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N A N O E X P R E S S Open Access
Nonlinear magnetic vortex dynamics in a circular
nanodot excited by spin-polarized current
Konstantin Y Guslienko
1,2*
, Oksana V Sukhostavets
1
and Dmitry V Berkov
3
Abstract
We investigate analytically and numerically nonlinear vortex spin torque oscillator dynamics in a circular magnetic
nanodot induced by a spin-polarized current perpendicular to the dot plane. We use a generalized nonlinear Thiele
equation including spin-torque term by Slonczewski for describing the nanosize vortex core transient and steady
orbit motions and analyze nonlinear contributions to all forces in this equation. Blue shift of the nano-oscillator
frequency increasing the current is explained by a combination of the exchange, magnetostatic, and Zeeman
energy contributions to the frequency nonlinear coefficient. Applicability and limitations of the standard nonlinear
nano-oscillator model are discussed.
Keywords: Magnetic nanodot; Nano-oscillator; Vortex; Spin torque transfer
Background
Spin torque microwave nano-oscillators (STNO) are in-
tensively studied nowadays. STNO is a nanosize device
consisting of several layers of ferromagnetic materials
separated by nonmagnetic layers. A dc current passes
through one ferromagnetic layer (reference layer) and
thus being polarized. Then, it enters to an active mag-
netic layer (so-called free layer) and interacts with the
magnetization causing its high-frequency oscillations
due to the spin angular momentum transfer. These oscil-
lation frequencies can be tuned by changing the applied
dc current and external magnetic field [1-3] that makes
STNO being promising candidates for spin transfer
magnetic random access memory and frequency-tunable
nanoscale microwave generators with extremely narrow
linewidth [4]. The magnetization in the free layer can form
a vortex configuration that possesses a periodical circular
motion driven by spin transfer torque [1,5-11]. For
practical applications of such nanoscale devices, some
challenges have to be overcome, e.g., enhancing the STNO
output power. So, from a fundamental point of view as
well as for practical applications, the physics of STNO
magnetization dynamics has to be well understood.
In the present paper, we focus on the magnetic vortex
dynamics in a thin circular nanodot representing a free
layer of nanopillar (see inset of Figure 1). Circular nano-
dots made of soft magnetic material have a vortex state
of magnetization as the ground state for certain dot radii
Rand thickness L. The vortex state is characterized by
in-plane curling magnetization and a nanosize vortex core
with out-of-plane magnetization. Since the vortex state
of magnetization was discovered as the ground state of
patterned magnetic dots, the dynamics of vortices have
attracted considerable attention. Being displaced from
its equilibrium position in the dot center, the vortex
core reveals sub-GHz frequency oscillations with a nar-
row linewidth [2,7,12]. The oscillations of the vortex
core are governed by a competition of the gyroforce,
Gilbert damping force, spin transfer torque, and restor-
ing force. The restoring force is determined by the vor-
tex confinement in a nanodot. Vortex core oscillations
with small amplitude can be well described in the linear
regime, but for increasing of the STNO output power, a
large-amplitude motion has to be excited. In the regime
of large-amplitude spin transfer-induced vortex gyr-
ation, it is important to take into account nonlinear
contributions to all the forces acting on the moving vor-
tex. The analytical description and micromagnetic simu-
lations of the magnetic field and spin transfer-induced
vortex dynamics in the nonlinear regime have been pro-
posed by several groups [12-22], but the results are still
* Correspondence: kostyantyn.gusliyenko@ehu.es
1
Depto. Física de Materiales, Facultad de Química, Universidad del País Vasco,
UPV/EHU, San Sebastián 20018, Spain
2
IKERBASQUE, Basque Foundation for Science, Bilbao 48011, Spain
Full list of author information is available at the end of the article
© 2014 Guslienko et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction
in any medium, provided the original work is properly credited.
Guslienko et al. Nanoscale Research Letters 2014, 9:386
http://www.nanoscalereslett.com/content/9/1/386
contradictory. It is unclear to what extent a standard
nonlinear oscillator model [13] is applicable to the vor-
tex STNO, how to calculate the nonlinear parameters,
and how the parameters depend on the nanodot sizes.
In this paper, we show that a generalized Thiele approach
[23] is adequate to describe the magnetic vortex motion in
the nonlinear regime and calculate the nanosize vortex core
transient and steady orbit dynamics in circular nanodots
excited by spin-polarized current via spin angular momen-
tum transfer effect.
Methods
Analytical method
We apply the Landau-Lifshitz-Gilbert (LLG) equation of
motion of the free layer magnetization _
m¼−γmHeff þ
αGm_
mþγτs,wherem=M/M
s
,M
s
is the saturation
magnetization, γ> 0 is the gyromagnetic ratio, H
eff
is the
effective field, and α
G
is the Gilbert damping. We use a
spin angular momentum transfer torque in the form
suggested by Slonczewski [24], τ
s
=σJm×(m×P),
where σ=ℏη/(2|e|LM
s
), ηis the current spin polarization
(η≅0.2 for FeNi), eis the electron charge, Pis direction of
the reference layer magnetization, and Jis the dc current
density. The current is flowing perpendicularly to the
layers of nanopillar and we assume P¼P^
z. The free layer
(dot) radius is Rand thickness is L.
We apply Thiele's approach [23] for the magnetic vortex
motion in circular nanodot (inset of Figure 1). We assume
that magnetization distribution can be characterized by
a position of its center X(t) that can vary with time
and, therefore, the magnetization as a function of the
coordinates rand X(t) can be written as m(r,t)=m(r,X(t)).
Then, we can rewrite the LLG equation as a generalized
Thiele equation for X(t):
Gαβ
_
Xβ¼−∂αWþDαβ
_
XβþFα
ST ;ð1Þ
where Wis the total magnetic energy, α,β=x,y,and
∂
α
=∂/∂X
α
. The components of the gyrotensor
^
G, damping
tensor
^
D, and the spin-torque force can be expressed as
follows [16]:
Gαβ XðÞ¼
Ms
γZd3r∂αm∂βm
⋅m
Dαβ XðÞ¼−αG
Ms
γZd3r∂αm∂βm
Fα
ST XðÞ¼MsσJP⋅Zd3rm∂αmðÞ:
ð2Þ
We assume that the dot is thin enough and mdoes
not depend on z-coordinate. The magnetization m(x,y)
has the components mxþimy¼2w=1þw
wðÞand
mz¼1−w
wðÞ=1þw
wðÞexpressed via a complex function
wζ;
ζ
[25]. Inside the vortex core, the vortex configur-
ation is described as a topological soliton, wζ;
ζ
¼fζðÞ,
|f(ζ)| ≤1, where f(ζ) is an analytic function. Outside the
vortex core region, the magnetization distribution is w
ζ;
ζ
¼fζðÞ=fζðÞ
jj
,|f(ζ)| > 1. For describing the vortex
dynamics, we use two-vortex ansatz (TVA, no side surface
charges induced in the course of motion) with function f(ζ)
being written as fζðÞ¼−iC ζ−sðÞ
sζ−1ðÞ=c1þs
jj
2
[26],
3.5 4.0 4.5 5.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
z
Reduced vortex orbit radius u
0
(J)
Current density J (MA/cm2)
FeNi cylindrical dot
L= 7 nm, R=100 nm
J
c1
=3.36 MA/cm
2
N(J
c1
)=0.369
Calculated
Simulated
Jc1
Figure 1 Magnetic vortex dynamics in a thin circular FeNi nanodot. Vortex core steady-state orbit radius u
0
(J) in the circular FeNi nanodot of
thickness L= 7 nm and radius R= 100 nm vs. current Jperpendicular to the dot plane. Solid black lines are calculations by Equation 7; red circles
mark the simulated points. Inset: sketch of the cylindrical vortex state dot with the core position Xand used system of coordinates.
Guslienko et al. Nanoscale Research Letters 2014, 9:386 Page 2 of 7
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where Cis the vortex chirality, ζ=(x+iy)/R,s=s
x
+is
y
,
s=X/R,c=R
c
/R,andR
c
is the vortex core radius.
The total micromagnetic energy W¼Wv
mþWs
mþ
Wex þWZin Equation 1 including volume Wv
mand sur-
face Ws
mmagnetostatic energy, exchange W
ex
energy,
and Zeeman W
Z
energy of the nanodot with a displaced
magnetic vortex is a functional of magnetization distri-
bution W[m(r,t)]. Using m=m(r,X(t)) and integrating
over-the-dot volume and surface, the energy Wcan be
expressed as a function of Xwithin TVA [16]. The Zeeman
energy is related to Oersted field HJ¼0;Hϕ
J;0
of the
spin-polarized current, W
Z
(X)=−M
s
∫dVm(r,X)⋅H
J
.We
introduce a time-dependent vortex orbit radius and phase
by s=uexp(iΦ). The gyroforce in Equation 1 is determined
by the gyrovector G¼G^
z,whereG=G
z
=G
xy
.Thefunc-
tions G(s)andW(s) depend only on u=|s| due to a circular
symmetry of the dot. G(0) = 2πpM
s
L/γ,wherepis the vor-
texcorepolarity.Thedampingforce
^
D
_
Xand spin-torque
force F
ST
are functions not only on u=|s|butalsoondirec-
tion of s. Nonlinear Equation 1 can be written for the circu-
lar dot in oscillator-like form
i_
sþωGuðÞs¼−duðÞ
_
sþiχuðÞs−idnsðÞ
_
s;ð3Þ
where ω
G
(u)=(R
2
u|G(u)|)
−1
∂W(u)/∂uis the nonlinear
gyrotropic frequency, d(u)=−D(u)/|G(u)| is the nonlinear
diagonal damping, D=D
xx
=D
yy
,d
n
(s)=−D
xy
(s)/|G(u)| is
the nonlinear nondiagonal damping, and χ(u)=a(u)/|G(u)|.
It is assumed here that F
ST
(s)=a(u)(z×s) [14], where
ais proportional to the CPP current density Jand a
(0) = πRLM
s
σJ.
To solve Equation 3, we need to answer the following
questions: (1) can we decompose the functions W(s), G
(s), D
αβ
(s), and F
ST
(s) in the power series of u=|s| and
keep only several low-power terms? and (2) what is the
accuracy of such truncated series accounting that u=|s|
can reach values of 0.5 to 0.6 for a typical vortex STNO?
Some of these functions may be nonanalytical functions
of u=|s|. If the answer to the first question is yes, then
we should decompose W(s)uptou
4
,F
ST
(s)uptou
3
,
and G(s), D
αβ
(s)uptou
2
-terms to get a cubical equation
of the vortex motion. The series decomposition of G(s)
does not contain u
2
-term; it contains only small c
2
u
2
-term,
G(u)=G(0)[1 −O(c
2
u
2
)], although G(u) essentially decreases
at large u, when the vortex core is close to be expelled from
the dot [16]. The result of power decomposition of the total
energy density wuðÞ¼WuðÞ=M2
sVis
wuðÞ¼w0ðÞþ
1
2κu2þ1
4κ′u4;u¼sjj;ð4Þ
and the coefficients are
1
2κβ;R;JðÞ¼8πZ
∞
0
dt fβtðÞ
tI2tðÞ−Le
R
2
þ2π
15c
JCRς
Ms
and
1
4κ
′
β;R;JðÞ¼2πZ
∞
0
dt fβtðÞ
tI2
2tðÞ−ItðÞI1tðÞ
þ1
2
Le
R
2
þπ
15c
JCR
Ms
;
where ItðÞ¼Z1
0
dρρJ1tρðÞ,I1tðÞ¼Z1
0
dρρ−11−ρ2
2J1tρðÞ,
I2tðÞ¼Z1
0
dρ1þρ2
J2tρðÞ,β=L/R,Le¼ffiffiffiffiffiffi
2A
p=Ms,and
ς=1+15(ln2−1/2)R
c
/8R.
There is an additional contribution to κ/2, 2(L
e
/R)
2
,
due to the face magnetic charges essential for the nano-
dots with small R[27]. The contribution is positive and
can be accounted by calculating dependence of the equilib-
rium vortex core radius (c) on the vortex displacement.
This dependence with high accuracy at cu < < 1 can be
described by the function c(u)=c(0)(1 −u
2
)/(1 + u
2
).
Here, c(0) is the equilibrium vortex core radius at s=0,
for instance c(0) = 0.12 (R
c
= 12 nm) for the nanodot
thickness L=7nm.
The nonlinear vortex gyrotropic frequency can be
written accounting Equation 4 as
ωGuðÞ¼ω01þNu2
;ð5Þ
where the linear gyrotropic frequency is ω
0
=γM
s
κ(β,R,J)/2,
and N(β,R)=κ′(β,R)/κ(β,R).
The frequency ω′
0¼γMsκβ;R;0ðÞ=2 was calculated in
[26] and was experimentally and numerically confirmed
in many papers. The nonlinear coefficient N(β,R)depends
strongly on the parameters βand R, decreasing with βand
Rincreasing. The typical values of N(β,R)atJ=0 are equal
to 0.3 to 1.
The last term in Equation 3 prevents its reducing to a
nonlinear oscillator equation similar to the one used for
the description of saturated STNO in [13]. Calculation
within TVA yields the decomposition dnsðÞ¼d0
nþd1
nsxsy,
where d0
n¼0 , i.e., the term containing d
n
(s)≈α
G
u
2
<<1
can be neglected. Then, substituting s=uexp(iΦ)to
Equation 3, we get the system of coupled equations
_
Φ−ωGuðÞ¼duðÞ
_
u
u;_
u¼χuðÞ−duðÞ
_
Φ
u:ð6Þ
Equation 3 and the system (6) are different from the
system of equations of the nonlinear oscillator approach
[13]. Equations 6 are reduced to the autonomous oscilla-
tor equations _
u=u¼χuðÞ−duðÞωGuðÞand
_
Φ¼ωGuðÞ
only if the conditions d
2
< < 1 and dχ<<ω
G
are satisfied
and we define the positive/negative damping parameters
[13] as Γ
+
(u)=d(u)ω
G
(u) and Γ
−
(u)=χ(u). We note that
Guslienko et al. Nanoscale Research Letters 2014, 9:386 Page 3 of 7
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reducing the Thiele equation (1) to a nonlinear oscillator
equation [13] is possible only for axially symmetric
nanodot, when the functions W(s), G(s), d(s)andχ(s)
depend only on u=|s| and the additional conditions
d
n
<<1, d
2
<<1, and dχ<<ω
G
are satisfied. The non-
linear oscillator model [13] cannot be applied for other
nanodot (free layer) shapes, i.e., elliptical, square, etc.,
whereas the generalized Thiele equation (1) has no
such restrictions.
The system (6) at _
u¼0 yields the steady vortex oscil-
lation solution u
0
(J) > 0 as root of the equation χ(u
0
)=d
(u
0
)ω
G
(u
0
) for χ(0) > d(0)ω
0
(J>J
c1
)andu
0
= 0 otherwise.
If we use the power decompositions ω
G
(u)=ω
0
+ω
1
u
2
,
d(s)=d
0
+d
1
u
2
,andχ(u)=χ
0
+χ
1
u
2
for the nonlinear
vortex frequency, damping, and spin-torque terms, re-
spectively, and account that the linear vortex frequency
contains a contribution proportional to the current density
ω0JðÞ¼ω′
0þωeJ,whereω
e
=(8π/15)(γR/c)ς[12,16], then
we get the vortex core steady orbit radius at J>J
c1
u0JðÞ¼λJðÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
J=Jc1−1
p;λ2JðÞ
¼d0ω′
0
d1ω0JðÞþd0ω1JðÞ−χ1JðÞ½
:ð7Þ
The model parameters are Jc1¼d0ω′
0=γσ=2−d0ωe
ðÞ
,
d
0
=α
G
[5 + 4 ln(R/R
c
)]/8, d
1
=11α
G
/6, χ
0
=γσJ/2. The ratio
χ
1
/χ
0
=O(c
2
u
2
) < < 1, therefore, the nonlinear parameter χ
1
can be neglected. The statement about linearity of the
ST-force agrees also with our simulations and the
micromagnetic simulations performed in [12,19]. The
coefficient λ(J) describes nonlinearity of the system
and decreases smoothly with the current Jincreasing.
Numerical method
We have simulated the vortex motion in a single
permalloy (Fe
20
Ni
80
alloy, Py) circular nanodot under
the influence of a spin-polarized dc current flowing
through it. Micromagnetic simulations of the spin-torque-
induced magnetization dynamics in this system were
carried out with the micromagnetic simulation package
MicroMagus (General Numerics Research Lab, Jena,
Germany) [28]. This package solves numerically the LLG
equation of the magnetization motion using the optimized
version of the adaptive (i.e., with the time step control)
Runge-Kutta method. Thermal fluctuations have been
neglected in our modeling, so that the simulated dynamics
corresponds to T= 0. Material parameters for Py are as
follows: exchange stiffness constant A=10
−6
erg/cm,
saturation magnetization M
s
= 800 G, and the damping
constant used in the LLG equation α
G
= 0.01. Permalloy
dot with the radius R= 100 nm and thickness L=5, 7, and
10 nm was discretized in-plane into 100 × 100 cells. No
additional discretization was performed in the direction
perpendicular to the dot plane, so that the discretization
cell size was 2 × 2 × Lnm
3
. In order to obtain the vortex
core with a desired polarity (spinpolarizationdirectionofdc
current and vortex core polarity should have opposite direc-
tions in order to ensure the steady-state vortex precession)
and to displace the vortex core from its equilibrium position
in the nanodot center, we have initially applied a short
magnetic field pulse with the out-of-plane projection
of 200 Oe, the in-plane projection H
x
= 10 Oe, and the
duration Δt= 3 ns. Simulations were carried out for
the physical time t= 200 to 3,000 ns depending on the
applied dc current because for currents close to the
threshold current J
c1
, the time for establishing the vortex
steady-state precession regime was much larger than for
higher currents (see Equation 8 below).
Results and discussion
Calculated analytically, the vortex core steady orbit radius
in circular dot u
0
(J) as a function of current Jis compared
with the simulations (see Figure 1). There is no fitting ex-
cept only taking the critical current J
c1
value from simula-
tions. Agreement is quite good, confirming that all the
nonlinear parameters of the vortex motion were accounted
correctly. The steady orbit radius u
0
(J) allows finding
the STNO generation frequency ωGJðÞ¼ω0JðÞþω1u2
0JðÞ,
which increases approximately linearly with Jincreasing up
to the second critical current value J
c2
when the steady
oscillationstatebecomesunstable(seeFigure2).Thein-
stability is related with the vortex core polarity reversal
reaching a core critical velocity or the vortex core expelling
from the dot increasing the current density J[12,16]. We
simulated the values of J
c2
= 2.7, 5.0, and 10.2 MA/cm
2
for
the dot thickness L= 5, 7, and 10 nm, respectively. The
calculated STNO frequency is 15 to 20% higher than
the simulated one due to overestimation of ω′
0within
TVA for β=0.1. The calculated nonlinear frequency part
is very close to the simulated one, except the vicinity of
J
c2
, where the analytical model fails.
Our comparison of the calculated dependences u
0
(J)
and ω
G
(J) with simulations is principally different from
the comparison conducted in a paper [19], where the au-
thors compared Equations 5 and 7 with their simulations
fitting the model-dependent nonlinear coefficients Nand
λfrom the same simulations. One can compare Figures 1
and 2 with the results by Grimaldi et al. [20], where the
authors had no success in explaining their experimental
dependences u
0
(J)andω
G
(J) by a reasonable model. The
realistic theoretical nonlinear frequency parameter Nfor
Py dots with L= 5 nm and R= 250 nm should be larger
than 0.11 that the authors of [21] used. N=0.25 can be
calculated from pure magnetostatic energy in the limit
β→0 (inset of Figure 2). Accounting all the energy contri-
butions in Equation 4 yields N= 0.36, which is closer to
the fitted experimental value N=0.50.
Guslienko et al. Nanoscale Research Letters 2014, 9:386 Page 4 of 7
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The system (6) can be solved analytically in nonlinear
case. Its solution describing transient vortex dynamics is
u2t;JðÞ¼ u2
0JðÞ
1þu2
0JðÞ
u20ðÞ
−1
hi
exp −t
τþJðÞ
;ð8Þ
where u(0) is the initial vortex core displacement and
1=τþJðÞ¼2d0ω′
0J=Jc1−1ðÞis the inverse relaxation time
for J>J
c1
(order of 100 ns). ut;Jc1
ðÞ∝1=ffiffi
t
pat t →∞and
J=J
c1
.IfJ<J
c1
, the orbit radius u(t,J) decreases expo-
nentially to 0 with the relaxation time 1=τ−JðÞ¼d0ω′
0
J=Jc1−1
jj
. The divergence of the relaxation times τ
±
at
J=J
c1
allows considering a breaking symmetry second-
order phase transition from the equilibrium value u
0
=0
to finite u0JðÞ∝ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
J=Jc1−1
pdefined by Equation 7. Equa-
tions 7 and 8 represent mean-field approximation to the
problem and are valid not too very close to the value of J=
J
c1
, where thermal fluctuations are important [13,21].
Equation 8 describes approaching of the vortex orbit
radius to a steady value u
0
(J) under influence of dc spin-
polarized current. We note that the corresponding relax-
ation time is determined by only linear parameters, whereas
the orbit radius (7) depends on ratio of the nonlinear
and linear model parameters. The solution of Equation 8
is plotted in Figure 3 as a function of time along with
micromagnetic simulations for circular Py dot with
thickness L= 7 nm and radius R= 100 nm. The vortex
was excited by in-plane field pulse during approximately
the first 5 ns, and then the vortex core approached the sta-
tionary orbit of radius u
0
(J). We estimated u(0) after the
pulse as u(0) = 0.1 and plotted the solid lines without any
Figure 2 The vortex steady-state oscillation frequency vs. current. The nanodot thickness Lis 5 nm (1), 7 nm (2), and 10 nm (3), and radius
is R= 100 nm. The frequency is shown within the current range of the stable vortex steady-state orbit, J
c1
<J<J
c2
. Solid black lines are calculations
by Equation 5; red squares mark the simulated points. Inset: the nonlinear vortex frequency coefficient vs. the dot thickness for R=100nm
and J= 0 accounting all energy contributions (1) and only magnetostatic contribution (2).
Figure 3 Instant vortex core orbit radius vs. time for different
currents. The results are within the current range of the stable
vortex steady-state orbit, J
c1
<J<J
c2
(5.0 MA/cm
2
). The nanodot
thickness is L= 7 nm and the radius is R= 100 nm. Solid lines are
calculations of the vortex transient dynamics by Equation 8, and
symbols (black squares, red circles, green triangles, and blue rhombi)
mark the simulated points.
Guslienko et al. Nanoscale Research Letters 2014, 9:386 Page 5 of 7
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fitting except using the simulated value of the critical
current J
c1
. Overall agreement of the calculations by
Equation 8 and simulations is quite good, especially for
large times t≥3τ
+
, although the calculated relaxation time
τ
+
is smaller than the simulated one due to overestimation
of ω′
0within TVA. The typical simulated ratio J
c2
/J
c1
≈1.5;
therefore, minimal τ
+
≈20 to 30 ns. But the transient time
of saturation of u(t,J) is about of 100 ns and can reach
several microseconds at J/J
c1
< 1.1. The simulated value of
λ= 0.83, whereas the analytic theory based on TVA yields
the close value of λ(J
c1
) = 0.81.
Typical experiments on the vortex excitations in nanopil-
lars are conducted at room temperature T= 300 K without
initial field pulse, i.e., a thermal level u(0) should be suffi-
cient to start vortex core motion to a steady orbit. To find
the thermal amplitude of u(0), we use the well-known
relation between static susceptibility of the system ^
χand
magnetization fluctuations M2
α
T−Mα
hi
2
T¼kBT=VðÞχαα .
The in-plane components are χxx ¼χyy ¼ξ2=2
γMs=ω′
0
,andM=ξM
s
s,whereξ= 2/3 within TVA [26].
This leads to the simple relation u2
hi
T¼kBT=M2
sV
γMs=ω′
0
.ItisreasonabletouseuT0ðÞ¼ ffiffiffiffiffiffiffiffiffiffiffi
u2
hi
T
pfor in-
terpretation of the experiments. u
T
(0) ≈0.05 (5 nm in ab-
solute units) for the dot made of permalloy with L=7nm
and R= 100 nm.
The nonlinear frequency coefficient N(β,R,J)=κ′(β,R,J)/
κ(β,R,J) is positive (because of κ,κ′>0 for typical dot
parameters), and it is a strong function of the dot geomet-
rical sizes Land Rand a weak function of J. For the dot
radii R>>L
e
,N(β,R,0)≈0.21 −0.25 (the magnetostatic
limit, see inset of Figure 2). If R>>L
e
and β→0, then N
(β,R,0)≈0.25 [14]. For the realistic sizes of free layer in a
nanopillar (Ris about 100 nm and L= 3 to 10 nm), this
coefficient is essentially larger due to finite βand exchange
contribution, and it can be of order of 1. The exchange
nonlinear contribution κ′
ex
is important for R< 300 nm.
However, the authors of [19-21] did not consider it at all.
Note that N(0.089, 300 nm,0)≈0.5 recently measured [29]
is two times larger than 0.25. The authors of [19] suggested
to use an additional term ~u
6
inthemagneticenergyfitting
the nonlinear frequency due to accounting a u
4
-contribu-
tion (N= 0.26) that is too small based on [14], while the
nonlinear coefficient N(β,R) calculated by Equation 5 for
the parameters of Py dots (L=4.8 nm, R= 275 nm) [19] is
equal to 0.38. Moreover, the authors of [19] did not account
that, for a high value of the vortex amplitude u= 0.6 to 0.7,
the contribution of nonlinear gyrovector G(u)∝c
2
u
2
to the
vortex frequency is more important than the u
6
-magnetic
energy term. The gyrovector G(u) decreases essentially for
such a large uresulting in the nonlinear frequency increase.
The TVA calculations based on Equation 5 lead to the
small nonlinear Oe energy contribution κ′
Oe
,whereas
Dussaux et al. [19] stated that κ′
Oe
is more important
than the magnetostatic nonlinear contribution.
Conclusions
We demonstrated that the generalized Thiele equation
of motion (1) with the nonlinear coefficients (2) considered
beyond the rigid vortex approximation can be successfully
used for quantitative description of the nonlinear vortex
STNO dynamics excited by spin-polarized current in a
circular nanodot. We calculated the nonlinear parameters
governing the vortex core large-amplitude oscillations and
showed that the analytical two-vortex model can predict
the parameters, which are in good agreement with the ones
simulated numerically. The Thiele approach and the energy
dissipation approach [12,19] are equivalent because they
are grounded on the same LLG equation of magnetization
motion. The limits of applicability of the nonlinear oscilla-
tor approach developed for saturated nanodots [13] to vor-
tex STNO dynamics are established. The calculated and
simulated dependences of the vortex core orbit radius u(t)
and phase Φ(t) can be used as a starting point to consider
the transient dynamics of synchronization of two coupled
vortex ST nano-oscillators in laterally located circular
nanopillars [30] or square nanodots with circular nano-
contacts [31] calculated recently.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
KYG formulated the problem and carried out the analytical calculations. OVS
and DVB conducted the micromagnetic simulations. KYG supervised the
work and finalized the manuscript. All authors have read and approved the
final manuscript.
Acknowledgements
This work was supported in part by the Spanish MINECO grant
FIS2010-20979-C02-01. KYG acknowledges support by IKERBASQUE
(the Basque Foundation for Science).
Author details
1
Depto. Física de Materiales, Facultad de Química, Universidad del País Vasco,
UPV/EHU, San Sebastián 20018, Spain.
2
IKERBASQUE, Basque Foundation for
Science, Bilbao 48011, Spain.
3
General Numerics Research Laboratory, Jena
07745, Germany.
Received: 15 June 2014 Accepted: 1 August 2014
Published: 8 August 2014
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Cite this article as: Guslienko et al.:Nonlinear magnetic vortex dynamics
in a circular nanodot excited by spin-polarized current. Nanoscale Research
Letters 2014 9:386.
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