Table Of ContentControlling the synchronization properties of two dipolarly coupled vortex
based spin-torque nano-oscillators by the intermediate of a third one
Flavio Abreu Araujo1,a) and Julie Grollier1
Unit´e Mixte de Physique CNRS, Thales, Univ. Paris-Sud, Universit´e Paris-Saclay, F91767 Palaiseau,
Franceb)
In this paper, we propose to control the strength of phase-locking between two dipolarly coupled vortex
based spin-torque nano-oscillators by placing an intermediate oscillator between them. We show through
micromagnetic simulations that the strength of phase-locking can be largely tuned by a slight variation
of current in the intermediate oscillator. We develop simplified numerical simulations based on analytical
expressions of the vortex core trajectories that will be useful for investigating large arrays of densely packed
spin-torque oscillators interacting through their stray fields.
Spin-torque nano-oscillators are magnetic auto- Oscillator Oscillator # Oscillator
7 oscillators of deep submicron dimensions. Made of spin- 1 2
1 valves1,2ormagnetictunneljunctions,3theycanbefabri-
0
catedontopofaplaneofCMOStransistorsandtheyop- 50 nm 50 nm
2
erateatroomtemperature. Thetorqueonmagnetization
n is generated by sending a spin-polarized current through
a the ferromagnetic layer. For high enough current densi- J1 = 1.25 x Jc J# = η x Jc J2 = J1
J
ties,thisspin-torquecaninducesustainedmagnetization
1 precessions that are then converted into voltage oscilla- FIG. 1. Schematic illustration of a three vortex spin-torque
3 oscillators chain where the edge to edge distance is 50 nm.
tionsbymagneto-resistiveeffects. Thefrequencyofthese
] microwave oscillators can be tuned over several GHz by Oscillators 1 & 2 are supplied with the same current density
(J =J )andthe”tuning”oscillator#issuppliedwithJ =
l changing the amplitude of the injected dc current or ap- 1 2 #
l η·J .
a plied magnetic field. Because to this high non-linearity, c
h
spin-torque nano-oscillators are sensitive to small vari-
-
s ations of magnetic field and electric current4. In par-
e amplitude of the current sent through the intermediate
ticular,severalspin-torquenano-oscillatorscanmutually
m oscillator modifies the coupling between the other two.
synchronize even if their individual frequencies are ini-
For this purpose, we perform full micromagnetic simu-
at. tially different5–11. Thanks to these features they are lations of the three coupled oscillators in order to have
excellent candidates for building computing systems in-
m an accurate estimate of the dipolar interactions. Then
spired from neural synchronization in the brain12–15. In-
we develop much faster numerical simulations based on
- deed bio-inspired computing with oscillators requires to
d analytical equations for the oscillators’ dynamics which
beabletofabricate verylargearrays ofinteractingoscil-
n willbeusefulforsimulatinglargescalearraysofdipolarly
o lators, and to be able to control the degree of coupling
coupled spin-torque oscillators.
c between the oscillators16,17. If several physical phenom-
[ ena can be used to couple spin-torque oscillators, such The system we consider is illustrated in Fig. 1. We
as spin waves5,6,18 or electric currents7,19, one of the study three identical circular nanopillars with diameters
1
200nm. Ineachoscillator, themagneticconfigurationof
v mostappealingtowardstherealizationofdensearraysis
thefreelayerisavortex,allwiththesamecorepolarities
1 the dipolar coupling. Indeed when oscillators are closely
andchiralities. Wefocusonvortexoscillatorsbecausethe
0 packed, with edge to edge distance below 500 nm, the
1 dipolar coupling becomes intense and can synchronize gyration of a vortex core through spin-torque in a single
9 their dynamics, as demonstrated theoretically20–22 and pillar is well understood. It has been shown that ana-
0 lytical descriptions of the dynamics match experiments
experimentally23. Whereasitispossibletotunethecou-
1. plingprovidedbyspinwaves9 andelectricalcurrents24,it quantitatively25. In addition vortex oscillators have a
0 low phase noise and have been shown to synchronize by
remainsachallengetomodifytheinteractionoriginating
7 dipolarcouplingexperimentally23. Inoursimulations,we
from the dipolar fields emitted by the oscillators. In this
1 consider that the magnetization of the polarizing mag-
letter,weproposetoadjustthedipolarcouplingbetween
:
v two close-by spin-torque oscillators by inserting a third netic layer is fixed, pointing out of plane, and that the
i magnetostatic field it emits is negligible. The geometri-
X oscillator between them. We study numerically how the
cal and magnetic parameters that we use are displayed
r in Table I.
a
In order to study the dynamics of the three dipo-
a)Electronicmail: abreuaraujo.fl[email protected] larlycoupledvortices, wehaveperformedfullmicromag-
b)Previous address: Institute of Condensed Matter and neticsimulationsaswellasnumericalsimulationssolving
Nanosciences, Universit´e catholique de Louvain, BE-1348 for analytical equations of vortex cores trajectories, and
Louvainla-Neuve,Belgium comparedbothmethods. Themicromagneticsimulations
2
h = 10 nm (dot thickness) sincewewanttogeneratesustainedgyrationsofthecore,
D = 200 nm (dot diameter) we choose the current sign so that the spin-torque force
M = 800 emu/cm3 (saturation magnetization) points opposite to the damping force. The effective spin
s
torque efficiency is given by κ = πa M h where a is
A = 1.3×10−6 erg/cm (stiffness constant) J s J
the spin torque amplitude a = P(cid:126)J/(2|e|M h) (with (cid:126)
α = 0.01 (Gilbert damping parameter) J s
the Planck constant and e the electron charge). Finally,
P = 0.2 (current spin polarization)
the last term in Eq. (1) accounts for the magnetostatic
interactionforcesduetostrayfieldsbetweenoscillatorsi
TABLEI.GeometricalandmaterialparametersforPermalloy
and j, the main contribution being dipolar. In contrast
(Ni Fe ) considered in the simulations.
81 19
to previous works, the analytical version of the magne-
tostatic interaction developed by Sukhostavets et al.30
areperformedusingtheGPU(GraphicsProcessingUnit) has been considered instead of evaluating it combining
basedmicromagneticcodecalledMuMax3,26withamesh micromagnetic simulations and analytical model. Fint is
ij
size2.5×2.5nm2. Thenumericalsimulationssolvingfor given by the following multipole approximation:
thevortexcoregyrotropicmotionarebasedontheThiele (cid:34) (cid:35)
equation20,22,27–29: Fint = ηxij 0 X (4)
ij 0 ηij j
(cid:16) (cid:17) y
G×X˙ +D·X˙ − kms+kOeJ X
i i i i
(1) where ηij are the magnetostatic interaction coefficients:
x,y
−κ(X ×zˆ)−Fint(X )=0
i ij j
(cid:32) (cid:33)
TcohriesoeqfupaotsioitniodnesXcribinesotshceilclairtcourlair. mTohteionfirosftttheermvoritsexa ηxij = hR2Ms2π2 9d43ij + 5d15ij + 56101d37ij + 8·156··12977d9ij ,
i (cid:32) (cid:33)
MItaagrnisuess-lifkroemfortchee, pfaositntuinpgwatrodwsarsdpsirtahleofedmgeagonfetthizeatdiootn. ηyij =−hR2Ms2π2 9d83 + 5d45 + 566·01d173 + 165··2179d79 ,
ij ij ij ij
in the core that generates a gyrovector G = −Gzˆ.27
i
The second term accounts for the damping force, tan- (5a)
gential to the core trajectory and opposite to the vor-
tex core velocity X˙i. The damping coefficient Di is (5b)
equal to D = αλG(1 + 0.6s2 − 0.2s4), where s is
i i i i with d = (2R + L )/R the reduced inter-distance
the normalized radius of gyration s = (cid:107)X (cid:107)/R and ij ij
i i
λ=0.5ln(cid:0)R/(2L )(cid:1)+3/8,withL =(cid:112)A/(2πM2)the between oscillators (Lij is the edge-to-edge distance be-
ex ex s tweentwooscillators). Inthiswork,thenon-linearitiesof
exchangelength. Thethirdtermistheconfinementforce
the gyrovector G and the spin-transfer-torque efficiency
pointing inwards the dot. It arises both from the mag-
κ have been neglected31.
netostatic energy (kms) and the current-induced Oersted
i The two extreme oscillators labeled 1 and 2 are set
field confinement (kOe). The magnetostatic contribution
i in a regime of sustained vortex oscillations by supply-
kms totheThieleequationhasbeencalculatedunderthe
i ing them with a dc current J = J above the thresh-
”Two Vortex Ansatz”28,29. We numerically evaluate the 1 2
old current for auto-oscillations J ≈−5.6×106 A/cm2:
energy (W ) and specialize our computation to the dot c
ms J = J = 1.25 · J . The edge to edge distance be-
aspect ratio of ε = h/(2R) = 0.05 and obtain after a 1 2 c
tween each oscillator is 50 nm, resulting in a separation
polynomial fit:
of 300 nm between oscillators 1 and 2. This distance is
small enough for oscillators 1 and 2 to interact strongly
8M2h2
kms(s )= s 1.594× through the dipolar fields they emit. In particular, in
i i R
the absence of the intermediate oscillator # they mutu-
(cid:16) (cid:17)
1+0.175s2+0.065s4−0.054s6 . (2) ally synchronize and lock their phases to the same value
i i i
(max(ϕ − ϕ ) < 2◦))20,22. We now study what hap-
2 1
The kOe coefficients are computed using the 10th order pens when the intermediate oscillator # is introduced,
i
Taylor expansion after evaluating the current-induced bylookingatthephasedifferenceϕ −ϕ extractedfrom
2 1
Oersted field contribution (W ) to the confinement en- micromagnetic and analytically based simulations. Fig.
Oe
ergy and is given by: 2(a)showsthemaximumvaluetakenbythephasediffer-
ence between oscillators 1 and 2, max(ϕ −ϕ ), during
2 1
8π2 vortex gyrations as a function of the dc current injected
kOe =JCM hR ×
i s 75 through oscillator #.
(cid:18) 4 1 16 125 (cid:16) (cid:17)(cid:19) As mentioned in the introduction, the simulations are
1− s2− s4− s6− s8+O s10 . (3)
7 i 7 i 231 i 3003 i i performed assuming that all the vortex core polarities
are parallel. Two regimes appear in Fig. 2(a). At low
The fourth term in Eq. (1) is the spin-torque in- currents,forη =J /J <0.75,max(ϕ −ϕ )takeslarge
# c 2 1
duced force exerted on the vortex core. In our case, values: the presence of the intermediate vortex destroys
3
120 Micromagnetics Vortex analytics
g) a) 100
e100
d a) Oscillator 1 c) Oscillator 1
φ1 - φ2 ) ( 468000 VMoicrrteoxm aanganleyttiiccss nm)6800 OOsscciillllaattoorr 2# OOsscciillllaattoorr 2#
x ( 20 us (
ma 0 adi40
480 R
z) b) Vortex analytics 20
H
M Micromagnetics
y (460 0
quenc440 deg)19305 b) φ2 - φ1 d) φ2 - φ1
Fre4200 .5 0.625 0.75 0.875 1.0 1.125 1.25 1.375 1.5 ence ( 45
η = J# / Jc ffer 0
di-45
FIG. 2. (a) Maximum amplitude of the phase difference be- e
s
a-90
otwsceiellnatoosrcsi,llabtootrhsa1saandfu2nc[mtioanx(oϕf2t−heϕc1u)]rr(ebn)tFtrherqouuegnhcyosocfitllhae- Ph-135 φ# - φ1 φ2 - φ# φ# - φ1 φ2 - φ#
tor #: J = ηJ . The results of micromagnetic simulations 0 50 100 150 200 0 50 100 150 200
# c
Time (ns) Time (ns)
(vortex analytics) are displayed as hollow red circles (small
blue disks).
FIG. 3. (a) and (b) (resp. (c) and (d)) show the radii and
phase difference (ϕ −ϕ ) evolutions of the three oscillators
2 1
illustrated in Fig. 1 for η =J /J =0.625 using micromag-
# c
thephaselockingbetweenoscillators1and2. Inthefirst netic simulations (resp. vortex analytics).
regime (0.5 < η < 0.86), modes from the different oscil-
lators can be observed (up to 3) but only the frequency
of the main common mode is shown in Fig. 2(b). Fig.
3 shows time traces of the vortex cores radius and phase the transition to phase-locking. Indeed, while the vortex
differences between each oscillator for J = 0.625·J . frequency is practically constant in the damped mode,
# c
Micromagnetic and core-dynamics-based numerical sim- it increases above the auto-oscillation threshold. In the
ulations indicate that large fluctuations of the phase dif- auto-oscillationregime,theorbitofthevortexcoregrows
ferences between the oscillators occur. They also both with current through spin torque leading to an increase
show that oscillator # oscillates with a lower amplitude of the confinement and larger frequencies. It should be
than oscillators 1 and 2. Indeed, in this regime, the cur- noted however that the threshold for auto-oscillations
rent through oscillator # is much lower than the current of oscillator # occurs for η < 1, in other words below
J leading to auto-oscillations. However, even if the vor- the critical current necessary to compensate the damp-
c
tex of oscillator # is damped its orbit fluctuates due to ing. Indeed, the magnetization dynamics of oscillator
the rotating microwave dipolar fields emitted by oscilla- # is driven by the dc current J# assisted by resonant
tors1and2. AsJ increasesthevortexinoscillator#is microwave excitations incoming from oscillators 1 and 2
#
lessandlessdampedanditsorbitgrows. Asaresult,the through their stray fields. As can be seen in Figs. 3 and
dipolar field generated by oscillator # increases with J 3, small amplitude oscillations of the vortex orbit radii
#
and disrupts the trajectories of oscillators 1 and 2 more appear. These oscillations are due to a slight shift of the
and more, leading to increasing values of max(ϕ −ϕ ) center of the vortex gyrotropic motion (from about 0.1
2 1
as can be seen in Fig. 2(a). to about 1 nm depending on the applied current). The
frequencyofthesmallamplitudefluctuationsistwicethe
However, for η >0.86, a second regime appears where
frequency of the main gyrotropic motion.
max(ϕ −ϕ ) is drastically reduced and phase-locking is
2 1
restored. Incontrasttothefirstmode,onlyonecommon To summarize, we shown for the first time that the
synchronization mode is observed for η > 0.86. Time strength of phase-locking between two close-by oscilla-
traces of the vortex cores radius and phase differences tors interacting via their dipolar fields can be tuned by
between each oscillator for J = 1.25J are shown in a slight variation (0.75 < η < 0.86 in Fig. 2(a)) of the
# c
Fig. 4. Now the radius of all three oscillators is much currentdensitysentthroughanintermediateone. Inad-
larger, around 60 nm. Indeed we observe from our sim- dition, we propose a new full analytical description of
ulations that the transition to the phase-locking regime the coupled dynamics. Furthermore, numerical simula-
coincides with the onset of self-sustained precessions of tions based on these analytical expressions of the vor-
oscillator #. This clearly appears in Fig. 2(b), which tex core dynamics are in excellent agreement with full
shows how the frequency of the three coupled oscillators micromagnetic simulations and several orders of magni-
varieswithJ . Akinkinthefrequencycurveappearsat tude faster, even with highly efficient solvers making use
#
4
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