OPENSUBJECT AREAS:
MAGNETIC DEVICESSPINTRONICS

Wide range tuning of resonant frequency
for a vortex core in a regular triangle
magnetSatoshi Yakata1,2, Terumitsu Tanaka3, Kohei Kiseki1,3, Kimihide Matsuyama3 & Takashi Kimura1,2,4Received
29 August 2013
Accepted
4 December 2013
Published
20 December 2013

Correspondence andrequests for materials
should be addressed toT.K. (t-kimu@phys.
kyushu-u.ac.jp)

1

Advanced Electronics Research Division, INAMORI Frontier Research Center, Kyushu University, 744 Motooka, Fukuoka, 8190395, Japan, 2CREST, Japan Science and Technology Agency, Sanbancho, Tokyo 102-0075, Japan, 3Graduate School ofInformation Science and Electrical Engineering, Kyushu University, 744 Motooka, Fukuoka, 819-0395, Japan, 4Department of
Physics, Kyushu University, 6-10-1 Hakozaki, Fukuoka, 812-8581, Japan.

A magnetic vortex structure stabilized in a micron or nano-sized ferromagnetic disk has a strong potential asa unit cell for spin-based nano-electronic devices because of negligible magnetostatic interaction andsuperior thermal stability. Moreover, various intriguing fundamental physics such as bloch point reversal
and symmetry breaking can be induced in the dynamical behaviors in the magnetic vortex. The static anddynamic properties of the magnetic vortex can be tuned by the disk dimension and/or the separation
distance between the disks. However, to realize these modifications, the preparations of other devices withdifferent sample geometries are required. Here, we experimentally demonstrate that, in a regular-triangle
Permalloy dot, the dynamic properties of a magnetic vortex are greatly modified by the application of thein-plane magnetic field. The obtained wide range tunability based on the asymmetric position dependenceof the core potential provides attractive performances in the microwave spintronic devices.

D

ynamic properties of the magnetizations in the ferromagnetic thin films have various potential applications because of their fast time scales ranging from sub-picoseconds to nanoseconds1–3. The time scales ofthe spin dynamics are mainly determined by the saturation magnetization and the effective magnetic field
due to the magnetic anisotropy. Particularly, in patterned ferromagnetic nanostructures, the intriguing magnetization dynamics can be obtained because of the significant modification of the spin excitation state compared tothe uniform excitation case4. As the representative structures, a considerable attention has been focused on amagnetic vortex stabilized in a micron or submicron dot because of its unique dynamic properties with size
tunability5–7. The domain structure of the magnetic vortex can be characterized by in-plane curling magnetization
and a nanometer-sized perpendicularly magnetized vortex core. The restoring force due to the demagnetizing
field and the gyrotropic force result in a spiral motion of the vortex core8.
The resonant frequency of the vortex core is in the sub GHz to GHz and can be excited by using a microwave
magnetic field9–12. The oscillation frequency of the vortex core is well characterized by the effective magnetic
susceptibility, which is roughly proportional to the dot aspect ratio t/D13. Here, t and D are the thickness and thediameter for the circular disk case, respectively. In the circular disk, the effective magnetic susceptibility isapproximately independent of the core position because the position dependence of the core potential can beapproximately described by a two-dimensional parabolic equation14–16. As a result, a weak field dependence of theresonant frequency is expected in a circular disk.
In the case of the elliptical disk instead of the circular disk, the resonant frequency is known to be tuned by theapplication of the magnetic field along the short axis for the ellipse because of the variation of the effective stiffness
constant with changing the core position17. Twofold enhancement of the oscillation frequency was demonstrated
by the ellipse with the aspect ratio of 2, which is defined by the ratio the major axis to the minor axis. To obtain thefurther increase of the frequency, the aspect ratio of the ellipse should be increased. However, the single vortex
domain structure becomes unstable in the ellipse with a large aspect ratio. The modulation of the resonant
frequency of the vortex core is found to depend on the core position also in the circular disk, when the pinningpotential is larger than the excitation power. However, it seems to be difficult to tune the resonant frequency
because the pinning potential is randomly distributed18.
To realize the large modification of the oscillation frequency with a reliable controllability, we focus on theregular triangle, in which the magnetic vortex structure is also stabilized19–22. In the triangular ferromagnetic dot, a
SCIENTIFIC REPORTS | 3 : 3567 | DOI: 10.1038/srep03567

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SCIENTIFIC REPORTS | 3 : 3567 | DOI: 10.1038/srep03567

NormalizedCore position yy1

0.8 0.5[Z0

0.6 - magnetic field+
[

0.4

Z

0.2 (a)
[
Z

1.0

y
1

0.8 0.66

[
Z

033

0.6

0

-

+

magnetic field

0.4

[
Z

0.2 (b)[Z

0.0-4 0 0 -2 0 0 0
200 400
magnetic field (Oe)q q q q q q q

Figure 1 | Numerically calculated equilibrium core positions as afunction of the in-plane magnetic field for (a) circular and (b) triangular
Permalloy dots. The schematics of the expected magneto-static potential asa function of the core position together with the micromagnetic structure
under each magnetic field are also shown in the right-hand side.

vortex have been investigated by simulating the temporal evolutionof the micromagnetic structure under the small ac magnetic fieldwith the magnitude of 2 Oe along the x axis. The steady-state vortex
oscillation is reproduced by the application of a small ac magnetic
field. We systematically study the amplitude of the vortex oscillation
with changing the frequency of the ac magnetic field in order to findout the resonant frequency of the vortex core numerically.
Figures 2(a) and 2(b) show the frequency dependences of theoscillation amplitudes for the circular and triangular dots, respectively, in the absence of the in-plane magnetic field. In the right-handside of the figures, the trajectories of the vortex core under the RF
magnetic field with several frequencies are plotted. Numerical simulations succeeded in capturing the resonant characteristics in both
Amplitude (arb. unit) Amplitude (arb. unit)

Results
Mircomagnetic simulation of vortex dynamics. First, we clarify the
relationship between the core position and the applied magnetic field
in a triangular Permalloy (Py) dot by using an originally developed
micromagnetic simulation24. Here, we focus on a regular triangle Py
dot with the 2 mm in diameter of the circumscribed circle and 40 nm
in thickness. In order to reflect the realistic shape of the experimentally produced triangular dot with the rounded corners, thetriangle is cut by the circle with 97.5% diameter of the circumscribed circle of the triangle. The exchange constant and thesaturation magnetization are, respectively, 1.0 3 10211 J/m and
1 T, which are typical values for Py film. The computational cellsize and the damping parameter a are 4 3 4 3 40 nm3 and 0.01,
respectively25. We confirmed that a single vortex is stabilized at theremanent state and the chirality of the vortex can be manipulated by
the application of the in-plane magnetic field, similarly to ourprevious study. Here, the vortex chirality was set to counter clockwise (CCW). The in-plane static magnetic field is applied along the x
direction in order to move the vortex towards the vertex of thetriangle. This means that the positive (negative) magnetic field
induces the positive (negative) shift of the vortex core along the y
direction. For a comparison, the micromagnetic configurations for acircular dot with 2 mm in diameter and 40 nm in thickness is also
simulated.
Figures 1(a) and 1(b) show the relative position of the core centeras a function of the in-plane magnetic field for the circular andtriangular dots, respectively. The representative magnetic domain
structures together with the schematics for the expected magnetostatic potential under the several magnetic fields are also shown in
the figures. For the circular dot, the linear field dependence isobtained, but it is limited only in the low magnetic field (jHj ,100 Oe). At the high magnetic field (jHj . 100 Oe), the dependence
becomes nonlinear. The symmetric field dependence with respect tothe polarity of the magnetic field originates from the perfect circularshape. On the other hand, for the triangular dot, the field dependenceof the position shift is almost linear in the wide field range from2300 Oe to 400 Oe although the slope of the position shift is slightly
different between the positive and negative magnetic fields. Since thestrong modification of the magnetic susceptibility is expected in thetriangular Py dot by changing the core position, the obtained result
implies that the core oscillation frequency is widely tuned by theapplication of the in-plane magnetic field. We also find that, forthe triangular dot with 3 mm or larger in diameter of the circumscribed circle, the moving direction of the vortex core deviates fromthe line perpendicular to the magnetic field because of the weak
confined potential. As a result, the core position cannot be manipulated by the simple in-plane magnetic field.
We then numerically study the position dependence of the vortexresonance characteristic. The dynamic properties of the magnetic

1.0

NormalizedCore position ysignificant increase of the core potential is expected when the coreapproaches the vertex of the triangle. Since the vortex chirality in atriangular micro magnet is precisely controlled by the application of
the initializing in-plane magnetic field21, the core position is wellmanipulated by applying the magnetic field. In fact, Vogel et al.
systematically studied the field dependence of the resonant frequency in the triangular vortex for the first time and demonstrated
that the resonant frequency varies with the amplitude and the direction of the external magnetic field23. However, the overall change ofthe resonant frequency is 20 MHz, which is smaller than 10% of theresonant frequency at the original position. In the present paper, weshow that, in a well-optimized device dimension and configuration,
the resonant frequency in a regular triangle micro magnet varies
greatly with the position of the vortex core. A wide range tuning of
the resonant frequency from 200 MHz to 600 MHz is experimentally demonstrated.

15

(a)

B

H = 0 Oe

A
[
Z

10

B

5

A

C

(b)

E

C

0
D

5

E

D
0
10 0

15 0
20 0
25 0
frequency (MHz)

F

F

30 0

Figure 2 | Numerically calculated oscillations amplitude of the vortex coreconfined in (a) circular and (b) triangular Permalloy dots under theconstant ac magnetic field as a function of the ac frequency at the remanentstate. The numerically calculated trajectory of the vortex core at each state
are also shown in the right-hand side.
2

(a)amplitude(arb. unit)

12
8
4

500

(b)

frequency (MHz)frequency (MHz)amplitude(arb. unit)

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400
300
200
100
-400

-200
0
200
magnetic field (Oe)

400

(c)
4
0

500

(d)

400
300
200
100
-4 0 0-2 0 00200magnetic field (Oe)40 0

Figure 3 | Numerically calculated resonant frequency of the magnetic vortex confined in (b) the circular and (d) triangular magnetic dots as a function of
the in-plane magnetic field. The field dependences of the oscillation amplitude of the vortex core for (a) the circular and (c) triangular dots are also shownin the top.

consisting of 1000 circular Py dots with 2-mm diameter and 40-nm
thickness is also evaluated experimentally.
Figures 4(b) and 4(c) show S11 spectra for the circular and triangular dots, respectively, in the absence of the dc magnetic field. In each
spectrum, a clear single dip due to the vortex resonance was observed.
The obtained resonant frequencies for the circular and triangular
dots are, respectively, 180 MHz and 230 MHz, which are highly
consistent with the numerical simulations described above.
We then study the field dependence of the resonant frequency by
performing the similar measurements under the various in-planemagnetic field along the x axis. As shown in Fig. 5(a), for the circulardot, a weak field dependence of the resonant frequency, which is
expected in the numerical simulation, is confirmed in the field range
from 2240 Oe to 1240 Oe. Moreover, the wide range modulation of
the resonant frequency is also reproduced in the S11 measurements
Vector Network Analyzer(a)

MFM images

cu

rre

nt
magnetic0-3

S11(dB x10 )the circular and triangular dots and found that the resonant frequencies for the circular and triangular dots are 170 MHz and 220 MHz,
respectively. It should be noted that the core motion makes a circular
trajectory also in the triangular dot. The resonant frequency of
170 MHz for the circular dot can also be analytically reproducedby the Thiele’s equation for the vortex core motion14,15. The resonant
frequency for the triangular dot is larger than that for the circularone. This is because the vortex core is confined in stronger potential
originating from the effectively larger surface magnetic charge in thetriangular dot26. By performing the similar calculation under various
dc magnetic field along the x direction, we can also evaluate the fielddependence of the resonant frequency.
Figures 3(b) and 3(d) show the field dependences of the resonant
frequencies for the circular and triangular dots, respectively. For thecircular disk, the resonant frequency is almost constant in the field
range from 2250 Oe to 1250 Oe. This indicates that the potential
profile for the vortex core in a circular disk is well expressed by a twodimensional harmonic oscillator14–16. On the other hand, for the
triangular dot, the resonant frequency asymmetrically changes withrespect to the polarity of the magnetic field. Especially, the resonant
frequency exceeds 550 MHz when the core is shifted to the corner of
the triangle by applying the in-plane magnetic field of 1300 Oe.
The field dependence of the oscillation amplitude of the core at theresonance is also plotted together on the same figures (Figs. 3(a) and3(c)). The oscillation amplitude gradually varies with changing themagnitude of the in-plane magnetic field (the core position). For thecircular disk, the amplitude monotonically decreases with increasing
the magnitude of the magnetic field. On the other hand, for the
triangular disk, the oscillation amplitude takes a maximum value
when the core is located at the center of the circumscribed circle ofthe triangle. At the corner, the amplitude of the core oscillation showsa significant reduction. These unique characteristics can be explainedby the fact that the equivalent parabolic potential in the triangular
dot changes with the core position. Thus, the resonant characteristicof the vortex confined in the regular triangle is greatly modified byapplying the in-plane magnetic field along the side of the triangle.

(b)

field

(c)

-20
-40
-60

Experimental demonstration of wide-range modulation of vortex
resonance. To demonstrate the expected attractive resonant
properties in the triangular dot experimentally, we have measured
the 1-port transmission impedance measurements (S11) of the micro
strip line with the triangular dots. Here, the size of the triangular Py
dot is the same as that in the simulation (2 mm in outer diameter and40 nm in thickness). As shown in the inset of Figs. 4(a), a magnetic
vortex confined in the triangular Py dot was clearly confirmed by themagnetic force microscope (MFM). For a comparison, the chain
SCIENTIFIC REPORTS | 3 : 3567 | DOI: 10.1038/srep03567

-80
0

200
400
frequency (MHz)200
400frequency (MHz)Figure 4 | (a) Schematic illustration of 1-port transmission impedance
measurement of a chain of the triangular Py dot together with MFM
images for the circular and triangular Py dots at the remanent state.
Experimentally obtained S11 spectra in the absence of the magnetic field for(b) the circular and (c) triangular magnetic dots.
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the wider frequency tuning range by increasing the film thickness inthe triangle. Thus, a variety of the field dependence of the resonant
frequency can be realized by choosing the shape and dimensions forthe patterned ferromagnetic dots.

S11 (dB)

0.2-120 Oe

0.1
0.0

0 Oe

-0.1

120 Oe-0.2

f (MHz)0.3 (a)H = -240 Oe

Methods
We have prepared a one-dimensional chain consisting of 1000 triangular Py dots byusing the conventional lift-off technique with electron beam lithography. Here, theouter diameter and the thickness of the Py dot are 2 mm and 40 nm, respectively. A
single Cu strip line, 500 nm in width and 200 nm in thickness, has been fabricatedalso by the electron-beam lithography, and was placed on the center of the Permalloy
dots for detecting the dynamic response of the magnetic vortices. The dynamic
properties of the magnetic vortex have been evaluated by performing the 1-port
transmission impedance measurements (S11) using the Vector Network Analyzer, as
schematically shown in Fig. 4(a)27. The position of the vortex core was controlled byadjusting the external magnetic field. The vortex chirality in the triangle dots was set
to CCW by using an initializing in-plane magnetic field.

H (Oe)

240 Oe-0.3-120 Oe

S11 (dB)

0.2

0 Oe

0.1
0.0

120 Oe-0.1

240 Oe-0.2

×3

f (MHz)0.3 (b)H (Oe)

330 Oe-0.3
150

300 450 600
frequency (MHz)750

Figure 5 | Experimentally obtained S11 parameters for (a) the circular and(b) triangular ferromagnetic dots for various magnetic fields. The inset
shows image plots of S11 parameter as functions of the in-plane magnetic
field and resonant frequency for each magnetic dots.

for the triangular microdots, as shown in Fig. 5(b). Especially, thehigh frequency resonance over 500 MHz is clearly observed in the
spectrum at H 5 300 Oe although the magnitude of the dip is oneorder smaller than that for Fig. 4(c). The magnitude of the dip takes amaximum value at H , 100 Oe and its field dependence is fully inagreement with the numerical results.

Discussion
Although the vortex dynamics in the triangular dot has been studied
in the previous experiment23, the wide range modulation of the resonant frequency from 200 MHz to 600 MHz has never beenreported. The present successful experiments have been achievedby optimizing the device geometry and dimension. We use a 500nm-wide narrow and 200-nm-thick strip line, in order to sensitively
detect the vortex oscillation especially at the corner of the triangle.
This is because the inductive coupling between the vortex and strip
line increases with increasing the ratio of the oscillation amplitude to
the width of the strip line. We also emphasize that the relative direction between the triangle and strip line rotates 90 degree from thedirection in Ref. 23. This is another important fact for detecting thevortex resonance at the corner of the triangle. This improved microstrip line enabled us to study the field dependence of the vortex
resonance in a circular dot more sensitively. The resonant frequencyof the vortex confined in the circular disk is found to show weak fielddependence, meaning that the vortex potential in a circular dot canbe described by a quasi-two-dimensional parabolic potential.
In summary, we have investigated the resonant properties of themagnetic vortex confined in the triangular and circular dots. A weak
field dependence of the resonant frequency, which is consistent witha quasi-two-dimensional harmonic oscillator, was observed in thecircular dot. On the other hand, the resonant properties for themagnetic vortex confined in the triangular dot showed a strong fielddependence because the core potential is effectively modified by thecore position. Especially, the resonant frequency is found to increasesignificantly when the core approaches to the vertex. We may realize
SCIENTIFIC REPORTS | 3 : 3567 | DOI: 10.1038/srep03567

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T.K. and K.M. supervised the experimental research and micromagnetic simulation. T.K.
wrote the main manuscript text and S.Y. & T.T. prepared all figures.

Additional information
Acknowledgments
The authors would like to thank M. Miyata for the experimental assistance. This work is
partially supported by Kurata-foundation, SCOPE and CREST.

Author contributions
S.Y. & K.K. carried out the experimental work, including the preparation of the sample. T.T.
carried out the micromagnetic simulation including the development of simulation codes.

SCIENTIFIC REPORTS | 3 : 3567 | DOI: 10.1038/srep03567

Competing financial interests: The authors declare no competing financial interests.
How to cite this article: Yakata, S., Tanaka, T., Kiseki, K., Matsuyama, K. & Kimura, T.
Wide range tuning of resonant frequency for a vortex core in a regular triangle magnet. Sci.
Rep. 3, 3567; DOI:10.1038/srep03567 (2013).
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