Spin torque and critical currents for magnetic vortex nano-oscillator in nanopillars

Abstract

We calculated the main dynamic parameters of the spin polarized current induced magnetic vortex oscillations in nanopillars, such as the range of current density, where a vortex steady oscillations exist, the oscillation frequency and orbit radius. We accounted for both the non-linear vortex frequency and non-linear vortex damping. To describe the vortex excitations by the spin polarized current we used a generalized Thiele approach to motion of the vortex core as a collective coordinate. All the calculation results are represented via the free layer sizes, saturation magnetization, Gilbert damping and the degree of the spin polarization of the fixed layer. Predictions of the developed model can be checked experimentally. Comment: 12 pages, 3 figures

Full text

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Spin torque and critical currents for magnetic vortex nano-oscillator in nanopillars

Konstantin Y. Guslienko1,2*, Gloria R. Aranda1, and Julian M. Gonzalez1

1Dpto. Fisica de Materiales, Universidad del Pais Vasco, 20018 Donostia-San Sebastian, Spain
2IKERBASQUE, the Basque Foundation for Science, 48011 Bilbao, Spain


We calculated the main dynamic parameters of the spin polarized current induced magnetic vortex
oscillations in nanopillars, such as the range of current density, where a vortex steady oscillation state
exists, the oscillation frequency and orbit radius. We accounted for both the non-linear vortex frequency
and non-linear vortex damping. To describe the vortex excitations by the spin polarized current we used
a generalized Thiele approach to motion of the vortex core as a collective coordinate. All the results are
represented via the free layer sizes, saturation magnetization, Gilbert damping and the degree of the spin
polarization of the fixed layer. Predictions of the developed model can be checked experimentally.


Key words: spin polarized current, magnetic nanopillar, nano-oscillators, magnetic vortex

*Corresponding author. Electronic mail: sckguslk@ehu.es

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Now excitations of the microwave oscillations in magnetic nanopillars, nanocontacts and tunnel
junctions by spin polarized current as well as the current induced domain wall motions in nanowires are
perspective applications of spintronics.1 A general theoretical approach to microwave generation in
nanopillars/nanocontacts driven by spin-polarized current based on the universal model of an auto-
oscillator with negative damping and nonlinear frequency shift was developed recently by Slavin and
Tiberkevich [see Ref. 2 and references therein]. The model was applied to the case of a spin-torque
oscillator (STO) excited in a uniformly magnetized free layer of nanopillar, and explains the main
experimentally observed effects such as the power and frequency of the generated microwave signal.
However, the low generated power ~ 1 nW of such STO prevents their practical applications. Recently
extremely narrow linewidth of 0.3 MHz and relatively high generated power was detected for the
magnetic vortex (strongly non-uniform state) nano-oscillators in nanopillars.3 The considerable
microwave power emission from a vortex STO in magnetic tunnel junctions was observed.4 It was
established that the permanent perpendicular to the plane (CPP) spin polarized current I can excite
vortex motion in free layer of the nanopillar if the current intensity exceeds some critical value, Ic1.5
Then, in the interval Ic1 <I <Ic2 (Ic2 is a second critical current) there is microwave generation at a
frequency which smoothly increase with the current increasing.6 This frequency corresponds to the
vortex oscillations with a stationary orbit determined by the current value. If the current exceeds a
critical value Ic27,8 the vortex steady state is not stable anymore, presumably because the vortex reaches a
critical velocity9 and reverses its core. The vortex with opposite core cannot be excited for the given
current sign I and the oscillations stop. Such excitation scheme is different from the current-in-plane
(CIP) case, where one needs to apply a.c. CIP of about the resonance frequency to excite the vortex
motion.10 Low value of Ic1 and high value of Ic2 make the vortex CPP nano-oscillators attractive for
applications as microwave devices. However, the calculations of the spin torque term (ST force) gave
contradictory results for the ST force and Ic1. The standard STO approach of Ref. 2 is not applicable to

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the vortex dynamics due to specific damping term. The problem was reduced to the problem of vortex
core reversal in the perpendicular to the layer plane magnetic field in Ref. 7 but vortex steady state
dynamics was not accounted. Using the Thiele approach the ST force was calculated in Refs. 5, 8 which
differs in 2 times from one calculated form the energy dissipation balance.6 Non-linearity of the main
governing parameters and the Oersted field of current were not accounted or accounted incorrectly. The
critical current Ic2 has not yet been calculated.
In this Letter we present a simple and effective approach to calculate the ST force and the critical
currents of the vortex STO in nanopillar. The approach is based on the Thiele formulation of the non-
uniform magnetization dynamics11 and conception of the linear spin excitations.12 The nanopillar device
consists of two ferromagnetic layers, typically FeNi or Co and a nonmagnetic metallic spacer, typically
Cu, arranged in a vertical stack (Fig. 1). Magnetization of one layer is fixed (this layer is the so called
polarizer), whereas the magnetization of the second layer of the nanopillar
() t , rM
is free to rotate. The
current of spin polarized electrons transfers some torque
s τ from the polarizer, which excites
magnetization dynamics of the free layer. We start from the Landau-Lifshitz equation of motion
sLLGeff
τm
?
mHmm
?
γαγ+×+×−=
, where
s
M/
Mm =
, Ms s is the saturation magnetization, γ>0 is the
gyromagnetic ratio, Heff is the effective field, and
LLG
α
is the Gilbert damping. We use the ST term in
the form suggested by Slonczewski,13
()
Pmmτ
××= J
σ
s
, where
()
s
LMe2/
η
?
σ=
, η is the current
spin polarization (η=0.2 for FeNi), e is the electron charge, L is the free layer (dot) thickness, J is the
current density, and
zP
P
=
is the unit vector of the polarizer magnetization (P=+1/-1). We assume the
positive vortex core polarization p=+1, P=+1 and define the current (flow of the positive charges) as
positive I>0 when it flows from the polarizer to free layer. The spin polarized current can excite a vortex
motion in the free layer if only IpP > 0 (only the electrons bringing a magnetic moment from the
polarizer to free layer opposite to the core polarization can excite a vortex motion). Except p, the vortex

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is described by its core position in the free layer, X=(X,Y), and chirality C=±1.14 Let denote the
Slonczewski´s energy density which corresponds to the spin polarized current as
s
w . Then, using the
Thiele approach and the ST field
Pmm
×=∂∂
aws/, the ST force acting on the vortex in the free layer
can be written as

⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∂
∂
×⋅=
∂
∂
X
−=
∫∫
αα
α
ST
X
daL dVwF
s
m
mρP
2
, (1)
where JMa
sσ=
, α=x, y,
()
ϕρ,
=
ρ
is the in-plane radius vector, the derivative is taken with respect
to the vortex core position X assuming an ansatz
() ( )
t
[]
tXρmρm,,
=
(m dependence on thickness
coordinate z is neglected). X has sense of the amplitude of the vortex gyrotropic eigenmode.
We use representation of m-components by the spherical angles
ΦΘ, (Fig. 1) as
)cos, sinsin,cos (sin
ΘΦΘΦΘ=
m and find the expression for the ST force

X
ρF
∂
Φ∂
Θ=
∫
22sindaL
ST
. (2)
In the main approximation we use the decompositions
()( )
ρ
( )(
ρ
)
ρ
ˆ
XXρ cos,
0
z
⋅+=Θ=
gmm
z
,
() ( )[
ρ
]
ϕϕ
cos sin,
00
YXm
−+Φ=Φ
Xρ , where
( )
ρ
0
z
m ,
0
Φ are the static vortex core profile and phase,
( )
ρ
() (
/
)
2
2222
14
ρρρ
++=
c pcg is the excitation amplitude of the z-component of the vortex
magnetization (RRc
c/
=
,
c R is the vortex core radius, ρ, X are normalized to the dot radius R) and
( ) (
ρ
) ρ
/
ρ
1
2
0
−=
m is the gyrotropic mode profile.12 One can conclude from Eq. (2) that only moving
vortex core contribute to the ST force because the contribution of the main dot area where
2/
π=Θ
is
equal to zero due to vanishing integrals on azimuthal angle φ from the gradient of the vortex phase
Φ∂X

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(it was checked accounting in ()
Xρ,
Φ
the terms up to cubic terms in Xα-components). This is a reason
why the ST contribution is relatively small being comparable with the damping contribution.
The integration in Eq. (2) yields the ST force
()
Xz
ˆ
F
×=
aL
ST
π
. This force contributes to the Thiele’s
equation of motion
ST
D
ˆ
WFX
?
X
?
G
X
++ −∂=×
, where
γπ
2/
ˆ
z
s
pLMG =
is the gyrovector, Dˆ is the
damping tensor. The vortex energy
( )
XW
and restoring force
W
XR
F
−∂=
can be calculated from an
appropriate model14 (the force balance is shown in Fig. 2). For circular steady state vortex core motion
the
XωX
?
×=
relation holds, which allows calculating Jc1. To calculate the vortex steady orbit radius
X
=
s R
we need, however, to account non-linear on Xα terms in the vortex damping and frequency (the
account only non-linear frequency as in Ref. 5 is not sufficient). The gyrovector also depends on X, but
this dependence is essential only for the vortex core p reversal, where G changes its sign. As we show
below, the most important non-linearity comes from the damping tensor defined as

( )
X
βα
αβ
γ
α
−
XX
dV
M
D
s
LLG
∂
∂
⋅
∂
∂
=
∫
mm
, (3)
or
()
[]
Φ∂Φ∂Θ+Θ∂Θ∂−=
∫
βαβα αβ
γα
22
sin/
ρ
dLMD
s LLG
in
ΦΘ,-representation Accounting
αβ αβ
δ
DD
=
and introducing dimensionless damping parameter
0/
>−=
GDd
15 we can write the
equation for a steady state vortex motion with the orbit radius
X
=
s R
: ( )
d
( )
s
ϕ
ST
ω
G
FsRGs
=
from
which RRs
s/
=
and the critical currents Jc1, Jc2 can be found. In the second order non-linear
approximation
( )
s
2
10
sddd
+=
,
( )
s
2
10
s
G
ωωω+=
and
aLRs
π
FST
ϕ=
, where
G
ω is the vortex
precession frequency,
0
1>
ω
is a function of the dot aspect ratio β=L/R calculated from the vortex
energy decomposition
( ) sW
in series of
Rs
/
X
=
. It can be shown that
( ) (
β
)
γ
[] 3 /41 9 /20
0
ββ
s
ω−=
M

and
( )
β
( )
βω
4
ω
01
≈
for quite wide range of β= 0.01-0.2 of practical interest, whereas considerably larger

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non-linearity
( )
β
( )
β
8 .42/
01
=ωω
was calculated in Ref. 5 due to incorrect account of the magnetostatic
energy. We use the pole free model of the shifted vortex
( )
t
[]
Xρm ,
, where the dynamic magnetization
satisfies the strong pinning boundary condition at the dot circumference16 R
=ρ
. The damping
parameters are
()() 2// ln8/5
0cLLG
RRd
+=α
,
() 4/3/8/
2
c
2
1
−=
LLG
RRd
α
. We need also to account for
the Oersted field of the current, which leads to contribution to the vortex frequency proportional to the
current density
() ( )
β
JJ
e
ωωβωω+==
000
,
, where
()() C
ξ
cR
e
γπ
8
ω
/15/
=
,
()
RRc8/2/ 1
−
2ln 151
+=ξ

is the correction for the finite core radius
c
R <<R. In the linear approximation we get the equality of the
damping force and the ST force (negative damping) as the condition to find the value of Jc1, with the
contribution of the Oersted field of the current accounted. The values of FST, Jc1 coincide with
calculation of Ref. 6 conducted by the method of the damping energy balance and differ by the
multiplayer 2 from Refs. 5, 8. The first critical current is
(
γσ
)
ec
ddJ
ω
0
ω
001
2//
−=
and the steady
state vortex orbit radius is

( )
1/
1−=
c JJJs
λ
,
1
c JJ >
,
()
[]
10101
1
+
2
2
1
ωω
γσ
J
λ
dd
J
c
c
=
(4)
In this approximation the vortex trajectory radius ( )
Js
increases as square root of the current
overcriticality ()
11/
cc
JJJ −
(for the typical parameters and R=80-120 nm we get λ=0.25-0.30) and the
vortex frequency
( )
β
()
ω
11
2
0
1/
λωωω−++=
ceG
JJJ
increases linearly with J increasing. The vortex
steady orbit can exist until the moving vortex crosses the dot border s=1 or its velocity X? reaches the
critical velocity
c
υ defined in Ref. 9. The later allows to write equation for the second critical current Jc2
as
( ) ( )
sJ
cG
RJ
υω=
. Substituting to this expression the equations for
( )
J
G
ω
and ( )
Js
derived above

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we get a cubic equation for Jc2 in the form
(
ω
)
[
γσ
]
RxxJdJ
ccec
λυλ
1
ω
/2/
2/ 12
101
=++
,
() 1
−
/
12
=
cc
JJx
. This equation has one positive root xc and the value of Jc2 can be easily calculated
(Fig. 3). The former condition (s=1) gives the second critical current
()
1
2
2
/ 1
+
1
cc
′
JJ
λ=
. More detailed
analysis shows that both the mechanisms of the high current instability of the vortex motion are possible
depending on the dot sizes L, R, and the critical current is the lower value of the currents Jc2, J’c2. The
vortex core reversal inside the dot occurs for large enough R (> 100 nm) and L. For the typical sizes
L=10 nm, R=120 nm and C=1, the critical currents are Jc1=6.3 106 A/cm2 (Ic1=2.9 mA), Jc2=1.13 108
A/cm2 (Ic2=51 mA), and for L= 5 nm, R=100 nm we get Jc1=1.8 106 A/cm2 (Ic1=0.56 mA), J’c2=2.7 107
A/cm2 (I’c2=8.4 mA).
In summary, we calculated the main physical parameters of the spin polarized CPP current induced
vortex oscillations in nanopillars, such as the critical current densities Jc1, Jc2, the vortex steady state
oscillations frequency and orbit radius. All the results are represented via the free layer sizes (L, R),
saturation magnetization, Gilbert damping and the degree of the spin polarization of the fixed layer.
These parameters can be obtained from independent experiments. We demonstrated that the generalized
Thiele approach is applicable to the problem of the vortex STO excitations by the CPP spin polarized
current. The spin transfer torque force is related to the vortex core only.
The authors thank J. Grollier and A.K. Khvalkovskiy for fruitful discussions. K.G. and G.R.A.
acknowledge support by IKERBASQUE (the Basque Science Foundation) and by the Program JAE-doc
of the CSIC (Spain), respectively. The authors thank UPV/EHU (SGIker Arina) and DIPC for
computation tools. The work was partially supported by the SAIOTEK grant S-PC09UN03.

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References
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2 A. Slavin and V. Tiberkevich, IEEE Trans. Magn. 45, 1875 (2009).
3 V.S. Pribiag, I.N. Krivorotov, G.D. Fuchs et al., Nature Phys. 3, 498 (2007).
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Captions to the Figures

Fig. 1. Sketch of the magnetic nanopillar with the coordinate system used. The upper (free) layer is in
the vortex state with non-uniform magnetization distribution. The polarizer layer (red color) is in
uniform magnetization state with the magnetization along Oz axis. The positive current I (vertical arrow)
flows from the polarizer to free layer.

Fig. 2. Top view of the free layer with the moving vortex. The arrows denote the force balance for the
vortex core. The spin torque (FST), damping (FD), restoring (
R
F ) and gyro- (FG) forces are defined in the
text. The vortex core steady trajectory Rs is marked by orange color. The vortex chirality is C=+1.

Fig. 3. Dependence of the critical currents Jc1 (solid red line), Jc2 (dashed green line) and J’c2 of the
vortex motion instability on the radius R of the free layer. L= 10 nm, Ms =800 G, η =0.2, 01. 0
=
LLG
α
,
γ/2 =2.95 MHz/Oe, Rc=12 nm. The vortex STO motion is stable at Jc1 < J < min(Jc2, J’c2).

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Fig. 1.

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Fig. 2.

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Fig. 3.


6080
Dot radius, R (nm)
100 120140
5
6
7
8
9
10
11
12
Jc2
J'c2
Current density, J (10
7 A/cm
2)
FeNi
Ms=800 G
L=10 nm
Jc1x10