisotropy. The latter includes the bulk
easy-plane anisotropy (with the anisotropy constant
> 0)K which favors magnetization distribution within xy-
plane, and the Néel surface anisotropy term (with the ani-
sotropy constant )sK ) [46],
22
an 2 2
( , )
ˆ= ( ) ( ) ,
2 2
sKK
⋅ − ⋅∑ ∑n l l
n l
m z m u
δ
δ
(1b)
where index l runs over the surface sites and the unit vec-
tor l
u
δ
connects the nearest neighbors of the lattice. Dy-
namics of magnetization is governed by the discrete ver-
sion of the Landau–Lifshitz–Gilbert equation
,
d d
d d
∂
= × + ε ×
τ ∂ τ
n n
n n
n
m mm m
m
H (2)
where = tτ Ω is the normalized time, = / ,KΩ
2= /( )K H is the normalized energy, is the Plank’s
constant, and ε is the Gilbert damping constant.
The continuum approach is based on smoothing of the
lattice model using the normalized magnetization
2 2( , ) = 1 cos , 1 sin , ,m m m τ − ϕ − ϕ
m r
where = ( , )m m τr and = ( , ).ϕ ϕ τr The total energy,
which corresponds to the Hamiltonian (1), takes the form
( )
2
3
2 2
2
2
= ,
ˆ( )= ,
2 2
= ( ) .
2
v s
v
s s
KE
a
dV
a dS
+
⋅
− ⋅∇ + − ⋅
⋅
∫
∫
m zm m m h
m n
E E
E
E
(3)
Here = a λ is the magnetic length, a is the lattice con-
stant and = / ,J Kλ = /sK K is the surface anisotropy
measured in units of K and sn is the normal to the sur-
face. We will consider cases of small surface anisotropy
468 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5
Vortex polarity switching in magnets with surface anisotropy
| |< 1. The parameter h is the field intensity, normalized
by the anisotropy field ,aH
= / , = /(2 ).a a BH H K µh H
The equilibrium magnetization distribution can be found
by variation of the energy functional (4) which results in
the following boundary-value problem [21,47]:
2 2 ˆ ˆ( ) = 0, × + − ⋅ m m h m z z ∇ (4a)
[ ]= ( ) ( ) .s s s
s S S
∂
⋅ ⋅ −
∂ λ
m m n m n m n
n
(4b)
Without the surface anisotropy ( = 0) the ground state of
the uniaxial magnet is determined by the field intensity. As
it follows from (4a), the magnetization z-component
= = const,m h the boundary condition (4b) has the Neu-
mann type, / = 0∂ ∂ sm n which is trivially satisfied for the
constant m solution.
The presence of the surface anisotropy drastically
changes the symmetry of the problem. The boundary con-
ditions (4b) becomes of the Robin type [48], which is the
source of the symmetry breaking of the magnetization
structure and causes the effect of magnetization curling.
There appears nonhomogeneous ground states with typical
examples as hedgehog, throttled, artichoke, onion, vortex
configurations, etc. [41,42,49–53].
Let us start with the easy-plane magnet which has the
shape of infinite film of a finite thickness = ( 1) ,L N a−
where N is the number of lattice sites along thickness
coordinate .z Without external field and the surface anisot-
ropy the ground state of such a magnet is an uniform easy
plane magnetization, = 0.m The same state takes place
with account of the weak surface anisotropy in the no-
magnetic field case.
The magnetization configuration under the action of
applied field is affected by the surface anisotropy influ-
ence: the magnetization distribution breaks its homogenei-
ty in z-direction due to the surface anisotropy influence
2 2
0 2
1 cosh( 1 / )( , ) = 1 ,
sinh( 1 /2 )
h h zm z h h
h L
− − −
λ −
(5)
see Appendix A for details. Typical magnetization profile
0m as function of z for EN and ES surface anisotropies is
shown in Fig. 1. Symbols correspond to the spin-lattice
simulations, see Sec. 3 for details. One can see that the
sign of the curvature 0 = ( )m f z is determined by the sign
of the product .h
In the following paper we consider the disk-shaped
samples with such parameters, that the surface anisotropy
favors the vortex configuration [41,42]. The in-plane struc-
ture of the stationary vortex state ( )ϕ r is determined by
the following magnetization distribution
= ,
2
C π
ϕ χ + (6a)
where ( , , )r zχ are the cylinder coordinates. Without sur-
face anisotropy effects, the vortex chirality C can take any
value due to the isotropy of Heisenberg exchange, while
the surface anisotropy on the edge surface fixes its value:
= 1C ± for ES magnets ( > 0) and = 0C or = 2C for
EN magnets ( < 0) [44].
Let us discuss the vortex out-of-plane structure, ( ).m r
Without magnetic field the out-of-plane vortex profile has
a bell-shaped structure [44].
2
2 2( , ) exp ,
2 ( )
rm r z p
w z
≈ −
(6b)
which generalizes a well-known Feldtkeller Ansatz [21,55],
originally used for magnets without surface anisotropy,
when the dimensionless core width 1.w ≈ Depending on
the surface anisotropy type there appears barrel- or pillow-
shaped deformation of the vortex core ( )w z for > 0 and
< 0 respectively [44].
Without magnetic field vortices with opposite polarities
( = 1)p ± are energetically equivalent. Under the action of
a transversal dc magnetic field the preferable polarity of
the vortex coinsides with the field direction (light vortex).
The vortex with opposite polarity (heavy vortex) turns into
so-called cone phase. Its structure is well-known [56,57].
While the in-plane magnetization distribution ϕ of the
cone phase coincides with (6a), the out-of-plane compo-
nent is deformed in the following way. The heavy vortex
becomes narrower with the typical core width
ch ( ) 1 | |w h h∝ − in contrast to the light vortex which
becomes broader with cl ( ) 1/ 1 | |w h h∝ − [57].
Fig. 1. (Color online) Magnetization profile 0( ; )m z h as a function
the thickness coordinate ,z = 0.7h (level is shown by dotted line)
and | |= 0.25.κ Symbols correspond to spin-lattice simulations and
solid lines to the Eq. (5). Schematics in insets show the magnetiza-
tion distribution in the film with surface anisotropy for the both
cases (red arrows show magnetization direction and blue arrows for
ES (a), EN (b) show field direction).
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5 469
Oleksandr V. Pylypovskyi, Denis D. Sheka, Volodymyr P. Kravchuk, and Yuri Gaididei
Instead of the exponentially localized structure (6b)
without field, the vortex profile has an algebraic decay in
the cone phase. Far from the origin it is characterized by
the asymptote 2 2= /m h h r+ [56,57]. For definiteness we
suppose that the vortex polarity = 1p + and negative field
intensity < 0h having in mind switching phenomenon.
With the field intensity increasing the heavy vortex
loses its stability [58]. It is important to stress that in
spite of the fact that the light vortex is energetically pre-
ferable, the heavy vortex cannot perform switching in
scope of continuous theory due to infinite barrier which
separates opposite polarized vortices. However the barri-
er becomes finite in the discrete spin lattice [59] and the
reversal process can occur [60,61]. The simple picture of
the vortex core switching provides the core model, origi-
nally introduced by Wysin [59] for the vortex instability
phenomenon. Because of the crucial role of exchange
interaction inside the vortex core for the switching mech-
anism, the core model, provides good qualitative descrip-
tion of the switching process [14,15]. Let us mention one
more possibility, the so called cutoff model [62]: model-
ing the discreetness effects by cutoff parameter it is pos-
sible to describe the vortex polarity switching under the
action of spin-polarized current [62], by dc [11] and ac
[14] magnetic fields.
3. Spin-lattice simulations of the vortex polarity
switching
The main purpose of the current research is to describe
the fine structure of the vortex core switching process. As
we discussed above the switching is possible only in the
discrete lattice. There appear serious difficulties in model-
ing of this process: the standard micromagnetic simulators
consider the numerically discretized Landau–Lifshitz equa-
tion, which is valid in continuum theory. Such approach
becomes mesh-dependent within the continuum description
of micromagnetism [10]. The switching is known to be
accompanied by the creation of micromagnetic singulari-
ties, Bloch points, which cannot be described adequately
within the continuum limit [10,33,34].
In order to overcome difficulties of continuous ap-
proach, we perform spin-lattice modeling using in-house
developed spin-lattice simulator SLaSi [63]. Numerical-
ly we solve Landau–Lifshitz–Gilbert Eqs. (2) for the
disk-shaped system. We consider two sets of parame-
ters: disk diameter 2 = 149 ,R a thickness = 49L a and
the magnetic length = 14a (disks A), 2 = 99R a with
the thickness = ( 1) ,L N a− where N is the number of
lattice sites along disk axis with = 11, 50N (disks B).
Chosen diameters allow vortex to be both in the center
of the sample and between lattice sites. Other parameters
are following: the initial vortex polarity = 1,p the Gilbert
damping constant = 0.5ε corresponds to the overdamped
regime. The surface anisotropy is chosen = 0.5ES and
= 0.5,EN − and the field intensity varies in the range
( 0.83;0).h∈ − The lattice planes are chosen parallel to the
face surfaces of the samples.
3.1. Magnets with bulk anisotropy only: = 0sK
Using the vortex distribution (6) as initial one for dif-
ferent field intensities < 0,h we relax numerically the
system to the static configuration, which is adopted to the
lattice. Starting from weak fields, we increase adiabatically
its absolute value up to the critical value .ch For stronger
field intensities, | | | |,ch h≥ the switching occurs.
We model the switching process for the vortex in the
disk A. Numerically we found that | | 0.83.ch ≈ The heavy
vortex is relaxed in the field with intensity = 0.82.h − It
demonstrates the narrowing of the core and shifting of the
background far from the origin described above. After that
we applied the field with intensity = 0.83h − which results
in the polarity switching. The heavy vortex losses its sta-
bility: the four central magnetic moments in the lattice
planes inside the vortex core rapidly switch their direction
Fig. 2. (Color online) Spatio-temporal picture of the vortex core switching for the magnet with bulk anisotropy only, the simulation data
(center). Left (a) and right (e) panels: schematic of the vortex profiles in three different horizontal cross-sections. During the switching
process the vortex polarity flips simultaneously in all lattice planes. The gray rectangle on background indicates the XZ cross-section of
the sample and the white boxes show the spatial profile of the z-component of magnetization ( , ).zm r z The magnetization distribution is
shown in three different lattice cross-sections ( = 24.5 ,tz a = 0.5cz a and = 24.5 )bz a− at three different time moments: before the
switching, = 6.107τ (b), at the switching moment, = 6.115τ (c), and just after the switching, = 6.121τ (d). The switching is accom-
panied by the linear singularity creation, see the green line. Parameters: disk A, = 0.83.h −
470 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5
Vortex polarity switching in magnets with surface anisotropy
to the opposite one. Another magnetic moments relax to
the light vortex distribution with a smaller speed. In gen-
eral, the background of the vortex remains the same, all
dynamics is observed only in the core region.
Essentially, the reversal process occurs simultaneously
in all lattice planes, hence the switching scenario is fully
two dimensional one. The temporal evolution during the
switching is shown in the Fig. 2. Schematics (a) and (e)
represent the 3D magnetization structure in the sample
before and after switching, respectively: vortex state within
three different horizontal cross-section (top, center and
bottom surfaces). Central figures (b)–(d) show the magnet-
ization distribution within the different cross-sections in
three different time moments: before the switching
( = 6.107),τ at the switching moment ( = 6.115),τ and
just after the switching ( = 6.121).τ During the switching
process, the linear singularity ( ) = 0m γ is created, see the
solid green vertical line γ in Fig. 2 [64].
In detailes the polarity switching occurs by the following
steps, shown in Fig. 2. To consider the spin-lattice system
we replace the vortex polarity = 1p ± by a dynamical polar-
ity = ( , )z tµ µ which represents the actual amplitude of
magnetization z-component in the vortex core center. The
magnetic moments in the central part of the initial heavy
vortex core are approximately perpendicular to the plane and
one can say that the dynamical polarity is 1,µ ≈ see curve
(1) in Fig. 3. When the applied field becomes stronger than
the switching threshold, µ rapidly decreases and changes its
sign passing through = 0.µ One can say about creation of a
linear singularity in this moment, see curve (2) in Fig. 3. The
dynamics of the central magnetic moments is going faster
than outer ones. Central moments rapidly flip their direction
from 1µ ≈ to 1,µ ≈ − and the overshooting in the vortex
profile appears, see curve (3) in Fig. 3. The final state is the
light vortex, see curve (4) in Fig. 3.
It should be noted that during the whole complicated dy-
namical process of the switching the system remains uniform
along the z-axis: the numerical difference for different lattice
planes is not larger than 10–6 for nm (corresponds to the ac-
curacy of the saved data) during relaxation in the external
field and polarity reversal. The same scenario takes place in
the case of non adiabatically applied external field.
3.2. Magnets with the surface anisotropy
Let us study the role of the surface anisotropy in the
vortex statics and dynamics, especially in the vortex
switching phenomenon. Very recently we have studied
how the surface anisotropy effects on the vortex core shape
without magnetic field: there appears the pillow- and the
barrel-deformation of the core for the ES and EN aniso-
tropies, respectively [44]. Qualitatively, the vortex core
width inside the sample volume is determined by the mag-
netic length, while the core width on a surface layer is
characterized by effective magnetic length [44]
eff = .
1+
(7)
In the case of ES surface anisotropy ( > 0), an effective
magnetic length eff < . Therefore the vortex core be-
comes more narrow near the face surfaces. This corre-
sponds to the barrel-shaped profile. In the same way, EN
surface anisotropy ( < 0) results in the pillow-shaped
profile because eff > , see Fig. 4 and schematics (c)
and (e) therein.
In weak fields (far from the switching threshold) the
vortex core looks qualitatively the same: barrel-shaped
profile for the ES case and the pillow-shaped one for the
EN case, while core becomes more narrow in agreement
with above mentioned estimations.
Under the action of intermediate field the vortex core
profile deforms. The further increase of the field intensity
(up to the switching threshold) drastically changes the vor-
tex core shape. In the ES case there appears a deformed
pillow core instead of original barrel one, see the Fig. 4
and schematic (f) therein. In the same way in the EN case
one has the deformed barrel core profile instead of original
pillow one, see the Fig. 4 and schematic (d) therein. Nu-
merically the vortex core width is computed as the 0 ( )w z
such that 0( ) = 0.5[ (0) ( )].m w m m R+
The inhomogeneous shape of the vortex core profile is a
key point for understanding the Bloch point mediated
switching scenario. Let us start with the case of EN surface
anisotropy. In this case the vortex core profile becomes
deformed barrel-shaped, see the Fig. 4 with schematic (d).
With the field increasing both bending points shift outside
to the ends of the Bloch line and they become closer to the
face surfaces when field approaches to the switching one.
This means that in a strong enough field the vortex profile
near the face surface is more narrow than in the bulk.
Fig. 3. (Color online) Vortex polarity reversal in the sequential
time moments: initial state of the heavy vortex before switching
(1); dynamics polarity in the center decreases due to discreetness,
linear singularity appears (2); central magnetic moments reverces
faster than other ones, overshooting appears (3); final state of the
light vortex (4).
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5 471
Oleksandr V. Pylypovskyi, Denis D. Sheka, Volodymyr P. Kravchuk, and Yuri Gaididei
This bulk deformation of the vortex core results in the
inhomogeneous vortex polarity switching: the Bloch line
breaks in its more narrow parts, i.e., on the face surfaces.
Each break of the Bloch line corresponds to the Bloch
point. Therefore in case of EN surface anisotropy two
Bloch points are nucleated simultaneously on the face sur-
faces. During the reversal process they move in opposite
direction to each other and finally annihilate at the sam-
ple’s center. The similar behavior occurs in the case of ES
surface anisotropy: the narrowest part of the vortex core is
situated in the center and a pair of Bloch points is nucleat-
ed in the center of the sample.
The schematics of the 3D polarity reversal accompanied
by Bloch points motion for EN surface anisotropy is shown
in Fig. 5. The simulation data are shown for the disk A
under the action of the external field = 0.85h − applied to
the sample relaxed in = 0.8.h − Bloch points always sepa-
rate vortices with opposite polarities. Therefore their exist-
ence can be easily detected by observing of the vortex po-
larities on the different lattice planes: face surfaces and in
the center of the sample, see Fig. 5(a). The temporal evolu-
tion of both Bloch points has the reflection symmetry with
respect to = 0z plane: curves which show polarity dynam-
ics of top and bottom (t, b) vortices coincide (solid curve).
The switching of the central (c) vortex occurs with delay
(dashed curve). In thicker samples more pairs of Bloch
points can be nucleated inside the sample volume, see be-
low (also see inset in Fig. 7).
The axial cut of the lattice during polarity switching is
shown in the Fig. 5(b) the vortex state is already reversed
( 1)µ ≈ − near the top and bottom surfaces, while its polari-
ty is still directed up ( 1)µ ≈ + in the sample’s center.
Fig. 4. (Color online) The reduced vortex core width w as a function of the sample thickness coordinate z for the disk B with thick-
ness L = 39a. Without field the vortex core has the pillow shape for the EN case, = 0.5− (a), see schematic (c), and the barrel one for
the ES case, = 0.5 (b) see schematic (e). Under the action of the field close to switching one ( = 0.6)h − there is an opposite behav-
ior: the vortex core profile becomes a deformed barrel one for the EN case, see schematic (d), and a deformed pillow one for the ES
case, see schematic (f). Arrows show transformation under the action of external field h which is increased by modulus.
Fig. 5. (Color online) Switching of the vortex polarity for 3D Heisenberg magnets with EN surface anisotropy (disk A, = 0.5,− simulations
data). Polarity dynamics for vortices with = 24.5z a and = 24.5z a− (solid curves which coincide), = 0.5z a (dashed curve) (a). Magnetiza-
tion distribution in the x-z plane near the center of the sample at = 9.07.τ Color indicates the m sign, thick arrows show the positions of the
Bloch points, thin vertical arrows indicate the direction of motion for the bottom and top Bloch points (b). Polarity switching in different x-y
planes, according to curves on (a). Green arrows show the direction of the Bloch points (BP) motion (c)–(e).
472 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5
Vortex polarity switching in magnets with surface anisotropy
There are two Bloch points in the system: their positions
are shown by thick white arrows and the direction of mo-
tion is shown by thin green arrows. The vortex profiles for
different times during polarity switching process are shown
in the Figs. 5(c)–5(e).
Let us discuss now the case of ES surface anisotropy,
see Fig. 6 (notations are the same as in Fig. 5). Here we
apply the external field = 0.85h − to the sample relaxed in
= 0.8.h − It results in the deformed pillow-shaped vortex
core profile. That is why the vortex core becomes more
narrow in the middle of the sample, hence the Bloch line
breaks in the middle and the switching process starts inside
the sample. There are two Bloch points which born in the
sample center. During the switching process they move
along z-axis in opposite direction to face surfaces.
Numerically we also analyze the case of the exchange
anisotropy; qualitatively the mechanism is very similar to
the EN surface anisotropy, see Appendix 7 for details.
To gain insight into the temporal evolution of the
switching process we perform a set of simulations for
fixed diameter of the sample (disk B) and varying thick-
nesses ( = 11,40).N The anistropy is = 0.5.− We ap-
ply the field = 0.8h − [65] to the vortex state relaxed
without field ( = 0).h Similarly to the previous case, the
inhomogeneous polarity reversal is observed. In all simula-
tions the reversal is accompanied by the nucleation of a
pair of Bloch points near face surfaces, their further dy-
namics inside the magnet and the final annihilation at the
point of contact in the middle. The temporal dynamics of
the Bloch points for different thicknesses is shown in
Fig. 7. We determine the position of Bloch points as a
cross-sections of three isosurfaces ( ) = 0mα r with
= , ,x y zα [32]. Each surface ( )mα r is computed as in-
terpolated functions for magnetization components based
on discrete values of .nm
A new feature appears for relatively thick samples
( 30).N ≥ During the switching process, additional pair of
Bloch points is nucleated in the sample center. Two points
of this pair repel each other moving in z-direction out from
the center. Finally the new born Bloch points annihilate
with originally nucleated ones, see the inset in Fig. 7. Note
that for the weaker field = 0.7h − the additional Bloch
points are not nucleated. One can conclude that there ap-
pears instability of the Bloch line in long enough samples,
which causes the breaking of the Bloch line inside the
sample and nucleation of additional Bloch points. Such a
picture is similar to the temporal formation of
micromagnetic drops in long nanocylinders [31].
Fig. 6. (Color online) Switching of the vortex polarity for 3D Heisenberg magnets with additional ES surface anisotropy ( = 0.5), the
simulations data. All notations and other parameters are the same as in Fig. 5.
Fig. 7. (Color online) Bloch point position as function of time for samples with different thickness. All curves are matched at = 0,τ for
EN surface anisotropy (a) and at = 0.548τ for ES surface anisotropy (b). Green circles indicate Bloch points creation event and crosses
indicate annihilation event. For 30N ≥ additional pair of Bloch points is nucleated shortly before annihilation. Arrows indicate direc-
tion of motion of Bloch points. Parameters: disk B, = 0.5− (a) and = 0.5 (b), = 0.8.h −
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5 473
Oleksandr V. Pylypovskyi, Denis D. Sheka, Volodymyr P. Kravchuk, and Yuri Gaididei
Numerically we also studied how the speed of the
Bloch point depends on the sample thickness. One can see
from the Fig. 7 that the Bloch point motion is almost a
steady-state one except the moments of nucleation and
annihilation. In this way we computed the speed of Bloch
points v in a wide range of the sample thickness, see
Fig. 8. The Bloch point speed rapidly decreases in thick
samples and tends to some constant value not dependend
on the sample thickness and surface anisotropy type.
4. Description of the polarity switching process
To describe analytically the observed vortex polarity
switching process we propose a wired core model. In
spite of simplicity this model is in a good qualitative
agreement with full-scale simulations. It generalizes the
discrete reduced vortex core model initially proposed by
Wysin [59] for the vortex instability phenomenon and
later extended for the description of the polarity switch-
ing in the two-dimensional case [15,60,61,66,67]. In the
current study we extend the model for 3D systems. A
wired core model allows one to take into account pro-
cesses of nucleation-annihilation of point singularities
during the vortex switching.
We start with the discrete Hamiltonian (1). The anisot-
ropy term (1b) can be written in the following form:
2
an 2= ,
2
, for = 2, 1=
, for =1 and .
n
n
s
K m
K n NK
K K n n= N
−
+
∑ n
n
(8)
Here = ( , , )x yn n nn and n enumerates the planes along z
direction.
We suppose that in each nth plane there exist only four
“free” magnetic moments, all other moments are “frozen”
into the background, see Fig. 9. The 3D vortex core is con-
sidered in our model as a wire of free magnetic moments.
In original core model [59] the discrete core magnetization
is matched with the continuous magnetization of the back-
ground (the ring), where the magnetization is fixed in the pure
planar vortex configuration. The inner radius inr of the ring
should be chosen from the condition that the ground state is a
pure planar vortex, which fulfills when in 0.3r ≥ [68]. The
situation drastically changes under the field action, the planar
vortex can never be a solution of the ring wire in the field
presence. Besides one has to take into account z-dependence
of the magnetization due to the surface anisotropy in the way
similar to (5). Finally, we get the background magnetization
distribution ( , ).rm r z see Appendix C for details:
( , , ) 2
( , , )
= ( ) = ( , ),
= , | | | |> 2,
2
n n n b n r nx y
n n n x yx y
m m z hm r z
n nπ
ϕ χ + +
(9a)
where 2r is the radius of the second coordination sphere,
function rm is defined by Eq. (C.3), nz is a z-coordinate
of nth lattice plane. For the numerical investigation of the
wired core model we use 2 0.35 .r ≈
All “free” moments come from the ring of principal
sites, they form the vortex core. Due to the axial symmetry,
all free moments have the same out-of-plane component
nµ and the same in-plane phase ψ which describes the
deviation from the vortex distribution (9a):
( , , )
( , , )
= ( ),
= ( ), , = 1, 1.
2
n n n nx y
n n n n x yx y
m t
t n n
µ
π
ϕ χ + +ψ −
(9b)
Fig. 8. (Color online) Bloch point speed as function of sample
thickness in units of lattice sites along disk axis for the EN and
ES surface anisotropies. Parameters are the same as in Fig. 7.
Fig. 9. (Color online) Schematic of the wired core model for nth
and (n + 1)th planes: Red arrows (1) indicate free magnetic mo-
ments (Eq. (9b)) and all other arrows (gray) indicate fixed ones
(Eq. (9a)). Layers are exchange-coupled with exchange integral J
and the each plane is characterized by own anisotropy constant
> 0,nK see Appendix D for details. Bloch point position is
marked by the green sphere.
474 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5
Vortex polarity switching in magnets with surface anisotropy
We consider the dynamical vortex polarity ( )n tµ for each
lattice plane, and the in-plane magnetization angle ( ),n tψ
which has the meaning of the turning phase, as two collec-
tive variables. In this manner we generalize the 2D ap-
proach, which we recently proposed in Ref. 15. Note that
the dynamical vortex polarity is mutually connected to the
effective vortex width in the full-scale model.
Now by incorporating the wired core Ansatz (9) into the
Hamiltonian (1) with account of the anisotropy term (8),
one can write down the model Hamiltonian, normalized by
2K as follows
( ) ( )
2
=1
= 4 1 2 2
2
N
n
c n b n
n
h m
µ
+ − λ − + λ µ −
∑ H
2 24 (1 )(1 ) cos
5 b n nm
λ
− − −µ ψ −
1
2 2
1 1
=1
4 (1 )(1 )
N
n n n n
n
−
+ +
− λ µ µ + −µ −µ ×∑
1cos ( ) .n n+
× ψ −ψ
(10)
Here = 0n for = 2, 1n N − and =n for = 1, ,n N
see Appendix D for details. The first sum in (10) describes
the intra-plane interaction and the second sum describes
the inter-plane interaction.
The equilibrium values of nµ and nψ can be found
analytically for the homogeneous case = 0,n = 1,n N
and = 0:h
( )
2
hom 02
16( = 0) = 1 , ( = 0) = 0,
5 2 1
h hλ
µ ± − ψ
λ −
where > 10λ for the out-of-plane vortex. For 0h ≠ , the
dependence hom ( )hµ can be found numerically as the so-
lution of the Eq. (D.3), see Fig. 15 in Appendix D.
When the field is applied one has to take into account
that the equilibrium z-components of magnetization are
different for the bulk and surface planes. For the fixed
magnetic moments the out-of-plane magnetization compo-
nent is taken equal to ( )bm z according to Eq. (C.3). As-
suming that the deviation hom( ) / ( )x z zλ ≡ µ −µ from
the equilibrium value homµ is small, | / | 1,λ << and
replacing the Hamiltonian (10) by its continuum version
we obtain the following boundary value problem for ( ):x z
2
1 2
2
hom hom
= /2
( ) ( ) = ( ),
(1 )
( ) = 0,
z L
a x z A x z A z
ax z
±
′′ −
µ −µ
′± +
λ
(11)
where coefficients 1A and 2 ( )A z are given by Eqs. (D.4),
see Appendix D for details. Function 2 ( )A z contains only
simple harmonics and hyperbolic sine and cosine, so the solu-
Fig. 11. (Color online) Bloch point position as function of time for samples with different thickness for the wired core model. All curves
are matched at = 0τ for EN surface anisotropy and at = 0.125τ for ES surface anisotropy. Green circles indicate Bloch points creation
event and crosses indicate annihilation event. For = 140N additional pair of Bloch points is nucleated shortly before annihilation. Ar-
rows indicate direction of motion of Bloch points. = 0.5,ε = 0.001,h − other parameters are the same as in Fig. 10. EN surface anisot-
ropy = –0.1 (a); ES surface anisotropy = 0.1 (b).
Fig. 10. (Color online) Dynamical polarity nµ for the discrete
model (10) (symbols labeled EN and ES model) and ( )zµ as
numerical solution of Eq. (11) (solid curves labeled EN and ES
linear approx.). Parameters: | | = 0.1, = 20,λ = 70,N
= 0.0009.h − Dashed line corresponds to polarity without sur-
face anisotropy. ES model (); ES linear approximation (red
line); EN model (); EN linear approximation (blue line).
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5 475
Oleksandr V. Pylypovskyi, Denis D. Sheka, Volodymyr P. Kravchuk, and Yuri Gaididei
tion of Eq. (11) can be found analytically, nevertheless it is
unwieldy and below we present only numerical solution.
The dynamical polarity as numerical solution of
Eq. (11) for both cases of surface anistoropy is plotted in
Fig. 10 by solid lines. One can see this result is in a good
agreement with a direct solution for Hamiltonian (10)
(symbols correspond to simulations of the system with
| | = 0.1, = 20λ and = 70).N Numerically we mini-
mize (10) with respect to nµ and nψ under the action of
field = 0.0009.h − In equilibrium = 0.nψ The corre-
sponding equilibrium polarity in the homogeneous system
hom ( ) = 0.266hµ is shown by the dashed line.
The dynamical polarity in the wired core model plays a
role of the vortex width ( )w z in the full-scale simulations.
They reproduce the same shapes near the switching field,
cf. Fig. 4 and 10, and the switching starts in places with
minimal .nµ The deformation of the vortex shape in the
wired core model is a result of the inhomogeneous ground
state ( )bm z and is highly sensitive to its exact form.
The switching occurs when field increases to
| | = 0.001.h As in the full-scale simulations discussed
above the Bloch points nucleate on the surface for EN sur-
face anisotropy, see Fig. 11(a), cf. Fig. 7(a); they propagate
from the lattice planes with = 1n and =n N to the center
of the axis. For a thick enough lattice systems additional
Bloch points are nucleated, see curve for = 140N in the
Fig. 11(a). Bloch points are nucleated in the center of the
system for ES surface anisotropy in the same way as in the
full-scale simulations, see Fig. 11(b), cf. Fig. 7(b).
We also studied thickness dependence of the Bloch
point speed in the same way as in full-scale simulations,
see the Fig. 11 for the field = 0.001h − (numerical solu-
tion of Eqs. (D.1) with = 0.5.ε The comparison with the
full-scale simulations (cf. Fig. 8) confirm the fast decay of
the Bloch wall speed with increasing of the sample thick-
ness with the further saturation (Fig. 12).
5. Discussion
To summarize, we describe different mechanisms of the
vortex polarity switching in Heisenberg magnets with and
without additional surface anisotropy. Under the action of
transversal dc magnetic field the vortex core is reversed
using the axially-symmetric scenario [11,14,15]. We con-
clude that without surface anisotropy the switching occurs
uniformly with respect to the thickness z-coordinate
through the transient linear singularity or the planar vortex.
In presence of the surface anisotropy the switching is ac-
companied by nucleation of point singularities (Bloch
points), their motion and final annihilation. The birthplace of
Bloch points and the direction of their motion depends on
the type of the surface anisotropy. The complicated vortex
dynamics including its switching obtained using full-scale
spin-lattice simulations can be described analytically by a
wired core model, which is elaborated in this work.
Qualitatively, the influence of the surface anisotropy
can be explained as follows. In terms of the surface anisot-
ropy the magnetization configuration is pinned near the
sample boundary. This effective pinning has to compete
with exchange interaction, which results in the Robin type
boundary condition (4b). That is why the homogeneous
magnetization distribution =m h is not possible, since it
does not satisfy the boundary condition; there appears
0 ( , )m z h background profile, see (5). Similar scenario
takes place for the vortex state particle, where the vortex
width becomes z-dependent, = ( ).w w z According to (4b)
the sign of the vortex core width gradient depends on the
surface anisotropy type: barrel-shaped for the ES surface
anisotropy and pillow-shaped for the EN one [44].
Let us sketch the physical picture of the influence of the
surface anisotropy on statics and dynamics using the par-
ticular case of ES surface anisotropy. The typical vortex
core width w on a surface layer without magnetic field is
determined by effective magnetic length eff , see (7). In
ES case, the value of effective magnetic length eff < ,
and the vortex core becomes more narrow near the face
surfaces, which corresponds to the barrel-shaped profile.
Under the action of magnetic field the vortex state under-
goes several changes. First of all, there appears a cone state
instead of the easy-plane one. Apart this the core width for
the heavy vortex becomes narrower, ( ) 1 | |.w h h∝ −
However there is a counteraction of the surface anisotropy,
which tries to fix the z-gradient of the core width ( ).w z As
a result of this competition, there appears a neck in the
vortex core profile, which results in the deformed pillow
core instead of original barrel one, Fig. 4 with schematics
(e) and (f). This static picture plays a crucial role for the
understanding the reversal mechanism. When the field
intensity approaches the critical value (switching field), the
vortex core width extremely decreases. However this hap-
pens nonhomogeneously: the Bloch line has a bottleneck in
its center, hence it breaks during the switching in this
Fig. 12. (Color online) Bloch point speed as function of sample
thickness in the wired core model, cf. Fig. 8. Parameters are the
same as in Fig. 11.
476 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5
Vortex polarity switching in magnets with surface anisotropy
place. Namely this point is the birthplace of two Bloch points,
which move along the Bloch line repelling each other until
they annihilate on the face surfaces of the sample.
In the current study we do not take into account the di-
polar interaction, which is of great importance for the real
magnets. Specifically, namely the dipolar interaction fa-
vors the magnetization curling and causes that the vortex
becomes the ground state of magnetically soft nanodisk
[21]. In our previous work on the influence of the surface
anisotropy on the vortex core we compared in details dif-
ferent limiting regimes when magnetostatics can lead to
effective anisotropy, see Ref. 44 and references therein.
In order to validate our results for nanomagnets with the
dipolar interaction, we also perform few simulations with
account of both surface anisotropy and dipolar interaction.
In the current study we do not use SLaSi simulator for the
dipolar interaction case, because of high numerical costs.
Instead we make micromagnetic OOMMF [45] simula-
tions. As we already discussed above the modeling of the
Bloch point is challenging due to singular magnetization
distribution, leading typically to the mesh size dependence
[10]. Nevertheless, according to the physical picture of the
described effect, the key role in the dynamic switching
phenomenon plays the static instability of the vortex core:
there appears a spatial nonhomogeneity of the Bloch line,
which causes breaks of Bloch line at the bottleneck places
and finally results in the Bloch point mediated switching.
That is why we perform OOMMF simulations of different
static vortex distributions which precede switching. Nu-
merically we model the Permalloy sample with account of
dipolar interaction without surface anisotropy (because the
magnetostatics produces the effective ES one [43]) and
additional EN surface anisotropy [69].
Similar to the Heisenberg magnet with EN surface ani-
sotropy the vortex core in the nanodisk with magneto-
statics and strong enough EN surface anisotropy has a pil-
low-shaped profile without external field, see Fig. 13(a)
[data for = 0H and schematic (c)]. Under the action of the
field there appears the curve bend of the vortex core pro-
file. Under the action of strong enough field the vortex
profile takes up a deformed barrel shape and barrel shape
in the stronger field just before switching field, see Fig. 13a
[data for = 3H − kG (0.3 T) with schematic (d) and data
for = 4.4H − kG (0.44 T)]. Even in the absence of addi-
tional ES surface anisotropy in the sample with large as-
pect ratio /(2 ) 1L R the initial barrel-shaped vortex core
deforms in the central part in the same case as it is shown
in Fig. 4(e). Such a bending is absent in the thin samples
where the creation of volume magnetostatic charges signif-
icantly influences to the surface ones. In the both cases of
surface anisotropies switching starts from the places,
where vortex core becomes more narrow under the action
of external field: from the surfaces for EN and from the
center for effective ES surface anisotropy. Thus, the influ-
ence of the external field on the vortex profile and switch-
ing mechanism for the EN and ES surface anisotropies is
in a good qualitative agrement with results of SLaSi simu-
lations, see Fig. 4 and wired core model, see Fig. 10.
Acknowledgments
All simulations results presented in the work were ob-
tained using the computing clusters of Taras Shevchenko Na-
tional University of Kyiv [70] and Bayreuth University [71].
Appendix A: Equilibrium magnetization in film with
surface anisotropy
Let us consider a film of thickness L with constant of
the easy-plane anisotropy > 0K and constant of the sur-
face anisotropy = .sK K The equilibrium magnetization
distribution is determined by the Eqs. (4). In order to de-
rive the magnetization distribution one can assume that
Fig. 13. (Color online) Simulations with dipolar interaction using OOMMF for EN and effective ES surface anisotropies. Schematics
show the change of the vortex profile in the same way as in Fig. 4. EN surface anisotropy, nanodisk of radius 100 nm and thickness
24 nm (a); effective ES surface anisotropy induced by dipolar interaction, nanodisk of radius 66 nm and thickness 100 nm (b).
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5 477
Oleksandr V. Pylypovskyi, Denis D. Sheka, Volodymyr P. Kravchuk, and Yuri Gaididei
2
0 0ˆ ˆ= 1 m m− +m x z without lost of generality, where
0 0= ( , ).m m z h In this case Eqs. (4) are reduced to
2 2 2
2 20 0
0 02 22
0
= 1 ,
1
m m
h m m
z zm
∂ ∂
+ − −
∂ ∂−
(A.1)
20
0 0
= /2
(1 ) = 0.
z L
m
m m
z ±
∂
± − ∂ λ
(A.2)
The analytical solution can be found in assumption that
| / | 1.λ << Assuming 0 ( , ) = ( )m z h h y z+
λ
the solu-
tion takes the form (5).
Appendix B: Simulations with exchange anisotropy
The surface anisotropy naturally appears in the Heisen-
berg ferromagnet with exchange anisotropy [72]. That is
why we additionally simulate the system, described by the
LLG equations (2) and Hamiltonian
2
2
, ,
=
2
zJJ m m+ +− ⋅ + −∑ ∑n n δ n n δ
n δ n δ
m m
H
2 ,B H m− µ ∑ n
n
(B.1)
where > 0zJ is a coefficient of the exchange anisotropy
for easy-plane magnet. The sum in the second term in
the Eq. (B.1) runs over 6 neighbors for volume sites and
over 3–5 for surface sites which can be interpreted as a
exchange-induced surface anisotropy due symmetry break-
ing of the lattice on the surface.
We use the same procedure of simulations as in previ-
ous sections with / = 0.005,zJ J which gives the vortex
polarity reversal under the action of the reduced magnetic
field = 2 /( ) = 2.2.B zh H Jµ The observed magnetization
dynamics is qualitatively similar to the case of easy-plane
volume anisotropy with EN surface anisotropy shown in
Fig. 5. In comparison with simulations with single-ion ani-
sotropy the difference in vortex core magnetization is
much smaller in this case. We compute the difference of
dynamical polarities in the center and face surfaces of the
sample top center= | | = 1.65EN∆µ µ −µ (see Fig. 5(d)) and
= 0.032ES∆µ which is about 50 times weaker than EN
surface anisotropy.
Appendix C: Ring under the action of DC magnetic
field
Let us consider a magnetic ring of inner radius
in 0.3r ≥ in the vortex state described by the Hamiltonian
(1). For such an inner radius without external fields the
out-of-plane magnetization component vanishes and planar
vortex appears [68] The Eq. (6a) takes the following form
after substitution of the vortex anzats (6(a)):
2
2 2 2 2= ,z
z z y
y
mm h m m
m
∇ + − ∇ (C.1)
where 2= 1 cos .y zm m− ϕ Let us consider the case of
weak fields | | 1.h << We find the solution of Eq. (C.1) in
the form
= .z rm hm
The Eq. (C.1) takes the form
2 2 2
2
2 2 2
1 1 1 = 0,r r r
r
m m m m
r rz r r
∂ ∂ ∂
+ + + − + ∂∂ ∂
(C.2a)
2
= /2
= 0,r
r
z L
m a m
z ±
∂
± ∂
(C.2b)
= ,in
= 0.r
r r R
m
r
∂
∂
(C.2c)
For 0≠ the solution of (C.2) can be found in the form
= ( ) ( ),r n n
n
m R r Z z∑ (C.3)
which results in the following boundary-value problem:
2
2 2
2 = 0,n
n n
d Z
Z
dz
+ σ (C.4a)
2 2
2 2
2 2
1 1 0,n n
n n n
d R dR
R F
r drdr r
+ + − −σ + =
(C.4b)
/2
0,n
n
z L
dZ
Z
dz =±
± =
λ
(C.4c)
,
0,n
r r Rin
dR
dr =
= (C.4d)
where nF are defined as coefficients of expansion
1 = ( ).n n
n
F Z z∑ (C.5)
Eigenfunctions nZ have the form
( / 2) ( / 2)
( ) = cos sin ,n n
n n
z L z L
Z z
σ + σ +
σ +
λ
with dispersion equation
2 2tan 2 = 2 , .
2
n
n
L
n +
σ
λ ∈
λσ −
For the case < 0, the first eigenvalue 0σ should be
found as solution of
0
0
tanh =
2
Lσ
−
λσ
478 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5
Vortex polarity switching in magnets with surface anisotropy
with eigenfunction
0 0
0 0
( /2) ( /2)
( ) = cosh sinh .
z L z L
Z z
σ + σ +
σ +
λ
Solution of the Eq. (C.4b) can be written in the following
form
2
1
( ) = K ( ) L ( ) K ( )
1
n
n i i i
n
F
R d B
ρ
ρ ρ ρ ρ ρ+ ρ +
+ σ
∫
L ( ) K ( ) ,i ix d
∞
ρ
+ ρ ρ ρ
∫ (C.6)
where 2= 1 / ,n rρ + σ L ( )i • and K ( )i • are real valued
solutions of modified Bessel equation of purely imarginal
order with Wronskian {K ( ),L( )} = 1/i ζ ζ ζ [73,74]. Con-
stant B can be found from the boundary conditions (C.4d)
at in= :r r
0
1 = 0
L /
( ) = L ( ) K ( )
K /dρ
i
i i
i
d d
B d d
d
ρ ∞
ρ ρ ρ
ρ ρ − ρ ρ ρ+ ρ ρ ρ ≈
∫ ∫
fit 0
0
0
0.557
( ) = ,
0.99 0.316
B
ρ +
≈ ρ
ρ −
(C.7)
where 2
0 in= 1 / .n rρ + σ The interpolation function
fit
0( )B ρ fits numerically calculated dependence 0( )B ρ in
the range 0 [0.34,0.8]ρ ∈ with an accuracy of about 5%.
Functions ( )rm r and 0( )B ρ are shown in Fig. 14. In the
absence of the surface anisotropy the solution of Eqs. (C.2)
can be found in the form = ( )r rhm m r which is the solu-
tion of the following equation:
2 2
2
2 2
1 1 1 0rh rh
rh
d m dm
m
r drdr r
+ + − + =
(C.8)
with boundary condition (C.2c). Its solution can be ob-
tained from Eq. (C.6) by replacing nR to rhm and substi-
tution = 1nF and = 0.nσ
For the calculations in scope of the reduced wired core
model we use 3 harmonics for smallest L and up to 7 for
the most thick samples.
Appendix D: Reduced wired core model
The Hamiltonian (1) with account of the anisotropy
term (8) can be rewritten in the following form:
2
,
=c J m m +
− +∑ n n δ
n δ
2 2(1 )(1 ) cos( )m m + +
+ − − ϕ −ϕ +n n δ n n δ
2
2
=( , , )
2 .
2
n
B
n n nx y
K
m H m
+ − µ
∑ n n
n
By incorporating the reduced vortex core ansatz (9) one get
2
2 2
=1
2= 4
2
N
n
c n b n n
n
J JK m
K K
µ
− µ − µ + −
∑
2
2 24 (1 )(1 ) cos
25
n
b n n n
J m h
K
µ
− − −µ ψ + −µ −
1
2 2 2
1 1
=1
4 (1 )(1 )
N
n n n n
n
J
−
+ +
− µ µ + −µ −µ ×
∑
1cos( ) .n n+
× ψ −ψ
Taking into account that = 0n for = 2, 1,n N −
1 = = ,N 2 2= / ,aλ = / /(1 )n n nJ Kλ ≡ λ + one
obtains the Eq. (10). The temporal evolution of ( )nµ τ and
( )nψ τ is governed by the Eq. (2), which results in
2
2 2
1 (1 ) ,
1 1
n c c
n
n n
d
d
µ ∂ ∂ε
= − −µ
τ ∂ψ ∂µ+ ε + ε
H H
2 2 2
1 1 .
1 1 1
n c c
n nn
d
d
ψ ∂ ∂ε
= − −
τ ∂µ ∂ψ+ ε −µ + ε
H H
(D.1)
The problem of magnetization dynamics in discrete model
can be analysed numerically. For analytical approach we
replace the discrete functions nµ and nψ using their con-
tinuous analogues ( )zµ and ( )zψ and replace summation
by integration. The volume contribution has the form
Fig. 14. (Color online) Magnetization distribution = ( , )z rm hm r z in
the center and on the face surfaces of the ring with in = 1.58r a
and ES surface anisotropy under the action of the field
= 0.0009h − (4 harmonics of the Eq. (C.4) are taken into ac-
count). Dotted curve corresponds to the function rhhm (absence
of surface anisotropy). Inset shows value of coefficient B as
function of 2
0 in= 1 /n rρ + σ = 0.1, other parameters are the
same as in Fig. 10.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5 479
Oleksandr V. Pylypovskyi, Denis D. Sheka, Volodymyr P. Kravchuk, and Yuri Gaididei
/2 2
/2
4[ ] = (1 2 ) ( 2 )
2
L
v b
L
h m
a
−
µ
µ − λ − + λ µ −
∫E
2 24 (1 )(1 ) cos
5 bm dz
λ
− − −µ ψ +
/2 2
2 2
2
/2
( )2 (1 )( ) ,
1
L
L
a dz
−
′µ ′+ λ + −µ ψ
−µ
∫ (D.2a)
with 0 0 ( , )m m z h≡ defined in Eq. (5). The surface contri-
bution is
2
= /2
[ ] = 2 .s z L
±
±
µ µE (D.2b)
We assume that the deviation from the equilibrium dynam-
ical polarity in absence of the surface anisotropy is small
and 0( ) = ( ) ( )z h x zµ µ +
λ
with | / | 1,λ << where
hom ( )hµ can be found numerically as solution of the fol-
lowing equation:
2
hom hom
1 (1 2 ) (1 2 )
2 rhh m− λ µ − + λ µ =
2 2 2
hom
4= (1 )(1 ),
5 rhh mλ
− −µ (D.3)
see Fig. 15. Taking ( ) = 0zψ and varying the functional
(D.2) one can find the equilibrium distribution of µ as
solution of the boundary-value problem (11) where
2
1 hom 2
hom
1 4= 2 (1 ) ,
5(1 )
A − −µ − λ −µ
(D.4a)
2
hom
2
1
( ) = 5 10 ( )
5 bA z h m z
−µ − λ +
λ
hom 2
hom
55 10 4 .
1
+ µ − λ + λ
−µ
(D.4b)
1. J. Stöhr and H.C. Siegmann, Magnetism: From Fundamentals
to Nanoscale Dynamics, Springer Series in Solid-State Sciences,
Springer-Verlag Berlin, Heidelberg (2006), Vol. 152.
2. Spin Dynamics in Confined Magnetic Structures III, Ser.
Topics Appl. Phys., B. Hillebrands and A. Thiaville (eds.)
Springer, Berlin (2006), Vol. 101.
3. Handbook of Magnetism and Advanced Magnetic Materials,
H. Kronmüller and S. Parkin (eds.), Wiley-Interscience
(2007), Vol. 2.
4. S.-K. Kim, K.-S. Lee, Y.-S. Yu, and Y.-S. Choi, Appl. Phys.
Lett. 92, 022509 (2008).
5. B. Pigeau, G. de Loubens, O. Klein, A. Riegler, F. Lochner,
G. Schmidt, L.W. Molenkamp, V.S. Tiberkevich, and A.N.
Slavin, Appl. Phys. Lett. 96, 132506 (2010).
6. Y.-S. Yu, H. Jung, K.-S. Lee, P. Fischer, and S.-K. Kim,
Appl. Phys. Lett. 98, 052507 (2011).
7. Y.B. Gaididei, V.P. Kravchuk, D.D. Sheka, and F.G.
Mertens, Fiz. Nizk. Temp. 34, 669 (2008) [Low Temp. Phys.
34, 528 (2008)].
8. T. Okuno, K. Shigeto, T. Ono, K. Mibu, and T. Shinjo,
Magn. Magn. Mater. 240, 1 (2002).
9. R. Höllinger, A. Killinger, and U. Krey, Magn. Magn.
Mater. 261, 178 (2003).
10. A. Thiaville, J.M. Garcia, R. Dittrich, J. Miltat, and T.
Schrefl, Phys. Rev. B 67, 094410 (2003).
11. V. Kravchuk and D. Sheka, Phys. Solid State 49, 1923
(2007).
12. R. Wang and X. Dong, Appl. Phys. Lett. 100, 082402 (2012).
13. M.-W. Yoo, J. Lee, and S.-K. Kim, Appl. Phys. Lett. 100,
172413 (2012)
14. O.V. Pylypovskyi, D.D. Sheka, V.P. Kravchuk, Y. Gaididei,
and F.G. Mertens, Ukr. J. Phys. 58, 596 (2013)
15. O.V. Pylypovskyi, D.D. Sheka, V.P. Kravchuk, F.G.
Mertens, and Y. Gaididei, Phys. Rev. B 88, 014432 (2013).
16. M. Helsen, A. Gangwar, J.D. Clercq, A. Vansteenkiste, and
B.V. Waeyenberge, (2014), http://arxiv.org/abs/1411.3470
1411.3470.
17. V.P. Kravchuk, Y. Gaididei, and D.D. Sheka, Phys. Rev. B
80, 100405 (2009).
18. Y. Gaididei, V.P. Kravchuk, D.D. Sheka, and F.G. Mertens,
Phys. Rev. B 81, 094431 (2010).
19. B. Van Waeyenberge, A. Puzic, H. Stoll, K.W. Chou, T.
Tyliszczak, R. Hertel, M. Fähnle, H. Bruckl, K. Rott, G.
Reiss, I. Neudecker, D. Weiss, C.H. Back, and G. Schütz,
Nature 444, 461 (2006).
20. R. Hertel, S. Gliga, M. Fähnle, and C.M. Schneider, Phys.
Rev. Lett. 98, 117201 (2007).
21. A. Hubertand and R. Schäfer, Magnetic Domains: the
Analysis of Magnetic Microstructures, Springer–Verlag,
Berlin (1998).
22. A. Arrott, B. Heinrich,and A. Aharoni, Magnetics, IEEE
Transactions on 15, 1228 (1979).
23. A. Hubert and W. Rave, J. Magn. Magn. Mater. 184, 67
(1998).
24. E. Feldtkeller, Z. Angew. Phys. 19, 530 (1965).
25. W. Döring, J. Appl. Phys. 39, 1006 (1968).
Fig. 15. (Color online) Equilibrium polarity hom( )hµ for the dis-
crete homogeneous model (10) ( = 0,n = 1,n N ) for = 20.λ
480 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5
http://www.amazon.com/Handbook-Magne%C2%ACtism-Advanced-Magnetic-Materials/dp/0470022175
http://dx.doi.org/%2010.1063/1.2807274
http://dx.doi.org/%2010.1063/1.2807274
http://dx.doi.org/%20DOI:10.1063/1.3373833
http://dx.doi.org/%2010.1063/1.3551524
http://dx.doi.org/%2010.1063/1.2957013
http://dx.doi.org/%2010.1063/1.2957013
http://www.sciencedirect.com/science/article/B6TJJ-447DCMV-3/2/84e9fdfbf9e0cab3d3182fc4db4b4032J
http://www.sciencedirect.com/%20science/article/B6TJJ-47F1D5K-1/2/02a9f675dac9cbb50d643664d50b98beJ
http://www.sciencedirect.com/%20science/article/B6TJJ-47F1D5K-1/2/02a9f675dac9cbb50d643664d50b98beJ
http://link.aps.org/abstract/PRB/v67/e094410
http://dx.doi.org/10.1134/S1063783407100186
http://dx.doi.org/%2010.1063/1.3687909
http://dx.doi.org/%2010.1063/1.4705690
http://www.ujp.bitp.kiev.ua/files/journals/58/6/580611p.pdf
http://dx.doi.org/%2010.1103/PhysRevB.88.014432
http://dx.doi.org/%2010.1103/PhysRevB.80.100405
http://dx.doi.org/%2010.1103/PhysRevB.80.100405
http://dx.doi.org/10.1103/PhysRevB.81.094431
http://dx.doi.org/10.1038/nature05240
http://link.aps.org/abstract/PRL/v98/e117201
http://link.aps.org/abstract/PRL/v98/e117201
http://ieeexplore.ieee.org/xpls/abs_all.jsp?tp=arnumber=1060342isnumber=22837
http://ieeexplore.ieee.org/xpls/abs_all.jsp?tp=arnumber=1060342isnumber=22837
http://dx.doi.org/%2010.1016/s0304-8853(97)01125-6
http://dx.doi.org/%2010.1063/1.1656144
Vortex polarity switching in magnets with surface anisotropy
26. A.B. Borisov and V.V. Kiselev, Nonlinear Waves, Solitons
and Localized Structures in Magnets, Russian Academy of
Science (2011), Vol. 2 (in Russian).
27. R.G. Elías, V.L. Carvalho-Santos, A.S. Núñez,and A.D.
Verga, Phys. Rev. B 90, 22 (2014).
28. Y.P. Kabanov, L.M. Dedukh, and V.I. Nikitenko, JETP Lett.
47, 737 (1988).
29. Y.P. Kabanov, L.M. Dedukh, and V.I. Nikitenko, JETP Lett.
49, 637 (1989).
30. S. Da Col, S. Jamet, N. Rougemaille, A. Locatelli, T.O.
Mentes, B.S. Burgos, R. Afid, M. Darques, L. Cagnon, J.C.
Toussaint, and O. Fruchart, Phys. Rev. B 89, 180405 (2014).
31. R. Herteland and J. Kirschner, J. Magn. Magn. Mater. 278,
L291 (2004).
32. R. Hertel and C. M. Schneider, Phys. Rev. Lett. 97, 177202
(2006)
33. O.V. Pylypovskyi, D.D. Sheka, and Y. Gaididei, Phys. Rev.
B 85, 224401 (2012).
34. C. Andreas, A. Kákay, and R. Hertel, Phys. Rev. B 89,
134403 (2014).
35. S.K. Kimand and O. Tchernyshyov, Phys. Rev. B 88, 174402
(2013).
36. V.S. Gornakov, V.I. Nikitenko, and I.A. Prudnikov, Pis’ma
Zh. Exp. Theor. Fiz. 50, 479 (1989)
37. A.P. Malozemoff and J.C. Slonzewski, Magnetic Domain
Walls in Bubble Materials, Academic Press, New York
(1979).
38. Y.A. Kufaev and E.B. Sonin, Phys. Solid State 30, 3272
(1988).
39. Y.A. Kufaevand and E.B. Sonin, JETP 68, 879 (1989).
40. E.G. Galkina, B.A. Ivanov, and V.A. Stephanovich, Magn.
Magn. Mater. 118, 373 (1993).
41. V.E. Kireevand and B.A. Ivanov, Phys. Rev. B 68, 104428
(2003).
42. S.A. Leonel, I.A. Marques, P.Z. Coura, and B.V. Costa, J.
Appl. Phys. 102, 104311 (2007).
43. J.-G. Caputo, Y. Gaididei, V.P. Kravchuk, F.G. Mertens, and
D.D. Sheka, Phys. Rev. B 76, 174428 (2007).
44. O.V. Pylypovskyi, D.D. Sheka, V.P. Kravchuk, and Y.
Gaididei, J. Magn. Magn. Mater. 361, 201 (2014).
45. The Object Oriented MicroMagnetic Framework, Developed
by M.J. Donahue and D. Porter mainly, from NIST. We used
the 3D version of the 1.2α5 release.
46. L. Néel, J. Phys. Radium 15, 225 (1954).
47. W.F. Brown, Jr., Micromagnetism, Wiley, New York,
(1963).
48. E.W. Weisstein, The CRC Concise Encyclopedia of
Mathematics, CRC Press (1998).
49. H. Kachkachi and M. Dimian, Phys. Rev. B 66, 174419
(2002).
50. H. Kachkachi and E. Bonet, Phys. Rev. B 73, 224402 (2006).
51. R. Yanes, O. Chubykalo-Fesenko, H. Kachkachi, D.A.
Garanin, R. Evans, and R.W. Chantrell, Phys. Rev. B 76,
064416 (2007).
52. L. Berger, Y. Labaye, M. Tamine, and J.M.D. Coey, Phys.
Rev. B 77, 104431 (2008).
53. S.-h. Noh, W. Na, J.-t. Jang, J.-H. Lee, E.J. Lee, S.H. Moon,
Y. Lim, J.-S. Shin, and J. Cheon, Nano Lett. 12, 37163721
(2012).
54. The sample of size 100 100 50× × sites is simulated with
magnetic length = 5a and = 0.25± under the action of
field = 0.7.h The magnetization profile along ẑ axis is
measured in the center of the sample.
55. E. Feldtkeller and H. Thomas, Zeitschrift für Physik B
Condensed Matter 4, 8 (1965).
56. A.M. Kosevich, V.P. Voronov, and I.V. Manzhos, Sov. Phys.
JETP 57, 148 (1983).
57. B.A. Ivanov and D.D. Sheka, Fiz. Nizk. Temp. 21, 1148
(1995) [Low Temp. Phys. 21, 881 (1995)].
58. B.A. Ivanov and G.M. Wysin, Phys. Rev. B 65, 34434
(2002).
59. G.M. Wysin, Phys. Rev. B 49, 8780 (1994).
60. Y. Gaididei, T. Kamppeter, F. G. Mertens, and A. Bishop,
Phys. Rev. B 59, 7010 (1999).
61. Y. Gaididei, T. Kamppeter, F. G. Mertens,and A. R. Bishop,
Phys. Rev. B 61, 9449 (2000).
62. J.-G. Caputo, Y. Gaididei, F.G. Mertens, and D.D. Sheka,
Phys. Rev. Lett. B 98, 056604 (2007).
63. SLaSi http://slasi.rpd.univ.kiev.ua
64. Because vortex center at r = 0 lies between lattice sites, the
length of spins in simulations remains constant.
65. Step-wise field turning-on decreases the switching threshold.
66. J.P. Zagorodny, Y. Gaididei, F.G. Mertens, and A.R. Bishop,
Eur. Phys. J. B 31, 471 (2003).
67. A.S. Kovalev and J.E. Prilepsky, Low Temp. Phys. 29, 55
(2003).
68. V.P. Kravchuk, D.D. Sheka, and Y.B. Gaididei, J. Magn.
Magn. Mater. 310, 116 (2007).
69. We simulate a nanodisks with cubic mesh of 2 nm. Magnetic
fiels are directed along the disk sample. Materials parameters of
Permalloy are used: exchange constant 6= 1.3 10A −⋅ erg/cm,
saturation magnetization = 68.7sM Oe (860 kA/m). The
surface anisotropy is simulated as nonzero easy-axis
anisotropy of value 2
bulk| | = 2.4 sMβ π in the cells on the
face surfaces which corresponds to the surface anisotropy
surf bulk| | = | | 19.7aβ β ≈ Merg/cm2.
70. http://cluster.univ.kiev.ua/eng/ “Kyiv National Taras Shev-
chenko University high-performance computing cluster”.
71. http://www.rz.uni-bayreuth.de/ “Bayreuth University com-
puting cluster.”
72. A.M. Kosevich, B.A. Ivanov, and A.S. Kovalev, Phys. Rep.
194, 117 (1990).
73. T.M. Dunster, J. Math. Anal. 21, 995 (1990).
74. A. Gil, J. Segura, and N.M. Temme, ACM Transactions on
Mathematical Software 30, 145 (2004).
Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5 481
http://dx.doi.org/%2010.1103/physrevb.90.224414
http://www.jetpletters.ac.ru/ps/1099/article_16614.shtml
http://www.jetpletters.ac.ru/ps/1099/article_16614.shtml
http://www.jetpletters.ac.ru/ps/1121/article_16982.shtml
http://www.jetpletters.ac.ru/ps/1121/article_16982.shtml
http://dx.doi.org/%2010.1103/physrevb.89.180405
http://www.sciencedirect.com/science/article/B6TJJ-4BYP1W4-4/2/f85731b704a46f19c401b90833e467b7
http://link.aps.org/abstract/PRL/v97/e177202
http://dx.doi.org/%2010.1103/PhysRevB.85.224401
http://dx.doi.org/%2010.1103/PhysRevB.85.224401
http://dx.doi.org/%2010.1103/PhysRevB.89.134403
http://dx.doi.org/%2010.1103/PhysRevB.88.174402
http://www.jetp.ac.ru/cgi-bin/e/index/e/68/4/p879?a=list
http://www.sciencedirect.com/science/article/B6TJJ-46FHH1R-CR/2/d01239b41a403665b912fdbe1c3ec94bJ
http://www.sciencedirect.com/science/article/B6TJJ-46FHH1R-CR/2/d01239b41a403665b912fdbe1c3ec94bJ
http://link.aps.org/abstract/PRB/v68/e104428
http://dx.doi.org/10.1063/1.2809408
http://dx.doi.org/10.1063/1.2809408
http://dx.doi.org/%2010.1103/PhysRevB.76.174428
http://dx.doi.org/10.1016/j.jmmm.2014.02.094
http://math.nist.gov/oommf/
http://dx.doi.org/10.1051/jphysrad:01954001504022500
http://www.amazon.com/The-CRC-Concise-Encyclopedia-Mathematics/dp/0849396409
http://www.amazon.com/The-CRC-Concise-Encyclopedia-Mathematics/dp/0849396409
http://dx.doi.org/%2010.1103/PhysRevB.66.174419
http://link.aps.org/abstract/PRB/v73/e224402
http://dx.doi.org/%2010.1103/PhysRevB.76.064416
http://dx.doi.org/10.1103/PhysRevB.77.104431
http://dx.doi.org/10.1103/PhysRevB.77.104431
http://dx.doi.org/%2010.1021/nl301499u
http://dx.doi.org/10.1007/BF02423256
http://dx.doi.org/10.1007/BF02423256
http://link.aps.org/abstract/PRB/v65/e134434
http://dx.doi.org/%2010.1103/PhysRevB.49.8780
http://link.aps.org/abstract/PRB/v59/p7010
http://dx.doi.org/%2010.1103/PhysRevB.61.9449
http://link.aps.org/abstract/PRL/v98/e056604
http://slasi.rpd.univ.kiev.ua/
http://dx.doi.org/10.1140/epjb/e2003-00057-y
http://dx.doi.org/%2010.1063/1.1542378
http://dx.doi.org/10.1016/j.jmmm.2006.07.035
http://dx.doi.org/10.1016/j.jmmm.2006.07.035
http://cluster.univ.kiev.ua/eng/
http://www.rz.uni-bayreuth.de/
http://www.sciencedirect.com/science/article/B6TVP-46SXP0P-19/2/d0a99bbbccf078602123db3e5d601202
http://www.sciencedirect.com/science/article/B6TVP-46SXP0P-19/2/d0a99bbbccf078602123db3e5d601202
http://dx.doi.org/10.1137/0521055
http://dx.doi.org/%2010.1145/992200.992203
http://dx.doi.org/%2010.1145/992200.992203
1. Introduction
2. The model and the vortex solution
3. Spin-lattice simulations of the vortex polarity switching
3.1. Magnets with bulk anisotropy only:
3.2. Magnets with the surface anisotropy
4. Description of the polarity switching process
5. Discussion
Acknowledgments
Appendix A: Equilibrium magnetization in film with surface anisotropy
Appendix B: Simulations with exchange anisotropy
Appendix C: Ring under the action of DC magnetic field
Appendix D: Reduced wired core model
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